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Implication methods for the determination of quadratic force constants Green, Raymond Winston 1971

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IMPLICATION METHODS FOR THE DETERMINATION OF QUADRATIC FORCE CONSTANTS by RAYMOND WINSTON GREEN B.S., Oregon State University, 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE. OF DOCTOR OF PHILOSOPHY in the Department of CHEMISTRY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1971 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver 8, Canada ii > ABSTRACT Currently the formulation of a valid force constant matrix poses the largest problem in the normal coordinate analysis or the mechanical interpretation of vibrational spectra. Usually a preselected set of trial force constants is iteratively corrected by means of first order perturbation theory and the principle of least squares. This thesis breaks that tradition and operates the normal coordinate analysis through an implied force constant matrix, F = L *"Aj_, \ where LL*" = G, the familiar Wilson G-matrix. The A-matrix is composed of the experimental vibrational frequencies for a selected basis molecule and the L-matrix is parameterized in a general way. The L-matrix parameters are varied until the implied force constant matrix generates an optimum mechanical picture of the basis molecule and its isotopic homologs. How ever this thesis emphasizes the vibrational fundamentals of isotopic homologs in specifying the implied force field. In application six L-matrix parameters encompass the sixty-three planar vibrational frequencies of ethylene and its deuterohomologs with slightly less error than traditional calculations using as many as fifteen potential energy parameters. As well, the implied force constants comply with the existing picture of chemical bonding without deliberate a priori reference to it. In particular, aspects of the hybrid orbital force field are confirmed without prior constraints. iii In more detailed computational studies the implied force field has revealed a systematic trend in anharmofiic effects which can he understood in terms of different vibrational amplitudes for different isotopic homologs. The influence of vibrational amplitude has been parameterized and included within the implication method as a simple anharmonicity correction. For example, one L-matrix parameter and three vibrational amplitude parameters encompass the nine observed vibrational frequencies of water and its deuterohomologs with -1 an average frequency error of 0.4 cm . Without amplitude corrections the average frequency error becomes 10.7 cm with one L-matrix parameter or 12.8 cm with four potential energy parameters.. It is particularly significant that this simple picture of anharmonicity employs the observed vibrational frequencies rather that the empirically derived harmonic frequencies. As well, the vibrational amplitude parameters comply with expected features of potential energy surfaces such as the dissociation limit. The principle advantage of the implication method is that there a fewer L-matrix parameters than F-matrix parameters. The principal disadvantage is that approximations and intuitive notations are not easily built into the implication method. However, as experimental information becomes more complete and better understood, the need for improved analytic foundations dominates the need for handy approximations. i v TABLE OF CONTENTS PAGE CHAPTER ONE: Introduction 1 (1-1) Molecular Structure(1-2) Molecular Vibrations 2 (1-3) Perturbation Methods 7 (1-4) Implication Methods 13 CHAPTER TWO: Simple Mixing Implication Methods 16 (2-1) The Implied force Field 19 (2-2) A Simple Anharmonicity Correction 23 (2-3) Applications of Simple Mixing Methods 25 (2-3a) H20, HDO and D20: Harmonic Fundamentals 28 (2-3b) H20, HDO and D20: Observed Fundamentals 31 (2-3c) HCCH, HCCD, DCCD: Observed Fundamentals 35 (2-3d) Methane and Deuterohomologs: Observed Fundamentals 39 CHAPTER THREE General Mixing Implication Methods 44 (3-D K Space 4(3-2) Isotopic Homologs in K-Space 46 (3-3) Implied Dispersion 48 (3-4) Selected Applications in K-Space 53 (3-4a) Formaldehyde and Its Deuterohomologs 55 (3~4b) Ethylene and its Deuterohomologs 59 (3-4c) Chemical Significance 68 v PAGE CHAPTER FOUR: Approximation Techniques 72 (4-1) The Weighted Trace Equation 72 (4-2) Participation and Molecular Partitioning 76 EPILOGUE 8Appendix I: The Vibrational Secular Equations 94 Appendix II: Redundant Coordinate Systems 107 Appendix III:The Exponential Matrix 113 References 88 vi LIST OF TABLES TABLE PAGE 1. Calculated Harmonic Fundamentals: HDO and D20 29 2.. Harmonic Force Constants: HgO, HDO and D20 30 3- Calculated Fundamentals: HDO and D20 32 4. Force Constants: HgO, HDO and' DO ' 33 5. Calculated Fundamentals with Amplitude Factors: 34 HDO and D20 6. Vibrational Fundamentals: Deuteroacetylenes 37 7. Force Constants: Acetylene 38 8. Vibrational Fundamentals: Deuteromethanes 4l 9. Vibrational Fundamentals: H2CO, HDCO and D2CO 56 10. The Force Constants of Formaldehyde 58 11-. The Force Constants' of Ethylene 64 12. Vibrational Fundamentals: Ethylene and Deuterohomologs 65 13. Similar Implied Force Constants 69 14. Participation Matrices for,Methylamine 82 ACKNOWLEDGEMENT To the novice, the line separating physical perspective and mathematical abstraction seems an unfocused domain with unresolved guidelines. The kindly leadership and secure insight of K.B. Harvey has been a welcoming guidepost throughout the course of these studies. DEDICATION to Professor K. B. Harvey In Memoriam CHAPTER ONE: INTRODUCTION (1-1) Molecular Structure Several branches of chemistry meet in the domain of molecular structure; here, chemists see an electronic structure superimposed on a rigid nuclear frame with specified geometry. This picture furnishes a simple basis for understanding chemical activity, for example, in terms of simple molecular orbital theory. Thus, to some extent, all measures of molecular structure belong to every branch of chemistry. Prom an interior position, the study of molecular structure joins rigorous physics, abstract mathematics, and experimental information into a self-contained body of knowledge. This joining process, the analysis of molecular structure, should not be viewed as a closed ring of things to do with data and theory. Rather, especially in chemistry, the analysis of molecular structure must contribute to a perview of the topic in a way that encompasses the various contributing branches. To achieve that objective, the chemist adds an exterior concept to his scheme of analysis. In effect, the chemist expects the parameters of molecular structure to exhibit a reasonable pattern which complies with the basic notions of chemical bonding. Existing studies confirm the expectation. - 2 -The electronic structure, when organized into chemical bonds, provides binding energy which is a function of the number of electrons, nuclear charge, and nuclear configuration. Nuclear mass does not significantly influence the binding energy. For stable molecules the binding energy is such that nuclear configuration is constrained to a neighbourhood of minimal potential energy (maximal binding energy). This picture does not imply a rigid structure. Rather, molecular structures are semirigid geometrical entities embedded within the potential energy surface imposed by the electronic superstructure. The rigidity depends upon the curvature of the surface and the inertial mass of the nuclei. Consequently, the analysis of molecular structure will involve much more than simple geometry. Further, if chemical bonding is recognized, more than pure physics will appear. In practice, the analysis is sufficiently cumbersome that some mathematical tools which are not part of the physics will enter the picture. (1-2) Molecular Vibrations Within limits, molecular in configuration is governed equations. The usual classi vibrations or the variation by the vibrational secular cal presentation of these - 3 -equations (1;2;3) expresses potential and kinetic energy as quadratic forms with translational and rotational kinetic energy removed. A modest quantum mechanical presentation, suitable for teaching purposes can be found in appendix one of this thesis. The dynamic variables, configuration displacement coordinates, measure the distortion of the molecule from its equilibrium configuration - the point of minimum potential energy. If these coordinates are defined with chemical bonding in mind, the definition of potential energy becomes greatly simplified in both approximative and interpretive aspects. Clearly bond stretching (an internuclear distance) and valence angle bending will be useful choices. In the harmonic oscillator approximation, dynamic distortions are assumed to be sufficiently small that potential and kinetic energy can be expressed as quadratic forms in configuration displacement coordinates, r^f , and their conjugate momenta, , respectively. kinetic energy = llz.HX-\ G\\V\.V[ (1.2.1) potential energy (1.2.2) - 4 -It is not possible to properly justify the kinetic energy expression in a few lines; see Wilson Decius and Cross (1) especially their appendix VII, or appendix one of this thesis for an account of the kinetic energy. Present purposes require only a clear definition of the G-matrix elements; these depend only upon the nuclear masses and the equilibrium geometry of the molecule. All possible G-matrix elements for bond stretching and valance angle bending have been tabulated in appendix VI of Wilson, Decius and Cross (1); otherwise they may be calculated from the following expression. Gij = (\&/OiWtfc/aj) (1.2.3) jU/£ is the reciprocal mass of the DC — nucleus (or atom) and \b is the cartesian gradient for the indicated atom. After the selected configuration coordinates have been written as functions of the cartesian coordinates of the individual atoms, the G -matrix follows; however, considerable labor is involved. Prom the outset little is known about the potential energy except that the concept of chemical bonding is involved. Currently potential energy is parameterized in various ways and experimental information is used to - 5 -specify the parameters. The principal parameters are the quadratic force constants - the F ,'s (F-matrix) of expression (1.2.2) above. Of the several types of experimental information dependent on potential energy, fundamental vibrational frequencies have provided the bulk of what is known about potential energy parameters. Quadratic force constants and fundamental vibrational frequencies are related through the vibrational secular equations. i. Q C = X the identity matrix (1.2.4) f l_ ~ /\ a diagonal matrix (1.2.5) Here kinetic and potential energy have been simultaneously diagonalized via the transformation A*» ~ 21 & Li*. Q»e_ (1.2.6) where the are normal coordinates. In classical mechanics each normal coordinate is a periodic function of time with period )/% . (See Wilson, Decius and Cross (1) or appendix one of this thesis for a development with physical substance.) - 6 -The diagonal A ~ matrix is related to the various periods (or frequencies) of vibration; 2. (1.2.7) where i^g is the k' th vibrational frequency in wavenumber units, potential energy is expressed in millidyne Angstroms, mass is expressed in atomic mass units (carbon-twelve = 12.0000), and distance is measured in Angstroms. Matrix multiplication of (1.2.4) and (1.2.5), respectively composed of the eigenvalues and eigenvectors of the matrix product GF; consequently the relation connecting potential energy parameters and vibrational frequencies is generally cumbersome. Usually the relation is described through perturbation techniques (1, 4, 5, 6) like those developed in the following section (1-3), Perturbation Methods. However, the bulk of this thesis seeks to establish an improved understanding of the relation by more abstract methods which are none the less more directly related to the physical problems. L'GFL = A shows that the - 7 -(1-3) Perturbation Methods In broad outline the calculation of potential energy-parameters by perturbation methods involves the variation of preselected parameters until calculated information agrees with experimental information. The information to be fitted includes vibrational frequencies, their symmetry type (1,6), and when known, related mechanical quantities such as mean square amplitudes (7,8), centrifugal distortion constants (8), and Coriolis coefficients (9»10). In rigorous applications, a given set of parameters span only the experimental information belonging to a series of isotopic homologs; moreover this information should be corrected for anharmonic effects. The work of Aldous and Mills (11) on the methyl halides represents a guidepost in rigorous applications for moderately complex molecules. (In triatomic molecules, rigor may assume meaning well beyond the scope of this thesis; see Suzuki's analysis (12) of carbon dioxide). When rigor is relaxed, a given set of parameters may span the vibrational frequencies for a series of related molecules. The analysis of the chlorinated benzenes delineated by Scherer (14) as well as the papers of Snyder and Schachtschneider (13,15,16,17) have added substantial credence to this approach - especially when valence force fields are employed. Very briefly, the valence force - 8 -field is built on patterns which describe the chemical bonding structure of the molecule. In this case, the potential energy function may be assembled from potential energy parameters belonging to a few simple bonding units. The perturbation methods begin with the selection of potential energy parameters, and some of these must be assigned nonzero initial values. Neither task should be considered as trivial (18,19), but both the set of parameters, , and adequate initial values, , will be assumed. Aldous and Mills (11) illustrate some of the problems. The G-matrix is constructed from geometric information and the F-matrix is constructed from the parameters. F=F° * r„ (dF/d<t>m)°(<t>m-4>m) The matrix ( dF/dfy}) ls a convenient notational device intended to cover various kinds of parameterization (or model-building) in quadratic potential energy functions. First order perturbation theory (1,4,5) provides a system of linear equations valid for small perturbations. where Ll GF"La - A0 - 9 -F; L0and A are cons tructed from the parameters either directly or by solving the above vibrational secular equation. (This latter problem, solving the vibrational secular equation by computer methods has been described by Shimanouchi and Suzuki (5). Schachtschneider's technical report (4) provides all relevant details for the application of perturbation methods .) The expressions (1.3.1) represent one useful linear equation for each secure assignment of an experimental fundamental frequency; clearly more than one molecule may be involved in the set of useful linear equations. It is assumed that experimental frequencies can be unambiguously assigned to the individual linear equations. Further it is assumed that the number of useful linear equations is larger than the number of potential energy parameters to be determined. In equation (1.3.1), a perfect fit of experimental frequencies, A RR. ) ^S n°t expected; consequently the errors, £R , defined by the identity are minimized as a weighted sum of squares, 21 K WR£R with respect to the parameter corrections, - 10 -The least squares solution (4,5,6,11), <j> = £ + zK \ (JVJ/'JVJ (A«-/&) (1.3.2) where JJ^ = ( (Of/d^JLc )**. and W is a diagonal matrix of positive weighting factors, provides improved estimates of the potential energy parameters; however, because higher order perturbation terms have been ignored, the above cycle of calculations requires iteration. The perturbation method outlined above suffers one major difficulty; both initial and target parameterization must exhibit a satisfactory formulation from the outset. Quite generally, initial parameterization is oversimplified and target parameterization involves the identification of next-most significant factors. Except (perhaps) for the hybrid orbital force field of Mills (11,20,21,22), the guidelines for ab initio target parameterization are seriously limited; see Aldous and Mills (11). - 11 -The validity of initial parameters is readily tested through the calculated vibrational frequencies and normal modes; unfortunately the value of a target parameterization is not easily tested. Schachtschneider's multiple regression analysis (4) has been applied with convincing success; see Gayles, King and Schachtschneider (19). To understand the results of a faulty target parameterization, let us define simple circumstances for illustrative purposes. Consider three linear equations in two parameters; each linear equation provides a straight line in parameter space (with allowance for error, each line becomes a band or strip). The intersection point(s) of the lines or bands specify the parameters; however a genuinely unique specification need not appear, as is illustrated in cases B and C below. CASE A CASE B CASE C - 12 -Case A, included for visual reference, indicates a physically significant specification of the parameters; the three lines intersect near a common point. Case B fails to specify significant parameters; the failure may be due to a poor choice of parameters or it may be due to more deeply rooted effects such as failure of the harmonic oscillator approximation itself. Here equation (1.3-2) provides a solution with little physical significance. In case B situation, the weighting factors will unduly influence the specification of parameters as has been shown by Nibler and Pimentel (23). Case C, ill-conditioning, provides no unique solution; with error bands, the three lines become a single hand -the equivalent of one equation. More experimental information may specify a unique solution; thus the parameters need not be incorrectly chosen. However, within the limitations imposed by the given experimental information, the parameterization must be considered faulty. Both cases B and C fail to provide a unique specification of the parameters, but only case C, ill-conditioning, has received deliberate attention in the literature (5,6,11,24) - probably due to the failure of equations (1.3-2). - 13 -In more general examples, similar uniqueness problems will appear but with complications that quickly become less and less tractable. In short summary, perturbation methods expect more fore-knowledge than is provided by the existing guidelines to chemical bonding. (1-4) Implication Methods If the normal coordinates or the L-matrix of equation 1.2.6 were specified by some means, then the complete set of quadratic force constants, the F-matrix would be specified by inversion of equation 1.2.5. F = L*A L" (i.4.i) where A is determined by experimental vibrational frequencies. This approach, the basis of implication methods was noted by William J. Taylor in 1950 (25). It is the objective of this thesis to devise both rigorous methods and intuitive grounds for specifying the L-matrix. The implication methods, developed in subsequent chapters have revealed aspects of potential energy, experimental information, and even geometric details (25) which have not or could not be determined by perturbation methods. These - Ill -methods appear to furnish a more definitive tool for the analysis of small molecule information; moreover, implication methods have indicated a fresh approach, partitioning and participation, to the analysis of larger molecules. Implication methods are not entirely new; Pulay and Torok (27), Freeman (28) and others (29,30,31) have discussed specific forms of the L-matrix which may serve useful purposes. In particular Strey's minimized bending force constants (32) indicate useful initial parameterization, but his technique is limited to simple molecules where initial parameterization is not needed. Though still in the formative stages, the implied force field obtained by implication methods offers several significant advantages over the parameterized force field obtained by perturbation methods. (1) Implication methods are constrained to positive definite force constant matrices as required by a minimum in the potential energy surface (3); perturbation methods need not obey this constraint. (2) The positive definite constraint improves uniqueness in a specification of the force constants. (3) An improved description of nonuniqueness can be obtained in that the range of possible solutions can be examined. - 15 -(4) Initial parameterization of the potential energy can be helpful, but it is not essential. (5) Implication methods help to identify essential force constants when target parameterization is involved. (6) The 'mixing parameters' of implication methods are fewer in number than the potential energy parameters of equivalent form. When applied to •.-••the molecules selected in this thesis implication methods have; (7) presented a simple account of major anharmonic effects; (8) confirmed some aspects of the hybrid orbital force field of Mills (21) without assuming it; (9) shown the harmonic bending fundamental of HOD (33) to be inconsistant with respect to the other harmonic fundamentals of water; (10) shown geometric distortions to be an important factor in the analysis of solid state spectra (see McQuaker (26) for a description of this application). However implication methods suffer one serious disadvantage. It is difficult, but not entirely impossible to invoke approximate descriptions involving the principles of chemical bonding. This is a natural feature of the perturbation methods. CHAPTER TWO: SIMPLE MIXING IMPLICATION METHODS If point group theory predicts n vibrations in the k— symmetry species, then the Ny distinct vibrational frequencies, the N distinct quadratic force constants, F and the NT distinct L-matrix elements are numbered as li follows (1) : NV 5 zk nR NF ='/z1.K nR(nK+f) The distinct L-matrix elements are constrained by Np equations of the form, thus the L-matrix exhibits N^ fewer degrees of freedom than the F-matrix. The basic hypothesis overlaying this entire thesis is that molecular mechanics can be securely analyzed in terms of the L-matrix belonging to a preselected basis molecule. The - 17 -main support for this hypothesis stems from the reduced number of parameters involved in the mechanical picture. The quadratic force constants are formed by implication through the observed vibrational frequencies and L-matrix belonging to the basis molecule. F = FA C Any mechanical quantity that is a function of the quadratic force constants is also a function of the L-matrix belonging to the basis molecule. These related mechanical quantities include the vibrational frequencies of isotopic homologs of the basis molecule (1), centrifugal distortion constants (8), Coriolis coupling constants (9, 10), and mean square amplitudes of vibration (7, 8). In - effect the Nv vibrational frequencies of the basis molecule eliminate Nv parameters from the mechanical picture. However, the L-matrix, for the basis molecule only, needs to be expressed in terms of the N„ independent parameters it contains as will be done in the following section. The case where = 1 has been designated as simple mixing; here the implication method is somewhat less abstract than the more general case dealt with in chapter three. - 18 -As yet the mechanical picture has been limited to isotopic homologs; here only deuterium substitution provides sufficient information to specify the L-matrix. Other isotopic substitutions appear to have too small an effect on vibrational frequencies to allow a proper interpretation of the mechanical picture by either implication or perturbation methods (3*0 without the aid of constraints (35) or mechanical information other than vibrational frequencies. However, the isotopic homologs of atoms heavier than deuterium have not been as. thoroughly studied by implication methods as the hydrogenic homologs. The failure of the harmonic oscillator approximation is already known through application of the product rules (1) and many previous force constant calculations. In section two of this chapter, we suggest a novel anharmonicity correction with an apparent physical basis which allows us to use the observed vibrational frequencies with almost as much success as derived harmonic frequencies (2). The final section of this chapter is given over to calculations involving the simple mixing implication method. - 19 -(2-1) The Implied Force Field When the L-matrix is expressed in the form, \A and P are orthogonal matrices if ^ is diagonal such that U6GU = r (2.1.2) The P-matrix encompasses exactly the NK parameters needed to span the family. Equation (1.2.4) is obeyed for all orthogonal P-matrices . C'GC = PY"lULOUr^P = PCP = I (2.1.3) But the implied force constant matrix, F - ii*AC = ur'*PAPtr*-ut li.i.o) will not exhibit its proper point group symmetry (1) unless the P-matrix exhibits the property: - 20 -"Pij ~ 0 unless An and designate vibrations belong ing to the same symmetry species. (2.1.5) The proof of this fact depends upon the application of point group theory in molecular vibrations (1). If a molecule exhibits symmetry, there exists a simple orthogonal transform ation matrix, S , such that and where (3* and F; are in a block diagonal form consistant with the symmetry of the molecule. Each distinct block belongs to a different symmetry species. Clearly there exists an orthogonal matrix ^JX also in block form, such that JJLGJJL Both (3 and G* have the same eigenvalues (1), and, because both and S are orthogonal, Consequently the equal matrices, - 21 -are also in block form consistant with the symmetry of the molecule. Notice the essential block form of the P-matrix. In combination (2.1.4) and (2.1.5) provide an NR para meter family of implied force constant matrices if P is not specified. The role of the P-matrix is to mix the A;'» belonging to the same symmetry species. Here the P-matrix contains NT elements constrained by N„ conditions of orthog-L r onality. N^. degrees of freedom remain. In general applications, the family is generated by the orthogonal matrix <2^ where K is a skew symmetric matrix. This will be developed in later chapters. However when exactly two fundamentals, 1/', and J belong to the same symmetry species, the mixing is simple. In simple mixing the family is generated by a single mixing para meter; here an alegraic form is preferred by the author. Pii = Pjj = (\+Xl) - pji = = xin-x1)'"2' P= I elsewhere (2.1.6) Substitution of (2.1.6) into (2.1.4) gives a one para meter family of force constant matrices generated by the mixing parameter X - 22 -0=0 el sewhere b~ 0 elsewhere (2.1.7) If the mathematical formalism of equations (2.1.6) and (2.1.7) were reduced to the form of 2 x 2 matrices, then iso-topic homologs of lesser point group symmetry could not be treated by convenient means. When the mixing fundamentals are degenerate, the CL and h matrix elements are repeated so as to comply with the degeneracy. As well, the corresponding columns of the U-matrix must be formed into properly oriented linear combinations so that the optimum symmetry factorization is obtained (1). It has been assumed that each normal coordinate defined by the L-matrix can be assigned an experimental vibrational frequency. With the implied force constant matrix now expressed in terms of an unspecified mixing parameter, it is possible to calculate the vibrational frequencies of any isotopic homolog of the basis molecule as a function of the mixing parameter. F = F° +- A Xd+X2-)"' F° = ur'Au* B -= uf'W Qij= Oji = Ajj-Aii - by = bv. = Ajj - An - 23 -The corresponding experimental frequencies then specify the mixing parameter and by implication, the force constants. However, various isotopic vibrational frequencies specify various mixing parameters and implied force constants. The range of specified values indicates the error or dispersion to be associated with the implied force constants. These errors are to be associated with the harmonic oscillator approximation rather than the implication method. (2-2) A Simple Anharmonicity Correction According to the Born-Oppenheimer approximation potential energy is independent of nuclear mass (2); consequently quad ratic (and higher) force constants are considered to be isotopic invariants. But the harmonic oscillator approximation includes only quadratic force constants and at least one very important feat ure of the potential energy surface is ignored. All stretching coordinates are expected to exhibit a dissociation limit. In this case the potential energy surface is expected to exhibit less-than-quadratic curvature. Consequently, for isotopic homolog.: studies within the harmonic oscillator approximation it is natural to associate smaller effective force constants with larger amplitudes of vibration. The same argument applies to valence angle bending - 24 -coordinates as well. One needs to consider the limiting values of potential energy for large distortions of the molecule. We have incorporated these qualitative features of potential energy into the harmonic oscillator approximation as follows. The effective quadratic force constants for different isotopic homologs are related through amplitude factors, the f^'s below. pjj (isotopic homolog) = ^'i^j Fij (basis molecule) (2.2.1) The j3'S can be treated as a diagonal matrix. To a first approximation, a distinct f^> is needed for each distinct isotopically substituted coordinate. For water and its deuterohomologs, only three amplitude factors are needed: |6>(0D), ^>(HOD) and j3(D0D). Here (OD) designates the OD stretching coordinate; (HOD) and (DOD) designate valence angle bending coordinates. The amplitude factors are easily calculated via the determinants of the vibrational secular equations for the isotopic homologs. The homolog is noted with a tilde. IGFIs Ml and \GF\= lAl Here both /\ and A are composed of experimental vibrational frequencies. Substitution of (2.2.1) for f gives the simple - 25 -equation: JAMS! IMIS1 (2.2.2) Each symmetry species of each isotopic homolog furn ishes an equation of the type (2.2.2); it is usually possible to obtain the individual amplitude factors by considering all equations of the type (2.2.2) in combination. Calculations presented in the following section show that : (1) Expected trends for less-than-quadratic curvature are confirmed. (2) A few amplitude factors enable the use of observed frequencies with nearly as much success as the harmonic freq uencies (2) which are not generally available. (3) The bending coordinates contribute almost as much to the anharmonicity as do the stretching coordinates. (2-3) Applications of Simple Mixing Methods In our initial work, simple mixing was to be no more than a prologue to general studies; however, simple mixing itself began to grow into a powerful analytic method. The advantage of simple mixing appeared from the outset. - 26 -Using perturbation methods Shimanouchi and Suzuki (5) report force constants for the harmonic frequencies of H^O, HDO, and D20 determined by Benedict, Gailer and Plyer (33). Unfortunately the symmetric stretching fundamental for D20 was misprinted. In their calculation Shimanouchi and Suzuki could not detect the rather large misprint error; they use the correct value in a later note (36). This simple oversight is very important; it unambiguously demonstrates the inadequacy of perturbation methods in the analysis of quantitative experimental information. By contrast, the simple mixing method showed a gross error which proved to be no more than the misprint already noted. More*over simple mixing methods show that the harmonic bending fundamental of HOD is uniquely inconsistent (see table one). This fact has not been previously noted elsewhere; a minor revision of the anharmonicity constants associated with this vibration is suggested. Harmonic frequencies are rarely available. A comparison of implied force fields for different basis molecules using the observed frequencies or zero-one transitions, and D^O for example, revealed the anharmonicity picture described in the previous section. Calculations will show that zero-one transitions can be used with nearly as much confidence as the harmonic frequencies. Finally, within our research group, simple mixing methods became sufficiently well understood that molecular distortions in the solid state were explored (26). Usually geometric para-- 27 -meters are not investigated by means of vibration spectros copy . Though a substantial number of simple mixing molecules have been studied with various objectives in mind, the pres entation here shall be limited to genuinely new information. In particular, the anharmonicity picture of the previous section will be established. A few preliminary notes and notational devices will simplify the presentation of the computations. (1) All frequencies are expressed in cm (2) A stretching coordinate is designated as (XY) and the associated bond length as r(XY) . Both are in A. (3) A valence angle bending coordinate is noted as (XYZ) and is expressed in radian measure. The equilibrium bond angle, ©(XYZ), will be expressed in degrees as is usual. Y designates the central atom. (4) Force constant units are as follows: Stretch-stretch millidynes/ A Stretch-bend millidynes/radian 2 bend-bend millidyne- A/(radian) (5) Force constants are noted with the appropriate co ordinate pair in parentheses and separated by a colon. Common atoms within the coordinate pair are underlined. Sometimes this provides a unique notation for all quadratic force constants. - 28 -For example, in methane, the stretch-stretch force constants are F(CH:CI1) and F(CH:CH), the stretch-bend constants are F(CH:HCH) and F(CH:HCH), and the bend-bend constants are F(H^:HCH), F(HCH:HCH) and F(HCH:HCH). For diagonal or prin cipal force constants, all atoms are underlined. If some atoms are underlined^ "Hi* force constant will be said to be connected. If no atoms are underlined, the force constant is unconnected. This notation devised here saves constant reference to related figures and diagrams; however,it is not yet sufficiently well defined for general usage. (2--3a) H^O, HDO and D20: Harmonic Fundamentals Benedict, Gailer, and Plyer (33) have analyzed the vib ration-rotation bands for water and its deuterohomologs; the harmonic frequencies were established by empirical methods (2) which involve only experimental data. For H20, the basis molecule, the two symmetric frequenc ies are , U)i = 3832.2 i \ and the antisymmetric frequency is I CJ3 = 3942.5 1 Analyses of the rotational bands furnish the geo metric parameters ••' '. - 29 -r(OH) = 0.9572 A Cos 0(HOH) = -0.25210 or 8(H0H) = 104.6° This information enables us to calculate the quadratic force constants and the harmonic frequencies of the isotopic homologs as a function of the mixing parameter. These quan-titles were calculated over a large range, but only the inter-esting portions shall be reported here. TABLE 1: Calculated Harmonic Fundamentals: HDO and Dn0. X HDO D?0* 0.000 1445.9 2822.7 3888.4 1208.8 2757.3 -0.050 1444.7 2824.1 3889.9 1205.8 2764.3 -0.100 1442.9 2827.2 3890.3 1202.6 2771.5 -0.150 1440.7 • 2832.0 3889.7 1199.5 2778.7 -0.200 1438.0 2838.4 3888.2 1196.4 2785.8 EXPTL. 1440.2 2824.3 3889.8 1206.4 2763-8** X=-0.045 1444.8 2824.0 3889.8 1206.1 2763.6 ERROR -4.6 0.3 0.0 0.3 0.2 * U)j for D20 is not a function of the mixing parameter. The calculated and experimental values agree exactly 2888.8. ** Misprinted as 2783.8 in (33). See (36). - 30 -Of the five fundamentals tabulated in table one, four of the experimental values appear in the domain predicted by a mixing parameter in the following interval. - 0.051$: X ^ - 0.037 The bending fundamental for HOD sits by itself at X -O.I56. It is clearly inconsistent with respect to the remaining fundamentals including those of the basis molecule. Excluding the outlier and including .the three basis frequencies, the mean error in fitting the experimental data is 0.1 cm \ Shimanouchi and Suzuki, who did not detect an outlier, report a mean error 1.3 cm"1 (36). The above interval for the mixing parameter places the implied force constants in the domain specified by table 2. TABLE 2: Harmonic Force Constants H^p, HDO, Dg0 force constants THIS WORK SHIMANOUCHI and SUZUKI F(0H:0H) 8.4534 ± 0.0002 8.4522 + 0.0079 F(OH:OH) -0.0999 ± 0.0002 -0.1053 ± 0.0079 F(0H:H0H) 0.2276 + 0.0160 0.1608 ± 0.0606 F(HOH:HOH) 0.6977 ± 0.0016 0.6929 ± 0.0019 Mean frequency error 0.1 m ^ 1,3 cm ^ Though the force constants of Shimanouchi and Suzuki (36) do not substantially differ from the implied force constants of this work, their dispersions do. The larger force constant - 31 -dispersions and larger mean frequency error are due to the fact that perturbation methods fail to detect inconsistent data. The implication method clearly identifies the inconsist ent frequency in this case. (2-3b) H^O, HDO, D^O: Observed Fundamentals Benedict, Gailer and Plyer (33) have reported the observed fundamentals, zero-one transitions, for H^O, HDO and D20. For HgO, the basis molecule, the two symmetric frequencies are = 3656.7 \)L = 1594.6 and the antisymmetric fundamental is I/3 = 3755-8 The bond length and valence angle are as before. Again the fund amental frequencies of HDO and D20 'are calculated as a function of the mixing parameter; however, the corresponding experimental values do not place the mixing parameter in a small interval. - 32 -TABLE 3: Calculated Fundamentals: HDO andD 0 X HDO D„0* 0.30 1395.1 2718.2 3679.1 d 1183.7 2599.0 0 . 20 1398.2 2703-2 3691.9 1179.8 2607 .6 0.10 1399-5 2694.0 3101.2 1174.9 2618.5 0.00 1398.6 2691.5 3707.0 1169-3 2631.0 -0.10 1395-7 2695.8 : 3708.8 1163.3 2644.6 -0.20 1391.0 2706.5 3706.8 1157.3 2658.3 -0. 30 1384.9 2722.4 3701.2 1151.6 2671.4 -0.40 1378.1 • 2 742.3 3692.3 1146.5 2683.2 EXPTL. 1402.2 2726.7 3707.5 1178.3 2671.5 X=-0.25 1388.0 2713.9 3704.4 1154. 4 2664 .9 error (cm 14.2 12.8 3.o 23-9 6.6 * \) (calculated) = 3 = 2752.1 1^ (exptl. ) = 2788.1 Beyond the fact that Table 3 fails to indicate a clearly significant mixing parameter, little can be said except that the errors appear to be systematic in that the bending frequencies fall together at one end of the range and the OD stretching fundamentals fall at the other end. The least squares fit, X= -0.25, gives errors of the same sign. - 33 -The implied force constants specified by the interval -0.30 £ X ± 0.10 are poorly defined but comparable to those reported by Shimanouchi and Suzuki (5) • TABLE 4: Force Constants: H„0, HDO, D^O F(0H:0H) F(0H OH) F(0H:H0H) F(HOH:HOH) mean frequency error Implied (H^O basis) 7.59 ± 0.10 -0.10 ± 0.10 0.13 ± 0.66 0.70 ± 0.06 10.7 cm Shimanouchi and Suzuki 7.67 ± 0.11 -0.15 ± 0.11 -0.17 ± 0.42 0.67 ± 0.04 12.8 cm The anharmonicity correction described in section two of this chapter improves the calculation by more than one order of magnitude. ^ (OD) can be calculated from the antisymmetric fundamentals of H20 and D^O through equation (2.2.2). Next ^ (DOD) is calculated from the symmetric frequencies of H^O and D._,0 and ^3(0D) calculated previously. Finally ^(HOD) is calculated from all of the vibrational frequencies of H^O and HDO. These amplitude factors (3(0D) = 1.013133 (3(D0D)= 1.009937 A(H0D)= 1.002664 - 34 -furnish the force constant trends F (OH:OH) < F(0J3:0D) F(HOT:HOH)< F(HOD:HOD) < F(DOD:DOD) which are consistent with less than quadratic curvature for both bending and stretching distortions of water. When vibrational frequencies are calculated via equation (2.2J.) as a function of the mixing parameter, the observed frequencies furnish a clear and distinct specification of the mixing parameter and subsequently implied force constants. TABLE 5: Calculated Fundamentals with Amplitude Factors HDO and D" 0. X HDO D20* 0.05 1403.2 2726.8 3704.8 1183.8 2659-0 0.00 1402.4 2726.6 3707.1 1180.9 2665. 6 -0.05 1401.1 2728.1 3708.5 1177.9 2672. 4 -0.10 1399-4 2731.1 3709-9 1174.9 2679. 3 EXPTL. 1402.2 2726.7 3707.5 1178.3 2671. 5 X=-0.04 1401.4 2727.8 3708.2 1178.5 2671. 0 error 0.8 -1.1 -0.7 -0.2 -0 . 5 * The experimental and calculated values for 1^3 agree exactly. These are used to calculate /3(OD) . - 35 -Table Five shows that the mixing parameter falls in the interval -0.05 < X< 0.00 which in turn provides implied force constants F(OH:'OH) = 7-681 ± 0.003 F(0H:0H) = -0.080 + 0.003 F(0H:H0H) = 0.274 ± 0.063 F(HOH:HOT)= O.663 ± 0.009 mean frequency error = 0.4 cm 1 In this calculation F(OH:HOH) is clearly positive as predicted by Mills' (21) hybrid orbital force field. Without anharmonicity corrections, the sign of this constant is not clearly defined (see table Four). (2-3c) HCCH, HCCD, AND DCCD -.Observed Fundamentals According to the symmetry of linear molecules, the longitudinal and transverse modes do not mix in the harmonic osc illator approximation, Here only the longitudinal modes will be considered. For the basis molecule, HCCH, the two symmetric long itudinal fundamentals have been placed at = 3372.9 )JL = 1974.0 and the antisymmetric fundamental has been placed at ])5 = 3285.8 - 36 -These values, and those of the isotopic homologs, were selected out of the literature by Pimentel and Nibler (23); the combin ation, +- \)s j is very close to and reported values for j^j vary to some extent. Without amplitude factors, the observed fundament als of HCCD and DCCD place the mixing parameter in the interval, 0.09 < X < 0.16 In the interval the minimum errors attainable cover age as 10.5 cm ^. This compared with the average error obtained by Nibler and Pimentel (23) which ranges from 8.2 to 10.1 cm ^ depending, on their choice of weighting factors. As well, the implied force constants agree with their force constants. The introduction of amplitude factors reduces freq uency errors and force constant dispersions by nearly an order of magnitude; however, amplitude corrections for the nonisotopically substituted coordinate (CC) are required. 1.011654 1.001905 1.004785 * Again the trend expected for a potential with a dissociation limit is obeyed. The effective quadratic force constant decreases with increasing amplitude. With these amplitude factors the observed fundamentals for HCCD for DCCD (3 (CD) = ^(CC) = ^(cc) = - 37 -of HCCD and DCCD place the mixing parameter in the interval: 0.112 < X < 0.118. — "i The minimum mean frequency error attainable is reduced to 0.9 cm TABLE 6 : Vibrational Fundamentals: deutero-acetylenes observed calculated with  l^(HCCD) = 3335.6 J4(HCCD) = 1853-8 V^(HCCD) = 2583.6 i^(DCCD) = 2705.3 i^(DCCD) = 1769.6 l/.(DCCD) = 2439.2 mean frequency error No amplitude factors 3333.1 1845.5 2562.4 2687.4 1752.6 2411.1 10.5 cm Amplitude factors 3333.4 1855•1 2583.6 2708.0 1768.0 2439-2 0.9 cm The implied force constants and their dispersions are given in table Seven along with the force constants calculated via perturbation methods (without anharmonicity corrections) by Nibler and Pimentel (23). Notice that the implied interaction constants are not clearly different from zero but the constants obtained by usual methods indicate, with limited confidence, otherwise for F(CH:CC). - 38 -TABLE 7: Force Constants: Acetylene Implied*: HCCH BASIS NIBLER & PIMENTEL, F(CH:CH) 5-920 ± 0.005 5.906 + 0.043 F(CH:CH) 0.014 ± 0.005 -0.033 ± 0.043 F(CC:CC) 15.677 ± 0.026 16.066 ± 0.222 F(CH:CC) -0.005 ± 0.020 -0.109 ± 0.069 mean frequency error 0.9 cm * The amplitude factors employed here will be found in the pre ceding text. This calculation shows that anharmonic effects of the type under consideration need not be confined to nonisotopically substituted coordinates. Implication methods confirm this with a certainty equal to the certainty in the assigned fundamentals. To be physically significant, amplitude factors should be subject to unambiguous interpretation. The amplitude factors associated with the (CC) - stretching force constants should not be interpreted via the amplitude of the (CC) - stretching coordin ate. In acetylene, the amplitude factors which modify F(CC:CC) belong more properly to the transverse modes of vibration. The appropriate limit to be considered is designated by large transverse displacements of acetylene. In this case it is reasonable to associate a (CC) - double bond with large trans verse displacements. For smaller displacements something less 8.2 cm - 39 -than triple bond strength can be assumed. Consequently it is plausible to expect F(CC_:CC_) to decrease as the average transverse displacements increase. The expected trend F(CC iCC)^. < F(CC:CC)Hcco <. F(CC:CC)0ccfi is confirmed by the calculated amplitude factors. Multiple bond anharmonicity of the kind described here may be expected for the multiple bonds contained in planar molecules. In the planar molecule, the amplitude of the out-of-plane modes will determine the multiple bond amplitude factors (2-3d) Methane and Deuterohomologs: Observed Fundamentals If the 29 distinct fundamentals of methane and its deuterohomologs can be encompassed by a single mixing parameter and three amplitude factors, (3 (CD), ^(HCD), and L3 (DCD) , then the power of implication methods and the simple picture of anharmonic effects will have withstood a very severe test. - 40 -Jones and McDowell (37) have reviewed, measured and assigned the fundamentals of methane and its deuterohomologs. The four fundamentals of CH^, the basis molecule, are ^(A..) = 2916.5 2/£(E) = 1534.0 2^(P2) = 3018.7 ]^(F2) = 1306.0 Rotational analysis provides the bond length. o r (CH) = 1.0936 A The tetrahedral symmetry defines the bond angles. Without amplitude corrections, the 25 observed fundamentals of the four deuterohomologs fail to imply a well defined solution. A least squares solution is included in table eight as a reference for the frequency errors. (S (DCD) was calculated from for CHi| and CEV ^5 (CD) was calculated from j/5 and llj for CH^ and CD^. ^3 (HCD) was calualated as an average from the observed fundam entals of the three mixed deuterohomologs (and methane). P (CD) = 1.01417 (DCD) = 1. 00635 ^(HCD) = 1.00421 These amplitude factors resemble those determined for water and they comply with the expected trend. - Ml -TABLE 8: Vibrational Fundamentals: Deuteromethanes Observed Calculated Calculated CDH3 1155 (E) 1155 1159 1300 (A ) 1303 1306 1471 (E) 1472 1473 2200 (A ) 2185 + 2215 + 2945 (A ) 2946 2946 3021 (E) 3017 3017 mean frequency error 4.0 4.CD2H2 1033 (A ) 1026 1032 1090 (B ) 1084 1089 1234 (B ) 1232 1237 1329 (A2) 1329 1334 1436 (A ) 1434 1436 2202 (A ) 2142 + 2172 + 2234 (B2) 2229 2260 2976 (A ) 2971 2972 3013 (B ) 3016 3016 mean frequency error 10.0 8.1 CD^H 1003 (A1) 997 1002 1036 (E) 1029 1036 1291 (E) 1289 1295 2142 (A ) 2101 + 2131 + 2263 (E) 2229 + 2259 2993 (A1) 2994 2994 mean frequency error 15-2 3-5 CD^ 997 (F2) 991 997 1092 (E) 1085 1092 2108 (Ax) 2063 + 2092 + 2259 (P2) 2226 + 2258 mean frequency error 22.9 4.3 overall mean frequency error 10.4 4 .1 * with amplitude corrections given in the text, •f" error larger than 10 cm - 42 -Comparison of calculated frequencies given in table Eight indicates a substantial reduction in the error following the inclusion of amplitude corrections but not as much as was obtained in the previous examples. Most of the persisting error is contained in the (CD) - stretching frequencies; apparently each homolog needs its own amplitude factor for this coordinate. A fact more significant than the error reduction is that most of the observed frequencies imply a mixing parameter in a small interval, - 0.18 < X< - 0.24, when the amplitude factors are employed. In turn this mixing parameter interval implies force constants as follows: F(CH:CH) = 4 .966 + 0 . 009 F(CH:CH) = 0 .028 + 0 .009 P(HCH:HCH)= 0 .443 + 0 .003 P(CH:HCH) = 0 .104 + 0 .020 F(CH:HCH) = -0 . 104 + 0 .020 F(HCH:HCH)= -0 .093 P(HCH:HCH)= -0 .072 + 0 . 003 The redundant coordinate system employed to describe methane, briefly described as 4(CH) -+• 6 (HCH), exhibits the symmetry coordinates, 2A + E + 2F2 (1). The genuine vibrations are A + E + 2F ; thus-two of the A force constants, _ 2,3 -F(CH:HCH,A1) = F(CH:HCH) + F(CH:HCH) F(HCH:HCH,A ) = F(H^H:HCH) + 4(HCH:HCH) + F (HCH:HCH), are often said to be indeterminate (1, 38, 39). We have reviewed the redundant coordinate system and have added new arguments in favour of the deterministic school of thought (40,41). These arguments (and counter arguments) are collected in appendix two of this thesis where it is shown that both of the force constants in question here are zero. In the present case F(CH:HCH) = - F(CH:HCH) which agrees with the hybrid orbital force field proposed by Mills (21). In simple mixing situations it is possible to express the vibrational frequencies belonging to isotopic homologs of a basis molecule as a function of a single mixing parameter. The corres ponding observed frequencies locate the mixing parameter in a specified interval which in turn places the implied force constants in specified intervals. As shown through the preceding examples, implication methods prove to be a powerful technique for the analysis of experimental information. CHAPTER THREE: GENERAL MIXING IMPLICATION METHODS Generalization of the simple mixing method depends on the completion of several tasks. (1) The orthogonal P-matrix of equation (2.1.4) must be generalized and parameterized. (2) The N^ parameters need to be expressed in terms of related mechanical information such as the vibrational freq uencies of isotopic homologs. (3) An estimate of error or dispersion within the implied force constants needs to be formulated. These aspects of the general problem are developed in the following three sections of this chapter; the fourth section is given over to applications. (3-D K- Space The desired parameterization of a generalized orthogonal matrix is achieved by the orthogonal form where Jo is orthogonal and denotes a convenient expansion point. The exponential matrix defined and reviewed in appendix three, is The task of determining a useful fo depends upon useful approximations. This topic is explored in chapter four of (3-1-0 orthogonal when the K-matrix is skew symmetric, Kii = this thesis; however, the identity matrix, is sometimes _ 2,5 -adequate. On substitution of (3.1.1) into (2.1.1), (2.1.4) and (2.1.5), the generalized implication equations are obtained. 1IT. K (Fo \j ~ ~ ^ unless Al'\ and Ajj designate genuine vibrations belonging to the same symmetry species (3.1.4) As in the simple mixing case, these equations refer only to^pecif ied basis molecule with well known vibrational funda mentals. Again, parameters belonging to the basis molecule are developed from related mechanical information,most conveniently the vibrational frequencies of its isotopic homologs. For present purposes, assume that the matrix, -l/z. Lo ~~ U P R> (3.1.5), dictates a specified ordering of the /\ - matrix. If symmetry species are identified by experimental means, then the problem is reduced. If qualitative normal coordinates are known then the - H6 -problem of ordering the f\- matrix is solved; however, such knowledge can be aafely assumed only in the case of small or very symmetrical molecules. This problem Is studied in greater depth in chapter four. As noted in chapter two, a K-space expansion contains fewer parameters than the F-space expansion; thus the magnitude of a general force field specification is reduced from the outset. (3-2) Isotopic Homologs in K-space The vibrational secular equation for an isotopic homolog can be arranged in a form most suitable for the problem in mind, where (AJCJ A and P belong to the basis molecule and Pj & and A belong to the isotopic homolog. However, P is not related to the isotopic homolog in the same way that P is related to the basis molecule. Rather it expresses the isotopic lj-matrix in terms of the basis L-matrix. L = LA PA (3.2.1) Equation (3-2.1) can be formulated from the vibrational secular equation in determinant form, JGF - *i\ = o - 47 -by substituting (2.1.4) and rearranging to yield lAVrVSurW1-^ 0 a contained matrix which is symmetric and diagonalized by an orthogonal P. A small perturbation in K-space is predicted by the partial derivative d^mn (o y/lrtrt/'rvim (3.2.3) where furnishes the K-space perturbation and P„/\ ?r a suf £/i r?=A (3.2.4) represents the expansion point. Equation (3-2.3) depends on properties of the expon ential matrix designated in appendix three. In application, equations (3-2.3) and (3.-2.4) provide the nucleus of an extremely efficient computational scheme. The U-matrix provides automatic symmetry factorization of' the G-matrix (see section 1 of chapter two). An equivalent formulation in F-space requires 2N^ matrix diagonalizations per isotopic homolog (N homologs) or large storage capabilities; the K-space formul ation required N-l matrix diagonalizations with partial pre-diagonalization provided by the U-matrix. - 48 -What remains is the solution of linear equations of the form (3.2.5) which has already been discussed in equations (1.3-1) and (1.3-2). is formed. But higher terms in the expansion of (3.2.1) are ignored; thus the P -matrix is replaced by ^(2* and the calcul ation is iterated until convergence is attained. (3-3) Implied Dispersion In simple mixing, the implication methods provide an easily understood projection scheme for determining the error associated with implied force constants. In general mixing situations, error or dispersion can be established by more general methods. Consider a force constant matrix, F0 , such that for a series of isotopic homologs: Equation (3.2.5) is a Taylor series expansion of A tffc -Given a sufficient body of experimental information, the K-matrix elements are calculated and the exponential matrix Q£* (3-3-D the corresponding implied force field F = L*A C is composed from the experimental frequencies. The scatter in the - 49 -implied force constants for a series of isotopic homologs defines our measure of implied dispersion. The dispersion can be concisely expressed in terms of the maximum frequency error. A = max I Pie-pal and •j "J The proof follows: (3.3-2) (3.3.3) (3-3.4) set where is calculated and is experimental. lFrR|l= 2(»«T]zR(in)k-.^)«j M Terms of the order Ap» (42) provides are ignored. The Cauchy inequality l'/2. 2. .i. All L\Q^ L\ consequently equation (3-3-3) is obtained. Recall - 50 -Polo (43) has written explicit expressions for G -matrix elements; however exact expressions are not needed to estimate force constant dispersions. Study of Polo's work will justify the following approximate expressions. For a stretching coordinate: where £ - f\ 6f 13 The sets of atoms >A and in the molecule. B are disjoint and cover all atoms For a valence angle bending coordinate: - 51 -where YQC is the distance from the central atom in the bending coordinate to the CC — atom. The set A designates all atoms connected to one branch of the angle and & designates all of the atoms connected to the second branch of the angle. Neither set includes the apex atom or atoms belonging to the other branches. In combination with implied force constants and a frequency error, the approximate inverse G-matrix elements give an estimate of the error or dispersion to be associated".with the implied force constants when isotopic frequency information is the determining factor. The implied dispersions of equations (3-3.2), (3-3-3) and (3-3-4) will not always agree with statistical dispersions calculated by means of the formulation of Overend and Scherer (46). Qualitative agreement between the two sets of dispersions can be expected for the principal force constants, but for some of the off diagonal force constants dispersions will be very significantly different. Most of the differences are due to the different meaning that the two expressions of dispersion carry. The implied dispers ions tell how much the implied force constants must be changed to cover all vibrational frequencies; by contrast, the dispersions of Overend and Scherer tell how much the force constants can be varied without exceeding a specified frequency error. However the statistical dispersions of Overend and Scherer may in some cases seriously overestimate the dispersion as will be shown by the following, argument. Since the potential - 52 -energy surface exhibits a minimum the force constant matrix is positive definite (3, 44). In turn, this requires the force constants to obey inequalities of the kind (45). Fu > 0 < F" Fjj (3.3.5) For dichloromethane Shimanouchi and Suzuki (5) report force constants and dispersions calculated by the Overend-Scherer method as follows: (only the magnitude of the numbers is import ant here. ) F22 = 3.8 ± 10.0 F2i| = 0.2 ± 32.1 F44 = 1.3 ± 23.3 Clearly the positive definite rules, equations (3-3-5), cannot be obeyed over the full range of the reported dispersions. Briefly perturbation methods assume force constants to be independent variables not constrained by the positive definite rules; consequently dispersion can be inflated. By contrast, implied force constants are always positive definite if the •matrix is positive definite (3). In the current context, positive definite means that every distortion from equilibrium configuration generates an increase in potential energy; otherwise equilibrium - 53 -configuration would not correspond to a minimum in potential energy. In summary neither measure of dispersion provides exactly what is wanted. Implied dispersion may be too small but the Overend-Scherer dispersions may be too large. Neither tells how unique the specified force constants are. Continued work in this area will reveal more appropriate measures of significance; the implied force field seems to be more subject to development along the lines of uniqueness. (3-4) Selected Applications in K-Space The interplay of calculation of experimental inform ation and of the connecting mathematical structure generally influences the development of the mathematical structure in a favorable manner. From the beginning of this project, calculat ions and mathematical structure were built in parallel. In this way several uninteresting notions were eliminated, and several interesting notions were uncovered. In the present situation the need for parallel calcul ations resulted in two limitations. Configuration coordinates were limited to bond stretching and valence angle bending. When isotopic substitution lowers the point group symmetry within a series of isotopic homologs special control mechanisms for degener ate fundamentals are required for the application of implication methods; these have not yet been included in the general mixing program. In effect, symmetric tops are excluded until a modified edition of the program is written. - 54 -A program designed for general application has not yet been written; it awaits the completion of several added tasks. The K-space picturesof centrifugal distortion constants (8), Coriolis coupling coefficients (9, 10), and the mean square amplitudes of electron diffraction experiments (7, 8) need to be developed and cast into the language of automatic computing. Further, the K-space picture awaits the development of various approximation techniques. To some extent, the groundwork for these tasks is established in the following chapter. Thus far only the vibrational fundamentals of isotopic homologs - as a means to specify the implied force field - have been discussed. Of three isotopic series studied, a) formaldehyde and its deuterohomologs b) ethylene and its deuterohomologs c) dichloromethane and its deuterohomologs, only formaldehyde and ethylene specify well defined implied force fields of the most general quadratic form. As could be expected in dichloromethane, the mixing of V (CC1, A ) and J^(C1CC1, A±) is not specified by the vibrations of the deuterohomologs. Per turbation methods also fail to specify a general force field for dichloromethane as has been shown by Shimanouchi and Suzuki (5) Though a complete mechanical picture of implication methods is not yet available, the present calculations demonstrate the effectiveness of more general Implication methods. The cal culations for formaldehyde and ethylene illustrate the positive cases. - 55 -In the case of dichloromethane, calculations will not be presented; here it is important that implication methods do not indicate uniqueness when the experimental information employed does not provide it. Several distinctly different force fields which provided very small frequency errors were determined for dichloromethane; additional experimental infor mation must be included to specify a single force field. (3-4a) Formaldehyde and its Deuterohomologs Shimanouchi and Suzuki (5) have reviewed the fundam entals of formaldehyde and its Deuterohomologs and calculated the general quadratic force constants. Their carefully executed perturbation study forms a reference for comparison with the implication method. The in-plane normal coordinates of the C2y molecules, briefly represented in the form 3A + 2B (see footnote), indic ate nine quadratic force constants but only four mixing para meters. Consequently, the implication method is easier to apply, faster to converge and initial estimates of the force constants are not needed. The calculated force constants and frequency errors differ somewhat from those of Shimanouchi and Suzuki (5). The symmetry species notation used here for formaldehyde and later for ethylene is used consistently throughout the liter ature even after the Joint Commission on Spectroscopy published its recommendations (47) for the selection of molecular axes. For small molecules the earlier rule (2), T T^, < » seems useful and clear enough. A 11 - 56 -TABLE 9: Vibrational Frequencies: H^CO, HDCO, DnCO. Observed CALCULATED Shimanouchi & Suzuki (10) This Work H2CO 2780 (Ax) 2796. 3 (-16.3)* 2780. 0 (0.0) 17^3. 6 (A1) 1752. 7 (-9.D 1743. 6 (0.0) 1503 (A2) 1510. 6 (-7.6) 1503. 0 (0.0) 2874 (B1) 2871. 9 (2.1) 2874. 0 (0.0) 1280 (V 1273. 2 (6.8) 1280. 0 (0.0) D2CO 2055- 8 (A1) 2049. 0 (6.8) 2045. 7 (10.1) 1700 (A1) 1694. 3 (5.6) 1679. 7 (20.3) 1105. 7 (A1) 1102. 0 (3.6) 1095. 8 (9.9) 2159. 7 (Bx) 2167. 6 (-7-9) 2178. 0 (18.3) 990 (B1) 994. 4 (-4.4) 995. 9 (-5.9) HDCO 2844. 1 (Ar) 2833. 7 (10.3) 2831. 8 (12.3) 2120. 7 (A' ) 2116. 6 (4.1) 2105 • 0 (15-7) 1723. 4 (AR ) 1720. 0 (3.4) 1717. 0 (6.4) 1400 (A' ) 1397. 1 (2.9) 1399. 1 (0.9) 1041 (A' ) ?,0 42. 1 (-1.1) 1038. 6 (2.4) mean error H2C0 8. 4 0 .0 HDCO + 5. 8 7 .5 D2CO 4.4 Overall mean error 6.2 12.9 6.8 cm -1 * (observed - calculated) - 57 -The implied force constants belong to the basis molecule, H^CO, and the corresponding vibrational frequencies are calculated without error. In terms of the mean error in the frequencies, the four parameter K-space fit is very nearly as good as the nine parameter F-space fit. The five configuration coordinates required to span the in-plane modes of formaldehyde can be represented in the form: 2(CH) + (CO) + (HCH) + (OCHH) Though appendix three shows that a redundant coordinate system can be used, non-redundant coordinate systems are used whenever possible. Consequently an in-plane-wag has been defined as follows: (OCXY) = (OCX) - (OCY). The geometric parameters selected by Shimanouchi and Suzuki (5) have been employed so that the calculations will be comparable in every way. o r(CH) = 1.1139 A o r(CO) = 1.2078 A Q(HCH) = 116.56° The implied d'l spersions for the implied force constants are based on the maximum frequency error, 20 cm-1. By contrast, the dispersions reported by Shimanouchi and Suzuki are, in several cases, much larger. The larger intervals specify the range of variation that an individual force constant can display without exceeding a specified frequency error. The smaller intervals specify the range of variation necessary to match exactly all of the frequencies employed. - 58 -(OCHH) contributes only to the modes while (HCH) and (CO) contribute only to the A-^ modes; consequently point group theory (1) requires that both F(OCHH:CO) and F(OCHH:HCH) be identically zero. As well, symmetry properties require F(OCHH:CH) = - F(OCHH:CH). TABLE 10: The force Constants of Formaldehyde. Force Constant Implication Perturbation F(CH:CH) 4.320 + 0.063 4.361 ± 0.084 F(CH:CH) 0.089 ± 0.063 0.092 ± 0.084 F(C0:C0) 13-415 ± 0-309 12.577 ± 0.271 F(C0:CH) 0.295 ± 0.140 0.704 ± 0.409 F(HCH:HCH) 0.819 + 0.022 0.840 ± 0.034 F(HCH:CH) 0.156 ± 0.037 -0.115 ± 0.255 F(HCH:C0) -0.920 + 0.082 -0.448 ± 0.146 F(0CHH:0CHH) 0.445 ± 0.032 0.432 ± 0.014 F(0CHH:CH) -0.115 ± 0.046 -0.071 ± 0.126 Neither the implied force constants nor the frequency errors calculated by implication methods differ greatly from those obtained by Shimanouchi and Suzuki (5). Consequently the four parameter implication method appears to be equivalent to the more cumbersome perturbation method. The implied force constants are heavily biased to.the basis molecule; whether or not this bias can be used to advan tage awaits further analysis. - 59 -Toward the end of this chapter it will be shown that implied force constants display a pattern of consistency which agrees with some of the principles of the hybrid orbital force field of Mills (21). The implication method generalizes without loss of significance - in terms of frequency errors or molecular structure. (3-4b) Ethylene and its Deuterohomologs. In conjunction with experimental work Crawford, Lancaster, and Inskeep have calculated the quadratic force constants of ethylene (48) from the vibrational frequencies of the two homologs, C2^4 ano" ^2D4' Tne^ imposed one constraint, F(HCH:CH) = 0, and did not attempt to adjust their force con stants to minimize frequency errors. Later Brodersen (49) repeated the determination with a deliberate attempt to minimize frequency errors and without constraints. Again the full symmetry homologs dominated the calculation; intermediate homologs were used to define one of the force constants. Though Brodersen's choice of fundamentals differed little from those of Crawford, Lancaster and Inskeep, the reported force constants differ significantly (see table eleven); however, the Brodersen force constants can be viewed as a refinement of the earlier work. Since automatic computing became available, a general valence force field for ethylene has not been reported. Scherer and Overend have reported a six parameter Urey Bradley force field (UBFF) (50). More recently Fletcher and Thompson have reported a ten parameter Hybrid Orbital force field (HOFF) (22). - 60 -Automatic computing would enable the use of data from all the deuterohomologs and provide minimum frequency errors in the least squares sense. In effect the implied force field reported here completes the refinement of the quadratic force constants for ethylene. The fifty-four fundamentals of the six deutero-ethylenes (51) are used to fix the six mixing parameters for the basis molecule ^C^H^. Though significant differences appear, the implied force constants appear to be a refinement of'the earlier work reported by Brodersen. It is possible to classify the vibrations of the deuteroethylenes under the point group of the potential energy, °2h ' ]/^aen isotopic substitution reduces the point group symmetry, ~~ species are scrambled but in accord with the lower point group. Consequently all vibrations can be classified by enclosing the scrambled D - species in parentheses. 2h H2C2H2 + D2C2D2 : 3Ae + 2Bn lg + 2B2u + 2B_ 3u HDC2H2 + HDC2D2 : (3A8 + lg + 2B2u + 2B_ ) 3u trans - HDC2HD (3Ag + 2Blg) + (2B 2u + 2V cis -HDC2HD : <3Ag + 2B2u> lg + 2V 1,1-dideuteroethylene : (3AG + B3u > lg + 2B2U> This notation immediately reveals the information needed to formulate the product rules. Further it shows at a glance the dependence of vibrational frequencies on symmetry factored force constants. - 61 -For example, in trans dideuteroethylene, the five symmetric vibrations, (3A + 2 EL ), deoend on the six A -force constants g lg g and the three EL -force constants. In K-space these five lg vibrations depend on the three A and the one B mixing para-g -Lg meters. This structure carries over to the K-space perturb ation equations, (3-2.5), and is of great help in assigning experimental frequencies to the individual linear equations. It should be realized that the classification under Dz\r\ , (3A + 2B-, ) + (2B~ + 2B_ ) , means exactly the same thing g lg 2u 3u as the classification, 5A + 4B , obtained under C~, ; however, 5 g u 2h' ' the Dgn notation emphasizes the underlying structure. The vibrations of the basis molecule, H^C^H^, indicate fifteen independent quadratic force constants but only six mixing parameters. The planar configuration coordinates adopted for this study, MCH) + (CC) + 2(HCH) + 2(CCHH), contribute to the vibrations as follows: 4(CH) A + B., + B0 + B_ g lg 2u 3u (CC) Ag 2(HCH) Ag + B^ 2(CCHH) B0 + B, 2u lg Fortunately the CH-stretching vibrations and the (CC) -stretching vibration are known to be characteristic; thus no problems are expected in structuring the implied force field. - 62 -For clarity let us note that (CCHH)1 = 2(CCHH) (CCHH) 2 = fa-fa The symmetry properties of the configuration coord inate impose five symmetry restrictions on the force constants, F(CCHH.-CC) = 0 F(CCHH:HCH) = 0 F(CCHH:HCH) = 0 F(CCHH:CH)- + .F(CCHH:CH) = 0 F(CCHH:CH, cis) + F(CCHH:CH, trans) = 0 In this calculation the geometric parameters selected by Herzberg (2) have been employed. o r(CH) = 1.086 A o r(CC) = 1.339 A 9(HCH) = 117.6° Beginning with PQ — I in equation (3.2.4), mean - 63 -error in the frequencies of 20.6 cm the K-space iteration sequence described in section (3-2) proceeds smoothly to the implied force constants given in table eleven and calculated frequencies given in table twelve. The largest frequency error appears in the (CC)-stretching frequency for 1,1-dideuteroethylene (the calculated frequency falls 33 cm ^ below the observed frequency). In fact all of the implied (CC) stretching fundamentals fall below the observed fundamental. The same discrepancy appears in formaldehyde for the (CO)-stretching fundamentals. In these cases; multiple bond anharmonicity like that described for acetylene is suspected. The remaining large frequency errors in the deutero-ethylenes appear, as expected, in the (CD)-stretching frequencies. The force constant dispersions, calculated by the implication technique of section (3-3), have not been included in table eleven. Dispersions based on twice the mean frequency error are reported separately below. All but eleven of/calcul ated frequencies fall within this range. For the principal force constants the implied dispersions are comparable to those implied Fletcher & Thompson (22) F(CC:CC) +0.18 ±0.15 F(CH:CH) ±0.044 ±0.036 F(HCH:HCH) +0.013 ±0.008 F(CCHH:CCHH) +0.018 ±0.009 determined by Fletcher and Thompson. The remaining implied dis-persions are geometric means/the above values, see equation (3-3-3) TABLE 11: The Force Constants of Ethylene - -Crawford Brodersen Scherer & Fletcher & This Lancaster & (49) Overend Thompson work Inskeep (48) (50) (22) F (CC_: CC_) 10.896 11.08 9.038 9- 305 11.184 F(CH:CH) 6.126 4.77 5.149 5.168 5.004 F(HCH:HCH) 0.731 0.708 0.661 0.683 0.725 F(CCHH:CCHH) 0.373 0.334 0.248 0.269 0.332 F(HCH:CC) -0.920 -0.826 -0.264 -0.273 -0.943 F(HCH:CH) 0.369 0.130 -0.123 0.087 0.182 F(HCH:CH) 0.000 0.141 0.000 0.000 0 .241 F(HCH:HCH) 0.035 -0.012 0.000 0.022 0 .066 F(CCHH:CH) 0.5H -0.163 0.123 0.098 -0.199 F(CCHH:CH, cis) 0.234 -0.332 0.000 0 .000 -0.313 F(CC_HH:CCHH) 0.035 0.006 -0.048 -0.048 0 .014 F(CH:CC) 0.000 0.00 0.367 0.000 -0.154 F(CH:CH) 0.043 0.02 0.000 -0.012 0.031 F(CH:CH1 Cis) -0.020 ' -0.06 0. 000 -0.018 -0.104 F(CH:CHj trans) 0.050 0.14 0.000 -0.018 0.089 Constraints F(CH:HCH) = = 0 NONE UBFF* HOFF NONE Parameters 14 15 6 10 6 mean error (cm-"'") 13.4 8.1 12.0 8.8 6.5 This is the valence bonding image of the Urey Bradley force field (14). - 65 -TABLE 12: Vibrational Fundamentals: ethylene and deuterohomologs SYMMETRY OBSERVED IMPLIED BRODERSEN MOLECULE 3026 3026 3026 1623 1623 1630 g 13^2 1342 1350 3103 3103 3110 2B lg 1236 1236 1238 3106 3106 3110 2B2u 810 810 815 2990 2990 3001 2B3u 1444 1444 1455 mean frequency error 0.0 6.0 H2C2H2 2660 2238 2253 1518 1508 1511 3A g 985 977 978 2310 2312 2308 1011 1007 1009 2345 2336 2340 584 576 580 2200 2169 2189 2B_ 3u 1078 1065 1067 mean frequency error 11.8 6.2 D2C2D2 3017 2998 3009 2230 2231 2236 1585 1552 1561 1384 1387 1402 1031 1024 . 1022. . . 3093 3096 3100 2334 2334 2336 1150 1153 1155 (2Blg + 2B2u' 660 664 668 mean frequency error 8.1 9.7 ASYM-H C D„ 2 2 2 - 66 -SYMMETRY OBSERVED IMPLIED BRODERSEN MOLECULE 3059 3069 3066 2299 2283 2300 1571 1562 1574 1218 1213 1209 (3Ag + 2B2u 646 647 652 3054 3051 3057 2254 2238 2253 1342 1336 1335 (2Bn + 2B0 ) 1039 1043 1048 lg 3u mean frequency error ... 7.8 5 ..8 CIS-HDC2HD 3045 3043 3069 2285 2285 2279 1571 1562 1572 1286 1281 1284 (3A + 2 B ) 1004 1000 1007 § lg 3065 3059 3040 2273 2268 2298 1299 1288 1280 (2B_ + 2B0 ) 678 659 664 mean frequency error 6.9 13.2 TRANS-HDC2HD - 67 -H2C2HD (3Ag + 2Bn . 2B0 + 2B0 ) lg + 2u 3u D2C2HD OBSERVED IMPLIED IMPLIED OBSERVED 3096 3101 3051 3049 3061 3062 2331 2332 3002 3000 2269 2281 2276 2278 2205 2222 1606 1589 1532 15^7 1401 1401 1280 1289 1290 1288 1043 1045 1129 1128 994 999 713 716 612 610 mean frequency error 3-7 7.2 Neither Crawford, Lancaster and Inskeep nor Brodersen pres ent calculated frequencies for the low symmetry homologs. Fletcher and Thompson calculate frequencies for H~2C2HD and report a mean frequency error 9«1 cm 1 for this molecule. The six parameter Urey Bradley force field Is too highly constrained to compare calculated frequencies. If the five calculations are intercompared by multiplying the mean frequency error and the number of adjustable parameters, then the Implied force field takes first rank. Moreover by this measure of merit, the hybrid orbital force field follows the Urey Bradley force field. - 68 -(3-4c) Chemical Significance On one hand quadratic force constants are mechanical parameters consistant with mechanical information of exper imental origin. On the other hand, as measures of bonding forces and interactions, quadratic force constants should comply with chemical bonding structure. Consequently the quadratic force constant serves both physical and chemical purposes. Physical significance, though subject to well defined measurement, is limited to pure mechanics. Chemical significance establishes itself through chemical bonding and is subject to a wider interpretation, but chemical sig nificance is not subject to well defined measurement. Because implied force constants are formulated with minimal reference to chemical bonding structures, their chemical significance needs deliberate emphasis. In some respects, it is remarkable that implied force constants exhibit any degree of chemical significance at all. Ethylene and formaldehyde are in fact closely related molecules; if one CH^-group of ethylene is regarded as a single atom, the resultant molecule would be an isotopic homolog of formaldehyde. Table thirteen shows that force constants for the two related molecules exhibit unexpected sign and size agreement. However the bonding unit, H2C= ^ does not possess a set of force constants which are largely independent of the substituent as has been shown (17,20) for the bonding unit H C- . - 69 -TABLE 13: Similar Implied Force Constants F(CH:CH) F(CX:CX) F(CH:CH) F(CH:CK) F(HCH:HCH) F(CH:HC_H) F(CX:HCH) F(XCHH:XCHH) F(XCHH:CH) Formaldehyde (X= oxygen) 4 . 320 13.415 0.089 0.295 0.819 0.156 -0.920 0.445 -0.115 Ethylene (X= Carbon) 5.004 11.184 0.031 -0.154 0.725 0 .182 -0.943 0.332 -0.199 The similarities confirm some of the principles used by Mills in the hybrid orbital force field (21). He suggests that F(HCH:CH) - k F(CH:CH) F(HCH:CX) = k1F(CX:CX) where the constants k and k1 depend mostly upon hybridization and to some extent on the nature of the substituent. If sp2" hybrid ization dominates other effects, then k1 = -2k. In both formaldehyde and ethylene the central atom is hybridized and the hybridization constants should be nearly the same. - 70 -HYBRIDIZATION CONSTANTS formaldehyde ethylene k 0.0361 0.0364 k 1 -0.0686 -0 .0841 -1.90 -2. 31 The hybridization constants, k , are nearly equal for 1 the two molecules. Those involving the double bonds, k , are comparable and related to the single bond hybridization constant, k, the expected way. It is gratifying to confirm these aspects of the hybrid orbital force field (21) without assuming it. However, the hybrid orbital force field predicts F(CCHH:CH) = F (HCH:CH), but the implied force constants agree more closely with a Urey Bradley force field here (14), As well, neither of the model force fields includes stretching Interactions but the implied force field indicates definite stretching interactions. As shown by Heath and Linnett (52), interaction force constants can be interpreted on strictly geometrical grounds. Let all coordinates except /O-ii and P-y be fixed at their equil ibrium values. Let ,Oj be assigned a definite displacement, &j These constrained conditions imply a pseudo-equilibrium value, /Qf- } for the internal coordinate associated with OA in that the potential energy is minimized. F(CCHH:CH) = - F(HCH:CH). - 71 -In formaldehyde, if the (CO) bond is stretched, the (CH) bond is shortened. In ethylene, if the (CC) bond is shortened the (CH) bond is lengthened. Consequently the psuedo-equilib-rium geometry of formaldehyde tends toward the equilibrium geometry of ethylene and conversely. In this sense the only signature disparity in table thirteen is quite creditable. The very few implied force constants now in hand do not provide a sufficient foundation for a general discussion of their special chemical significance; however, these examples indicate the definite value of continued studies with implic ation methods. CHAPTER POUR: APPROXIMATION TECHNIQUES Thus far the implication method has been limited to the case where there exists sufficient experimental information to specify all quadratic force constants. Clearly larger molecules with low symmetry, methylamine for example, need not submit to such a general approach. Consequently, further progress with implication methods will depend upon approximation techniques which either operate entirely within the implication scheme or cooperate with the more traditional methods of parameterized potential energy. The weighted trace equations delineated in section one of this chapter provide some information about the latter objective: cooperation of implication methods and traditional parameterization of molecular potential energy. Fortunately the implication scheme lends itself to a unique approximative technique whenever some vibrations of a molecule can be said to be characteristic vibrations. The second section of this chapter describes the role of characteristic vibrations in the analysis of the vibrational secular equations entirely within the implication scheme. (4-1) The Weighted Trace Equations Consider the function (4.1.1) - 73 -where F=ur\<£lAd*P?rILUT (4.1.2) as defined in equations (3-1-3) and S is an ordering parameter - see appendix three. The weighting matrix W is some specified matrix with the same symmetry as the F-matrix. In essence's/forms various linear combinations of force constants; explicit choices for the VV-matrix will follow from the general development. Notice that Thus /"L is defined such that where JUT is a diagonal matrix. The diagonal.JfoP-matrix entries are the eigenvalues of the matrix product , (3 W The techniques described in appendix three give the general weighted trace equation: J^B A^R. Jjf^ = Xcife/c + higher terms (4.1.4) - 74 -Consequently Trace Wf- is maximum if AJX- < py-Trace lA/P is minimum if J/X\ ^> j^j". Trace W~ is saddle point if JjS^ ~JjXi with respect to the mixing parameter y^. Suppose that the three force constants P P and ^ mm, mn ^nn dominate the two vibrational frequencies Xv\ and (or the mixing parameter K is most important). Then mn equations (4.1.3) and (4.1.4) provide bounds for the force constants. ( /{^ > ^ _(4.1.5) £ Finn £ Gwn(/U + 'M~ Am)~yGmm Gw Equations (4.1.5) are formed from three different W-matrices -one to select each different force constant. The inequalities (4.1.5) are exact for the vibrational secular equation of order two. Otherwise they are not exact but they may indicate important interaction force constants. For example, the two B2u modes of benzene and the inverse G-matrix elements (symmetry factored) (1) give the following bounds for the symmetry factored force constants. - 75 -3-353 - F(CC:CC,B ) £ 4.375 O.916 ^ F(HCCC :HCCC ,B ) 6z 1.195 -I.658 ^ F(HCCC:CC,B2u) ± -0.593 Here the application of (4.1.5) is exact and limits for an essential interaction constant have been set by simple means. Thus, F(HCCC:CC,B2u) has not been specified but it cannot be zero. The Importance of this interaction constant has already been established (53). Equations (4.1.5) may prove most useful in selecting important interaction constants. Another way of identifying important interaction force constants by implication methods follows. Assume an approximate force constant matrix f-Q and solve the. secular equations where Aa represents calculated frequencies. The associated implied force field F = JL~* A C may well indicate important corrections to the approximate force constants used to assemble The A -matrix is composed of experimental frequencies. In this application the dispersion equations, (3.3-3) and (3-3-4), show that if the calculated and observed frequencies are close, so are the trial and implied force constants close. Consequently like - 76 -the perturbation method itself, this approach may assume more fore-knowledge than is provided by the existing guidelines to chemical bonding. Yet it is believed that the implied force constant matrix associated with approximate force constants will provide a less cumbersome approach than perturbation methods when the problem is too complex to solve entirely within the perturbation scheme. (4-2) Participation and Molecular Partitioning It is often possible to associate selected vibrational frequencies with selected parts of the whole molecule - usually chemical groups such as -NH^ or -CH^ If a vibrational frequency belongs to a chemical group, then the remaining atoms participate to a much lesser extent in that vibration. This section considers the partitioning of a molecule into two sets of atoms and describes the participation of the individual sets in the various vibrations. Characteristic frequencies are closely related to mass dependence within the vibrational secular equations. Characteristic frequencies will depend mostly on selected atoms and consequently their masses. Differentiation of the matrix equations G - LlS and L?FL. =A - 77 -th with respect to the reciprocal mass of the ^ — atom leads to the equations: Both /I and QA/c^j^are diagonal. Consequently (4.2.1) (4.2.2) (4.2.3) Where is a skew symmetric matrix. In combination (4.2.1) and (4.2.3) provide the fundamental mass equation. jj Because CA, KAT. = O />' 3£ = /C = The G-matrix is of the form (4.2.4) - 78 -therefore (see appendix one) which is positive semi definite (44), and moreover £> = Za; (QS/d/U<) (1.2.5) Thus the equation gives Zx /Joe = i (4.2.6) Consequently the quantity Sin Aii th can be viewed as the participation of the QC atom in the . th »J vibrational frequency. - 79 -As well, it can be established that (4.2.8) The latter equation follows from the identity (54). Partitioning. J Let a molecule be partitioned such that each of its atoms belongs to either the set J\ or the set B Define the participation matrices: yUcC ( / TTfA) = £^{L m«L ) (4.2.9) Both 777/1) and 7T/B) are square positive semidefinite matrices and their diagonal elements are sums of atom-participations. Through (4.2.5) they sum to the identity matrix Ti(A) + 7T(B) = I (4.2.10) - 80 -and they obey positive semidefinite inequalities (45) hi For a nonplanar molecule riOMfl TTffii) $ 3A/A where is the number of atoms in A (44,45). The quantity defines the participation of the A— set of atoms in the J — vibration. If the A — set of atoms does not participate in the ft — vibration, then TT(A)„n = O TT(B)nn = 1 and the positive semidefinite inequality gives TT(A)n^ - O for all j again TT(&>)flj = O for all j except j =n recall TT(A) + TJ(B) = I moreover note the strong implication of null participation TTMnn=0 implies M*^~^=0 for all CC contained in the A-set. - 81 -Consequently Isotopic substitution within the r\ -set leaves the characteristic vibrations of the E> - set invariant. This fact provides a test for the validity of null participation ! As an example of partitioning and participation let us consider methylamine. Let there be six modes characteristic of the methyl-set, three modes characteristic of the amine-set, and six mixed modes. The principles just described require TTXCH^and 7T(NH2) to be of the form given by table fourteen. Unity designates the identity matrix and the arrows show the inter relationships used to formulate 77" (CH) and 7T(NH ). - 82 -TABLE 14 Participation Matrices for Methylamlne The distribution of unit, nonzero, and zero blocks in the participation matrices for (CH ) - (NH^); the arrows summarize the derivation of the zero-blocks for limited participation. TT(CH ) (CH^)-MODES UNITY kA' + 2 A" ZERO ZE i RO ZERO MIXED MODES NONZERO 3A» + 3A" ZE i RO RO H^)-MODES a£jR\J (N <0-ZERO — (CH^)-MODES ZERO ZERO ZERO MIXED MODES NONZERO 3A' + 3A" ZERO ZERO ZERO (NH )-MODES UNITY 2A' + A" Tt (NH2) - 83 -The symmetry species show the further factorization obtained when methylamine is given a plane of symmetry. The fundamentals assigned by Dellepiane and Zerbi (55) support the hypothesis of complete separation. The three modes characteristic of amine are easily identified. CH3NH2 3361 A' 1623 A' 3427 cm-1 A" It is important to note that these invariant fundamentals support the hypothesis of null participation over the methyl group. Similarly for the six methyl modes: CH3NH~2 2961 A' 2820 A' 1473 A' 1430 A' 2985 A" 1476 cm-1 A" And these fundamentals support the hypothesis of null participation over the amine group. Note the violation of the rule of monotony (1) which states that all vibrational frequencies must decrease upon heavier isotopic substitution (see equation 4.2.7). CD3NH2 3361 1624 3427 cm CH3ND2 2961 2817 1468 1430 2985 1485 cm' - 84 -The question as to how limited participation infor mation can be used in connection with implication methods now arises. TT(A) and/T(&)can be written in the form: 4 and <pA is a diagonal matrix The matrices and sum to the identity matrix, therefore they commute and are simultaneously diagonalized by a single orthogonal According to the principles set down in appendix three for the exponential matrix and the principles of limited par ticipation, the skew-symmetric K-matrix of equation (4.2.11) exhibits the mixing property! - 85 -|^ •» —Q unless and represent genuine vibrations of the same symmetry species and cover the same set of atoms. In the present example, the selected forms of the TT — matrices cause the methyl-modes to mix only with themselves, the amine-modes to mix only with themselves and the mixed-modes to mix only with themselves. The mixed modes are not coupled to any of the characteristic modes; this is a remarkable and unexpected property ! With a plane of symmetry methylamine exhibits 66 independent possible quadratic force constants or 51 possible general mixing parameters. But if limited participation is assumed - and experimental evidence validates the assumption - the number of mixing parameters is reduced to 14. Through molecular partitioning it seems quite likely that reasonable implied force constants could be obtained for a class of more complex molecules. As well partitioning applies in other senses. The G-matrix can be partitioned into two parts, <3(A) and (3(B)such that Gift) +-G(B) = G with much the same results. The atom-participation densities, foCj > provide an interesting description of vibrational modes. The equation / / \ provides meaning in that foij is directlv related to small isotopic shifts. EPILOGUE Unfortunately the exploration of implication methods in the description of molecular mechanics and chemical bonding structure is not yet complete but this thesis has established the basic approach to a cumbersome problem by new methods. The ingredients of the implication method include two important factors. An experimentally well known basis molecule is adopted and its vibrational frequencies remove a corres ponding number of parameters from the pure mechanical description. The reduced parameterization has been cast into a mathematical form handy in computations as well as analysis, as delineated in chapter four. Application of the Implication technique has led to a simple account of some dominating anharmonic effects. If continued application of the method is as rewarding, then further studies are definitely warranted.^ To this end, the concepts of participation and molecular partitioning appear to offer the optimum prospects. If molecules not rich in hydrogen are to be considered in more rigorous terms, the role of mechanical information other than vibrational frequencies must be included in the implication scheme, Jones, Asprey and'Ryan (3*0 have shown that a complete picture of mechanical information is essential in the perturbation method. - 87 -Therefore, the limitation to hydrogen rich molecules is not specific for the implication method. The limitation is merely one, of development within the implication method. When the molecular system precludes a more rigorous approach,it is plausible that implication tech niques may contribute substantially to a simple but proper parameterization of the potential energy function. BIBLIOGRAPHY 1. E.B. Wilson, Jr., J.C. Decius, and P.C. Cross, Molecular Vibrations, McGraw-Hill Book Company, Inc., Toronto, Canada, 1955-2. G. Herzberg, Molecular Spectra and Molecular  Structure, Volume 1 (1945), Volume 11 (1950), Volume 111 (1966), D. Van Nostrand Company (Canada), Limited, Toronto, Canada. 3. S. Bhagavantam and T. Venkatarayudu, Theory of  Groups and Its Application to 'Physical Problems, Third Edition, Bangalore Press, Mysore Road, Bangalore City, India, 1962. 4. J.H. Schachtschneider, Vibrational Analysis of  Polyatomic Molecules, Technical Report No. 57-65, Shell Development Company, Emeryville, California, U.S.A., 1966. 5. T. Shimanouchi and I. Suzuki, The Journal of Chemical Physics 42_, 296 (1964). 6. J.M. Freeman and T, Henshall, Journal of Molecular Spectroscopy 25_, 101 (1968). 7. S.J. Cyvin, Molecular Vibrations and Mean Square  Amplitudes, Elsevier Publishing Company, Amsterdam, The Netherlands, 1968. 8. J.C. Decius, The Journal of Chemical Physics 3_8, 24l (1963). - 89 -9. J.H. Meal and S.R. Polo, The Journal of Chemical Physics 24, 1119 (1956) . 10. J.H. Meal and S.R. Polo, The Journal of Chemical Physics 24., 1126 (1956). 11. J. Aldous and I.M. Mills, Spectrochimica Acta 19., 1567 (1963). 12. I. Suzuki, Journal of Molecular Spectroscopy 25, 479 (1968). 13- R.G. Snyder and J.H. Schachtschneider, Journal of Molecular Spectroscopy 30, 290 (1969). 14. J.R. Scherer, Spectrochimica Acta 20., 345 (1963). 15- R.G. Snyder and J.H. Schachtschneider, Spectrochimica Acta 19, 85 (1963)• 16. R.G. Snyder and J.H. Schachtschneider, Spectrochimica Acta 19, 117 (1963)• 17. R.G. Snyder and J.H. Schachtschneider, Spectrochimica Acta 21, 169 (1965) . 18. J.H. Schachtschneider and P. Cossee, The Journal of Chemical Physics 44_, 97 (1966). 19. J.H. Schachtschneider, J.N. Gayles, Jr., and W.T. King, Spectrochimica Acta 23, 703 (1967). - 90 -20. J.L. Duncan, Spectrochimica Acta 20, 1197 (1964). 21. I.M. Mills, Spectrochimica Acta 19., 1585 (1963) 22. W.H. Fletcher arid W.T. Thompson, Journal of Molecular Spectroscopy 25, 240 (1968). 23. J.W. Nibler and G.C. Pimentel, Journal of Molecular Spectroscopy 26, 294 (1968). 24. N.A. Narasimham and C.V.S. Ramachandraras, Journal of Molecular Spectroscopy 30, 192 (1969). 25- W.J. Taylor, The Journal of Chemical Physics 18, 1301 (1950). 26. N.R. McQuaker, "Vibrational Spectra of the Ammonium Halites and the Alkali-Metal Borohydrites". Ph.D. Thesis, The University of British Columbia, 1970. 27- F. Torok and P. Pulay, Journal of Molecular Structure 1, 1 (1969). 28. D.E. Freeman, Journal of Molecular Spectroscopy 2_7, 27 (1968). 29. F. Billes, Acta Chim, Acad. Sci. Hung. 4_7, 53 (1966). 30. C.J. Peacock, U. Heidborn and A. Muller, Journal of Molecular Spectroscopy 30., 338 (1969). 31. B.S. Averbukh, L.S. Mayants and G.B. Shaltuper, Journal Of Molecular Spectroscopy 30, 310 (1969). - 91 -32. G. Strey, Journal of Molecular Spectroscopy 2_4, 87 (1967). 33- W.S. Benedict, N.Gailer and E.K. Plyler, The Journal of Chemical Physics 2_4, 1139 (1956). 34. L.H. Jones, L.B. Asprey, and R.R. Ryan, The Journal of Chemical Physics 47, 3371 (1967). 35. L.H. Jones, R.S. McDowell and M. Goldblatt, The Journal Of Chemical Physics' '48, 2663 (1968). 36. T. Shimanouchi and I. Suzuki, The Journal of Chemical Physics 43, 1854 (1965). 37- L.H. Jones and R.S. McDowell, Journal of Molecular Spectroscopy 3, 632 (1959). 38. I.M. Mills, Chemical Physics Letters 3, 267 (1969). 39- B. Crawford, Jr., and J. Overend, Journal of Molecular Spectroscopy 12, 307 (1964). 40. R.L. Hubbard, Journal of Molecular Spectroscopy 6_, 272 (1961). 41. R. Gold, J.M. Dowling and A.G. Meister, Journal of Molecular Soectroscopy 2, 9 (1958) 42. W. Kaplan, Advanced Calculus, Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1952. 43. S.R. Polo, The Journal of Chemical Physics 24., 1133 (1956) - 92 -44. S. Perlis, Theory of Matrices, Addison-Wesley Publishing Company, Inc., Reading Massachusetts, 1952. 45. M. Marcus, Basic Theorems in Matrix Theory, National Bureau of Standards Applied Mathematics Series 57, I960. 46. J. Overend and J.R. Scherer, The Journal of Chemical Physics 3_2, 1289 (I960). 47. Joint Commission for Spectroscopy, The Journal of Chemical Physics 23, 1997 (1955). 48. B.L. Crawford, Jr., J.E. Lancaster, and R.G. Inskeep, The Journal of Chemical Physics '21, 678 (1953). 49. S. Brodersen, Matematisk-fysiske skrifter udgivet af Det Kongelige Dansk Videnskabernes Selskab 1, no. 4 (1957). 50. J.R. Scherer and J. Overend, The Journal of Chemical Physics 33., 1681 (i960). 51. S. Brodersen and A. Langseth, Matematisk-fysiske Skrifter udgivet af Det Kongelige Danske Videnskabernes Selskab 1, no. 5 (1958) 52. D.F. Heath and J.N. Linnett, Transactions of the Faraday Society 44, 556 (1948). 53- R.A. Kydd, "The Vibrations of Some Aromatic Molecules", Ph.D. Thesis, The University of British Columbia (1969). - 93 -54. H. Margenau and G.M. Murphy , The Mathematics of  Physics and Chemistry, Toronto, Canada, 19 43. 55. G. Dellepiane and G. Zerbi, The Journal of Chemical Physics 4_8, 3573' (1968) . 56. L.J. Bodi, The Theory of Vibrational-Rotational Interaction in Polyatomic Molecules, University of Wisconsin, Naval Research Laboratory, Technical Report Wis-OOR-11, Madison, Wisconsin, 1954. 57. ' E.C. Kemble, The Fundamental Principles of Quantum Mechanics with Elementary Applications, Dover Publications, Inc., New York, New York, 1937-58. R.J. Malhiot and S.M. Ferigle, The Journal of Chemical Physics 22, 717 (1954); 23., 30 (1955). 59- L.L. Oden and J.C. Decius, Spectrochimica Acta 20, 667 (1963). 60. C. Chevally, Theory of Lie Groups, Princeton University Press, Princeton, New Jersey, 1946. APPENDIX ONE: THE VIBRATIONAL SECULAR EQUATIONS A Simplified Quantum Mechanical Description -The concepts of classical mechanics, mainly well defined trajectories, fail to provide a genuine description of microsystems (56, 57)- In some examples such as the hydrogen atom, the failure of classical concepts is sub stantial; in other cases such as the harmonic oscillator, classical concepts remain adequate within limits, especially for polyatomic molecules. Consequently, a classical picture of molecular vibrations has persisted while the necessary concepts of quantum mechanics grew to dominate the description of microsystems, especially molecular structure. Though chemists are not quantum mechanicians, they are becoming more and more quantum oriented; classical mechanics has in fact almost entirely disappeared from the latticework of chemical logic. Thus, to the student of physical chemistry, a classical picture of molecular vibrations is built upon unfamiliar, almost irrelevant, foundations. A quantum mechanical description of molecular vibrations falls within the quantum picture usually presented to chemists; moreover both molecular vibrations and the existing quantum picture would benefit from a connected development. - 95 -Quantum descriptions including molecular vibrations already exist but these (1, 56, 57) are written for the experienced spectroscopic specialists or the pure mechanicians. In short, a handy reference for molecular vibrations, written in quantum mechanical language suitable for introductory purposes, is not known (to the author). the above-stated purposes; it includes all classical information^ captures the flavor of rigor in quantum mechanics and provides a firm foundation for further rigor and detail. (I-l) Transformation of the Schrodinger Equation The Cartesian coordinate Schrodinger equation for potential energy function V, governs the total energy of The following description is intended to serve N particles under a 0 (T.1.1) the system;E, and the probability, of a configuration specified by the Cartesian coordinates (>Gj^ Vct,^*) where CC counts particles. In the Cartesian S'chrodinger equation (T.1.1) ~p] is Planck's constant; 3jp the Cartesian wave function, is a function of Cartesian coordinates. - 96 -V the potential energy, is a function of the configuration of the molecule (in the absence of significant external influences). It is invariant with respect to rotations and translations. For present purposes electronic motion is considered to be embedded within the potential energy function; see the Born-Oppenheimer separation of electronic and nuclear motion (2) . At Whether or not the electronic masses should be included in the masses of the individual particles under consideration is not completely clear. The Born-Oppenheimer separation calls for nuclear masses but detailed studies usually employ atomic masses with satisfactory agreement (2). The invariance of the potential energy with respect to rotation and translation, as well as the principles of chemical bonding, suggests a transformation of coordinates. New coordinates would include three translation coordinates, three rotation coordinates and (3N-6) coordinates of configuration - bond lengths, valence angles, and related geometric measures. The transformation is curvilinear. - 97 -To this end let the transformation properties of (T;l.l) be established; let the new coordinates be designated as t^ . The rules of partial differentiation (42) establish the identity: 55?^" 1 UxJlati (1.1.2) Differentiation of (T. 1.2) with respect to Xoc provides a second identity: d>a - t-z-i (axJi SJJ aha* iri3)  J dxi a*j When cast into generalized coordinates, the kinetic part of the Cartesian coordinate Schrodinger equation becomes where the following quantities have been defined for brevity, (T.1.6) - 98 -With reflection, paper, and pencil, the first of these quantities (T.1.5) will be recognised as the familiar G-matrix of Wilson, Decius and Cross (1); this form, but not the derivation, of the G-matrix has been previously written by Malhiot and Ferigle (58). However, in the present context, the G-matrix elements may be dynamic variables. From the outset, quantum mechanics furnishes a natural and efficient origin for the G-matrix; in classical mech anics the origin of the G-matrix can be cumbersome. As could be expected, quantum mechanics leads to new terms without simple classical analogs; the ^ 's of (1.1.6) - local angular momentum? - contribute to the zero point energy and generate distortions in the average geometry of the molecule depending on the vibrational state of the molecule (within the harmonic oscillator approximation.) (We are not prepared to endow these higher order kinetic energy terms with physical significance.) Equation (1.1.4) is nothing more than a generalized Laplacian operator; the simple derivation and form of (1.1.4) gives it some advantage, depending on objective, over the equivalent expression found in Margenau and Murphy (54) and the usual quantum mechanical textbooks (1, 57). In our notation, the usual generalized Laplacian reads: (1.1.7) - 99 -where \(3\ is the determinant of the G-matrix (including rotation and translation). The equivalence of (T.1.4) and (T.1.7) can be seen through the action of the operator (T.1.7) on the generalized coordinates themselves. With reflection, (T.1.8) brings (T.1.7) into the form (T.1.4); the converse, (T.1.4) to (T.1.7) is much more difficult. The transformation is not yet complete. The transformed wavefunction carries a density factor if it is to preserve its meaning as a probability distribution when squared (57). in (T.1.8) (the identity matrix) Consequently the generalized Schrodinger equation is frequently expressed in the form (1, 36). But the Cartesian wave function can be retained if its meaning and the role of the density factor is recalled. - 100 -(T-2) Separation of Rotation, Translation and Configuration The three following translational coordinates define the centre of mass of. the molecule; these are dynamic variables. A ?' v~ IAS. v Zo = /v\' Hoc /rioc^oc (1.2.1) where and Z??^ is the mass of the CC~ particle. The three rotational coordinates for a semirigid body appear to be difficult to define (56); however the 3N Cartesian vector quantities below will be shown to exhibit desired mechanical properties. -- /fo{(&-£),° ] Here, R , R , and R are presumed to be rotational ' x y z ^ coordinates. (T.2.2) - 101 -With these rotational and translational coordinates, the G-matrix takes the form: G = /G(translation) 0 0 \ 1 .0 G(rotation) 0 \ \ 0 0 G(configuration) I (1.2.3) When the potential energy is a function of configuration only, the Schrodinger equation, according to the above representation, separates into translational, rotational and vibrational parts. The separation of translation is rigorous but the separation of rotation depends upon the rigidity of the molecule. For rotation and translation the higher order kinetic terms, (T.1.6), are all identically zero; thus the decoupling is complete. The matrix form (T.2.3) is most easily derived by considering the translational (and rotational) G-matrix elements in a Cartesian vector form. Let t denote a generalized coordinate. [G(Xo,t), G(y.,i)j 6(i.,t)] = M' z« v«t [GM ,6M,G(H^i)] = E* fot)x(A*) where /[^ = ( Xoc-X0 ^ & — Y0 ; 2d J - 102 -Rotational and all configurational coordinates exhibit the property: ZK %t = (0,0,0) And all configuration coordinates exhibit the property EJ&£WA*) = 10,0,0) These properties can be verified by carrying out the indicated operations for the various coordinates. With very little labor it can be shown that G(translation) = KA' G(rotation) Ixx -I IxY I, "2£ and the most common configuration G-matrix elements have been tabulated elsewhere; otherwise these can be written via (1.1.5). The moment of inertia tensor, G(rotation), depends upon the orientation of the molecule and its instantaneous configuration; thus the separation of rotation and configur ation (or vibration) is not complete. Vibration-rotation - 103 -interactions are discussed by Wilson, Decius and Cross (1, see Chapter 11) and in greater detail by Bodi (56); however, both references employ the Eckart conditions and differ from this development in that respect. Because the Eckart conditions have not been invoked here, linear molecules are not exceptional cases (from the outset); moreover the equations derived thus far need not be constrained to rigid systems. However, the physical substance of this development, which depends on equations (T.2.2), the rotational coordinates, remains largely unexplored. Equations (1.2.2) have not been previously written but here they indicate a useful role in the analysis of molecular mechanics. The author suggests that these equations be named "the Harvey conditions" in memory of K.B. Harvey, director of this research. (1-3) The Vibrational Schrodinger Equation After the separation of rotation and translation the vibrational Schrodinger equation remains to be solved. An approximate solution adequate for many purposes can be obtained by replacing kinetic energy coefficients by either their average of equilibrium values and expressing the potential energy as a quadratic form in configuration-displacement coordinates, /Ot . Here it is assumed that - 104 -the molecule remains near its equilibrium configuration while it vibrates. To solve the vibrational Schrodinger equation, L>L^xl dfrdft dps V J/ Yr.3.1) when <Jij } ^-i and are all constant coefficients linear transformation of the form, Aft = Zj L*j (T.3-2) (T.3.3) such that (the identity matrix) (3*.3-4) f~ [_ — /\ (a diagonal matrix) (T.3-5) reduces (T.3-T) to a sum of independent quantum oscillators in the normal coordinates, Qj - 105 -Except for the terms J?j - \T.n $n Ljn d.3.7) the solution of (1.3-6) is a product of harmonic oscillator wavefunctions and the energy is a sum of harmonic oscillator energies (1). With nonzero S , standard methods for solving differential equations (42) indicate a solution as follows: (1.3.8) _ M s~\ //. —^ n fh y is the normalizing factor i\y is the ^ hermite polynomial (1.3-9) (1.3-10) (T. 3.1D (T.3.12) - 106 -For the wavefunctions (T.3-10) the expected value of the normal coordinates is not zero. For example. and the magnitude of the shift depends on the quantum numbers. The added zero-point energy is simply the potential energy of the molecule when it assumes its expected ground state configuration. However the physical significance of these higher order kinetic terms, given the approximations invoked, is not yet certain. - 107 -APPENDIX TWO: REDUNDANT COORDINATE SYSTEMS Redundant coordinate systems are sometimes needed for the analysis of molecular structure in familiar geometrical terms. This appendix deals with the question as to whether or not redundant coordinates imply related interdependencies for mechanical quantities such as force constants as has been suggested by Hubbard (40) and Gold, Dowling and Meister (4l). Crawford and Overend (39) have attempted to show that such relations for force constants are in fact arbitrary. All of these authors have treated the redundant coordinate system as though the coordinates were independent variables subject to constraints or redundancy conditions. Potential energy is expressed as a Taylor series in redundant coordinates. In their approach it is not clear that the Taylor series coefficients, formed by partial differentiation, can be called upon without substantial justification when the coordinates cannot be independent. Unfortunately the previously cited authors have not considered the propriety of the Taylor series coefficients when redundant coordinate systems are employed. Further, there is something intrinsically unsatisfying with an ab-initio expression of potential energy in terms of redundant coordinates. If necessary, one can accept the - 108 -notion of potential energy in terms of redundant coordinates but only as a derived or auxiliary quantity - not as an initial assumption or intrinsic property. In this appendix, the redundant coordinate system and all related quantities shall be strictly referred to a non redundant coordinate system. The significant step here is that the transformation matrix is not square and therefore not invertible in the usual simple way. Our approach provides the supplementary information needed to resolve the indeterminacy noted by Crawford and Overend (39). The two approaches are infact complementary rather than contradictory. (II-1) Transformations involving redundant coordinates As geometrical quantities defined on a set of points or atoms, the redundant coordinates ( p ) can be defined without question in terms of nonredundant coordinates ( "fc ). higher terms (II.1.1) Clearly every configuration of the molecule specifies exactly all of the non .redundant coordinates and all of the redundant coordinates. Consequently the relation between redundant coordinates and nonredundant coordinates is one-to-one, in terms of the molecular configuration. This means - 109 -that the inverse transformation exists even though the matrix (®^fc)0 no^ square! For infinitesmal distortions, equations (11.1.1) define an overdetermined system of linear equations; thus the nonredundant coordinates can be expressed in terms of the redundant coordinates. Let The matrix is the left hand inverse of the matrix «S , j/s's)VJs - Gofers) - x ( S^s) is a square matrix of full rank and (3N-6) x (3N-6), where N is the number of atoms. Therefore If the potential energy is properly defined in terms of nonredundant coordinates, 2V - F.jtitj substitution of (IT.1.3) provides an equivalent expression in terms of redundant coordinates. - 110 where the -matrix, (ir.i.4) represents the force constant matrix for redundant coordinates. As well G*= SG^ (II.1.5) ^-K- ~~ SL (II.1.6) where and are redundant coordinate representations of L, and c3 for nonredundant coordinates. Pseudo inverses of F% 5 and i—£ , noted as P]?. and respectively, can be written in the form <3^ - sistsyG (s%) s (ir.1.7) (II.1.8) (II.1.9) -111-Decius (8) has shown that r gives the potential energy in terms of generalized forces' - a fact later iterated by Cyvin (7). Decius and Oden (59) , because the inverse transformation is not needed to formulate f~~/z , have used the above relation to specify all of the constants involved in F~/z The eigenvectors of which correspond to zero eigenvalues span the null space of the S-transformation (impossible configurations of the molecule). Let Ny = 3N-6 the number of genuine vibrations Np. be the number of redundant coordinates be the orthogonal matrix with diagonalizes where j"-1 is diagonal but of the diagonal entries are zero. Let XJ% be partitioned as a matrix "Uk= (U\U«^ such that G^U-UP and 0KUZ=0 where LA.^. are eigenvectors with zero eigenvalues and are eigenvectors with nonzero eigenvalues noted by the diagonal matrix ' (see reference 42) (II.1.10) therefore (II.1.11and UtFtU^o (ii.1.12) - 112 -Equations OCT. 1.11) and (II.1.12) which follow from (11.1.4) and OCT. 1.10) provide the relations needed to define the so called indeterminate force constants arising from the redundant coordinate system. It will be observed that the equations of this thesis are correct as written if the U-matrix is understood to be composed of the eigenvectors of the C3^_ matrix with nonzero eigenvalues. Here the U-matrix need not be square but its columns remain orthogonal. In summary once it is realized that all coordinate transformations covering the possible configurations of a molecule are one-to-one whether the transformation generates redundant coordinates or not, then the redundancy conditions on force constants follow via the inverse transformation. - 113 -APPENDIX THREE: THE EXPONENTIAL MATRIX The exponential matrix is defined like the exponential function, 0Ks= Z Knsrt/ni where K is a square matrix and 5 is an ordering parameter. The ordering parameter enables us to establish the essential properties of the exponential matrix with little algebraic labor. For example, the formula ^ = K<Z*S = <ZKsK o) s is easily derived. More general properties can be established by considering the matrix function ^ ~Ks HB .<zr*_a<z where is, for the present purpose, a square matrix; K and s are as defined above. Matrix differentiation gives m = LKjH]= sue," The (H+l )th derivative can be expressed in terms of the Y\ — derivative, - 114 -Consequently In our applications we are interested in the H-matrix as a perturbed form of the _fL -matrix and conversely. The H and jfL matrices are related through the K-matrix and ordering parameter by the expansion about S=0. + higher terms recall that: Limit H — JS-*O When is the identity matrix, the above equations show that (& — X" (the identity matrix) or (<z.K0"'= <2>*s - 115 -As well, when the K-matrix is skew symmetric (skew hermitian) , 4-K = -K the exponential form is orthogonal (unitary). Consequently the __fl_-matrix can be viewed as a diagonal matrix composed of the eigenvalues of H; in this case the orthogonal matrix C^/ represents the eigenvectors of H. The exponential matrix may appear in the literature of mathematics (60) usually in connection with the theory of lie groups. Applications of the exponential matrix have not been extensive in chemistry nor in any area where mathematics is more a tool than a logical art. Yet its properties indicate that it can be a useful tool. 


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