Spin-orbit vibronic coupling in 3 states of linear triatomic molecules

Transcription

1 THE JOURNAL OF CHEMICAL PHYSICS 16, Spin-orbit vibronic coupling in 3 states of linear triatomic molecules Sabyashachi Mishra a Department of Chemistry, Technical University of Munich, D Garching, Germany Leonid V. Poluyanov Institute of Chemical Physics, Academy of Sciences, Chernogolovka, Moscow 143, Russia Wolfgang Domcke Department of Chemistry, Technical University of Munich, D Garching, Germany Received 8 November 006; accepted 8 February 007; published online 5 April 007 The Renner-Teller vibronic-coupling problem of a 3 electronic state of a linear molecule is analyzed with the inclusion of the spin-orbit coupling of the 3 electronic state, employing the microscopic Breit-Pauli spin-orbit coupling operator for the two unpaired electrons. The 66 Hamiltonian matrix in a diabatic spin-electronic basis is obtained by an expansion of the molecular Hamiltonian in powers of the bending amplitude. The symmetry properties of the Hamiltonian with respect to the time-reversal operator and the relativistic vibronic angular momentum operator are analyzed. It is shown that there exists a linear vibronic-coupling term of spin-orbit origin, which has not been considered so far in the Renner-Teller theory of 3 electronic states. While two of the six adiabatic electronic wave functions do not exhibit a geometric phase, the other four carry nontrivial topological phases which depend on the radius of the integration contour. The spectroscopic effects of the linear spin-orbit vibronic-coupling mechanism have been analyzed by numerical calculations of the vibronic spectrum for selected model examples. 007 American Institute of Physics. DOI: / I. INTRODUCTION Vibronic coupling in degenerate electronic states of linear molecules has been discussed in the pioneering work of Renner 1 and later been termed Renner-Teller RT effect. After the first experimental detection of the RT effect in the NH radical,,3 Pople and Longuet-Higgins revisited the original work of Renner with a model Hamiltonian approach. 4 Since then, significant progress has been made in the understanding of the RT effect both in theory as well as in experiment, see Refs. 5 1 and references therein. Most of the above cited works ignore the spin-orbit SO coupling. Pople was the first to consider the SO coupling in electronic states of linear triatomic molecules. 13 He provided a perturbative analysis of the vibronic energy levels of a electronic state with the inclusion of both RT and SO interactions, resulting in a clear picture of the perturbations caused by SO coupling in RT spectra. Pople assumed the simplified phenomenological form of the SO operator, H SO = AL z S z, based on the argument that the x and y components of the electronic orbital angular momentum operator are effectively quenched for linear molecules. 13 Since then, most of the investigations on the RT effect with SO coupling have been based on this approximation. 5,9,14 18 While there has been extensive research on the RT effect with SO coupling in the degenerate electronic states of linear The purpose of the present work is a systematic study of the RT effect in 3 electronic states of linear triatomic molecules by employing the microscopic expression for the SO operator the Breit-Pauli operator 6 9. We determine the via Author to whom correspondence should be addressed. Electronic mail: sabyashachi.mishra@ch.tum.de 1 triatomic molecules of doublet spin multiplicity i.e., with one unpaired electron, studies on or electronic states with higher spin multiplicities are rather limited. This is partly due to the fact that high spin multiplicities are rather infrequent in nature with the exception of transition-metal and rare-earth compounds and partly due to the difficulties associated with the theoretical analysis of many-electron systems. Hougen performed the first systematic perturbative analysis of the vibronic and rotational energy levels of linear triatomic molecules in a 3 electronic state, taking both RT and SO interactions into account. 19 This analysis, which can be considered as an extension of Pople s work on the RT-SO problem of states, 13 was based on the analogous phenomenological form of the SO operator. An essential conclusion of Hougen s work was that the vibronic energy levels of a 3 electronic state can be divided into two sets: the energies of one set are like those of a 1 electronic state, as the energies of the other set are like those of a electronic state. 19 The same phenomenological form of SO coupling has been employed in subsequent studies of the spectra of specific systems. Perić et al. have investigated the RT effect in the 3 electronic state of NCN with high-level quantum chemical methods, employing Hougen s SO operator. 0 In recent work by Carter et al., a variational calculation of spinrovibronic energy levels has been performed for the 3 state of CCO. 1 Other examples of vibronic and SO coupling in 3 states are SiCO, CSiO, CCO,,3 CNN, 4 and InOH /007/1613/13431/10/$ , American Institute of Physics

2 Mishra, Poluyanov, and Domcke J. Chem. Phys. 16, bronic Hamiltonian by expanding the matrix elements of the electrostatic potential and the SO operator in a diabatic electronic basis in powers of the bending coordinate, taking into account the symmetry selection rules. This results in a 6 6 vibronic matrix which, unlike the Hamiltonian in Ref. 19, cannot be block-diagonalized. In particular, this Hamiltonian exhibits a linear that is, of first order in the bending mode vibronic-coupling term of SO origin which is absent when the phenomenological form of SO coupling is assumed. We derive the vibronic Hamiltonian in the adiabatic representation and discuss the geometric phases of the adiabatic electronic wave functions. The spectroscopic effects of RT and SO couplings in 3 states are investigated by variational calculations of the vibronic energy levels for selected models. It will be shown that the effects of linear relativistic vibronic coupling are particularly interesting when the bending vibrational frequency and the SO splitting of the 3 state are in resonance. II. VIBRONIC HAMILTONIAN IN THE DIABATIC REPRESENTATION Let us consider two electrons moving in the field of three linearly arranged nuclei of electric charges Q 1, Q, and Q 3. Let us focus on the situation the components of a 3 state are coupled by the degenerate bending vibrational mode. The Hamiltonian of this two-electron system, in atomic units, can be written as H = H el T N = H es H SO T N, H es = 1 k k=1 H SO = k=1 3 Qn n=1 H k SO H 1 SO, r kn 1 r 1, T N = Here H es is the electrostatic part of the electronic Hamiltonian that includes the kinetic energy of the two electrons, the electron-nuclear interaction term, as well as the electronelectron repulsion term. H SO represents the SO interaction and consists of two parts, the one-electron SO operator H k SO, k=1 and, and the two-electron contribution H 1 SO. These are given by 9 H SO 3 4 k =.03i 3 Qn 4c S k 3 n=1 r r kn k, k = 1,, 6 kn 1 =.03i 4c 3 r S 1r S r H SO Denoting by r 1 and r the radius vectors of electrons 1 and, respectively, and by R n n=1,,3 the nuclear radius vectors, the vectors in Eqs. 6 and 7 are defined as r kn = r k R n, k = 1,, r 1 = r 1 r =r 1, S k = iˆ xk jˆ yk kˆ, k = 1,. zk Here ˆ k x, ˆ k y, and ˆ k z are the Pauli spin matrices acting on the spin eigenstates of the first k=1 or second k= electron. c is the speed of light, r kn =r kn, and r 1 =r 1. T N is the nuclear kinetic-energy operator in polar coordinates,, which are defined as e ±i = Q ± = Q x ± iq y, Q x and Q y are the Cartesian components of the dimensionless bending normal coordinate. We have omitted the totally symmetric stretching modes in Hamiltonian, since these are decoupled from the degenerate bending mode as long as the degenerate 3 state can be assumed to be isolated from other electronic states. The inclusion of these modes is straightforward. 30,31 The electronic Hamiltonian H el has the following symmetry properties: and H el,ĵ z = H el,tˆ =0, 11 Ĵ z = Lˆ z Ŝ z =i i ˆ 1 1 z ˆ z 1 is the z projection of the electronic orbitalspin angular momentum operator and Tˆ =4ˆ 1 y ˆ y c.c. ˆ 13 is the time-reversal symmetry operator. 3 Here c.c. ˆ stands for the operator of complex conjugation. Tˆ is antiunitary, and for systems with an even number of electrons, as is the case here, Tˆ =1. 14 The full Hamiltonian H of Eq. also possesses the time-reversal symmetry and commutes with the total electronicnuclear angular momentum operator Ĵ z = Ĵ z i. 15 The eigenvalues of Ĵ z are integers =0,±1,±,... Let us consider electronic diabatic basis functions associated with the two components of the degenerate 3 electronic state with electronic orbital angular momentum quantum numbers = ±1 and spin angular momentum quantum numbers S z =0, ±1. These basis functions refer to the linear geometry of the triatomic system in the absence of the interelectronic interactions and SO effects. We introduce two molecular orbitals, one of type Qr,z and the other of type Pr,zexp±i, r,, and z are cylindrical elec-

3 SO coupling in linear triatomic molecules J. Chem. Phys. 16, tronic coordinates. The molecular orbitals Qr,z and Pr,z are solutions of a one-electron Schrödinger equation. Twoelectron basis functions Jz, J z =S z which are orthonormal, antisymmetric with respect to electron exchange and simultaneous eigenfunctions of S, S z, and L z are defined as follows: = 1 QP1expi 1 Q1Pexpi 1, 1 = 1 QP1expi 1 Q1Pexpi 1 1, 0 = 1 QP1expi 1 Q1Pexpi 1, 16 0 = 1 QP1exp i 1 Q1Pexp i 1, 1 = 1 QP1exp i 1 Q1Pexp i 1 1, = 1 QP1exp i 1 Q1Pexp i 1, Pk= Pr k,z k and Qk=Qr k,z k, k=1,. The two-electron basis functions are eigenfunctions of Ĵ z, Jˆ zjz = J z Jz. 17 The time-reversal operator Tˆ has the following effect on the diabatic electronic basis functions: 1 0 Tˆ = c.c. ˆ Using the six diabatic spin-electronic basis functions Eq. 16, the electronic Hamiltonian can be written as a 6 6 vibronic matrix. The derivation of the 66 spinvibronic Hamiltonian is similar to the procedure outlined in Refs The determination of the matrix elements of H el can be simplified by using relations 17 and 18, and the Hermitian property of H el. A Taylor series expansion of the matrix elements of the electronic Hamiltonian in powers of the bending distortion is performed, retaining the lowestorder nonvanishing term for each of the matrix elements. Further simplification of the vibronic Hamiltonian is achieved by the direct calculation of the individual coefficients, performing the integration over the angular variables 1 and as well as over the discrete spin variables. The derivation of the 66 spin-vibronic Hamiltonian is given in Appendix A. The result is E, H = T N 1 cei dei c e i E,0 0 de i 0 0 de i 0 E,1 c e i de i 0 0 de i c e i E,1 0 de i de i 0 E,0 c e i de i c e i E, 1 is the 66 unit matrix. E,0, E,1, and E, represent the energies of the three SO components 3 0, 3 1, 3 of the 3 state at the linear geometry. From the time-reversal symmetry, it is obvious that E, =E,0 and E,1 =E,0 E, /. In terms of the SO splitting of the 3 state, we can write E, =, E,1 =0, and E,0 =. The relative ordering of the states depends on the sign of the SO splitting. 9 The coupling term c is the well-known quadratic RT parameter of nonrelativistic purely electrostatic origin. All other coupling terms in Eq. 19 are of SO origin. While the nonrelativistic RT effect results in and coupling, the linear SO coupling term de ±i is responsible for the coupling between and The linear SO coupling term is absent when the simplified phenomenological SO operator is employed. 19,36 The vibronic Hamiltonian 19 cannot be transformed to block-diagonal form by a constant unitary transformation. When d=0, then Hamiltonian 19 decouples into three

4 Mishra, Poluyanov, and Domcke J. Chem. Phys. 16, blocks which represent the pure orbital RT effect. In this case, the 3 RT-SO coupling problem with SO splitting decomposes into a RT-SO coupling problem with SO splitting and a 1 RT-SO coupling problem with vanishing SO splitting. 19,36 III. VIBRONIC HAMILTONIAN IN THE ADIABATIC REPRESENTATION For fixed nuclei i.e., T N =0, the vibronic Hamiltonian 19 can be diagonalized analytically, yielding the following expressions for the eigenvalues: U 1, = c, U 3,4 = W = c 4 d, 0 U 5,6 = W = c 4 d. In the nonrelativistic limit i.e., =0 and d=0, two distinct adiabatic potential-energy terms are obtained which are degenerate at the linear geometry, i.e., =0. This degeneracy is lifted in second order by the nonrelativistic RT coupling. Both potential-energy terms are threefold degenerate, representing the spin degeneracy. In the limit of vanishing RT coupling and nonzero SO coupling, the adiabatic terms are represented by three parabolas corresponding to the three SO components of the 3 state, see Fig. 1a. Upon introduction of RT coupling c/0, the degenerate 3 1 state splits into two components similar to the RT effect in a state without SO splitting, as the 3 and 3 0 states exhibit a change of their curvatures similar to the RT effect in a state with SO splitting, see Fig. 1b. At strongly bent geometries, the splitted potential-energy curves of the 3 1 state FIG. 1. Adiabatic potential-energy curves of the 3 0 solid, 3 1 dashed, and 3 dotted states for /=0.8 and a c=0, d=0, b c/=0., d =0, and c c/=0., d/=0.3, as a function of one of the dimensionless Cartesian bending coordinates. come close to the 3 and 3 0 potential-energy curves, see Fig. 1b. This reflects the fact that the RT splitting of the degenerate 3 1 state is stronger than that of the SO-splitted 3 and 3 0 states. The near degeneracy of the and the potential-energy curves for large bending angles is lifted by the linear coupling term of SO origin, see Fig. 1c. The orthonormal eigenvectors V 1,...,6 of the fixed-nuclei Hamiltonian are given in Appendix B. The V i, i=1,...,6, form the 66 eigenvector matrix W which is unitary. It transforms the potential matrix of Hamiltonian 19 to diagonal form. The same unitary matrix transforms the timereversal operator Tˆ to block-diagonal form, i.e., Tˆ ad = W Tˆ W = cos sin e i sin e i cos e 4i 4ic.c. ˆ cos sin e i sin e i cos e 1 The two -dependent angles ± are defined by 1 cos 0 cos 1. 3 cd 4 cos ± = d W d and The matrices in the adiabatic representation of the time-reversal operator, given by Eq. 1, are unitary, as Tˆ ad itself is antiunitary, and Tˆ ad = 1. The operators H el and Tˆ cannot be simultaneously diagonalized because of the presence of the c.c. ˆ in Eq. 1. The vibronic Hamiltonian 19 is characterized by an

5 SO coupling in linear triatomic molecules J. Chem. Phys. 16, additional symmetry, which is connected to the existence of two pairs of doubly degenerate eigenvalues. This symmetry property is restricted to the six-dimensional subspace of the eigenvectors. The existence of additional symmetry operators giving rise to doubly degenerate eigenvalues can be seen as follows. It is evident that the 66 block-diagonal matrix 1 S ad =1 4 ˆ ˆ with ˆ ± = C 1 ± ˆ x C ± ˆ y C 3 ± ˆ z, 5 commutes with the diagonal Hamiltonian W H el W and contains six real parameters C ± j, j=1,, and 3, which can be arbitrarily chosen. This proves that the vibronic Hamiltonian 19 possesses six matrix symmetry operators in addition to the time-reversal symmetry operator Tˆ. Each of these operators can be obtained from the general form of the matrix symmetry operator S = WS ad W, 6 if one of the constants C j ± is chosen to be 1 and the others are zero. IV. TOPOLOGICAL PHASES OF THE ADIABATIC ELECTRONIC STATES The adiabatic electronic states are obtained as j ad = V dt V j,, j =1,...,6, 7 V T d =, 1, 0, 0, 1,. The well-known expression for the Berry phase of the adiabatic electronic wave functions in, space can be written in the simplified form j = ic V j, V j,d, 8 d = d,d, 9 =, 1. Since the result of the integration does not depend on the form of the integration contour C, 37 we can perform the integration along a circle with radius R around the origin. Thus Eq. 8 reduces to j = V j i0, V j,d. 30 For the eigenvectors V j, j=1,...,6, Eq. 30 gives the following results: 1 = =0, 3,5 = c R 4 c d R 6 d R W ± WW d R ± d R, 4 4,6 = c NR 4,6 1 W dr ± c R 3 d d cr SR ± DR ± ± SR 3c R d 1WSR ± SR ± 1c R 4 SR ± The functions DR ±, SR ±, NR 4,6 are defined in Appendix B. The phases 3, 4, 5, and 6 exhibit a complicated dependence on the integration radius R. It can be seen from Eqs. 3 and 33 that the geometric phases 3 6 vanish both in the limit R 0 as well as in the limit R. Such R-dependent geometric phases have previously been found for the EE Jahn-Teller problem with SO coupling The result of integration does not depend on the form of the closed contour for a given area. 37 This implies that the whole area covered by the integration contour contributes to the geometric phase. V. CALCULATION OF VIBRONIC SPECTRA We consider a linear triatomic molecule which is initially in the vibrationless nondegenerate ground state 0 and is excited by some operator T to an isolated 3 electronic state. According to Fermi s golden rule, the excitation spectrum is given by PE = 0 T E E 0 E, 34 E 0 denotes the energy of the reference state 0 and E is the energy of the final vibronic state. The final vibronic states are determined by H = E, 35 with the Hamiltonian H of Eq.. We solve Eq. 35 by the expansion of in a complete basis which is constructed as the product of electronic,sz and vibrational nl basis functions. The,Sz, =±1,S z =0, ±1, are the six diabatic electronic basis states which correspond to representation 19 of H el. The nl, n=0,1,,..., l=n,n,...,n,n, are the eigenfunctions of the two-dimensional isotropic harmonic oscillator. 43 Owing to the existence of the total angular momentum operator Ĵ z which commutes with the Hamiltonian H, the Hamiltonian matrix decouples into submatrices corresponding to different eigenvalues of Ĵ z. The real symmetric Hamiltonian matrix is constructed and diagonalized by standard routines for a given value of. The vibrational basis is increased until convergence of the eigenvalues of interest has been achieved. These eigenvalues represent the vibronic en-

6 Mishra, Poluyanov, and Domcke J. Chem. Phys. 16, FIG.. Energy levels of a 3 RT system with SO coupling in the resonant case in the absence of linear SO vibronic coupling d=0. Solid, dashed, dotted, and dashed-dotted lines represent energy levels with =0, 1,, and 3, respectively. FIG. 3. Effect of the linear SO vibronic-coupling term on the vibronic levels for /=0.8 and c/=0.. Solid, dashed, dotted, and dashed-dotted lines represent energy levels with =0,1,, and 3, respectively. ergy levels E with vibronic angular momentum quantum number. The square of the first component of the eigenvector represents the spectral intensity within the Condon approximation, assuming that the optical excitation takes place from the vibrational ground state VI. GENERIC ASPECTS OF LINEAR SO VIBRONIC COUPLING IN A 3 STATE In this section, we analyze the influence of the linear coupling term de ±i on the vibronic spectrum of a 3 electronic state. The effect of this coupling term is expected to become significant when the potentially interacting unperturbed levels are nearly degenerate. Since de ±i couples, in first order, levels with different spin quantum numbers which differ by one quantum of the bending mode, we expect pronounced effects when. We shall refer to the case as the resonant case in the following. For small values of the SO splitting and the RT coupling, the perturbative treatment of vibronic energy levels by Hougen 19 provides excellent insight to the splitting pattern of the vibronic levels of the 3 state. This analysis fails, however, when the SO coupling is either comparable to or larger than the bending frequency and/or the RT coupling c. We shall consider here, as an example, the case and c. The vibronic energy levels of such a system are shown in Fig. for the parameter values d=0, /=0.8, and c/ =0.. Upon introduction of the SO splitting, each of the unperturbed bending levels shown in the left column splits into three SO components, see the middle column of Fig.. The relative ordering of the spin multiplets depends on the sign of the SO splitting. Here we assume to be positive. The energy levels obtained with the inclusion of RT coupling are shown in the last column of Fig.. The vibronic energy levels are assigned by the conventional spectroscopic terms. 36 The vibrational level 1,n, 3, 1,3 are quantum numbers of the stretching modes and n is the occupation number of the degenerate bending mode of the linear triatomic molecule, is denoted as n, since the stretching modes are absent in the present work. The degeneracy of the vibronic levels of 0 3 0± symmetry has been discussed in Refs. 19 and 44. It should be mentioned that the assignment of individual vibronic levels becomes ambiguous for higher energies, especially in the case when several vibronic energy levels of the same value are close-by. The example of Fig. shows that the energy levels and are close in energy, as a consequence of. The same is true for the 0 3 and 1 3 levels. These quasidegenerate energy levels interact with each other when the linear vibronic-coupling term is taken into account. Figure 3 displays the energy levels of this system as a function of the dimensionless parameter d/. The energy levels with different values of are shown by different line types, see caption of Fig. 3. The figure reveals several significant level repulsions with increasing d/, for example, the pair and On the other hand, the level remains nearly unperturbed. The levels 0 3, 1 3, 3, and 3 exhibit a complicated interaction pattern. An avoided crossing between 3 and 3 can be seen at d/0.08. An important observation is that the degenerate 3 0± splits as a function of d/. While one of the components remains nearly unaffected, the other appears to interact with the high-lying state, resulting in a lowering of its energy with increasing d/. Unlike the 3 0± level, the 0 3 0± level exhibits a minor splitting even for large values of d/. Figure 3, in general, illustrates that even a small value of d/ can lead to significant changes of the RT spectrum of a 3 state. The mixing of the zero-order energy levels also has a significant effect on the intensity of the spectral lines. We

7 SO coupling in linear triatomic molecules J. Chem. Phys. 16, FIG. 4. Redistribution of the spectral intensity by the linear SO vibroniccoupling parameter d for /=0.8 and a c=0, d=0, b c/=0., d=0, and c c/=0., d/=0.15. Solid, dashed, and dotted lines represent energy levels with =0, 1, and, respectively. shall consider here the case of electronic excitation from the 0 vibrational level of some nondegenerate electronic state into the 3 manifold. In the Condon approximation and in the absence of RT coupling of the 3 state, only the 0 3 0, 0 3 1, and 0 3 zero-order levels carry spectral intensity, see Fig. 4a. The three lines of equal intensity are separated by the SO splitting /=0.8. As is well known, 9,10,1,14 the RT coupling leads to a redistribution of intensity, but only among levels with an even number of quanta in the bending mode, as a consequence of the secondorder quadratic in character of the RT coupling. This is illustrated in Fig. 4b c/=0., d=0 the levels 3 and 3 1 acquire nonzero spectral intensity. Similar to the case of a state, 45 the relativistic vibroniccoupling term de ±i, being of first order in, transfers spectroscopic intensity to vibronic levels with an odd number of quanta in the bending mode, see Fig. 4c c/=0., d/ =0.15. Figure 4c reveals, in addition to the transfer of intensity from the level to the level, the presence of a complicated interaction among the levels 0 3, 1 3, 3, and 3 also seen in Fig. 3, which leads to a redistribution of intensity among them. VII. CONCLUSIONS The effect of SO coupling on the RT effect in a 3 electronic state of a linear molecule has been analyzed by employing the microscopic expression for the two-electron Breit-Pauli SO operator. The microscopic RT-SO vibroniccoupling problem involves six coupled electronic states with electronic angular momentum projections ±, ±1, and 0 ±. The 66 vibronic Hamiltonian has been derived in the diabatic representation, employing a Taylor expansion up to second order in the bending displacement. It has been found that in addition to the well-known nonrelativistic quadratic in the bending coordinate RT coupling term, there exists a linear vibronic-coupling term of SO origin. While the nonrelativistic coupling term c e ±i couples the diabatic electronic states and , the linear coupling term de ±i is responsible for and coupling. This linear coupling term is absent when the phenomenological SO operator 19 is used. In the latter case, the six-state problem reduces to three decoupled two-state problems. In other words, the presence of the linear SO vibronic-coupling term prevents the decoupling of the 3 RT-SO problem into a and a 1 RT-SO problem. The symmetry properties of the 3 RT-SO problem have been analyzed. The Hamiltonian commutes with the total electronicspinvibrational vibronic angular momentum operator Ĵ z. As a result, the vibronic eigenstates can be classified in terms of an angular momentum quantum number which takes integer values. The time-reversal symmetry reduces the number of independent matrix elements of the vibronic Hamiltonian significantly. The vibronic Hamiltonian has been transformed to the adiabatic electronic representation. The adiabatic potentialenergy functions and the transformation matrix W have been obtained in analytic form. The effect of the SO splitting, the RT coupling parameter c, and the linear SO coupling parameter d on the adiabatic potential-energy functions has been discussed. The topological phases of the six adiabatic electronic wave functions have been analyzed. While two of them exhibit no geometric phase, the other four exhibit complicated geometric phases which depend on the radius of integration in a nonmonotonic fashion. The spectroscopic effects of the linear SO coupling term have been analyzed by variational calculations of the vibronic energy levels. It has been shown that the linear SO vibronic-coupling mechanism can lead to significant perturbations of the vibronic spectra when the SO splitting and the bending frequency are of similar magnitude. It is well known that the lowest order in the bending amplitude for nonrelativistic RT coupling in a degenerate electronic state of electronic angular momentum is. 5,9,10 The existence of the linear SO vibronic-coupling term de ±i, on the other hand, is independent of the angular momentum of the degenerate electronic state, since it couples the diabatic electronic states with same orbital angular momentum quantum number and spin angular quantum number differing by one unit see Appendix A. This implies that for or electronic states, the nonrelativistic RT coupling term is of fourth or sixth order in the bending amplitude, respectively, the vibronic-coupling term of SO origin will be comparatively more important. In such cases, the SO vibronic coupling may dominate over the nonrelativistic RT coupling even in molecules with light atoms. APPENDIX A: DERIVATION OF THE 3 VIBRONIC HAMILTONIAN The electronic Hamiltonian in the diabatic electronic basis Eq. 16 can be written as follows:

8 Mishra, Poluyanov, and Domcke J. Chem. Phys. 16, H el = H 1 H 1 H 11 0 H 0 H 10 H 00 0 H 0 H 10 H 00 H 00 1 H 1 H 11 H 01 H 01 H 11 H H 1 H 0 H 0 H 1 H. A1 The lower triangular block of the vibronic Hamiltonian is the Hermitian conjugate of the upper triangular block. H 1 =H 1 = D 1, A10 H 10 =H 01 = D, A11 H 10 =H 01 = D 3, A1 H 00 = G, A13 1. Diagonal elements Let us expand the diagonal matrix element H in a Taylor series around the reference geometry up to second order in the degenerate bending coordinate Q ±, H = H 0 H Q ± 0 Q ± 1 H Q Q. Q Q 0 H Q ± Q ± 0 A Using Eq. 17 it can be shown that only the totally symmetric terms in the above expansion can contribute to the matrix element H. Thus, only the first and last terms survive. Furthermore, each of these two terms has an electrostatic as well as a spin-orbit part. By using the following definitions: H es 0 =0, H SO 0 =, H es Q Q = 0, and neglecting the quadratic SO coupling term, we have H = 1 = E,. A3 Similarly, H 11 =0 1 = E,1, A4 H 00 = 1 = E,0. A5 Using time-reversal symmetry relation 18, we have H = H, H 11. Off-diagonal elements = H 11, H 00 = H 00. A6 Using time-reversal symmetry 18 and employing the following definitions: H = A, A7 H 1 H 0 =H 1 = B, A8 = H 11 = H 0 = C, A9 H 0 = H 0 = H, A14 Eq. A1 can be written as H el = E, D 1 H C B A E,1 D D 3 C B E,0 G D 3 C E,0 D H E,1 D 1 E,. A15 The matrix elements of the electronic Hamiltonian are expanded in a Taylor series in powers of the degenerate bending coordinate Q ±. The terms with appropriate symmetry with respect to the symmetry operation Ĵ z on the corresponding electronic matrix elements survive. Using Eq. 17, it is found that the matrix element G is of zeroth order in the expansion; D 1,,3 are of first order, C and H are of second order, B is of third order, while A is of fourth order. Introducing the following abreviations: G 0 = g, D 1,,3 = d 1,,3, Q 0 C H = c, Q Q 0 = h, A16 Q Q 0 B Q Q Q 0 A = b, = a, Q Q Q Q 0 we obtain the 66 vibronic Hamiltonian

9 SO coupling in linear triatomic molecules J. Chem. Phys. 16, H el = E, d 1 e i h e i c e i b 3 e 3i a 4 e 4i E,1 d e i d 3 e i c e i b 3 e 3i E,0 g d 3 e i c e i E,0 d e i h e i E,1 d 1 e i E,. A17 The electrostatic part of the electronic Hamiltonian H es contributes to matrix elements with basis functions involving identical spin eigenstates. Hence only the matrix element C will have contributions from the electrostatic Hamiltonian. Since the matrix elements G, H, and A couple two orthogonal spin eigenstates 1 and 1, the relativistic contribution to these elements vanishes. Therefore, a = h = g =0. A18 On the other hand, the matrix elements D 1,,3 and B have a nonzero contribution from the SO operator. Let us consider the matrix element H 10. Using Eqs. A1 and 16, we have and d 3 e = i d 1 d 1 * H 1 el 0 = 1 d 3 r 1 d 3 r QP1exp i 1 Q1Pexp i ds 1 z ds z 1 1 H 1 el 1 QP1exp i 1 Q1Pexp i, A19 d k =ds z k d 3 r k =ds z k r k dr k dz k d k, k =1, H 1 el = H el. Q 0 Performing the integration over the spin variable ds z k and the angular variable d k, we obtain d 3 e i = c ei r 1 r dr 1 dr dz 1 dz n=1 n=1 3 r1 Q n 3 r Q n 5 r n 5 r 1n r r 1 z 1n z 1 z n z z 1nP 1Q r 1 r 1 z np Q 1. r r A0 Similarly, the matrix element H 01 can be calculated as d 1 d 0 * H 1 el 1 =d 3 e i. A1 The Hermiticity requirement gives H 10 * = H 01. A For the real parameter d 3, from Eqs. A19, A1, and A it is obvious that d 3 =0. In a similar way it can be shown that d 1 = d. A3 A4 Using relations A18, A3, and A4 in Eq. A17 and ignoring the third-order relativistic vibronic-coupling term b, we obtain final form 19 of the 66 vibronic Hamiltonian. APPENDIX B: ORTHONORMAL EIGENVECTORS ±expi 1 V 1, = 1 N 1, expi/d, B1 exp 1 expi N 1 = N = 1 /d are the normalization factors. B V 3,5 = 1 N 3,5 expic/d D ± expic/d, B3 exp i ± W/d 0 expi

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