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Extracting molecular information from spectroscopic data Menzel-Jones, Cian John 2014

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Extracting Molecular Information fromSpectroscopic DatabyCian John Menzel-JonesB.Sc., Queen’s University, 2004M.Sc., The University of British Columbia, 2007A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2014c© Cian John Menzel-Jones 2014AbstractThis thesis explores new ways with which to utilize molecular spectroscopic datain both the time and frequency domain. Operating within the Born-Oppenheimerapproximation (BOA), we show how to obtain the signs of transition-dipole ampli-tudes from fluorescence line intensities. Using the amplitudes thus obtained we givea method to extract highly accurate excited state potential(s) and the transition-dipole(s) as a function of the nuclear displacements. The procedure, illustrated herefor the diatomic and triatomic molecules, is in principle applicable to any polyatomicsystem. We, also, extend this approach beyond the BOA and demonstrate applica-tions involving bound-continuum transition, and double-minimum potentials.Furthermore, by using as input these measured energy level positions and thetransition dipole moments (TDMs), we derive a scheme that completely determinesthe non-adiabatic coupling matrix between potential energy surfaces and the coor-dinate dependence of the coupling functions. We demonstrate results in a diatomicsystem with two spin-orbit coupled potentials, whereby experimentally measured in-formation along with TDMs computed for two corresponding diabatic potentials tothe fully spin-orbit coupled set of eigenstates, are used to extract the diagonal andoff-diagonal spin-orbit coupling functions.Using time-resolved spectra, we show that bi-chromatic coherent control (BCC)enables the determination of the amplitudes (=magnitudes+phases) of individualtransition-dipole matrix elements (TDMs) in these non-adiabatic coupling situation.The present use of BCC induces quantum interferences using two external laser fieldsto coherently deplete the population of different pairs of excited energy eigenstates.The BCC induced depletion is supplemented by the computation of the Fourierintegral of the time-resolved fluorescence at the beat frequencies of the two statesinvolved. The combination of BCC and Fourier transform enables the determinationof the complex expansion coefficients of the wave packet in a basis of vibrationalenergy eigenstates, from simple spontaneous fluorescence data.iiPrefaceChapter 3 is based on the work first developed by the author and colleagues atthe Institute of Molecular Science (IMS), Okazaki, Japan, between October andDecember 2009 by invitation of Dr. Kenji Ohmori. Dr. Moshe Shapiro suggestedthe direction of the project, which was jointly researched by the author and visitingscience Dr. Xuan Li. This research spawned several publications:(I) X. Li, C. Menzel-Jones, D. Avisar and M. Shapiro, Solving the spectroscopicphase: imaging excited wave packets and extracting excited state potentials fromfluorescence data, PCCP 12, 15760-15765 (2010)(II) X. Li, C. Menzel-Jones and M. Shapiro, Spectroscopic Phase and the Extrac-tion of Excited-State Potentials from Fluorescence Data, J. Phys. Chem. Lett. 1,3172-3176 (2010)(III) C. Menzel-Jones, X. Li and M. Shapiro, Extracting double minima excitedstate potentials from bound-continuum spectroscopic data, J. Mol. Spectrosc. 268,221-225 (2011)For manuscripts (I) and (II), the calculations, first draft, revisions, and editing,were jointly performed with Dr. Li, while Dr. Li acted as the principal in the reviewprocess. D. Avisar had originally tackled the problem and provided some initialinsight into its direction. The calculations for (III) were simultaneously developedwith Dr. Li, while the first draft, revision, editing and review process were carried outby the author. Dr. Shapiro was seminal to the revisions and edits of the manuscripts(I,II,III) as well as being involved in monitoring and providing advice on the results.The central idea of Chapter 4 was provided by Dr. Shapiro. The theoreticalsolution and numerical calculations were developed by the author. This work lead toa publication (IV) which was written and submitted by the author with supervisionfrom Dr. Shapiro:(IV) C. Menzel-Jones and M. Shapiro, Complex Wave Function Reconstructionand Direct Electromagnetic Field Determination from Time-Resolved Intensity Data,J. Phys. Chem. Lett. 3, 3353-3359 (2012)iiiPrefaceThe material in Chapter 5 was jointly devised by the author and Dr. Shapiro.The author performed the theoretical analysis and numerical calculations, then com-posed and submitted the resulting two manuscripts (V,VI). Dr. Shapiro providedimportant draft revisions and references.(V) C. Menzel-Jones and M. Shapiro, Using Time-Resolved Experiments andCoherent Control to Determine the Phase of Transition Dipole Moments betweenIndividual Energy Eigenstates J. Phys. Chem. Lett. 4, 3083-3088 (2013)(VI) C. Menzel-Jones and M. Shapiro, Using coherent control to extract the phasesof electronic transition-dipole matrices : The LiRb case Can. J. Chem. 92, 94-99(2014)Chapter 6 is based on an idea initially proposed by Dr. Shapiro, the authordeveloped the work and performed numerical calculations. The first draft, revisions,and editing, of the manuscript (VII) were done in collaboration with Dr. Shapiro.The work lead to a manuscript that is currently in the review process.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Units and Numerical Simulations . . . . . . . . . . . . . . . . . . . . 82 Background Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Molecular Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.1 Born-Oppenheimer Approximation . . . . . . . . . . . . . . . 122.1.2 Diatomic Molecules . . . . . . . . . . . . . . . . . . . . . . . 142.1.3 Polyatomic Molecules . . . . . . . . . . . . . . . . . . . . . . 222.2 Light-Matter Interaction . . . . . . . . . . . . . . . . . . . . . . . . 242.2.1 Semi-Classical Theory . . . . . . . . . . . . . . . . . . . . . . 242.2.2 Molecular Transitions . . . . . . . . . . . . . . . . . . . . . . 262.3 Laser Excitations in Molecules . . . . . . . . . . . . . . . . . . . . . 292.3.1 Molecular Ensemble . . . . . . . . . . . . . . . . . . . . . . . 302.3.2 Excitation Laser Fields . . . . . . . . . . . . . . . . . . . . . 312.3.3 Quantum Control Schemes . . . . . . . . . . . . . . . . . . . 352.4 Molecular Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 37vTable of Contents2.4.1 High-Resolution Rovibrational Spectroscopy . . . . . . . . . 382.4.2 Time-Resolved Fluorescence of Vibrational States . . . . . . 432.4.3 Bound-Continuum Transition . . . . . . . . . . . . . . . . . . 453 Inverions of Potential Energy Surfaces . . . . . . . . . . . . . . . . 473.1 Potential Inversion Methods . . . . . . . . . . . . . . . . . . . . . . 473.1.1 Dunham Expansion . . . . . . . . . . . . . . . . . . . . . . . 473.1.2 Rydberg-Klein-Rees (RKR) Method . . . . . . . . . . . . . . 493.1.3 Direct-Potential Fits . . . . . . . . . . . . . . . . . . . . . . . 503.1.4 Reflection Method . . . . . . . . . . . . . . . . . . . . . . . . 523.2 A New Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2.1 Potential Inversion . . . . . . . . . . . . . . . . . . . . . . . . 543.2.2 Transition-Dipole Function . . . . . . . . . . . . . . . . . . . 563.2.3 Going Beyond the FC Approximation . . . . . . . . . . . . . 573.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.3.1 Bound-Bound Diatomic Model . . . . . . . . . . . . . . . . . 583.3.2 Na2 Single Well . . . . . . . . . . . . . . . . . . . . . . . . . 603.3.3 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.4 Continuum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.4.1 Box Normalization . . . . . . . . . . . . . . . . . . . . . . . . 713.4.2 Extraction of the Entire Na2 C(1Πu) Potential . . . . . . . . 743.5 Double Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.5.1 Morse Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.5.2 Extraction of the Na2 21Σ+u (3s+4s) Double Well Potential . 753.6 Rotational States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.6.1 Inversion Formula for R and P Branches . . . . . . . . . . . 783.6.2 Inversion Formula for Different Rotational States, J1 and J2 783.7 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.7.1 Two Dimensional Potentials . . . . . . . . . . . . . . . . . . 813.7.2 Dipole Correction Extension . . . . . . . . . . . . . . . . . . 873.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.8.1 A Brief Comparison . . . . . . . . . . . . . . . . . . . . . . . 954 Molecular Wavefunction Imaging . . . . . . . . . . . . . . . . . . . . 974.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.1.1 Quantum State Tomography . . . . . . . . . . . . . . . . . . 97viTable of Contents4.1.2 Algebraic-Inversion . . . . . . . . . . . . . . . . . . . . . . . 994.1.3 Interferometric Approaches . . . . . . . . . . . . . . . . . . . 994.1.4 CARS Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.1.5 Kinetic Energy Distribution and Coulomb Explosion . . . . . 1014.1.6 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.2.1 Imaging Method . . . . . . . . . . . . . . . . . . . . . . . . . 1034.2.2 Electromagnetic Field Determination . . . . . . . . . . . . . 1054.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.3.1 Transform-Limited Pulse . . . . . . . . . . . . . . . . . . . . 1064.3.2 Linearly-Chirped Pulse . . . . . . . . . . . . . . . . . . . . . 1084.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095 Extraction of Transition Dipole Moments . . . . . . . . . . . . . . 1105.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.3.1 Isolated Potential . . . . . . . . . . . . . . . . . . . . . . . . 1165.3.2 Coupled Excited State Potentials . . . . . . . . . . . . . . . 1185.3.3 LiRb Coupled Potentials . . . . . . . . . . . . . . . . . . . . 1215.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286 Extraction of Non-Adiabatic Couplings . . . . . . . . . . . . . . . . 1296.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.2.1 Weak Coupling - NaK . . . . . . . . . . . . . . . . . . . . . . 1356.2.2 Strong Coupling - RbCs . . . . . . . . . . . . . . . . . . . . . 1376.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397 Final Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144viiList of Tables3.1 RMS errors (in cm−1) of the A(1Σ+u ) potential in different regions,extracted using the FC approximation and beyond, where LTP(ν)and RTP(ν) denote the left-turning-point and right-turning-point ofa given νth vibrational state. . . . . . . . . . . . . . . . . . . . . . . . 613.2 RMS errors (in cm−1) of the B(1Π+u ) potential in different regions,extracted using the FC approximation and beyond, where LTP(ν)and RTP(ν) denote the left-turning-point and right-turning-point ofa given νth vibrational state. . . . . . . . . . . . . . . . . . . . . . . . 633.3 Global RMS errors of the A(1Σ+u ) potential with different number oftransition bands from only a few highly excited vibrational states νA. 66viiiList of Figures2.1 Hund’s case (a) angular momentum and good quantum numbers.[1]. 162.2 Ground electronic state potential energy curve of the beryllium dimer.The vibrational wave functions for ν = 0, 3 are also shown for refer-ence. The dashed curve is a Morse potential constructed to reproducethe experimental dissociation energy and harmonic vibrational con-stant [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Diabatic (crossing) potential energy curves cross at Rc as a resultof neglecting the part of He that causes the adiabatic (non-crossing)curves’ avoided-crossing by 2He [3]. . . . . . . . . . . . . . . . . . . . 212.4 Jablonski diagram where: S = single state, T = triplet state, A =absorbance, F = fluorescence, P = phosphorescence, IC = internalconversion, ISC = intersystem crossing, and VR = vibrational relax-ation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5 Implementation of bichromatic control (BCC) in a three state systemwhere the initial population resides in states E1 and E2. Two continu-ous wave (CW) lasers with frequencies ω1 and ω2 respectively, couplethese states to a final state E. The amplitude and phase relation be-tween the two laser fields will determine the population of the finalstate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.6 (a) A schematic illustration of the emission process between two di-atomic potentials. (b) A typical spectrum of fluorescence lines calcu-lated between the rovibrational states of the two electronic potentialsshown in (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39ixList of Figures2.7 (a) Wavepacket decaying from an excited PES to the ground state.The most probable decay routes are shown for when the wavepacketis at its turning points. Image taken from Ref. [4]. (b) Temporalfluorescence captured over several picoseconds. This picture is takenfrom Ref. [5] where the number of counts are captured after theupconversion of the raw fluorescence. The two lines capture the resultswithout (a) and with (b) a monochromator inserted after the crystal. 433.1 The beginning of the target potential extraction process near the min-imum region, using a Morse potential as an initial guess. . . . . . . . 593.2 The simulated emission spectrum from the Na2 A(1Σ+u ) and B(1Πu)potentials to the X(1Σ+g ) potential, where we display the transitionsbetween νA = [0− 25] (νB = [0, 25]) and νX = [0, 60] states. . . . . . 593.3 The A(1Σ+u ) Na2 potential extracted piece by piece by including anever increasing number of excited vibrational states. . . . . . . . . . 603.4 Number of wrong signs of transition dipole moments for the A(1Σ+u )Na2 potential by including an ever increasing number of excited vi-brational states, where the black circle, red square, and blue diamondlines denotes signal strength greater than 1 × 10−2, 1 × 10−4, and1 × 10−6 of the peak signal, respectively. The small inset describesthe same physical argument with different uncertainties in the mea-sured signal strength (green: 5% uncertainty; orange: 10% uncertainty). 623.5 The extracted potential from Eq. (3.21) for the B(1Π+u ) state. . . . . 633.6 (a) The extracted dipole function µX,A(R) for transitions betweenthe X(1Σ+g ) and A(1Σ+u ) potentials. (b)The extracted dipole functionµX,B(R) for transitions between the X(1Σ+g ) and B(1Π+u ) potentials. 653.7 Computed A 1Σ+u potentials with νA = [0− 25], [1− 25] and [2− 25]states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.8 (a) The deviations of the average extracted A(1Σ+u ) potential relativeto the true ab-initio potential as a function of R, for different RMSerrors of the simulated fluorescence line strengths. (b) Root-mean-square (RMS) errors of the constructed B 1Π+u potential with a varyingdegree of errors in the experimental fluorescence data. |di,s|2. . . . . 67xList of Figures3.9 (Black line) - the ground X(1Σ+g ) Na2 state. (Red dots) - ab initio energiesof the excited C(1Πu) potential; (Green dashed line) - the initial Morsefit. (Brown line) - the partial Na2 C(1Πu) potential extracted using onlythe s = [0, 32] excited bound states and the i = [0, 63] vibrational groundstates. (Violet) - the highest vibrational states used. (Blue) - the entireexcited C(1Πu) potential extracted using the s = [0, 110] states. (Cyan) -the highest vibrational state used. This extraction can only be done whenwe incorporate transitions to the continuum of the ground X(1Σ+g ) state.The RMS deviation of the potential extracted in this way from the ab initiopotential is 0.1 cm−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.10 The ds,k/µ Franck-Condon factors from the s = 40 state of the Na2 C(1Πu)state to the discretized continuum states of the X(1Σ+g ) state for two dif-ferent boxes: Blue stars - the R = [0 − 15] box results; black circles - theR = [0− 25] box results. The “exact” ds(k)/µ values, as obtained using theACM scheme, are marked by a dashed green line for the [3.5 − 15] Bohrrange and as a red line for the [3.5− 25] Bohr range . . . . . . . . . . . . 723.11 The values of Cs of Eq. (3.29), representing the completeness condi-tion, for each of the φs states of the C(1Πu) potential, as more andmore X(1Σ+g ) (χi+χ(k)) vibrational states are included in the expan-sion. Shown are Cs values for i = 1, ..., 63 (black circles); when weadd to these states all the χ(k) continuum states below 516 cm−1 (redsquares); all the continuum states below 1244 cm−1 (blue triangles);and all the continuum states below 2018 cm−1 (green x’s). The insetshows these energy levels relative to the dissociation energy of theX(1Σ+g ) potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.12 Step-wise construction of a model double-well potential. . . . . . . . 75xiList of Figures3.13 The wave functions of the fifth vibrational state (dashed) shows the maximalenergy to which we are able to re-construct the Na2 21Σ+u (3s+4s) potentialusing only the bound states of the X(1Σ+g ) state. In contrast, the inclusionof continuum states up to the 2300 cm−1 above the dissociation thresholdallows for a complete re-construction of the potential curve. The s = 13 vi-brational level (dot-dashed) exhibits the first indication that an additionalwell (brown line) might exist. Using this information to smoothly extrapo-late the next potential iteration reveals more of the second well (black line).The two wells and the barrier between them assume their fully developedforms (dotted line) when the s = 27 state is introduced. The inset showsthe complete re-construction (blue line) of the 21Σ+u (3s+4s) potential usingup to the s = 110 state (black line). The RMS deviation of this potentialfrom the ab initio one (red dots) is less than 1 cm−1. . . . . . . . . . . . 763.14 (a) A schematic illustration of two 2D PES. (b) Fluorescence linesassociated with a typical spectrum of the system in (a). Image takenfrom Ref. [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.15 A comparison between a PES extracted using Eq. (3.21), using onlythe s = 1− 10 states, with E(s = 10) = 3.602× 10−2 a.u. (full lines),and the “true” PES (points). Image taken from Ref. [6] . . . . . . . 853.16 Extraction of a double-well PES: (a) A comparison between the PESextracted using Eq. (3.21) (full lines), and the “true” PES (points),having used only the s = 1 − 6 states, with E(s = 6) = 3.54 × 10−2a.u.; (b) |φ6(R, r)|2 - the probability-density for the highest state used. 863.17 The average transition dipole moment µ¯e,g of Eq. (3.13) for transitionsfrom |s = 0− 10, J ′ = 45, A 1Σu〉 to the |i = 0− 54, J ′ = 44, X 1Σg〉states as a function of the transition magnitude, |di,s|. Figure andcaption taken from Ref. [7]. . . . . . . . . . . . . . . . . . . . . . . 883.18 A comparison of the “dipole correction” and the FCA-based inversionsof the Na2 A 1Σ+u potential and the A 1Σ+u → X1Σ+g electronic tran-sition function. (a) Differences (in cm−1) between the inverted andthe true potential; (b) Differences (in Debye) between the invertedand the true electronic transition dipole function. Figure and captiontaken from Ref. [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 91xiiList of Figures3.19 A comparison between the accuracies of the computed vibrationalwave functions as derived by the FCA-based inversion and the “dipolecorrection” inversion for (a) φs=0(R) and (b) φs=5(R). Figure andcaption taken from Ref. [7]. . . . . . . . . . . . . . . . . . . . . . . 923.20 A comparison between the “dipole correction” and FCA-based inver-sions of the Na2 B 1Πu potential and the B 1Πu → X1Σ+g transitiondipole function. (a) Differences (in cm−1) between the inverted andthe true potential; (b) Differences (in Debye) between the invertedand the true electronic transition dipole function. Figure and captiontaken from Ref. [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.1 Gating fluorescence of an excited vibrational wavepacket in Na2 forthe gate positions t = O, 37, 75, 112, and 150 fs relatively to thepump pulse. Taken from Ref. [8] . . . . . . . . . . . . . . . . . . . . 984.2 (a) Temporal fluorescence captured over 10 ps. (b) Actual and imagedexcited state wave packets on A1Σ+u potential of Na2 at t=0 and 1 ps.(c) Spectrum of the initial excitation pulse, showing absorption linescorresponding to the excited rovibrational eigenstates. (d) Temporalpulse profile of 50 fs width. The displayed real part of the electricfield demonstrates good phase extraction. . . . . . . . . . . . . . . . 1074.3 (a) Temporal fluorescence captured over 10 ps. (b) “True” (circles) andimaged (full lines) excited state wave packets moving on the A1Σ+upotential of Na2 at t = 0 and t = 1 ps. (c) The spectrum of theinitial excitation pulse, showing absorption lines corresponding to theexcited rovibrational eigenstates. (d) Temporal pulse amplitude andthe -π and π phase of the field at each instant. . . . . . . . . . . . . 108xiiiList of Figures5.1 (Main drawing) The ground, X1Σ+g , and first excited, A1Σ+u , potentialenergy surfaces of Na2 and a schematic description of the light pulses.Marked as (1) is the pulse exciting the ground vibrational eigenstates|0〉 to a set of excited vibrational eigenstates |s〉; (2) the BCC pulsecoupling two excited state |s〉 and |s′〉 to a ground state |f〉; (3) thespontaneous emission from the |s〉 vibrational states of A1Σ+u to the|f〉 vibrational states of X1Σ+g . (Inset) The BCC stimulated emissionprocess which couples states |s〉 and |s′〉 to |f〉. Some ancillary cou-plings of adjacent |s′ + 1〉 and |s − 1〉 states to different final states|f〉 may also result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.2 (a) R(t) captured over 10 ps and strobed every 60 fs in the presenceand absence of the BCC field which removes populations from theexcited state. (b) |R(ω3,4)| as a function of the relative phase φ3,4;5ranging over −π to π. Full black line - f including only states |3〉 and|4〉; dashed blue line - the result of including in addition to states |3〉and |4〉 all ancillary states. (c) The contrast, C(0, π), of Eq.(5.8), asa function of |ǫ(ω3,5)/ǫ(ω4,5)| - the ratio between the two interferingcomponents of the BCC field, for fixed |ǫ(ω3,5)|2 + |ǫ(ω4,5)|2 sum ofintensities. The line codes are as in (b). The maximum contrast marksthe point at which |ǫ(ωs,f )/ǫ(ωs′,f )| = |as′µs′,f/asµs,f |, allowing usto extract |as′/as| and (by varying |f〉) |µs′,f/µs,f |. (d) |R(ω3,4)| atdifferent φ3,4;0 - φ3,4;f phase differences for various final states |f〉. . 1195.3 (a) Schematic of the RbCs system showing the four PES and thelocation of some vibrational eigenstates. (b) The resulting temporalfluorescence captures over several picoseconds for when the spin-orbitcoupling (SOC) is excluded (blue-square) and included (black-circle). 120xivList of Figures5.4 (a) The BCC pulse intensity (black bars) and bandwidth (blue circles)used to achieve sufficient C(0, π) contrast (of at least 20%). The rangeof the pulse intensity used was varied as a function of the final X1Σ+state (f = 20 − 29) probed. (b) Extraction of the singlet and thetriplet components of an excited wave packet Ψ(r, t) at t = 0, theexcitation pulse center (“true” wave packet - blue points) and after 10ps (“true” wave packet - red points). The imaged wave packet, givenat both times as a thick black line, faithfully reproduces the ”true”values. Shown also are the A1Σ+u (thin black line) and b3Π (dashedline) PES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.5 (a) (Main) Schematics of the LiRb system showing four PES’s andselect rovibrational eigenstates. (Inset) The BCC stimulated emissionprocess which couples states |ψs〉 and |ψs′〉 to |χf 〉. Some ancillarycouplings of adjacent |ψs′+1〉 and |ψs−1〉 states to different final states|χf 〉 may also result. (b) Temporal fluorescence captured over 10 psand strobed every 60 fs for two different relative phase choices of theBCC field which either leaves or removes populations from two (ormore) of the excited states. . . . . . . . . . . . . . . . . . . . . . . . 1225.6 (a) |R(ω17,18)| as a function of the relative phase φ17,18;10 rangingover −π to π. Full black line - including only states |17〉 and |18〉;dashed blue line - the result of including in addition to states |17〉 and|18〉 all ancillary states. (b) The contrast, C(0, π), of Eq.(5.9), as afunction of |ǫ(ω17,10)/ǫ(ω18,10)| - the ratio between the two interfer-ing components of the BCC field, for fixed |ǫ(ω17,10)|2 + |ǫ(ω18,10)|2sum of intensities. The line codes are as in (a). The maximum con-trast marks the point at which |ǫ(ωs,f )/ǫ(ωs′,f )| = |as′µs′,f/asµs,f |,allowing us to extract |as′/as| and (by varying |χf 〉) |µs′,f/µs,f |. (c)|R(ω17,18)0 −R(ω17,18)pi| values at different φ17,18;0 - φ17,18;f phasedifferences for various final states |χf 〉. (d) The BCC pulse intensity(line-points) and bandwidth (blue circles) used to achieve sufficientC(0, π) contrast (of at least 20%). The range of the pulse intensityused is shown for three final a3Π state (χf ) probed as a function ofthe lower A1Σ+u ∼b3Π coupled excited state (ψs). The red diamonds(black squares) depict what happens when µs,f and µs+1,f have iden-tical (opposite) signs. . . . . . . . . . . . . . . . . . . . . . . . . . . 124xvList of Figures5.7 (a) Absolute difference of |R(ω17,18)| values for 0 and π cases at dif-ferent φ17,18;0 - φ17,18;f phase differences for various final states |χf 〉in X1Σ. (b) The BCC pulse intensity (line-points) and bandwidth(blue circles) used to achieve sufficient C(0, π) contrast (of at least20%). The range of the pulse intensity used is shown for four finalX1Σ state (χf ) probed as a function of the lower A1Σ+u ∼b3Π coupledexcited state (ψs). The black squares (red diamonds) represent whenµs,f and µs+1,f have similar (opposite) signs. . . . . . . . . . . . . . 1265.8 Extraction of the singlet and the triplet components of an excitedwave packet Ψ(r, t) at t = 2 ps, the excitation pulse centre (“true”wave packet - red points) and after 10 ps (“true” wave packet - bluepoints). The imaged wave packet, given at both times as a thick blackline, faithfully reproduces the ”true” values. Shown also are the A1Σ+u(thin black line) and b3Π (dashed line) PES. . . . . . . . . . . . . . 1276.1 (a) (Main) Four NaK potentials in the diabatic representation: Solidlines - the X1Σ+ and D(3)1Π singlet states; dashed lines - the a3Σ+and d(3)3Π triplet states [9–11]. (Inset) The dispersed emission spec-trum from the coupled D(3)1Π/d(3)3Π state to the lowest singlet andtriplet states. (b) The “true” (Ref. [12–15]) and extracted SO func-tions in NaK, exhibiting good agreement for the off-diagonal (VOD(R))term as well as for the diagonal (VD(R)) term. . . . . . . . . . . . . 1366.2 (a) (Main) Solid black lines - the Morse RbCs PES in the diabaticrepresentation [16, 17]; (Inset) The off-diagonal (V OD(R)) and diag-onal (V D(R)) SO functions between the diabatic potentials [18, 19].(b) Percentage difference between the true and computed SOC eigen-values calculated in the A1Σ+ and b3Π diabatic crossing region, for adifferent number of total included eigenvalues. . . . . . . . . . . . . 138xviin memory,Moshe ShapiroxviiChapter 1Introduction1.1 MotivationMolecules are an exciting platform for fundamental and applied research due to theircomplex quantum mechanical structure. In addition to the degrees of freedom of anatom, a molecule also possesses vibrational and rotational motions that can be har-nessed [20–22]. This added complexity gives molecules interesting characteristicsand behaviors, and opens the door for widespread applications, particularly in thenew field of ultracold chemistry [23]. In this regime, molecular systems would be-come good candidates for performing large scale quantum information processing,and acting as quantum simulations machines. Ultracold molecules would allow forthe realization of spin-lattice models and the study of exotic few- and many-bodyquantum mechanics. In addition, they could become a test bed for fundamentalphysics, including parity violation, time variation of constants, and the search fora permanent electric dipole moment of the electron [24]. Speculatively, moleculesmay even allow for the development of quantum sensors, cloaking devices, and novellasers [25].All of the above hoopla assumes that we have very accurate knowledge of thestructure and coherence properties of the molecules in our system. Acquiring this in-formation, which requires observing the interactions between photons and molecules,concerns the field of molecular spectroscopy. The laser, as a monochromatic, intenseand coherent light source, has made possible high-resolution spectroscopic (HRS)measurements from which molecular constants and discrete energies levels can bedetermined. The absorption/emission spectra tools such as observing the fluores-cence of electromagnetic radiation from molecules, typically in their gas phase, leadsto information about the internal molecular structure (electronic distributions, nu-clear vibrations and rotations). Combining the ability of light to be used as probewith the richness of molecular spectroscopy data in both the frequency and timedomains, we constantly seek to improve on and develop new methods for extract-11.1. Motivationing information from the spectroscopic data in order to better deduce the physicalor chemical properties of a molecular species. A recent quote from a 2013 reviewarticle in Molecular Physics titled Manipulation of Molecules with ElectromagneticFields states: “Theoretical simulations of experiments at ultracold temperatures areimpeded by the lack of numerical methods to produce intermolecular potentials withsufficient accuracy. It is necessary to develop approaches for inverting the scatteringproblem in order to fit intermolecular potentials,...” [24].Alkali-metal diatomics are the simplest molecules consisting of two atoms havingonly one valence electron and thus serve as excellent test objects to probe differ-ent theoretical approaches that extract spectroscopic information. Experimentally,alkali-metal diatomics are also relatively easy to produce and can be addressed us-ing conventional laser sources (Ar+,He-Ne,dye laser, etc.). The research of diatomicmolecule spectroscopy is very old but still active and innovative [26], since under-standing the structure of these diatomic molecules is an essential step in the movefrom atoms to larger molecules. Central to the study of diatomic molecules is theconcept of electronic potential energy surface (PES) which represents an effectivepotential energy function for the nuclear degrees of freedom due to the electronicmotion. The electronic PES’s are key to our current understanding of the structureand dynamics of molecules, especially that of collisional phenomena and chemicalreactivity. Currently, there is an additional motivation for obtaining highly accu-rate diatomic potentials due to our emerging ability to synthesize ultracold diatomicmolecules from ultracold atoms [27–29]. This can be done by, e.g., photoassociation[30], which provides a highly accurate ground potential close to the dissociation limit[31–33].So far, knowledge of PES with many degree of freedoms has relied mainly onab initio [34] or semiempirical [35–38] quantum chemistry computations. Despitethe simple structure of alkali-metal dimers, ab initio (fully-theoretical) calculationsare not sufficient, their results don’t match spectroscopically derived informationof the PES, electronic transition dipole moments or lifetimes, etc. This is becausethe Schrödinger equation cannot be solved analytically, and approximations mustbe used, leading to discrepancies between the empirical and theoretical PES forheteronuclear dimers that can reach several hundred cm−1 while experimental er-rors in the energy eigenvalues are only 0.1cm−1. The accuracy of computationaltechniques is still limited relative to the accuracy of measurements of spectroscopicline positions. Thus, a direct procedure for the extraction (“inversion”) of PES21.1. Motivationfrom spectroscopic data is highly desirable. Though the semiclassical method (theRydberg-Klein-Rees (RKR) method) exists for (one-dimensional) diatomic systems,no direct inversion method has so far been developed for higher dimensional sys-tems. Although many approximate schemes do exist, they only extract potentialswhen the BO approximation is valid (Sec. (4.1)). Fortunately, a lot of effort hasbeen put towards determining the lowest (ground) singlet and triplet PES, since allmolecules typically reside in these states, and thus these states will serve as knownstructure on which any new method can be based. We have developed a fully quan-tum mechanical inversion procedure, based on both frequencies and intensities, thatis really remarkable, valuable, unique and builds a solid foundation. The PES areextracted from experimental observations by elevating relative intensity informationto the status of primary input to the spectrum-to-potential inversion alongside en-ergy level information need for assignment of observed spectra. Implementations areperformed on some of the best studied homonuclear (e.g. Na2) and heteronuclear(e.g. NaK) alkali dimers. This molecular spectroscopic technique looks to fill thevoid where accuracy is limited in regards to knowledge about structure and prop-erties of many other diatomic species. The exact quantum procedure is valid forpolyatomics and may also be generalized to liquid systems in terms of mean forcepotentials.Spectroscopy in the frequency domain produces energy spectral lines which, how-ever, yield no dynamical information. A complementary picture to HRS, is obtainedwith time-resolved spectra which records total absorption/emission intensity as afunction of time. Time-resolved spectroscopy (TRS) has been used to determinelifetimes of excited states (related to transition probabilities and line intensities)used for the study of chemical reactions [39], radiative lifetimes and orbital mix-ing coefficients [40]. Moreover, these measurements allow one to observe quantumstate dynamics operating on short time scales, in which vibrations (100s of fs) androtations (1-1000 picoseconds (ps)) occur [41]. In particular, modern experimentalstudies have been made possible with the recent developments in ultrafast laser tech-nology (e.g. 10 fs pulse in the visible) capable of exciting molecular rovibrationalwavepackets (WP) consisting of a superposition of a set of stationary eigenstateseach with a given phase. Unlike stationary wavefunctions, the molecular WP has awell defined position and group velocity within the uncertainty principle and requiresa coherent excitation to instantiate phase relation between states (using a broadbandfemtosecond pulse). The vibrational WP in diatomics represents the probability of31.1. Motivationthe system having a given internuclear distance and typical oscillates with a periodof a few hundred fs. Thus the time-dependent spectrum is governed by the motionof the WP and reveals information about the internal molecular dynamics.A lot of effort has been put toward the active control of molecular quantumdynamics, especially in regards to state-selective control schemes and for realizingmolecular quantum computations. In addition to developing realistic logic gatesand controlling long-range dipole interactions, it will be imperative to extract stateinformation without loss of amplitude or phase information. In the quantum infor-mation community the issue of phase reconstruction of an unknown quantum stateis recognized as a crucial problem, and the ability to do so in spite of the non-cloningtheorem lies in the multi-replicas generation of a given quantum state as found ina gaseous ensemble. In this work we present a scheme capable of reading out amolecular ensembles state properties, however the approach is reliant on accurateknowledge of the phases of the electronic transition dipole moments.For homonuclear diatomics with no permanent dipole moment, electronic dipoletransitions between rovibrational states are induced by electric fields and their strengthare thus governed by this induced electric dipole moment. This is also true for polar(or heteronuclear) dimers in field-free environment, because the permanent dipolemoment from the non-uniform distribution of charge does not play a role in tran-sitions between different electronic states. Well-known to scattering theory is the“phase problem”, where the objective is the inversion of experimental atomic colli-sion cross sections to yield the interaction potential or repulsive behavior (see reviewin Ref. [42]) by determining the phases of the complex-valued scattering amplitudes[43, 44]. In bound-bound spectroscopy this phase problem is less severe as comparedto the scattering case because here we deal with real electronic transition dipolematrix elements so we only need to determine their signs. Nevertheless, these signsaffect experimental observations, for example, in the short pulse excitation of a wavepacket, the fluorescence signal is composed of the beatings between many transitions,any change in whose signs fundamentally affecting the observations. Researchers,such as J. Tellinghuisen [45, 46], have detailed what determines the magnitudes ofthe Franck-Condon factors (not overall signs), and shows where and why vibrationaloverlap integrals accumulate. However, lacking a potential, Tellingheusen’s workdoes not solve for the signs of the FC factors, since accumulation of overlap inte-grals does not affect the sign of the integral itself because the wavefunction itself canpossess an undetermined global phase in coordinate space. For instance, when the41.2. Thesis Outlinewavefunction of some state |v′ > is unknown, it is not trivial to draw lines betweenbasis wavefunctions |v′′ > and the Frank-Condon factors, < v′′|v′ >. We argue thatthis consistency is important in any theoretical/experimental study when both thewavefunction and the Frank-Condon factors are used: in short pulse excitation ofa wavepacket, both the wavefunction and the Frank-Condon factors influence thepropagation of the system and thus the quantum echo phenomena.Nowhere do these relative phases (or signs) play a greater role than in coher-ent control, whereby one can tailor the amplitude and phase of an electric field atspecific spectral components of the optical field to manipulate the desired complexcoefficients of the eigenstates. This interference between quantum routes involvingdifferent vibrational states are very important to understand/control the quantumdynamics, and we harness this technique to extract relative transition dipole phaseinformation between select states.A fundamental property of molecular electronic states is that they have an asso-ciated spin angular momentum quantum number, S. Similar to most operators inquantum mechanics, the transition probabilities are diagonal in S, when working inthe diabatic representation of the Born-Oppenheimer approximation. Thus, to firstorder, electronic transition between different spin states is forbidden. For example,a diatomic molecule residing in a singlet (S = 0) electronic state would not be ableto access a triplet (S = 1) electronic state to first order. However, we find that ex-tensive mixing between different spin states does occur and its effects on both HRSand TRS can be described by a spin-orbit coupling (SOC) Hamiltonian. This SOCmixing of singlet-triplet bare states is one of the most important mechanisms forextending radiative lifetime of a molecule in a collision free environment. Moreover,it allows for direct access from the scattering continuum to the ground PES; thishas been utilized in approaches geared towards the generation of ultracold molecules[23]. Understanding and describing this coupling mechanism is possible by analyz-ing differences in the HRS (or perhaps TRS) data and gives vital information into amolecule’s behavior.1.2 Thesis OutlineThe following (and last) section of this chapter will mention the units and numericalmethods used in the simulations throughout this work. The next chapter will thenprovide a basic review of the background theory necessary for understanding the51.2. Thesis Outlinesubsequent research. For completeness, we cover fundamental molecular structureincluding the Born-Oppenheimer approximation, and an introduction to diatomicand polyatomic molecules. Light-matter interaction from a semi-classical perspectivefollows with application to molecular transitions. Next, we present laser excitationand control schemes directed at diatomic molecules in an ensemble, before coveringmolecule spectroscopy in the frequency and time domains.In Chapter 3, we first discuss currently existing methods for determining poten-tial energy surface from spectroscopic data. Then, we develop an inversion schemefor obtaining the signs of transition-dipole amplitudes from fluorescence line inten-sities. Using the amplitudes thus obtained we show how to extract highly accurateexcited state potential(s) and the transition-dipole(s) as a function of inter-nucleardisplacements. The same dipole amplitudes can also be used to extract the phaseand amplitude of unknown time-evolving wave packets, in essentially a quantumnon-demolition manner. The procedure, which is demonstrated for the A(1Σ+u ) andB(1Πu) states of the Na2 molecule, is shown to yield reliable results even when weare given incomplete or uncertain data.Next, we present an extension to the bound-continuum cases of our bound-boundinversion scheme for extracting excited state potentials and transition-dipoles fromfluorescence data. The procedure involves the discretization of the continuous spec-trum using box-normalization. The addition of the continuous spectrum guaranteescompleteness of the basis set used in the implicit expansion of the unknown excitedstate vibrational wave functions. The method, which is found to be robust withrespect to missing data or uncertainties in the line strengths, is also capable of in-verting polyatomic fluorescence data. We demonstrate the viability of the method bysuccessfully generating the potential energy curve (PEC) for the C(1Πu) state of theNa2 molecule using a fraction of the total transitions to/from the X(1Σ+g ) groundstate, and by extracting the double-well 21Σ+u (3s+4s) curve of Na2, assuming noprior knowledge of its structure.Lastly, we present two extensions of this work. First, we demonstrate a success-ful extraction of a model two-dimenionsal (2D) PES using our inversion procedure.This follows along the same route as in the one-dimensional (1D) case, by using themagnitudes and positions of a set of frequency-resolved fluorescence (or absorption)lines we extract the relative phases of the transition-dipole matrix elements. Withthis information together with the (ground) PES to (from) which emission (absorp-tion) occurs, we reconstruction a point by point two dimensional excited state PES.61.2. Thesis OutlineThe method is applied to 2D PES with multiple minima and many missing lines, andachieves typical (root-mean-square) RMS errors of < 0.002 cm−1 in the classicallyallowed region, and < 0.02 cm−1 in the classically forbidden region.Next we present a new (“dipole correction”) inversion scheme for the accurate ex-traction of excited state potentials from fluorescence line positions and line strengthswhich does not require the Franck Condon Approximation (FCA). The accuracy ofthe potential energy surfaces (PES) thus extracted is much higher than that of theFCA derived PES because we make use of the coordinate dependence of the electronictransition-dipoles. Using Na2 we find the A→ X electronic transition dipole functionto accuracies better than 1 × 10−3 Debye. Then we use the A(1Σ+u )→X(1Σ+g ) P-branch emission to extract the excited state potential to global errors of less than 0.1cm−1, Moreover, we demonstrate that using emissions data from only the s = 0− 5low-lying levels or the s = 20 − 23 states of the Na2 B(1Πu) PES, we can obtaininversion results with global errors as small as 0.08 cm−1.In Chapter 4, we describe methods that one can use to image molecular wave-functions and discuss their associated drawbacks. Then, we present a reference-freerobust method for the non-destructive imaging of complex time-evolving molecularwave functions using as input the time-resolved fluorescence signal. The method isbased on expanding the evolving wave function in a set of bound stationary states,and determining the set of complex expansion coefficients by calculating a series ofFourier integrals of the signal. As illustrated for the A1Σ+u electronic state of Na2,the method faithfully reconstructs the time-dependent complex wave function ofthe nuclear motion. Moreover, using perturbation theory to connect the excitationpulse and the material expansion coefficients, our method is used to determine theelectromagnetic field of the excitation pulse. Thus providing a simple technique forpulse characterization which obviates the additional measurements and/or iterativesolutions that beset other techniques. The approach, which is found to be quiterobust against errors in the experimental data, can be readily generalized to thereconstruction of polyatomic vibrational wave functions.To address the “phase problem”, to which no direct approach exist, in Chapter 5we develop an imaging method which uses Bi-chromatic Coherent Control (BCC) inconjunction with time-resolved fluorescence to extract the complex amplitudes (mag-nitudes and phases) of individual Transition Dipole matrix elements (TDM’s) as wellas the amplitude of time-evolving wave packets. The method relies on determiningthe phase relation between the BCC fields, which look to deplete the population of71.3. Units and Numerical Simulationsdifferent pairs of excited energy eigenstates, through the computation of a Fourierintegral of the time-resolved fluorescence at the beat frequencies of these pairs ofstates. We illustrate our procedure by determining the amplitudes of the TDM’slinking the vibrational states of the A1Σ+u and those of the X1Σ+u electronic states ofNa2. Furthermore, we demonstrate its broad applicability to systems in which thereexist interacting potential energy surfaces by extracting the expansion coefficients ofa wave packet in the basis of vibrational energy eigenstates in the strong spin-orbitcoupling potentials in RbCs.In addition, we illustrate our method by determining the amplitudes of the TDMslinking the vibrational states of the A1Σ+u ∼ b3Π spin-orbit coupled potentials toboth the singlet X1Σ+u and triplet a3Σ+u electronic ground states in LiRb. The ap-proach, which is found to be quite robust against errors in the BCC procedure andexperimental data, can be readily generalized to the imaging of wave packets ofpolyatomic molecules.Lastly, Chapter 6 develops the first direct extraction of non-adiabatic couplingsfrom HRS data. We show that it is possible to extract spin-orbit and non-adiabatic(non Born Oppenheimer) coupling terms, as well as potential curves and transitiondipole moments from fluorescence line positions and line strengths. We demonstratethe viability of our method by extracting the spin orbit couplings of the weakly cou-pled D(3)1Π and d(3)3Π electronic states of NaK, and the strongly coupled A1Σ+and b3Π electronic states of RbCs. The method can be applied to the extraction ofnon-adiabatic couplings in multi-dimensional systems and for more than two inter-acting electronic states.To end we provide a final conclusion summarizing the key results of this thesis,and their applicability. The bibliography can be found thereafter.1.3 Units and Numerical SimulationsThroughout the theoretical work of this thesis we will be using atomic units, namely,a0 = me = e = ~ =14πǫ0= 1 (1.1)where a0 is the Bohr radius, me and e is the mass and elementary charge of theelectron respectively. ~ represents Dirac’s constant and Coulomb’s constant includesthe permittivity of free space ǫ0. The energy in these units is given by Hartrees81.3. Units and Numerical Simulations(labelled as H or a.u.) and the speed of light c ≈ 137.Atomic units were used in all numerical calculations, however, for presentationpurposes the final units were occasionally converted to a more recognized form.Spectroscopist typically use units of wavenumber (cm−1) and Angstroms (Å) forenergies and distances, with conversions as follows:1 H ≈ 21947 cm−1 1 Bohr ≈ 0.5292 Å . (1.2)Time is always converted back to the SI units of seconds.The simulations themselves were generally done using very well known diatomicmolecules. One of the most explored molecules and theoretically simplest after H2,due to small number of valence electrons, may be Na2 (after Li2). References willshow that Na2 has thoroughly explored low-lying single and double well electronicstates, including spectroscopically measured adiabatic curves and ab initio diabaticpotentials and non-adiabatic (spin-orbit) couplings functions. For each method, the“experimental” data has been simulated from the given electronic structure. It will beshown how this information can be easily attained from laser fluorescence via high-resolution Fourier transform spectroscopy or time-resolved upconversion techniques.It will be assumed that the frequency-resolved spectra can be correctly assigned toeigenenergies of electronic PES (e.g. via combination differences or LoomisWooddiagrams).Numerical solution to the time independent Schrödinger were found using a ba-sic finite difference approach. Typical higher order methods for solving ordinarydifferential equations, such as 4th-order Runge-Kunta or Numerov’s method, werefound to be unnecessary. Using a finite difference grid by discretizing space intoN distinct points, Ri, for i = {1, N}, allows for simply evaluation of functions andtheir derivatives. For example using a one-dimensional (1D) Taylor expansion of thesecond derivative operator about a point Ri, we getd2dR2ψ(R) ≈ψ(Ri + h)− 2ψ(Ri) + ψ(Ri − h)h2 (1.3)where h = Ri+1 − Ri represents the distance between adjacent points. A 1D treat-ment of Schrödinger equation can now solved asψ(Ri + h)− 2[1 + h2V (Ri)]ψ(Ri) + ψ(Ri − h) = Eψ(Ri) (1.4)91.3. Units and Numerical Simulationsfor eigenvalues E with eigenstates ψ(R) at each point Ri. The N above equationscan be succinctly expressed as a tridiagonal matrix containing the kinetic and po-tential energy terms, with column vectors ψ(R) solving the equality for some givenenergy E. Using common matrix diagonalization routines, either from LAPACK inFORTRAN or eigen3 in C++, a set of N eigensolutions can be found matching thegrid dimensions.An alternative approach used to solve Eq. (1.4) for large grid sizes, when thediagonalization becomes cumbersome, is known as the shooting method. We treatthe Schrödinger equation as a two-points boundary value problem where ψ(R)→ 0at the ends of the range (for bound states) and re-express Eq. (1.4) asψ(Ri + h) = 2[1 + h2(V (Ri) + E)]ψ(Ri) + ψ(Ri − h) . (1.5)Using two initial values ψ(R0) = 0 and ψ(R1) = 1 and starting at the minimumpossible energy, E = 0 or E = −De, where De is the dissociation limit of a PES,we can propagate the solution to ψ(Ri) along the range until reaching the end.Our objective is to increment E until we find ψ(RN ) ≈ 0, which yields a solutionsatisfying the two boundary conditions. This convergent solution gives the eigenvalueE and corresponding eigenvectors one at a time without large computational expense.10Chapter 2Background TheoryHere we provide an introduction to molecular structure based on the time-independentSchrödinger equation (TISE). We wish to understand the basic principles govern-ing molecules, especially diatomics, and the origin of their spectroscopic properties.This theoretical background of their energy structure will be necessary for the under-standing of molecular excitations, wavepackets, and light-matter interactions. For amore detailed review of the physics of diatomic molecules please see Refs. [47, 48].A directly computational approach to the TISE for molecular system will befound to be unfeasible, and several approximations will be discussed. Most impor-tantly, is the assumptions that the electrons adiabatically follow motion of nucleiand the nuclei rotate and vibrate within effective field of electrons. This allows for adefinition of the potential energy surface, however, breakdowns of this approximationwill become important, particularly when considering non-adiabatic couplings.2.1 Molecular StructureIn order to describe a molecular system we look towards solving the TISEH(q,R)Ψ(q,R) = EΨ(q,R) (2.1)where H(q,R) is the quantum mechanical Hamiltonian for the molecule, E andΨ(q,R) are the eigenvalues and corresponding eigenstates with q and R represent-ing the collective electronic and nuclear coordinates, respectively. The molecularHamiltonian H(q,R) can be written as a sum of its electronic and nuclear partsH(q,R) = He(q,R) +HN (R) , (2.2)withHe(q,R) = Te(q) + Vee(q) + VeN (q,R) (2.3)112.1. Molecular StructureandHN (R) = TN (R) + VNN (R) , (2.4)where Te(R) and TN (R) are the kinetic energy operators for the electronic andnuclei respectively. The V terms are the electrostatic potential energies representing:the sum of all electron-electron repulsionsVee(q) =12∑i 6=j1|qi − qj |,the sum of all nuclear-nuclear interactionsVNN (R) =12∑α 6=βZαZβ|Rα −Rβ |,and the sum of all electron-nuclei attractionsVeN (R) = −∑α 6=iZα|Rα − qi|.The coordinate vectors Rα designates the position of a nucleus α and qi are the elec-tronic coordinate of the ith electron. The strength of these Coulombic interactionsare quantified by the nuclear charges Zα of the α nucleus and the elementary chargeof the electrons (e = 1).2.1.1 Born-Oppenheimer ApproximationEven for small molecules, treated non-relativistically, it is not possible to obtain anexact solution to the Schrödinger equation in Eq. (2.1). The complexity of the Hamil-tonian even prevents one from performing direct numerical (ab initio) molecularstructure calculation. As a result, progress must be made through approximations,the most fundamental of which, known as the Born-Oppenheimer approximation(BOA), enables one to separate the nuclear from the electronic coordinates. Basedon the large mass difference between the electrons (m = 1) and nuclei (µ ' 1836),it is assumed that the electronic motion occurs on a much faster timescale than thenuclear motion, therefore the electronic configuration quickly adjusts itself to anynuclear motion. By assuming that this adjustment occurs instantaneously we may122.1. Molecular Structurewrite the total wavefunction asΨ(q,R) = Ψe(q;R) · ψN (R) (2.5)where Ψe(q;R) are the electronic eigenstates of the electronic Hamiltonian in Eq.(2.3)He(q;R)Ψe(q;R) = Ee(R)Ψe(q;R) , (2.6)and ψN (R) are the nuclear wavefunctions for the corresponding Hamiltonian in Eq.(2.4)HN (R)ψN (R) = ENψN (R) . (2.7)Notice in Eq. (2.6) that under the BOA it was assumed that Te ≫ TN (by a factor ofm/µ), thus the effect of the nuclear kinetic energy operator TN (R) on the electronicwavefunctions Ψe(q;R) has been ignored. Corrections to this approximation will beaddressed in Sec. (2.1.2).By omitting the couplings between the nuclear and electronic parts, the fullSchrödinger equation can be written in terms of the nuclear wavefunctions(TN (R) + VNN (R) +Ee(R))ψN (R) = EψN (R) , (2.8)where the electronic wavefunctions, Ψe(q;R), normalized to unity (〈Ψe |Ψe 〉 = 1),have dropped out upon integration over q. The sum of the electronic energy andCoulomb potential terms of the nuclei in Eq. (2.8),V (R) = VNN (R) +Ee(R) , (2.9)represent the effective (or averaged) potential on which the the nuclei move, and arereferred to as (electronic) potential curves (or surfaces). In other words, it representsthe potential energy governing the dynamical equations of nuclear motion. Thesepotential energy surfaces (PESs) of a molecule, V (R), do not represent any physicalobservable, but are merely convenient mathematical constructions derived from aspecific set of assumptions, namely the separation of variables (Eq. (2.5)) and that〈Ψe |TN |Ψe 〉 ∼ TN 〈Ψe |Ψe 〉. When the eigenvalues of the full Hamiltonian, Eq.(2.1), which are now a sum of the nuclear-coordinate dependent electronic eigenvalue,132.1. Molecular StructureEe(R), and the nuclear energy ENE = Ee + EN (2.10)do not exactly match observed energy levels, it points to a failure in one of these ap-proximations (see discussion in Sec. (2.1.2)). By treating molecules in this way, theelectronic Schrödinger equation, Eq. (2.6), can be solved for various electronic con-figuration at different values of the inter-nuclear separation R. Then each electroniceigenenergy defines a different Born-Oppenheimer potential energy surface (PES)which defines the allowable vibrational and rotational motions and energy states ofthe nuclei.2.1.2 Diatomic MoleculesWe now continue the discussion with an emphasis on diatomic molecules, thoughmany of the following concepts extend to polyatomic systems (see Sec. (2.1.3)).A diatomic can easily be cast into a central-potential problem, where the frameof reference of the nuclear Schrödinger equation of Eq. (2.8) is moved into thecenter-of-mass (CM) coordinates and the variables are separated into radial andangular spherical components. Namely, the nuclear wavefunction is expressed asψN (R) = R−1ψν,l(R)Yl,m(θ, φ) where ψν,l represents the radial wavefunction andYl,m are the spherical harmonic functions. By omitting the CM motion, the problemreduces to the familiar radial Schrödinger equation (with ~ = 1), whose eigenstatesrepresent the possible vibrational and rotation (i.e. rovibrational) states available tothe molecule under the influence of the central potential,(− 12µd2dR2 + V (R) +l(l + 1)2µR2)ψν,l(R) = Eν,lψν,l(R) , (2.11)where µ is the reduced mass, ν and l label the vibrational and rotation quantumnumbers respectively, and R is a scalar parameter representing the separation be-tween the atoms. One should see that the form of the above equation is identical tothat for a single particle in a one dimensional (1D) effective potential given byV effl (R) = V (R) +l(l + 1)2µR2 (2.12)142.1. Molecular Structurewhere the second (centrifugal) term gives the energy associated with the rotationalmotion, and V (R) represents the radial dependent potential curve (or PES). Asexpected, when values of l, the quantized amount of spatial angular momentum, getlarger, the atoms experience a stronger repulsive force that stretches the bond, thusalso affecting the vibrational modes of the molecule. This behavior is a manifestationof rotational-vibrational coupling. For an extensive description of diatomic moleculesplease refer to Ref. [47].Electronic StatesThe electronic eigenstates or PESs in the Born-Oppenheimer approximation caneither be obtained from direct ab initio calculations of the electronic Schrödingerequation (Eq. (2.6)), or from inversion techniques based on the analysis of exper-imental data (Sec. (3.1)). The classification of these molecular electronic states isdone according to the quantum numbers of the angular momenta operators thatcommute with the electronic Hamiltonian (He(q,R)). In diatomic molecules thereexist angular momenta associated with the electron orbital motion (L), the electronspin (S), the nuclear rotation (R), and the nuclear spin (I). Ignoring the small nu-clear spin component, the total angular momentum is written as J = L+S+R, andthe total exclusive of spin as N = J − S. However, not all these angular momentaare conserved and they may interact together in various combinations. In particular,there are five idealized coupling cases which may occur between these angular mo-menta [1, 49], and the choice of which coupling pair (i.e. {L,S}) dominates definesa given basis set of H(q,R) (Eq. (2.2)).The preferred basis throughout this thesis will be that of Hund’s case (a) (seeFig.2.1) in which the strongest coupling occurs between the two angular momentaL and S and the inter-nuclear axis (A), namely the energies associated with the{L,A} and {S,A} interactions are much larger than the other pairs. This is themost common choice in most diatomic molecules and allows for the definition oftwo new “good” quantum numbers (i.e. operators that commute with He(q,R)).We define the L and S projections onto the inter-nuclear axis, A, by Λ and Σrespectively. In addition, this defines another conserved quantity known as the totalmolecular angular momentum given by Ω = Λ + Σ. The set of good quantumnumbers {J, S,Λ,Σ,Ω} now define a molecular basis set, and can be used to label152.1. Molecular Structureelectronic eigenstates as follows:(n)2S+1Λ±Ω,g/u, (2.13)where S = {0, 12 , 1, ...} is the electronic spin quantum number and we describe thecorresponding molecular state as either a singlet, doublet, triplet,... respectively. Theelectronic label n denotes the eigenstate of He(q,R) similar to the atomic principlequantum number. These are typically labelled, in terms of increasing energy, n =X,A,B,C, ... (for experimentalist) or n = 0, 1, 2, 3, ... (for theorist), while excitedstates of different spin multiplicity to that of the ground electronic state use lowercases letters n = a, b, c, .... The states corresponding to the value of the projection ofFigure 2.1: Hund’s case (a) angular momentum and good quantum numbers.[1].the orbital angular momentum on the molecular axis Λ = 0,±1,±2, ... are designatedas Σ, Π, ∆,... respectively. The double degeneracy of these states (whenever Λ 6=0) may be lifted through the interaction of R and L (Λ-doubling), however thisonly becomes important for high rotational speeds (large J values), and will notbe encountered in this thesis. The ± and g/u in Eq. (2.13) specifies symmetries162.1. Molecular Structureof the electronic wavefunction. For Λ = 0 (i.e. Σ states), the + (even) or −(odd) represent the symmetry with respect to a reflection in the plane containingthe inter-nuclear axis. Then for homonuclear diatomics, there exist a symmetryof the wavefunction about the inversion center which can either be gerade (g,even)or ungerade (u,odd). These symmetries will become important when consideringtransitions between different electronic states (in Sec. (2.2.2)).Another important basis set which will be encountered arises from Hund’s case(c). In this situation the spin-orbit interaction, {L,S}, is stronger than their indi-vidual interactions with the internuclear axis (A). The projections, Λ and Σ, cannot longer be defined, and it is the projection of the Jα = L+ S vector onto A thatdefines Ω. Just as in case (a), the doubly degenerate Ω 6= 0 states are split withthe inclusion of rotation (Ω-doubling), though these effects won’t be considered here.Most notable, is that these electronic states now exhibit avoided-crossings, a factwhich will be returned to in Sec. (2.1.2).Lastly, it should be mentioned that because the couplings associated with the nu-clear spin angular momentum (I), which give rise to hyperfine structure in molecules,are much smaller than all other couplings, they will be ignored throughout this work.See Ref. [1] for more details and a descriptions of the Hund’s cases.Vibrational and Rotational EnergiesEach distinct electronic eigenenergy, (n)2S+1Λ±Ω,g/u, defines a PES in the nuclearcoordinates for a molecule’s vibrations and rotations. In order to study these vibra-tional and rotational states we begin with two further approximations. Assumingthat there exist a minimum to the electronic PES (V(Re)), instead of an unstableelectronic state in which the two atoms repel each other for all values of R, then onecan Taylor expand about this equilibrium distance (Re) to second order givingV (R) ≈ V (Re) +ke2 (R−Re)2 . (2.14)This is known as the harmonic approximation where ke is the force constant of themolecular bond which relates to the oscillation frequency through ωe = (ke/µ)1/2.Then using the centrifugal potential term (Eq. (2.12)) the rotational frequencyof the molecule can also be estimated as ωr ≈ 1/(µR2). Therefore, we find that fora typical molecule the ratio between these frequencies is ωe/ωr ∼ O(102− 103), so itcan be assumed that a molecule generally rotates at the midpoint of its vibrational172.1. Molecular Structureoscillation, namely at the internuclear distance Re. This allows us to apply therigid rotor approximation, replacing R with Re in the centrifugal potential. Now bysolving the radial Schrödinger equation (Eq. (2.11)) with these simplified potentialswe find the rovibrational energiesEν,l = V (Re) +Bel(l + 1) + ωe(ν + 1/2) , (2.15)where Be ≡ 1/(2µR2e) is known as the rotational constant, l = {0, 1, 2, ...} is therotation quantum number, and ν = {0, 1, 2, ...} is the vibrational quantum number.Higher order corrections in both the vibrational and rotational potentials can beused to improve the above result, namely,Eν,l = V (Re) +G(ν) +Bν l(l + 1)−Dν [l(l + 1)]2 +Hν [l(l + 1)]3 − ...(2.16)where G(ν) = ωe(ν + 1/2)− ωexe(ν + 1/2)2 + ωeye(ν + 1/2)3 + ... (2.17)The unfamiliar coefficients in Eν,l are the so-called band constants which account forcentrifugal distortions in the rotational motion. The additional coefficients in G(ν),called the vibrational constants, provide anharmonic corrections to the quadraticapproximation. Together these series of constants which can be obtained from ex-perimental measurements are known as the spectroscopic constants. In Sec. (3.1)this model will be further discussed in the context of determining electronic potentialenergy surfaces.Another common approach to account for the anharmonicity of a molecular PESsis through modeling the electronic potential using a Morse functionV (R) = De(e−2β(R−Re) − 2e−β(R−R2))+De (2.18)where the (Morse) parameters β = √2µωeχe and De = ωe/(4ωeχe) control the widthand depth of the well, respectively. In Fig. (2.2) ([2]), a typical Morse potential(dashed-curve) is shown together with a realistic PES of Be2. As expected, the curvesagree very well about the equilibrium position (minimum of the potential) and alongthe repulsive wall but diverge at longer range. Alternatives to the Morse functions fordescribing PESs will be discussed in Secs. (2.4,3.1). However, this form captures theeffect of the anharmonicity of the PES, whereby rovibrational levels get increasinglycloser together as the vibrational quantum number ν increases. Eventually, whenthe potential energy function reaches zero (at the dissociation energy), the molecule182.1. Molecular StructureFigure 2.2: Ground electronic state potential energy curve of the beryllium dimer.The vibrational wave functions for ν = 0, 3 are also shown for reference. The dashedcurve is a Morse potential constructed to reproduce the experimental dissociationenergy and harmonic vibrational constant [2].192.1. Molecular Structureis no longer bound and the difference in energy between successive quantum statesvanishes forming an energy continuum. Note that the dissociation energy may bemeasured either from the bottom of the potential (De, as in Eq. (2.18)), or from thezero point vibrational energy, in which case it is called D0.Continuum StatesAbove the dissociation limit the vibrational energy of the continuum state(s) areno longer quantized; consequently, these vibrational wave functions (ψE,l(R)) arelabeled by the quantum number, E (for energy). Unlike the bound state eigenstateswhich can be “space-normalized” since ψν,l(R) R→{0,∞}−−−−−−→ 0, the continuum functionmust be “energy-normalized” because ψE,l(R) 6= 0 for R → ∞. A general treat-ment of continuum (scattering) states is beyond the scope of this work (see Ref.[50] on Scattering Theory), instead these states will be approximated in terms ofquasi-continuum states using the method of Box-Normalization. By adding a verti-cal and infinite barrier at large internuclear distance above the dissociation energythe continuous spectrum can be artificially discretized into a set of quasi-continuouslevels. These quasi-continuous states, now “bound” within a region of space, can nowbe space-normalized to unity. However, notice that the location of this outer wallwould affect the amplitude and density of these discrete levels, thereby influencingtheir behaviors and couplings within the system. So care must be taken in charac-terizing these states if they are to be treated as ordinary vibrational bound states.In particular, note that the product of the amplitude times the density of statesremains a constant, finite value as the barrier goes to infinite internuclear distance;even when the amplitude of each wavefunction goes to zero, and the density of statesbecomes infinite, thereby returning to a continuous spectrum. See Sec. (2.4.3) fora comparison of the transition dipole couplings between bound–continuum statesderived using the artificial channel method [51–53], and the bound–quasi-continuumstates.Diabatic and Adiabatic CurvesIn this section we will go beyond the Born-Oppenheimer(BO) approximation anddiscuss the perturbations that arise from the off-diagonal elements of the molecularHamiltonian H(q,R) in the BO basis. The exact eigenvalues and eigenfunction ofH(q,R) can always be expressed as an infinite linear combination of any set of basis202.1. Molecular Structurestates, however, as Lefebvre-Brion and Field state: the BO wavefunctions of Eq.(2.5) are the “only available type of complete, rigorously definable basis set” [48].There are two different representations of the BO basis that will be most useful.We have already encountered one, known as the diabatic basis, in which the electronicwavefunctions, |Ψde,i 〉, are defined such that〈Ψde,j |TN |Ψde,i 〉 = 0 (2.19)In this case, the off-diagonal elements of He give rise to either electrostatic perturba-tions from Vee for states with identical electronic symmetry (equal Λ,Σ, S), or spin-orbit couplings, upon inclusion of a relativistic perturbative operator, HSO(q,R), tothe non-relativistic Hamiltonian He, between states of different Λ, S, but the sameΩ = Λ + Σ. Because the diabatic functions are not exact solutions of the electronicHamiltonian (i.e. 〈Ψde,j |He|Ψde,i 〉 6= 0), these PESs are able to exhibit crossings.Figure 2.3: Diabatic (crossing) potential energy curves cross at Rc as a result ofneglecting the part of He that causes the adiabatic (non-crossing) curves’ avoided-crossing by 2He [3].Alternatively, the adiabatic functions, |Ψade,i 〉, which are defined as the exact212.1. Molecular Structuresolutions to He,〈Ψade,j |He|Ψade,i 〉 = 0, (2.20)take into account the electronic couplings between the diabatic functions and leadto PESs with an avoided-crossing proportional to the strength of the interaction (seeFig.2.3). This occurs due to the non-crossing rule of von Neumann and Wigner [54](initially proposed by Hund [55]), which states that for diatomic molecules therecannot exist a degeneracy in the electronic states (which would occur at a crossingpoint). The perturbations in the adiabatic basis, known as non-adiabatic inter-actions, now arise due to the off-diagonal elements of the nuclear kinetic energyoperator, TN (R).We leave this discussion here (see Ref. [3] for more details), noting, as in Fig.2.3,how PES with double minima may arise. Unless stated otherwise, this thesis willwork in the diabatic picture in which interactions between curves (when they oc-cur) are introduced as off-diagonal elements in the electronic Hamiltonian in theunperturbed basis. However, the diabatic states are not unique, unlike in the adia-batic case, their form can change along with the interaction terms yielding the sameelectronic Hamiltonian. Lastly, because this presentation will focus on rotationlessstates (J = 0), the rotational and spin-rotation perturbations will not be addressed.In addition, the matrix elements associated with spin-spin couplings will be ignoreddue to their relative weakness to those of the spin-orbit terms.2.1.3 Polyatomic MoleculesDetermining the molecular structure of polyatomic molecules becomes increasinglymore difficult as one increases the number of constituent atoms. This is a conse-quence of the many more nuclear degrees of freedom (DOF) that become availablesuch as additional stretching, bending, and torsional motions. In general, the po-tential energy of a polyatomic molecule is given as a function in 3N − 3 dimensions,where 3N defines the number of independent coordinates available to N ≥ 3 num-ber of atoms in free space, and, three coordinates are required to specify the centerof mass of the body. By solving separately for the rotational motion through in-troducing constants for the angular momentum, the DOF of the effective potential(potential plus centrifugal terms) reduces to 3N − 6 since three more DOF are usedto define the object’s orientation in space. The remaining coordinates for specify-ing the molecular structure will now consist of internuclear distances and/or bond222.1. Molecular Structureangles, and it should be evident that for molecules with more than three atoms (orthose triatomics in which the bond lengths and angles are independent), it will notbe possible to plot and fully visualize the potential energy surface (PES).Another complication in the description of polyatomics is that the internal co-ordinates (distances, angles) do not generally form an orthogonal set. This impliesthat there exist non-zero off-diagonal terms in the Hamiltonian for the kinetic energy,i.e. couplings between the normal modes. However, it is possible to abandon thenormal modes picture and define a new set of coordinates which are linear combina-tions of the internal coordinates where the kinetic energy operator becomes diagonal[56] (although there exist an infinite number of these sets). The only polyatomicmolecules with a reasonable simple form of the kinetic energy operator are the lineartriatomics, which can be described by a set of three coordinates.As we saw for diatomic molecules, with a single vibrational DOF, when two elec-tronic states possess the same symmetry, we encounter an avoided crossing. However,for polyatomic molecules, which have multiple vibrational DOF, the crossing of suchelectronic states is possible, and leads to a structure known as a conical intersection(CI). The CI is a 3N − 8 dimensional subspace in which the two electronic statesof the same spatial/spin symmetry are allowed to cross, because their energies re-main degenerate. The two-dimensional space that lifts this degeneracy is spannedby two vectors relating the two intersecting electronic states: namely, the differenceof their energy gradient vectors, and their non-adiabatic coupling vectors. Whenthe PES is plotted as a function of these two vectors, we observe a pair of conesmeeting at the degeneracy point, and separated by the branching plane that liftsthe degeneracy, leading to the name “conical intersection”. The occurrence of conicalintersections is often a result of the Jahn-Teller theorem [57] which states that anonlinear molecular with a spatially degenerate electronic state will spontaneouslydistort its configuration to that of a reduced symmetry in order to lower its overallenergy.When two adiabatic PES come close to each other, as they do in the vicinity ofconical intersections, the vibronic coupling becomes large leading to a breakdown ofthe BOA (and giving rise to such non-adiabatic phenomena as radiationless decay).The coupling of different electronic states through nuclear vibration occurs often inpolyatomics due to the many nuclear DOF and large number of energetically closeelectronic states.Now we begin to appreciate the complexity of performing spectroscopic studies232.2. Light-Matter Interactionon polyatomic systems. Even for the simplest case of linear triatomics, there ex-ist three internal DOF, and many of the electronic states are two fold degenerate,leading to the Renner-Teller effect [58] in which strong mixing occurs between Born-Oppenheimer terms. In lieu of the difficulties associated with polyatomic molecules,this thesis will proceed with diatomics in mind. However, the work to come, will, inprinciple, be application to both diatomic and polyatomic systems.2.2 Light-Matter InteractionIn this section we provide a basic overview of the theory of light-matter interactions,in particular, as it pertains to diatomic molecules. The most general approach usesthe quantum electrodynamics HamiltonianH = H0 +Hf +Hint (2.21)where the system, H0, the electromagnetic (EM) field, Hf , and their interaction,Hint Hamiltonians are all treated quantum mechanically. However, the majority ofmolecular experiments involve high photon densities and thus it can be assumedthat the EM Hamiltonian can be described classically (i.e. by Maxwell’s equations).This opens the door for a semi-classical treatment of the light-matter interactions,in which the matter is treated as an unperturbed quantum mechanical (QM) system(e.g. the Born-Oppenheimer rovibrational Hamiltonian HN (R)) and the influenceof the classical EM field becomes a QM perturbation.2.2.1 Semi-Classical TheoryThe semi-classical approach provides use with the ability to develop an intuitivepicture of the light-matter interaction, and allows use to develop a theory necessaryfor manipulating and studying the internal structure of molecules. For our purposeswe will use only the first (and in our cases dominate) term of the dipole expansionof Hint (e.g. ignoring higher multipole terms such as the electric-quadrupole ormagnetic-dipole); this is known as the electric-dipole approximation. Moreover, wewill make use of the large-wavelength approximation, assuming that our EM field ishomogeneous field over the molecular dimensions, and thus the spatial variation offield can be ignored. With these simplifications, the interaction Hamiltonian canbe written as a scalar product of the electric transition dipole moment operator,242.2. Light-Matter Interactionµe = −∑j eqj , and the time-dependent electric field, ε(t),Hint(q, t) = µe(q) · ε(t) , (2.22)where qi specifies the position of the jth electron and t represents time. Here wehave ignored the nuclear contribution to the dipole moment because it drops out ofthe molecular transition matrixH i,f (t) = 〈Ψf (q,R) |µe(q) · ε(t)|Ψi(q,R) 〉 , (2.23)which quantifies the transition amplitude between some initial, Ψi(q,R), and somefinal, Ψf (q,R), eigenstate of the full molecular Hamiltonian H0. Upon applying theBOA, we can simplify Eq. (2.23) toH i,f (t) = 〈ψfN (R) |µi,fe (R) · ε(t)|ψiN (R) 〉 (2.24)where the electronic transition dipole surface, µi,fe (R), is defined asµi,fe (R) = 〈Ψfe (q;R) |µe(q)|Ψie(q;R) 〉 (2.25)The series of approximations leading to this expression of the probability ampli-tude for a transition from an initial state | i 〉 to a final state | f 〉 is known as theFranck-Condon (FC) principle. In regards to the interaction, this assumes that thetransition time is short relative to changes in the nuclear configuration, thus theamplitude of the transition is related to the degree with which both the initial andfinal electronic (Ψie(q;R)) and nuclear (ψiN (R)) wavefunctions overlap respectively.As a result, electronic transition are more likely to occur at the classical turningpoints where the momentum is zero, and where the nuclei spend most of their time.In addition, it is often assumed (in the Condon approximation) that the nuclearcoordinate dependence of transition dipole surface (TDS) is rather smooth, thus itcan be written as an average value over the nuclear coordinates R, µi,fe (R) ≈ µi,fe ).Then, the strength of a transition within molecules becomes modulated by only theFC factor 〈ψfN (R) |ψiN (R) 〉 (up to O(1)). However, the largest determinant of thetransition probability is found through molecular selection rules (see Sec. (2.2.2)),though we will find that these selection rules are not always valid, either due to thebreakdown of the BOA or due to contributions from weaker multipole moments.252.2. Light-Matter Interaction2.2.2 Molecular TransitionsBefore presenting the selection rules that largely govern molecular transitions, wewill first review the internal transitions available within molecules. These can bedivided into two main groups: radiationless transitions resulting from collisions orcouplings to the physical environment, and radiative transitions which evolve aninteraction of electromagnetic (EM) fields with the eigenstates of a molecule.Radiationless TransitionsA radiationless transition can occur as a result of an inelastic collision of a moleculewith its surroundings. Often this is a de-excitation process in which internal molec-ular energy (electronic, vibrational or rotational) is transferred into kinetic energy.An excited electronic state of a molecule in a given vibrational level (rotational statesare not considered for simplicity) may undergo three sorts of transitions. These areconveniently represented in the Jablonski diagram Fig. (2.4) as internal conversion(IC), intersystem crossing (ISC), and vibrational relaxation (VR), where the Si andTi represents the ith singlet and triplet electronic states respectively. Within a givenelectronic state, Si or Ti, a molecule’s vibrational energy will gradually decreaseover time as a result of thermal collisions causing VR. Once either in the groundvibrational state or before of a given Si (Ti) state, collisional energy transfer mayresult in a change of the electronic state to one of the same spin multiplicity Si−1(Ti−1) for IC, or one of a different spin multiplicity Tj (Sk) as in ISC. As a result, adense ensemble of excited molecules, uninfluenced, will usually decay by a series ofradiationless transitions from excited electronic and vibrational states to a groundstate Boltzmann distribution (see Sec. (2.3.1)). Note that nearly any energeticallyallowed transition is possible between electronic, vibrational and rotational statesas a result of an inelastic collision. The well-known molecular selection rules whichlimit allowed transitions, occur due to electric-dipole couplings in the BOA, thisdiscussed in Sec. (2.2.2).Because these nonradiative processes cannot be quantified or measured, they aredetrimental to molecular spectroscopist or those designing quantum control schemes(See Sec. (2.3.1) on dephasing). Fortunately, the mean time between collisions ofparticles in a gas can be controlled via the pressure, density, and temperature, andin this work we will assume that the collisions occur on a nanosecond (ns) timescalefor the small molecular species of interest [59, 60].262.2. Light-Matter InteractionFigure 2.4: Jablonski diagram where: S = single state, T = triplet state, A =absorbance, F = fluorescence, P = phosphorescence, IC = internal conversion, ISC= intersystem crossing, and VR = vibrational relaxation.Radiative Transitions and Selection RulesRadiative transitions result from the absorption or emission of one (or more) pho-tons, these are shown in Fig. (2.4) as: absorption(A), fluorescence(F) and phospho-rescence(P). The mathematical details of these transitions will be discussed in Sec.(2.4).At the moment our interest will be in estimating which transitions are permittedwithin a diatomic molecule interacting with an EM field in the dipole approximation.From Eq. (2.23), we write the probability amplitude for a transition in the BOA asA ∝ 〈ΨfeψfN |µe · ǫˆ|ΨieψiN 〉 (2.26)A ≈ 〈ψfeχfsψfrotψfvib |µe · ǫˆ|ψieχisψirotψivib 〉 (2.27)where we have expanded the electronic wave function, Ψe, to include both the spatial,ψe, and the electron spin, χs, components. And, the nuclear wave function, ψN ,has been separated into rotational, ψirot, and vibrational, ψivib, parts. The canonicalselection rules are expressed from the point of view of the Hund’s case (a) basis which272.2. Light-Matter Interactionexclude spin-orbit coupling, thus we can separate the electronic wavefunction intothe orbital and spin terms (|nΛ 〉|Σ, S 〉). By omitting the rotational-vibrationalcoupling we can specify the vibrational and rotation eigenstates in terms of theirquantum numbers ν and J respectively (i.e. ψvib = | ν 〉 and ψrot = | J 〉), where Jshould not be confused with the total electronic angular, but equals the l rotationalquantum number from earlier (following convention).Thus we rewrite Eq. (2.26) in terms of these quantum numbers correspondingto their appropriate wavefunctionsA ≈ 〈ψfe |µe|ψie 〉〈χfs |χis 〉〈ψfrot |µˆe · ǫˆ|ψirot 〉〈ψfvib |ψivib 〉= 〈nfΛf |µe|niΛi 〉〈Σf , Sf |Σi, Si 〉〈 Jf ,mfJ |µˆe · ǫˆ| J i,miJ 〉〈 νf | νi 〉 ,(2.28)where we have included the quantum number mJ = {−J, ..., J} labelling the rota-tional sublevels.The first term in Eq. (2.28) yields the electronic orbital selection rules associatedwith transitions between nΛ±g/u states:∆n 6= 0 ∆Λ = 0,±1 ± ↔ ± g ↔ u . (2.29)Recall that the ± refer to the reflection symmetry of Λ = Σ state, and g/u rep-resents a symmetry with respect to inversion for homonuclear diatomics. For thespin component of the electronic wavefunction we have the simple requirement that∆S = 0, since operators governing transitions are diagonal in S in the absence ofspin-orbit coupling. The third term, known as the Hönl-London factor, involvingthe rotational wavefunctions gives rise to the rules:∆J = ±1(∆Λ = 0) ∆J = 0,±1(∆Λ = ±1) . (2.30)Moreover, there is a dependence of the mJ sublevel couplings on the polarizationof the electric field ǫˆ. Linear polarized light (e.g. ǫˆ = zˆ), which we will usedthroughout this thesis, requires that ∆mJ = 0, whereas a circularly polarized field(e.g. ǫˆ = xˆ ± yˆ) permits only ∆mJ = ±1 transitions. Lastly, the inner product ofthe vibrational wavefunction do not define a selection rule and simply provides thepreviously discussed Franck-Condon (FC) factor.Thus, we find that a single photon electric dipole transition will occur betweentwo different electronic states for all ∆ν, however if ∆Λ = 0 then there must be a282.3. Laser Excitations in Moleculeschange in the rotational state, ∆J 6= 0, as shown in Eq. (2.30). This restriction re-sults from parity conservation, whereby the absorption or emission of a photon (withparity -1) must change the parity of the overall wavefunction. We will refer to thesebound-bound transitions between rovibrational states where ∆J = -1, 0, or +1 as P,Q, or R branches respectively. We must keep in mind that these selection rules, andthe concept of the Franck-Condon factor, come about from a series of assumptions,most notably through application of the BOA which permits a factorization of thetotal wavefunction into nuclear, electronic spatial and spin wavefunctions, and theomission of vibrational and rotational coupling between states. As a result, theserules may not always be strictly observed, however they will serve suitable well forour investigation.Before continuing, we make a quick comment on the phenomenon of phospho-rescence. In particular, for phosphorescence to occur there must exist spin-orbitcoupling between some excited diabatic singlet (Si) and triplet (Tj) states (whichis not necessary for nonradiative ISC). This “singlet-triplet transition” in fact is “al-lowed” only because the triplet state obtains some singlet character as we shall see inSec. (5.3.2). Together with nonradiative effects, phosphorescence can result in thevery long lifetimes (' 1s), compared to the relatively short lifetimes (100ps-10ns) offluorescent states. Studying this form of radiation is beyond the scope of this work,however, it plays an important role in acting as metastable states, capable of storingenergy, for instance, where it can act as reaction intermediate in photosystem II inelectron-transfers reactions [61].2.3 Laser Excitations in MoleculesUnless stated otherwise, for the remainder of this thesis we will assume that we areworking with a gaseous ensemble of molecules. In this section we discuss the natureof such systems and discuss their behavior in the presence of electric fields. Inparticular, we focus on the internal dynamics of molecules by looking at the originof coherence and the behavior of molecular wavepackets. This will set the stagefor understanding the time- and frequency-dependent molecular spectroscopy in thesection to come.292.3. Laser Excitations in Molecules2.3.1 Molecular EnsembleIn the BOA there exist a set of nuclear wavefunctions, ψN (R), for each electronicPES, Ee(R). The amplitude of these quantized stationary eigenstates reflect theprobability distribution, in the coordinates R, of observing a nuclei at that posi-tion. Under typical conditions a molecule doesn’t exist in a single eigenstate butin a thermal ensemble of multiple internal quantum states following the Boltzmanndistributiongie−Ei/(kbT ), (2.31)where gi and Ei is the degeneracy and energy of a state i relative to the overallground state of the molecule respectively. kb is the Boltzmann factor and T is thetemperature.Given that for the typical diatomic molecule the spacings between vibrationallevels is O(103) Kelvin (K), and the separation between rotational levels is only afew K, then at ordinary temperatures (100’s K), a molecule will reside in its groundvibrational state ν = 0, and in a distribution of several rotation levels J following Eq.(2.31). For instance, we can calculate and find the rotational state with the maximumpopulation as Jmax = (√2kbT/B − 1)/2, where B is the rotational constant.Due to the nature of thermal excitations, such superpositions of eigenstates wouldbe incoherent, that is, there wouldn’t be a fixed phase relation between the individ-ual, ψvibψrot = | ν = 0, J 〉, levels. An example is an ensemble of molecules at roomtemperature in which all of the diatomic molecules will be rotating out of phasewith each other. To addresses this fundamental problem of controlling a systemhaving an initial state that is an incoherent thermal mixture of different states, wecan prepare a molecule in a single quantum state 〈R | ν, J 〉 ≡ ψ(ν,J)N (R), by insuringthat the translational (kinetic) energy be reduced to below 1 K (for a relatively lightmolecule) such that the thermal energy kbT is smaller than the rotational energyspacing. In this way our ensemble of molecules will begin in a single | ν = 0, J = 0 〉state. Note that throughout this thesis we will average over all hyperfine levels whichhave spacing of O(10−2) K. From this initial state, a coherent process can be used,such as from a laser field, to transfers coherence to the ensemble of molecules.302.3. Laser Excitations in Molecules2.3.2 Excitation Laser FieldsWe define our laser (light amplified by stimulated emission of radiation) field asε(t) = ε(t)ǫˆ= E(t) cos(ω0t)ǫˆ , (2.32)where E(t) represents the (complex) pulse envelope of the field, and ω0 is the carrierfrequency. Unless stated otherwise the polarization of the field, ǫˆ, will be linearand along the zˆ direction in the laboratory frame. It will be useful to represent thecomplex-valued electric field also in frequency-space by its inverse Fourier integralε(t) =∫ ∞−∞dωε(ω)e−iωt=∫ ∞−∞dω|ε(ω)|eiφ(ω)e−iωt (2.33)where |ε(ω)| and φ(ω) are the magnitude and phase of the field at the frequencyω, respectively. In addition, we define the time averaged intensity and the periodaveraged spectral intensity of a slowly varying pulse as follows:〈 I(t) 〉 = c8π |ε(t)|2 , 〈 I(ω) 〉 = c16π2 |ε(ω)|2 . (2.34)Ultrashort Pulses and Pulse ShapingA property of the Fourier transform is that the time-bandwidth product of a Gaussianlaser pulse is approximately fixed at∆τ∆ω ≈ 0.441 , (2.35)where ∆τ and ∆ω are the full-width-half-maximum (FWHM) of the two intensityprofiles of Eq. (2.34) respectively. So we observe a broadening occurring to a pulse’sfrequency bandwidth upon a reduction of its temporal width, with the precise valueof Eq. (2.35) depending on the shape of the pulse envelop (e.g hyperbolic-secant-squared = 0.315) [62].It has become common to generate pulses of only a few femtoseconds (fs), knownas ultrashort pulses, consisting of only a few optical cycles. The standard approachinvolves pumping a mode-locked oscillator with a continuous-wave (cw) laser field.312.3. Laser Excitations in MoleculesIn this case there exist a fixed phase between the modes of the optical cavity, andthe modes will periodically all constructively interfere with one another leading toan intense pulse of light with a duration determined by the number of modes whichare oscillating in phase [63]. For more information on ultrashort laser fields refer toRef. [62].Another tool which has become vital to study and observe molecular quantumdynamics on short time scales is pulse shaping. Given a femtoseconds pulse in thevisible spectrum, the electric field can be modulated in the frequency domain likeε(ω)mod = M(ω)ε(ω) , (2.36)where M(ω) is the modulation function of a spatial light modulator capable of ma-nipulating the spectral amplitude, phase or polarization, with common resolutionsof approximately 0.5 nm (or 10 cm−1) [64]. Note that this energy is at least an orderof magnitude smaller than spacing between vibrational levels. Finally, returning thisshaped pulse back into the time domain requires an inverse Fourier transform (IFT)ε(t)mod =12π∫ ∞−∞M(ω)ε(ω)eiωtdω . (2.37)These ultrashort pulses with large bandwidth allow for the simultaneous excitationof several molecular eigenstates, and by tuning pulses’ temporal and spectral profileswe are able to prepare a molecule in very particular electronic, vibrational, rotationalstates.Perturbation TheoryUsing the assumptions and results of Sec. (2.2), we are ready to solve the time-dependent Schrödinger equation for a molecule in the presence of an EM fieldi∂Ψ(t)∂t = (HN +Hint(t))Ψ(t) , (2.38)where HN represents the nuclear Hamiltonian in the Born-Oppenheimer approxima-tion with (rovibrational) eigenstates ψN = | ν, J 〉, and Hint is given by Eq. (2.22).The wavefunction of the interacting system can be expanded in the unperturbed322.3. Laser Excitations in Moleculesbasis (omitting electronic spin states) as|Ψ(t) 〉 =∑q∑j∑kcq,j,k(t)e−iEq,j,kt|nqΛqνjJk 〉 (2.39)where Eq,j,k is the energy of the (j, k)th rovibrational level, and the square of thecoefficients cq,j,k(t) give the population in a given state | νjJk 〉, within the qth elec-tronic state. The elements of the interaction Hamiltonian between some initial statei and a final state f can be written as (from Eq. (2.22))H i,fg,e(t) ≈ ǫ(t)di,fg,e (2.40)where we define the electronic transition dipole moment (TDM) asdi,fg,e = 〈 νf , Jf |µg,e · ǫˆ| νi, J i 〉 . (2.41)Here we have simplified the notation by defining µg,e as the radial dependent elec-tronic transition dipole function between a ground, g, and an excited, e, electronicstate (different n’s and Λs). In the case of electronically forbidden transitions thisdipole function will thus tend to zero. For simplicity, the initial, i, and final, feigenstates of these different electronic potentials will now only be specified by theirrovibrational quantum numbers ν and J . The MJ indices are not included since wewill assume that the rotational sublevels will remain approximately degenerate, andthe Σ, S labels become superfluous in the absence of spin-orbit coupling.Now, if we take θ to be the angle between µe(q) and ε(t), where we assume thatthe EM field is linearly polarized, thenH i,fg,e(t) = ε(t) cos θ〈 νf , Jf |µg,e| νi, J i 〉 . (2.42)In which case, if the field interacts with an ensemble of randomly oriented moleculesin the gas phase, then the angle θ between µe(q) for each molecule and the fixedfield ε(t) can be averaged (〈 cos2 θ 〉 = 1/3), and we find that the probability of anelectronic transition to be given byPi,f (t) = |H i,fg,e(t)|2 ∝ |ε(t)|2|µi,fg,e|2|〈 νf , Jf | νi, J i 〉|2 , (2.43)where first term is proportional to the intensity of the laser field, the second term332.3. Laser Excitations in Moleculesprovides us with the orbital selection rule, and the last term is known as the Franck-Condon (FC) factor. As stated above, the coupling of an individual molecule with theradiative field (Eq. (2.41)) is a function of the angle, cos θ, between the internuclearaxis and zˆ. Thus every molecule in the ensemble with a different spatial orientationwill experience a different overall strength of the electric field, |E(t)|. Fortunately,our procedures are fairly robust to variations in the pulse intensity. Although, it ispossible to prepare the sample as a volume of (non-isotropically) aligned moleculesusing an initial pump pulse [65]. In this way the molecular axes of all the moleculesin the ensemble can become oriented parallel to the electric field of the laser beforestarting an experiment.Molecular WavepacketsUnder perturbation theory (PT), a monochromatic or narrowband laser field willexcite a single eigenstate νj from an initial eigenstate νi according to Eq. (2.43).However, a broadband Gaussian laser pulse, which in PT can be thought of as acoherent sum of monochromatic light fields with varying wavelengths, can accessmany different energy levels from a given initial state. It is this coherent propertyof the laser field that allows for the accurately manipulation of molecular statesand the control of quantum processes (see also Sec. (2.3.3) on Coherent Control).By using such a laser excitation we can create a coherent superposition of a set ofeigenfunctions (stationary states) known as a molecular wavepacket. Assume thatwe are only dealing with vibrational states (ψvibk (R)) in a given electronic surface,we may write the wavepacket asΨ(R, t) =∑kckψvibk (R)e−iEkt (2.44)where ck are the coefficients or amplitudes of the kth vibrational state with eigenen-ergy Ek. In first-order PT, these coefficients become [66]ck = 2πiǫ(ωk,i)dk,i (2.45)where ǫ(ωk,i) is the field strength at the frequency ωk,i, and dk,i is the electronicTDM between the initial and final states. We see from Eq. (2.45) that the initialform of the wavepacket Ψ(R, t) will be determined by the spectral width of the laserfield and the strength of the electric dipole couplings. However, because each of342.3. Laser Excitations in Moleculesthe individual stationary eigenstates evolve in time with a complex phase the initialwavepacket will quickly delocalize. This delocalization or de-phasing of vibrationaleigenstates, is due to the interferences between the constituent eigenstates whichevolve with different periods. The wavepacket probability density,|Ψ(R, t)|2 =∑k,k′ckc∗k′ψvibk (R)ψvibk′ (R)∗ cos(ωk,k′t) (2.46)shows that the predominant frequencies in wavepacket motion are the beat frequencies(ωk,k′ = Ek−Ek′) between individual eigenstates. To first order, ∆k = ±1, the semi-classical vibrational period of the wavepacket is given as Tcp = 2π/ωk,k′ . Of courseall the higher order terms along with the weighting aνaν′ describing the magnitudeof the oscillations are required to fully describe a wavepacket’s behavior. Thoughwe can roughly state that a wave packet revivals occur with period Tr ≈ 2Tcp/ωk,k′ ,so for example in Na2, we find Tcp ≈ 300 fs and Tr = 29 ps. In Sec. (2.4.2) we willdiscuss the future of wavepackets when left untouched, particularly in the context ofmolecular fluorescence (see Sec. (2.4)).2.3.3 Quantum Control SchemesOnly upon the invention of coherent fields (lasers) have quantum control schemesbeen able to accurately specify the interaction and control the final states of molec-ular systems. In the current subsection we will briefly outline one quantum controlscheme which harnesses the complex phases inherent in electric fields and quantumstates; this has become known as coherent control (CC).Coherent control [66, 67] of quantum systems is very active area of researchin physics and chemistry. The mechanism for control is the interference betweenmultiple pathways from some initial state(s) to some final state(s). The nature ofthe interference is manipulated by changing the relative complex phase of the laserfields which are exciting the different paths.For simplicity and for our purposes, we will use weak-fields and can thus un-derstand the process in the context of perturbation theory (PT). In general, CCbecomes more difficult in the strong field case once AC Stark shifts arise and theperturbations of the quantum system becomes dynamical.352.3. Laser Excitations in MoleculesFigure 2.5: Implementationof bichromatic control (BCC)in a three state system wherethe initial population residesin states E1 and E2. Twocontinuous wave (CW) laserswith frequencies ω1 and ω2 re-spectively, couple these statesto a final state E. The am-plitude and phase relation be-tween the two laser fields willdetermine the population ofthe final state.Bi-Chromatic ControlOne of the simpliest implementations of coherent control is a scheme known as Bi-Chromatic Control (BCC) [66], depicted in Fig. (2.5). In this case, there are twoground states which are initially populated, say E1 and E2, and we wish to excite athird state, E. Take two CW fields with frequencies ω1 and ω2 which independentlycouple the two initials states to the one final state, then using perturbation theory(or Eq. (2.45)) the amplitude of state E can be written ascE ∝ ǫ(ω1)d1 + ǫ(ω2)d2 (2.47)where the complex quantities, ǫ(ωk) and dk (see Sec. (2.3.2)), can be written in termsof a real-valued amplitude and phase as: ǫ(ωk) = |ǫ(ωk)|eiδk and dk = |dk|eiφk whereφk = {0, π} for bound states {E1, E2, E}. With this representation we can write theprobability amplitude of the final state as|cE |2 ∝ |ǫ(ω1)d1|2 + |ǫ(ω2)d2|2 + 2|ǫ(ω1)||ǫ(ω2)||d1||d2| cos(∆δ1,2 +∆φ1,2), (2.48)where ∆δ1,2 = δ1− δ2 and ∆φ1,2 = φ1 − φ2. By varying the relative laser intensities(ǫ(ωk)) and phases (δk) between the two pathways 1 and 2, such that |ǫ(ω1)||d1| =|ǫ(ω2)||d2| and ∆δ1,2 + ∆φ1,2 = {0, π}, the final population residing in state E362.4. Molecular Spectroscopyafter the BCC process can be tuned to be anywhere between 0 and 4|ǫ(ω1)||d1|.This process is analogous to controlling the appearance of fringes in the well-knowndouble slit experiment, in which two coherent beams interfere either constructivelyor destructively on a particular target depending on their relative amplitude andphase relations.Although we have presented the BCC technique using two independent CW laserfields, it is possible to stimulate the two excitation pathways using a single broadbandlaser pulse [68–70]. By using pulse shaping techniques (Sec. (2.3.2)) to manipulatethe magnitude and phase at the two frequencies components of the field (ω1 andω2) corresponding to the transition energies we can apply BCC in the femtosecondregime. In Chapter 5 we will harness this technnique for determining the relativephases of the transition dipole matrix elements between rovibrational eigenstates.2.4 Molecular SpectroscopyHere we extend the discussion of the light-matter interactions to include the radiativeemissions of diatomic molecules in the gas phase. This emitted fluorescence providesus with information about a molecules internal structure and dynamics, and theability to verify theoretical ideas and models.For a single excited rovibrational state, | i 〉, with population |ci|2, we define theradiative lifetime as τi = 1/γi where γi is the decay rate and the population aftertime t is|ci(t)|2 = |ci(t = 0)|2e−γit. (2.49)In the absence of collisions, nonradiative decay processes or stimulated emission, thedecay rate is given asγi =∑fAf,i (2.50)where the summation occurs over all (allowed) final states of transition, and Af,i isthe Einstein spontaneous emission coefficient. The Einstein A-coefficient is a first-order decay constant defined (in atomic units) asAf,i =4ω3f,i3c3 |µf,i|2 , (2.51)where ωf,i are the transition frequencies and µf,i are the transition dipole matrixelements. Therefore, the decay of a state can be quantified by a single exponential372.4. Molecular Spectroscopydespite the summation over all Einstein coefficients. Note that for rotating moleculesa sum over the mJ quantum number is implicit in |µf,i|, namely,|µf,i|2 =mi+1∑mf=mi−1|〈 f,mf |µ| i,mi 〉| (2.52)Experimentally, the fluorescence or phosphorescence (e.g. the decay to a tripletstate in the presence of spin-orbit coupling) emitted by molecules can be viewed ineither the frequency- or the time-domain. We will briefly discuss these two picturesbelow, though for a detailed review see Ref. [71].2.4.1 High-Resolution Rovibrational SpectroscopyHigh-resolution molecular spectroscopy is principally used for determinating the in-ternal structure of a molecule, this includes its rotational, vibrational, and electronicenergies. In this thesis we will be interested only in the visible or ultraviolet (UV)rovibrational spectrum of, predominantly, diatomic molecules. In Fig. (2.6)(a) weshow a pair of Born-Oppenheimer potential energies curves and a few of the eigenen-ergies associated with each. Population in the excited potential will spontaneouslydecay through the emission of radiation to the levels of the ground state. A spectrum,I(ω), shown in Fig. (2.6)(b), is a plot of of the signal strength versus frequency, wherethe signal strength is often based on detection of photons recorded as an electricalcurrent or voltage. The difference magnitudes of the spectral lines across frequen-cies correspond to the difference strengths of the transition dipole matrix elements(TDMe) coupling excited and ground rovibrational eigenstates. Spectroscopist willalmost always write the frequency in units of cm−1 (or wavenumbers), however, as atheorist, atomic units (Hartree) will be used interchangeably (1 Ha = 2.1947 × 105cm−1).There are many approaches that can be used for collecting such frequency-resolved spectra, this thesis will not deal with these practical techniques but will fo-cus on the application of this data. Common methods such as Raman-spectroscopy,Fourier-transform interferometry and CARS spectroscopy, and others can be re-viewed in Refs. ([71],[48]).382.4. Molecular Spectroscopy(a)(b)Figure 2.6: (a) A schematic illustration of the emission process between two di-atomic potentials. (b) A typical spectrum of fluorescence lines calculated betweenthe rovibrational states of the two electronic potentials shown in (a).392.4. Molecular SpectroscopyLaser-Induced Fluorescence and Fourier Transform SpectrometryIn brief, laser-induced fluorescence (LIF) involves exciting a state or states of aquantum system with an electromagnetic field, such that it will undergo spontaneousde-excitation (or fluorescence), emitting radiation which can be then captured. LIF iswidely used in the spectroscopy of diatomic molecules since the lasing wavelength canbe tuned to a particular excitation, which allows for the determination of intensitiesassociated with a particular vibrational or rotation level. This method is applicableto very weak transitions with Franck-Condon factors below 10−4, and capable ofresolving high values of vibrational and rotational quantum numbers. Advantagesover absorption spectroscopy are the very high signal-to-noise ratios and the abilityto capture radiation in all directions since the fluorescence signal is often isotropic.Note that, as a result of the low excitation energies, stimulated emission processesare typically ignored in the data analysis due to their negligible contribution to thefluorescence.To characterize the spectrum of the fluorescence radiation, the simplest approachwould be to use a monochromator and measure the intensity of the light at all therelevant wavelengths. However, the full spectrum can be generated much more effi-ciently, and with less sensitivity to noise, using a technique known as Fourier trans-form spectroscopy (FTS) [72]. FTS is based on the principles of interferometry, inwhich coherent waves from at least two sources or paths are combined and theirsuperposition measured. In particular, a Fourier transform spectrometer is designedsimilar to a Michelson interferometer, except that one of the mirrors can move rapidlyback and forth, and the recombined beam is detected synchronously with the mo-tion of this mirror. This allows for the temporal correlation function of the light tobe measured at each different time delay, thus converting the time domain into aspatial coordinate. The measured outcome, known as an interferogram, is an inten-sity measurement as a function of the retardation path length. The interferogramrecorded by the detector can be thought of as a sum over weighted monochromaticinterferograms, in which we think of each spectral component of the radiation asproducing its own interferogram with an amplitude weighted by the relative spectralintensity. From this perspective, we can see that by making measurements of thesignal at many discrete mirror positions, we can use a Fourier transform to pickout the intensities of each frequency and produce a spectrum. Or, in other words,the interferogram (intensity of recombined beam as a function of path length) is402.4. Molecular Spectroscopysimply the Fourier transform of the intensity of the light source. In practice, thistechnique has become very reliable and can provide very high resolution (∼ 0.005cm−1) [16, 73].Phase of TDMeBefore continuing on we will comment on the phases of the TDMe; particularly,since the spectral data only provides information about their magnitudes. Thesematrix elements are linked to the potentials and the nuclear wave functions that arederived from them. In bound-bound transitions, both initial and final eigenstates in〈 ν ′′ |µ| ν ′ 〉 are real, thus this becomes a simple sign (±), unlike the usual complexphase encountered in scattering theory. When dealing with transition between nexcited states and m ground states there will exist n×m TDMe and thus seemingly2nm sign possibilities. However, because each individual wavefunction | ν 〉 can havean arbitrary phase, we are free to choose the signs of n+m− 1 TDMe at will. Forexample, all m TDMe of sgn(〈 ν ′′ |µ| ν ′ = 0 〉) for ν ′′ = {0,m−1} can be chosen to bepositive, where sgn is the sign function. This fixes the arbitrary phases of the final| ν ′′ 〉 states and the one initial | ν ′ = 0 〉 state. With these states fixed, we can nowchoose sgn(〈 ν ′′ |µ| ν ′ 〉) for all other ν ′ = {1, n − 1} by selecting any overall phasefor each | ν ′ 〉. This gives us the n +m − 1 arbitrary chosen phases of the TDMe,and the rest of the (n− 1)× (m− 1) + 1 are then fixed and need to be determinedby performing the integration over the relevant wave function or extracted from theexperimental data (Chapter 3).A basic argument towards obtaining the relative phase (sign) information involvesusing a semi-classical stationary phase point approximation. Namely, it states thatwhenever the observed transition intensities along a fluorescence progression (sameν ′, series of ν ′′) depart from monotonic behavior, the phase of the transition am-plitude has reversed. It is true that the fluorescence progression shows a trend andthe nodal structures reveals the wavefunction of the ν ′ state. However, the fixing ofphases of the transition dipole amplitudes is strictly tied to the fixing the relativephases of wavefunctions of ν ′′ states. Let us take the simplest case: ν ′ = 0 vibra-tional state. If one expands |ν ′ = 0 > in terms of the |ν ′′ > states, the formula willbe: φ(ν ′ = 0) = ∑ν′′ |ν ′′ >< ν ′′|ν ′ = 0 >. The second term is the Frank-Condonoverlap while the first |ν ′′ > is the basis wavefunction for the expansion. We knowthe fluorescence spectrum, | < ν ′′|ν ′ = 0 > |2, shows no nodal structures. Therefore,one can choose to assign < ν ′′|ν ′ = 0 > to be all positive. However, this choosing412.4. Molecular Spectroscopysigns fixes the relative phases of |ν ′′ > simultaneously. Therefore, one then needs tobe able to use/find these relative phases of |ν ′′ > as well, instead of blindly assigningthem inconsistently. Equivalently speaking, after choosing < ν ′′|ν ′ = 0 > to be allpositive (or all negative), there is no freedom to choose the signs for each |ν ′′ >except for an overall sign, while the relative signs are fixed! It is, in general, verydifficult to search for the correct combination of relative signs for the basis wave-function: it is a 2Ng−1 problem where Ng is the number of basis functions for evenjust one ν ′ state (not 2Ng because of the arbitrary overall sign). Furthermore, thechoice of using the simple semi-classical wavefunction as the basis wavefunction doesnot give all positive/all negative signs for < ν ′′|ν ′ = 0 >, which confirms the factthat the solution for these relative signs are non-trivial.Also, we need to point out that for diatomics these semi-classical methods areat best approximate. In fact, as is well known in semi-classical theories, the singlestationary phase approximation often fails, especially in the long wavelength regimeor when the main contributions to the dipole matrix elements come from the vicinityof the classical turning points. For any procedure to be useful it has to be completelygeneral and highly reliable. The fact there are some cases in which a simple argu-ment based on a single stationary phase approximation might work is not sufficientfor a general procedure. So although fluorescence progression ’might’ indicate signreversal, this is certainly not the case when double minima exist, or in polyatomiccase. In particular, two or more phase reversals (nodes) can occur in the regionbetween the ν ′, ν ′′ and ν ′, ν ′′ + 1 transition, for example when the nodes are closelyspaced (due to high kinetic energy) or when the transition amplitudes are small.As will be seen in Chapter 3, determination of these relative phases can be doneaccurately and efficiently because, instead of searching for various combinations ofrelative signs of basis wavefunctions, knowledge of the energy levels of the excitedstates can be used to “guess” smartly what the signs of < ν ′′|ν ′ = 0 > are whenthe relative signs of the basis wavefunctions, |ν ′′ >, are pre-chosen. This is possiblebecause interference effects are experimentally observable; hence the correct sign ofan interference effect can be obtained from a computation that embodies internalconsistency.In Chapter 5, we present another approach which directly extracts the com-plex amplitudes (magnitudes and phases) of transition dipole matrix elements. Themethod uses Bi-chromatic Coherent Control (BCC) in conjunction with time-resolvedfluorescence to uniquely determine the transition properties between individual rovi-422.4. Molecular Spectroscopybrational eigenstates without any a priori assumptions. See Sec. (3.1) for otheralternative approaches that use spectral information for uncovering the structureand properties of a molecule.2.4.2 Time-Resolved Fluorescence of Vibrational StatesHigh-resolution measurements for gases using time-resolved picosecond Raman spec-troscopy were first demonstrated by Graener et al. in 1984 [74]. Modern approaches(see Ref. [71]) now include using femtosecond Four-wave Mixing Spectroscopy to-gether with optical gating [75] and up-conversion [4, 5] techniques to achieve tem-poral resolution of less than 60 fs. In Fig. (2.7)(a) we show an excited wavepacket(a) (b)Figure 2.7: (a) Wavepacket decaying from an excited PES to the ground state. Themost probable decay routes are shown for when the wavepacket is at its turningpoints. Image taken from Ref. [4]. (b) Temporal fluorescence captured over severalpicoseconds. This picture is taken from Ref. [5] where the number of counts arecaptured after the upconversion of the raw fluorescence. The two lines capture theresults without (a) and with (b) a monochromator inserted after the crystal.at the two end points of its oscillations in the excited PES. The two vertical linesdepict the most probable decay path for when the wavepacket is in these positions,432.4. Molecular Spectroscopynamely, when the wavepacket has the greatest overlap (Franck-Condon factor) witha vibrational state in the ground potential. Although the typical lifetimes of vi-brational states for diatomics, which can be calculated using Eq. (2.50), are in thenanoseconds (ns), the probabilistic behavior of the spontaneous fluorescence processmeans that a measurable fraction (∼ 10−8) of the excited molecules in a gas (of say1023 particles) will decay after a few femtoseconds (fs). Unfortunately, the fastestelectronics operate on a picosecond timescale, and thus capturing time-dependentprocesses at any shorter times requires the application of alternative techniques; onesuch approach is up-conversion.Up-ConversionThis method focuses the spontaneous fluorescence captured over a fixed solid angle(say, 0.03sr) onto a nonlinear crystal [4]. A gate-pulse (or pulse-train) is also sentinto the crystal such that whenever the radiation temporally and spatially overlapsit leads to sum-frequency generation. The resulting radiation (often in the ultra-violet (UV)) is sent through a double-prism monochromator and measured using aphotomultiplier [5]. For single gate-pulses, the time dependence of fluorescence oc-curs by varying the timing of the gate pulse relative to the pulse causing the initialexcitation (in a similar manner to pump-probe absorption measurements).In a mathematical sense we define time-dependent spectrum in terms of theenergy spectrum (S(T, ωF )) over finite-time and frequency. Following Walmsley inRefs. ([5],[4]):S(T, ωF ) =∫ ∞−∞dt|EFB(t)|2 (2.53)where the electric field is given asEFB(t) =∫ t−∞dt′H(t− t′, ωF )B(t′, T )E(t′) . (2.54)The gate-pulse acts as the time gate such thatB(t, T ) = e(−|t−T |Γ) (2.55)where Γ relates to the pulse duration, while the double monochromotor acts as a442.4. Molecular Spectroscopyfrequency filter and can be modeled byH(t− t′, ωF ) = γ2[γ2 + (ω − ωF )2]−1 . (2.56)In this way, temporal resolutions of at least 60 fs with energy ranges of 4 nm can beattained.2.4.3 Bound-Continuum TransitionThe discussion of molecular state emissions has so far only included bound-boundtransitions. However, as an excited bound state decays, there will also be a weakcoupling to lower continuum states. The dipole matrix elements associated with thesetransitions are complex due to the nature of the continuum states. We demonstratein Sec. (2.1.2) how to represent these continuum states as pseudo-bound statesand calculate an effective real-valued TDMe. To be sure of the accuracy of thisapproximation, the effective eigenstates and TDMe representing the continuum spacemust be compared with the results of a proper calculation. One method to obtainthese complex TDMe involving continuum levels is known as the artificial channelmethod.Artificial Channel MethodThe quality we are interested in determining is the matrix elementfm,i = 〈E,m− |µe|Ei 〉 (2.57)where |Ei 〉 is the initial bound excited state, |E,m− 〉 represents an incoming (−)scattering states with energy E and collective quantum number m, and µe is theusual electric dipole function [51–53]. Note that we do not use the outgoing scatter-ing states because they correspond at early times to a well-defined fragment state|E,n0 〉 and not a molecular bound state |Ei 〉 as in our case. On the other-hand, theincoming solutions, |E,− 〉e−iEt, approach in the infinite future a single well-definedasymptotic state |E,n0 〉. The fm,i integral, which is often described as a photodis-sociation amplitude, is difficult to calculate due to the highly oscillatory scatteringfunction 〈R |E,m− 〉.Shapiro [51] developed an approach to calculate fm,i by treating the problem us-ing time-independent scattering theory with a source term, namely, |Ei 〉. However,452.4. Molecular Spectroscopyinstead of solving a set of inhomogeneous differential equations (DEs) (with sourceterm), he re-expressed these as a set of coupled homogeneous DEs. This allows forgreater generality of the solution in regards to different ground state PESs, but leadsto a great difficulty in integrating over the bound wavefunction for all energies, be-cause its solutions diverge asymptotically at all energies other than at the particulareigenenergies. This problem can be stabilized by including one extra “artificial” con-tinuum channel which serves as a source for the bound manifold. The final solutionsto fm,i using the artificial channel method (ACM) can be found in Ref. [52], thereinalso contains several alternative approaches so such calculations. For our applicationin Sec. (2.1.2) we use a FORTRAN subroutine written by M. Shapiro in 1972, en-titled: “Program for Quantum Mechanical Solutions of Photo(Pre)dissociation andBound State Problems, using the Artificial Channel Method”.46Chapter 3Inverions of Potential EnergySurfaces3.1 Potential Inversion MethodsThe Born-Oppenheimer approximation gives rise to the concept of molecular poten-tial energy surfaces/curves (PES) which then governs a molecule’s behaviors andinteractions. However, even for large-scale ab initio calculation on small molecularsystems, these electronic eigenvalues are difficult to obtain within spectroscopic ac-curacy [76, 77]. Therefore, it is necessary to determine these potentials (or PES)empirically from experimental observations. In this section we outline several cur-rently existing methods of extracting the PES from spectroscopic data, and mentionthe drawbacks associated with each.3.1.1 Dunham ExpansionNearly all approaches that determine an electronic PES from experimental data useknowledge of the rovibrational energies of the desired potential. These eigenen-ergies can be obtained directly by analyzing the transitions appearing in a givenspectrum. By taking the differences of spectral lines relating a common state(ωi+1,j − ωi,j = ωi+1,i = ∆Ei+1,i) and the combination of differences between twosuch results ∆(∆Ek,i) = ∆Ek+1,k −∆Ei+1,i, we can use a least-squares approach tofind the spectroscopic constants {ωe, ωexe, Bν , Dν , ...} used in the expansion of therovibrational energies (Eq. (2.16) from Sec. (2.1.2)). Given these constants it isthen possible to obtain an expression for an ordinary single minimum PES.In the simplest case, we found in Sec. (2.1.2) that these constants can be relatedto those of a Morse function (Eq. (2.18)). In particular, the eigenvalues of the473.1. Potential Inversion Methodsvibrational states for a rotationless Morse potential are given byEν = ωe(ν + 1/2)− ωexe(ν + 1/2)2 (3.1)= β√2Deµ (ν + 1/2)− 2µβ2(ν + 1/2)2 , (3.2)where β and De are two of the Morse parameters. This form of the PES is usuallynot sufficient for realistic applications, however, it serves very-well in providing arudimentary model of the PESs in a diatomic for testing purposes.Often, a better expression can be obtained using the work of Dunham [78]. Heprovided an expression for the rovibrational energies in terms of spectroscopic con-stantsE(ν, J) =∑i,jYi,j(ν +12)i[J(J + 1)]j , (3.3)where the Dunham coefficients, Yi,j , are given by e.g. Y1,0 ≈ ωe, Y2,0 ≈ −ωeχe,Y0,1 ≈ Be, Y0,2 ≈ −De, etc. In his treatment, Dunham modeled a diatomic asa vibrating rotor in which the electronic PES, U(x), is given as a Taylor seriesexpansion about the minimum, Re, where x = (R−Re)/Re,U(x) = a0x2(1 +∑i≥1aixi) , (3.4)and the {ai} potential parameters can also be related to the Dunham coefficients(e.g. a0 = −Y 21,0/(4Y0,1), a1 = Y1,1Y1,0/(6Y 20,1),...). In the same way as earlier,these Dunham coefficients can be determined directly from a global fit of the ex-perimentally assigned spectral lines to the analytical energy levels using a least-squares fitting routine. The solution Dunham provided is found using the first-orderWKB (Wentzel-Kramers-Brillouin) approximation, thus the model suffers for lightmolecules and at long-ranges where the potential function diverges. Many groupshave provided corrections using higher-order WKB and Born-Oppenheimer break-down terms involving adiabatic and non-adiabatic effects [79, 80]. However, in theend this approach cannot make prediction beyond the range of the available data[81], and is complicated by the multitude of corrections necessary for reasonableresults and the strong inter-parameter correlation in the Dunham expansion.483.1. Potential Inversion Methods3.1.2 Rydberg-Klein-Rees (RKR) MethodGiven knowledge of the vibrational, G(ν), and rotational, Be, constants obtainedfrom fitting to spectral lines there is an another approach known as the Rydberg-Klein-Rees (RKR) method [82–85] for determining PESs. Similar to Dunham’s ap-proach, but with the advantage that one is not fitting a given functional form (Eq.(3.4)), this method also applies the first-order WKB (Wentzel-Kramers-Brillouin)approximation in the form of the quantization condition∫ R+R−pdR = π(ν + 1/2) (3.5)where p = 2µ√E − V (r) is the momentum and R− and R+ are the classical innerand outer turning points of the nuclear motion. However, instead of starting with amodel and finding some expression for V (r), the RKR method relies on a clever trickwhich avoids creating any definition for the potential. This requires the definitionof a special function A(E, J) with partial derivatives, ∂/∂E and ∂/∂J that can beexpressed as both a function of the two turning points, R+ and R−, and as anintegral over the vibrational quantum number, ν, where the integrand depends onlyon the spectroscopic functions G(ν) and Bν [86].R±(ν) =(f(ν)2 + f(ν)g(ν))2± f(ν) (3.6)wheref(ν) =√12µ∫ ν− 12|G(ν)−G(ν ′)|− 12dν ′g(ν) =√2µ∫ ν− 12Bν′ |G(ν)−G(ν ′)|−12 dν ′ . (3.7)Note that corrections to the rotational parameter Be to account for rotational-vibrational interactions are given by the rotational function,Bν = Be +∑i=1(−1)iαe(ν + 1/2)i (3.8)where αe are higher order constants.Although the RKR method lacks self-consistency in the vibrational and rotational493.1. Potential Inversion Methodsconstants, researcher, such as Tellinghuisen [45, 46], have developed iterative schemesto converge the results. And, standard programs such as LeRoy’s RKR1 [87] nowexist to construct RKR curves from spectral data. The procedure uses the initialestimation of the G(ν) and Bν harmonic constants to calculation V (R) using theRKR method. The RKR potential, V (r), is then used to calculate the centrifugaldistortion (spectroscopic) constants De and He which are then used as correctionsto the measured (raw frequencies) wavenumbers. With these adjusted wavenumbers,we iteratively return and obtain a least-squares estimation of the two harmonic termsto generate a new potential. This technique and many other methods [88–91] refineand expand the use of the RKR approach, however, several drawbacks appear to beinherent within its application. Namely, that its validity is generally restricted toonly when the BOA is valid, and its foundation in the WKB approximation limitsits accuracy for very light dimers. And similar to Dunham, the potentials can onlybe constructed up to highest observed vibrational state, so it becomes unreliable atenergies near and above the dissociation limit (e.g. it cannot resolve the repulsivebarrier). Lastly, RKR cannot address exotic potentials with multiple minima andstruggles with multidimensional (polyatomic) systems.3.1.3 Direct-Potential FitsThe most common approach for obtaining accurate PESs in diatomic molecules inrecent years [92–97] is through using direct potential fits (DPFs).Inverted Perturbation ApproachThe modern DPF technique is based on the original inverted perturbation approach(IPA) of Kosman and Hinze [98], which is a fully quantum mechanical method (con-trary to the semiclassical Dunham or RKR schemes) for defining diatomic PESs.The method seeks to find a linear correction δU(R) to some initial approximationof the potential curve U(R) such that the eigenvalues of the molecular Schrödingerequation with a potential term U(R) + δU(R) best agree with the measured spec-tra. Using a parametrized function for δU(R), the numerically solutions of theSchrödinger equations can be designed to converge, in the least-squares sense, to theexperimental eigenvalues. Initially [98], a series of global Legendre polynomials wereused to describe δU(R), however, it was later found [99] that IPA gave more realisticresults when the potential energy correction term was expanded in terms of local503.1. Potential Inversion Methodsdistributed Gaussian functions. These local functions were evaluated directly on aset of discrete variable representation (DVR) points and the least-square fitting tothe spectroscopic data of eigenenergies was done using singular value decomposition(SVD). However, because the success of IPA depends strongly on the choice of basisfunctions, it is difficult to guess which expansion will fit the correction better andwith less coefficients; particularly when the the real shape V (R) is quite exotic.To avoid this functional dependence, Pashov [100] proposed a modified-IPA ap-proach which expressed the correction to the potential curve as a set of n equidistantpoints and then afterwards connected these points with a spline function. Other au-thors [101–103] have extended this program, however more recently, the preferredapproach has been to use direct fits to determine parameters characterizing the po-tential energy functions.DPF AnalysisThe more general DPF analysis allows one to create model Hamiltonians of molecu-lar systems that also take into account atomic-mass-dependent radial and centrifugalpotential corrections due to the breakdown of the BOA. The unknown functions forthe PES, spin-orbit couplings or non-adiabatic corrections are parametrized usingsome initial estimates. Similar to IPA, the theoretical eigenenergies from a numericalsolution of the radial Schrödinger equation are compared with the experimental spec-tral lines, and the error between these transition energies are minimized by iterativelyoptimizing the values of the parameters in the unknown functions. Researchers suchas Coxon [104, 105], Bergeman [19, 93, 95, 96, 106] and LeRoy [80] have been veryactive in applying this approach. In particular, LeRoy [107] has developed and pro-vided free publicly available computer programs (DPotFit [108]) for implementingsuch routines.However, because IPA is based on the first-order perturbation approach to itera-tively correct the potential energy and DPF involves minimizing a multi-dimensionalnonlinear system, it becomes essential to have a good prior knowledge of the poten-tial surface. Without starting with very good initial estimations neither method willconverge to the experimental data; often because, the system of linear equations isusually overdetermined and it becomes difficult to find the global minimum. Un-fortunately, such prior estimates are not always available, particularly for the casesof double minimum potentials, and thus a lot of variables must be included in theparametrization; greatly increasing the computational costs.513.1. Potential Inversion MethodsModel PotentialsModel potentials are often used when given a set of conventional spectroscopicconstants or when fitting spectral lines from, say, FT-spectroscopy measurements[32, 73, 109]. As mentioned earlier, to represent ordinary single minimum PES, thesimplest option is the Morse function whose vibrational eigenvalues can be relatedto the spectroscopic constantsEν = ωe(ν + 1/2)− ωexe(ν + 1/2)2 (3.9)= β√2Deµ (ν + 1/2)− 2µβ2(ν + 1/2)2 . (3.10)These analytical energies can be used to fit the molecular spectra to find the optimumcoefficients, and thus give a Morse representation of the potential.Other multi-parameter functions such as the Lennard-Jones, Generalized Morse,or Morse-Lennard-Jones [110] provide more flexibility in their application, particularfor non-linear direct potential fitting [78, 80, 105, 111], however in the past severalyears the Morse/long-range (MLR) model, first introduced by LeRoy [81], has provedvery successful in representing PESs of diatomic and polyatomic molecules (see Ref.[112] and references therein). The MLR potential is a single analytical functionthat accurately describes both the deep-well region and long distance behavior ofmolecular PESs. Fitting experimental data to a single function avoids the problemsassociated with interpolation and yields reasonable results even with vacancies inthe spectra, see the work of Madison [112] for a recent application.Tiemann [113] also provides another expression, known as the “Hannover” form,which has been used a lot in the DPF papers of Bergeman [19, 93, 95, 96, 106]. Theirform will be used in subsequent Chapters as a theoretical model of our PESs.3.1.4 Reflection MethodNone of the above potential inversion techniques provide reliable information aboutthe repulsive wall at and above the dissociation energy. In an attempt to charac-terize this centrifugal barrier comes the reflection method, which is an approximateprocedure related to photodissociation dynamics, namely the “photofragmentationmapping” [114]. This approach, also known as collision-induced dissociation (CID)[115, 116], relies on collecting the kinetic energy distribution to determine repul-sive wall of a potential. However, theoretically the approach relies on semi-classical523.2. A New Approachassumptions and fitting restrictions; and experimentally the research has been lim-ited to the study of ions due to the need of large molecular currents for good CIDmeasurements [116].3.2 A New ApproachGiven the ever increasing pool of high-quality spectroscopic and scattering data,our understanding of intra- and inter-molecular dynamics is slowly becoming moreand more detailed. The main missing element appears to be the lack of systematicmethods for obtaining accurate potential energy surfaces and dipole moment sur-faces (DMS) from such data. Though ab initio computational methods and moreapproximate methods which are suitable for larger systems [35–38, 117–119] arequite successful, in most cases the line positions predicted by such computations arenot yet of “spectroscopic accuracy”. Present day analysis of spectroscopic data areinvariably performed in the “forward direction”, using analytic functional forms torepresent the PES, and optimizing their parameters to replicate to the extent possiblethe experimental values. Such parameter fitting methods are, however, inherentlydeficient due to our rather arbitrary choice of the functional forms used and the lim-ited number of parameters we can consider. As we saw the RKR method [82–85] is adirect inversion approach available for diatomic molecules, but it can only generatepotentials below the dissociation energy and is limited by the range of validity of theWKB approximation. Moreover, potentials possessing two or more minima cannotbe inverted by this method. A few modified RKR methods have been developedfor polyatomic molecules [88–91], but these methods are approximate as they arebased on the adiabatic separation of the molecular coordinates and the vibrationalself-consistent-field (SCF) approximations. Overcoming some of the restrictuions ofthe RKR method are the fitting approaches such as inverse perturbation analysis[98, 99, 120], IPA for bound-continuum [101], DPF [19, 93, 95, 96, 106], and oth-ers [102, 103], but these methods usually rely on having a good prior knowledge ofthe potential surface, which may not always be available. Other authors [121–123]have developed procedures which determine the PES numerically using Tikhonovregularization. Though, this requires substantial number of iterations and computerresources to deal with slow convergence, the method is under-determined due tothe small number of data points used relative to the large number of unknowns,resulting in multiple solutions and non-unique PES. As in other inversion schemes533.2. A New Approach(e.g. that of scattering cross sections data[50, 124]) the stumbling block appearsto be the extraction of the relative phases of the relevant (e.g. fluorescence, photo-absorption) amplitudes. Given these phases, it was previously shown [125, 126] thata point-by-point extraction of the excited state PES is possible.The inversion procedure presented herein, based upon the work of Shapiro [126],avoids using SCF or WKB approximations and is theoretically applicable to poly-atomic systems [6]. The non uniqueness of the associated inverse problem is solvedby introducing a priori restrictions on the form of the PES using experimental in-formation. The method as developed so far suffers however from the need to firstsolve the phase problem, namely to determine the relative phases of the transition-dipole amplitudes from the (absolute-value) squares of these amplitudes provided byexperiments.In the following we show how the extraction of the phases (for bound-boundtransitions - the signs) of transition-dipole amplitudes from their experimentally-measured absolute-value-squares, can be done in an iterative manner. Concurrentwith the iterative phase extraction we generate, in an ever increasing range, the ex-cited state potential(s) from which emission occurs. We demonstrate this procedurefor several excited states of the Na2 molecule. We are able to extract the potential(s)below and above the dissociation threshold with accuracies that are percentage-wisesubstantially better than, though proportional to, the accuracy of the experimentaldata. Our results for the repulsive regions are obtained with greater accuracy andare beyond the assumptions and fitting restrictions inherent of procedures basedupon the reflection method [115, 116].As a by-product, we also generate the transition-dipole function and go beyondthe Franck Condon (FC) approximation[47]. We see no inherent restrictions, givendata of sufficient quality and completeness, in successfully applying this method toany polyatomic molecule.3.2.1 Potential InversionOur aim is to compute a “target” potential Ve(R), assuming that we already know the(reference) potential Vg(R) to which emission occurs, where R ≡ (R1, R2, · · · , RN )designates a collection of the internuclear coordinates of the (polyatomic) moleculeof interest.The time-independent Schrödinger equations associated with the two Born Op-543.2. A New Approachpenheimer potentials are,[Kˆ(R) + Vg(R)− Ei]χi(R) = 0 (3.11)and[Kˆ(R) + Ve(R)− Es]φs(R) = 0, (3.12)where Kˆ(R) denotes the kinetic energy operator for the nuclear coordinates R, χiand φs are, respectively, the bound (rovibrational) wave functions of the ground andexcited electronic states. Ei and Es are the energies of these rovibrational states.We begin by assuming that we already know the transition-dipole amplitudes,di,s ≡∫dRχ∗i (R)µe,g(R)φs(R) ≈ µ∫dRχ∗i (R)φs(R), (3.13)where the last equation spells out the FC approximation [47]. We now rewrite Eq.(3.12) asφs(R)Ve(R) = [Es − Kˆ(R)]φs(R), (3.14)and multiply both sides from the left by φ∗s. Summing over all discrete states s, anddividing by ∑s |φs|2, we obtain thatVe(R) =1∑s |φs(R)|2∑sφ∗s(R)[Es − Kˆ(R)]φs(R). (3.15)The unknown excited states φs(R) are now expanded in the basis set of groundrovibrational wave functions χi(R),φs(R) =∑iχi(R)〈χi | φs〉, (3.16)By employing the FC approximation we can write the 〈χi|φs〉 overlaps in terms ofthe transition-dipole matrix elements,〈χi|φs〉 =1µ〈χi|µe,g(R)|φs〉 =di,sµ . (3.17)After replacing 〈χi|φs〉 of Eq. (3.15) with di,s/µ and using Eq. (3.16), we obtain553.2. A New Approachthat,Ve(R) =1∑s |∑i χi(R)di,s|2∑s(∑iχidi,s)∗[Es − Kˆ(R)](∑jχjdj,s). (3.18)Use of Eq. (3.11) allows to express the action of the Kˆ(R) operator on the groundstate wave function χi(R) as,Kˆ(R)χj(R) = [Ej − Vg(R)]χj(R). (3.19)Insertion into Eq. 3.18 yields,Ve(R) =1∑s |∑i χi(R)di,s|2∑s∑i,j(χidi,s)∗ [ωi,s + Vg] (χjdj,s), (3.20)where ωi,s ≡ Es − Ej are the transition energies. Finally, by pulling Vg(R) out ofthe double summation, we arrive at the expressionVe(R) =∑s∑i,j d∗i,sdj,sωi,sχ∗i (R)χj(R)∑s |∑i χi(R)di,s|2+ Vg(R). (3.21)Irrespective of any experimental source of error, discussed in section 3.3.3 below,this formula is expected to be accurate only for R values for which the expansion ofφs(R) according to Eq. (3.16) converges well. An additional source of error occurswhen the numerator and denominator of Eq. (3.21) approach zero in the far tail ofthe classically forbidden region. This point is discussed further below.3.2.2 Transition-Dipole FunctionThe Ve(R) potentials extracted as described above, enable us to obtain the electronictransition dipole function µe,g(R). Starting from the definition of the transitiondipole matrix elements (Eq. (3.13)), we multiply both sides by χ∗i (R) and performa summation over the index i. Assuming that the set of ground states {χi} forms acomplete basis in which we can expand φs, we apply the completeness condition∑iχi(R)χ∗i (R′) = δ(R−R′), (3.22)563.2. A New Approachand obtain that∑iχi(R)di,s = µe,g(R)φs(R). (3.23)Using the redundancy of Eq. (3.23) with respect to the choice of φs, we multiply itby φ∗s(R) on both sides and sum over the index s. Upon rearrangement, our formulafor the dipole function becomesµe,g(R) =∑i,s φ∗s(R)χi(R)di,s∑s |φs(R)|2. (3.24)The s summation in Eq. (3.24) guarantees (as in Eq. (3.21)) that the denominatordoes not vanish at the zeroes of any of the wave functions.As in the potential extraction, the accuracy in calculating the dipole function ata given position R depends on the completeness of the eigenstates of our referencepotential in that region and having non-vanishing excited states φs(R) amplitudes.3.2.3 Going Beyond the FC ApproximationIn deriving Eq. (3.21) we have invoked the FC approximation[125, 126], havingreplaced the overlap matrix elements fis ≡ 〈χi | φs〉 that should have been used in Eq.(3.21), by dis/µe,g. (The unknown constant µe,g is unimportant because it cancels outin both the denominator and numerator.) We had to make this replacement becauseprior to the extraction of Vex all we had were the, experimentally derived, |dis|matrix elements. Having completed the above iteration procedure obtaining a goodrepresentation of Vex(R) (because the FC approximation is usually well justified), wecan now go beyond the FC approximation by computing the overlap matrix elements,fis, and using them in Eq. (3.21) instead of dis. As will be shown, this results in aslightly improved potential, most noticeably near the Re region.Explicitly,φs(R) =∑i χi(R)di,sµe,g(R). (3.25)We can now recompute the overlap integrals as,fi,s ≡ 〈χi | φs〉 =∫dR∑j χj(R)dj,sχi(R)µe,g(R), (3.26)and use these new values in Eq. 3.21. We will return to this approach later inSec. Application3.3 Application3.3.1 Bound-Bound Diatomic ModelTo better explain how the phase extraction procedure works we simplify the treat-ment by concentrating on bound-bound transitions of diatomic molecules, reportingabout the use of continuum states in a later section (Sec. 3.4). In order to “jump-start” the inversion scheme, given that neither the signs of di,s, nor Ve(R), nor φs(R)are initially known, we rely on the fact that the excited potential can be parametrizednear its minimum position Re as an harmonic, or better still, a Morse potential,VM (R) = De [exp(−β(R−Re))− 1]2 −De + Te. (3.27)We can make a rough guess as to the value of Re based on some measured intensityratios, such as |d0,0|2/|d1,0|2. Likewise, we can estimate De, Te, and β, based on thethree lowest transition frequencies ωs=0,1,2;i=0, to the i = 0 ground vibrational level.It turns out that a rough initial estimate is all we need, as the final result is ratherinsensitive to it. Once we make any reasonable initial guess for Ve, we can obtainthe signs of the transition-dipole matrix elements according to Eq. (3.13) (note thatthe magnitudes are already known from experiment), and substitute the results inEq. (3.21) using only low-lying vibrational states, to obtain an improved estimateof Ve(R) near Re. This procedure can then be iterated until convergence.As shown in Fig. (3.1), at this stage the potential is only known over a small[(Re −∆2)−(Re +∆2)]region, since we have used data pertaining to only its lowest lying φs(R) vibrationalstates. We now augment this region by smoothly extrapolating the potential usingtwo exponential functions, one pertaining to R < Re − ∆2 and the other pertainingto R > Re + ∆2 . We then compute a few higher vibrational states of the augmentedpotential, and re-calculate the signs of the di,s matrix elements for all states con-sidered thus far. We substitute the matrix elements thus obtained in Eq. (3.21)and iterate for Ve(R) until convergence. We repeat this procedure anew, each timeslightly increasing the ∆ range of the potential for which Eq. (3.21) is used.583.3. ApplicationFigure 3.1: The beginning of the target potential extraction process near the mini-mum region, using a Morse potential as an initial guess.Figure 3.2: The simulated emission spectrum from the Na2 A(1Σ+u ) and B(1Πu)potentials to the X(1Σ+g ) potential, where we display the transitions between νA =[0− 25] (νB = [0, 25]) and νX = [0, 60] states.593.3. Application3.3.2 Na2 Single WellAs a realistic application of our procedure we now present the extraction of theA(1Σ+u ) and B(1Π+u ) excited state potentials of Na2 from fluorescence data. We sim-ulate the experimental fluorescence data by calculating |di,s|2 for various transitionsbetween the A(1Σ+u ) and B(1Πu) states and the ground X(1Σ+g ) state, using the abinitio potentials and transition dipole functions as provided by Schmidt et al.[127].In Fig. 3.2, we show the emission transition signals from the A(1Σ+u ) and B(1Πu)potentials to the X(1Σ+g ) potential.Figure 3.3: The A(1Σ+u ) Na2 potential extracted piece by piece by including an everincreasing number of excited vibrational states.Given the above |di,s|2 and ωi,s line positions, we have computed the A(1Σ+u )potential from Eq. (3.21) using the νX = [0 − 60] and νA = [0 − 20] states. InFig. 3.3, we see that the computed potential matches the ab initio potential veryaccurately, extending well beyond the turning points of the highest φs(νA = 20)state. As we venture more and more into the classically forbidden regions, themagnitudes of the φ(νA = 0−20) states start to diminish rapidly, eventually causingthe error in the extracted potential to be unacceptable. As the number of includedstates is increased, a highly accurate excited potential appears to “spread out” tothe right and left of Re at an ever increasing ∆. At the end of the procedure,603.3. ApplicationRegion Rleft Rright RMS RMSwith FCA Beyond FCADeep Well LTP(0) RTP(0) 0.19 0.01Well LTP(5) RTP(5) 0.22 0.11Left end LTP(20) LTP(18) 1.69 1.10Right end RTP(18) RTP(20) 0.16 0.03Global LTP(20) RTP(20) 0.33 0.55Table 3.1: RMS errors (in cm−1) of the A(1Σ+u ) potential in different regions, ex-tracted using the FC approximation and beyond, where LTP(ν) and RTP(ν) denotethe left-turning-point and right-turning-point of a given νth vibrational state.the range for which the potential is accurate is limited only by the range of thehighest φs(R) vibrational state used. It is not possible to further increase the rangeof the extracted potential by including even higher (νA > 25) states without alsoincluding the X(1Σ+g ) continuum states [128]. The ground continuum states areneeded to satisfy the completeness condition of Eq. (3.16) for the νA > 25 high lyingvibrational states.We supplement the figures by providing in Table 3.1 the details about the root-mean-square (RMS) errors of the calculated A(1Σ+u ) potential in the FC approxima-tion and beyond. The various regions of interest are defined via the left (LTP(ν)),and right (RTP(ν)) turning points of a given vibrational level ν. The RMS errors arecalculated between the inverted curve and the “true” ab initio potential within thesedifferent ranges of the potential. As expected, our accuracy decreases as we expandthe well region where we calculate the RMS error from between the turning points ofthe ν = 0, to the ν = 5, to the ν = 20 states. This is due to the increasing difficultlyof expanding the higher excited states in the ground state basis of eigenstates. Wealso find larger errors appearing on the left (repulsive) wall arising from the inaccu-racies in this expansion. Although we use this RMS error between the inverted andreal curves throughout this chapter to quantify the accuracy of our inversion proce-dure, experimentalist may be more interested in how well we reproduce the spectrallines (i.e. how accurately do we find the energy eigenvalues). Unfortunately, thereis no direct way to compare the RMS error of the extracted potential curve to theRMS error between the calculated and actual eigenenergies since it depends on howthe discrepancies in the curves arise. Instead, the (global) RMS curve error can bethought of as an upper limit to the errors in line positions. In general, the latter arefound to be at least half that due to oscillations in the calculated potential about613.3. Applicationthe “true” curve.Figure 3.4: Number of wrong signs of transition dipole moments for the A(1Σ+u )Na2 potential by including an ever increasing number of excited vibrational states,where the black circle, red square, and blue diamond lines denotes signal strengthgreater than 1× 10−2, 1× 10−4, and 1× 10−6 of the peak signal, respectively. Thesmall inset describes the same physical argument with different uncertainties in themeasured signal strength (green: 5% uncertainty; orange: 10% uncertainty).As shown in Fig. (3.4), the number of wrong computed signs of transition dipolemoments (TDMs) for the A(1Σ+u ) Na2 potential decreases to zero as the number ofexcited vibrational states increases. The total number of transition lines betweenνA = [0 − 25] and νX = [0, 60] states is 1586. In Fig. (3.4), black circle denotesthe data for signals strength greater than 1 × 10−2 of the peak signal, for whichthere are 623 transition lines. By increasing number of excited vibrational statesand extending ∆, the calculated potential and the associated wavefunctions becomemore accurate, which directly improves the calculated signs of TDMs. We noticethat, in Fig. (3.4), there are a total of 908 transitions whose signals strength aregreater than 1×10−6 of the peak signal, signs for all of which are computed correctlywhen all νA = [0− 25] states are included.The results of our calculation for the B(1Πu) potential, using the same number ofstates, are shown in Fig.3.5. Table 3.2 shows the RMS errors of the calculated B(1Πu)potential in the different regions using the FC approximation and going beyond it.623.3. ApplicationFigure 3.5: The extracted potential from Eq. (3.21) for the B(1Π+u ) state.Quite naturally, the FC approximation is most accurate near the equilibrium regionwhere the wave functions are more tightly localized.Na2 Transition Dipole FunctionAs an application of the above procedure we have calculated the transition-dipolefunction µe,g(R) for the X(1Σ+g )←A(1Σ+u ), B(1Π+u ) transitions. Figure 3.6a showsthe dipole function µX,A(R) produced using the νX = [0 − 60] and νA = [0 − 25]states. The extracted µX,A(R) function is in excellent agreement with the ab-initioRegion Rleft Rright RMS RMSFCA Beyond FCADeep Well LTP(0) RTP(0) 0.11 0.07Well LTP(5) RTP(5) 0.12 0.08Left end LTP(25) LTP(23) 2.63 2.19Right end RTP(23) RTP(25) 0.88 0.75Global LTP(25) RTP(25) 0.33 0.29Table 3.2: RMS errors (in cm−1) of the B(1Π+u ) potential in different regions, ex-tracted using the FC approximation and beyond, where LTP(ν) and RTP(ν) denotethe left-turning-point and right-turning-point of a given νth vibrational state.633.3. Applicationone. It starts deviating from it only in the R < 5.5 Bohr and R > 9.5 Bohr deeptunneling regions.Figure 3.6b shows the dipole function µX,B(R) using of the νX = [0 − 60] andνB = [0 − 25] states, where, similarly, µX,B(R) becomes less accurate in the deeptunneling region of the high-lying states of φs(νB).3.3.3 RobustnessIn order to test the robustness of this procedure against insufficient or inaccuratefluorescence data, we performed several calculations. We considered the followingsources of errors:1. Missing lines, such as those connected with the low lying excited vibrationalstates.2. Errors in the magnitudes of the measured |di,s|2 line strengths.Missing Lines.As described above, in order to build the potential we start with the lowest vi-brational states, e.g. νA = 0, and construct the well region. Then, by iterativelyincluding higher states, we extend the range of the inverted potential in a step-wise fashion. Note, not all transition data are required to be available because only39.3% of the transitions are of real usefulness because the intensities of all the othertransition are less than 1% of the highest transition intensity.If the fluorescence lines associated with the lowest states are unavailable it wouldseem that this procedure might run into difficulties. This however is not the case,as we show by performing the potential extraction of the A(1Σ+u ) state using onlythe νA = [1 − 25] or νA = [2 − 25] states. Fig. 3.7 shows the potentials extractedwith the deficient data as compared to the complete νA = [0− 25] case. Figure 3.7clearly shows that this procedure is stable even when the fluorescence data for theνA = 0, 1 states are unavailable.One extreme case is that only a few emission bands from highly energetic vi-brational states are available. In this case, it is still possible to generate a globalpotential with an RMS error of a few cm−1. Table 3.3 shows the RMS error anal-ysis for the global potential with the use of different number of transition bands,where we assume the available emission data are from νA = [20 − 25] highly ener-getic vibrational states. It is obvious that, even with only four vibrational states,643.3. Application(a)(b)Figure 3.6: (a) The extracted dipole function µX,A(R) for transitions between theX(1Σ+g ) and A(1Σ+u ) potentials. (b)The extracted dipole function µX,B(R) for tran-sitions between the X(1Σ+g ) and B(1Π+u ) potentials.653.3. ApplicationFigure 3.7: Computed A 1Σ+u potentials with νA = [0 − 25], [1 − 25] and [2 − 25]states.the global potential can be generated with an impressive accuracy with a 2.6 cm−1error. This can be easily applied to experiments to locate the optimal intermediatestate, which has good Frank-Condon overlaps with both the initial and final states,in the creation of low-vibrational states of diatomic molecules [30, 129, 130].Errors in |di,s|2.To address this situation we have introduced random errors to the |di,s|2 data andrepeated the extraction of the potentials in the presence of such errors. The errorsνA used RMS (cm−1)[24− 25] 7.8[23− 25] 4.1[22− 25] 2.6[21− 25] 2.0[20− 25] 1.7Table 3.3: Global RMS errors of the A(1Σ+u ) potential with different number oftransition bands from only a few highly excited vibrational states νA.663.3. Application(a)(b)Figure 3.8: (a) The deviations of the average extracted A(1Σ+u ) potential relativeto the true ab-initio potential as a function of R, for different RMS errors of thesimulated fluorescence line strengths. (b) Root-mean-square (RMS) errors of theconstructed B 1Π+u potential with a varying degree of errors in the experimentalfluorescence data. |di,s|2.673.4. Continuum Statesin |di,s|2 were generated in a random fashion of 1%, 2%, 5% and 10% RMS errorsrelative to the average |di,s|2 values. In Fig. (3.8a) we show the deviation from theab-initio potential of the average A(1Σ+u ) numerical potential at different R-valuesextracted from such data for the above magnitudes of errors. In Fig. (3.4) we alsoshow the number of wrong calculated signs for the TDMs for the A(1Σ+u ) potentialwith 0% (black), 5% (green), and 10% (orange) RMS errors in the measured signalstrength, |di,s|2, where signs for all 623 transition lines for signals strength greaterthan 1 × 10−2 of the peak signal are computed correctly! Figure (3.8a) and Fig.(3.4) demonstrate a remarkable robustness against inaccuracies in the experimentalfluorescence data: Percentage-wise the deviations from the ab initio potential aremuch smaller than the relative experimental RMS error. We note that, in addition,no attempt was made to smoothly interpolate the numerically obtained averageextracted potential values. It stands to reason that when such interpolations areintroduced, the deviations from the true potential would be further reduced.Similarly, in Fig. (3.8b) we show the deviation from the ab-initio potential of theaverage B 1Π+u numerical potential at different R-values extracted from such data,for the above magnitudes of errors. Figure (3.8b) demonstrates the remarkablerobustness against inaccuracies in the experimental fluorescence data: Percentage-wise the deviations from the “true” potential are much smaller than the relativeexperimental RMS error.3.4 Continuum StatesWe present an extension to the bound-continuum cases of our bound-bound inver-sion scheme for extracting excited state potentials and transition-dipoles from fluo-rescence data. The procedure involves the discretization of the continuous spectrumusing box-normalization. The addition of the continuous spectrum guarantees com-pleteness of the basis set used for the implicit expansion of the unknown excited statevibrational wave functions. Here we show how to extend these ideas by incorporat-ing spectral information of bound-continuum transitions. We do so by discretizingthe continuum, thereby circumventing the need for solving for the general phasesassociated with continuum-to-bound transitions.We first show that the use of bound state information alone is insufficient inmany cases, including the extraction of the Na2 C(1Πu) electronic potential. In thisdemonstration we use an ab initio Vg(R) potential [127], which is in agreement with683.4. Continuum Statesother sources [131–133]). Given Vg(R), we generate χi, and Ei. We then use theab initio [127] excited state potential, Ve(R), to simulate the experimental emissionspectrum to the ground state, consisting of ωi,s and |di,s|. In order to determinethe phases (signs, in the case of bound-bound transitions) of di,s, which enter Eq.3.21, we begin with an initial estimate of Ve(R). Given this estimate, we calculatean initial estimate of the signs of di,s. Since near the equilibrium position Re, mostdiatomic potentials can be approximated by the Morse potential,VM (R) = De(e−2β(R−Re) − 2e−β(R−Re))+ Te, (3.28)we perform a least-squares fit of the Morse parameters: Te, De, and β to match afew of the transition frequencies ωs;i=0 to the (Es−Ei)/~ analytical values generatedby VM (R).A rough approximation as to the value of Re can be made by observing thestationary-phase point associated with the transition intensity |ds=0,i| from the ex-cited ground state s = 0, the flexibility allowed for in this value is shown below.Using this method we have computed the C(1Πu) potential. The end result shownin Fig. 3.9 matches the ab initio potential to a root-mean-squared (RMS) accuracyof 0.1 cm−1 in the ∆ region. The potential extends well beyond the turning points ofthe highest φs(νC) state, and fails only when the magnitudes of all the φ(νC = 0−32)states have become so small in the classically forbidden region that the calculationbecomes numerically unstable.RestrictionsIn view of the above one would like to use as many φs(R) states as possible. Howeverthe number of φs(R) states is restricted because we cannot always maintain Cs ofthe following completeness condition close enough to unity1 ≥ Cs ≡imax∑i=1|〈χi|φs〉|2, (3.29)where imax is the highest ground vibrational state included in the expansion (Eq.3.16).Figure (3.9) clearly shows the limited extent for which the Na2 C(1Πu) potentialcan be computed based solely on bound-bound transition data. Of the approximately73 bound states in C(1Πu) we are only able to accuracy include up to state s = 32693.4. Continuum StatesFigure 3.9: (Black line) - the ground X(1Σ+g ) Na2 state. (Red dots) - ab initio energiesof the excited C(1Πu) potential; (Green dashed line) - the initial Morse fit. (Brown line) -the partial Na2 C(1Πu) potential extracted using only the s = [0, 32] excited bound statesand the i = [0, 63] vibrational ground states. (Violet) - the highest vibrational statesused. (Blue) - the entire excited C(1Πu) potential extracted using the s = [0, 110] states.(Cyan) - the highest vibrational state used. This extraction can only be done when weincorporate transitions to the continuum of the ground X(1Σ+g ) state. The RMS deviationof the potential extracted in this way from the ab initio potential is 0.1 cm−1.703.4. Continuum States(upper grey line) using the i = [0, 63] bound states of X(1Σ+g ) in the expansion.Thus, it is not possible to further increase the range of the extracted potential bysimply including higher (s > 32) states without first increasing the size of our basisset for expansion. Thus the only solution is to enlarge our basis by including theX(1Σ+g ) continuum states. We discuss how to implement this approach and dealwith the continuous spectrum in the next section.3.4.1 Box NormalizationAs mentioned above, we wish to include the continuum states in our procedure inorder to satisfy the completeness condition [134, 135]imax∑i=1|χi(R)〉〈χi(R)|+∫|χ(k,R)〉〈χ(k,R)| dk = 1 (3.30)over as large a range of R values as possible. However, dealing directly with thecontinuum states is difficult because the transition dipole matrix elements, ds,k,are complex numbers when |k〉 are scattering states. To avoid this difficulty wereplace the true scattering states, χ(k,R), with discretized box-normalized [136–139] scattering states, χk(R). In this way we are in effect binning the continuousspectrum into discrete intervals, each being represented by a box-normalized state.We have that,∫|χ(k,R)〉〈χ(k,R)| dk ≈kmax∑k|χk(R)〉〈χk(R)| (3.31)In order to bin ds(k), the “true” bound continuum dipole matrix-elements, wecompare dk,s, the dipole matrix elements for the box-normalized states, with integralsover a range of continuous ds(k) values,dk,s ≡∫ k+δk+k−δk−ds(k′) dk′ (3.32)where δk− = (Ek − Ek−1)/2 and δk+ = (Ek+1 − Ek)/2. In order to do that wefirst compute the “true” bound-continuum ds(k) matrix elements using the artificialchannel method (ACM) [51–53], for all the transitions between the s = 1, ..., 40vibrational states of the Na2 C(1Πu) to continuum states whose energies start at713.4. Continuum States0 0.001 0.002 0.003 0.004 0.005Energy (Hartree) overlap  (a.u.) ACM Integration R = [3.5,15]Wavefunction overlap (Rbarrier = 15)ACM Integration R = [3.5,25]Wavefunction overlap (Rbarrier = 25)Figure 3.10: The ds,k/µ Franck-Condon factors from the s = 40 state of the Na2 C(1Πu)state to the discretized continuum states of the X(1Σ+g ) state for two different boxes: Bluestars - the R = [0− 15] box results; black circles - the R = [0− 25] box results. The “exact”ds(k)/µ values, as obtained using the ACM scheme, are marked by a dashed green line forthe [3.5− 15] Bohr range and as a red line for the [3.5− 25] Bohr range .just above the dissociation limit of X(1Σ+g ).Using two normalization boxes, the inner wall of both of which is placed at R = 0and the outer walls - at either R = 15 Bohr or at R = 25 Bohr, we calculate dk,saccording to Eq. (3.32) and compare them to dk,s. The results are shown in Figs.3.10. We see that the smaller normalization box results in substantial inaccuraciesdue to the very sparseness of the box-normalized levels. As we increase the size ofthe box, the density of the box-normalized states increases, and so does the accuracyof the binning, resulting in an excellent agreement for R = 25 Bohr (and even forR = 20 Bohr) between dk,s and dk,s.The success of the box normalization procedure allows us to use the set of dk,smatrix elements in exactly the same way we use any set of bound-bound matrixelements in our bound-bound inversion procedure, enabling us to use the informa-tion contained in the bound-continuum transitions. In this way we have effectivelyconverted the difficult bound-continuum phase problem into the sign-determinationproblem of the bound-bound case.We can therefore extend the excited wave function expansion of Eq. (3.16) over723.4. Continuum StatesFigure 3.11: The values of Cs of Eq. (3.29), representing the completeness condition,for each of the φs states of the C(1Πu) potential, as more and more X(1Σ+g ) (χi+χ(k))vibrational states are included in the expansion. Shown are Cs values for i = 1, ..., 63(black circles); when we add to these states all the χ(k) continuum states below 516cm−1 (red squares); all the continuum states below 1244 cm−1 (blue triangles); andall the continuum states below 2018 cm−1 (green x’s). The inset shows these energylevels relative to the dissociation energy of the X(1Σ+g ) potential.733.5. Double Wellall available states. The quality of this expansion for each individual excited statecan be expressed via the Cs completeness parameter of Eq. (3.29), which approachesunity when the basis becomes complete. In Fig. 3.11 we plot the value of Cs foreach of the bound states φs of C(1Πu) when we include 63 vibrational bound state inX(1Σ+g ) (◦), and when we add box-normalized continuum states covering an energyrange of 516 cm−1 (✷), 1244 cm−1 (△), and 2018 cm−1 (×). As expected, the numberof excited states that can be accurately represented increases with the increase inbasis set, until all the bound states of C(1Πu) can be well reproduced by the groundstates expansion.3.4.2 Extraction of the Entire Na2 C(1Πu) PotentialIn Fig. 3.9 we have shown how the inclusion of continuum state allows us to extractthe entire Na2 C(1Πu) potential curve. In order to do so we have supplementedthe bound-bound transitions (including the i = 1, ..., 70 bound states) with the box-normalized continuum states whose energies reach 2000 cm−1 above the dissociationlimit. The box-normalized continuum state of the highest energy considered is alsoshown in the figure. We have incorporated only half of the transitions (those withprobability > 0.1%) and obtained RMS deviation from the exact potential of lessthan 0.1 cm−1.3.5 Double Well3.5.1 Morse ModelIn order to show that our approach is capable of extracting potential energy curveswith several minima, we have examined a model double-Morse potentials of the form,V (R) = f(R)V1(R) + [1− f(R)]V2(R), (3.33)where f(R) is a smooth switching functionf(R = (1 + tanh[(R− 9.12)/0.3])/2 (3.34)and V1 and V2 are Morse potentials whose parameters are, De1 = 8 × 10−3 a.u.,β1 = 0.2952 a.u., Re1 = 6.94 Bohr, De2 = 6 × 10−3 a.u., β2 = 0.29 a.u., andRe2 = 10 Bohr. By adopting the same approach and using the FCA, we extract743.5. Double Wellthe potential by adding vibrational states ν in Eq. (3.21). Fig. 3.12 shows thebuilding process of the double-well potential obtained by considering the transitionsassociated with the ν = [0−11] states. The reason behind the successful constructionlies in the extension of the wave function beyond the classical turning points. Thus,though the ν = 5 vibrational state is well localized in the left well, the wave functionφ(ν = 5) spills over to the second well, thereby enabling the accurate extraction ofthe second minimum.Figure 3.12: Step-wise construction of a model double-well potential.3.5.2 Extraction of the Na2 21Σ+u (3s+4s) Double Well PotentialAs a more sophisticated demonstration of the use of continuum states, we presentthe extraction of a double well potential, that of the Na2 21Σ+u (3s+4s) state [100,127, 140]. In this Σ+g ⇋ Σ+u transition we use for simplicity only the rotation-less 21Σ+u (3s+4s) states. These states are coupled optically to the J = 1 X1Σ+grovibrational levels, hence we add a centrifugal potential of 1/(MR2) to the electronicground state potential.With the use of only the bound states of X(1Σ+g ), we are restricted by the lackof completeness to reconstructing the 21Σ+u (3s+4s) potential at energies below Es=5753.5. Double Wellof the deeper well. In contrast, the addition of continuum transitions over a rangeof about 2300 cm−1 above the ground state dissociation limit, allows for over 110bound states of the 21Σ+u (3s+4s) potential to be accurately expanded in the groundstate basis set.In Fig. 3.13 we show how the piecewise potential re-construction proceeds. Westart with the s = 5 bound state (blue dashed line) and proceed to the s = 13level (green dot-dashed line). This level yields the first indication that an additionalwell might exist (brown colored lines): The inclusion of the s = 13 state causes thepotential to exhibit a new oscillatory behavior at a region which up till now wassmoothly increasing.Figure 3.13: The wave functions of the fifth vibrational state (dashed) shows the maximalenergy to which we are able to re-construct the Na2 21Σ+u (3s+4s) potential using only thebound states of the X(1Σ+g ) state. In contrast, the inclusion of continuum states up to the2300 cm−1 above the dissociation threshold allows for a complete re-construction of thepotential curve. The s = 13 vibrational level (dot-dashed) exhibits the first indication thatan additional well (brown line) might exist. Using this information to smoothly extrapolatethe next potential iteration reveals more of the second well (black line). The two wellsand the barrier between them assume their fully developed forms (dotted line) when thes = 27 state is introduced. The inset shows the complete re-construction (blue line) of the21Σ+u (3s+4s) potential using up to the s = 110 state (black line). The RMS deviation ofthis potential from the ab initio one (red dots) is less than 1 cm−1.We thus proceed by averaging over the highly oscillating region on the left and763.6. Rotational Stateslinearly extrapolating the inner and outer regions of the two wells. The new set ofeigenstates thus obtained account to some extent for the contribution of the secondwell. As we introduce new eigenstates and iterate this procedure, the potentialbegins to converge in the two well regions (see black line in Fig. 3.13). Uponreaching level s = 27 (dotted-line) we have fully resolved the potential hump betweenthe wells due to the information contained in the tunneling tails of the vibrationalwave functions. Continuing onward, we end the construction of this potential at thes = 110 vibrational state (E110 ∼ 0.116833 a.u.) finding that we have extractedthe repulsive wall of the double well potential nearly 1000 wave-numbers above theknown atomic asymptote of Na(3s) + Na(4s) (0.11728 Hartree at about 56.7 Bohr[100, 140]). The blue curve in Fig. 3.13 displays the final result constituting theextraction of the double well potential to within 1 cm−1 RMS error.One drawback of the present inversion is that our procedure so far assumes theFCA which is not justified in the present case because there is a substantial difference[141, 142] between the dipole function at each well. Thus one has to go beyond theFCA, as we will show in Sec.3.7 (see Ref. [7]). However, first, we also introduce anapproach to image diatomic potentials using different rotational bands.3.6 Rotational StatesHere we demonstrate how to image diatomic potentials using data from differentrotational bands. We will assume that we have emission spectrum for only twoexcited rovibrational states exist: (s, J1) and (t, J2) where s 6= t and J1 6= J2. Recallthat we need at least two states in our fundamental expression to kill the nodalbehavior in the denominator. Thus, we wish to express the excited state electronicPES as:Vex(R) = f(Vgr(R), χi[J′1], χm[J′2], ω, J1, J2)(3.35)where Vgr(R) is the ground state PES, ω is the transition frequency, and χi[J1] andχm[J2] are ground state rovibrational wavefunction with angular momentums of J ′1and J ′2, respectively. For simplicity, we will use i, j, s to denote the quantities associ-ated with transitions from (s, J1) and use m,n, t to denote the quantities associatedwith transitions from (t, J2).773.6. Rotational States3.6.1 Inversion Formula for R and P BranchesConsider when we use only one state, (s, J1), and we take the R branch first:(s, J1)→ (i1, J1 − 1). The previous inversion formula for the Q branch [143, 144]Vex(R)(Q branch) =∑i,j d∗i,sdj,sωi,sχ∗i (R)χj(R)|∑i χi(R)di,s|2+ Vgr(R), (3.36)then becomes[Vex(R) +J1(J1 + 1)2µR2](R branch) =∑i,j d∗i,sdj,sωi,sχ∗i (R)χj(R)|∑i χi(R)di,s|2+[Vgr(R) +J1(J1 − 1)2µR2], (3.37)and thusVex(R)(R branch) =∑i,j d∗i,sdj,sωi,sχ∗i (R)χj(R)|∑i χi(R)di,s|2+ Vgr(R)−2J12µR2 . (3.38)As before, we add a summation over all s to express a more general term whichavoids nodal structure in the denominator of Eq. [3.38]Vex(R)(R branch) =∑s∑i,j d∗i,sdj,sωi,sχ∗i (R)χj(R)∑s |∑i χi(R)di,s|2+ Vgr(R)−2J12µR2 . (3.39)And, similarly for the P branch transition the inversion formula is given byVex(R)(P branch) =∑s∑i,j d∗i,sdj,sωi,sχ∗i (R)χj(R)∑s |∑i χi(R)di,s|2+ Vgr(R) +2J12µR2 . (3.40)3.6.2 Inversion Formula for Different Rotational States, J1 and J2Now we consider the probable scenario that there are only two sets of emission dataready to be used: (s, J1) and (t, J2). However, with limited source of experimentalmeasurements, as in Ref. [145], there is often not much freedom to choose twoexcited states within the same branch and the same rotational quantum number.Let us take the case that J1 6= J2 but they both correspond to the R branches, i.e.transitions for (s, J1)→ (i, J1 − 1) and (t, J2)→ (m,J2 − 1). The inversion formula783.6. Rotational Statesusing these two excited states is derived similar to before (Eq. (3.39))Vex(R)(R branch) =∑i,j d∗i,sdj,s(ωi,s − 2J12µR2)χ∗i (R)χj(R)|∑i χi(R)di,s|2 + |∑m χm(R)dm,t|2+∑m,n d∗m,tdn,t(ωm,t − 2J22µR2)χ∗m(R)χn(R)|∑i χi(R)di,s|2 + |∑m χm(R)dm,t|2+ Vgr(R). (3.41)Again, we use i, j, s to denote the quantities associated with transitions from (s, J1)and use m,n, t to denote the quantities associated with transitions from (t, J2).Again, the general form for using more vibrational states with the same J1 and/orJ2 numbers can be derived by including summations over these labels to improveextraction accuracy:Vex(R)(R branch) =∑s∑i,j d∗i,sdj,s(ωi,s − 2J12µR2)χ∗i (R)χj(R)∑s |∑i χi(R)di,s|2 +∑t |∑m χm(R)dm,t|2+∑t∑m,n d∗m,tdn,t(ωm,t − 2J22µR2)χ∗m(R)χn(R)∑s |∑i χi(R)di,s|2 +∑t |∑m χm(R)dm,t|2+ Vgr(R). (3.42)Likewise, the inversion formula can be derived in a similar fashion for pure P branchtransitions as wellVex(R)(P branch) =∑s∑i,j d∗i,sdj,s(ωi,s + 2J12µR2)χ∗i (R)χj(R)∑s |∑i χi(R)di,s|2 +∑t |∑m χm(R)dm,t|2+∑t∑m,n d∗m,tdn,t(ωm,t + 2J22µR2)χ∗m(R)χn(R)∑s |∑i χi(R)di,s|2 +∑t |∑m χm(R)dm,t|2+ Vgr(R). (3.43)Simple extensions include finding mixed branch transitions for different rotational793.6. Rotational Statesstates: namely for R+P branches:Vex(R)(R+P branches) =∑s∑i,j d∗i,sdj,s(ωi,s − 2J12µR2)χ∗i (R)χj(R)∑s |∑i χi(R)di,s|2 +∑t |∑m χm(R)dm,t|2+∑t∑m,n d∗m,tdn,t(ωm,t + 2J22µR2)χ∗m(R)χn(R)∑s |∑i χi(R)di,s|2 +∑t |∑m χm(R)dm,t|2+ Vgr(R); (3.44)for R+Q branches:Vex(R)(R+Q branches) =∑s∑i,j d∗i,sdj,s(ωi,s − 2J12µR2)χ∗i (R)χj(R)∑s |∑i χi(R)di,s|2 +∑t |∑m χm(R)dm,t|2+∑t∑m,n d∗m,tdn,t (ωm,t)χ∗m(R)χn(R)∑s |∑i χi(R)di,s|2 +∑t |∑m χm(R)dm,t|2+ Vgr(R); (3.45)and for Q+P branches:Vex(R)(Q+P branches) =∑s∑i,j d∗i,sdj,s (ωi,s)χ∗i (R)χj(R)∑s |∑i χi(R)di,s|2 +∑t |∑m χm(R)dm,t|2+∑t∑m,n d∗m,tdn,t(ωm,t + 2J22µR2)χ∗m(R)χn(R)∑s |∑i χi(R)di,s|2 +∑t |∑m χm(R)dm,t|2+ Vgr(R). (3.46)Lastly, a generalized inversion formula of Eq. (3.21) using all three branches canbe given, where the R branch involves the J ′ = J1 → J ′′ = J1 +1 transitions, the Qbranch would correspond to the J ′ = J2 → J ′′ = J2 transitions, and the P branch- to the J ′ = J3 → J ′′ = J3 − 1 transitions. Using standard spectroscopic notationground states are denoted by J ′′, and we use J ′ for the excited state.803.7. ExtensionsVex(R)(R+Q+P branches) = 1A∑s∑i,jd∗i,sdj,s(ωi,s −2J12µR2)χ∗i (R)χj(R)+∑t∑m,nd∗m,tdn,t (ωm,t)χ∗m(R)χn(R)+∑u∑k,ld∗k,tdl,u(ωk,u +2J32µR2)χ∗k(R)χl(R)+ Vgr(R), (3.47)where A ≡∑s |∑i χi(R)di,s|2 +∑t |∑m χm(R)dm,t|2 +∑u |∑l χl(R)dl,u|2. Note,a summation over J1, J2 and J3 can be made when more than one rotational excitedstates exist within the same branch.3.7 ExtensionsIn this section we present two works which extend our inversion method. The firstto multi-dimensional potentials [6]:(A) X. Li and M. Shapiro, Inversion of two-dimensional potentials from frequency-resolved spectroscopic data, J. Chem. Phys. 134, 094113 (2011) ,and the second to obtaining solutions beyond the FCA [7]:(B) X. Li and M. Shapiro, The Dipole Correction Method for Extracting ExcitedState Potentials and Electronic Transition Dipoles from Fluorescence Data, Isr. J.Chem. 52, 1-7 (2012) .The author was intimately involved in the discussions and verification of theresults leading to manuscript (A), although, the writing and processing was solelycarried-out by Dr. Xuan Li with the support of Dr. Moshe Shapiro. The workcontained in (B) is an elaboration of results already presented (see Sec. (3.2.3)).The author participated closely in discussions leading to the manuscript, which wascomposed and submitted by Dr. Xuan Li and Dr. Moshe Shapiro3.7.1 Two Dimensional PotentialsTo date, the determination of multi-dimensional PESs have relied on ab initio [34]or semiempirical [35–38, 117–119] quantum chemistry computations. Unfortunately,813.7. Extensionsthese calculations are not capable of finding the spectroscopic line positions to thesame accuracy as experimental measurements, this is in particular true for PESwith many degrees of freedom (DOF). A lot of effort [81, 89, 98, 146–149, 149, 150].has been put forth into studying polyatomics, however, the Hamiltonians based oninternal modes becomes complicated due to distortions in the normal mode structuredue to anharmonic couplings, Duschinksy rotations [151], and rotational (Coriolis)couplings [152]. Even though new coordinates which diagonize the kinetic energyoperator may simplify matters [56], no systematic approach exists for extracting aPES for experimental data.Here, we review the extension of the inversion procedure that was used to obtainone-dimensional PES from the spectroscopic line positions and line strengths fordiatomic molecules. The model is based on representing a linear triatomic moleculewith two degrees of vibrational freedom, or, a torsion and bending mode within apolyatomic system. For complete details see Ref. [6]Linear Triatomic ModelFor modeling a linear triatomic molecule with two vibrational DOF, we can definetwo independent coordinates:R = x1 −m2x2 +m3x3m2 +m3, and,r = r1 = |x3 − x2| , (3.48)where mi and xi are the masses and one-dimensional positions of the three atomsrespectively. The two-dimensional potential model, V (R), will be represented as asum of independent, one-dimensional, potentials between each atom. For instance,the pairwise potentials between atoms i and j will be given as Vk(rk), k 6= i, j whererk, k 6= i, j is the distance between the two atoms i and j. Therefore, the groundstate PES will be given asVg(r1, r2, r3) = V g1 (r1) + Vg2 (r2) + Vg3 (r3) , (3.49)where the V gk (rk), k = {1, 2, 3}, potentials can be taken as three Na2 X(1Σ+g ) groundstate potentials [127].The excited state potential, Vex(R), will be modeled in the same way except fora three-body term to account for distortions, as well as, a constant term giving the823.7. ExtensionsFigure 3.14: (a) A schematic illustration of two 2D PES. (b) Fluorescence linesassociated with a typical spectrum of the system in (a). Image taken from Ref. [6]833.7. Extensionsenergy shift between the ground and excited state PES. Similar to the ground state,Vex(r1, r2, r3) = V e1 (r1) + V e2 (r2) + V e3 (r3)+ V3b(r1, r2, r3)− Eshift, (3.50)where the V ek (rk) represent pairwise potential between atoms i and j, however wenow use the Na2 B(1Πu) excited state potentials ([127]) for their expression. Thethree-body potential will be defined asV3b = D3b e[−∑k(rk−reqk )2/∆R2], (3.51)where D3b = −0.02 a.u., req3 = req1 = 0.5req2 = 6 Bohr, and ∆R = 8 Bohr. And, theenergy shift will be equal to Eshift = E(Na3p)− E(Na3s).Fig. (3.14a) gives a schematic illustration of the two-dimensional PES as a func-tion of the two internal coordinates. The simulated frequency-resolved spectrumbetween rovibrational eigenstates of these two surfaces is shown in Fig. (3.14b).With our objective being to determine the excited state potential from the intensityfluorescence data, we follow our inversion procedure, and first make an initial guessfor the excited state potential. For this we use a sum of Morse potentials,VMk = Dek[exp(−βk(rk − reqk ))− 1]2 + T ek . (3.52)where, as before, we can estimate the Morse parameters Dek, T ek , reqk and βk usinglower transition frequencies, ωi,s, to some, say i = 1, ground vibrational level, wehope to roughly estimate Vex about its minimum position (Req, req), asVex ≈ VM1 (r1) + VM2 (r2) + VM3 (r3)− Eshift . (3.53)And as before, the distribution of |di=1,s|2 for the low lying excited φs states can beused to estimate reqk .This approximation for Vex now allows for an initial estimate of the (unknown)signs of the transition dipole matrix elements, di,s, using Eq. (3.13) (recall that themagnitudes of the real bound-bound matrix elements) are already known from exper-iment). Then, using Eq. (3.21) including only a few low-lying rovibrational excitedstates, a new estimate of Vex(R, r) about (Req, req) can be calculated. Iterating thisprocedure until convergence, and then extrapolating the potential in both R and r843.7. Extensionsdirections allows for the entire surface of the electronic excited state to be generated.We show the results of the extraction in Fig. (3.15) against the “true” PES. TheFigure 3.15: A comparison between a PES extracted using Eq. (3.21), using onlythe s = 1− 10 states, with E(s = 10) = 3.602× 10−2 a.u. (full lines), and the “true”PES (points). Image taken from Ref. [6]calculated potential, which used the ten lowest rovibrational states in the excitedpotential, required 335 of the low lying ground vibrational states We find excellentagreement between the two surfaces, with average RMS errors of approximately 0.01cm−1. (see Ref. [6] for more details of the results).The two dimensional inversion approach also works for more complicated excitedstate potentials, Vex, such as those possessing multiple minima. Using the sameexcited state PES as earlier (Eq. (3.50)), we can add an additional three body term,V ′3b, with parameters given by D′3b = −0.003 a.u., req3 = req1 = 0.5req2 = 7.2 Bohr,and ∆R = 0.5 Bohr, to generate a second local minima. Shown in Fig. (3.16a)is this “true” potential compared with the results of the extraction using excitedrovibrational levels s = 1 − 6. First, we see that the results match very well, withRMS errors below 0.1 cm−1 (see Ref. [6] for more details). However, it is importantto note that the energy level of the highest rovibrational state, E(s = 6) = 0.03539a.u., is in fact lower than the energy of the local minimum (0.03554 a.u.). Thusthe ability to resolve the second local minima occurs solely due to the informationcontained in the classically-forbidden region of the vibrational wavefunctions. In853.7. ExtensionsFigure 3.16: Extraction of a double-well PES: (a) A comparison between the PESextracted using Eq. (3.21) (full lines), and the “true” PES (points), having usedonly the s = 1 − 6 states, with E(s = 6) = 3.54 × 10−2 a.u.; (b) |φ6(R, r)|2 - theprobability-density for the highest state used.863.7. ExtensionsFig. (3.16b), we plot the probability-density associated with the highest energeticstate (s = 6) used in the extraction. This illustrates that the tunneling tails ofthe vibrational wave functions spread across the molecular space and allows for theaccurate determination of the PES well beyond the classical allowed region. Finally,out of the 4800 possible emission lines of non-negligible strengths which could havebeen used, only one fourth were included in the calculation. As we have encounteredearlier, weak transitions lines, particularly those with large uncertainties [143, 144]can be removed from the computation without detrimental effects.This inversions of a two-dimensional excited state PES from emission (or absorp-tion) line positions and intensities opens up the door to extracting more complicatedstructures, such as three-dimensional PES and those of nonlinear molecules. Also,similar to the diatomic case [143, 144], given knowledge of the excited PES, it willbe of great interest to determine (the R coordinate dependence of) the electronictransition dipole function. Having the transition dipole function, then permits us tomove beyond the Franck-Condon approximation (FCA) to obtain a more accuratesolution to the PES.3.7.2 Dipole Correction ExtensionAs we have shown in this chapter, we are able to extract excited state electronicpotentials and the nuclear coordinate dependence of the transition dipole functionfrom spectroscopic data by using the Franck-Condon approximation (FCA). How-ever, for many situations [141, 142], the FCA may not be justified because therecould be a substantial variation in the dipole function across the region spannedby the potential. In Sec. 3.2.3, we briefly suggested an approach for extending ourinversion procedure beyond the FCA. Here, we review the work of collaborators (seeRef. [7]) which continued on this idea and gives the specific accuracy improvements.It should be noted that this technique is applicable to any other inversion schemeswhich are based on the FCA (e.g. method of Avisar and Tannor [153, 154]).In the following presentation, we also further address an inherent limitation oftenfound in experimental data, namely, that not all of the desired data is available. Inparticular, it can be easier to label transition lines in a spectrum to/from low-lyingstates and highly-excited states [33]. Thus, in the case when no information isavailable for the majority of the intermediate states of an PES, inversion methodssuch as RKR are not possible. However, as we demonstrate below by using theQ-branch transitions from only a few low-lying states (s = 0− 5) and some highly-873.7. Extensionsexcited states (s = 20− 23) with large rotational constants (J ′ = 44), our approachis still capable of obtaining great results.Na2 A StateFigure 3.17: The average transition dipole moment µ¯e,g of Eq. (3.13) for transitionsfrom |s = 0 − 10, J ′ = 45, A 1Σu〉 to the |i = 0 − 54, J ′ = 44, X 1Σg〉 states as afunction of the transition magnitude, |di,s|. Figure and caption taken from Ref. [7].As mentioned, the accuracy of the inverted potential using Eq. (3.21) and thetransition dipole function using Eq. (3.24) is limited by the validity of the FCA.In order to test this assumption, we would like to look at the true variation ofthe average dipole, µ¯e,g, for different vibrational quantum numbers of the ground(g), and excited (e) state potentials. Using the |ν = 0 − 10, J ′ = 45, A 1Σu〉 to|ν = 0 − 54, J ′ = 44, X 1Σg〉 transitions in Na2, we plot, in Fig. 3.17, µ¯e,g as afunction of the transition magnitudes, |di,s|. Instead of these plotted points for µ¯e,gfalling on a horizontal line (as would be the case when FCA is valid), we find astandard deviation of the points from the average value of the dipole is ∼ 6.6%. Totake this variation into account, our objective will be to use the information of the883.7. Extensionsµe,g(R) function derived in Eq. (3.24) to improve our inversion method.By rearranging Eq. 3.13, we can get an expression for the excited state rovibra-tional eigenstatesφs(R) =∑i χi(R)di,sµe,g(R), (3.54)in terms of the ground rovibrational states, χi(R), the TDMe, di,s, and the transi-tion dipole function, µe,g(R). These eigenstates can then be used to recompute theoverlap integrals with the ground states as followsfi,s ≡ 〈χi | φs〉 =∫dR∑j χj(R)dj,sχi(R)µe,g(R). (3.55)Then these new values allow for us to write the potential inversion formula withoutemploying the FCA asVex(R) =∑s∑i,j f∗i,sfj,sωi,sχ∗i (R)χj(R)∑s |∑i χi(R)fi,s|2+ Vg(R). (3.56)Now we have a dipole correction method to Eq. (3.21) which proceeds in thefollowing manner:1. Compute Vex(R) and µe,g(R) using our original approach within the FCA2. Compute the lowest eigenfunction φs=0(R) of Vex(R) and use Eq. (3.13) im-prove µe,g(R).3. Use the improved µe,g(R) and Eq. (3.55) to improve the fi,s overlap integralsfor s = 0, 1.4. Finally, use the overlap integrals and Eq. (3.56) to improve the inverted po-tential, Vex(R).5. Iterate steps 2–4 for an increasing number of the excited states (e.g. s = 0, 1),until all available excited rovibrational states, φs have been included.Notice, that this iterative process uses both the intensities and frequencies of theexperimental data to improve the potential and the transition dipole function simul-taneously.The success of this approach is demonstrated with corrections to the extractedA(1Σ+u ) excited state potential of Na2 using fluorescence data to the ground X(1Σ+g )893.7. Extensionsstate. Note that we use ab initio potentials and transition dipole functions of Schmidtet al.[127] to simulate the experimental fluorescence data. In particular, we use therovibrational state s = [0−10] and i = [0−54], in the J ′ = 45→ J ′′ = 44 P-branch,where s and i label the excited and ground states respectively. In Fig. 3.18a,we plot the difference between the computed and the true potentials as a functionof the internuclear distance for our original procedure and after using the dipolecorrection method. It can clearly be seen that going beyond the FCA improves theaccuracy of the extracted A(1Σ+u ) potential by up to a factor of 20. As expected thedetermination of the transition dipole function is improved by a similar factor, asshown in Fig. 3.18b.Unlike our previous FCA-based inversion schemes [143, 144], this study itera-tive improves the signs and magnitudes of the overlap integrals, fi,s, instead of onlyfinding the signs of the transition dipole matrix elements. This occurs as a result ofiteratively obtaining more and more accurate representations of the excited rovibra-tional eigenstates due to utilizing the transition dipole function. We demonstratedthis in Fig. 3.19a and Fig. 3.19b, where we plot the errors in the extracted wavefunctions, φs=0(R) and φs=5(R), as a function of the radial distance, R, for the twocases (with and without FCA). And, as anticipated a factor of 10 improvement isseen in the wavefunction accuracy.Na2 B StateWe now demonstrate how the dipole correction procedure improves the inversionresults even when limited spectroscopic data is available. Using the excited B 1Πuelectronic potential of Na2, we consider only the Q branch transitions (J ′ = 44 →J ′′ = 44) between the rovibrational states in the X1Σ+g state and the six low-lyingstates (s = 0 − 5) and the four most highly-excited states (s = 20 − 23) of the Bstate. In Fig. 3.20 we see the same results as earlier, the differences between thecomputed and the true potential (a) and transition dipole function (b) as a functionof the internuclear distance is improved by a factor of 10-20 when performing thedipole correction. Similar to before, we obtain a global RMS error of 0.08 cm−1 overan energy range of 2300 cm−1, and an RMS error about the minimum of 0.02 cm−1,using only 10 rovibrational states of the B potential.As with our original formulation, the accuracy of the inversion is reliant on thedegree with which we satisfy the completeness condition. Namely, how well can theexcited rovibrational wave functions, φs, be expressed as a linear combination of the903.7. Extensions(a)(b)Figure 3.18: A comparison of the “dipole correction” and the FCA-based inversionsof the Na2 A 1Σ+u potential and the A 1Σ+u → X1Σ+g electronic transition function.(a) Differences (in cm−1) between the inverted and the true potential; (b) Differences(in Debye) between the inverted and the true electronic transition dipole function.Figure and caption taken from Ref. [7].913.7. Extensions(a)(b)Figure 3.19: A comparison between the accuracies of the computed vibrational wavefunctions as derived by the FCA-based inversion and the “dipole correction” inversionfor (a) φs=0(R) and (b) φs=5(R). Figure and caption taken from Ref. [7].923.7. Extensions(a)(b)Figure 3.20: A comparison between the “dipole correction” and FCA-based inversionsof the Na2 B 1Πu potential and the B 1Πu → X1Σ+g transition dipole function. (a)Differences (in cm−1) between the inverted and the true potential; (b) Differences(in Debye) between the inverted and the true electronic transition dipole function.Figure and caption taken from Ref. [7].933.8. Summaryground vibrational states, χi. To this end, it may be necessary to include continuumstates to satisfy completeness [155], a point which has already been address in Sec.3.4 (and Ref. [128]).3.8 SummaryWe have presented a way of solving the “spectroscopic phase”, namely the extrac-tion of the signs of transition-dipole matrix elements from their measured absolute-squared values. We have shown that given the transition-dipole amplitudes it ispossible to generate the excited state potential(s) and transition dipole moment(s)within (to global RMS error of 0.11 cm1) and beyond (0.03− 0.1 cm−1) the FC ap-proximation. This implies that line positions can, in general, be calculated to within0.01 cm−1. The procedure, which was demonstrated numerically for the diatomic(Na2) case, is in principle valid for any polyatomic molecule. It has been shown thatthe inversion formula, as well as the procedure, works for a modeled two-dimensionalpotential [6].One strong requirement of our approach is the ability to correctly assign thespectral lines to the right pair of ground and excited state eigenvalues. Otherwise,the fundamental inversion equation (Eq. (3.21)) which includes a double summationover the frequencies and transition dipole moments between particular ground andexcited eigenstate doesn’t make sense. Fortunately, not all lines need to be assigned.As we have shown, only a few excited eigenstates are necessary to extract the fullpotential curve. A more critical restriction is that those excited states included inthe calculation can be nicely expanded in the GS basis (e.g. that the completenesscondition is fulfilled). This is because the errors in the potential will grow quadrat-ically with errors in the completeness condition (see Eq. (3.21)). Therefore, ourprocedure works best when the electronic GS spans a similar or greater radial rangethat the electronic excited state. For instance, it is preferable for the long rangebehavior of the GS to be C3R3 and that of the excited state to be C6R6 rather than viceversa. However, one advantage to this summation over transition dipole moments(TDMs) in the basis set expansion is the robustness of our procedure to randomerrors in the spectroscopic data. We have shown that we can achieve great resultswith even 10 % random errors in the transition magnitudes, and we attribute thisto the summation averaging over the random fluctuations. Though in the generalcase, the tolerable size of these random errors is difficult to define since it depends943.8. Summaryon the system’s characteristics such as the number of GS levels included in the basisset. One should note that any GS eigenstates having weak TDMs to an excitedstate included in the calculation can be omitted in the expansion (and thus does notrequire assignment). Lastly, although we haven’t explicitly proven that our methodwill converge to a unique solution, we expect this to be the case due to the unique-ness of the expanded excited eigenstates and knowledge of their eigenvalues, whichtogether with a Hamiltonian of a given form fully specifies the potential term.3.8.1 A Brief ComparisonWhere our method excels over other approaches is in cases when the excited statepotential is not well represented by a standard model or when only a few energylevels are known (but most of their TDMs have been resolved).RKR methods perform well for heavier molecules where the WKB approximationis most valid, and when the PES is low dimensional and simple in form. It is knownto break down for irregularly shaped PESs, such as those with a “barrier”, “shelf” ormultiple minima. RKR results are given in tabular form and defined only for regionsof the PES covered by the spectroscopic data. Moreover, the RKR approach doesn’tprovide any information in how to extrapolate beyond regions lacking experimentaldata. Determining all the energy levels can be a cumbersome task which requires arelatively large number of empirical molecular constants with most of them havinglittle physical significance.DPF approaches require a well-behaved analytical potential model that can beused in the fitting procedure. Poorly chosen functions result in nonphysical behavioroccurring in either the short-range or long-range regions. To accurately describe non-standard potential curves with irregular portions, higher order polynomials must beused, which leads to greater difficultly in converging the results. This high degreeof parametrization, which is also necessary as the dimensionality of the system in-creases, tends to become very computationally expensive. In contrast, our inversionprocedure performs equally well for irregularly shaped potentials, and scales muchbetter for multidimensional cases. Our results typically converge for each iterationin the construction after a few cycles, independent of the potential form or dimen-sionality. Thus we require far fewer time-consuming diagonalizations of the systembefore converging.Ab-initio quantum chemistry calculations which have begun to yield reasonableresults for small molecular systems, are limited in their availability and are difficult953.8. Summaryto implement individually for a case of interest. Also, they tend to struggle eitherat short or large interatomic distances due to the balance between accounting forthe electron exchange and charge overlap at short ranges, and the effects of weaklong-range forces which should be treated perturbatively. These calculation becomegreatly complicated as the number of atoms in polyatomic molecules increases andrequire significant computer resources. In contrast, our procedure is simple to im-plement and can reproduce equally accurate results at all ranges of the potential,given that the excited states can be well described in the ground state basis. Thecomputational time for the FCA approach is estimated to scale as Nb× tdiag, whereNb is the total number of excited rovibrational states used in Eq. [3.47], and tdiagis the time required to calculate the eigenstates from the constructed PES. Thedipole correction method requires additional effort which is estimated to scale asNb ×Nb/2× tdiag. Thus, for the one dimensional diatomic case, where the value oftdiag is on the order of seconds, the required computational time is rather short. Formulti-dimensional case, past work (see Ref. [38]) involving the realistic calculationof several hundred eigenstates, has required computational times on the order ofhours, making the inversion process more expensive, yet more efficient than otherapproaches which require a greater number of diagonalizations.96Chapter 4Molecular Wavefunction Imaging4.1 IntroductionAs the field of quantum information processing progresses into the experimental do-main, it becomes increasingly necessary to develop techniques for the non-destructivedetection and reconstruction of (complex) wave functions. Such reconstruction isneeded for the verification of the initial preparation; the detection of errors due todecoherence and noise; the determination of the fidelity of the logical operations andfor the read-out of the final state. In the last few years, molecules have emerged aspromising media for large-scale quantum computations, partly due to the existenceof strong dipole-dipole interactions that serve as a resource for two-qubit operations[156–161]. A number of theoretical and experimental publications demonstratingfeasible read/write operations [75, 162–166] and logic gates [167–173] have beenpublished, with the read-out amounting to the retrieval of both the amplitude andthe phase of a molecular wave packet [125, 174–177]. There have been several tech-niques for imaging molecular wave packets, each with their own drawbacks, alreadysuggested.4.1.1 Quantum State TomographyOne such technique based on quantum state tomography, [75, 174, 175, 178] whichhas been mainly applied to photonic states, uses a sequence of identical measure-ments within a series of different bases to completely reconstruct the quantum wave-function.Developed by Walmsley [8], molecular emission tomography measures the spec-trum of fluorescence from a molecule at many different times for many differentorientations of the phase space coordinates. For an excited vibrational wavepacket,this measurement obtains a quasi-probability distribution of the vibrational mode ofa molecule.974.1. IntroductionFigure 4.1: Gating fluorescence of an excited vibrational wavepacket in Na2 for thegate positions t = O, 37, 75, 112, and 150 fs relatively to the pump pulse. Takenfrom Ref. [8]In Fig. (4.1), the spectra of the gated fluorescence, calculated using the up-conversion technique (Sec. (2.4.2)), is shown for an excited wavepacket in Na2 [8].Semi-classically we would expect that a molecule will emit a photon at time T ofenergy equal to the difference between potential energy at an internuclear distance,R(T ), of the position of the initial wavepacket and the final electronic state. Thus,as the wavepacket oscillates in the potential, the fluorescence intensity as a functionof wavelength should change correspondingly in time. Assuming, for a molecularensemble, that the total fluorescence intensity is constant in time (i.e the Franck-Condon approximation where value of the transition dipole moment is constant),then the results of Fig. (4.1) should be a good representation of the shape of thewavepacket as a function of internuclear distance at different times. As can be seen,the wavepacket only samples a small range of difference potential when near theturning points leading to the narrow spectrum. Whereas, when the the wavepackethas a higher velocity it traverses over a wider range of potential energies causingthe broader spectrum. By assuming that the wave packet propagates in a harmonicpotential it is possible to use these probability distributions to generate the Wignerfunction of the state of the molecule [174]. Unfortunately, molecular imaging tomog-984.1. Introductionraphy requires a very large number of measurements, such as different phase-spaceorientations, and is restricted to working well only in the harmonic approximation.4.1.2 Algebraic-InversionAnother technique [125, 176, 177] involves the (algebraic) inversion of the time-dependent fluorescence intensity. It is also possible[125, 126] to reconstruct an un-known excited state Ψ(R, t) by assuming that it can be expanded in terms of N ofthe excited vibrational eigenstates,Ψ(R, t) =N∑s=1asφs(R) exp(−iEst),using the fluorescence signal F (t), given[125, 126] as,F (t) =∑i∑s,s′(ωi,sωi,s′)32 ds,idi,s′a∗sas′e−i(Es−Es′ )t,provided we know the di,s amplitudes (including their signs). According to Eq.(3.21) this knowledge is enough to determine Vex(R), which in turn determinesφs. The expansion coefficients as can then be determined by “strobing” the fluo-rescence at N(N − 1)/2 time points and solving a set of quadratic[125, 126] orlinear, “holographic”[179], equations. Under conditions pertinent to most electronictransitions, namely, when the Franck-Condon approximation is justified, the previ-ous procedure of extracting the dipole moment phases [126] did not work well. Themain limitation of this method is that the conversion of the bi-linear a∗s′as productsto individual as coefficients is very sensitive to experimental errors.4.1.3 Interferometric ApproachesThis inordinate sensitivity to experimental errors can be overcome by linearizing theproblem, as done in the “quantum state holography” method [179, 180] and otherlinear interferometric approaches [165, 166, 181–183]. However, these linear wave-packet interferometry (WPI) techniques encounter limitations related to evolving thetarget and reference wave packets on similar potential energy surfaces (PES) [184].In addition, they are insensitive to any target probability amplitudes resulting fromnonlinear excitation process involving eigenstates with small Frank-Condon factorsto the initial states [184]. Using a non-linear WPI approach [185, 186], Cina and994.1. IntroductionHumble proposed a remedy to these problems and a resolution to the hindranceof linear-WPI methods associated with the reconstruction of unknown target wavepackets on ill-characterized high-lying potentials. Their approach is based on usinga pair of phase-related pulse pairs which produce a signal that is quadrilinear in theincident field amplitudes. Though, success of their approach relies on the existence,and detailed information, of an auxiliary intermediate electronic state used for thepropagation of the reference wave packet. Moreover, it is not clear how uncertaintiesin this reference wave packet affect the fidelity of their reconstruction. Thus, wefind that hampering all wave-packet interferometry (WPI) approaches remains thenecessity, and accurate characterization, of some reference electronic state.4.1.4 CARS ImagingCorrelation functions and/or wave-packet overlaps have also been accessed withoutthe need of reference wave packets, instead through the use of Coherent Anti-StokesRaman Scattering (CARS) spectroscopy [153, 154, 187–190].The CARS process is a third-order nonlinear optical process involving three laserbeams which interact with molecular (vibrational) states in much the same way as inRaman spectroscopy. However, unlike Raman whose signal relies on a spontaneoustransition, CARS provides a coherently driven transition giving rise to a signal thatis quadratically stronger and can be collected much faster in practical situations(∼ 105) than that of Raman transitions. Moreover, because CARS uses two laserfrequencies to interact resonantly with a specific molecular vibration the signal isrelatively high frequency (thus minimal fluorescence interference) and provides betterresolutions. A good review of CARS can be found in Ref. [191].Milner et. al. [192, 193] developed a technique for characterizing the molecularvibrations of polyatomic molecules by performing a cross-correlation frequency re-solved optical gating (XFROG) of the CARS signal. XFROG is an extension, basedoff the original FROG method of Trebino [194], which was initial designed for thecomplete temporal and spectral characterization of ultrashort laser pulses. The pro-cedure measures the cross-correlation between a known and unknown electric fieldby focusing the two (pulses) together in a nonlinear medium. The spectrum of thenonlinear signal, which manifest itself as nonresonant sum frequency generation ina non-linear crystal, is then measured with a spectrometer for many delay points.Using the data from both the frequency and time domains, XFROG is capable of pro-viding accurate spectral intensity and phase retrieval of the unknown electric field.1004.1. IntroductionThe combined XFROG CARS approach thus allows for the extraction of the ampli-tude and phase information of molecular vibrations in a iterative process. However,the technique has only been demonstrated from ground electronic state vibrations,and is sensitive to one-photon resonances effecting the reference field.Recently, Avisar and Tannor [153, 154, 190] have also applied a CARS schemeto imaging multi-dimensional wave packets. In their approach the first of the CARSpulses acts as the excitation field, creating wavepacket in an excited PES. Then, thesecond pulse dumps the population back into the ground state, before the third pulsereturns that population to the excited state again. By performing a Fourier transformalong the time delay of the second field, the excited wave packet’s cross-correlationfunction with the vibrational eigenfunction of the ground state PES can be relatedto the third order polarization given as the CARS signal. Since their relationsinvolves a direct correspondence with elements of the raw CARS signal, significantdiscrepancies arises once reasonable errors are included in the experimentally data[195]. In addition, their method has so far been applied only to the special case ofa δ(t − t0) pulse, where the wavepacket Ψ(r, t0) is known at t0 as being a simplereplica of the ground energy eigenstate, χg(r). This is only reasonable once a largenumber of excited states have been included, which is at odds with the solution theyprovide to the unknown phase problem of the dipole matrix elements that requiresexponential time with the number of states.4.1.5 Kinetic Energy Distribution and Coulomb ExplosionLastly, kinetic energy distributions of nuclei and photo-electrons or coulomb ex-plosion [196–200] techniques provide a good basis for wave function reconstruction[201, 202]. These methods are however destructive, as they involve the ionizationand/or the break-up of molecules.4.1.6 PreambleIn many approaches the stumbling block appears to be the extraction of the phases ofthe relevant (e.g. fluorescence, photo-absorption) amplitudes. This problem is alsorelated to the reconstruction of unknown time-evolving quantum states [125, 126,179, 184–186, 203]. Given that such states can be prepared (e.g., by photo-excitationwith an ultrashort pulse of light) in multiplicity of replicas, then the relatively tinyfraction of the replicas that manage to fluoresce during the 100fsec-1psec time scales1014.1. Introductionof interest ought to yield information about the state of the majority of the (non-fluorescing) replicas in an essentially quantum-non-demolition manner. Tomographicmethods that have been tried [8, 75, 204] appear to be inaccurate far from theharmonic limit. Again, the crux to such state reconstruction is the extraction of thephases of the fundamental transition-dipole amplitudes that mediate the informationfrom the few fluorescing replicas of the unknown state to the observer.We describe a method that extracts the excited wave packet amplitudes andphases from the time dependent intensity of fluorescence. The method works bytaking a series of finite time Fourier transforms at the (ωs′,s) beat frequencies ofthe data [48]. In this way one performs a one-by-one extraction of the expansioncoefficients of the unknown wave packet. The method requires no reference state,yet because of the relatively long time averaging, it is extremely robust and is muchless sensitive to experimental errors compared to the other mentioned schemes. Ourapproach also avoid the application of the Franck-Condon approximation, which isa necessary assumption in nearly all tomographic, interferometric and CARS imag-ing proposals. In contrast to destructive methods, because, typically spontaneousemission lifetimes are relatively long (10−8− 10−6 seconds), only 10−6− 10−5 of theexcited molecules fluoresce over the 1 - 10 ps time span of interest, thus makingfluorescence based methods essentially non-destructive over such short time spans.In addition to molecular wave packet imaging, by using the well known perturba-tive regime connection between the field and the material expansion coefficients, ourmethod (following a single calibration run), can be used to extract any electromag-netic (e.m.) field whose time dependent intensity function is known. The problem ofextracting the phases of the e.m. fields has received much attention in recent years.There now exist a panoply of methods [205–208] which can be roughly categorizedas interferometric [194, 209–211] and non-interferometric [212–215]. These methodswhich involve non-trivial non-linear mixing and iterative solutions of integral equa-tions, work best for ultrashort laser pulses of large bandwidths. They do not workas well for relatively longer (10−11 − 10−10 seconds) pulses of narrower bandwidths,where our method remains successful.1024.2. Theory4.2 Theory4.2.1 Imaging MethodWe consider the short laser pulse excitation of χg(r), a rovibrational energy eigenstatebelonging to the ground electronic state, where r is a (collective) nuclear coordinate,to form a wave packet Ψ(r, t) moving on an excited electronic state, given asΨ(r, t) =∑sasφs(r) exp(−iEst/~− γst/2). (4.1)In the above, φs(r) are the excited rovibrational eigenstates of energies Es, as, arethe expansion coefficients of Ψ in terms φs, and γs are the (spontaneous emission)decay rate of each φs state, which are expressed as [125]γs =∑fAf,s (4.2)where Af,s are Einstein A-coefficients, given, in atomic units, asAf,s =4ω3f,s3c3 |µf,s|2 , (4.3)and µf,s ≡ 〈χf |µ|φs〉 are the dipole matrix elements between the electronically ex-cited energy eigenstates, φs, and the ground energy eigenstates, χf .We have shown in the past [6, 128, 143, 144] that knowledge of |µf,s|2, themagnitude-squared of the dipole matrix elements, which can be determined directlyfrom the frequency-resolved fluorescence (absorption) spectrum, coupled with knowl-edge of the ground state potential to (from) which emission (absorption) occurs,enables the extraction of the phases of µf,s. Thus, the only remaining unknowns arethe amplitudes |as| and phases δs of the as expansion coefficients. As shown below,these quantities can be extracted from the time-dependent fluorescence of Ψ(t) intovarious rovibrational ground states.Following Ref. [125] we write the rate of the time dependent fluorescence asR(t) =∑s,s′a∗sas′Cs,s′ exp(−iωs,s′t− (γs + γs′)t/2) (4.4)where ωs,s′ are “beat” frequencies [48], defined as, ωs,s′ = (Es − Es′)/~, and Cs,s′ is1034.2. Theorya time-independent molecular matrix, given asCs,s′ =23c3∑f(ω3f,s + ω3f,s′)〈φs|µ|χf 〉〈χf |µ|φs′〉. (4.5)Over typical short time scales of interest (e.g., ∼ 1 ps) the decline in populationdue to fluorescence is negligible, allowing us to disregard γs. Writing as explicitly asas = |as| exp(iδs) we obtain thatR(t) =∑s,s′Cs,s′ |as||as′ | exp(iωs,s′t+ iδs,s′) (4.6)where δs,s′ = δs−δ′s. We now select a specific (s, s′) component of the R(t) signal bycalculating at each beat frequency its “fFT” - the ωs,s′ Fourier integral calculated overlong finite time T. The filtering method works because all the ω 6= ωs,s′ componentsdecay as 1/T. Explicitly,1T∫ T0 dtR(t) exp(−iωt) = 1T∫ T0 dt∑s,s′ Cs,s′ |as||as′ | exp(iδs,s′) exp(i(ωs,s′ − ω)t)=Cs,s′ |as||as′ | exp(iδs,s′), for ωs,s′ = ωiCs,s′ |as||as′ | exp(iδs,s′ )(ωs,s′−ω)T(exp[i(ωs,s′ − ω)T ]− 1), for ωs,s′ 6= ω .(4.7)Thus, by choosing sufficiently long integration times T , satisfying T ≫ 1/min |ωs,s′−ω| for ωs,s′ 6= ω, our method amplifies just one pre-chosen (s, s′) beat term out of theR(t) signal, while eliminating all others. The absolute value of each (s, s′) term canbe taken together with the knowledge of Cs,s′ to generate a set of equations containingproducts of the unknown amplitudes |as||as′ |, which can then be solved, in the least-squares sense, to obtain the coefficient amplitudes |as|. Using Eq. (4.7) we determineδs,s′ for all s and s′, given which, we extract the individual state phases δs (up toan arbitrary overall phase). For diatomic vibrational spacings of ∼ 1000 cm−1, theintegration time must be at least 1 ps to ensure a filtering accuracy of ∼ 1%. We findthat the long integration times tend to wash out measurement errors in the time-dependent fluorescence, rendering the method, as we show in detail below, quiterobust against such errors. Because most systems will be over-determined (havingroughly s2/2 equations for s unknowns), it is possible to increase the accuracy andreduce the integration time by using as many redundant equations as possible. Inaddition, by subtracting out of R(t) the (s, s′) terms whose values have already1044.2. Theorybeen determined, we highlight the remaining (possibly weakly contributing) terms,thereby increasing their accuracy.4.2.2 Electromagnetic Field DeterminationOur ability to determine (the amplitude and phase of) excited state coefficients makesit possible to use the same setup to extract the excitation pulse’s electric field. Weagain envision a molecule, initially in one of the electronically ground vibrationalstates χg, with energy Eg, being excited by an electric field~ε(t) = ǫˆ Re {E(t) exp(−iωt)} (4.8)where ǫˆ is the polarization direction. Our aim here is to determine E(t) - the (com-plex) pulse envelope, and ω - the carrier frequency of the pulse. Provided that theRabi angle θ = ∫ dtΩ(t) << π, where the Rabi frequency Ω(t) = ~µ · ǫˆ E(t)/~,the light-matter interaction can be treated perturbatively. Thus, a time-dependentwave packet will be created in the excited state as in Eq. (4.1) with excited statecoefficients given byas = 2πif(ωs,g)µs,g, (4.9)where µs,g = 〈φs|µ|χg〉 are the known dipole matrix elements between the ground χgand excited φs rovibrational states and f(ωs,g) ≡∫dtE(t) exp[i(ωs,g − ω)t]. Usingthe method of the previous section to determine the excited state coefficients as, weobtain the complex values of the frequency spectrum f(ωs,g) from Eq. (4.9) as,f(ωs,g) =as2πiµs,g. (4.10)By tuning the pulse to excite multiple vibrational (or rotational) excited states, wehave devised a complex-valued spectrometer with a resolution given by the spacingbetween energy levels. For pulses within the limit set by one over the samplinginterval (up to several picosecond for rotational spacing) we can perform a discreteFourier transform to obtain an image of its temporal profile. The bandwidth of thepulses must, of course, be restricted by the energy span of the optically accessibleexcited states.1054.3. Results4.3 ResultsThe wave packet imaging and pulse characterization techniques are demonstrated byperforming numerical simulations on sodium dimer (Na2) wave packets. We assumethat a molecules, initially in the X1Σ+g ground rovibrational state (v = 0; J = 0)gets excited by a short laser pulse to a J = 1 wave packet of vibrational eigenstatesbelonging to the A1Σ+u electronic state. These states decay to either J = 0 orJ = 2 (the P- and R-branches), generating fluorescence to all the vibrational statesof X1Σ+g that enter the summation implicit in Cs,s′ of Eq. (4.5). Below we presentthe results for two types of (“transform-limited” and “non-trivial linearly chirped”)excitation pulses, omitting from the results excited states with populations < 0.1%because their effect on the fluorescence intensity is negligible.4.3.1 Transform-Limited PulseWe first consider the case in which a 50fs Gaussian pulse of ∼ 1012 W/cm2 withcarrier frequency of 15802 cm−1 excites the v = (2− 12); J = 1 rovibrational statesof the A state. The time-resolved fluorescence, shown in Fig. 4.2a, is generatedby time-averaging the output of Eq. (4.4) over 70 fs intervals, providing a temporalresolution that has been experimentally demonstrated using both optical gating [75]and up-conversion [4, 5] techniques. In Fig. 4.2b, displaying two different times,t = 0 (red circles) and t = 1ps (blue squares), we demonstrate that the imagedwave packet (solid lines) reproduces perfectly the “true” wave packet. Even whena random 10% Gaussian noise is introduced into the fluorescence data, the imagedwave packet is seen to coincide to better than 1% root-mean squared (rms) withthe “true” one. This robustness is likely due to the averaging over by the long timeintegration of error-generated signal fluctuations.Using Eq. (4.10) we can now deduce the frequency spectrum of the pulse ata discrete set of frequencies corresponding to the location of the absorption lines.Figure 4.2c exhibits excellent agreement between the “true” real part (blue circle),imaginary part (green squares) and absolute value (black bars) relative to the cor-responding values of the imaged pulse. The displayed temporal pulse was generatedfrom the discrete Fourier components as obtained by our method by performing dis-crete Fourier transforms at the discrete frequencies. Figure 4.2d shows the amplitude(solid line) and real part (dashed-line) of the result against that of the “true” electricfield (red circles and green crosses, respectively).1064.3. Results(a)6 7 8Radial Distance (Bohr) (Hartree)A1Σ+ut = 0t = 1 psImaged(b)(c) (d)Figure 4.2: (a) Temporal fluorescence captured over 10 ps. (b) Actual and imagedexcited state wave packets on A1Σ+u potential of Na2 at t=0 and 1 ps. (c) Spectrumof the initial excitation pulse, showing absorption lines corresponding to the excitedrovibrational eigenstates. (d) Temporal pulse profile of 50 fs width. The displayedreal part of the electric field demonstrates good phase extraction.1074.3. Results4.3.2 Linearly-Chirped Pulse(a)6 7 8 9Radial Distance (Bohr) (Hartree)A1Σ+ut = 0t = 1 psImaged(b)(c) (d)Figure 4.3: (a) Temporal fluorescence captured over 10 ps. (b) “True” (circles)and imaged (full lines) excited state wave packets moving on the A1Σ+u potentialof Na2 at t = 0 and t = 1 ps. (c) The spectrum of the initial excitation pulse,showing absorption lines corresponding to the excited rovibrational eigenstates. (d)Temporal pulse amplitude and the -π and π phase of the field at each instant.We have performed calculations similar to those of the previous section for lin-early chirped pulses. The results are shown in Fig. 4.3d. The frequency pro-file, seen in Fig. 4.3c, has sufficient bandwidth to excite ∼ 18 rovibrational states(v = (2− 19), J = 1) with population > 0.1%. We have computed the time-resolvedfluorescence signal, adding 5% of random Gaussian noise to each time-averaged (∼ 70fs) data point, from which, using our method, we have extracted the excited statecoefficients. The results of the wave packet imaging are found to be within 1% rmsof the “true” values. Figure 4.3b compares the “true” (discrete points) wave functionat the pulse center (t = 0) and at some later time (t = 1 ps), with the imaged values1084.4. Summary(solid lines).The f(ωs,g) discrete frequency components of the laser field, extracted from theas coefficients via Eq. (4.10), are displayed in Fig. 4.3c. The Fourier transformof these points gives us the temporal shape of the pulse. Figure 4.3d exhibits anexcellent agreement between the “true” and imaged time dependent pulse amplitudeand phase, correctly identifying the linear chirp of the field.4.4 SummaryWe have presented a method which reconstructs excited-state molecular wave pack-ets (and e.m. field amplitudes) by selecting each beat component of the fluorescencesignal using finite-time Fourier transforms calculated at each beat frequency. Theprocedure requires a pre-calibration involving the frequency resolved spectrum fromwhich we extract the relevant dipole matrix elements to the final states. The inte-grating time over the temporal fluorescence is bound between two limits: It must belonger than the inverse of the energy splitting between adjacent vibrational states(∼ 50fs); and it must be shorter than the average decay time of the rovibrationallevels (∼ 10ns). Within this range our method exhibits impressive robustness torandom errors in the experimental signal.Using the acquired knowledge of the excited coefficients to characterize the exci-tation pulse is a novel approach especially suitable for femtosecond pulses. Contraryto currently available methods, the experimental procedure and mathematical anal-ysis needed to reconstruct the pulse is simple and straightforward.109Chapter 5Extraction of Transition DipoleMoments5.1 IntroductionMolecules interact with light via their electric (or magnetic) multipole moments,however largely we may use the dipole approximation, and only consider theirtransition dipole moments (TDM). Whereas the |〈χf |µ|ψs〉|2 between |χf 〉 and|ψs 〉 energy eigenstates can be deduced from the strength of frequency-resolvedspectral lines, it is much more difficult to use such data to determine the phasesof the TDMs. Knowledge of these signs is however vital in many applications[125, 163, 164, 168, 170, 172, 173, 216]. For example, in the short pulse produc-tions of wave packets [125, 216], the fluorescence signal is composed of the beat-ings between many transitions whose signs fundamentally affect the observations.The determination of the TDM’s phases is highly desirable for such applicationsas these wave packet dynamics[125, 216] and quantum computation operations onatomic/molecular systems [163, 164, 168, 170, 172, 173].The well-known “inverse scattering” problem [50] involves determining the phaseof the complex-valued scattering amplitudes [43, 44] from (differential) cross sectionmeasurements. In bound-bound molecular spectroscopy the analogous “phase prob-lem” is less daunting because the relevant amplitudes are 〈χf |µ|ψs〉 - the transition-dipole matrix elements (TDMs) which are real. The “phase problem” is thus reducedto a sign determination.We have previously shown [128, 143, 144] in Chapter 3, that given the squares ofthe TDMs, we are able to derive the TDMs amplitudes and perform a point-by-pointconstruction of the excited Born-Oppenheimer potential energy surface (PES) fromwhich the emission occurs. In Chapter 4, we have also shown [176, 177, 217] that wecan image time-evolving wave packets. However, the use of frequency resolved dataentails satisfying certain “completeness conditions” [128, 143, 144]. Operationally,1105.1. Introductionthis means that we need to know the magnitudes of spectral lines linked to a largenumber of final (electronically-ground) rovibrational states. At present such exten-sive data is not yet available.An alternative method for TDM sign determination proposed in the past is basedon the semi-classical stationary phase approximation. It assumes that 〈χf |µ|ψs〉in a progression of |χf 〉 states, must go through zero and change sign whenever|〈χf |µ|ψs〉|2 hits a minimum (node) as s is varied. Unfortunately, this approach isoften unreliable because the nodes can be very closely spaced (due to high kineticenergy or small transition amplitudes). In addition, the single stationary phaseapproximation fails for many cases, especially in the long wavelength regime, orwhen the main contributions to the TDMs come from the vicinity of the classicalturning points.In the present work we consider using time-resolved data [216], containing in prin-ciple information about many spectral transitions, as an effective way of overcomingthe dearth of frequency resolved data noted above. We show that it is possible touse such data to derive the phases of individual TDMs between energy eigenstates.The method uses Bi-chromatic Coherent Control (BCC) [218] in conjunction withthe performance of a finite-time Fourier transform [217] at various (ωs′,s) beat com-ponents [48] of time-resolved fluorescence data. In the present use of BCC, oneinterferes between the stimulated emissions to a pre-determined ground state of achosen pair of states that are part of the wave packet. When the relative phasebetween the two external light fields matches the relative phase of the two TDMslinked to a common ground state, the depletion of the Fourier transform of the signalat the ωs,s′ beat frequency, giving the (ψs, ψs′) pair contribution, is maximal.One advantage of this method is that it does not require explicit knowledge of theχf (r) or ψs(r) eigenfunctions. Another advantage is that the method is essentiallynon-destructive: Typically, spontaneous emission lifetimes are 10−8 − 10−6 seconds[48], hence, over the 1 - 10 ps time span of interest only 10−6 − 10−5 of the excitedstate molecules are destroyed via their decay.In contrast to frequency-resolved experiments, time resolved experiments[216]that might in principle contain information about many spectral transitions, appearto be available. In the present paper we show how to use such data to derive the phaseinformation for individual transitions between energy eigenstates. The method usesan approach developed for the imaging of molecular wave packets whose constituentenergy eigenstates are known[217], in conjunction with Bi-chromatic Coherent Con-1115.2. Theorytrol (BCC)[218], thereby obviating the need to know the energy eigenstates. In thisapproach[217], one extracts the excited wave packet amplitudes and phases by takinga series of finite time Fourier transforms at the various (ωs′,s) beat frequencies[48]contained in the time resolved fluorescence of the entire wave packet.The way BCC helps obviate the need to know the constituents energy eigenstatesis by inducing stimulated emission from a chosen pair of states comprising the wavepacket to a pre-determined ground state. By tuning the relative phase between twoexternal light fields to match the relative phase of two TDM’s linked to a commonground state, one can attain maximum depletion of the population of the pair ofstates (s, s′) whose Fourier transform at the beat frequency ωs,s′ is being computed.5.2 TheoryWe consider exciting χg(r), a vibrational energy eigenstate belonging to the groundelectronic state, with r being a (collective) nuclear coordinate, by a short laser pulseto form a wave packet Ψ(r, t) moving on an unknown excited potential energy surface(PES), given asΨ(r, t) =∑sasψs(r) exp(−iEst/~− γst/2). (5.1)Our objective is to determine as, the expansion coefficients of Ψ in terms of ψs(r),the excited vibrational eigenstates. Using frequency-resolved spectroscopic data, wemay assume that we know the energy Es, and the (spontaneous emission) decay rate,γs, for each ψs state, written as [125]γs =∑fAf,s (5.2)where Af,s are Einstein A-coefficients, given, in atomic units, asAf,s =4ω3f,s3c3 |µf,s|2 , (5.3)with |µf,s| ≡ |〈χf |µ|ψs〉| being a set of transition dipole matrix elements (TDM’s)linking the electronically excited energy eigenstates and the ground energy eigen-states, χf .In the past we have shown [6, 128, 143, 144] that knowledge of |µf,s|2 and theground PES, enable the extraction of the phases of µf,s as well as the excited PES,1125.2. Theoryknowing which, we can calculate the excited state eigenstates ψs(r). This infor-mation, together with knowledge of the excitation laser field, is sufficient for thedetermination of the expansion coefficients as, from which we can obtain the entireexcited state wave packet Ψ(r, t). There are however difficulties in implementing thisprocedure in practice due to the scarcity of frequency-resolved studies that recordall the transitions mandated by the completeness requirement of the method.For this reason we now introduce an alternative approach that uses time-resolveddata for the determination of the phases of µf,s. The method, which by its verynature requires a less complete set of measurements, harnesses a technique recentlydeveloped by us for the imaging of excited wave packets moving on known PES[217].It is based on writing the rate of the time-resolved fluorescence from the excitedstate wave packet of Eq. (5.1) as[125]R(t) =∑s,s′a∗sa∗s′Cs,s′ exp(−iωs,s′t) (5.4)where ωs,s′ = (Es−Es′)/~ are “beat” frequencies [48], and Cs,s′ is a time-independentmolecular matrix, given asCs,s′ =23c3∑f(ω3f,s + ω3f,s′)〈ψs|µ|χf 〉〈χf |µ|ψs′〉. (5.5)In the above we have neglected the decay due to spontaneous emission because for τ(the time scale of interest) of ∼ 1 ps, γsτ << 1. In other words, due to the relativelylong spontaneous emission lifetimes of 10−8− 10−6 seconds, only 10−6− 10−5 of theexcited molecules decay over the 1 - 10 ps time span of interest.We now select[217] a specific (s, s′) component of the R(t) signal by calculatingover time T the finite time Fourier transform (fFT) of R(t) at each ωs,s′ beat fre-quency. In this way we filter out all other ω 6= ωs,s′ components, which decay as1/T. We have that,R(ω) ≡ 1T∫ T0dtR(t) exp(iωt) ≈ |Cs,s′ ||as||as′ | exp[i(δs,s′ + ξs,s′)] for ω = ωs,s′ ,(5.6)where ξs,s′(= 0, π) is the phase of the real matrix Cs,s′ , and δs,s′ = δs− δ′s is a result1135.2. Theoryof expressing as as as = |as| exp(iδs). In contrast, for ω 6= ωs,s′ ,R(ω) = iT∑s,s′Cs,s′asa∗s′(exp[i(ωs,s′ − ω)T ]− 1)ωs,s′ − ω= O(1/T ) for ω 6= ωs,s′ .Thus, in order to filter out all the Fourier components save for the ωs,s′ one, we needto choose T ≫ 1/min |ωs,s′−ωt,t′ | where ωt,t′ 6= ωs,s′ . The relatively long integrationtimes (typically T ∼ 1 ps) tend to average out random measurement errors in thetime-resolved fluorescence, rendering the method quite robust.Since only the exponential contributes to the complex nature of R(ω), we can usethe outcome of Eq. (5.6) to determine δs,s′ for all states s and s′ up to ξs,s′ = 0, π.In order to fix the sign of Cs,s′ , (or ξs,s′) thereby resolving the π uncertainty inthe phase of as, we use BCC[218] to extract the individual TDM’s, from which wecan, using Eq. (5.5), calculate Cs,s′ . With knowledge of both Cs,s′ and δs,s′ we havealready shown[217] how to determine the magnitudes of as through solving a seriesof equations for each set of states (s, s′).The present application of BCC[68, 219, 220] consists of applying shortly afterthe formation of the excited wave packet of Eq. (5.1), a single broadband laserpulse ǫ(ω), containing the ǫ(ωs,f ) and ǫ(ωs′,f ) Fourier components that couple apair of excited states ψs and ψs′ to a single ground state |f〉. By manipulating themagnitudes |ǫ(ωs,f )| and phases φ(ωs,f ) of these two components we can coherentlycontrol the population transferred from the ψs and ψs′ components of Ψ(t) to | f 〉.The maximum effect will occur when the ǫ(ωs,f )asµs,f product will be identical tothe ǫ(ωs′,f )as′µs′,f product which will then interfere constructively. Under theseconditions knowledge of µs,f will enable us to extract µs′,f according to,µs′,f =asǫ(ωs,f )as′ǫ(ωs′,f )µs,f =∣∣∣∣asǫ(ωs,f )as′ǫ(ωs′,f )∣∣∣∣exp[i(δs,s′ + φs,s′;f )]µs,f . (5.7)Due to the reality of µs′,f , φs,s′;f = φ(ωs,f ) − φ(ωs′,f ) must be equal to −δs,s′or to π − δs,s′ , depending on the sign of µs′,f relative to µs,f . Although at thispoint we do not know the magnitudes |as| and |as′ |, provided that neither |as| >>|as′ | nor |as| << |as′ |, the population transfer resulting from complete constructiveinterference will be distinguishable from the case of destructive interference occurringwhen φs,s′;f → φs,s′;f + π.Noting that we are allowed to choose the phases of all (Nf ) | f 〉 states and the1145.2. Theoryphases of all (Ns) |ψs 〉 states at will, then of the Ns × Nf transitions, we are freeto choose the phases (or signs) of Ns + Nf − 1 of the TDM’s. Therefore, we takeall the µs,f1 associated with one, | f1 〉, state and all the µs1,f associated with one,|ψs1 〉, state to be positive. The phases of the remaining TDM’s depend, in additionto the overall phase factor multiplying each individual |ψs 〉 or | f 〉 wavefunction, ontheir detailed shape (e.g. nodal structure). Given this choice, it follows from Eq.(5.7) that the sign of every µs,f TDM’s relative to the sign of µs,f1 is determined bywhether φs,s1;f1 − φs,s1;f is 0 or π.Once the relative TDM signs are obtained in this manner, we use Eq. (4.5)to compute the Cs,s′ amplitudes. This allows us to use Eq. (5.6) to lift the πambiguity in δs,s′ and obtain the complex as amplitudes. We have thus chartereda way, using BCC and fFT, of obtaining both the amplitudes of the TDM’s andthe characterization of the time evolving wave packet Ψ(t), without ever havingcomputed the basis wavefunctions ψs!The maximum in the population depletion resulting from the above applicationof BCC is detected as a minimum in |R(ωs,s′)| - the fluorescence Fourier componentof Eq. (5.6). We choose this method in preference to other experimental techniquesfor population detection, such as laser-induced fluorescence(LIF) [221], resonantlyenhanced multiphoton ionization (REMPI) [222], or, cross-correlation frequency-resolved optical coherent anti-Stokes Raman scattering (XFROG CARS) [192, 193],because each of the latter approaches requires additional laser fields and might befurther complicated by other excitation pathways.One possible complication that might result is that due to its large bandwidththe BCC pulse might affect other states that are close in energy to the targeted ones.If the BCC Fourier components couple (s, s+1)→ f , then they might also stimulate,although to a lesser extent, the (s+ 1, s+ 2)→ f + 1, and (s− 1, s)→ f − 1 tran-sitions. In practice however, as we show below, these ancillary stimulated emissionprocesses, which occur at different frequencies of the pulse, (ωs+1,f+1, ωs+2,f+1) and(ωs,f−1, ωs−1,f−1) respectively, do not affect the signs of the µs,f derived from themaximal depletion in |R(ωs,s′)|.If cases arise where we find that the integrated signal for the two relative phasecase of the BCC field are too similar, and thus indistinguishable, we may addi-tionally tailor the ancillary components of the BCC field. This involves tuning themagnitude and phase of the pulse at the known frequencies {ws+1,f+1, ws+2,f+1}and {, ws,f−1, ws−1,f−1} in order to reduce the effects of the adjacent BCC processes1155.3. Simulationson the two excited states ψs and ψs+1 of interest. Again, the field magnitudes attheir components will be determined from the spectroscopic values of |µs,f | while therelative phases between the pairs of frequencies that minimize the coupling will befound to be either −δs+1,s+2 or π − δs+1,s+2 and −δs−1,s or π − δs−1,s respectively.5.3 Simulations5.3.1 Isolated PotentialWe demonstrate our imaging procedure using time-resolved fluorescence for an ex-cited sodium dimer (Na2) wave packet, in a process depicted schematically in Fig.5.1,according to which a molecule, initially in the X1Σ+g ground rovibrational state (|0〉= |v = 0; J = 0〉) gets excited by a short (δt ∼ 20fs) laser pulse to a wave packetof vibrational eigenstates belonging to the A1Σ+u electronic state. As illustrated inthe inset of Fig.5.1, after the excitation pulse is over (at t ∼ 300fs) we activate aweaker and longer “bichromatic coherent control”(BCC) pulse, dominated by twodistinct frequencies that mainly couple two vibrational levels (|s〉, |s′〉) in A1Σ+u tothe vibrational state |f〉 of X1Σ+g .After the BCC pulse, which we take to be a Gaussian of time-averaged intensityof 3 × 1010W/cm2 and bandwidth of 88cm−1, is over (at t ∼ 500fs), we collect thetime-resolved fluorescence resulting from the decay of the population still residing inthe excited states.The fluorescence is recorded every ∼ 60fs over a period of ∼ 10ps, aresolution that is attainable experimentally using up-conversion techniques [4, 5, 75].As an example we choose a particular case of {s = 3, s′ = 4, f = 5}, andµ3,5µ4,5 > 0. In this case φ3,4;5 = −δ3,4 corresponds to constructive interference,thereby maximizes the |R(ω3,4)| fluorescence depletion rate, whereas φ3,4;5 = π−δ3,4minimizes the fluorescence depletion. In Fig.5.2a we compare the time-resolved R(t)fluorescence rate of Eq. (5.4) with BCC relative phase of φ3,4;5 = −δ3,4, to theφ3,4;5 = π − δ3,4 case. As shown in the Fig.5.2a, the result is a small (∼ 5%)reduction in the fluorescence signal which would be difficult to detect experimentally.In contrast, as shown in the Fig.5.2b, the variation of |R(ω3,4)| of Eq. (5.6) withφ3,4;5 is very significant. When we ignore the ancillary transitions (black line), thefluorescence minimum, occurring exactly at φ3,4;5 = −δ3,4, is 250% smaller than thefluorescence maximum, occurring at φ3,4;5 = π−δ3,4. The relative difference remainshigh even when we do take into account the ancillary transitions (blue dashed line),1165.3. SimulationsFigure 5.1: (Main drawing) The ground, X1Σ+g , and first excited, A1Σ+u , potentialenergy surfaces of Na2 and a schematic description of the light pulses. Marked as(1) is the pulse exciting the ground vibrational eigenstates |0〉 to a set of excitedvibrational eigenstates |s〉; (2) the BCC pulse coupling two excited state |s〉 and |s′〉to a ground state |f〉; (3) the spontaneous emission from the |s〉 vibrational states ofA1Σ+u to the |f〉 vibrational states of X1Σ+g . (Inset) The BCC stimulated emissionprocess which couples states |s〉 and |s′〉 to |f〉. Some ancillary couplings of adjacent|s′ + 1〉 and |s− 1〉 states to different final states |f〉 may also result.1175.3. Simulationsalthough the maximum gets shifted slightly to the left. In order to quantify theeffect we plot in Fig. 5.2c the contrast ratioC(0, π) = |R(ωs,s′)0 −R(ωs,s′)pi||R(ωs,s′)0 +R(ωs,s′)pi|, (5.8)where subscript 0 denotes the φ3,4;5 = −δ3,4 phase choice, and subscript π denotesthe φ3,4;5 = π − δ3,4 phase choice. In Fig.5.2c, we display the contrast C(0, π)as a function of the |ǫ(ωs,f )/ǫ(ωs′,f )| ratio. Since we know from Eq. (5.7) that theconstructive interference occurs when ǫ(ωs,f1)asµs,f1 = ǫ(ωs′,f1)as′µs′,f1 , the contrastmaximum as a function of |ǫ(ωs,f1)/ǫ(ωs′,f1)| immediately yields the |as/as′ | ratio(relative to |µs,f1/µs′,f1 |). Once the |as/as′ | ratios are known, we can, by varyingthe state |f〉, obtain all other |µs,f/µs′,f | TDM ratios.This procedure is illustrated in Fig.5.2d where we present |R(ω3,4)0 − R(ω3,4)pi|for various final states |f〉, for a fixed |ǫ(ω3,5)/ǫ(ω4,5)| ratio. The variation in thecontrast with |f〉 yields the variation in the |µs,f/µs′,f | TDM ratios.5.3.2 Coupled Excited State PotentialsA case of great importance and much greater complexity is a wave packet evolvingon two coupled electronic states. Here the frequency-resolved methods used by usin the past to extract the TDM’s[128, 143, 144], do not work because they rely onthe existence of only one excited PES. In this section we demonstrate the viabilityof the present time-resolved method for such a case by determining the TDM’s forthe spin-orbit (SO) coupled RbCs system. Here the coupling occurs between the twolowest excited electronic states [223–225].We simulate the intersystem dynamics assuming that we know the PES of theexcited A1Σ+ singlet state and the b3Π excited triplet state[17]. In addition weassume that the electronic transition-dipole[226] and the A1Σ+/b3Π spin-orbit (SO)coupling [18, 19] functions are known. The above information allows us to calculatethe frequency-resolved fluorescence line strengths |µs,f |2 of RbCs, where s denotesthe vibrational states of the SO coupled A1Σ+/b3Π excited states, and f denotes thevibrational states of the X1Σ+ ground state[16] (or possibly the a3Σ+ lowest tripletstate).In order to produce the input time-resolved fluorescence we envision using anultrashort broad-band pulse to excite the X1Σ+(|v = 0〉) state to a wave packetcomposed of the |s = 24− 33〉 vibrational eigenstates of the SO-coupled A1Σ+/b3Π1185.3. Simulations0.5 5 10Time (ps)Amplitude (arb. units)BCC destructive interferenceBCC constructive interference(a)- pi  pi0 -δ3,4pi-δ3,4Relative Phase (∆φ 3,4,5  + ξ3,4)|R(ω3,4)| (arb. units)Single BCC process+ ancillary effects(b)0 |a3µ3,5| / |a4µ4,5| 20.5 2.5Relative magnitude ( |ε(ω4,5)| / |ε(ω3,5)| ) of |R(ω3,4)| Signals Isolated BCC process+ ancillary couplings(c)0 5 10 15Final state (f)0piPhase relation φ ,3,4; f to φ 3,4; 02550050757525Relative difference in |R(ω3,4)|(%)Single BCC process+ ancillary couplings(d)Figure 5.2: (a) R(t) captured over 10 ps and strobed every 60 fs in the presenceand absence of the BCC field which removes populations from the excited state.(b) |R(ω3,4)| as a function of the relative phase φ3,4;5 ranging over −π to π. Fullblack line - f including only states |3〉 and |4〉; dashed blue line - the result of includ-ing in addition to states |3〉 and |4〉 all ancillary states. (c) The contrast, C(0, π),of Eq.(5.8), as a function of |ǫ(ω3,5)/ǫ(ω4,5)| - the ratio between the two interfer-ing components of the BCC field, for fixed |ǫ(ω3,5)|2 + |ǫ(ω4,5)|2 sum of intensities.The line codes are as in (b). The maximum contrast marks the point at which|ǫ(ωs,f )/ǫ(ωs′,f )| = |as′µs′,f/asµs,f |, allowing us to extract |as′/as| and (by varying|f〉) |µs′,f/µs,f |. (d) |R(ω3,4)| at different φ3,4;0 - φ3,4;f phase differences for variousfinal states |f〉.1195.3. Simulations5 10 15Distance (Bohr) (Hartree)X1Σ+ (v=0; J=0)A1Σ+ + b3Π (s = 24-33)X1Σ+ (f = 20-29)a3Σ+(a)0.5 5 10 15Time (ps)Amplitude (arb. units)Omitting SOC couplingSOC included(b)Figure 5.3: (a) Schematic of the RbCs system showing the four PES and the locationof some vibrational eigenstates. (b) The resulting temporal fluorescence capturesover several picoseconds for when the spin-orbit coupling (SOC) is excluded (blue-square) and included (black-circle).electronic states. As shown in Fig. 5.3a, these eigenstates straddle the region ofcrossing between the A1Σ+ and b3Π PES, and in Fig. 5.3b we see the effect ofthe spin-orbit coupling (SOC) on the time-dependent fluorescence (black-circle) incontrast to the outcome when this coupling is omitted (blue-squares). We nextsimulate applying a BCC pulse whose φs,s′;f phase differences between the ωs,fand ωs′,f frequency components affect the fluorescence to one of the |f = 20 − 29〉vibrational states of the X1Σ+ PES.In Fig. 5.4a we display as blue circles the BCC pulse bandwidth (ranging over∼ 14 − 22cm−1), and as black bars - the intensities (ranging between 106 − 1012W/cm2) used for each |s〉 and |s′〉 excited pair and the |f = 20−29〉 final vibrationalstates for which we extract the TDM’s.We demonstrate, in Fig. 5.4b, a very successful extraction of the SO-coupledsinglet/triplet components (as) of the excited wave packet for two time points at theexcitation pulse center t = 0 (blue points), and after 10 ps (red points). The imagedvalues (thick black line) appear to faithfully reproduce the ”true” values, where wehave assumed knowledge of the SO-coupled eigenfunctions for presentational pur-poses. Figure 5.4b exhibits a transition, taking place over the 10 ps time span, froma wave packet (blue curve) centered about the singlet potential minimum, to a wavepacket centered about the triplet potential minimum (red curve). A behavior of thissort is only possible if the blue curve is mostly singlet and the red curve is mostly1205.3. Simulations24 25 26 27 28 29 30 31 32 33SO excited state (s)56789101112Log Pulse Intensity (log W/cm-1 )1416182022Pulse Bandwidth (cm-1 )(a)7 8 9 10 11Radial Distance (Bohr)0.050.06Energy (Hartree)A1Σ+ub3Πt = 0t = 10 psImaged(b)Figure 5.4: (a) The BCC pulse intensity (black bars) and bandwidth (blue circles)used to achieve sufficient C(0, π) contrast (of at least 20%). The range of the pulseintensity used was varied as a function of the final X1Σ+ state (f = 20−29) probed.(b) Extraction of the singlet and the triplet components of an excited wave packetΨ(r, t) at t = 0, the excitation pulse center (“true” wave packet - blue points) andafter 10 ps (“true” wave packet - red points). The imaged wave packet, given at bothtimes as a thick black line, faithfully reproduces the ”true” values. Shown also arethe A1Σ+u (thin black line) and b3Π (dashed line) PES.triplet. The wave packet, composed of the SO coupled eigenstates, starts out as asinglet because the selection rules of the optical transition from the ground singletcreate at t=0 only the singlet part of the wave packet. The temporal evolution ofthe wave packet which changes the extracted as expansion coefficients to asexp(iEst)gradually builds up a triplet component which, as shown in Fig. 5.4b, becomes veryprominent at t=10 ps.5.3.3 LiRb Coupled PotentialsIn this section, we demonstrate our TDM extraction procedure using time-resolvedfluorescence for an excited lithium-rubidium dimer (LiRb) wave packet evolving onthe spin-orbit (SO) coupled A1Σ+u and b3Π electronic states [33, 227–229]. In Fig.5.5a, we depict the ground state singlet, X1Σ+g , and triplet a3Σ+, and two lowestexcited PES of LiRb. We also schematically present the light pulses. Marked as(1) is a short (δt ∼ 20fs) pulse exciting the ground vibrational eigenstates |0〉 =|v = 0; J = 0〉 to a set of SO-coupled excited vibrational eigenstates |ψs〉. Markedas (2) is the BCC pulse, dominated by two distinct frequencies, coupling two excitedstate |ψs〉 and |ψs′〉 to a ground state |χf 〉 in X1Σ+g (or a3Σ+). Marked as (3) is the1215.3. Simulations(a)0.5 5 10Time (ps)Amplitude (arb. units)Destructive BCC FieldConstructive       "  (b)Figure 5.5: (a) (Main) Schematics of the LiRb system showing four PES’s andselect rovibrational eigenstates. (Inset) The BCC stimulated emission process whichcouples states |ψs〉 and |ψs′〉 to |χf 〉. Some ancillary couplings of adjacent |ψs′+1〉 and|ψs−1〉 states to different final states |χf 〉 may also result. (b) Temporal fluorescencecaptured over 10 ps and strobed every 60 fs for two different relative phase choicesof the BCC field which either leaves or removes populations from two (or more) ofthe excited states.1225.3. Simulationsfluorescence of the molecules remaining in the |ψs〉 vibrational states of A1Σ+u ∼b3Π.The fluorescence is to the |χf 〉 vibrational states of the X1Σ+g and a3Σ+ states. Thefluorescence is collected every∼ 60fs over a period of∼ 10ps. This degree of temporalresolution is possible due to the use of up conversion techniques [4, 5, 75].BCC to a3Σ StateAs an example we choose a particular case of {ψ17, ψ18, χ10}, where µ17,10µ18,10 > 0,and |χf 〉 lies in a3Σ. In this case φ17,18;10 = −δ17,18 corresponds to constructiveinterference, thereby minimizing the fluorescence depletion rate (appearing as amaximum in |R(ω17,18)|), whereas φ17,18;10 = π − δ17,18 maximizes the fluorescencedepletion (reducing the signal |R(ω17,18)|). In Fig. 5.2a we compare R(t), the time-resolved fluorescence rate of Eq. (5.4) after a BCC pulse (time-averaged intensity of1×1010W/cm2 and bandwidth of 36cm−1,) with relative phase of φ17,18;10 = −δ17,18,to the case when φ17,18;10 = π − δ17,18. As shown in the Fig. 5.2a, there is ∼ 20%reduction in the fluorescence signal between the two cases. For such large differencesthat can confidently be distinguished experimentally, we immediately know whichBCC phases choice lead to constructive (and which to destructive) interference, andtherefore have determined the relative phase (or sign) between two TDMs (µ17,10and µ18,10).Often the primary time-dependent signals may not be so easily discriminateddue to different TDM magnitudes. In this case we turn to the Fourier transformfiltering technique. As shown in the Fig. 5.2b, the variation of |R(ω17,18)| of Eq.(5.6) with φ17,18;10 is very significant. We find the minima and maxima occurring atthe expected values of φ17,18;10, the ancillary BCC processes having the minor effectof narrowing the range or slightly shifting the extrema points. When we ignore theancillary transitions (black line), the fluorescence maximum, occurring exactly atφ17,18;10 = −δ17,18, is several times larger than the fluorescence minimum, occurringat φ17,18;10 = π − δ17,18. This difference remains high even when we do take intoaccount the ancillary transitions (blue dashed line). In order to quantify this effectwe plot in Fig. 5.2c the contrast ratioC(0, π) = |R(ωs,s′)0 −R(ωs,s′)pi||R(ωs,s′)0 +R(ωs,s′)pi|, (5.9)where subscript 0 denotes the φ17,18;10 = −δ17,18 phase choice, and subscript π de-notes the φ17,18;10 = π − δ17,18 phase choice. In Fig. 5.6b, we display the contrast1235.3. Simulations- pi- pi/2  pi/2  pi0-δ17,18 pi-δ17,18Relative Phase (∆φ 17,18,10  + ξ17,18)|R(ω17,18)| (arb. units)Isolated BCC+ ancillary states(a)0 1 |a17µ17,10| / |a18µ18,10| 3 4Relative magnitude ( |ε(ω18,10)| / |ε(ω17,10)| ) of |R(ω17,18)| Signals (arb. units) Isolated BCC process+ ancillary couplings(b)0 5 10 15 20 25 30a3Σ Final state (f)0piPhase relation φ ,17,18; f to φ 17,18; 02550050757525Relative difference in |R(ω17,18)|(%)Single BCC process+ ancillary couplings(c)11 13 15 16 17 18 20 21 22 23SO excited state (s)9101112Log Pulse Intensity (log W/cm-1 )f = 5f = 10f = 20µs,f µs+1,f > 0µs,f µs+1,f < 02030405060Pulse Bandwidth (cm-1 )(d)Figure 5.6: (a) |R(ω17,18)| as a function of the relative phase φ17,18;10 ranging over−π to π. Full black line - including only states |17〉 and |18〉; dashed blue line - theresult of including in addition to states |17〉 and |18〉 all ancillary states. (b) Thecontrast, C(0, π), of Eq.(5.9), as a function of |ǫ(ω17,10)/ǫ(ω18,10)| - the ratio betweenthe two interfering components of the BCC field, for fixed |ǫ(ω17,10)|2 + |ǫ(ω18,10)|2sum of intensities. The line codes are as in (a). The maximum contrast marks thepoint at which |ǫ(ωs,f )/ǫ(ωs′,f )| = |as′µs′,f/asµs,f |, allowing us to extract |as′/as|and (by varying |χf 〉) |µs′,f/µs,f |. (c) |R(ω17,18)0 −R(ω17,18)pi| values at differentφ17,18;0 - φ17,18;f phase differences for various final states |χf 〉. (d) The BCC pulseintensity (line-points) and bandwidth (blue circles) used to achieve sufficient C(0, π)contrast (of at least 20%). The range of the pulse intensity used is shown for threefinal a3Π state (χf ) probed as a function of the lower A1Σ+u ∼b3Π coupled excitedstate (ψs). The red diamonds (black squares) depict what happens when µs,f andµs+1,f have identical (opposite) signs.1245.3. SimulationsC(0, π) as a function of the |ǫ(ωs,f )/ǫ(ωs′,f )| ratio. Since we know from Eq. (5.7)that the constructive interference occurs when ǫ(ωs,f1)asµs,f1 = ǫ(ωs′,f1)as′µs′,f1 , thecontrast maximum as a function of |ǫ(ωs,f1)/ǫ(ωs′,f1)| roughly yields the |as/as′ | ratio(relative to |µs,f1/µs′,f1 |). This should be in agreement with the |as/as′ | ratios ob-tained from the finite time Fourier transform after we have compute the Cs,s′ ampli-tudes using Eq. (4.5). Recall, that we first lift the π ambiguity in δs,s′ by extracting aseries of µs,fµs′,f relations for fixed (ψs, ψs′) over a range of final states f . This proce-dure is illustrated in Fig. 5.6c where we present |R(ω17,18)0−R(ω17,18)pi|/R(ω17,18)|for various final states |χf 〉 in a3Σ, for a fixed |ǫ(ω17,f )/ǫ(ω18,f )| ratio. The vari-ation in the contrast with |χf 〉 is related to the variation in the |asµs,f/as′µs′,f |TDM ratios and the BCC field intensity. Due to the nature of the PES, we findµs,fµs′,f > 0 for nearly all lower levels in a3Σ, with exception to the ambiguouscases where |µs,f | becomes very small (<5% of the average of the TDMs) and wecannot resolve the sign relation with greater than 10% difference (shown in Fig. 5.6cfor f = {0, 15, 16, 23, 24, 30}). The effect of including the ancillary states, which isgenerally unpredictable without complete knowledge of the system, does not greatlyalter the behavior of the integrated fluorescence signal.In Fig. 5.6d we display as blue circles – the BCC pulse bandwidth (rangingover ∼ 14 − 65cm−1), and line-points – the intensities (ranging between 109 − 1012W/cm2) used to achieve sufficient C(0, π) contrast (of at least 20%). For each |ψs〉and |s′ = s + 1〉 excited pair and three |f = 5, 10, 20〉 final vibrational states forwhich we extract the TDMs, we label the phase relation between the µs,f and µs+1,fterms as black squares (same sign) or red diamonds (opposite sign). All phases wereextracted using experimentally reasonable parameters and found to be in agreementwith actual values.BCC to X1Σ StateSimilar to the previous example, we produce the input time-resolved fluorescenceby using an ultrashort broad-band pulse to excite the X1Σ+(|χ0〉) state to a wavepacket composed of the |ψs=11−23〉 vibrational states of the SO-coupled A1Σ+ ∼b3Πelectronic states. As shown in Fig. 5.5a, these eigenstates straddle the region ofcrossing between the A1Σ+ and b3Π PES. We next simulate applying a BCC pulsewhose φs,s′;f phase differences between the ωs,f and ωs′,f frequency componentsaffect the fluorescence to one of the |f = 0 − 20〉 vibrational states of the X1Σ+PES. In Fig.5.7a we demonstrate the same calculation as that shown in Fig. 5.6c,1255.3. Simulations0 5 10 15X1Σ Final state (f)0piPhase relation φ ,17,18; f to φ 17,18; 02550050757525Relative difference in |R(ω17,18)|(%)Single BCC process+ ancillary couplings(a)11 13 15 16 17 18 20 21 22 23SO excited state (s)89101112Log Pulse Intensity (log W/cm-1 )f = 8f = 9f = 10f=11µs,f µs+1,f > 0µs,f µs+1,f < 0102030405060Pulse Bandwidth (cm-1 )(b)Figure 5.7: (a) Absolute difference of |R(ω17,18)| values for 0 and π cases at differentφ17,18;0 - φ17,18;f phase differences for various final states |χf 〉 in X1Σ. (b) The BCCpulse intensity (line-points) and bandwidth (blue circles) used to achieve sufficientC(0, π) contrast (of at least 20%). The range of the pulse intensity used is shownfor four final X1Σ state (χf ) probed as a function of the lower A1Σ+u ∼b3Π coupledexcited state (ψs). The black squares (red diamonds) represent when µs,f and µs+1,fhave similar (opposite) signs.though now, with the BCC fields coupling rovibrational levels (χf ) in the groundsinglet state X1Σ. We observe that the relative difference between the |R(ω17,18)0and R(ω17,18)pi| values is generally significant enough for reliable differentiation at afixed field intensity of 1010 W/cm2. That is until the magnitudes of the TDM becomevery small when as f becomes large (f & 20). The location of the points relativeto the zero line indicate the phase relation between the µs,f and µs+1,f values. Ourresults are in complete agreement with the expected signs for µs,fµs+1,f , and thesedo not change noticeably with the inclusion of ancillary BCC processes.Similar to Fig. 5.6d, we provide Fig. 5.7b which depicts the BCC pulse band-widths (blue circles) and intensities (points) required to achieve sufficient C(0, π)contrast (of at least 20%). The pulse parameters are shown for the ψs and ψs′=s+1excited pairs coupling to four consecutive final vibrational states χf=8,9,10,21 in theX1Σ PES. We label the phase relation between the TDMs, sgn(µs,fµs+1,f ), as before(black square – same, red diamond – opposite), and find complete agreement withthe expected signs. Only in the cases in which these magnitudes are very small (e.g.µ13,10, µ18,9, µ23,9), and thus negligible, is the BCC pulse intensity, which is directlyrelated to the strength of the TDMs, beyond that of our perturbative approach.With knowledge of the magnitude and signs of the TDMs from rovibrationallevels in X1Σ to the A1Σ+ ∼b3Π spin-orbit coupled states, we can extract the as1265.3. Simulations6 8 10 12Radial Distance (Bohr)0.050.06Energy (Hartree)t = 2 pst = 10 psImaged |Ψ|2        "A1Σ+ub3ΠFigure 5.8: Extraction of the singlet and the triplet components of an excited wavepacket Ψ(r, t) at t = 2 ps, the excitation pulse centre (“true” wave packet - red points)and after 10 ps (“true” wave packet - blue points). The imaged wave packet, givenat both times as a thick black line, faithfully reproduces the ”true” values. Shownalso are the A1Σ+u (thin black line) and b3Π (dashed line) PES.1275.4. Conclusionexcited state coefficients of a time-evolving wave packet [176, 177, 217]. In Fig. 5.8 weshow results for our wave function imaging performed in this way. We demonstratea very successful extraction of the complex amplitudes (with average error ≈0.2%)at two time points, when t = 2 ps (red points), and after 10 ps (blue points). Theimaged values (thick black line) appear to faithfully reproduce the ”true” values. Wesee very clearly how a wave packet predominately in the singlet state region leaksinto the triplet state after several picoseconds.5.4 ConclusionWe have shown how time-resolved fluorescence data in conjunction with Bi-chromaticCoherent Control (BCC) can be used to derive the phases as well as the amplitudesof the as expansion coefficients of a wave packet Ψ(r, t) =∑s asψs(r) exp(−iEst/~),where ψs(r) are vibrational eigenfunctions. The method can also be used to extractthe magnitudes and phases of µs,f , the individual transition dipole matrix-elements(TDM’s) between energy eigenstates, ψs and ψf . The extraction of as and µs,f doesnot necessitate having prior knowledge of the PES and/or SO-coupling terms, nor dowe need to know the ψs(r) functions. Imaging of Ψ(r, t) in coordinate space, wouldappear to necessitate knowledge the PES because it necessitates knowing ψs(r), butas we showed in the past [128, 143, 144, 176, 177, 217], it is possible, using the |µs,f |magnitudes, to extract the excited states PES from which we can calculate ψs(r).128Chapter 6Extraction of Non-AdiabaticCouplingsRecent advances in chemical reaction dynamics [230, 231]; the imaging of electronicwave functions [232]; the creation of ultracold molecules [25, 233, 234]; and the useof molecules for quantum information applications [156–159, 161, 161, 167, 169],have stimulated a drive towards a better (preferably representation free) quantifica-tion of molecular properties beyond the Born Oppenheimer approximation (BOA).In particular, one would like to develop methods for a point-by-point extraction(“inversion”) of excited state potential energy surfaces (PES) and the non-Born-Oppenheimer terms that couple them. So far, such inversion efforts [6, 7, 125, 128,128, 143] have been applicable only to isolated electronic states where the BOA isvalid. The above methods are inapplicable when electronic states interact, givingrise to “avoided crossings” or “conical intersections”.One of the simplest mechanisms for electronic state interactions is that of spin-orbit coupling (SOC) [235, 236]. SOC has been intensely studied in alkali-metaldiatomic molecules and has been found to give rise to interesting phenomena suchas inter-system crossing [235], “window” states [236], and Feshbach resonances [237–239]. An accurate description of the SOC in a molecule is of vital importance, as itallows for better understanding the internal dynamics and for developing accuratequantum control schemes within these systems.Traditional ab initio methods are not effective at providing this information, sincethey often ignore the diagonal and off-diagonal corrections to the BOA. Althoughmodern ab initio calculations are beginning to reach higher levels of accuracy (within5 cm−1) [76, 77], many spectroscopic experiments require knowledge of the relevantenergies to a much greater precision in order to practically perform the desiredmeasurements [112, 240] (especially in the case of [112] where the measurements weremade in a regime where no prior experimental knowledge existed). An alternative,is to iteratively construct a multi-parameter fit of the SOC term to accommodate1296.1. Theoryexperimental data, such as, from laser-induced fluorescence and Fourier-transformspectroscopy. Considerable progress in executing this strategy has been made byBergeman and others [92–97], however the SOC term obtained in this manner is non-unique. Moreover, the procedure is very difficult to execute because one must obtainthe line positions by diagonalizing a large set of differential or (even larger set of)algebraic equations, in each iteration step. Bussery and Aubert-Frécon have derivedthe full SOC matrix, analytically, for alkali dimers at long-range where SOC effectsemerge [112, 241]. Their approach has been used by Le Roy and company [107, 242]to determine for certain states of Li2, the resonance dipole-dipole interaction termC3 which is the leading long-range contribution to the spin-orbit coupling matrix.Unfortunately, their method suffers from requiring a great amount of spectroscopicinformation in order to compute the non-adiabatic (spin-orbit) couplings.The present chapter attempts to remedy many of these drawbacks: We derivea strict inversion procedure which extracts the radial dependence of the SOC fromexperimental data via an analytic formula which directly ties the SOC terms withthe observed line positions and their transition-dipole matrix elements (TDMs).6.1 TheoryWe first consider two non-interacting singlet and triplet excited PES within theBOA. The time-independent Schödinger equation for each potential is written asHS |ES 〉 = ES |ES 〉 and HT |ET 〉 = ET |ET 〉 , (6.1)where HS , ES and |ES 〉 represent the decoupled singlet Hamiltonian, its respectiveenergy eigenvalues, and their corresponding eigenstates. An analogous definition ap-plies to HT , ET and |ET 〉 for the triplet state. Each of the HS or HT Hamiltoniansare given asHS = TN +WS , HT = TN +W T , (6.2)where TN is the nuclear kinetic energy operator and WS(T ) are the singlet (triplet)diabatic, or decoupled, potentials. The interaction between these potentials is as-sumed to arise from the SO term, HSO [235]; similar to Bergeman el al. [17, 19] wehave neglected other, smaller, non-adiabatic effects such as: hyperfine, Zeeman, spin-spin, etc. In the diabatic representation this SO term results in diagonal, V D(R),and off-diagonal, V OD(R), SO functions of the nuclear coordinate, R.1306.1. TheoryFollowing other authors [17, 19, 106, 243], we write the full SOC Hamiltonian inthe decoupled singlet and triplet diabatic basis asH =(EˆSVODVOD† EˆT + VD)(6.3)where EˆS and EˆT are diagonal matrices of the rovibrational eigenvalues for the singletand triplet potentials. The dimensions of these matrices are determined by thenumber of singlet (NS) and triplet (NT ) eigenvalues, namely, EˆS is NS × NS , EˆTis NT × NT , and VOD is NS × NT . The rovibrational matrix elements of the SOcomponents are defined as(V OD)i,j = 〈ESi |V OD(R)|ETj 〉 , and (V D)i,j = −〈ETi |V D(R)|ETj 〉 , (6.4)where the integration over the radial-dependent SOC functions, V OD(R) and V D(R),is taken over R. We diagonalize the full SOC Hamiltonian asEˆ = U†HU (6.5)where Eˆ is an N × N diagonal matrix composed of the full N = NS + NT SOCeigenvalues. The unitary matrix, U, is constructed from the corresponding fullyinteracting eigenstates, in the given basis, arranged in columns. Multiplying the twosides of Eq. (6.5) by U we represent Eq. (6.5) in a 2× 2 block form as(USu USlUTu UTl)·(Eˆu 00 Eˆl)=(EˆSVODVOD† EˆT + VD)·(USu USlUTu UTl)(6.6)where the subscripts u and l denote respectively the the upper and lower elementsof the diagonal matrix Eˆ. Writing two equations for the upper portion of Eq. (6.6)givesUSu · Eˆu = EˆS · USu + VOD · UTu and USl · Eˆl = EˆS · USl + VOD · UTl , (6.7)either of which could be solved directly for the off-diagonal SOC matrix, VOD, assum-ing that we know the U’s and E’s. Note that the dimensions of the first equations areNS×NS , which can be seen from the dimensions of the sub-matrices USu (NS×NS),Eˆu (NS × NS), and UTu (NT × NS). On the other-hand, the second equation has1316.1. Theorydimensions of NS × NT since its constituent sub-matrices are USl (NS × NT ), Eˆl(NT ×NT ), and UTl (NT ×NT ). Consider the case in which the dimensions of thesetwo equations are the same, namely, that the number of singlet (NS) and triplet(NT ) eigenvalues are equal, where the total number of SOC states N = 2NS = 2NT .Now, we may sum the matrix equation in Eqs. (6.7) and rearrange to yieldVOD ={USu · Eˆu + USl · Eˆl − EˆS ·(USu + USl)}·{UTu + UTl}−1 . (6.8)This expression now has the advantages of averaging the result for VOD over moredata, and improving the condition number of the matrix targeted for inversion,which would otherwise suffer when a particular SOC state doesn’t contain muchtriplet character (if the columns of Uu or Ul were nearly zero, there may be troublesolving by the use of a matrix inverse). Once V OD is known, we can solve for V Dby using the lower portion of Eq. (6.6), yielding two similar equations,UTu ·Eˆu = VOD† ·USu+EˆT ·UTu−VD ·UTu and UTl ·Eˆl = VOD† ·USl +EˆT ·UTl −VD ·UTl ,(6.9)with dimensions NT ×NS and NT ×NT respectively. These can easily be combined(when NS = NT ) to giveVD ={VOD† ·(USu + USl)+ EˆT ·(UTu + UTl)− UTu · Eˆu − UTl · Eˆl}·{UTu + UTl}−1 .(6.10)It follows from Eq. (6.8) and Eq. (6.10) that we can extract the SO matricesusing the diabatic singlet energies EˆS and the set of fully interacting eigenvalues Eˆuand Eˆl, along with their corresponding eigenvectors in the diabatic basis (arrangedin USu , USl , UTu and UTl ). We now show how to obtain the latter quantities fromphoto-absorption or photo-emission data. We start by assuming: (i) that WS andW T , the “zero-order” decoupled singlet and triplet PES of Eq. (6.2), are known tous, and (ii) that we know the TDMs between the physical (fully interacting) excitedmanifold to some set of Ns lower singlet (s) and Nt lower triplet (t) states that arephysically decoupled from one another. The TDMs between the excited, coupledsystem and the lower states, |Es 〉 and |Et 〉, are given as(ds)k,i = 〈Esk |d · ǫˆ|Ei 〉 and (dt)m,i = 〈Etm |d · ǫˆ|Ei 〉 , (6.11)where d is the electric dipole operator and ǫˆ is the polarization direction of the1326.1. Theoryemitted radiation. Using the definition of U, we can write each fully interactingeigenstates as a sum of excited singlet (S) and excited triplet (T ) states|Ei 〉 =NS∑j=1(US)j,i|ESj 〉+NT∑j=1(UT )j,i|ETj 〉 , (6.12)where the summation is over the row (j) index of each i column (here we havedropped the superfluous u and l labeling) in the U matrix. Using Eq. (6.12), we canrewrite Eq. (6.11) as(ds)k,i =NS∑j=1(ds,S)k,j(US)j,i and (dt)m,i =NT∑j=1(dt,T )m,j(UT )j,i (6.13)where (ds,S)k,j = 〈Esk |d · ǫˆ|ESj 〉 and (dt,T )m,j = 〈Etm |d · ǫˆ|ETj 〉, are the TDMsbetween the excited and lower singlet, and triplet, states, respectively. The diago-nalizing transformation U, can now be expressed in terms of the TDMs asUS =(ds,S)−1 · ds and UT = (dt,T )−1 · dt . (6.14)Note that these equations hold true for both the upper (u) and lower (l) labelledblocks. For example, when solving for USu , which has dimensions of Ns × NS , it isbest to take Ns = NS to ensure that ds,S is square. In fact, the initial partition intofour blocks need not be symmetric, as it is simply a convenient tool. The matrices inthe two sets of equation in Eq. (6.7) will initially be of different dimensions, however,we can still proceed with their summation provided that we first expand the vectorspace of the smaller dimensioned equation to the same size as the large one, placingzeros in its new elements. In determining the “inverses” of non-square matrices,the system will be either over- or under-determined, and we must use mathematicaloptimization methods to find their pseudo-inverse [244–246] (where only the left (orright) inverse may be defined). This will yield reasonable results provided that theasymmetry of the matrices isn’t too great and that their dimensionality is sufficientlylarge (i.e. NS ≈ NT ≈ Ns ≈ Nt ≫ 1).Substituting the above form of U (Eq. (6.14)) lets us express Eq. (6.8) in terms ofthe physical, fully interacting, energies and the TDMs, note that the superscripts Suand Sl will denote singlet states corresponding to their arrangement in the original1336.1. TheoryU matrix given in Eq. (6.6),VOD ={(ds,Su)−1 · ds · Eˆu +(ds,Sl)−1 · ds · Eˆl − EˆS ·[(ds,Su)−1 +(ds,Sl)−1] · ds}·{[(dt,Tu)−1 +(dt,Tl)−1] · dt}−1. (6.15)Similar to the off-diagonal case, we can substitute the expressions for the U’s interms of dipole matrices for the case of the diagonal SO matrix into Eq. (6.10) toobtainVD ={VOD† ·[(ds,Su)−1 +(ds,Sl)−1] · ds + EˆT ·[(dt,Tu)−1 +(dt,Tl)−1] · dt−(dt,Tu)−1 · dt · Eˆu −(dt,T)−1 · dt · Eˆl}·{[(dt,Tu)−1 +(dt,Tl)−1] · dt}−1.(6.16)These expressions are the desired solution for the SO matrix, given in terms ofEk and (d)k,i. The energy levels, Ek, and |(d)k,i|2 are routinely measured in highresolution spectroscopy. As we have shown [247, 248], knowledge of these quantitiesallows us to extract the desired (d)k,i amplitudes. The remaining information neededfor Eq. (6.15) is the diabatic energy levels, (EˆS and EˆT ), and the TDMs of thedecoupled states (ds,S and dt,T ). This information can be computed from the excited,decoupled, diabatic singlet and triplet PES.In the case of weak SOC, these potentials could be obtained from the groundelectronic state and the measured |(d)k,i|2 values for vibrational states which aresufficiently removed from the (near) crossing regions [7, 125, 128, 128, 143]. Sincesuch states will not be as affected by the SO-coupling and retain mostly their singletor triplet character, their dipole couplings to either the ground singlet or tripletstates can be used to reconstruct an accurate representation of the correspondingdiabatic potential. As demonstrated by Li et al. [7], one is able to extract an excitedPES using dipole data from only a few (≈ 5) highly excited rovibrational states ofthe unknown PES. After determining these PES we can calculate the singlet staterovibrational energies (EˆS) and, within the Franck-Condon approximation, the ds,Sand dt,T TDMs. When the SOC is strong, and the above inversion does not work,we can either obtain diabatic singlet and triplet potentials from traditional ab initiocalculations or use those calculated by spectroscopic methods [104, 107, 242].Lastly, using our calculated results of the two sets of diabatic eigenstates (after1346.2. Resultshaving found their potentials), we may solve Eq. (6.4) in the least-squares sense toobtain a polynomial form for the VOD(D)(R) function. Or alternatively, we can usecompleteness (for details see derivation of Eq. (16) in Ref. [144]) to write Eq. (6.4)asV OD(D)(R) =∑i,kφ∗Si (R)VOD(D)i,k φTk (R)/∑j|φTj (R)|2, (6.17)where φSj (R) ≡ 〈R|ESj 〉 and φTj (R) ≡ 〈R|ETj 〉. The resulting SOC function and theknown eigenvalues thus lead to the extraction of the full SOC Hamiltonian.6.2 Results6.2.1 Weak Coupling - NaKIn this section we describe a set of calculations where we determine the SO couplingfunction between low-lying electronic states of NaK [249]. We use data from Ref.[11], where it was demonstrated that for the weakly coupled D(3)1Π and d(3)3Πelectronic states, the Rydberg-Klein-Rees (RKR) curves are in good agreement withexperimental data and can thus be used to define our excited decoupled potentialsin the diabatic representation. We have used the ground state singlet(X1Σ+) andtriplet(a3Σ+) states of Ref. [9, 10], together with the transition dipole moments [250,251] and the SO function [12–15] to produce realistic spectroscopic line-positions andamplitudes, (|dsk,i| and |dtm,i|), shown in Fig. 6.1a.We first use the decoupled potentials in the diabatic representation of Ref. [11],to calculate ds,S , the TDM matrix between the two decoupled singlets, X1Σ+ andD(3)1Π, and dt,T the TDM matrix between the two decoupled triplets, a3Σ+ andd(3)3Π. Armed with these matrix elements and using a recent development (fromChap. 5 and Refs. [247, 248]) in which we have shown how to extract the signsof the ds and dt TDMs from their experimentally derived magnitudes, we can useEq. (6.14) to extract the SOC eigenstates in the diabatic basis, thus leading usto determine the SOC functions. We illustrate this procedure here by extractingthe SOC functions using simulated TDMs between 40 SOC excited eigenstates andtwo sets of low lying rovibrational eigenstates of the X1Σ+ and the a3Σ+ electronicmanifolds. With all the ingredients in place, we now use Eq. (6.15), Eq. (6.16) andEq. (6.17) to perform a point-by-point extraction of the two SOC terms. We haveverified that 40 is the smallest number of states needed to maintain an accuracy of1356.2. Results(a)(b)Figure 6.1: (a) (Main) Four NaK potentials in the diabatic representation: Solidlines - the X1Σ+ and D(3)1Π singlet states; dashed lines - the a3Σ+ and d(3)3Πtriplet states [9–11]. (Inset) The dispersed emission spectrum from the coupledD(3)1Π/d(3)3Π state to the lowest singlet and triplet states. (b) The “true” (Ref.[12–15]) and extracted SO functions in NaK, exhibiting good agreement for the off-diagonal (VOD(R)) term as well as for the diagonal (VD(R)) term.1366.2. Results< 1% in the extracted SOC functions in the (avoided-)crossing region.The SOC extracted terms, fitted in Eq. (6.4) by a 5th order polynomial, areshown in Fig. 6.1b where excellent agreement between the “true” and extractedSOC terms is clearly in evidence. As expected, the accuracy of the extraction beginsto decrease outside the region spanned by the spatial extension of the eigenfunctionsincluded in the calculation.6.2.2 Strong Coupling - RbCsWe now address the situation of strong SO-coupling between electronic states, oc-curring for example in the RbCs molecule [223–225]. To simulate the rotationlessfluorescence spectrum of RbCs, we use the X1Σ+ and a3Σ+ low lying PES of Ref.[16], the A1Σ+ and b3Π excited PES in the diabatic representation of Ref. [17],the transition dipole moments of Ref. [226], and the SO functions of Ref. [18, 19](see Fig.6.2a for diagram). Due to the strong coupling between the A1Σ+ and b3Πstates, the RKR method or other inversion schemes [6, 7, 125, 128, 128, 143] fail togenerate reasonable candidates for the excited potentials, so instead we use the abinitio diabatic potentials from Ref. [17] for this demonstration.As in the weak coupling case we extract the SOC functions, V OD(R) and V D(R),from Eq. (6.15), Eq. (6.16) and Eq. (6.17), using the simulated and computed TDMsand eigenenergies. Then, we take these determined SOC functions and the diabaticeigenstates, |ES 〉 and |ET 〉, to re-generate the SO-coupling matrix, VOD and VDusing Eq. (6.4). This allows us to re-create the full SOC Hamiltonian (Eq. (6.3))from the extracted information and thus compare the calculated SOC energies (i.e.the eigenvalues of Eq. (6.3)) obtained through the extraction of the SOC couplingfunctions with those computed directly from the ab initio data.In Fig. 6.2b we plot the percentage difference between the eigenvalues obtained inthese two cases when the total number of SO eigenstates included in the calculation isvaried. The comparison is performed over the SO eigenvalues that would experiencethe greatest energy shifts due to the coupling – those about the (avoided)-crossingregion of A1Σ+ and b3Π states. We see that when too few states (NSO = {60, 80})are included in the calculation, the coupling between the potentials is not fullycharacterized and large errors arise in the extracted eigenvalues. However, with 120SO eigenstates the accuracy becomes quite good (≈ 1-2 cm−1), and the inclusionof more functions doesn’t significantly improve the results, which is good news forexperimentalists who often don’t have all the transition data. The oscillations seen1376.2. Results(a)0.0455 0.046 0.0465 0.047 0.0475Energy (Hartree) Error (%)NSO=60NSO=80NSO=100NSO=120(b)Figure 6.2: (a) (Main) Solid black lines - the Morse RbCs PES in the diabaticrepresentation [16, 17]; (Inset) The off-diagonal (V OD(R)) and diagonal (V D(R))SO functions between the diabatic potentials [18, 19]. (b) Percentage differencebetween the true and computed SOC eigenvalues calculated in the A1Σ+ and b3Πdiabatic crossing region, for a different number of total included eigenvalues.1386.3. Conclusionin the results are due to changes in the singlet-triplet composition of the different SO-eigenstates. We suspect that those states comprising of more triplet character havegreater errors due to the additional diagonal coupling term, VD, being computed inthe extraction.This demonstrates that we can accurately characterized the coupling betweenthe WS(R) (A1Σ+) and W T (R) (b3Π) potentials such that we can express the SOcoupled eigenstates in the WS/T (R) basis states, enabling the accurate imaging ofexcited wave packets in non-adiabatically coupled systems [217].6.3 ConclusionWe have performed a point-by-point extraction of the SO-coupling as a function ofthe nuclear coordinates R using experimental data. Knowledge of the R dependenceof these coupling terms enables the generation of the eigenstates of the coupledsystem and the reconstruction of excited wave packets generated in ultrashort pulseexcitation experiments [217]. The procedure uses a formula that expresses the spinorbit coupling matrix in terms of the TDMs and the fully interacting energy levels.Our method is not limited to this type of interaction: any interaction betweenelectronic states can be extracted in a similar manner. We intend to apply thismethod to the mapping of non-adiabatic effects, including conical intersections, inpolyatomic molecules.139Chapter 7Final ConclusionsWe have outlined several original approaches for extracting molecular informationfrom spectroscopic data. With the increasing breadth of spectral information andimprovements in temporal and spatial resolutions, original techniques such as thosepresented here, may help to better characterize a molecule’s structure and behav-ior. There are now many avenues of research in the molecular sciences, all of whichhowever, require as detailed information as possible on the molecular species athand. For instances, the formation of ultracold molecules (e.g. via the photoas-sociation of ultracold atoms) relies on accurate molecular information and internallevel structure and dynamics. In particular, this requires very good information nearthe dissociation limit for Feshbach resonances, and accurate knowledge of stronglymixed (singlet-triplet) complexes for transferring molecules to bound states via Ra-man process. A good description of a system would facilitate studying the propertiesof molecules such as in: Bose-Einstein condensates and Fermi degenerate gases [25],controlling chemical reactions and collisions [252], and, the quantum computationon aligned molecular dipoles [157]. There has been even been renewed interest inthe usage of heteronuclear diatomics for sensitive non-contact probing and mappingof external electric field distribution via changes in LIF [253, 254] to which precisestate information would be essential.The first method involves determining one of the most important concepts ofmolecular physics [48], the potential energy surface (PES). These structures are fun-damental to spectroscopy, chemical kinetics and to the study of the bulk propertiesof matter. It is used as a concept for both qualitative and quantitative descriptionof molecular properties. Our approach is capable of extracting excited diatomicpotentials over energy ranges spanning thousands of wave numbers using only tran-sition spectral data from a few (∼ 10) excited rovibrational states. We developedour procedure to include transition information to/from continuum states, addressmultidimensional surfaces and operate beyond the Franck-Condon approximation(FCA). The ten-fold improvements in accuracy observed when including a dipole140Chapter 7. Final Conclusionscorrection is also applicable to any other inversion scheme which are based on theFCA (such as those of Avisar and Tanner [153, 154]). The key requirements fora successful PES construction are, first, sufficient data with reasonably small errorbars of relative intensities (∼ 5%) Ref. [143], and next, fulfillment of the complete-ness condition in order to expand the excited state rovibrational wave function interms of the ground state wave functions (Ref. [128]). Recently, there have beeninteresting studies measuring the transition dipole moments over a wide range oftransition frequencies to very high accuracy (e.g. Ref. [33] by Dutta et al). Thesestudies would serve as a good source for future work to validate our method usingexperimental data.On route to inverting the PESs we solved the spectroscopic phase problem.Whereas the magnitudes of the transition dipole matrix elements (TDMe) betweenenergy eigenstates can be deduced from the strength of frequency-resolved spectrallines, yielding the absolute-value-squared of the TDMs, it has much more difficultto use such data to determine the phases of the TDMs. This current approach,however, fails when there exist couplings between the unknown potentials, for in-stance, due to spin-orbit coupling (SOC). In molecules, SOC causes inter-systemcrossings (ISC), the nonradiative transition between different spin (e.g., singlet andtriplet) states [235]. SOC also gives rise to the formation of “doorway” or “window”states [236], called such, because they enable electric-dipole-induced optical transi-tions between different spin states. Its determination is vital for many applications,such as the laser cooling of molecular species, which leads towards the formationof Bose-Einstein condensates. In RbCs, SOC is known to promote Feshbach typeresonances [237–239] and to affect the “permanent” dipole moment of this molecule,allowing for the tuning of the (dipole-dipole) interactions between trapped RbCsmolecules, potentially enabling the performance of quantum simulations in such sys-tems. Using a simulated frequency spectrum of a spin-coupled diatomic system, wehave demonstrated the extraction of these coupling matrix elements, as well as, theirradial functional dependence. This work opens the door for extracting PESs in thepresense of non-adiabatic couplings.We have also shown how the amplitudes of the electronic TDMs linking the ex-cited and ground rovibrational states, in addition, to the amplitude of time-evolvingwave packets can be found from time resolved fluorescence data. By assuming thatthe time-dependent decay signal is given as R(t) = ∑s′,s a∗s′asCs′,s exp(−iωs′,st),where as are the desired expansion coefficients, ωs′,s are (known) beat frequencies,141Chapter 7. Final Conclusionsand Cs′,s are molecular response matrix elements, we have shown how a finite timeFourier transform over the fluorescence data at a given beat frequency will extractout the component of the signal pertaining to the bi-linear a∗s′as product. Usingthis result which assumes knowledge of the amplitudes and phases of µs,f (the indi-vidual transition-dipole matrix-elements (TDMs) between energy eigenstates), onecan derive the phases as well as the amplitudes of the as expansion coefficients of awave packet Ψ(r, t) = ∑s asψs(r) exp(−iEst/~), where ψs(r) are vibrational eigen-functions. We have demonstrated these imaging results in basic diatomic systems,and also applied this technique to SOC systems where we have seen very clearly howa wave packet, which starts out in the singlet state region, leaks out to the tripletstate after several picoseconds.Although we have shown that for an electronic transition between two Born-Oppenheimer potential energy surfaces (PES), knowledge of one PES allows one toderive the other PES, and the TDM’s phases [128, 143, 144], as well as imaging ofunknown time-evolving wave packets [176, 177, 217], the method requires data of alarge number of spectral lines and of sufficient quality and completeness, which atthis stage is only slowly becoming available [33]. Since obtaining the amplitudes ofthe electronic TDMs linking the excited and ground vibrational states, is the crucialstep in both the PES inversion and wavepacket imaging procedures, we developed analternative approach for their determination. First, we discussed how a semiclassicalstationary phase calculation does not give the unknown phases of the TDMs whenthe wavefunction of the excited state is unknown. Then, we presented an originalapproach using ultrashort pulses in the molecule-field interaction. The method isbased on using bichromatic coherent control (BCC), which uses quantum interfer-ences between different pairs of transitions induced by two external laser fields tocoherently deplete the population of (hence the fluorescence from) different pairs ofthe excited energy eigenstates. The BCC induced depletion is supplemented by theFourier transform technique on the time-dependent spontaneously emission data toobtain the sign relation between pair of TDMs. Armed with this combined informa-tion, of all pairs of sign relations between TDMs, one can conclusively extract theindividual phases of the TDMs.Our various methods, which in general, are found to be quite robust against er-rors can be readily generalized to other systems, such as polyatomic molecules. Itwould be interesting to apply our work to multi-dimensional problems and to useour procedures to obtained a detailed image of nuclear motions associated with “in-142ternal conversions” (transitions between different electronic states of the same spinmultiplicity), and “intersystem crossings” (transitions between electronic states ofdifferent spins). More effort can be put towards the treatment of curve crossing situ-ations, particularly in the case of singlet/triplet interactions. 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