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Spin-Orbit Vibronic Coupling in States of Linear Triatomic Molecules

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Spin-Orbit Vibronic Coupling in States of Linear Triatomic Molecules THE JOURNAL OF CHEMICAL PHYSICS 126, 134312 ͑2007͒ Spin-orbit vibronic coupling in 3⌸ states of linear triatomic molecules ͒ Sabyashachi Mishraa Department of Chemistry, Technical University of Munich, D-85747 Garching, Germany Leonid V. Poluyanov Institute of Chemical Physics, Academy of Sciences, Chernogolovka, Moscow 14232, Russia Wolfgang Domcke Department of Chemistry, Technical University of Munich, D-85747 Garching, Germany ͑Received 28 November 2006; accepted 8 February 2007; published online 5 April 2007͒ 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-Pauli͒ spin-orbit coupling operator for the two unpaired electrons. The 6ϫ6 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. © 2007 American Institute of Physics. ͓DOI: 10.1063/1.2713382͔ I. INTRODUCTION triatomic molecules of doublet spin multiplicity ͑i.e., with one unpaired electron͒, studies on ⌸ or ⌬ electronic states Vibronic coupling in degenerate electronic states of lin- with higher spin multiplicities are rather limited. This is ear molecules has been discussed in the pioneering work of partly due to the fact that high spin multiplicities are rather 1 ͑ ͒ Renner and later been termed Renner-Teller RT effect. Af- infrequent in nature ͑with the exception of transition-metal ter the first experimental detection of the RT effect in the and rare-earth compounds͒ and partly due to the difficulties NH radical,2,3 Pople and Longuet-Higgins revisited the 2 associated with the theoretical analysis of many-electron sys- original work of Renner with a model Hamiltonian tems. Hougen performed the first systematic perturbative approach.4 Since then, significant progress has been made in analysis of the vibronic and rotational energy levels of linear the understanding of the RT effect both in theory as well as triatomic molecules in a 3⌸ electronic state, taking both RT in experiment, see Refs. 5–12 and references therein. and SO interactions into account.19 This analysis, which can Most of the above cited works ignore the spin-orbit ͑SO͒ be considered as an extension of Pople’s work on the RT-SO coupling. Pople was the first to consider the SO coupling in problem of 2⌸ states,13 was based on the analogous phenom- 2⌸ electronic states of linear triatomic molecules.13 He pro- enological form of the SO operator. An essential conclusion vided a perturbative analysis of the vibronic energy levels of of Hougen’s work was that the vibronic energy levels of a 3⌸ a 2⌸ electronic state with the inclusion of both RT and SO electronic state can be divided into two sets: the energies of interactions, resulting in a clear picture of the perturbations one set are like those of a 1⌸ electronic state, whereas the caused by SO coupling in RT spectra. Pople assumed the energies of the other set are like those of a 2⌸ electronic simplified phenomenological form of the SO operator, state.19 The same phenomenological form of SO coupling ͑ ͒ has been employed in subsequent studies of the spectra of HSO = ALzSz, 1 specific systems. Perić et al. have investigated the RT effect based on the argument that the x and y components of the in the 3⌸ electronic state of NCN with high-level quantum electronic orbital angular momentum operator are effectively chemical methods, employing Hougen’s SO operator.20 In quenched for linear molecules.13 Since then, most of the in- recent work by Carter et al., a variational calculation of spin- vestigations on the RT effect with SO coupling have been rovibronic energy levels has been performed for the 3⌸ state based on this approximation.5,9,14–18 of CCO.21 Other examples of vibronic and SO coupling in While there has been extensive research on the RT effect 3⌸ states are SiCO, CSiO, CCO,22,23 CNN,24 and InOH.25 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 mol- ͒ a Author to whom correspondence should be addressed. Electronic mail: ecules by employing the microscopic expression for the SO 26–29 [email protected] operator ͑the Breit-Pauli operator ͒. We determine the vi- 0021-9606/2007/126͑13͒/134312/10/$23.00126, 134312-1 © 2007 American Institute of Physics Downloaded 06 Apr 2007 to 129.187.254.46. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 134312-2 Mishra, Poluyanov, and Domcke J. Chem. Phys. 126, 134312 ͑2007͒ bronic Hamiltonian by expanding the matrix elements of the rkn = rk − Rn, k = 1,2, electrostatic potential and the SO operator in a diabatic elec- ͑ ͒ tronic basis in powers of the bending coordinate, taking into r12 = r1 − r2 =−r21, 8 account the symmetry selection rules. This results in a 6 ϫ6 vibronic matrix which, unlike the Hamiltonian in Ref. ␴͑k͒ ␴͑k͒ ␴͑k͒ Sk = i ˆ x + j ˆ y + k ˆ z , k = 1,2. 19, cannot be block-diagonalized. In particular, this Hamil- ␴͑k͒ ␴͑k͒ ␴͑k͒ tonian exhibits a linear ͑that is, of first order in the bending Here ˆ x , ˆ y , and ˆ z are the Pauli spin matrices acting on mode͒ vibronic-coupling term of SO origin which is absent the spin eigenstates of the first ͑k=1͒ or second ͑k=2͒ elec- ͉ ͉ ͉ ͉ when the phenomenological form of SO coupling is as- tron. c is the speed of light, rkn= rkn , and r12= r12 . TN is the sumed. We derive the vibronic Hamiltonian in the adiabatic nuclear kinetic-energy operator in polar coordinates ͑␳,␹͒, representation and discuss the geometric phases of the adia- which are defined as batic electronic wave functions. The spectroscopic effects of ␳e±i␹ = Q = Q ± iQ , ͑9͒ RT and SO couplings in 3⌸ states are investigated by varia- ± x y tional calculations of the vibronic energy levels for selected where Qx and Qy are the Cartesian components of the dimen- models. It will be shown that the effects of linear relativistic sionless bending normal coordinate. vibronic coupling are particularly interesting when the bend- We have omitted the totally symmetric stretching modes ing vibrational frequency and the SO splitting of the 3⌸ state in Hamiltonian ͑2͒, since these are decoupled from the de- are in resonance. generate 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 II. VIBRONIC HAMILTONIAN IN THE DIABATIC The electronic Hamiltonian Hel has the following sym- REPRESENTATION metry properties: ͓ ˆ ͔ ͑ ͒ Let us consider two electrons moving in the field of three Hel,Jz =0 10 linearly arranged nuclei of electric charges Q1, Q2, and Q3. Let us focus on the situation where the components of a 3⌸ and state are coupled by the degenerate bending vibrational ͓H ,Tˆ ͔ =0, ͑11͒ mode. The Hamiltonian of this two-electron system, in el atomic units, can be written as where ͑ ͒ ͒ ͑ ͒ ͑ ץ ץ H = Hel + TN = Hes + HSO + TN, 2 Jˆ = Lˆ + Sˆ =−i − i + ␴ˆ 1 + ␴ˆ 2 ͑12͒ ␸ z zץ ␸ץ z z z 2 3 1 2 1 Q 1 ͒ ͑ ͪ n ١2 ͩ Hes = ͚ − − ͚ + , 3 is the z projection of the electronic ͑orbital+spin͒ angular 2 k r r k=1 n=1 kn 12 momentum operator and ͑ ͒ ͑ ͒ 2 Tˆ =4␴ˆ 1 ␴ˆ 2 c.c.ˆ ͑13͒ ͑k͒ ͑12͒ ͑ ͒ y y HSO = ͚ HSO + HSO , 4 k=1 is the time-reversal symmetry operator.32 Here c.c.ˆ stands for the operator of complex conjugation. Tˆ is antiunitary, and for 2ץ 1 ץ ץ ␻ 1 ͩ ␳ ͪ ͑ ͒ systems with an even number of electrons, as is the case TN =− + . 5 ,␹2 hereץ ␳ ␳2ץ ␳ץ ␳ 2 Here Hes is the electrostatic part of the electronic Hamil- Tˆ 2 =1. ͑14͒ tonian that includes the kinetic energy of the two electrons, the electron-nuclear interaction term, as well as the electron- The full Hamiltonian H of Eq. ͑2͒ also possesses the electron repulsion term. HSO represents the SO interaction time-reversal symmetry and commutes with the total ͑k͒ ͑electronic+nuclear͒ angular momentum operator and consists of two parts, the one-electron SO operator HSO, ͑12͒ ץ k=1 and 2, and the two-electron contribution HSO . These are 29 JˆЈ = Jˆ − i . ͑15͒ ␹ץ given by z z 3 ͑ ͒ 2.023i Q .͒...,k = 1,2, ͑6͒ The eigenvalues ͑␮͒ of JˆЈ are integers ͑␮=0,±1,±2 ,͒ ١ H k =− S ͚ n ͑r ϫ SO 4c2 k 3 kn k z n=1 rkn Let us consider electronic diabatic basis functions asso- ciated with the two components of the degenerate 3⌸ elec- ͑ ͒ 2.023i -tronic state with electronic orbital angular momentum quan ͔͒ −2١ ١͑ H 12 = ͓S ͓r ϫ SO 2 3 1 12 1 2 ⌳ 4c r12 tum numbers = ±1 and spin angular momentum quantum numbers S =0, ±1.
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