spin-vibronic angular momenta

Spin-orbit coupling and the Jahn-Teller effect: vibronic and spin-vibronic angular momenta Timothy A. Barckholtz and Terry A. Miller Laser Spectroscopy Facility Department of Chemistry The Ohio State University 120 W. 18th Avenue Columbus, Ohio 43210 January 26, 1999 1 Introduction The concept of symmetry and its breaking holds a special and important place in the physical sciences. One of the best known examples of such symmetry breaking is the Jahn-Teller effect. In their original paper, Jahn and Teller demonstrated(jahn & Teller 1937) that a symmetry-dictated electronic degeneracy will always be raised by a non-totally symmetric distortion linear in nuclear coordinates(bunker & Jensen 1998). If the Jahn-Teller distortion is large it will result in a wavefunction localized around the minimum (minima) of the resulting potential energy surface (PES), which occurs (occur) at other than the symmetrical nucelar configuration. In this case, frequently referred to as the static Jahn-Teller effect, the vibronic and spin-vibronic eigenvalues and eigenfunctions can be calculated much as for any other asymmetric molecule. A rather more interesting case and the one we consider herein occurs when the Jahn-Teller distortion is not large enough to localize the molecule about a single minimum on the PES for the duration of the 1

experimental measurement. This case is often called the dynamical Jahn-Teller effect. In this latter case the calculation of the vibronic and spin-vibronic eigenvalues and eigenfunctions is considerably more complex. Many of the princples of such calculations were laid out(child 1963, Child & Longuet-Higgins 1961, Longuet- Higgins, Öpik, Pryce & Sack 1958) in the late 1950 s and early 1960 s. While pioneering, these calculations were perturbative in nature and performed only for molecules with a single Jahn-Teller active vibrational mode. They were therefore of limited value to an experimentalist attempting to decipher the spectra of a molecule with more than three atoms and various strengths of Jahn-Teller coupling. The next advancement in the calculation of the vibronic levels of a Jahn-Teller state came in the early 1980s, when the introduction of high speed computers made possible large calculations of the vibronic energy levels. The first applications of these calculations were the ground states of the halogen-substituted benzene ions, for which new jet-cooled laser induced fluorescence spectra and related experiments offered the first reliable data(cossart-magos, Cossart & Leach 1979a, Cossart-Magos, Cossart & Leach 1979b, Cossart- Magos, Cossart & Leach 1979c, Cossart-Magos & Leach 1980a, Cossart-Magos & Leach 1980b, Cossart- Magos, Cossart, Leach, Maier & Misev 1983, Sears, Miller & Bondybey 1980, Sears, Miller & Bondybey 1981b, Sears, Miller & Bondybey 1981a, Yu, Foster, Williamson & Miller 1990, Bondybey, Sears, English & Miller 1980, Miller & Bondybey 1983) on isolated molecules, with several Jahn-Teller active modes, which could test quantitative Jahn-Teller calculations. Since these states exhibited both linear and quadratic Jahn-Teller coupling, in multiple active modes, these calculations were quite substantial in size, requiring the earliest versions of the Cray supercomputers to produce solutions. One simplifying feature of the calculations on the substituted benzene ion ground states was that spinorbit coupling was negligibly small. Because Jahn-Teller coupling and spin-orbit coupling both involve the electronic angular momentum of the state, a true solution for the vibronic structure of any non-singlet degenerate state must simultaneously include both of these effects. The most common approach has been to first solve the Jahn-Teller problem, and then apply several approximations to deduce the spin-orbit splitting of the levels(ham 1965, Ham 1972, Child 1963). During the course of the analysis of the vibronic structure of the ground states of the CF 3 OandCF 3 S radicals(barckholtz, Yang & Miller 1999), we realized that separating the Jahn-Teller and spin-orbit cou- 2

pling into two separate problems has some inherent deficiences. We therefore reinvestigated the Jahn-Teller problem, including spin-orbit coupling directly in the Hamiltonian, rather than as an afterthought to the Jahn-Teller solutions. We have recently published this research(barckholtz & Miller 1998), which we will refer to as Paper I throughout this chapter. In that work, we described in some detail calculations of this kind and reported on several features of Jahn-Teller and spin-orbit coupling that had, for the most part, been neglected. It has generally not been recognized that the operator that couples the vibrational and electronic angular momenta depends on the symmetry species of the electronic state and point group of the molecule. As we showed in Paper I, this has a profound effect on the definition of the conserved quantum number for linear Jahn-Teller coupling. As we will discuss quantitatively in this chapter, the Coriolis coupling is significantly different between these two general cases of linear Jahn-Teller coupling. A second effect we will discuss in this chapter is a direct result of the combination of spin-orbit and Jahn- Teller coupling. The most noticeable effect of the addition of spin-orbit coupling to the Jahn-Teller problem is a splitting of the vibronic energy levels into two spin-vibronic energy levels. At higher resolution, though, as we showed in Paper I, the effective spin-rotation constant of the molecule can be drastically affected by the presence of both spin-orbit and Jahn-Teller coupling. The two spin-components of a vibronic level have usually been assumed to have identical Coriolis coupling constants. However, they in fact can have quite different Coriolis constants, the difference of which is manifested in the observed spin-rotation constant of the state. In this chapter, we will present detailed calculations of this effect for an example state. 2 Hamiltonian, basis set, and eigenvectors The appropriate Hamiltonian operator for the components of a degenerate electronic state is the sum of a number of terms, Ĥ = ĤT + ˆV + ĤSO + Ĥrot, (1) where ĤT is the vibrational kinetic energy of the nuclei. The sum of Ĥ SO and the potential ˆV define the potential energy surface (PES). We refer the reader to our previous papers(miller & Bondybey 1983, 3

Barckholtz & Miller 1998) on this topic for the terms of the power series expansion of the potential ˆV and for the details of the PES. For our purposes here, it suffices to say that the potential has, in addition to the standard harmonic oscillator terms for the degenerate modes, terms linear and quadratic in the vibrational coordinates of the degenerate modes that account for linear and quadratic Jahn-Teller coupling. In this chapter, we will consider only linear Jahn-Teller coupling in a single active mode with a harmonic vibrational frequency ω e and a linear Jahn-Teller coupling constant D (see Paper I for precise definitions of these parameters). The spin-orbit Hamiltonian ĤSO is parameterized by the product aζ e,whereζ e is the projection of the electronic orbital angular momentum on the principal axis and a is the spin-orbit coupling constant. The details of the rotational Hamiltonian have been presented elsewhere(barckholtz & Miller 1998). The basis set used to compute the Hamiltonian matrix is the product of electronic, vibrational, and spin functions: Λ p v i,l i Σ. (2) i=1 Λ=±1; v i =0, 1, 2,...; l i = v i,v i 2,v i 4,..., v i +2, v i ;Σ= S, S +1,...,S 1,S We include in the basis set only those vibrational modes that are Jahn-Teller active (see Tables 2 and 3 of Paper I). Under the harmonic oscillator approximation, each of the v i can take any positive integer value, which makes the basis set infinitely large. Therefore, the basis set for each Jahn-Teller active mode must be truncated to a manageable level. In general, the larger the Jahn-Teller coupling constants are for each mode, the larger the basis set must be for that mode. In practice, a relatively small basis set is typically used to do some initial calculations. The basis set is then expanded until additional basis functions have a negligible effect on the eigenvalues. In Paper I, we developed a general selection rule for which symmetries of vibrational modes of a point group will be Jahn-Teller active for a given electronic state s symmetry. In our notation, s e and s v are the numerical labels from the irreducible representations of the electronic and vibrational symmetries, respectively (see Eqs. 3 and 5 of Paper I). As we showed, the following equality must hold true for the 4

k th -order Jahn-Teller term in the Hamiltonian to be non-zero, (2s e +( 1) s k ks v )modn =0, (3) where s k must be either 0 or 1. The choice of s k is dictated by the symmetry of the state and its point group, and has a direct impact on the relative phases of the electronic and vibrational terms in the Hamiltonian. As we show in the remainder of this chapter, the value of s k has no effect on the eigenenergies of the vibronic or spin-vibronic levels, but has a direct impact on the Coriolis and spin-rotation parameters determined via their eigenfunctions. Jahn-Teller coupling destroys the electronic and vibrational angular momenta quantum numbers Λ and l i, respectively. (Strictly speaking, Λ is not a quantum number, but for most purposes can be treated as one.) However, a new, conserved quantum number, j, can be defined by the linear combination j = l t + 1 2 ( 1)s1 Λ, (4) where l t = i l i and s 1 = 0 or 1 is determined from the symmetry of the state (see Table 2 of Paper I). The Jahn-Teller quantum number j is conserved for linear coupling (k = 1), though not for quadratic coupling. We have used two general forms for the eigenvectors resulting from the diagonalization of the Hamiltonian matrix, depending on whether spin-orbit coupling is or is not included directly in the vibronic Hamiltonian. For the approximation when spin-orbit coupling is not included directly in the calculation, the form of the eigenvectors is j, n j,α Σ = i ( ) p c i,nj Λ i v m,i,l m,i Σ, (5) m=1 where α is the symmetry of the level under the molecules nominal point group. When spin-orbit coupling is included in the Hamiltonian, the eigenvector notation is n j,j,ω,α = ( ) p c i,nj,ω Λ i v m,i,l m,i Σ i, (6) i m=1 where Ω = Λ l t + Σ and the summation over i runs over all of the basis functions used in the calculation. Foracoefficientc i,nj,ω to be nonzero, the equality of Eq. 4 for j must be satisfied by the basis function. Each eigenvector n j,j,ω,α has an associated eigenvalue E j,nj,ω. The notation n j,j,ω,α indicates to which j-block the level corresponds and which eigenvector, n j, it is from that symmetry block, with the lowest 5

energy solution of each symmetry block being n j = 1. In the strong spin-orbit coupling limit the vibrational quantum number v again becomes useful and it can be used in place of n j. The symbol α represents the symmetry of the level under the spin-double group appropriate for the molecule. 3 Computational Details A Fortran90 computer program, SOCJT (pronounced sock-it ), has been written to calculate the eigenenergies and eigenfunctions of a molecule described by the Hamiltonian of equation (1), excluding Ĥrot. The program can handle an arbitrary number of active modes and the truncation of the basis set is limited only by available memory and computer time. The program has been written for the Cray T90 computer and for DOS on personal computers. It is this program that has provided the numerical results presented in the remainder of this chapter. The program is available upon request from the authors. One of the features of this program that makes it amenable to realistic systems is its ability to handle an arbitrary number of Jahn-Teller active vibrational modes. The basis set, Eq. 2, is the set of all possible combinations of v and l for each mode. The generation of the basis set uses a recursive subroutine so that any number of vibrational modes can be included in the basis set. In this way, only one program had to be written, which had no limit on the number of active modes, a significant improvement over previous programs written for this purpose. To make the diagonalization routine most efficient, the symmetry blocking of the Hamiltonian matrix is utilized to the full extent possible. For linear Jahn-Teller coupling, the basis set and Hamiltonian matrix are constructed in blocks diagonal in j (defined by Eq. 4) and Σ, while for quadratic coupling the selection rule on j of j = ±3 is utilized. While all of the examples discussed in Paper I were appropriate for doublet electronic states S =1/2, we have written the program to be generalized for any half-integer or integer value of S. The program is not applicable to the cubic point groups, as the higher symmetry of these point groups would alter the structure of the program significantly. The program can be used for Renner calculations of linear molecules by setting the linear coupling constant to zero and treating the quadratic Jahn-Teller coupling constant as the Renner coupling constant ɛ. 6

The efficiency in the generation of the Hamiltonian matrix is maximized by the utilization of the selection rules on the basis set. Because the matrix is extremely sparse ( 5% of the matrix elements are non-zero for a single mode linear Jahn-Teller coupling case; 0.001% of the matrix elements are non-zero for a 5- mode linear Jahn-Teller coupling case), only the non-zero matrix elements are stored in memory. Three one-dimensional arrays hold the value, row index, and column index of each of the non-zero matrix elements. In this way, the memory requirements of the calculation scale linearly rather than quadratically with basis set size. The sparseness of the Hamiltonian matrix is also utilized in the diagonalization routines, which have been optimized for sparse matrices. Furthermore, the Hamiltonian matrix is a positive-definite symmetric Hermitian matrix, which allows even more highly optimized diagonalization routines to be used. Because in spectroscopic problems only the lowest energy levels are usually observed, diagonalization routines that find only the lowest eigenvalue solutions are used. The version of the program written for the Cray T90 uses algorithms from the NAG libraries, which are written and optimized specifically to take advantage of the Cray s vector capabilities. The DOS version uses the Underwood(Underwood 1975) method of diagonalizing sparse matrices using freely available code available over the Internet at http://www.netlib.org. One limitation of the Cray version of the program is that it is not able to use the eigenvectors from a previous calculation to begin the iterations of the diagonalization routine. The DOS version does have this capability, so that even though the DOS version will not be as fast as the Cray version, it will be relatively more efficient for the calculation of the eigenvalues and eigenvectors when a parameter (such as D i or aζ e ) has changed only slightly from one calculation to the next. One limitation on this capability is that the basis set must be exactly the same from one calculation to the next; i.e., that an additional mode has not been added to the calculation, or that the maximum values of v on each vibrational mode have not been changed. Table 1 illustrates the size of the basis set, number of non-zero matrix elements, and approximate compuational times for several different coupling combinations. As the first several rows indicate, increasing the basis set by an additional vibrational mode dramatically increases the size and computational cost of the calculation. Quadratic Jahn-Teller coupling also significantly increases the size of the calculation, though not as dramatically as does the addition of another vibrational mode. As can be seen from the Table, the 7

computational costs on the Cray T90 are minimal for even relatively large calculations. The computational time required for the DOS version, while approximately 25 times longer than on the Cray T90, are still reasonable for most situations. One of the advantages of the DOS version over the Cray version is that the DOS version is able to use as initial guesses to the iterative procedure the final results from a previous calculation. The program also calculates several quantities directly related to the spectroscopy of the state. When spin-orbit coupling is not directly included in the Hamiltonian, the Ham quenching parameter d j,nj (see Eqs. 80-83 of Paper I) is computed for each vibronic energy level. The Coriolis coupling constant (vida infra) is also calculated for each level. Perhaps most relevant to experimentalists, the relative transition intensities for E A electronic transitions are calculated, both for cold and for hot-band spectra. Lastly, to aid in the fitting of the energy levels of the state, the derivatives of the energies with respect to the parameters of the coupling calculation (all ω e,i, D i, K i,andaζ e ) are computed. 4 Energies of the vibronic and spin-vibronic levels In the remainder of this chapter, we will focus on the differences (or lack thereof) between the two general symmetry cases, s 1 = 0 (appropriate for the most well-studied case, a 2 E state of C 3v symmetry) and s 1 = 1 (appropriate for many states of molecules that belong to point groups with C 5 or higher axes). Three properties will be discussed: the energies of the vibronic levels under linear Jahn-Teller coupling, the Coriolis coupling constants under linear Jahn-Teller coupling, and the effective spin-rotation constant induced by the presence of spin-orbit coupling. For illustration, we present only the case of a single Jahn-Teller active mode; results for more than one active mode are easily calculated using our computer program. 4.1 Energies of the vibronic levels under linear Jahn-Teller coupling We show in Fig. 1 the energies, as a function of D, of the lowest several vibronic energy levels of a Jahn- Teller state. There is in fact no difference in the energies of the vibronic levels between the two general symmetry cases. Thus, if all one is interested in is the energies of the levels, the symmetry of the state can be completely neglected. 8

Fig. 1 shows the absolute energies of the vibronic levels as a function of D, relative to the energy of the symmetric point of the PES, which is defined as the zero of energy for Fig. 1. If, instead, the zero of energy is defined as the energy of the lowest-energy vibronic level, the energies of the levels are given by Fig. 2. Fig. 2 is a more useful diagram when examining the spectroscopy of these levels. Because the lowest energy vibronic level, denoted 1 2, 1,e Σ=±1/2 in the notation of equation (6), is the zero-point energy level of the molecule, the differences between its energy and the other energy levels will be the energies measured in spectroscopic experiments. In particular, two common techniques, laser-induced fluorescence and infrared absorption, most often observe the second j = ± 1 2 level ( 1 2, 2,e, Figure 2). The energy difference between these two levels is the fundamental transition frequency of the vibrational mode, ω 0,i. However, as Figure 2 shows, this energy difference will be greater than the equilibrium vibrational frequency ω e,i. This holds true even when spin-orbit coupling is included in the calculation, provided that the spin-orbit coupling splitting of the electronic potential is not larger than the harmonic frequency of the vibrational mode. The difference between the two frequencies ω e,i and ω 0,i is governed by the magnitude of D i. In many published papers of the spectroscopy of Jahn-Teller active states, ω 0,i is given as the vibrational frequency of the Jahn-Teller active modes. In the limit of very weak Jahn-Teller coupling, ω 0,i and ω e,i will be comparable to each other, but Figure 2 shows that even a relatively small D i can cause ω 0,i to be greater than ω e,i by 25% or more. A true description of the vibrational structure of a Jahn-Teller active molecule must include both ω e,i and D i. While the value of ω 0,i is useful, by itself it is not an accurate representation of the vibrational potential of the molecule. While the Jahn-Teller Hamiltonian does not include any matrix elements that are non-diagonal in more than one mode, Jahn-Teller coupling does mix the levels of each active mode with the others. To see how this can be the case, consider, for a two-mode calculation, the three basis functions 0 = Λ =+1 v 1 =0,l 1 =0 v 2 =0,l 2 =0 (7) 1 = Λ = 1 v 1 =1,l 1 =1 v 2 =0,l 2 =0 (8) 2 = Λ = 1 v 1 =0,l 1 =0 v 2 =1,l 2 =1. (9) While the two basis functions 1 and 2 do not have a non-zero off-diagonal matrix element between them, they each do have a non-zero off-diagonal matrix element with the basis function 0. Thus, while direct 9

coupling between the two v = 1 basis functions is not present, they are indirectly coupled via the v =0basis function. Therefore, a calculation of the vibronic energy levels must include both modes simultaneously. The impact of mode mixing on the calculation of the vibronic energy levels can be significant for even modest Jahn-Teller effects and represents a substantial complicating factor to the analysis of experimental spectra. 4.2 Energies of the spin-vibronic levels with spin-orbit and linear Jahn-Teller coupling One of the principal aims of this work was to develop a computational method for the accurate calculation of the energy levels for a molecule for which both Jahn-Teller and spin-orbit coupling are both non-neglible. Figure 3 shows the calculated energy levels from the diagonalization of the full spin-vibronic Hamiltonian for an example set of parameters. Previous treatments(ham 1965, Child 1963, Child & Longuet-Higgins 1961) of spin-orbit coupling for the Jahn-Teller problem had only considered approximate formulations of the spin-vibronic energy levels, which had several deficiencies. In our approach, the addition of spin-orbit coupling to the Hamiltonian properly describes the avoided crossings of states of the same symmetry under the spin double group. As spin-orbit coupling begins to quench the Jahn-Teller coupling (the right-hand side of Figure 3), the vibrational spacings of a harmonic oscillator for a pair of 2 E 3 and 2 E 1 electronic 2 2 states begin to be recovered. Notice, however, that Jahn-Teller coupling still has two significant effects on the energy levels: the vibrational spacing has yet to recover its harmonic value (100 cm 1 in this example) and the individual v i levels are still split according to their l i values, much as they are in the absence of spin-orbit coupling. 5 Coriolis coupling in the absence of spin-orbit coupling Up to this point, we have considered only the eigenvalues of the vibronic and spin-vibronic levels of an orbitally degenerate state. However, as we show in this and the following section, the corresponding eigenfunctions are very important. They determine several experimentally observable quantities, such as the vibronic and spin-vibronic angular momenta of the state. These properties are drastically altered by the magnitude of the Jahn-Teller coupling and the symmetry of the state. 10

In Paper I, we gave the following expression the Coriolis coupling operator, Ĥ COR = 2A j, n j,α= e ˆL z + Ĝz j, n j,α= e ˆN z (10) = 2Aζ t ˆNz, (11) where ζ t = ( )] p [c 2i,nj Λ i ζ e + l m,i ζ m, (12) i m=1 where ζ e and ζ i are defined by v, l Λ Ĝz Λ v, l = lζ i (13) v, l Λ ˆL z Λ v, l = Λζ e, (14) The parameter ζ t is the Coriolis constant for the level of interest. In Fig. 4 we show for an example set of parameters how ζ t varies as a function of D for both Jahn-Teller cases s 1 = 0 (solid line) and s 1 = 1 (dotted line). In the absence of Jahn-Teller coupling, the eigenfunction of the lowest energy level of the electronic state contains no admixture of vibrational basis functions with v 0andl 0. Hence, the value of ζ t at D =0is thesameasζ e, which we assumed to be unity. As the linear Jahn-Teller coupling constant D i is increased, the vibronic angular momentum is decreased by the addition to the eigenfunction of basis functions with non-zero l i,asbothcurvesoffig.4show. The limits at large D represent a distinct difference between the two symmetry cases. We showed in Paper I that the limit for ζ t at large D for the case s 1 = 0 is (see Eq. 110 of Paper I) lim ζ t = jζ i. (15) D Because j = 1 2 for the lowest energy level, ζ t converges to 1 2 ζ i (= 0.125 in our example). A more general form of Eq. 15 is lim D ζ t =( 1) s1 jζ k. (16) For the symmetry case of s 1 = 1, then, the limit at large D has the same magnitude as for s 1 = 0, but the opposite sign. Because in each case the value in the absence of linear Jahn-Teller coupling is the same (ζ e = 1), the two curves of Fig. 4 have distinctly different shapes to arrive at their respective limits for large D. 11

6 The effective spin-rotation coupling constant induced by spinorbit coupling The Coriolis constant ζ t is often of sufficient magnitude that only moderate resolution is needed to resolve Coriolis structure in experimental spectra. In nearly all rotationally-resolved spectra analyzed to date, the analyses assumed that the two spin-orbit components of the level had identical Coriolis constants ζ Σ t (where the superscript Σ = ± 1 2 indicates a particular spin-component). As we showed in Paper I, the two spin-components do not necessarily have identical values of ζ Σ t. The ramification of different values of ζ Σ t for a pair of Σ = ±1/2 spin components is that if they are analyzed using a model that assumes ζ 1 2 t = ζ 1 2 t, then a very large spin-rotation coupling constant must be used to satisfactorily fit the rotational levels. As we showed in Paper I, the effective spin-rotation constant ɛ eff aa contains contributions from both the inherent spin-rotation term, ɛ0 aa, of the state, as well as a second term, denoted ɛ 2v aa, ɛ eff aa = ɛ 0 aa + ɛ 2v aa. (17) We further showed that ɛ 2v aa is proportional to the difference between the Coriolis coupling constants of the two spin components, ζ t, where ɛ 2v aa 2A = ζ t, (18) ζ t = ζ + 1 2 t ζ 1 2 t, (19) and A is the rotational constant along the symmetry axis. This formulation of the spin-rotation constant in terms of the Coriolis coupling constants of the two spin components, which can be calculated using the Jahn-Teller and spin-orbit computer program, was sufficient to explain the formerly anomalously large spin-rotation constants in the ground state of the methoxy radical (-1.3533 cm 1 )(Liu, Damo, Lin, Foster, Misra, Yu & Miller 1989, Liu, Foster, Williamson, Yu & Miller 1990a, Liu, Yu & Miller 1990b) andthat observed in the excited state of the CdCH 3 radical (15.8 cm 1 )(Cerny, Tan, Williamson, Robles, Ellis & Miller 1993, Pushkarsky, Barckholtz & Miller 1999). 12

In Fig. 5 we show accurate calculations of ζ t as a function of aζ e, for both symmetry cases for an example set of parameters. While the difference between the two symmetry cases for ζ t is not as pronounced as for ζ t, Fig. 4, the difference is non-negligible. The graph clearly shows that the larger the spin-orbit coupling is for the system, the larger the effective spin-rotation constant will be. 7 Conclusions In this chapter, we have presented accurate calculations of the energy, Coriolis coupling constant, and the effective spin-rotation constant induced by spin-orbit coupling for the vibronic and spin-vibronic levels of Jahn-Teller active states. While we chose to show results for various parameters for a molecule with only one active mode, the principals of these calculations can be extended to any other combination of active modes and coupling constants, using the computer program, SOCJT. In the past, it has generally been the case that only the energies and electronic transition intensities have been used to deduce the Jahn-Teller and spin-orbit coupling constants of a Jahn-Teller active molecule. The results exemplified in Figs. 4 and 5 show that Jahn-Teller and spin-orbit coupling also have significant effects on the expectation values of these vibronic and spin-vibronic levels. We therefore expect in the future that the coupling parameters of Jahn-Teller active states will be deduced not just from the energies of the vibronic and spin-vibronic levels, but also from the molecular parameters obtained from their rotationally resolved spectra. 8 Acknowledgments We thank the National Science Foundation for support of this work, in the form of a grant to TAM (# 9320909) and a fellowship to TAB. We are also grateful to the Ohio Supercomputer Center (Grant # PAS540) for a generous allotment of computer time. 13

References Barckholtz, T. A. & Miller, T. A., 1998, Int. Rev. Phys. Chem. 17, 435 524. Barckholtz, T. A., Yang, M. C. & Miller, T. A., 1999, Mol. Phys. accepted. Bondybey, V. E., Sears, T. J., English, J. H. & Miller, T. A., 1980, J. Chem. Phys. 73, 2063 2068. Bunker, P. & Jensen, P. (1998), Molecular Symmetry and Spectroscopy, 2nd edn, NRC Press, Ottawa. Cerny, T. M., Tan, X. Q., Williamson, J. M., Robles, E. S. J., Ellis, A. M. & Miller, T. A., 1993, J. Chem. Phys. 99, 9376 9388. Child, M. S., 1963, J. Mol. Spectrosc. 10, 357 365. Child, M. S. & Longuet-Higgins, H. C., 1961, Proc. Roy. Soc. 245A, 259 294. Cossart-Magos, C. & Leach, S., 1980a, Chem. Phys. 48, 329 348. Cossart-Magos, C. & Leach, S., 1980b, Chem. Phys. 48, 349 358. Cossart-Magos, C., Cossart, D. & Leach, S., 1979a, Mol. Phys. 37, 793. Cossart-Magos, C., Cossart, D. & Leach, S., 1979b, Chem. Phys. 41, 345 362. Cossart-Magos, C., Cossart, D. & Leach, S., 1979c, Chem. Phys. 41, 363 372. Cossart-Magos, C., Cossart, D., Leach, S., Maier, J. P. & Misev, L., 1983, J. Chem. Phys. 78, 3673 3687. Ham, F. S., 1965, Phys. Rev. A 138, 1727 1740. Ham, F. S. (1972), In Electron Paramagnetic Resonance, Plenum, New York, pp. 1 119. Jahn, H. A. & Teller, E., 1937, Proc. Roy. Soc. A 161, 220 235. Liu, X., Damo, C., Lin, T.-Y. D., Foster, S. C., Misra, P., Yu, L. & Miller, T. A., 1989, J. Phys. Chem. 93, 2266 2275. Liu, X., Foster, S. C., Williamson, J. M., Yu, L. & Miller, T. A., 1990a, Mol. Phys. 69, 357 367. Liu, X., Yu, L. & Miller, T. A., 1990b, J. Mol. Spec. 140, 112 125. Longuet-Higgins, H. C., Öpik, U., Pryce, M. H. L. & Sack, R. A., 1958, Proc. Roy. Soc. A 244, 1 16. Miller, T. A. & Bondybey, V. E. (1983), In Molecular Ions: Spectroscopy, Structure, and Chemistry, North- Holland, pp. 201 230. Pushkarsky, M., Barckholtz, T. A. & Miller, T. A., 1999, J. Chem. Phys. 110, 2016 2028. Sears, T., Miller, T. A. & Bondybey, V. E., 1980, J. Chem. Phys. 72, 6070 6080. Sears, T., Miller, T. A. & Bondybey, V. E., 1981a, Discuss. Farad. Soc. 71, 175 180. Sears, T., Miller, T. A. & Bondybey, V. E., 1981b, J. Chem. Phys. 74, 3240 3248. Underwood, R. R. (1975), PhD thesis, Stanford University. Yu, L., Foster, S. C., Williamson, J. M. & Miller, T. A., 1990, J. Chem. Phys. 92, 5794 5800. 14

Figure Captions 1. Absolute energies of the vibronic levels for linear coupling in a single Jahn-Teller active mode over the range of D =0toD =0.5. The ordinate scale is in units of ω e while the abscissa is the dimensionless linear Jahn-Teller coupling constant D. The levels are labeled by their value of j. 2. Relative energies of the vibronic levels, j, n j,α, for linear Jahn-Teller coupling in a single active mode over the range of D i =0toD i =0.5. The ordinate scale is in units of ω e,i while the abscissa is the dimensionless linear Jahn-Teller coupling constant D i, with the zero of energy set as the energy of the lowest level. 3. Calculated relative energy levels for a 2 E state of C 3v symmetry including linear Jahn-Teller coupling and spin-orbit coupling for a single Jahn-Teller active mode. Only those levels originating from v =0, 1 and 2 are shown. The energy levels were calculated assuming D i =0.125 and ω e,i = 100 cm 1 and varying aζ e.ataζ e = 0, the lowest several energy levels are labeled by the ket j, n j,α and have components, Σ = ±1/2. At aζ e = 200 cm 1 the energy levels are labeled by the ket v, j, Ω,α. In the strong spin-orbit limit, the value of Ω = Λ l t + Σ determines(barckholtz & Miller 1998) the C 3v double group symmetry α. The vibrational quantum number v is also included since it again becomes meaningful when spin-orbit coupling dominates Jahn-Teller coupling. 4. Coriolis coupling constant for the lowest energy level, calculated for a single Jahn-Teller active mode over the range of D =0toD =2.0. The solid line is the result for the symmetry case s 1 = 0 while the dotted line is for s 1 = 1. The parameters used the calculation were ω e = 500 cm 1, ζ e =1.0, and ζ i =0.25. 5. The difference, ζ t, of the Coriolis coupling constant of the two spin components of the lowest energy level, calculated for a single Jahn-Teller active mode over the range of aζ e =0toaζ e = 100. The solid line is the result for the symmetry case s 1 = 0 while the dotted line is for s 1 =1. The parameters used the calculation were ω e = 500 cm 1, D =0.25, ζ e =1.0, and ζ i =0.25. 15

Table 1: Size of basis set, number of non-zero matrix elements, and approximate timings for several different types of spin-orbit/jahn-teller calculatoins. Linear Quadratic Spin-orbit # modes [v max ] a N b n c Time (Cray) d Time (Dos) e Y N N 1 [35] 36 71 1 6 Y N N 2 [7,5] 133 345 1 21 Y N N 3 [7,5,3] 1246 3983 23 400 Y N N 4 [7,5,3,2] 7236 26 767 166 4100 Y N N 5 [7,5,3,2,2] 42 098 176 876 6000 150 000 Y N Y 1 [35] 36 71 1 11 Y N Y 2 [7,5] 133 345 2 38 Y N Y 3 [7,5,3] 1246 3983 25 800 Y Y N 1 [35] 164 508 1 23 Y Y N 2 [7,5] 465 1498 2 50 Y Y N 3 [7,5,3] 4516 17 247 35 770 Y Y Y 1 [35] 164 508 1 36 Y Y Y 2 [7,5] 465 1498 3 70 Y Y Y 3 [7,5,3] 4516 17 247 50 2100 a Maximum values of v included in the basis set for each vibrational mode. b Size of basis set for the j =1/2 block. c Number of non-zero matrix elements in the lower triangle of the Hamiltonian matrix. d Approximate time, in seconds, for finding the 15 lowest energy solutions of each block up to j =15/2, on a Cray T90. e Same as footnote d except for execution under DOS on a 200 MHz Cyrix-686. 16

Absolute Energy 3ω e 2ω e ω e 3 /2 1 /2 5 /2 1 /2 3 /2 1 /2 0 0.0 0.1 0.2 0.3 0.4 0.5 Figure 1: Absolute energies of the vibronic levels for linear Jahn-Teller coupling in a single active mode over the range of D =0toD =0.5. The ordinate scale is in units of ω e while the abscissa is the dimensionless linear Jahn-Teller coupling constant D. The levels are labeled by their value of j. D 17

2ω e Relative Energy 1 /2, 2, e ω e 3 /2, 1, a 1 +a 2 ω 0 1 /2, 1, e 0 0.0 0.1 0.2 0.3 0.4 0.5 D Figure 2: Relative energies of the vibronic levels, j, n j,α, for linear coupling in a single Jahn-Teller active mode over the range of D i =0toD i =0.5. The ordinate scale is in units of ω e,i while the abscissa is the dimensionless linear Jahn-Teller coupling constant D i, with the zero of energy set as the energy of the lowest level. 18

1, 3 /2, - 1 / 2, e 1/2 1, 1 /2, - 3 / 2, e 3/2 3ω e 3, 3 /2, 1 /2, e 1/2 3, 1 /2, - 5 / 2, e 1/2 E 1/2 3, 5 /2, - 9 / 2, e 3/2 3 /2, 2, a 1 /a 2 0, 1 /2, 1 /2, e 1/2 Relative Energy (ω e ) 1 /2, 3, e 5 /2, 1, e 1 /2, 2, e 5 2, /2,- 1 / 2, e 1/2 1 3 2, /2, /2, e 3/2 3 2, /2,- 7 / 2, e 1/2 E 3/2 ω e 3 /2, 1, a 1 /a 2 3 1 1, /2, /2, e 1/2 1 1, /2,- 5 / 2, e 1/2 1 /2, 1, e 0 0 j, n j,α -50-100 aζ e (cm -1 ) -150-200 0, 1 /2, 3 /2, e 3/2 v, j, Ω, α / Figure 3: Calculated relative energy levels for a 2 E state of C 3v symmetry including linear Jahn-Teller coupling and spin-orbit coupling for a single Jahn-Teller active mode. Only those levels originating from v =0, 1 and 2 are shown. The energy levels were calculated assuming D i =0.125 and ω e,i = 100 cm 1 and varying aζ e. At aζ e = 0, the lowest several energy levels are labeled by the ket j, n j,α and have components, Σ = ±1/2. At aζ e = 200 cm 1 the energy levels are labeled by the ket v, j, Ω,α. In the strong spin-orbit limit, the value of Ω = Λ l t + Σ determines(jahn & Teller 1937) the C 3v double group symmetry α. The vibrational quantum number v is also included since it again becomes meaningful when spin-orbit coupling dominates Jahn-Teller coupling. 19

1.0 0.9 0.8 0.7 0.6 ζ t 0.5 0.4 0.3 0.2 0.1 0.0-0.1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 D Figure 4: Coriolis coupling constant for the lowest energy level, calculated for a single Jahn-Teller active mode over the range of D =0toD =2.0. The solid line is the result for the symmetry case s 1 = 0 while the dotted line is for s 1 = 1. The parameters used the calculation were ω e = 500 cm 1, ζ e =1.0, and ζ i =0.25. 20

0.00-0.02-0.04 ζ t = ζ t + - ζt - -0.06-0.08-0.10-0.12-0.14-0.16-0.18-0.20 0 20 40 60 80 100 aζ e Figure 5: The difference, ζ t, of the Coriolis coupling constant of the two spin components of the lowest energy level, calculated for a single Jahn-Teller active mode over the range of aζ e =0toaζ e = 100. The solid line is the result for the symmetry case s 1 = 0 while the dotted line is for s 1 = 1. The parameters used the calculation were ω e = 500 cm 1, D =0.25, ζ e =1.0, and ζ i =0.25. 21