Study of strong R P and spin orbit vibronic coupling effects in linear triatomic molecules

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1 Chemical Physics 327 (26) Study of strong R P and spin orbit vibronic coupling effects in linear triatomic molecules Syashachi Mishra a, *, Wolfgang Domcke a, Leonid V. Poluyanov b a Department of Chemistry, Technical University of Munich, D Garching, Germany b Institute of Chemical Physics, Academy of Sciences, Chernogolovka, Moscow 14232, Russia Received 1 March 26; accepted 25 May 26 Availle online 9 June 26 Abstract The vibronic coupling between 2 P and 2 R electronic states of a linear molecule is considered with the inclusion of the spin orbit coupling of the 2 P electronic state, employing the microscopic (Breit Pauli) spin orbit coupling operator in the single-electron approximation. The 6 6 Hamiltonian matrix in a diatic spin-electronic basis is derived by an expansion of the molecular Hamiltonian in powers of the bending amplitude up to second-order. The symmetry properties of the Hamiltonian are analyzed. It is pointed out that there exist zeroth-, first-, and second-order R P vibronic-coupling terms of spin orbit origin, which are sent when the usual phenomenological form of the spin orbit coupling operator is used instead of its microscopic form. The influence of the R P and spin orbit vibronic-coupling terms on the adiatic potential energy curves as well as on the vibronic spectra is analyzed for selected models. It is demonstrated that the interplay of strong R P vibronic-coupling and strong spin orbit splitting of the P state can result in unexpectedly complex vibronic spectra. Ó 26 Elsevier B.V. All rights reserved. Keywords: Vibronic coupling; Spin orbit coupling; R P coupling; Time-reversal symmetry; Renner Teller effect 1. Introduction The original paper of Renner [1], describing the vibronic interactions in degenerate P electronic states of linear molecules, has been fundamental for molecular spectroscopy. Experimental observations of Renner s predictions in the spectrum of the NH 2 radical [2,3] stimulated early methodological developments for the treatment of the Renner Teller (RT) effect [4 6]. These early theoretical descriptions, which were based on perturbation theory, were later supplemented by variational methods for the calculation of RT spectra [7 9]. For a detailed survey of the RT effect, we refer to the review by Rosmus and Chambaud [1]. * Corresponding author. Tel.: ; fax: address: syashachi.mishra@ch.tum.de (S. Mishra). The spectra of isolated 2 P electronic states in many triatomic and tetra-atomic molecules have been quantitatively analyzed by these methods. Quite often, however, the interaction of a degenerate 2 P state with other closelying electronic states is also important. Examples are the sorption spectra of NCO [11] and NCS [12] and the photoelectron spectra of HCN [13,14], N 2 O [], and C 2 N 2 [16]. In radicals like C 2 H [17 2], C 2 F [21], and C 2 Cl [22,23], the vibronic interaction of closely spaced 2 P and 2 R states leads to very complex vibronic energy-level spectra. Some selected effects of R P vibronic coupling (VC) have been calculated by perturbation theory, e.g. corrections to rotational energy levels [11,24], modifications of Zeeman orbital g-factors [24,25], and intensity borrowing effects [24]. A comprehensive discussion of the effects associated with the weak perturbation of a P electronic state by a R electronic state has been given by Aarts [26]. A detailed analysis of strong VC of closely spaced 2 R and 2 P states of 31-14/$ - see front matter Ó 26 Elsevier B.V. All rights reserved. doi:1.116/j.chemphys

2 458 S. Mishra et al. / Chemical Physics 327 (26) a linear molecule has been given by Köppel et al. [27], using variational methods. The inclusion of the spin of the unpaired electron in the analysis of vibronic spectra of linear molecules was first considered by Pople [28]. He treated the spin orbit (SO) coupling as a perturbation of the RT Hamiltonian of a 2 P state and obtained the spin-vibronic energy corrections up to second-order. Since then, numerous studies on the interplay of RT and SO coupling effects have been performed, see Refs. [29 31] for reviews. The majority of the existing RT SO treatments, however, have been based on the simplified phenomenological form of the SO operator introduced by Pople [28], H SO ¼ AL z S z ; ð1þ rather than on the microscopic form of SO coupling given by the Breit Pauli Hamiltonian [32,33]. Here, L z and S z are the projections of the electronic orbital and spin angular momenta on the molecular axis, respectively, and A is a phenomenological constant. An expansion of the matrix elements of the Breit Pauli SO operator for an isolated 2 P electronic state in a quasilinear molecule yields, in addition to the well-known quadratic nonrelativistic RT coupling, a linear (that is, of first-order in the bending coordinates) VC term of SO origin [34]. This term is sent when the simplified phenomenological form (1) of the SO operator is employed. The spectroscopic effects of this linear-relativistic VC term have been discussed for a series of triatomic radicals and radical cations [35]. For GeCH, in particular, it has been shown that the effect of the SO induced VC on the vibronic energy levels is rather pronounced [35]. A more recent RT SO analysis of the X e 2 P state of ClCN predicts the observility of this novel VC mechanism in its photoelectron spectrum: it leads to significant intensity transfer to vibrational levels with odd quanta of the bending mode [36]. In this work, we have extended the analysis of SO induced VC effects to vibronically coupled 2 P and 2 R electronic states. The SO coupling is described by the Breit Pauli operator. The matrix elements of the SO operator in a diatic electronic basis have been expanded in powers of the bending coordinate up to second-order. This results in a 6 6 vibronic Hamiltonian, which contains several new VC terms of SO origin. The effect of these R P SO VC mechanisms on the vibronic energy levels and spectral intensity distribution corresponding to the transition from an unperturbed initial state to the vibronically coupled 2 R and 2 P states has been investigated by variational calculations for selected models. The phenomena are particularly interesting when the R P coupling as well as the SO coupling are strong. In these cases, the Born Oppenheimer and Franck Condon approximations break down completely and perturbation theory fails. Therefore, we employ the variational approach, that is, the numerical diagonalization of large secular matrices. 2. Vibronic Hamiltonian in the diatic electronic basis In this section, we formulate the Hamiltonian of a single unpaired electron in a quasi-linear molecule. We focus on the situation where a degenerate 2 P electronic state and a nondegenerate 2 R state of a linear triatomic molecule are closely spaced and coupled by the degenerate bending mode. The vibronic Hamiltonian of this system is written as (h = 1), H ¼ T N þ H el ¼ T N þ H es þ H SO ; T N ¼ x2 1 o 2 q oq q o oq þ 1 o 2 q 2 o/ 2 þ x 2 q2 ð2þ ; ð3þ H es ¼ 1 2 r2 X3 eq n ; r n¼1 n ð4þ H SO ¼ A x r x þ A y r y þ A z r z : ð5þ Here, H es is the electrostatic part of the electronic Hamiltonian that includes the kinetic energy of the unpaired electron and the electronic nuclear interaction term. The Q n are effective nuclear charges. The coupling of the spin and orbital motion of the electron in the field of three nuclei is given by the Breit Pauli SO operator, H SO, A x, A y, and A z being differential operators in electronic coordinate space, the explicit form of which is given in Appendix A. The r x, r y,andr z are the Pauli spin matrices. x is the harmonic bending frequency. T N is the nuclear kinetic energy operator, described in polar coordinates (q,/), where q and / are the radial and angular part of the degenerate bending vibrational mode, respectively. In terms of the Cartesian bending normal coordinates Q x and Q y, they are defined as qe i/ ¼ Q ¼ Q x iq y : It should be noted that we have suppressed the totally symmetric stretching modes in the Hamiltonian (2). The stretching modes generally tune the energy gap of the vibronically coupled 2 R and 2 P states and are, therefore, not separle from the bending motion in the R P VC problem [37]. Since the aim of this work is to understand the generic effects of R P VC as well as SO coupling rather than to calculate spectra of specific molecular systems, the omission of the stretching modes is appropriate for simplicity and clarity. The extension of the present formalism to include the stretching modes is straightforward. The electronic Hamiltonian H el can be shown to commute with the z-component of the electronic angular momentum operator j z ¼ i o oh þ r z; ð7þ where h is the angular electronic coordinate and r z is the spin angular momentum. j z describes a continuous symme- ð6þ

3 S. Mishra et al. / Chemical Physics 327 (26) try, generating an one-parametrical group of unitary transformations J z ðþ ¼e ij z : Here, is the angular group parameter. It is evident that ½H el ; J z ðþš ¼ : The electronic Hamiltonian H el also possesses time-reversal (TR) symmetry. For odd-electronic systems, the TR operator bt is an antiunitary operator that satisfies [38,39] hbt w 1 jbt w 2 i¼hw 1 jw 2 i : ð8þ ð9þ ð1þ The full Hamiltonian H of Eq. (2), satisfies, in addition to the TR property, ½H; j z Š¼ ð11þ with j z ¼ j z i o o/ : ð12þ Here, j z is the z-component of the total angular momentum operator (including the nuclear angular momentum). The eigenvalues (l) of j z are half-integral (l = ±1/2, ±3/ 2,...). The eigenvalues of H are doubly degenerate (Kramer s degeneracy [4]), which is a consequence of the TR symmetry. Lets define w ± i and w i as diatic [41 43] electronic basis set associated with the two components of the degenerate 2 P electronic state (with electronic orbital angular momentum quantum numbers K = ±1) and the nondegenerate 2 R state (K = ), respectively. The coupling of the spin motion with the orbital motion of the electron gives rise to six SO-coupled states. Hence, a complete diatic electronic basis set is given by w ±a i, w ±b i, w a i, and w b i, where the notation w ±a i stands for w ± i ai. Here, a and b represent the two spin eigenstates of the single unpaired electron. For linear molecules with cylindrical symmetry, the following relationship is fulfilled by the diatic electronic basis, J z ðþjw K;r i¼e iðkþrþ/ jw K;r i: ð13þ The TR operator bt has the following effect on the diatic electronic basis functions: bt jw a i¼jw b i; bt jw a i¼jw b i; ð14þ bt jw b i¼ jw a i; bt jw b i¼ jw a i: Using the ove defined six diatic spin-electronic basis functions, the electronic Hamiltonian can be written as a 6 6 vibronic matrix. The determination of the vibronic matrix elements can be simplified by using the TR symmetry, Eqs. (1) and (14), 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 up to second-order in the bending coordinate. The derivation of the 6 6 spin-vibronic Hamiltonian is given in Appendix B. The result is H ¼ T N 1 E P;3=2 kqe i/ cq 2 e 2i/ dqe i/ hq 2 e 2i/ 1 kqe i/ E R;1=2 kqe i/ g hq 2 e 2i/ cq 2 e 2i/ kqe i/ E P;1=2 g dqe i/ þ dqe i/ g E P;1=2 kqe i/ cq 2 e 2i/ : B hq 2 e 2i/ g kqe i/ E R;1=2 kqe i/ A hq 2 e 2i/ dqe i/ cq 2 e 2i/ kqe i/ E P;3=2 ðþ Here, 1 is the six-dimensional unit matrix and E P,3/2, E P,1/2, and E R,1/2 are the electronic energies at the reference geometry (q = ). To simplify the notation, we introduce the R P energy gap D and the SO splitting f as follows D ¼ E P E R ; f ¼ E P;3=2 E P;1=2 : ð16þ ð17þ Here, E P and E R are the electrostatic energies at the reference geometry, while E P,1/2 and E P,3/2 are the reference energies with inclusion of SO coupling. c is the well-known nonrelativistic quadratic RT coupling constant [1 4], k is the linear R P coupling constant [27], while g and h are purely relativistic R P coupling constants of zeroth- and second-order in the bending coordinate, respectively. The parameter d is the relativistic linear RT coupling constant introduced in Ref. [34]. The zeros appearing along the cross diagonal are a consequence of the TR symmetry. The two 3 3 diagonal blocks of the vibronic Hamiltonian () have the same form as the nonrelativistic R P Hamiltonian (f = ) [27]. The off-diagonal 3 3 blocks of the vibronic Hamiltonian () are, necessarily, of purely relativistic origin. For linear geometry (q = ), all off-diagonal elements of the Hamiltonian matrix () vanish, with the exception of the zeroth-order off-diagonal coupling SO matrix element g. This special case of the R P SO Hamiltonian has been considered in Ref. [44]. It has been shown that the 6 6 matrix can be block-diagonalized in this case to two 3 3 matrices by a constant (q independent) unitary transformation. It should be realized, however, that this transformation mixes the 2 R and 2 P diatic states. The transformed basis states thus are no longer diatic electronic states, if, for example, a R P crossing occurs as a function of the stretching coordinates. In the general case (q 5 ) the Hamiltonian matrix () cannot be decoupled into smaller submatrices. Although the adiatic approximation fails to describe the vibronic energy levels of nearly degenerate electronic states, the adiatic potential energy (PE) surfaces are nevertheless very helpful for the qualitative interpretation of the vibronic spectra. The adiatic PE surfaces are obtained by diagonalizing H T N at a fixed nuclear geometry. The derivation of analytic expressions for the eigenvalues of the 6 6 matrix is unfeasible even with the help

4 46 S. Mishra et al. / Chemical Physics 327 (26) of symbolic-mathematical tools. The adiatic PE surfaces therefore have been obtained by numerical diagonalization of the 6 6 matrix. 3. Calculation of spectra For a given value of the good quantum number l, we expand jw l m i in a complete basis, which is constructed as the product of electronic (jw l K;ri) and vibrational ( nli) basis functions, where K = ±1, (for the 2 P and the 2 R state, respectively) and r = ±1/2. The nli, n =,1,2... and l = n, n +2,...n 2, n, are the eigenfunctions of the two-dimensional isotropic harmonic oscillator [45]. Within the Condon approximation, only the vibronic levels with l = ±1/2 and ±3/2 can carry intensity, assuming optical excitation from a closed-shell and vibrationless ground state. The real symmetric Hamiltonian matrix is constructed and diagonalized for a given value of l. The vibrational basis is increased until convergence of the eigenvalues of interest has been achieved. A standard diagonalization method for real symmetric matrices has been used. The eigenvalues represent the vibronic energy levels for the given vibronic angular momentum quantum number l. While l = 3/2 levels gain intensity only from the 2 P 3/2 state, l = 1/2 levels gain intensity from both the 2 R 1/2 and 2 P 1/2 states (we assume equal oscillator strengths of all three states). For l = 3/2, the square of the eigenvector element corresponding to the 2 P 3/2 i i basis state gives the intensity. For l = 1/2, on the other hand, two eigenvector components, corresponding to the 2 P 1/2 i i and 2 R 1/2 i i basis states, may give intensity. In the case of strong R P mixing, both components of the eigenvectors become comparle. In this case, we assume, for clarity, that either the 2 R 1/2 or the 2 P 1/2 oscillator strength is zero. 4. Results and discussion In this section, we analyze the influence of the R P VC and SO coupling on the energy levels and the spectral intensity distribution for the photoinduced transition from an unperturbed initial state into the vibronically coupled 2 R and 2 P final states. For brevity, we consider only the case where the 2 R state is higher in energy than the 2 P state. Since the 6 6 R P vibronic Hamiltonian contains many coupling parameters, a systematic study of the effects of all parameters is beyond the scope of this article. We will limit ourselves to the investigation of a few selected and particularly interesting cases. The effect of nonrelativistic RT coupling (parameter c) and linear relativistic VC (parameter d) within a 2 P state is well understood. Therefore, we do not consider these coupling mechanisms in the present study. The coupling term h is purely relativistic and quadratic in the bending distortion and hence will be ignored in the following. It should be kept in mind that all coupling parameters, in particular D and f, may be functions of the stretching modes, which is not taken into account here Variation of the R P coupling strength The nonrelativistic R P VC mechanism has been analyzed in detail in Ref. [27]. For weak SO coupling, it is straightforward to include the SO effects by perturbation theory. Hence we shall focus here on the cases where the SO splitting (f) ofthe 2 P state is relatively large, being comparle to the R P energy gap (D) Case I Here, we discuss a system with D/x = 5. and f/x = 2.. This represents a typical case of a relatively large R P gap and moderate SO splitting of a 2 P state, which can be found in many linear molecules with moderately heavy atoms. Fig. 1a c shows the adiatic PE curves of this system for k/x =, 2., and 4., respectively. All other coupling parameters in Eq. () are set to zero. The solid, dashed, and dotted lines correspond to the 2 P 3/2, 2 P 1/2, and 2 R 1/2 states, respectively. For k =, the adiatic PE curves are parolae which are separated by the zerothorder splittings (D and f). With increasing k/x, the 2 P 3/2 PE functions develop a double-minimum shape, while the PE function of the 2 P 1/2 state becomes very flat in the vicinity of q =. The corresponding vibronic spectra are shown in Fig. 2a c. For k/x =, the result is trivial, showing three purely electronic transitions of equal intensity, see Fig. 2a. The intensities of these lines get distributed over vibronic levels when k/x becomes nonzero. The increase in the curvature of the uppermost adiatic PE curve (the 2 R 1/2 state), leads to a substantial increase of the zero-point energy, thus shifting the corresponding lines to higher energy. The VC effects are most pronounced in the 2 P state, where the adiatic PE function develops a bent geometry. Strong R P coupling (k/x = 4.) leads to very complicated vibronic structures of the 2 P 3/2 and 2 P 1/2 states, see Fig. 2c. It is noteworthy that the vibronic spectra of the two components of the 2 P state are very different from each other in this case. This possibility of very different vibronic structures of the two SO components of a 2 P state apparently has never been considered so far in the assignment of observed spectra Case II Here, we investigate the case of very large SO splitting (f/x = 2.) and moderate R P gap (D/x = 5.). This combination of parameters leads to the interesting situation where the nondegenerate 2 R state lies between the two SO components of the 2 P state. This situation may arise for excited states of molecules containing relatively heavy atoms, as is well known for diatomic molecules [46]. Fig. 3a c shows the adiatic PE curves for k/x =, 2., and 4.. For k/x =, the adiatic PE curves are harmonic and the 2 R 1/2 state lies in between the two SO

5 S. Mishra et al. / Chemical Physics 327 (26) Fig. 1. Adiatic PE curves of the 2 P 3/2 (solid), 2 P 1/2 (dashed), and 2 R 1/2 (dotted) states for D/x = 5., f/x = 2., and k/x =, 2., and Intensity (arbitrary units) Energy (units of ω) Fig. 2. R P vibronic spectra for D/x = 5., f/x = 2., and k/x =, 2., and 4.. The solid, dashed, and dotted lines represent the vibronic levels which gain intensity from the 2 P 3/2, 2 P 1/2, and 2 R 1/2 states, respectively.

6 462 S. Mishra et al. / Chemical Physics 327 (26) Adiatic PE Curves (units of ω) Dimensionless Bending Coordinate (ρ) Fig. 3. Adiatic PE curves of the 2 P 3/2 (solid), 2 P 1/2 (dashed), and 2 R 1/2 (dotted) states for D/x = 5., f/x = 2., and k/x =, 2., and Intensity (arbitrary units) Energy (units of ω) Fig. 4. R P vibronic spectra for D/x = 5., f/x = 2., and k/x =, 2., and 4.. The solid, dashed, and dotted lines represent the vibronic levels which gain intensity from the 2 P 3/2, 2 P 1/2, and 2 R 1/2 states, respectively.

7 S. Mishra et al. / Chemical Physics 327 (26) components of the 2 P state, see Fig. 3a. For k/x = 2., the PE function of the 2 P 3/2 state becomes very flat near q =, while the PE function of the 2 R 1/2 state has developed a slight double minimum. With further increasing k/x, both the 2 P 3/2 as well as the 2 R 1/2 PE functions develop minima at bent geometries, see Fig. 3c. Fig. 4a c shows the corresponding sorption spectra. For k/x = 2., the close-lying 2 R 1/2 and 2 P 1/2 states show weak indications of vibronic interaction, while the far-lying 2 P 3/2 state remains nearly unperturbed. Compared to Fig. 2b, the effect of R P coupling is less pronounced here because of the lager SO splitting. For k/x = 4., the spectrum of the 2 P 3/2 state exhibits a pronounced progression in the bending mode, which reflects the nonlinear equilibrium geometry, see Fig. 3c. The 2 R 1/2 and 2 P 1/2 states exhibit rather complex vibronic spectra which are dominated by quasi-degeneracy effects, see Fig. 4c. This again shows that the two SO components of the 2 P state can exhibit completely different vibronic spectra The 2 P 1/2 2 R 1/2 resonance case Here, we consider the case of accidental degeneracy of the 2 P 1/2 and 2 R 1/2 states. The R P gap (D) and the SO splitting (f) are adjusted such that the 2 P 1/2 component of the 2 P state and the 2 R 1/2 state are degenerate at the reference geometry. For a sufficiently large value of D, the 2 P 3/2 can be considered to be decoupled from the 2 P 1/2 and 2 R 1/2 states. The 6 6 vibronic Hamiltonian can thus be truncated to 4 4 form E R;1=2 kqe i/ 1 g kqe i/ E P;1=2 g H ¼ T N 1 þ B g E P;1=2 kqe i/ A ; ð18þ g kqe i/ E R;1=2 where 1 is the four-dimensional unit matrix. The ove Hamiltonian is isomorphic to the Hamiltonian of the linear E E Jahn Teller (JT) effect with SO coupling in trigonal symmetry, see Ref. [47]. k is equivalent to the linear JT coupling parameter, while g is equivalent to the matrix elements of the A x r x + A y r y term of the SO coupling operator Case I In this special case of accidental degeneracy of the 2 R 1/2 and 2 P 1/2 states, we have investigated the effect of the zeroth-order coupling parameter g. Fig. 5a c exhibits the adiatic PE curves of a system with D/x = 5., f/x = 1., k/x = 1., while g/x takes the value,.2, and.5. Fig. 5a shows that the 2 P 1/2 and 2 R 1/2 PE functions touch each other at the reference geometry (q = ). With increasing -1 1 Adiatic PE Curves (units of ω) Dimensionless Bending Coordinate (ρ) Fig. 5. Adiatic PE curves of the 2 P 3/2 (solid), 2 P 1/2 (dashed), and 2 R 1/2 (dotted) states for D/x = 5., f/x = 12., and k/x = 1. with g/x =,.2, and.5.

8 464 S. Mishra et al. / Chemical Physics 327 (26) Intensity (arbitrary units) Energy (units of ω) Fig. 6. R P vibronic spectra for D/x = 5., f/x = 12., and k/x = 1. with g/x =,.2, and.5. The solid and dashed lines represent the vibronic levels gaining intensity from the 2 P 3/2 and 2 P 1/2 states, respectively Adiatic PE Curves (units of ω) Dimensionless Bending Coordinate (ρ) Fig. 7. Adiatic PE curves of the 2 P 3/2 (solid), 2 P 1/2 (dashed), and 2 R 1/2 (dotted) states for D/x = 5., f/x = 12., and k/x = 4. with g/x =,.5, and 1..

9 S. Mishra et al. / Chemical Physics 327 (26) g, this degeneracy is removed, see Fig. 5b and c, while the adiatic PE curve of the 2 P 3/2 state remains essentially unchanged. Fig. 6a c shows the corresponding vibronic spectra, assuming vanishing oscillator strength of the 2 R 1/2 state (the alternative case will lead to a spectrum with same line positions, but different intensities). In Fig. 6a, the 2 P 1/2 state exhibits a vibronic spectrum which corresponds to a moderately strong E E JT effect, each vibronic line being doubly degenerate. With increasing g/x, these degenerate vibronic lines split proportional to g/x. The coupling parameter g thus removes the accidental degeneracy of the vibronic levels (due to the 2 R 1/2 2 P 1/2 accidental degeneracy) in zeroth-order Case II In this final example, we investigate the resonance case in the limit of very strong R P nonrelativistic coupling (k/x = 4.). Fig. 7a c shows the adiatic PE functions for D/x = 5., f/x = 1., while g/x takes the values,.5, and 1.. The lowest adiatic PE curve develops a minimum at a strongly bent geometry as a consequence of the strong R P coupling, see Fig. 7a. The accidental degeneracy of the 2 P 1/2 and 2 R 1/2 states in Fig. 7a is lifted by finite values of g/x, see Fig. 7c. Fig. 8a c shows the corresponding vibronic spectra. For g/x =, the 2 P 1/2 state shows vibronic structure which is typical for strong JT coupling. Note in particular, the double-hump shape of the spectral envelope. The well separated 2 P 3/2 state, on the other hand, exhibits the characteristic extended Franck Condon progression of a linear-to-bent transition. With increasing g/x, each of the 2 P 1/2 lines splits into two, as described earlier. However, in addition to changes in the vibronic structure of the 2 P 1/2 state, the parameter g also strongly affects the vibronic structure of the 2 P 3/2 state, see Fig. 8b. In this case of large k/x, the 2 P 3/2 state cannot be decoupled from the 2 R 1/2 and 2 P 1/2 states. For g/x =.5, the density of intensity carrying lines in the 2 P 3/2 state is doubled, see Fig. 8b. Interestingly, further increase of g/x to 1. essentially restores the line density observed for g/x =, see Fig. 8c. In the latter case, the levels of the 2 P 3/2 state are again doubly degenerate. 5. Conclusions The combined effects of strong SO coupling and the strong VC of closely spaced 2 P and 2 R electronic states of a linear molecule has been investigated, employing the Breit Pauli SO operator. The 6 6 vibronic Hamiltonian has been derived in the diatic representation up to second-order in the bending displacement. It has been found that there exists, in addition to the well-known nonrelativistic (quadratic) RT coupling term and nonrelativistic (linear) R P coupling term, three coupling terms of relativistic origin. While the two components of the 2 P state are coupled by a relativistic linear term, the 2 R and 2 P states are relativistically coupled in zeroth- and second-order of the bending displacement. The quadratic coupling term of SO origin has been ignored in the present study, while the effects of the zeroth-order term have been studied explicitly..2 Intensity (arbitrary units) Energy (units of ) Fig. 8. R P vibronic spectra for D/x = 5., f/x = 12., and k/x = 4. with g/x =,.5, and 1.. The solid and dashed lines represent the vibronic levels gaining intensity from the 2 P 3/2 and 2 P 1/2 states, respectively.

10 466 S. Mishra et al. / Chemical Physics 327 (26) The combined effects of nonrelativistic R P coupling and strong SO splitting of the 2 P state have been investigated by variational calculations of the vibronic energy levels. It has been shown that this problem is very rich and that very complex vibronic spectra can arise. For example, the 2 P 1/2 and 2 P 3/2 states can exhibit completely different vibronic structures when both the SO splitting and the nonrelativistic R P coupling are strong. The effect of zeroth-order relativistic R P coupling has been shown to be important when 2 R 1/2 2 P 1/2 near degeneracies occur. In this case, for a sufficiently large SO coupling, the 2 P 3/2 state gets approximately decoupled, and the 2 P 1/2 and 2 R 1/2 states exhibit a linear JT type interaction. Acknowledgement This work has been supported by the Deutsche Forschungsgemeinschaft (by a research grant as well as by a visitor grant for L.V.P) and the Fonds der Chemischen Industrie. Appendix A. The Breit Pauli SO operator The differential operators A x, A y,anda z of Eq. (5) in cylindrical coordinates are given by A x ¼ igb 2 X3 Q n r sin / o r 3 n oz z n sin / o or þ cos / o r o/ n¼1 A y ¼ igb 2 X3 n¼1 A z ¼ igb 2 X3 n¼1 Q n r 3 n Q n r 3 n z n cos / o sin / o or r o/ o o/ r cos / o oz ða:1þ ða:2þ ða:3þ where g 2.23 is the g-factor of the electron and b ¼ eh 2m ec is the Bohr magneton. Appendix B. Derivation of the R P spin-vibronic Hamiltonian The electronic Hamiltonian in the diatic electronic basis can be written as follows B.1. Diagonal elements Let us expand the diagonal matrix element in a Taylor series around the reference geometry up to second-order in the degenerate bending coordinate (Q ± ), ¼ þ 1 2 ðþ þ þþ oh oq o 2 oq þ oq Q þ 1 2 Q þ Q þ o 2 oq 2! Q 2 ðb:2þ Using Eq. (13) it can be shown that only the totally symmetric terms in the ove expansion can contribute to the matrix element. Thus, only the first and last terms survive. Furthermore, each of these two terms have an electrostatic as well as spin orbit part. By using the following definitions; ðesþðþ ¼ D 2 ; ðsoþðþ ¼ f 2 ; 1 o 2 ðesþ ¼ x 2 oq þ oq 2 ; and neglecting the quadratic SO coupling term, we have, ¼ D 2 þ f 2 þ 1 2 xq2 ¼ E P;3=2 þ 1 2 xq2 : ðb:3þ Similarly, H ¼ D 2 f 2 þ 1 2 xq2 ¼ E P;1=2 þ 1 2 xq2 ; ðb:4þ H ¼ D 2 þ 1 2 xq2 ¼ E R;1=2 þ 1 2 xq2 : ðb:5þ Using the TR symmetry relations of Eqs. (1) and (14), we have ¼ H bb ; H B.2. Off-diagonal elements H þ ¼ bb ; H ¼ H bb : ðb:6þ Using the TR symmetry, Eqs. (1) and (14), we have We define ¼ H þ ¼ H ¼ : ðb:7þ ðb:1þ H þ H H þ H þ H þ ¼ðHbb Þ ¼ H þ bb ¼ðH Þ ¼ þ ¼ðHbb Þ ¼ H þ bb ¼ðH Þ ¼ L; ¼ðH þ bb Þ ¼ H þ þ bb ¼ðH Þ ¼ C; ¼ ðhba Þ ¼ H þþ ¼ðHba Þ ¼ D; ¼ ðhba Þ ¼ H þ ¼ðHba Þ ¼ G; ¼ ðhba Þ ¼ H þ ¼ðHba Þ ¼ H: ðb:8þ ðb:9þ ðb:1þ ðb:11þ ðb:12þ

11 S. Mishra et al. / Chemical Physics 327 (26) Using the ove definitions, Eq. (B.1) can be written as ðb:13þ The electrostatic part of the electronic Hamiltonian H es contributes to matrix elements with the basis functions involving identical spin eigenstates. Hence the matrix elements L and C will have contributions from the electrostatic Hamiltonian. The contribution of H SO to the matrix elements of the electronic Hamiltonian is determined in the following way, L SO ¼hw þa jh SO jw a i¼ 1 2 hw þja z jw i¼; C SO ¼hw þa jh SO jw a i¼ 1 2 hw þja z jw i¼; D SO ¼hw þa jh SO jw þb i¼ 1 2 hw þja x ia y jw þ i 6¼ ; G SO ¼hw a jh SO jw þb i¼ 1 2 hw ja x ia y jw þ i 6¼ ; H SO ¼hw a jh SO jw b i¼ 1 2 hw ja x ia y jw i 6¼ : ðb:14þ ðb:þ ðb:16þ ðb:17þ ðb:18þ The matrix elements of the electronic Hamiltonian are expanded in a Taylor series up to second-order in the degenerate bending mode (Q ± ). The terms with appropriate symmetry with respect to the symmetry operation J z () on the corresponding electronic matrix elements survive. Using Eq. (13), it is found that the matrix element G is of zeroth-order in the expansion; L and D are of first-order; while C and H are of second-order. The following breviations are introduced to arrive at the final form of the 6 6 vibronic Hamiltonian (): G ðþ ¼ g ol od ¼ k ¼ d oq þ oq þ oc oh ¼ c oq þ oq þ oq þ oq þ References ¼ h: ðb:19þ [1] R. Renner, Z. Phys. 92 (1934) 172. [2] K. Dressler, D.A. Ramsay, J. Chem. Phys. 27 (1957) 971. [3] K. Dressler, D.A. Ramsay, Phil. Trans. R. Soc. Lond. A251 (1959) 69. [4] H.C. Longuet-Higgins, in: H.W. Thompson (Ed.), Advances in Spectroscopy, vol. II, Interscience, New York, 1961, p [5] J.A. Pople, H.C. Longuet-Higgins, Mol. Phys. 1 (1958) 372. [6] J.T. Hougen, J. Chem. Phys. 36 (1962) 519. [7] G. Duxbury, The electronic spectra of triatomic molecules and the Renner Teller effect, in: Molecular Spectroscopy: Specialist Periodical Report, vol. III, Chemical Society, London, 1975, p [8] A.J. Merer, C. Jungen, in: K.N. Rao (Ed.), Molecular Spectroscopy: Modern Research, vol. II, Academic Press, New York, 1976, p. 127 (Chapter 3). [9] J.M. Brown, F. Jørgensen, Adv. Chem. Phys. 52 (1983) 117. [1] P. Rosmus, G. Chambaud, The Renner Teller effect and the role of electronically degenerate states in molecular ions, in: C.-Y. Ng (Ed.), Photoionization and Photodetachment, vol. 1A, World Scientific, Singapore, 2 (Chapter 5). [11] P.S.H. Bolman, J.M. Brown, A. Carrington, I. Kopp, D.A. Ramsay, Proc. R. Soc. Lond. A343 (1975) 17. [12] R.N. Dixon, D.A. Ramsay, Can. J. Phys. 46 (1968) [13] D.C. Frost, S.T. Lee, C.A. McDowell, Chem. Phys. Lett. 23 (1973) 472. [14] H. Köppel, L.S. Cederbaum, W. Domcke, W. von Niessen, Chem. Phys. 37 (1979) 33. [] P.M. Dehmer, J.L. Dehmer, W.A. Chupka, J. Chem. Phys. 73 (198) 126. [16] L.S. Cederbaum, W. Domcke, J. Schirmer, H. Köppel, J. Chem. Phys. 72 (198) [17] M. Perić, S.D. Peyerimhoff, R.J. Bunker, Mol. Phys. 71 (199) 693. [18] A. Mebel, M. Bare, S. Lin, J. Chem. Phys. 112 (2) 173. [19] R. Tarroni, S. Carter, J. Chem. Phys. 119 (23) [2] W. Chen, S.E. Novick, M.C. McCarthy, C.A. Gottlieb, P. Thaddeus, J. Chem. Phys. 13 (1995) [21] R. Tarroni, Chem. Phys. Lett. 38 (23) 624. [22] R. Tarroni, S. Carter, J. Chem. Phys. 123 (25) [23] Y. Sumiyoshi, T. Ueno, Y. Endo, J. Chem. Phys. 119 (23) [24] P.S.H. Bolman, J.M. Brown, Chem. Phys. Lett. 21 (1973) 213. [25] J.M. Brown, J. Mol. Spectrosc. 68 (1977) 412. [26] J.F.M. Aarts, Mol. Phys. 35 (1978) [27] H. Köppel, W. Domcke, L.S. Cederbaum, J. Chem. Phys. 74 (1981) [28] J.A. Pople, Mol. Phys. 3 (196) 16. [29] M. Perić, C.M. Marian, S.D. Peyerimhoff, J. Mol. Spectrosc. 166 (1994) 46. [3] M. Perić, C.M. Marian, S.D. Peyerimhoff, J. Chem. Phys. 114 (21) 686. [31] M. Perić, M. Mladenović, K. Tomić, C.M. Marian, J. Chem. Phys. 118 (23) [32] H.A. Bethe, E.E. Salpeter, Quantum Mechanics for One- and Two- Electron Atoms, Springer, Berlin, [33] W. Pauli, Z. Phys. 43 (1927) 61. [34] L.V. Poluyanov, W. Domcke, Chem. Phys. 31 (24) 111. [35] S. Mishra, V. Vallet, L.V. Poluyanov, W. Domcke, J. Chem. Phys. 123 (25) [36] S. Mishra, V. Vallet, L.V. Poluyanov, W. Domcke, J. Chem. Phys. 124 (26) [37] H. Köppel, W. Domcke, L.S. Cederbaum, Adv. Chem. Phys. 57 (1984) 59. [38] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Academic-Verlag, Berlin, [39] C.A. Mead, J. Chem. Phys. 7 (1979) [4] H. Kramer, Acad. Sci. Amsterdam 33 (193) 959. [41] W. Lichten, Phys. Rev. 164 (1967) 131. [42] F.T. Smith, Phys. Rev. 179 (1969) 111. [43] T. Pacher, L.S. Cederbaum, H. Köppel, Adv. Chem. Phys. 84 (1993) 293. [44] M. Alexander, D. Manolopoulos, H. Werner, J. Chem. Phys. 113 (2) [45] L. Schiff, Quantum Mechanics, third ed., McGraw-Hill, [46] E.P.F. Lee, T.G. Wright, J. Chem. Phys. 123 (25) [47] W. Domcke, S. Mishra, L.V. Poluyanov, Chem. Phys. 322 (26) 45.