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Pseudorotational energy spectrum of an octahedral molecule with 1/2-integer spin Douglas, Jessica Lynn Rogers 1998

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T H E P S E U D O R O T A T I O N A L E N E R G Y S P E C T R U M OF A N O C T A H E D R A L M O L E C U L E W I T H 1 / 2 - I N T E G E R S P I N by JESSICA LYNN ROGERS DOUGLAS B.Sc. (Mathematics) University of British Columbia, 1996 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF M a s t e r p f . S c i e r i . e e _._..:' in THE FACULTY OF GRADUATE STUDIES Department of Physics and Astronomy We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1998 © Jessica Lynn Rogers Douglas, 1998 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for refer-ence 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 or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University of British Columbia Vancouver, Canada Date OcUlaer S~, frta-Abstract The Jahn-Teller effect, which underlies non-linear molecular energy spectra, has been the focus of many studies and measurements since its discovery in the 1930's. In this work we theoretically calculate the pseudorotational energy spectrum of an octahedral molecule using a two parameter fit. In particular we calculate the doubly degenerate energy levels belonging to the Fg <S>(e©£2g) problem. The Born-Oppenheimer Hamiltonian is constructed and solved on the canonical line bundle over HP(l) . According to the strong Jahn-Teller approximation, the non-Jahn-Teller active vibrations are built upon the solutions to the Born-Oppenheimer Hamiltonian. This complete spectrum is compared to spectral data taken using IrFg. 11 Table of Contents Abstract ii Table of Contents iii List of Tables v List of Figures vi Acknowledgement vii Chapter 1. Introduction 1 Chapter 2. The Born-Oppenheimer Approximation 3 2.1 The Total Hamiltonian 3 2.2 Redefine Hmoi acting on bundle sections 4 2.3 The Born-Oppenheimer Hamiltonian 6 2.4 The Strong Jahn-Teller Approximation 7 Chapter 3. The Jahn-Teller Effect 9 Chapter 4. Analyzing the electronic Hamiltonian 15 Chapter 5. The Spectrum of the Laplacian over certain Bundles 23 5.1 The Complex Projective Space 24 5.1.1 The Pullback of a vector bundle over its principal bundle 24 5.1.2 Two Theorems 26 5.1.3 The Spectrum of the Laplacian on CP(n) 29 5.1.4 Completeness of the spectrum on CP(n) 31 5.1.5 Comparison of the spectrum on CP(1) with that o n 5 2 33 5.1.6 The Pullback of a connection over CP(n) 36 5.1.7 The Spectrum of the Laplacian on the canonical bundle 39 5.1.8 Completeness of the spectrum over the canonical bundle 40 5.2 The quaternion Projective Space 41 5.2.1 The Spectrum of the Laplacian on HP(n) 41 5.2.2 Completeness of the spectrum on MP(n) 43 5.2.3 Comparison of the spectrum on HP(1) with that of S4 44 5.2.4 The Pullback of a connection over HP(n) 45 5.2.5 The Spectrum of the Laplacian on the canonical bundle 46 5.2.6 Completeness of the spectrum over the canonical bundle 48 Chapter 6. Fitting of Theory to Existing Data 49 Chapter 7. Conclusion 55 iii Table of Contents Bibliography 57 Appendix A. Basic Group Theory 59 Appendix B. Fibre Bundles 66 B. 1 Basic Definitions 66 B.2 Constructing new vector bundles 69 B.3 Characteristic Classes 72 Appendix C. The quaternion Numbers 80 iv List of Tables 3.1 Character table of 0' 12 3.2 Determining g' — g1 14 6.1 Non-Jahn-Teller active frequencies 49 6.2 The First Five Calculated Jahn-Teller Frequencies 50 6.3 Calculated frequencies less than 1100 cm - 1 51 6.4 Major Energy Bands measured by Bernstein and Webb with assignments 54 A.l A 3 Element Multiplication Table 59 A.2 The Csv character table 62 v List of Figures 4.1 Octahedral normal mode coordinates 16 B.l A real line bundle over Sl 68 VI Acknowledgement I would like to offer grateful thanks to Dr. Gordon Semenoff and Dr. Alexander Rutherford, for their guidance and support from the initiation to the finish of this project, and of course many thanks to Linda, Diane and my dear Sandy M. for just being there. Most importantly, this work is dedicated to my late father, Roy, who was by far the most swinging mathematician I have ever known. vn Chapter 1 Introduction In 1937, H. A. Jahn and E. Teller [14] showed by exhaustive group theoretic calculation that any symmetric atomic configuration of a non-linear molecule is not at an energy minimum. The following year, H. A. Jahn [15] showed that the same is true for any non-linear molecule with spin. For each molecular symmetry group, Jahn and Teller calculated normal mode coordinates on which the molecule finds a minimum potential energy. These normal mode coordinates transform according to one or more irreducible representation of the symmetry group. In particular, an octahedral molecule with half integer spin exhibiting four-fold degeneracy is referred to as the Ts <8> (eg © tig) problem. Tg is the irreducible representation to which the potential energy transforms and eg and tig are the two irreducible representations to which the normal mode coordinates transform. The Jahn-Teller effect has been studied extensively over the years, and both the Tg <S> eg and Tg ® tig problems have been solved exactly [3, 16, 19, 4]. However, the equal coupling case of Tg <g> (eg © tig) has only been approximated and modeled on computer. A complete solution has not been mathematically calculated to this date. The goal of this work is to theoretically calculate the vibronic energy spectrum of an octahedral molecule with half integer spin and four-fold degeneracy, solving the Tg ® (eg © tig) problem, and compare the result to existing data. The Jahn-Teller normal mode coordinates determine a nuclear coordinate space in which the molecule's minimum potential energy submanifold exists. The Born-Oppenheimer approxima-tion is applied in order to break the total molecular Hamiltonian into an electronic part which 1 Chapter 1. Introduction is parametrically dependent on nuclear position and a nuclear part. The problem then becomes one of determining electronic eigenvalues on a four dimensional configuration space. This calculation is carried out in the quaternion number field, which has been previously asso-ciated with physical problems involving half integer spins [7], in order to reduce the number of degrees of freedom. Furthermore, due to the global twisting of the electronic eigenspace, fibre bundles are employed to solve the eigenvalue problem in a more geometrically simple manner. Pseudorotational oscillations occur when the nuclear coordinates remain on the previously determined minimal energy submanifold. Transverse oscillations occur when the molecule's nuclear positions cause it to oscillate in the configuration space perpendicular to the minimal submanifold. In the strong Jahn-Teller coupling approximation, the transverse oscillations are at high frequencies. Thus, their contribution to the total spectrum is solely in the high energy range. It has also been found that these transverse energy levels can be modeled well using computer algorithms [20]. This work, therefore, concentrates on determining the low energy pseudorotational spectrum only. Finally, the solution to the electronic Hamiltonian is multiplied by a scale factor corresponding to the radius of the submanifold and added to an arbitrary zero point potential energy. This gives the pseudorotational energy levels which are fitted to known data [2] to determine these two parameters. 2 Chapter 2 The Born-Oppenheimer Approximation 2.1 The Total Hamiltonian The Born-Oppenheimer approximation is useful when dealing with a many body problem with a very large mass ratio, such as that of nuclei to electrons. Rather than attempt to calculate the solution of a complex Schroedinger equation on a relatively straightforward Hilbert space, we will instead use a less complicated Born-Oppenheimer Hamiltonian consisting essentially of a Laplacian and an effective potential, on sections in a particular fibre bundle. The total molecular Hamiltonian for a molecule with K nuclei of mass, M, and k electrons of mass, m, is, K k H™i = ^ E AP. - h E Arj + V(p, f) + Vnuc{p). (2.1) i=l j= l The coordinates are defined as follows: P={PI,P2,---PK)£NCR3K where p is the position vector of all K nuclei in three dimensional Euclidean space and N is the subspace of nuclear configurations in R3^. Similarly, f is the k-vector defining the configuration of the k electrons, and X is the subspace of R3k corresponding to electronic configurations. In Equation 2.1, V(p, r) is the potential function incorporating to all pairwise Coulomb interactions between two electrons or between a nucleus and an electron. Vnuc(p) is the potential function corresponding to all nuclei-nuclei interactions. 3 Chapter 2. The Born-Oppenheimer Approximation Since we are using Euclidean spaces it is convenient to define the following metrics for the configuration spaces; N has the metric g^ = 2M5ij X has the metric hij = 2mSij. Substituting the new metrics into Equation 2.1 we have, Hmoi — -A(w) - A(x) + V + Vnuc. (2.2) The Hamiltonian in Equation 2.2 acts on eigenfunctions, 4> € L2(N x X;C); Hmol : L2{N xX;C)^ L2{N x X;C). L2(N x X; C) is defined as the set of all square integrable functions, <fi : N x X —> C. A function, f, with domain, D, is square integrable if fD | / | 2 = 1. We are now in a position to separate the total Hamiltonian into a nuclear and an electronic component. The latter is denoted, Hei, and is parametricaily dependent on nuclear coordinates; Hel(p) = - A W + V(p,-) : L2(X;C) -+ L2(X;Q. (2.3) Thus, Hmoi has been decomposed into, Hmol = -A{N) + Vnuc + / Het(p)dp (2.4) j x where fx refers to a direct integral over the electronic configuration space, X. 2.2 Redefine Hmoi acting on bundle sections We wish to define Hm0\ as acting on sections of a trivial fibre bundle. In order to accomplish this goal, first define the canonical isomorphism, h, between L2(N x X; C) and L2(N; L2(X; C)). For any function, f €L2(NxX; C) the isomorphism, h : / ^ h{f) where h(f) € L2(N; L2{X; Q ) . 4 Chapter 2. The Born-Oppenheimer Approximation h(f) is defined by h(f) : x t-> f(x, •) Vx € N such that the inner product of < / , g >€ L2(N; C) for &llg€L2(N;L2(X;C))-Second, define the trivial fibre bundle, e: e: L2{X;C) L2(X;C) x N N Third, construct an isomorphism, h, between square integrable functions in L2(N;L2(X;C)) and square integrable sections of e. Let CT be a square integrable section of e; a : x i-» (x, CT). For all x € N, where a 6 p~1(x) is some function in L2(X; C), define the isomorphism, h : a i—> a. Thus, there exists an isomorphism, h o h : L2(N x X;<C) —> L2(e) and we can view Hmoi a s acting on square integrable sections of the the trivial bundle; Hmot : L2(e) - L2(e). (2.5) Next we wish to define a sub-bundle, rj C e which will be more topologically useful. Suppose that Hei(p) has one or a direct sum of eigenspaces, Z{p) with associated eigenvalues which are bounded away from the rest of the total spectrum. Then we can define a new fibre bundle, 77: 77: Z ( - ) c >E N where E = {{p,4>) E N x L2(X;C)\^) € Z{p)} <mdp{p,cf)) = p. Each fibre of rj is a subspace of a fibre of e and thus rj is a sub-bundle of e, 77 C e. However, it is important to note that since iV is not necessarily contractable and Z is assumed to be finite dimensional, 77 is in general a non-trivial finite dimensional fibre bundle. Since 77 C e, it follows that the set of square integrable sections over 77 is similarly, L2(r)) C L2(e). 5 Chapter 2. The Born-Oppenheimer Approximation 2.3 The Born-Oppenheimer Hamiltonian We know that Hmoi acts on L2(e). The Born-Oppenheimer Approximation states that Hmoi restricts to L2(rj) C L2(e). This means that if P is the orthogonal projection operator onto L2(rj) then the construct, P^HmdP, will be small. There have been some approximations made of P - L i J m o ; P by Hagedorn [12] for diatomic molecules, but it has been found that this approximation is valid at low energies and becomes increasingly worse at higher energies. Assuming that the Born-Oppenheimer approximation is valid, we construct the Born-Oppenheimer Hamiltonian, HBO'I HBO = PHmolP : L2(rj) - L2(r?). (2.6) where P = f P{p)dp and Hmoi is defined in Equation 2.2. Thus, HBO = P(p)(-d} -% + V + Vnuc)P(p) = P(p)(-d} + 2P(dp~P(p))dp- + d2P{p)) + Hel + Vnuc Thus, HBO 1S a- second order differential operator with leading symbol, —gij, (from the —d2~ term) where gtj is the metric on N. L e m m a : [10] If D : C°°(£) —> C°°(^) is a second order differential operator on sections of a bundle, £, and D has the leading symbol, g^, where gij is some Riemannian metric on the Base space, then there exists a unique connection, a;, on £ such that D = - A w + V. V is locally a n n x n matrix of potential functions if the fibres of f are n-dimensional. By this lemma, it is clear that the Born-Oppenheimer Hamiltonian can be expressed as HBO = - A „ + Veff (2.7) Choose a local orthonormal basis for sections of 77, {4>m}. Then the connection, LO, can be calculated; Chapter 2. The Born-Oppenheimer Approximation where A? is the d x d matrix whose elements are < </>m, -^j4>n >L2(X;C) a n d 4>m-, fin are eigen-kl values of He{. TJkl are Christoffel symbols of the base space of r/ which are for most smooth Base spaces, zero. Thus, the effective potential in Equation 2.7, Ve/f, is the sum: I d <92 d Veff = [Hel] + Vnuc + [--gVUjg-l(j-:g) - glJ < (f>m, Q-JgTJ^n > + < ^ ( ^ W j ) + gtJUiUJj] (2.8) where [Hel] =< (f>m,Hei4>n > is a d x d, p-dependent matrix, g — det[gij] and {pi} are the coordinates on N. 2.4 The Strong Jahn-Teller Approximation We now apply the strong Jahn-Teller approximation to the problem, which means that we view the first term in Equation 2.8 as dominant, and Veff becomes, Veff = [Hel(pj] (2.9) Let the symmetric nuclear configuration be p~o € N. Then [Hei(p~b)] =< 4>m(pb), Hei{po)(f)n{pQ) >. If N is a contractable neighborhood of p"5 then r\ over N is a trivial bundle. Note that [Hei(p)] is not necessarily generally diagonalizable over N. Now we puncture iV by removing p"o to get N' = N- {pb} C N. Veff will be minimized on a sub-manifold of non-zero dimension and non-trivial topology. A direct result of this non-trivial topology is the existence of pseudorotations on this minimum energy submanifold. It is these pseudorotational energy levels that this work attempts to calculate, and thus we wish to find the submanifold on which Veff is minimized. The strong Jahn-Teller approximation means that we find the minimum eigenvalue and thus the minimum Veff and assume an infinite potential well elsewhere. Therefore, we have a successfully restated the problem of finding the spectrum corresponding to an octahedral molecule. We now wish to find the eigenvalues of the negative Laplacian, — Aw, on the non-trivial fibre bundle 7 Chapter 2. The Born-Oppenheimer Approximation over the minimum energy submanifold. In this case Vefj is a constant zero point energy and can be easily added to the calculations in the final stage. The solution to this simplified problem corresponds to the pseudorotational spectrum, referring to pseudorotations of the nuclear coordinates on the minimum energy submanifold. The re-mainder of the complete molecular spectrum is made up by the transverse oscillations, referring to nuclear coordinate oscillations perpendicular to the minimum energy submanifold. However, the transverse oscillatory spectrum have been shown to be well modeled to a first order approximation by Simple Harmonic Oscillators [20], with characteristic frequency de-pendent on potential. In the strong Jahn-Teller approximation it is therefore reasonable to assume that there frequencies are quite large and thus the lower energy range is dominated by pseudorotational energy levels. The complete molecular energy spectrum is composed of the transverse oscillations built in SHO towers on each pseudorotational energy level. Given that our two assumptions of the Born-Oppenheimer approximation and the strong Jahn-Teller approximation are valid, the solution to the problem of the molecular spectrum corre-sponds to finding eigenvalues of HBO o n the square integrable sections of 77' over N'. From Equation 2.7, HBO is the sum of —A and an effective potential, Veff- If we now separate our restricted nuclear configuration space, N', into submanifolds on which the eigenvalues of Hei are constant, then from Equation 2.9 Vefj is also necessarily constant on each of these submanifolds. In Chapter 3 the specific geometry of these submanifolds is determined for the problem of an octahedral molecule, by calculating the eigenvalues of the electronic Hamiltonian, He[, and the manner in which the eigenvalues depend on nuclear coordinates. 8 Chapter 3 The Jahn-Teller Effect In 1937 it was proven by Jahn and Teller [14] that a polyatomic molecule with orbital degeneracy arising solely from symmetry will not be energetically stable. Later that year H. A. Jahn generalized this theory [15] to include molecules with spin degeneracy. This section serves as a summary of their work. Initially, it must be demonstrated that if the molecule's electronic energy is linearly dependent on nuclear displacement from symmetry in any direction, then the symmetric configuration cannot be stable. In other words, linear dependence on any spatial variable implies a non-zero slope of the energy curve at symmetry, and thus a perfectly symmetric molecule cannot be at minimum energy. Let the nuclear configuration of the molecule be denoted p, and the symmetric nuclear config-uration, pQ. p can then be expressed as a linear combination of displacements about po; P = Po + J^PiVi, (3-1) i where r\i are small normal displacements of the nuclei which can be chosen to transform ac-cording to irreducible representations of the molecular symmetry group [27]. A Taylor expansion of the electronic Hamiltonian from Equation 2.4 gives, Hei = Ho + J2 Vii^i + Yl VijWmrij + ••• (3-2) i ij where f represents the electron coordinates, Vj and Vy are entirely dependent upon electron 9 Chapter 3. The Jahn-Teller Effect positions, and electronic kinetic energy is included in Ho. H is, of course, dependent on nuclear configuration, p and HQ = Hei(po). At the degenerate symmetric state we have the following eigenvalue equation; Ho0„ = EQ(/)U, (3.3) where {$„} is a set of eigenfunctions for H(po). At any nuclear configuration, p^po, the energy levels may be multiply degenerate, split into a set of energies, {Ea}. The only way po will be stable is if every Ea has a minimum at p = pb in which case Ea becomes a perturbed energy level; Ea = EQ + E1a + --- (3.4) where the first order perturbation term, Ea, is due to the linear terms, ]T^ V ^ r ) ^ , in the total electronic energy. It follows from perturbation theory [5] that these terms, Ea, are the diagonal elements of the perturbation matrix; Mva = Y,m [KVitadT, (3.5) or, Mul/ = ^ 2ru fVil^dr. (3.6) i •* Unless the integrals in Equation 3.6 are zero, the perturbation energies will be linearly dependent on m. However, at a minimum energy, -^-H = 0. Therefore, if the perturbation terms, {Ea} (and therefore the matrix elements, M w ) are linearly dependent on some rn the symmetric state is non-stable. By showing that the integrals in Equation 3.6 are non-zero, we prove that the symmetric state is non-stable and thus demonstrate the existence of the Jahn-Teller effect. A background summary of the necessary group theory and derivations is provided in Appendix A. From [6, Chap. 5], if we have an integrand consisting of two factors, fAfBdT, (3.7) 10 Chapter 3. The Jahn-Teller Effect the integral will be non-zero if it is invariant under all symmetry transformations which will be the case if it forms a basis for the totally symmetric representation of the group. It follows that the integrals in Equation 3.7 will be non-zero if the representation of the direct product, T/tBj contains the total symmetry representation. We know from Equation A.6 that the number of times the total symmetry irreducible repre-sentation occurs in TAB is, ^ = \Y,^B{R)XI{R), (3.8) R where h is the order of the group and Xi{R) IS the character of the group element, R, in the Ith irreducible representation, xi (R) — 1 f° r &U group operations, R, where xi is a character of the totally symmetric representation. Further, from Equation A.7; XAB(R) = XA(R)XB(R) (3-9) for. direct product representations. Thus, ai = \TjXA{R)XB{R). (3-10) hR However, from Equation A.4, Y,Xi{R)x0{R) = h5lj (3.11) R and substituting Equation 3.11 into Equation 3.10 we find that, ai = 8AB (3.12) Therefore, in order for the direct product, /A/B, to contain the total symmetry representation, the representation of one factor, TA, must be or contain the same representation as TB- We now apply this result to the matrix elements from Equation 3.6 which we wish to be non-zero. In this case \4>\2 will transform as the symmetric product of the representation of <fi with it-self, [4>1]. Therefore, if Vj(f) and [</£] contain the same irreducible representation then the 11 Chapter 3, The Jahn-Teller Effect matrix elements are non-zero and the symmetric configuration is non-stable. In [14] all irre-ducible representations of [(f)2] are calculated, for <j> transforming according to any irreducible representation of any molecular (point) symmetry group. In the follow-up paper to [14], H. A. Jahn considers degeneracy arising from electron spin. It is shown that if the molecule has an even number of electrons then the above argument from [14] applies and the symmetric state is unstable. However, if the molecule has an odd number of electrons, degeneracy will still be broken by moving away from symmetry, unless the degeneracy is a twofold one of the sort introduced by Kramers [17], known as Kramers doublets. The molecular symmetry group under study in this work is the double group, 0'h belonging to an octahedral molecule with spin. The addition of a group operation of a rotation by TT generates the double group from Oh and corresponds to the operation of reversing spin. Table 3.1 is the complete character table for Oh [22]. 0' Ax A2 E Tx T2 r6 r7 r8 (xrs(g))2 Xrs{g2) ix(g)2} Ax+E + T2 E 1 1 2 3 3 2 2 4 16 4 6 6 R 1 1 2 3 3 -2 -2 -4 16 4 6 6 4C3 4C|i2 1 1 -1 0 0 1 1 -1 1 1 0 0 ACi 4C3R 1 1 -1 0 0 -1 -1 1 1 1 0 0 3C2 3C2R 1 1 2 -1 -1 0 0 0 0 -4 2 2 3C4 oO^R 1 -1 0 1 -1 V2 -V2 0 0 0 0 0 3Q3 3C4R 1 -1 0 1 -1 -V2 ^/2 0 0 0 0 0 6C2 GC^R 1 -1 0 -1 1 0 0 0 0 -4 2 2 Table 3.1: Character table of O' 12 Chapter 3. The Jahn-Teller Effect The integrand of the perturbation matrix elements (Equation 3.6) are found to transform ac-cording to V[(f>2} for an even number of electrons, and as V{</>2} for an odd number of electrons, where {4>2} refers to the antisymmetric product of 0's representation with itself. Eigenfunctions of a molecule with spin, {^}, are not L2 functions like {</>}. The Hilbert space containing {cf>} is anticommuting. From Equation A.9, the antisymmetric product of an irreducible represen-tation, T, with itself is defined by its characters as; {x(g)2} = \[(x(g))2-x(92)} (3-13) for any group element, g. Since we are interested in the molecule, IrF6, and Ir has atomic number, 77, IrF6 has an odd number of electrons. Thus we need to find the antisymmetric product of greater than twofold degenerate representations of the octahedral double group. We want more than twofold degenerate representations since unsplitable Kramers doublets may well exist. It should be noted here that 0'h is O' with a center of inversion, thus the total number of classes of 0'h is 16, the second 8 of which are simply the first 8 followed by an inversion, i. In the irreducible representations, F\ through Fs of 0'h, the characters of the second 8 classes are duplicates of the characters of Ti through F$, for O'. Also, since performing the operation of inversion twice is equal to E, the identity element, the following calculations on O' apply identically to 0'h. The characters of the square of the group elements (xr8 (#2)) a r e found using the g2,s determined in Table 3.2. The group element, R, refers to a rotation by 27r radians, which is differentiated from the identity element, E, which refers to a rotation by 0 or 4TT radians. For each group element, g, in a given class, Table 3.2 determines g' where, g' = 92 by calculating the total rotation of g2 and any integer number of 4ir rad rotations. 13 Chapter 3. The Jahn-Teller Effect 92 E2 R2 (Cz)2 (elf (C2)2 (C4)2 cl)2 (C2)2 total rotation (rad) 8TT 4ir 3 8TT 3 2TT TV 3TT 2vr reduced rotation (rad) 0 0 4-7T 3 2n + f 2TT 7T 27T + 7T 2TT resulting g' E E Q2 C3i? R c2 C2R R Table 3.2: Determining g' = g2 {T82} is thus calculated in Table 3.1 and is found to be reducible to; {Yl2} = Alg + Eg + T2g. (3.14) as shown by the equality of the last two lines of Table 3.1. Therefore, if Vi(f) transforms according to any of these irreducible representations contained in {02} then the integrals in Equation 3.6 are non-zero and the symmetric molecule is unstable. However, since Vi(f) transforms according to the molecular symmetry group, if {4>2} contains one of the irreducible representations of the molecular group then the integral is non-zero. Equation 3.14 shows that this is the case. Furthermore, the above calculation determines the normal mode coordinates which span a configuration space in which the molecule may reach a lower energy level than that at symmetry. For an octahedral molecule with spin, this space is a 5 dimensional nuclear configuration space with two coordinates, ei and e2 transforming according to the two dimensional representation, Eg, and three coordinates, T\, T2, and T3 transforming according to the three dimensional representation, T2g. 14 Chapter 4 Analyzing the electronic Hamiltonian We have already determined in Chapter 2 that an octahedral molecule will lower its poten-tial energy by distorting along normal mode coordinates which transform according to the 2-dimensional E and 3-dimensional Ti representations. Octahedral molecules with spin still vibrate along these normal coordinates, however, we are now concerned with a four-fold degen-eracy of two unsplitable Kramers doublets. This is known as the Tg <8> (e © t-i) problem referring to the Jahn-Teller effect for an electronic Hamiltonian transforming according to the T% representation with coupled linear vibronic in-teractions between the E and T2 representations. Let us denote the symmetrized normal coordinates transforming according to the E and T2 representations as {Qr7} where each coordinate possesses the following transformation proper-ties [3]; Q , ~ i = ( 3 2 2 - r 2 ) , Q e ~ x 2 - i / 2 , Qt ~ 2yz, Qn ~ 2xz, Q^ ~ 2xy. Figure 4.1 displays corresponding cartesian coordinates as follows; (i) Eg displacements {Qe), (ii) Eg displacements (Q£), (hi) T2g displacements (Qf), (iv) T29 displacements {Qn), (v) T29 displacements (Q^). We recombine the total Hamiltonian from Equation 2.2 in the following form; Hmol = H{r) + V(r,p) + f(p) (4.1) 15 Chapter 4. Analyzing the electronic Hamiltonian 1 Figure 4.1: Octahedral normal mode coordinates. where H(r) includes electronic kinetic and electron-electron potential terms, V(f, p) how con-tains electron-nuclei and nuclei-nuclei Coulomb interactions, and T(p) is the nuclear kinetic energy term, —A<Ny Now we expand V(f,p) in terms of the symmetrized displacements, {Qr7}> to second order; V{r,p) ~ V{r,pQ) + £ ( —)0(pi - p0i) + - Y,ij-7-)o{pi - P0z)(pj - POj) (4.2) ^Pi 2^8(n5pj-where the zero subscript means evaluate at the symmetric nuclear configuration. By substituting Equation 4.2 into Equation 4.1, Hmoi becomes, Hmol=f(p) + U(f,p) (4.3) U(f,p) = H(f,p-0) + ^Vrj(f)QTy + ^ E E ^r l 7 1r2 7 2(*0Qr l 7 1Qr a 7 2 (4-4) r7 1 1 7 I 1 2 7 2 SVfopb) _ S2V(f,pb) where Hftfc) = H(r) + V(f,p0), VTi(f) = ™ 2 l a n d W^r^ (f) - 6Qr^6Qr 71 ^72 16 Chapter 4. Analyzing the electronic Hamiltonian Suppose now that we classify the electronic basis states, ipnir), by the irreducible representa-tions, T7 where the total eigenvalues are made up by, / k=l In the case where all f electronic wave functions belong to the same irreducible representation, the Wigner-Eckart theorem states that matrix elements of any operator with these functions are not independent. Thus, <^71(Ol^r7(*0l#72(r! >=<r|Fr |rxr7r72 |r71 > (4.5) where < r |Vr | r > is the reduced matrix element independent of 71, 72, or 7, which shall be denoted, Vr, for convenience. < r 7 r 7 2 | r 7 l > are the Clebsch-Gordan coefficients. Substituting Equation 4.5 into Equation 4.4 and summing over the appropriate normal mode coordinate representations; U ^Y^VrQrA, + \ E E ^ F i x T2){Qri x QrJrA-, (4"6) r7 riT2 r7 where Cr7 are matrices composed of the Clebsch-Gordan coefficients, Wr(r i x ^ ) is the reduced matrix element of the operator, WYX r2 from Equation 4.4, and the zero point energy is taken from Equation 4.4, < tp^\H{r,pb)\ip^- >= 0. Separating the totally symmetric tensors from Equation 4.6, i.e. those proportional to CAXI and adding them to the kinetic term from Equation 4.4 we get the totally symmetric part of the Hamiltonian; Ho = ( S 2iw7TT^7 + ^E E W ^ x W r x Q r , ) CM. (4.7) \r7 [ ' r7 1^2 / Changing to mass weighted coordinates to remove the M(V)'s and denoting the eigenvalue of | |W/li(ri x T2)|| in the new coordinates by u>r we obtain, #o = ^ E ( A ( * ) r +^Q2TJ)CA1. (4:8) 1 r . 17 Chapter 4. Analyzing the electronic Hamiltonian Finally, adding the totally symmetric term from Equation 4.8 to the rest of the Hamiltonian from Equation 4.6 and taking into account only linear vibronic couplings we find the electronic Hamiltonian, tfe/ = ^ ( A ( x ) r +u:lQli)CAl+YJVvQvA, (4.9) The Clebsch-Gordan coefficients given by a basis of electronic states of the Ts term have been determined as follows [3]. The basis of states are: |ei > = 7 ^ ( ^ 1 / 2 + ^ - 3 / 2 ) |e2 > = -y=(—0-i/2 - #3 /2) |e3 > = - ^ ( - ^ 1 / 2 + ^ - 3 / 2 ) |e4 > = -T=W>- l /2 -#3 /2 ) and the Clebsch-Gordan coefficients are, 0 0 - 1 0 Ci = Cz = C5 = 0 0 0 - 1 - 1 0 0 0 0 - 1 0 0 0 1 0 0 1 0 0 0 0 0 0 - 1 0 0 - 1 0 1 0 0 0 0 - 1 0 0 0 0 - 1 0 0 0 0 1 c2 = c4 = CA = 0 0 -1 0 0 0 0 -1 % 0 0 0 0 2 0 0 0 - i 0 0 1 0 0 0 0 0 0 1 0 0 - 2 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 (4.10) (4.11) It has been demonstrated [7] that time-reversal invariant fermionic systems have a natural quaternion basis, and thus we wish to make the following basis change [23]. If \J, m > is the 18 Chapter 4. Analyzing the electronic Hamiltonian basis for a spin J multiplet, m = —J. . . J, then the time reversal operator acts as follows: e[a\J,m>] = (-l)J-ma*\J,-m> V a <E C (4.12) which leads to a canonical quaternion basis of the Hilbert space with half-odd integer spin, J; et = | J, J - 2/ + 2 > V / = 1, 2 . . . J + ]-. (4.13) For example, if we wish to study a Hilbert space of a molecule with spin J = | , then / = 1, 2 and the quaternion basis is; ,3 3 i 3 ! ,A ^A^ e i = l 2 ' 2 > ' e2 = l 2 ' ~ 2 > ( 4 1 4 ) where, ee,=e|§, !>-|§,-§> (4.15) ^ ,3 1 i3 1 Therefore, we now change from the {|e; >} basis, to a basis consisting of {'03/2) "0-1/2} along with the Time Reversal Operator, 9 . In this way we have i^xj2 — ©^-1/2 a n d "0-3/2 = ©^3/2-19 Chapter 4. Analyzing the electronic Hamiltonian From Equation 4 we have the new basis in terms of four-vectors in {|ej > } , i x ^3/2 = ~7|(le2 > +l e4 >) = ~T= ^1/2 = —7^(|ei > - | e 3 > ) = ~7f 1 1 ^ - i / 2 = " ^ ( l e 4 > —|e2 > ) = —7= V2 V2 ^-3/2 = ^ ( | e i > + h » = ^ 0 1 0 1 1 0 - 1 0 0 - 1 0 1 1 0 1 0 (4.16) Now, using Equation 4.11 and Equation 4.16 we change bases of the Clebsch-Gordan coefficients in the following manner: C l ^ / 2 = ^ = C l l ^ / 2 = ^ 0 - 1 0 - 1 0 - 1 0 1 -^3/2 = ^-1/2-20 Chapter 4. Analyzing the electronic Hamiltonian Thus, in the new basis we have, - 1 0 0 1_ where Herrno(2,M) is the set of 2 x 2 Hermitian quaternion matrices with zero trace. Ci = eHerm0{2,M) (4.17) Following the example of Equation 4.17, the remaining four Clebsch-Gordan matrices are cal-culated in the new basis and are all elements of Hermo(2,M): C2 = CA = 0 1 - 1 0 0 3 -J 0 C'3 = c5 = 0 k -k 0 0 % -i 0 It is easily verified that in both the quaternion and real bases, the Clebsch-Gordan coefficients behave consistently. That is, C;Vj = CiV'j for all i £ {1, 2, 3} and j = 3/2,1/2. The span of the Clebsch-Gordan matrices over R is now, ^2 QiCi = i = l 91 92 + <?3« + Q4j + q$k 92 - 93« - QiJ - 95& ~9i which is the general form of any element of Hermo(2,M). Constructing the Hamiltonian now from Equation 4.9 we have, Hei = 5^1(9? + 92) + huH<ll + 94 + 95) + 91 Vk92 + VT(93* + 94j + 95*0 VE92 - Vr(93« + 94j + 95*0 \u2E{q\ + ql) + 5^ (93 + 94 + 9s (4.18) When one coupling constant is dominant, the calculation is carried out assuming the dominant constant is one, and the other is zero. However, it has been found that equal Jahn-Teller coupling holds true for many systems [20]; LOE = LOT = LO and VE = VT = V. Once the equal coupling calculation is successfully carried out, it is assumed that further results between equal and zero coupling can be extrapolated from the existing calculations. 21 Chapter 4. Analyzing the electronic Hamiltonian The eigenvalues from Equation 4.18 are now £«?• \ti Thus, with the eigenvalues depending solely on YM=I ih o r *n e radius of an S4, we have a configuration space of 5 4 C Hermo(2,M). Further, S*4 is isometric to HP(1), which is the quaternion Grassmanian manifold, consisting of one-dimensional quaternion lines in H2. Also, Hermo(2, H) ~ M5, and so we have a configuration space of a round HP(l) with a variable radius canonically embedded in R5. 22 Chapter 5 The Spectrum of the Laplacian over certain Bundles It has been determined in Chapter 1 that we wish to find the eigenvalues of the Laplacian, A, where the eigenfunctions are sections of the canonical quaternion line bundle, rj, over the quaternion Projective Space, MP(n). Initially the necessary steps towards the solution of this problem will be taken using the canon-ical line bundle over the Complex Projective Space, CP(n), since this is a somewhat simpler task. Also the spectrum of the Laplacian on CP(1) itself, can be compared to known results on the isomorphic space, S2 and the spectrum over the canonical bundle may be compared with calculated spherical harmonics and their eigenvalues, known as Monopole Harmonics. The first step to solving the eigenvalue problem is to pullback rj over its associated principal bundle and prove that the resulting line bundle, rj* is equivalent to a trivial bundle. (For an explanation of the terms pullback, or principal bundle see Appendix B.) Sections on rj* are then associated with real functions on its base space, g 2"+ 1 in the Complex case and 54 n+3 in the quaternion case, with respect to a choice of a basis frame of sections. We begin by finding all U(l) or SP(1) invariant functions on the spheres, S2n+l or Sin+3, and then determine the transformation properties of the pullback connection on rj. It then becomes possible to construct eigenfunctions which are non-invariant in such a way that they cancel the non-invariance of the chosen frame of sections and thus push down to sections on rj with eigenfunctions making up the spectrum of the Laplacian on rj. 23 Chapter 5. The Spectrum of the Laplacian over certain Bundles In Section 6.1 we perform the above calculations in the Complex case and compare to known results, and in Section 6.2 we complete these calculations in the quaternion case. 5.1 The Complex Projective Space 5.1.1 The Pullback of a vector bundle over its principal bundle We seek first to prove that the pullback of the canonical line bundle, rj, over its associated principal bundle results in a line bundle, denoted r/*, which is identical to a trivial bundle; E-(*t)p C/(l) • + £ p ( t < r ) • 5 2 " + 1 -?-* CP( n In this case, sections on r/*, which solve the Laplacian may be "pushed down" by the projec-tion map, q, to sections on r/ which are solutions to the Laplacian and thus are the desired eigenfunctions. Lemma 1: The pullback of a vector bundle over its associated principal bundle is equivalent to a trivial bundle [25]. Proof: The following diagram depicts a vector bundle, 77, pulled back over its associated principal bundle, where G is the structure group of 77; 24 Chapter 5. The Spectrum of the Laplacian over certain Bundles rj : F E G' •E- B Consider the trivial bundle, rj*, over the principal total space and construct the following pullback square: rj* : F F :r) ExF •E-E B We wish to define the Principal Map, P : E x F —• E which will make rj* the trivial pullback of rj over its associated principal bundle. If b G Vi C B where b = q(e), e € E, then define for all y € F, Pi(e,y) — <pi(b,qi(e) • y) where {qi} are functions with range in the subset Vi C B and {4>i} are the coordinate functions (defined in Appendix B). Now, for all b G Vi fl Vj we have <f>i{b,qi{e) • y) = <i>j(b,gji(b) • qi{e) • y) = (j>j(b,qj{e) • y). where gij are the change of coordinate functions (defined in Appendix B). Thus, {Pi} is con-tinuous on the intersections of two coordinate subsets of B and a unique function, P, is defined as the union of {Pi}-Commutivity in the pullback square is then straightforward; p o P(e, y) = b = q(e) = qo yr(e, y)U 25 Chapter 5. The Spectrum of the Laplacian over certain Bundles 5.1.2 T w o Theorems Theorem 1: Let / be a homogeneous harmonic polynomial of degree k on Mn+1, where a harmonic function is once which satisfies, A / — 0. Then f\$n is an eigenfunction of the Laplacian on the sphere, with eigenvalue k(k + n — 1). Proof: Let p be any point on Sn with associated unit vector, xp € Rn+1. Then let {x2, • • • xn+i} be a frame of unit vectors forming a basis for the tangent space, Tp(Sn). Define a set of geodesies based on the frame of vectors, 7i(a) = (cosa)x + (sina)xj, (5.1) where each % is a great circle around Sn in the X{ direction and a is the angle. Now take a function on M"+1, / , with partial derivatives with respect to the basis, (^ J , i = 1 , . . . n + 1 where The first derivative of / o 7, with respect to a is then, d(f o 7 i) . df df oa ox 1 axi and the second derivative evaluated at the point, q = 7i(0); d 2 ( /o 7 , ) _ df a2/ 5M0) " dxx{q} + dx*{q}- ( 5 , 3 j Thus we can calculate, A^/ls-X?) = - E ^ ( / ° 7i)(0) = ~ E ^ + " ^ ( 9 ) (5-4) i=2 i=2 * i = l * i=2 l i and therefore, (A«"+1/)(?) - A5"(/|S.)(,) = - g f o ) - nj^fo) or, since x\ is in fact a radial vector of 5™, dr2 |l5 '"9r (AK"+1/)|5n = A5n(/|5.) - ft5„ - „^|5- (5.5) 26 Chapter 5. The Spectrum of the Laplacian over certain Bundles Now, suppose that / is a homogeneous harmonic polynomial of degree k on Mn+1. This means that / can be written as a sum of terms of the following form; mo mi mn where for each term, Y17-0 mi = ^- By changing to spherical coordinates it follows that from each term there may be factored rk, leaving a multiplication of sin's and cos's. Thus, / can be written as rkf\$^ and Equation 5.5 becomes, 0 = ASn(f\Sn) ~ k(k ~ l)f\Sn - nk(f\Sn). Therefore, Asn(f\Sn) = k(k + n-l)f\Sn, (5.6) and it is proven that f\$n is an eigenfunction of A5" with eigenvalue, k{k + n — 1) • Theorem 2: For all k > 0, the following orthogonal decomposition holds: V2k = W-2k © r2H2k-2 © • • • © r2kUo (5.7) V2k+i = ^k+i © r2H2k-i © • • • © r2kUx (5.8) where Vk is the set of all homogeneous polynomials of degree k on Rn + 1 and Uk is the set of all homogeneous harmonic polynomials of degree k on Rn + 1 . Proof: It is clear that Theorem 2 holds for VQ and V\ since VQ = HQ and V\ =T-Li. By induction, if we show that for all k>0/Ph = 'Hk@ r2Vk-2 then it follows that Vk+2 = Hk+2 © r2Vk and so on, built up from VQ and V\. To prove Theorem 2 inductively, it is sufficient to show that if there exists an element, P G Vk+2 which is orthogonal to Vk then P must be harmonic. P is a harmonic function if and only if A P = 0. However, since P G Vk+2, A P G Vk, and A P is therefore equal to zero if and only if it is orthogonal to all r2lT-Lk-2i (0 < 21 < k) or, identically, if A P is orthogonal to all i-Lk-2i-Here, ~refers to restriction to Sn. Let A / refer to AEn / and A / refer to A5" f\sn-27 I Chapter 5. The Spectrum of the Laplacian over certain Bundles Suppose there exists some P G Pk+2 a n d H G T-Lk~2i for some I. As well, suppose that P is orthogonal to Vk and thus also to Vk- Then, APH = {AP)H + 2<dP\dH >+P{AH). (5.9) Integrating Equation 5.9 over the Sn sphere, 0 = / APH = f (AP)H + 2[ <dP\dH>+ f P(AH). (5.10) JSn Jsn JSn JSn Since H G Hk-21, AH = (k-2l)(k-2l + n-l)H, (5.11) (from the results of Theorem 1). Therefore the third term becomes, f P(AH) = (k-2l)(k-2l + n-l) f PH = 0, Jsn J Sn the last integral being equal to zero since P is assumed to be orthogonal to Vk-Thus, 0 = f {AP)H + 2 f <dP\dH>. (5.12) JSn JSn However, from Equation 5.4, ~ -pj: &P OP AP = AP + — r + n— dr2 dr (5.13) = AP + {k + 2){n + k + 1)P, and the first term in Equation 5.12 becomes, / {AP)H = [ APH + (k + 2)(n + k + l) f PH = f APH, (5.14) Jsn Jsn Jsn Jsn again, since P is orthogonal to Vk- Therefore, Equation 5.12 is rewritten as, / APH = - 2 I < dP\dH >= - 2 < P\AH > JSn Jsn = -2(k -2l)(k-2l + n-l)< P\H >= 0, (5.15) 28 Chapter 5. The Spectrum of the Laplacian over certain Bundles from Equation 5.11 and the fact that P is orthogonal to Vk-We have shown that f$n APH = 0 and thus, that A P is orthogonal to Vk for all k > 0 and therefore, A P = 0 • 5.1.3 The Spectrum of the Laplacian on CP(n) Consider once more the pullback of 77, the canonical line bundle over CP(n), over its associated principal bundle: C2 c2 V : :rj E- •E (*T)p p(t<r) 17(1)< • S2n+l -?-> CP(n) From Lemma 1, proved in Sec. 4.1.1, we know that fj is identical to a trivial complex line bundle whose fibres are great circles, and thus its sections a can be represented by complex functions oh S2n+l. The projection function, q, takes the great circles in S2n+l to the complex line in CP(n) in which the circle lies. Thus, q is invariant under unit complex (U(l)) multiplication. In order to determine the spectrum of the Laplacian on 77 we first wish to determine the spectrum of the Laplacian on functions on CP(n). To find these functions, we initially construct the spectrum of the Laplacian on functions on g 2 n + 1 and then q projects these functions down to CP(n). Consider the set of homogeneous, harmonic polynomials of degree k on M2n+2. From Theorem 1, these polynomials are eigenfunctions of the Laplacian on S2n+1 with known eigenvalues. If we then determine those homogeneous harmonic polynomials which are also invariant under unit complex multiplication, the projection map, q, takes these functions to eigenfuctions on CP(n) with the same eigenvalues. 29 Chapter 5. The Spectrum of the Laplacian over certain Bundles Homogeneous polynomials of degree k in M2n+2 are in 2n+2 real variables, (XOJ^I • • • £271+1) Now construct n+1 complex variables from pairs of these real variables as follows: Z0 = XQ + IX\ Z\ = X2 + 1X3 Zn = X2n + ™1n+\ z = (z0,zi...zn) Let Vk,i be the set of polynomials of degree k in z and degree / in z, where z = (Z~Q,Z\ ... zn) is the complex conjugate of z. Since all Zj's commute, we can classify polynomials by degrees of z and z rather than worry about individual z^s. If P £ Tk,i and we multiply z by a unit complex number, el&; P(z, z) - • P(el0z, e^ez) - P(el0z, e~l6z) = e<k~l)eP{z, z). Therefore, P will be an invariant polynomial ii k — I. Now we have Pk,k, the set of bihomogeneous polynomials of degree k in z and z (which are also invariant under 17(1) multiplication). Let W.'k,k be the set of harmonic polynomials satisfying the above conditions. From Theorem 2 we have, V2k=n2k®r2V2k-2. Thus, by turning these polynomials into bihomogenous polynomials of complex degree k we have, Vk,k = W-k,k © r2Pk-i,k-i, and then restricting all of these polynomials to the sphere, S2n+1: Vk,k = 'Hk,k@r2'Pk-i,k-i- (5-16) 30 Chapter 5. The Spectrum of the Laplacian over certain Bundles Therefore, dim?4 i fc = dimVk,k - dim-pfc_liA;_i = dim7\ f e - dim7V-i,fc-i- (5.17) It follows from Equation 5.16, ®Vk,k = ®nk,k. (5.18) k k It only remains to show that (BVk,k is dense in C°°(CP(n)), or dense up to unit complex mul-k tiplication in C°°(S2n+1). Then, by the Stone-Weierstrass theorem and Equation 5.18, ®Hk,k k is the complete set of eigenfunctions on CP(n). This is proven in Section 6.1.4 . The following is a determination of the spectrum of the Laplacian on CP(n) assuming the Stone-Weierstrass proof. The dimension of Vk,k will be the square of the dimension of Vk-, which is simply the number of polynomials of degree k in n + 1 variables. Thus, from Equation 5.17, ^M"r)2-("t-iT- <5-i9) Therefore, we have found the spectrum of the Laplacian on CP(n) whose eigenfunctions are harmonic, bihomogeneous polynomials of degree k in z,z € C"+1 , restricted to S2n+1. The eigenvalues are from Equation 5.6, Afc = 2k(2k + (2n + 1) - 1) = 4k(k + n), (5.20) with multiplicity, mk equal to the dimension of lik,k which is simplified from Equation 5.19; t , ^{n(n + l)...{n + k-l)\2 mk = n(n + 2k) I — - j . (5.21) 5.1.4 Completeness of the spectrum on CP(n) To prove that we have constructed a complete set of eigenfuctions on CP(n) we use the Stone-Weierstrass Theorem which states that if the calculated functions on S2n+1 separate points on 31 Chapter 5. The Spectrum of the Laplacian over certain Bundles CP(n) then the set of eigenf unctions is complete. Separation of points on CP(n) is equivalent to separation of points on S2n+l up to unit complex multiplication (or U(l)) invariance. Suppose z and z' are two non-identical points on S2n+1. Separation of points means that if z ^ z' • u for some u € ^ ( l ) then there must exist some function, / € ~Hk,k for some k, such that f(z) ± f(z'). Let us define the points as follows; z = (z0,zi,...zn) z = (^0)^1) • • • zn) where zk = Pke%6k and z'k = p'kelBk. One of the following three cases will hold: 1) There exists some i € {0 . . . n} such that pi =£ p\. 2) pi = p\ for alii € {0 . . . n} and, e'i-ei = e (5.22) is true for all i € {0 . . . n} and for some constant 9. 3) Pi = p[ for all i € {0 . . . n} and Equation 5.22 is not true for alH € {0 . . . n}. Then there exists some j , k € {0 . . . n} such that 9j — 9k ^ 9\ — 9'k. If case (1) is true, then the function, / = ZiZi <E 7^i , i will separate the points, z and z', even if #j = 9\ V i since in that case, Z{Z~i = Yll=i \Pj\2 a n d there is some i such that pi ^ p\ so f(qi) ^ / f e ) -If case (2) is true, then 2' • el° = z and the two points differ only by a unit complex and are therefore the same point on CP(n). 32 Chapter 5. The Spectrum of the Laplacian over certain Bundles If case (3) is true, then / = Zifj separates the points since, f{qi) = pie*eipje-*J=piPje«0i-W f{Q2)=fe*p>je-*,i=PipjeM-V and thus f(z) ^ f{z'). 5.1.5 Comparison of the spectrum on CP(1) wi th that on S2 For n = 1, CP(1) is the sphere, S2, therefore the previous calculation of the spectrum with n = 1 must agree with the known spectrum of the Laplacian on S2. The spectrum of the Laplacian on Sn with radius 1 has been calculated in Sec. 4.1.2 to be: \k = k(k + n-l), (5.23) with multiplicity, (n + k-2)(n + k-3)--(n + l)n. mk = - - —(n + 2 / c - l ) . (5.24) ft! However, since CP(1) and S2 do not have the same radius the following calculations determine the relative radius of CP(1). We define the complex projective plane, CP(n), in terms of the bundle, [/(!)< >S2n+1 CP(n) where the projection, q, takes great circles in S2n+l to the plane (or complex line) in CP(n) in which they lie. If q E S2n+l such that p(q) = m € CP(n) then decompose the tangent plane, Tq(S2n+1) into two orthogonal subspaces; the first subtended by i • q which p projects entirely to zero in Tm(CP(n)), and the second orthogonal to the first subspace. p will thus project the second 33 Chapter 5. The Spectrum of the Laplacian over certain Bundles subspace isometrically onto Tm(CP(n)). We now have an almost complex structure on CP(n) created by multiplying by i in the second subspace. Definition: An almost hermitian manifold with a complex structure, J, which is invariant under parallel translation is known as a Kahler manifold. It is possible to show [1, F.34] that a Kahler manifold with constant holomorphic curvature has sectional curvature described by, / R(x, -)x = < 0 for K • x, 4k x Id for R • Jx, (5-25) k x Id for vectors 1 to f and Jx. Clearly, the almost complex structure previously constructed on CP(n) respects parallel dis-placement and thus CP(n) with this structure is a Kahler manifold. The group of unitary transformations on C"+ 1 , U(n + 1), is transitive and isometric on CP(n). That is, for any two points, m,n € CP(n) with associated 2-planes, Tm and Tn, there exists an element, u G U{n+1) which takes (m,Tm) H-> (n,Tn) isometrically. It follows that CP(n) has constant holomorphic sectional curvature, since sectional curvature on a 2-plane is invariant under isometric unitary transformations. Therefore, we may suppose that this constant holomorphic curvature of CP(n) is equal to 4k for some k. Definition: A continuous vector-valued function, Y, along a geodesic, c : [a, b] —> M (where a, 6 G K and M is the manifold in question) is a Jacobi Section if (Dd)2Y + R(c,Y)c = 0, (5.26) where, once again, R is the sectional curvature. Let {ei (s) . . . en(s)} be a set of vector-valued functions (s G [a,b]) parallel along c which form a basis for each TC^M. If we now express Y as a linear combination of functions: Y(s) = Yi(s)ei(s), 34 Chapter 5. The Spectrum of the Laplacian over certain Bundles then Equation 5.26 becomes d2Y + a)Yj = 0, (5.27) ds2 - " ^ where a1, is the sectional curvature of ej(s) with respect to ei{s): R(c(s),ej(s))c(s) = a)ei(s). Now, let x be a tangent vector to CP(n) at the point m, and let 7(f) be the associated geodesic along x. Define a Jacobi section, Y, along 7, such that, y(0) = 0 and Y'(0) = y where y is orthogonal to x. We can decompose y into y\ and y2, where yi is the projection of y onto J • 2? and y2 is orthogonal to J • a? (and to x). Since y must satisfy Equation 5.27 where the sectional curvature is already known from Equa-tion 5.25 we may express Y as, Yi{t) = y{ sin(t(2>/fc)) (5.28) Y2(t)=y2sm(tVk) (5.29) Y\(t) has one half the period of I2M) and since they are orthogonal to each other, they meet at zero for the first time when t = n/y/k. However, it has been shown [1, CIV.11] that any two geodesies, cx and c'x, along two tangent vectors to CP(n), x and x', meet for the value IT if x and x' do not lie in the same orthogonal complex line. Thus, from Equation 5.29, Y\ and I2 should meet at t = ir and k must equal 1. Therefore, the holomorphic curvature of CP(n) is 4. Since the square of the radius of a sphere is proportional to its holomorphic curvature, and the S2 mentioned previously has radius 1, we may consider CP(1) to be an S2 with radius 2. We must, therefore, divide its spectrum calculated in Sec. 4.1.3 by the square of its radius, 4. 35 Chapter 5. The Spectrum of the Laplacian over certain Bundles From Eqns 5.20 and 5.21 the spectrum of the Laplacian on CP(1) is, Afc(CP(l)) =4fc(fc + l), m*(CP(l)) = {2k + l). (5.30) Dividing these eigenvalues by 4 we get, X'k = k{k + 1), m'k = (2k + l). From Eqns 5.23 and 5.24 the spectrum of the Laplacian on 5 2 is, Xk(S2) = k{k + 1) mk(S2) = (2k + 1) (5.31) (5.32) which is identical to the eigenvalues and multiplicities calculated in Equation 5.31 which are those of CP(1), adjusted for the radius difference. 5.1.6 The Pullback of a connection over CP(n) Consider the pullback square from Sec. 4.1.1 where n = 1: C2 C2 f] : E •E U(l) ?3 1 •CP(1) The canonical line bundle over CP(1), 77, is pulled back over its principal bundle to get the bundle, ??*, which was shown to be equivalent to a trivial bundle in Sec. 4.1.1. CP(1) is the set of all complex lines (or planes) through the origin in C2 (or R4). E — {(I, z) € CP(1) x <£?\z £ I}. The projection, p : (l,z) i-» I. The projection, q takes x € 5 3 to the plane 36 Chapter 5. The Spectrum of the Laplacian over certain Bundles through the origin which is defined by the great circle on which x lies. Clearly, different points which lie on the same great circle of S3 will be projected by q to the same complex line in CP(1), thus the fibre of this bundle is U(l), representing multiplication by unit complex numbers, el6, around the given great circle. E = {{x, {I, z)) E S3 x E\p o q = q o p} or, E = {(x, q(x), z) G S3 x CP(1) x C2 \z € q(x)}. The projection map, p : (x,q(x),z) f-> x and the projection map, q : (x,q(x),z) i—•  (q(x),z). Therefore, sections of 77* behave as, <7 : x H->• {x,q(x),z)\zeq(x) for all x € S3. (5.33) Thus, sections on 77* are equivalent to functions from S3 to C where C is the complex line, q(x). We wish to find sections, a, on 77 which are eigenfunctions of the Laplacian. However, since rj* is equivalent to a trivial bundle, it is far easier to find sections, a, on rj* which push down to appropriate sections on rj. We have found in Sec. 4.1.3 U(l) invariant functions mapping S3 —* C which are eigenfunctions of the Laplacian. Since the map, q, is invariant under unit complex multiplication, these U(l) invariant functions push down to form the eigenspace of the Laplacian on CP(1). However, a function which maps S3 —> C is not equivalent to its corresponding section. For any trivial bundle there exists an orthonormal frame of sections, (</>i,... <j>m). In this case, m = 2, and the functions are of the form, / : S3 —> C ~ M2. Each function can be expressed as / = ( /b/2) where / ; : S3 —»• M and fi : x 1—>< <pi(x),a(x) > . Thus a section on 77* corresponding to the function, / , may be expressed as 2 i=i For this reason we must determine how a normalized basis of sections on 77* transforms under U(l) multiplication. Clearly, this frame cannot be invariant under U(l) multiplication, other-wise the frame would push down to a continuous frame of sections on 77, implying that 77 be a trivial bundle, which it is not. Once it is understood how the sections on 77* transform, it will 37 Chapter 5. The Spectrum of the Laplacian over certain Bundles be apparent how to construct non-invariant functions on S3 which cancel the non-invariance of the frame of sections and thus push down to eigensections of the Laplacian on 77. Lemma 2: For an arbitrary normalized real section, &R on 77* the following equation is true: GR{UX) = UGR{X) for all x € S3,u£ U(l). (5.34) Proof: We modify the definition of a section on rj* from Equation 5.33 to the following definition of a normalized real section; &R(X) — (x,q(x),ax) where a G R, \a\ = 1. Beginning with the left hand side of Equation 5.34, OR(UX) — (ux,q(ux),a(ux)). Let u = el6 for some 0 < 9 < 1-n. Multiplication by e%e will move the point a; to a different point of S3 on the same great circle. q(ux) = q(x) by its fundamental U(l) invariance. a{ux) = el6x, and thus, &R(UX) = (ux, q(x),u • ax) Now, assuming we define multiplication by unit complex numbers in E as multiplication in the great circle, we may quickly evaluate the right hand side of Equation 5.34, U&R(X) = u(x,q(x),ax) = (ux,uq(x),uax) — (ux, q{x), u • ax), since multiplication by el& in the great circle which defines the plane, q(x), will not change the plane • Thus, Equation 5.34 is correct and real sections of rj* will transform by one full rotation for every full U(l) rotation on S3. As a result, we must construct functions which rotate once in the opposite direction for every rotation on S3, i.e.: f(elBx) = e~%ef(x) for all x € 53 ,0 < B < 2TT. (5.35) 38 Chapter 5. The Spectrum of the Laplacian over certain Bundles 5.1.7 The Spectrum of the Laplacian on the canonical bundle As determined in Sec 6.1.6, the spectrum of the Laplacian on the canonical line bundle, 77, will consist of those sections on 7) which are not pullback sections, but instead satisfy the following, o(uz) = u~la(z) for all u 6 U(l). That is, the corresponding functions on S2n+1 will be non-invariant under unit complex multi-plication and will instead satisfy Equation 5.35. The set of functions satisfying Equation 5.35 will consist of an invariant harmonic bihomogenous function as defined in Sec 6.1.3, multiplied by a single factor of Z{. This set of functions is simply the set of all bihomogenous harmonic functions with one more factor of z than z, {©'Hfc,/c+i}-k From Theorem 1, the eigenvalues of these functions will be once again, k(k + n — 1) where in this case, k is odd and h is the dimension of S2n+l, k = 2k + l, n = 2n + 1. The possible eigenvalues of the Laplacian on r? are therefore, Ek = {2k + l){2k + 2n + l), k = 0 ,1 ,2 . . . (5.36) The solution to this problem relates to the physical problem of monopole harmonics on the S sphere. Wu and Yang [28] calculated the eigenvalues of these harmonics to be, Ei = 1(1 + 1), 1 = 0,1,2... (5.37) In comparison, we may substitute n = 1 into Equation 5.36 (since CP(1) is homomorphic to S2) and divide by the square of the relative radius of CP(1) which is 2 (determined in Sec. 4.1.5). These energy levels become: ^ = i(2fc + l)(2^ + 3 ) = ( ^ ± 1 ) ( ( ^ ± i ) + l ) , £ = 0 ,1 ,2 . . . (5.38) 39 Chapter 5. The Spectrum of the Laplacian over certain Bundles which match the monopole harmonic eigenvalues in Equation 5.37 exactly. In order to determine the multiplicity of non-invariant harmonic functions on CP(1), we return to the number of invariant 2k dimensional polynomials in n + 1 variables, A- v fn + k^2 dunVk,k = y k These polynomials are multiplied by one complex factor of ZQ or z\ and the dimension is thus multiplied by 2. If we denote the set of non-invariant 2k + 1 dimensional functions as Vk,k+\ then, dim7\fc+i = 21 and thus, n + k\ n (n + k — 1N dJmHktk+i = 2[ k ) - 2 ( k_1 ) . (5.39) For CP(1) we substitute n = 1 into Equation 5.39 and find the multiplicity of the spectrum of the Laplacian on 77, m£(l) =2m*(l) =2{2k + l). 5.1.8 Completeness of the spectrum over the canonical bundle In the same manner as Sec. 4.1.4, we seek to prove that the spectrum determined for the Laplacian on the canonical complex line bundle over CP(n) is complete. The Stone-Weierstrass theorem states that the spectrum is complete if and only if the eigenfunctions separate points on CP(n). The set of eigenvectors in question is, {^fc,fc+i}-Suppose that z,z' € S2n+1 C C"+1 , where z = (zo,...zn) = {poel8o...pnel0n) z' = (z'0,...z>n) = (p'0e<...p'ne<). Then one of the following cases must be true: 40 Chapter 5. The Spectrum of the Laplacian over certain Bundles 1) There exists some i € {0 . . . n} such that p, ^ p\. Then the function, / = ZiZiZi separates the points, z and z'. 2) pi = p\ for all i € {0 . . . n} and there exists some j G {0 . . . n} such that 0j — 0Q ^ 0' — 9'0. Then the function, / = (ZJZO)ZJ separates the points, z and z'. 3) pi = p\ and &% — QQ = Q\ — 6'0 for all i 6 {0 . . . n}, in which case there exists an a, with 0 < a < 2ir, such that z' = zeia, thus the projection, g, projects z and z' to the same point in CP(n) and need not be separated. The details of the calculations of f(z) and f(z') for the above three cases is left out as they are essentially identical to those carried out in Sec. 4.1.4. 5.2 The quaternion Projective Space 5.2.1 The Spectrum of the Laplacian on HP(n) As in the Complex case we construct invariant functions on 5"4n+3 which are harmonic, and thus determine the spectrum of the Laplacian on HP(n). Homogeneous polynomials in R4n+4 are in real variables, (xo,x\,... X4n+s). We equate groups of four real variables with quaternion variables as follows; qo = xo + ix\ + jx2 + kxz qi—x^ + iXf, + jx$ + kxj Qn = Xin + IXAn+l + 3xA.n+1 + kXin+z Now we seek homogeneous quaternion polynomials in (go, qi, • • • qn) G 1HP+1 which are invariant under unit quaternion multiplication. 41 Chapter 5. The Spectrum of the Laplacian over certain Bundles For any pair of quaternion variables, qm and qi, there are six invariant terms with which to construct invariant polynomials: qmqm, qm, qmqi, qiqm, qmqi, qmqi-We are using the notation, q and q in order to create a fully one to one correspondence between (:ro • • • £471+3) and (qo • •. qn)- To define these terms, let q = x + ty + jz + kw. Then, q = x — ly — jz — kw, q = x — ty + jz + kw, q = x + ly — jz + kw, and thus, q = x + ly + jz — kw. The terms, qiqm and q~iqm are not included in the list of invariant terms since they are not independent of the six invariant terms; Mm = qmqi, Mm = QmQl-Thus the number of possible invariant pairs of (qo • • • qn) is ^ ( " J 1 ) - Since the number of poly-nomials of degree k in M variables is (M_A.1+A;). It is also important to note that all terms including, qiqjqjqi = qmqjqj = qmqjqj are redundant since qiq~i = qiqi = \qi\ which is a real number, and thus qiqiqjqj is also a real number. The dimension of all degree 2k polynomials in (qo ... qn) which are invariant under unit quater-nion multiplication, (denoted Nf.), is the number of degree k polynomials in the allowed pairs 42 Chapter 5. The Spectrum of the Laplacian over certain Bundles of qt's minus the number of redundant degree k — 2 polynomials in the same pairs: which simplifies to, (a + 2k - 1) • (a + k - 2)(a + k - 3) • • • {k + 1) Nk = ( ^ where a = 3n(n+l) — 1. The number of harmonic homogeneous invariant quaternion polynomials in W+l is therefore, d i m ? ^ . = Nk - Nk-i-From Theorem 1, the spectrum of the Laplacian on HP(n) is k'(n' + k' — 1) where k' is even and n' = 4n + 3. So the eigenvalues and multiplicities are; Ek = 4k(k + 2n + 1) A = 1,2,3. . . (5.41) (a + A - 3 ) ( a + A;-4) ••• (fc + l ) r o „ „ , „ , „ • , mk = Nk - Nk^ = ± A _ ; i '-[a2 - 3a + 2aA; - 2fe + 2] (a 1J. where a = 3n(n + 1) — 1. 5.2.2 C o m p l e t e n e s s of t h e s p e c t r u m on ETP(n) The Stone-Weierstrass argument proving that {®ii^k} is the complete spectrum of the Lapla-k ' cian on HP(n) runs along similar lines as that of the Laplacian on CP(n). Since the sets of (quaternion) functions restricted to the sphere are equal; ®Vj£k = ®T^kk we wish to show that {®Vkk\ separates points on the base space, MP(n). This is equivalent k ' to showing that for any two non-equivalent points in different fibres on 5 4 n + 3 , there exists a function in Pkk which separates these points. Let Q, Q' 6 5 4 n + 3 be two non-equal unit length variables in n + 1 quaternion dimensions, and let us denote, Q = (qo,Qi,---Qn) Q' = (q'0,q[, •••q'n) 43 Chapter 5. The Spectrum of the Laplacian over certain Bundles where each quaternion variable, qi = p^i or q\ = p'^. pi is then the real norm of qi, and gi € SP(1) is a unit quaternion. There are now three cases to consider; 1) There exists some i such that pi ^ p[, 2) pi = p\ for all % but there exists some i such that gig$l ^ g'iid'o)^1 i-e., the relative angles between the variables of Q differ from those of Q' 3) pi = p\ and g^1 = s ^ d ) - 1 for all i. In case (1), the polynomial / = g ^ separates the points Q and Q'. Clearly / G ®^k and since k ' Pi ¥" Pi, J{Q) = \Pi\2 which is not equal to f(Q') = \p'i\2. In case (2), the polynomial, h = qoq~i separates Q and Q'. Again, h € ®fif?k and k ' h{Q) = Po9o{Pi9i) = PoPigom HQ') = pWoiM=PoPi9o9,i-PoPi — PoP'i, and gog~i ^ g'^g'i thus h separates Q and Q'. In case (3), pi = p\ for all i and there exists some g G SP(1) such that g[ = gig for all i. However, this is equivalent to the statement that Q and Q' differ only by the multiplication of a unit quaternion, Q' = Qg, and the two points are therefore in the same fibre. As previously stated, it is not necessary to separate points in the same fibre as they are projected to the same point in EBP (n). Therefore, the Stone-Weierstrass conditions are satisfied and the spectrum calculated in Sec. 4.2.1 is the complete spectrum of the Laplacian on HP(n) • 5.2.3 Comparison of the spectrum on HP(1) wi th that of S4 As the spectrum of the Laplacian on CP(1) was compared to that on S2 in Sec. 4.1.5, we now compare the spectrum of the Laplacian on HP(1) calculated in Sec. 4.2.1 to that on S4. 44 Chapter 5. The Spectrum of the Laplacian over certain Bundles From Equation 5.23 and Equation 5.24, the spectrum of the Laplacian on a unit radius S4 is; Xk(S4) = k(k + 3), mk{SA) = \{2k + 3)(k + 2)(k + 1). (5.42) The argument in Sec. 4.1.5 may be exactly duplicated in the case of HP(1), to prove that the holomorphic curvature of the quaternion Projective Space is 4, and its relative radius is therefore, 2. Substituting n = 1 into Equation 5.2.1, the spectrum of the Laplacian on IHTP(l) is, Afc(MP(l)) = 4k(k + 3), m*(HP(l)) - i(A + 2)(fc + l)(2fc + 3). (5.43) Dividing these eigenvalues by 4 we get, \'k(MP(l))=k(k + 3), m'k{EP(l)) = l(2k + 3)(k + 2)(k + 1). 6 (5.44) which match the eigenvalues and multiplicities of the spectrum of the Laplacian on S4. 5.2.4 T h e Pullback of a connect ion over HP(n) Consider the canonical line bundle over HP(1) pulled back over its principal bundle: SP(1) M E-M' E ?7 1_ •HP(1) HP(1) is equal to the set of all quaternion lines through the origin in tf (or M4). E = {(P, Q) € HP(l) x H2 \Q E P}. The projection, p : (P,Q) ^> P and q takes a point x G S7 to 45 Chapter 5. The Spectrum of the Laplacian over certain Bundles the quaternion plane containing the "great circle" (S3) on which x lies. E — {(x,q(x),Q) E S7 x HP(l) x E P | Q 6 q{x)}, p : {x, q(x),Q) •-» x, and g : (x, q{x), Q) i-> (q(x), Q) E £ . As in Sec. 4.1.6 we wish to determine how a frame of orthonormal sections on £* will transform under SP(1) (unit quaternion) multiplication. Let us construct a frame of orthonormal sections, {4>q, 02> 03i 04} such that any section, a = Yli=i fi4>i where fi : S7 —*• R. Then let; 01 : x H-> (x,5(a;),x), 02 : x H-> (x,q(x),ix), 03 : a; H-> (x,g(x), j f ) , 04 : a: 1—> (x, g(a;), fcx), Clearly all vectors lie in the same plane, x,ix,jx,kx E q{x) and they form an orthonormal frame of the quaternion plane, q[x). If g E SP(1), we wish to show that, ga(x) = a(gx) (5.45) 94>\{x) =g(x,q(x),x) = {gx,q{x),gx) and 4>i(gx) = {gx,q(gx),gx) = (gx,q(x),gx) thus Equa-tion 5.45 holds for 0i and is similarly true for 02, 03 and 04. Therefore, the sections on £* undergo a rotation equal to the SP(1) rotation by which they are multiplied, and thus we will construct functions on S7 which undergo an equal but opposite rotation for every SP(1) multiplication. 5.2.5 T h e Spec trum of the Laplacian on the canonical bundle As in the complex case, the desired non-invariant functions on S4n+3 which make up the spec-trum of the Laplacian on £ over HP (n) are the invariant functions multiplied on the right by a quaternion factor. Let us denote such harmonic functions as 'H^+ik4 From Theorem 1, the eigenvalues of these 46 Chapter 5. The Spectrum of the Laplacian over certain Bundles functions will be, k{k + n — 1) where k = 2k + 1 and n = An + 3. Thus, Afc = (2A; + l)(2A; + 4n + 3). (5.46) To construct the non-invariant functions on SAn+3, we take the invariant functions in n + 1 quaternion dimensions and multiply them on the right by a quaternion factor. For simplicity's sake, we carry out the calculation of multiplicities over HP(1). When n — 1, the functions are in two quaternion dimensions, 9o,9i- From Equation 5.40 the number of invariant 2k quaternion dimensional polynomials is, 'k + 5 \ (k + 3N Nk = , , , "Jfe-2 Each of these functions ends with one of the six invariant factors listed in Sec. 4.2.1. However, when multiplying on the right by either qo or q\ the following redundancies occur: 9292 • <?i = <?i<?2 • <?2 92ft • 9 i = 9 i9 i • 92-Therefore, the number of non-invariant 2k + 1 quaternion dimensional polynomials is; Nk+hk = 2Nk-2Nk„u and the number of non-invariant harmonic 2k+ 1 quaternion dimensional polynomials is there-fore, d im^?+i,fc = Nk+i,k ~ Wfc,fc_i = 2(7Vfc - 2Nk_! + Nk_2) (5.47) Therefore, the eigenvalues and multiplicities of the Laplacian on the canonical fibre bundle over HP(1) are; \k = (2k + l)(2k + 7), (5.48) mk = 2{Nk-2Nk_l+Nk_2), where Nk — ( £ ) — (^2) . It remains to be proven that these functions form the complete spectrum. This proof is carried out in Sec. 4.2.6. 47 Chapter 5. The Spectrum of the Laplacian over certain Bundles 5.2.6 Completeness of the spectrum over the canonical bundle As in Sec. 4.1.8, the Stone-Weierstrass theorem is applied to the set of eigenvectors of the Laplacian on the line bundle determined in Sec. 4.2.5. This set is, {n^fc+1}-Let Q, Q' G 5 4 n + 3 C HP+1 where, Q = (qo • • • qn) = (Pogo • • • Pn9n) where &, sj G SP(1). One of the following three cases will then apply; 1) There exists some i G {0 . . . n) such that pi ^ p\. Then / = qiqiqi separates Q and Q'. 2) pi = p\ V % E {0 . . . n} and there exists some j € {0 . . . n} such that gig^1 = g'ig'o~ • Then / = qi%qi separates Q and Q'. 3) pi = p'i and gig^1 = g\gQ~ Vi G {0 . . . n}, in which case there exists some g G SP(1) such that Q' — Qd a n ( i t n u s Q a n d Q' project to the same point on HP(n) and need not be separated. Once again, the above calculations are not carried out in detail, as they are almost identical to those in Sec. 4.2.5. 48 Chapter 6 Fitting of Theory to Existing Data In 1978, Bernstein and Webb [2] measured the Ts <8> {eg © £23) spectrum of the octahedral molecule, IrF6- This experiment involved measuring the low temperature, high energy absorp-tion spectrum of IrF6 crystals slightly doped with another transition metal hexafluoride. In analysis of their data, Bernstein and Webb attempted to identify Jahn-Teller active frequencies, vi and z/5 corresponding to Ts <8> eg and Ts <8> tig respectively. They further pointed out that this method ignores the known coupling between the two modes in the Ts <8> (eg + t2g) problem which is what this work has attempted to calculate. Bernstein and Webb were able to confidently identify values for z/5 as well as values for non-Jahn-Teller active vibrations, 1/1,1/3,1/4 and v§ which couple with the Jahn-Teller active vibrations, however, no reasonable fit was made for the v<i frequency. These frequencies determined were identified as, Constant V\ "3 1/4 v% Value 682cm"1 287cm-1 217cm"1 696cm -1 Table 6.1: Non-Jahn-Teller active frequencies The data collected by Bernstein and Webb shows crystal field splitting in many places which corresponds to a spacing of about ±17cm_ 1 for most major lines. 49 Chapter 6. Fitting of Theory to Existing Data The eigenvalues calculated in Sec. 6.2.5, Xk = (2k + l)(2k + 7), are theorized to be energy levels of IrF6 with a scale factor and a zero-point energy: Ek = (2k + l){2k + 7)-A + B (6.1) If we equate the value for u^ determined by Bernstein and Webb with E\, then subsequent energy levels (displayed in Table 6.2) fit will with the existing data, where {Ei} are the Jahn-Teller pseudorotational energies on which the frequencies, u\, V3, U4 and VQ will be built. Due to the crystal field splitting we have taken the middle of the first three lines identified as 1/5 and thus designated, E\ — 223 cm""1. Substituting this value of E\ into Equation 6.1 and taking Bernstein and Webb's chosen origin line at zero energy, the scale factors are A = 11.2 c m - 1 and B = —78.05 cm - 1 , and the first five Jahn-Teller energy levels are, Jahn-Teller Energy EQ Ei E2 E3 E4 Calculated Frequency 0cm - 1 223cm -1 535cm_1 936cm -1 1427cm-1 Table 6.2: The First Five Calculated Jahn-Teller Frequencies 50 Chapter 6. Fitting of Theory to Existing Data These energy levels from Table 6.2 are coupled with the non-Jahn-Teller active frequencies, v\, vZi u4 and v& from Table 6.1 and the resulting frequencies less than 1100 cm - 1 are calculated; Calculated Coupled Energy Level Ex . E2 Ez E0 + v\ #0 + 3^ -Eo + ^4 EQ + VQ Ei +vx Ex + V3 Ei + Vi Ei + v& E2 + Vi E2 + v% E0 + 2vi E0 + 2i/6 Ei + 2^4 Ei + 2v& E2 + 2^6 Frequency (cm - 1) 223 535 936 682 696 287 217 905 919 510 440 822 752 574 434 797 657 969 Table 6.3: Calculated frequencies less than 1100 cm 51 Chapter 6. Fitting of Theory to Existing Data Table 6.4 displays the major bands identified by Bernstein and Webb as strong in intensity or as Jahn-Teller active, along with the assignment given them by Bernstein and Webb as well as the assignment given them as a result of this work. Crystal field splitting is considered to be active throughout these energies. Fitting was not attempted for those frequencies greater than 1100 cm - 1 since almost all measured lines above this frequency were of weak intensity as well as the fact that at greater energies the approximation of a perfectly round HP(l) becomes less accurate. Freq. Assignment of Bernstein & Webb <i(a) i/6(a) 1" I "1(a) r L r 2^5 L Frequency cm - 1 202.7 208.9 215.8 218.2 241.5 254.4 271.8 284.7 292.9 416.6 419.1 452.3 461.4 Assignment of Calculated Freq. 1 1 El Eo + ve, J J 1 E0 + v4 J 1 1 EQ + 2^6 Ei + ve J J 52 Chapter 6. Fitting of Theory to Existing Data Freq. Assignment of Bernstein & Webb V5 + v± (a) VA + u6 (a) i>5 + ^ (b ) 1/4 + f6(b) 2^4 (a) 2^4(b) "2(a) ^3 (a) Frequency cm - 1 487.8 494.9 521.7 528.1 532.9 538.1 554.3 561.5 612.2 645.9 665.5 682.1 696.3 721.0 724.2 Assignment of Calculated Freq. 1 Ex + v4 ] J 1 1 ^4 + "6 E2 J J ~\E0 + 2v4 J ]E1+2v6 J 1 1 Eo + vi E0 + v3 J J Chapter 6. Fitting of Theory to Existing Data Freq. Assignment of Bernstein & Webb 3^4 (a) 3i/4(b) [ L \ v\ + ^5(b) L phonons Frequency cm - 1 734.9 742.5 764.8 804.1 806.5 831.7 845.5 884.0 890.3 923.2 927.1 933.2 964.4 965.6 991.9 1024 Assignment of Calculated Freq. 1 E2 + u& }E1+2lS4 J E2 + V4 Ei + Vi 1^1+^3 1 J ]E2 + 2v6 phonons Table 6.4: Major Energy Bands measured by Bernstein and Webb with assignments Bernstein and Webb theorized that the strong line at 1024 cm - 1 was due to a charge transfer origin and belonged in the F^g or Fjg spectrum. The previous calculations agree with this assessment. Furthermore, based on our calculations we were able to successfully identify the J ahn- Teller excitations in the spectrum based on an equal coupling case of T$ <g> (eg ®t2g). 54 Chapter 7 Conclusion The Jahn-Teller active pseudorotational energy spectrum of an octahedral molecule was suc-cessfully calculated to a two parameter fit. This calculation depends on the use of the Born-Oppenheimer approximation, which states that the pseudorotational spectrum is equivalent to the spectrum of the Laplacian on a specified fibre bundle with the addition of a zero-point energy. The completion of this work marks an important step in theoretical molecular spectra, due to the success of the assumption of equal Jahn-Teller coupling between the eg and tig modes in the Y% ® (eg © *2g) problem, whereas in the past, calculations and experimental assignments have focused on either the F$ <g) eg or the Fg <g> tig modes. Further, the geometry of the eigenspace of this Laplacian is entirely determined by the electronic Hamiltonian of the given molecule. The group theoretic symmetry properties of an octahedral molecule cause the electronic Hamiltonian to act on a round S4. Thus, the required calculation is that of the spectrum of the Laplacian on the canonical line bundle over S4. Because S4 is isomorphic to HP(l) it was found to be more convenient to find the spectrum of the Laplacian on the line bundle over HP(1). Since the pullback of this bundle over its principal bundle is a trivial bundle, the final calculation was made on this trivial pullback bundle which then induced the desired spectrum on the line bundle over HP(1) ~ 5 4 . The final calculation of this spectrum produced the following eigenvalues, Afc = {2k+ l){2k+ 7)-A + B k = 0 , 1 . . . (7.1) where A is a real constant corresponding to the radius of the 5 4 sphere and B is a real constant 55 Chapter 7. Conclusion corresponding to the zero-point potential energy of the molecule. These eigenvalues were then fit to data taken using IrF6 by Bernstein and Webb [2]. Given the inherently uncertain nature of experimentalist assignments of Jahn-Teller active or inac-tive modes to the measured data, the fit obtained was encouraging. The constants used in Equation 7.1 were A = 11.2 cm - 1 and B = -78.05 cm - 1 . While Bernstein and Webb hypothesized the existence of two Jahn-Teller active modes corre-sponding to T8 <8> e and Fg <S> tig representations, they were unable to assign a clear value to one of these modes, that of Ts <8> e. The Jahn-Teller theory proven in Chapter 2 implies that there exists only one vibrational mode for the coupled modes, T$ <8> (e © t^g)-In this work the experimentally assigned vibrational frequency for Fg <8> tig was used as the second (A; = 1) energy level and further energies as well as non-Jahn-Teller active modes were then calculated and fit to the data. These assignments using equal Jahn-Teller coupling between eg and tig modes were in reasonable agreement with the data as well as with the assignments of Bernstein and Webb. 56 Bibliography [I] M. Berger, P. Gauduchon and E. Mazet. Le Spectre d'une Variete Riemannienne. Springer Verlag, 1971. [2] E. R. Bernstein and J. D. Webb. Molec. Phys. 37 No. 1, 191, 1979. [3] I. B. Bersuker and V. Z. Polinger. Vibronic Interactions in Molecules and Crystals. Springer Verlag, 1989. [4] C. C. Chancey and M. C. M. O'Brien. J. Phys. A: Math. Gen. 21, 3347, 1984. [5] C. Cohen-Tannoudji, B. Din and F. Laloe. Quantum Mechanics, Vol. 2. John-Wiley and Sons, 1977. [6] F. Albert Cotton. Chemical Applications of Group Theory. Interscience Publications, 1963. [7] F. J. Dyson. J. Math. Phys. 3, 140, 1962. [8] J. Englman. The John-Teller Effect in Molecules and Crystals. Wiley, New York, 1972. [9] H. Eyring, J. Walter and G. E. Kimball. Quantum Chemistry. John Wiley and Sons, New York, 1944. [10] P. B. Gilkey. Invariance Theory, The Heat Equation, and the Atiyah-Singer Index Theorem. Publish or Perish, Inc., 1984. [II] Michel Goosens, Frank Mittelback and Alexander Samarin. The 3TgX Companion. Addison-Wesley, 1994. [12] G. A. Hagedorn. Ann. Inst. H. Poincare Sect. A. 47, 1, 1987. [13] D. Husemoller. Fibre Bundles. Springer, New York, 1975. [14] H. A. Jahn and E. Teller. Proc. Roy. Soc. London, Ser A. 161, 220, 1937. [15] H. A. Jahn. Proc. Roy. Soc. London, Ser A 164, 117, 1938. 57 Bibliography [16] B. R. Judd. Adv. Chem. Phys. 57, 247, 1984. [17] H. A. Kramers. Proc. Acad. Sci. Amst. 33, 959, 1930. [18] J. W. Milnor and J. D. Stasheff. Characteristic Classes. Annals of Mathematics Studies, No. 76. Princeton University Press, 1974. [19] M. C. M. O'Brien. Phys. Rev. 187, 407, 1969. [20] M. C. M. O'Brien. Private Communication. [21] Joseph J. Rotman. An Introduction to Algebraic Topology. Springer-Verlag, 1988. [22] J. A. Salthouse and M. J. Ware. Point Group Character Tables and Related Data. Cam-bridge University Press, 1972. [23] J. E. Avron, L. Sadun, J. Segert and B. Simon. Commun. Math. Phys. 124, 595, 1989. [24] Edwin H. Spanier. Algebraic Topology. Springer-Verlag, 1966. [25] Norman Steenrod. The Topology of Fibre Bundles. Princeton University Press, 1957. [26] R. O. Wells. Differential Analysis on Complex Manifolds. Springer Verlag, New York, 1980. [27] E. P. Wigner. Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. New York Academic Press, 1959. [28] T. T. Wu and C. N. Yang. Nucl. Phys. B 107, 365, 1976. 58 Appendix A Basic Group Theory Definition: A Group, G, is a set of elements with a multiplication operation which obeys the following four axioms: 1) G is closed under the multiplication operation. That is, for all g,h € G, gh € G. 2) G contains a two-sided unit. That is, there exists some E € G, such that Eg = gE — g for all g € G. 3) The multiplication operation is associative. That is, for all g,h,k € G, (gh)k = g(hk). 4) Every element of G has a two-sided inverse. That is, for all g € G, there exists some h G G such that gh = hg — E. For a finite group, the group's order is equal to the number of elements belonging to the group. The distinction between groups is determined solely on the basis of their multiplication tables. Example: There is only one group of order 3, and its multiplication table is: e b c e e b c b b c e c c e b Table A.l: A 3 Element Multiplication Table. 59 Appendix A. Basic Group Theory which means that be = e, b2 = c and so on. Definition: If two elements of a group, A,B € G are related as, A = X~lBX for some X 6 G then they are said to be conjugate to each other. A complete set of all elements in a group which are conjugate to each other is called a class. The elements of a group may be represented by matrices which multiply and commute in the same way that the group elements do. This set of matrices is not necessarily unique. Definition: A representation of a group is a set of matrices corresponding to the set of elements such that if M(g) is the matrix corresponding to the element, g, then M(g)M{h) = M(gh) for all g,heG. This set of matrices will obey the four axioms of a group. In order that they are able to be multiplied in arbitrary order, the matrices of a representation must be square and of the same size, n x n. The dimension of a representation is defined as the dimension of the matrices, n. Two representations, {M} and {M'} of G are equivalent if one is obtained by a similarity transform of the other; M'(g) = SM(g)S~1 for all g G G for some n x n invertible matrix, S. It is easy to show that the {M'} representation will follow the same multiplication rules and be of the same dimension as {M}. A representation is reducible if there exists some similarity transform which, when applied, turns each matrix of the representation into a block diagonal matrix. For example, SM(g)S~1 = for all g G G Ml(g) 0 0 M2(g)_ where M\{g) is an m x m matrix for all g G G and M2{g) is an / x I matrix for all g G G. If this is the case then we write, M{g)=Mi{g)@M2{g) 60 Appendix A. Basic Group Theory and the n-dimensional representation, M, is reduced to two m and / dimensional representations, Mi and M2, where m + l = n. An irreducible representation is one for which no such similarity transformation exists. The totally symmetric representation is a one-dimensional irreducible representation which consists entirely of one-dimensional unit matrices. Each group contains this representation. Rather than write out a set of matrices for each representation, all the necessary information is contained in the set of characters (or traces) of these matrices. It is easily shown that the character of a matrix of a representation is unchanged by a similarity transformation, and thus the characters of conjugate elements are equal. Symmetry operations on molecules form a natural group. A symmetry operation on a molecule can be a rotation, reflection or an inversion of that molecule, which leaves every atom of the molecule spatially unchanged. The set of all symmetry operations on a given molecular shape forms its molecular symmetry group. All possible symmetry operations are now listed with their Schonflies notation: E: identity element. C™: rotation around an axis by m • —^ radians, n — 2,3,4,6 and m — 1,2,... n — 1. a: reflection in a plane. ah-, reflection in a horizontal plane (referring to a plane through the origin perpendicular to the axis of highest symmetry). av: reflection in a vertical plane (containing the axis of highest symmetry). a^. reflection in a plane diagonal to the horizontal and vertical planes. S1™: rotation around an axis by m • "^— and reflection in a plane perpendicular to that axis. i: complete inversion through the origin. 61 Appendix A. Basic Group Theory Clearly, some of these operations may be constructed from others. For example, S™ = Cn when m is even, 5*2 = i. In general symmetry operations do not commute (just as group operations do not necessarily commute). However, the following types of symmetry operations do commute: 1) rotations about the same axis 2) reflections through planes perpendicular to each other 3) inversion and any reflection or rotation 4) C2 rotations about perpendicular axes 5) rotation and reflection in a plane perpendicular to the rotation axis. A group is often described by a character table instead of a multiplication table. For example, the group, C$v which contains an identity, two C3 rotation axes in one class and three uv reflection planes in another class, has the following character table; Czv A, A2 E E 1 1 2 2C3 1 1 -1 3av 1 -1 0 Table A.2: The Czv character table The numbers in the character table are the characters (or Traces) of the matrix belonging to the row's representation and corresponding to a group element in the column's class. In general, A\ will refer to the total symmetry representation and further, ^4j's will refer to other 1-dimensional representations. i?'s will refer to 2-dimensional representations, and T's will refer to 3-dimensional representations. The character of the identity in a given representation will 62 Appendix A. Basic Group Theory be the dimension of that representation, since any matrix corresponding to the identity element must be the n x n identity matrix. Many important theorems and rules concerning the construction of a character table are derived from the following theorem. The proof is not included but may be found in [9, p.371]. Theorem: (A.l) R yjkh where h is the order of the group, Ti(R)mn is the element in the mth row and the nth column of the matrix in the ith irreducible representation corresponding to the group operation, R. Zj is the dimension of the ith irreducible representation and * is complex conjugation. This theorem essentially states that any set of corresponding matrix elements of two matrices in an irreducible representation behave as orthogonal vectors in h-dimensional space. One important result of Equation A.l which is not proven here is that the sum of the squares of the irreducible representations' dimensions is equal to the order of the group: i Substitute m = n and m' = n' into Equation A.l; / j 1 i\**-)mm\ i\"')mlm! — , v-mm'• R % Then, summing over m and m', L H S = J2J2J2T^R^mri(R)m'm' = ^HT^R^m^i(R)m'm') = 1 > ( # ) ] 2 m m1 R R m m1 R RHS = - Y^ ^2 5mm' = yh = h k m m> k and therefore, £[*(*) ]* = *• (A.2) R 63 Appendix A. Basic Group Theory Suppose that i / j in Equation A.l. Then we have, ^r i ( i? ) m „r i ( J R) m n = o i^j. R Now let m = n, 5^r i(JR)mmr i(JR)mm = o i^j. R Summing over m, Y^Xi{R)Xj{R) = ^2(J2r^R^rnTj(R)mm) = 0 R m R and therefore, Y,K(R)XJ(R) = O i^j. (A.3) R Combining Equations A.2 and A.3 we have, Y,Xi(R)Xj(R) = h6ij. (A.4) R Reducible representations are composed of a certain number of irreducible representations. Suppose that a,- is the number of times that the block consisting of the j t h irreducible repre-sentation appears in a given reducible representation. If x(R) 1S the matrix corresponding to the operation, R in the reducible representation then, X(R) = ^ajXj(R). (A.5) 3 Let us multiply Equation A.5 by the character from the ith irreducible representation and sum over all operations; Y,x(R)xi{R) = ^2J2a^jiR)xi(R) R R j = ^aj^Xi(R)xJ(R) j R = 2, Q-jhSij = aih j using Equation A.4 64 Appendix A. Basic Group Theory Therefore, the number of times a given irreducible representation appears in the reduction of a reducible representation is, ai = lj2^R^i(R)- (A-6) R If there exist two irreducible representations, X and Y, with basis functions, X\... Xm and Y\ ... Yn respectively (these may be eigenfunctions of the wave equation) then we construct the direct product representation which has the basis functions, XiYj. Let R be a symmetry operation, then define Xij and y^ as; m RXi = y xjiXj n RYk = Y,yikYL. Then, m n RXiYk = ] P ^2 XjiyikXjYi = ^ zjl:ikXjY[. i=i 1=1 hi If we denote the direct product group with the index, z, then, m n Xz{R) = Y^ ziUi = ^2J2xHyu = XX{R)XY{R)-Therefore, xi2(R) = xi(R)x2(R)- (A-7) Finally, the characters of a symmetric product of a representation with itself are denned; ix(R)2} = \lx(R)2 + x(R2)} (A.8) and the characters of an antisymmetric product are; ix(R)2} = \[x(R)2 - x(R2)}- (A.9) 65 Appendix B Fibre Bundles The following is a brief summary of Fibre Bundles and their Characteristic Classes. Those familiar with these concepts may skip this appendix. Alternatively for a more thorough coverage the reader is directed to [18] or [25] for bundles and to [24, 21] for Topological material. B.l Basic Definitions Definition: A manifold is an n-dimensional space which is locally homeomorphic to W1. That is, a manifold, M, is covered by a (not necessarily finite) union of subspaces, {U a}, such that for each XJa there exists a homomorphism, ha : Va —> XJa, where V a is an open subset of W1. Furthermore, for any two subspaces where U a n U ^ 0 there exists a set of continuous change-of-basis functions from Ua to Up. Example: The simplest example of a smooth manifold is Rn, which is covered by itself and is homomorphic to Rn by the identity map. Example: The two dimensional sphere, S2, embedded in 1R3 is a two dimensional manifold which can be broken into the top half plus a small strip (defined by the azimuthal angle; 0 < ' 0 < f + e ) e > O ) and the bottom half plus a strip; (f - e < if) < TT). Each "half" is homomorphic to the plane, R2. Definition: A tangent vector to a manifold, M, at the point x 6 M is a vector in Rn which can be expressed as the instantaneous velocity vector of some path through x in M. In this way the path defines the vector. The vector space consisting of all vectors tangent to M at x is called 66 Appendix B. Fibre Bundles the tangent space of M at x, denoted T^M). Definition: The normal space of M is the space of all vectors normal to TX(M) for all x € M, where normal to M implies, i f M c t m for some m € Z then TXM C TxRm, x € M C Rm. Definition: A real vector bundle, £, consists of a total space, E, a projection map, n : E —> B, as well as a vector space structure on the subset 7T~1(b) C E for all b € B. B is called the base space and E is called the total space of the bundle. The set rr~l(b) = F^ is called the fibre over b. The vector bundle, ^ is displayed below, £ : F< >E TT B where the space F is homomorphic to Fj, for all b € B. If the dimension of F& is n, for all b € B then the fibre bundle is said to be n dimensional. Every vector bundle satisfies the following local triviality condition; for every b € B, there is a neighborhood, U C B, and a bundle isomorphism, h, such that the following square commutes: UxRk ^->p-l(U) v'. O P / Identity / U > U The restriction of h to each fibre; h : bxRk —> p~l(b) is a vector space isomorphism for all b £ U. If a fibre bundle consists of p : E —> B and B is covered by open spaces, {Vi} then we denote the local trivialization maps as {<&}; & : F i x K n - > p - 1 ( V j ) (pi : V H-+ E x En 67 Appendix B. Fibre Bundles The transition functions in the base space from Vi to Vj are denoted {gji}; 9ij •• Vi n Vj x Rn — Vi n Vj x Rn gij : (6, x) H-> (6, y) such that, 9ij = <t>J °'t>i-Definition: A cross-section of a bundle is a continuous function, a : B —> E taking each b 6 B to a vector in Ft. Example: An infinitely tall cylinder is a fibre bundle over a circle: s1 Figure B.l: A real line bundle over S1 where S1, the horizontal circle, is the base space and each fibre over a given 6, (0 < 9 < 2ir) is simply the line passing vertically through the circle at 9. A point in the total space, (9, x), is a point on the cylinder, defined by both a point on the circle at angle, 9, and the distance,^, 68 R1 Appendix B. Fibre Bundles along the line running through that point. This bundle is trivial. The fibres are lines, each homeomorphic to R1 and sections are just curves around the cylinder. In other words, a section, a : S1 —y Sl x Rl takes a point, 9, on S1 to a point on the cylinder directly "above" 9, and thus a traces out a path around the cylinder which is periodic, with period 2ir. Example: The Mobius band is a non-trivial bundle over S1. That is, the total space, E ^ 5 1 x R1. The fibres are each homomorphic to R1 but each section, a : Sl —> E must be an anti-periodic function. In other words, once we go around the S1 by 2-7T, the space has undergone a twist and the function a(2ir) = —<r(0). Definition: If £ and 77 are fibre bundles over the same base space, B, they are isomorphic if and only if there exists a homomorphism, h : E(0 -> E{rj) which takes Fb(£) isomorphically to ^(77) for each b € B. Definition: We define the Tangent Bundle of a manifold M, denoted TM, as the bundle over M with fibres, Fb(TM)=Tb(M). (B.l) B.2 Constructing new vector bundles 1) Let £ be the bundle, IT : E —> B and let B' c B. The restriction of £ to the new base, B', is IT' : E' —> 5 ' where E' = 7r_1(i?') and ir' = TT\E>- This restricted bundle is written, £!#/. It should also be noted that the restricted bundle is simply a special case of the pullback bundle, (defined below). 2) Let B2 be an arbitrary topological space and £ be the bundle, n : E —> B. If / : B2 —> B, 69 Appendix B. Fibre Bundles then the pullback bundle, /*£, over B2 is : / * £ : E2 E :£ # 9 / • s where E2 = {(b,e)\b G i?2,e G E,f(b) = 7r(e)} and 772(6,e) = 6. Thus, the following diagram commutes; where f(b, e) = e. 3) The Cartesian Product of two bundles, £1 and £21 1S £1 x £2 a n d corresponds to; £1 x £2 : Fi x F2 --> Ex x £ 2 5 i x B 2 where (TTI X TT2) H & I . M = *l>i(£i) x ^62 (6)-5) The Whitney Sum of two bundles, £1 and &, both over the same base space is £1 © £2 = d*(£i x £2), where d : B -+ B x B takes 6 •-> (6, b). Each fibre, F6(fi © &) ^ ^(£1) © ^(62)-Definition: If £ and 77 are bundles over the same base space and E(£) C E{rj) then £ is a sub-bundle of 77 and we write £ C 77. 6) If £ C 77 and 77 (and thus -£(??)) has a Euclidean metric, then £ x is the orthogonal compliment of F, where F^) = {v G ^ ( T ? ) ^ • «; = 0 for all u; G i ^ f ) } and E(^) = U6 W f X ) } -7) Following from the definition of the Tangent Bundle is the Normal Bundle, v. If the base 70 Appendix B. Fibre Bundles space, B' <-> B for some B and TB' is a sub-bundle of TB, then the normal bundle over B' is the orthogonal complement of TB' in TB. 7) If 77 is a bundle with projection map, p : E —• B and £ is a bundle with projection map, q : X —> B with the same base space, then the pullback bundle, 77*, of 77 over £ is defined by the following commutative square: F F •V where the total space of 77* is E = {{x,y) € XxE\p(y) = q(x)} and the projections q : (x,y) >-+ y and p : (x,y) H-> x. 77* is simply the induced bundle of 77 over another Fibre bundle. 8) The following is a definition of an associated principal bundle. Let £ be a k-dimensional fibre bundle, £ : F< >E B with transition functions, g^ : Ui D Uj x l ' —» C/j (~l C/j x Efc for subspaces (7j, C7j G B. Now restrict the transition functions to each fibre, 7r_1(6), for every b € B. g\y : b x Mfc -» 6 x 52- is now an isometry of Kfc and is therefore and element of 0(k). We may now define a 71 Appendix B. Fibre Bundles continuous map, gij :Uif\Uj^G = 0{k) b^gffeOik). G is denoted as the structure map of the bundle. We now define £, as the associated principal bundle of £. £ has the same base space, B, with fibres, G equal to the structure group of £. £ also has the same trivialization neighborhoods, with transition functions, g^ -.UinUjxG^UinUjXG (b,g)~(b,gV.g) The total space of £ will be denoted, E. Let Z be the union of the locally trivial fibres, Z — \Ji{Ui x G} and let q± be the inclusion map, qi-.UixG^Z. Two point in Z are equivalent, (bi, g\) ~ (62,52) if and only if 61 = 62 G Ui<T\Uj and 52 = g\j 9i-The total space of the associated principal bundle is then the quotient space, E — Zj ~ . where q : Z —> E. Finally, the projection map of £ is simply, p : (b,g) *—> b, which is well defined, since if (bi,9i) ~ (62,^2) then 61 = 62-B.3 Characteristic Classes Fibre bundles over a given base space are characterized by naming them according to their characteristic classes. Every real vector bundle, £, of dimension, n, has n + 1 Stiefel-Whitney classes which are cohomology classes of the base space, wi(Z)£H*(B({;);Z2). Following are the seven defining properties of Stiefel-Whitney classes. #*(!?(£);Z2) is the cohomology group of the base space, #(£), with coefficients in the group, Z2. A good background on the subject of homology and cohomology theory is given in [24, 21]. 72 Appendix B. Fibre Bundles 1) The first Stiefel-Whitney class of any bundle is always 1; WQ — 1 E H°(B(£); Z2) 2) All Stiefel-Whitney classes of index greater than the bundle's dimension are zero; Wj = 0 for all j > n. 3) If two bundles have the same base space; B{£) — B(rj) then the direct sum of these bundles have Stiefel-Whitney classes, k 3=0 4) If / maps the base space of one bundle to another, / : B(£) —> B{rj) then Wi(Q=fwi(ri) (B.3) where /* is the pullback of / . 5) If two bundles are isomorphic, £ ~ 77, then their Stiefel-Whitney classes are identical. Similarly, it is important to note that if two bundles are not isomorphic then their Stiefel-Whitney classes cannot all be equal. However, identical Stiefel-Whitney classes does not imply isomorphic vector bundles, with the exception of line bundles (one dimensional vector bundles). 6) If £ is a trivial bundle then all Stiefel-Whitney classes are zero. ioi(£) = 0 f o r a l H > 0 : (B.5) 7) If £ is a trivial bundle then Wi{£@i]) =Wi{rj) for alii. (B.6) Definition: The total Stiefel- Whitney class of £ is defined as, n H0 = J2M0- (B.7) 73 Appendix B. Fibre Bundles Beyond simple real vector spaces are oriented spaces: Definition: A vector space, V, has two distinct orientations, each an equivalence class on the bases of V. Suppose {vi,... ,vn} and {w\,... ,wn} are two ordered bases for V. These bases are related by; n and are considered equivalent if and only if det [ajj] > 0. It is important to note that the choice of an orientation for V is equivalent to the choice of a preferred generator for the singular homology group, Hn(V,Vo;Z) and thus a preferred generator, uy, for Hn(V, VQ; Z). VQ is simply the set of non-zero vectors in V. Definition: An orientation of a fibre bundle, £, is defined by a function which assigns an orientation to each fibre, F^, in £. This orientation must be preserved by the local triviality homomorphism between JVxM" and n~1(N) for all neighborhoods, iV C B(£),determined by beN. An alternate definition of an orientable vector bundle is one which has a trivialization atlas (the spanning set of open spaces and transition functions) such that all transition functions are elements of the group, SO(N). For an n-dimensional oriented fibre bundle, (, with projection map, ir : E —> B, Hl{E,Eo;Z) = 0 for alii < n and there exists only one class in Hn(E, EQ\ Z) which is u, such that; u\{FbiFo} = uFb € Hn(Fb, i f ; Z) for all b € B. (B.9) Definition: The Euler Class of an n-dimensional oriented fibre bundle, £, is defined by; e(0 = (K*)-1f(u)eHn(B;Z) (B.10) where j * : Hn(E,E0) —> Hn(E) in the cohomology exact sequence, and n* : Hn(B) —> Hn(E). ir* is the pullback of the projection map, ir : E —> B and is an isomorphism since E is continu-ously deformable into B. 74 Appendix B. Fibre Bundles The Euler class of a fibre bundle has the following properties: l)The Euler class of a bundle changes sign if the orientation of the bundle is reversed. 2) If £ is an n-dimensional, oriented bundle and n is odd then, e(£)+e(O = 0. 3) If / : B —> B' is covered by an orientation preserving bundle map, F : £ —> £' then e(0 = /MO-4) The homomorphism, /i : Hn(B;Z) -+ Hn(B;Z/2) takes e(£) ^ wn(£) which implies the following property; 5) e(f @r))= e{i)e{rf) and e(£ x 77) = e(£) x efo). We may also begin with a real n-dimensional fibre bundle, £, and complexify it, creating £' = £<8>C, a complex n-dimensional bundle. Each fibre, Fb, of £ is a real n-dimensional vector space. The corresponding fibre, Fb' of £' is a complex n-dimensional vector space such that, Fl = FbxC = Fb@iFb. (B.ll) Of course, for any complex n-dimensional bundle, 77, there is a canonical underlying real 2n-dimensional bundle, 77^ , arrived at by simply ignoring the complex structure on the fibres of V-With a complex n-dimensional bundle, 77, we can construct the following new bundles: 1) The conjugate bundle, 77, to 77 is the complex n-dimensional bundle such that, m - rjR. 75 Appendix B. Fibre Bundles fJR has the opposite complex structure to 77 so that the map, / : E{rj) —> E{fjn) is defined by / : Ae i—*• A/(e) for all A S C. In this case, / is fibrewise complex conjugation, the equivalent of multiplying by i. 2) The dual bundle, Homc(rj,C), of 77 is the bundle over the same base space, B(rj), whose fibre corresponding to Fb(rj) is Homc(Fb,C). Now we may introduce a new kind of characteristic class for complex fibre bundles. Definition: The Chern classes of the complex n-dimensional bundle, 77, are; Ci{V) € H2l(B;Z) (B.12) where cn(rj) = e„ (??#), the top Euler class of the underlying real bundle. Ci(rj) = 0 for all i > n , 00(77) = 1 and (B.13) ci(V)=7r*0-1ci(rl0). (B.14) where TTQ is the isomorphism, TTQ : H2l(B) —• H2I{EQ). 770 is the (n-l)-dimensional fibre bundle whose base space is Eo, the set of non-zero vectors in E(rj). (v,F) specifies a point in EQ and thus the fibre of 770 over a vector, v, is the orthogonal complement to v in the fibre, JP, and is an n-dimensional vector space, as shown in the following diagram, 770: e - 1 ' >£(77) 77: s2n~1=cn^ >EQ TO B 7IQ is an isomorphism for % < n due to the homotopy type of the fibres of 77 which are that of S2n~1. Thus, cn_ 1(770) = en(r]oR) since cn_ 1(770) is the top Chern class of 770 and cn_i(77) 76 Appendix B. Fibre Bundles is well defined as 7rQ_1cn_i(7?o) = 7To_1en_i(77oi{). cn-2(vo) an<3 therefore c„_2(r?) can be found inductively just as cn-i(rj) was. This process may be repeated until we find Ci(rj) = 0 for all i < 0. Alternatively, the Chern classes of a complex n-dimensional bundle, 77, may be defined in terms of the Chern classes of the universal bundle, 7™, over Gn(C°°). The Infinite Grassmanian manifold, Gn(C°°), is the union of all finite Grassmanian manifolds, Gn(C°°)= \jGn(Ck), k>n where Gn(Ck) is the set of all complex n-planes through the origin in Ul+k. The topology on GntC00) is the inductive topology, which means that a subset, U C Gn(C°°) is open if and only if U n Gn(Ck) is open for each k eZ, k > n. The universal bundle, 7", has total space, £ ( 7 n ) = Gn x C°° (B.15) where Gn refers to Gn(C°°) and n : E(-yn) —> Gn takes (n-plane X, vector x g X) 1—> n-plane X £ Gn. The universal bundle over Gn(B.°°) is defined identically using real n-plane bundles. Theorem B . l : if*(Gn(C°°); Z) is the cohomology ring over Z freely generated by the Chern classes c i (7 n ) , . . . c n (7 n ) . Furthermore, Theorem B.2: For any complex n-dimensional fibre bundle, rj, there exists a function, / : r\ —> 7 n such that the Chern classes of 77 are determined by C{{r\) — f*Ci(-yn). Definition: The total Chern class of the fibre bundle, 77, is defined; n 77 Appendix B. Fibre Bundles Just as for the Stiefel-Whitney classes or the Euler classes, the Chern classes obey the following properties: 1) For any two complex n-dimensional fibre bundles, £ and rj, c(Z®v) = c(Oc(ri). (B.16) 2) For any complex n-dimensional bundle, £, «($ = ( - 1 ) ^ ( 0 (B.17) where £ is the dual bundle to £. Definition: For a real n-plane bundle, £, the Pontrjagin Classes, Pi(0eH4i(B;Z) . (B.18) are defined as, Pi(£) = ( - l o c a t e ®C). (B.19) Po(£) = l ,P i (0 = 0 for all i > n/2 and the total Pontrjagin class is defined as [n/2] i = l where [n/2] is the largest integer less than or equal to n/2. The following properties are true for Pontrjagin classes: 1) If f is a bundle map, / : £ —-> n then, Pi(0 = f*Pi(v)- (B.20) 2) If e is a trivial n-plane bundle then, p(£ © e) = p(£) for all bundles, £. (B.21) 78 Appendix B. Fibre Bundles 3) For any two bundles, £ and rj, Pit®v)=P(0p{v)Mod2. (B.22) By Theorems B.l and B.2 it is easily shown that the Pontrjagin classes are defined by the Chern classes and are equal to: PkiVR) = Ck(v)2 ~ 2cfc_i(7?)cfc+i(77) + - . . . ± 2ci(77)c2ifc_i(r/) =F 2c2k(ri). (B.23) 79 Appendix C The quaternion Numbers The following conventions and theorems may be found in [23]. The set of quaternion numbers, denoted EI to honor William Hamilton, is a non-commutative field over the real numbers, generated by four elements, l,i,j and k, such that; t*=f = k2 = -l (C.l) and ij = —•}% = k, jk = —kj = i, ki = — ik = j . (C2) For a given element of the quaternions, x = XQ + x\i + x^j + xzk GH, i i £ M, the conjugate of x is x = XQ — x\i — X2J — x%k and the norm of x is |a;|2 = xx. The following convention of multiplication on the right is used to define multiplication in a quaternion vector field. Let v and w be elements in a quaternion vector space, V, and x, y G H be quaternion numbers. Then, (v + w)x — vx + wx, v(x + y) = vx + vy, v(xy) — (vx)y. (C3) Another way to look at a quaternion vector space is as a complex vector space with an operator, 6 , called the quaternion structure map. Q is anti-linear and 0 2 = — 1. A complex n-dimensional 80 Appendix C. The quaternion Numbers vector space, W, is then transformed into a quaternion n-dimensional space by the following conventions; vi = iv, vj = Qv, vk = (vi)j = Q(iv) = —%Qv (C4) where v G W. Multiplication on the right refers to multiplication by a unit quaternion and multiplication on the left is just by the complex number i or by the structure map, 0 . We can now view VT as a quaternion space and give it a basis, {ej, ©e^} over C, where i = 1 . . . n and n is the quaternion dimension of W. 81 

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