Renner-Teller and Spin-Orbit Vibronic Coupling Effects in Linear Triatomic Molecules with a Half-ﬁlled ␲ Shell ͒ ͒ Ilias Sioutis,1,A Sabyashachi Mishra,2 Leonid V

THE JOURNAL OF CHEMICAL PHYSICS 128, 124318 ͑2008͒

Renner-Teller and spin-orbit vibronic coupling effects in linear triatomic molecules with a half-ﬁlled ␲ shell ͒ ͒ Ilias Sioutis,1,a Sabyashachi Mishra,2 Leonid V. Poluyanov,3 and Wolfgang Domcke1,b 1Department of Chemistry, Technical University of Munich, D-85747 Garching, Germany 2Department of Chemistry, University of Basel, CH-4056 Basel, Switzerland 3Institute of Chemical Physics, Academy of Sciences, Chernogolovka, Moscow 14232, Russia ͑Received 18 December 2007; accepted 14 January 2008; published online 31 March 2008͒

The vibronic and spin-orbit-induced interactions among the 3⌺−, 1⌬, and 1⌺+ electronic states arising from a half-ﬁlled ␲ orbital of a linear triatomic molecule are considered, employing the microscopic ͑Breit-Pauli͒ spin-orbit coupling operator. The 6ϫ6 Hamiltonian matrix is derived in a diabatic spin-orbital electronic basis set, including terms up to fourth order in the expansion of the molecular Hamiltonian in the bending normal coordinate about the linear geometry. The symmetry properties of the Hamiltonian are analyzed. Aside from the nonrelativistic fourth-order Renner-Teller vibronic coupling within the 1⌬ state and the second-order nonrelativistic vibronic coupling between the 1⌺+ and 1⌬ states, there exist zeroth-order, ﬁrst-order, as well as third-order vibronic coupling terms of spin-orbit origin. The latter are absent when the phenomenological expression for the spin-orbit coupling operator is used instead of the microscopic form. The effects of the nonrelativistic and spin-orbit-induced vibronic coupling mechanisms on the 3⌺−, 1⌬, and 1⌺+ adiabatic potential energy surfaces as well as on the spin-vibronic energy levels are discussed for selected parameter values. © 2008 American Institute of Physics. ͓DOI: 10.1063/1.2840356͔

I. INTRODUCTION system for the development of ab initio electronic-structure methods. From an electronic-structure point of view, CH2 is It has previously been shown that the inclusion of the a linear molecule with a half-filled ␲ shell, since two of its microscopic Breit-Pauli spin-orbit ͑SO͒ operator in the mo- valence electrons occupy a degenerate ␲ orbital. This elec- lecular Hamiltonian of linear triatomic molecules gives rise tronic configuration gives rise to the 3⌺−, 1⌬ and 1⌺+ elec- 2⌸ g g g to novel vibronic coupling terms for degenerate electronic states, the first one being the electronic ground state of tronic states.1,2 The vibronic coupling terms arising from the CH2 at the linear configuration, as expected from Hund’s two-electron Breit-Pauli operator have also been discussed rule. However, the lower-energy Renner-Teller-split compo- 3⌸ 3 for electronic states of linear triatomic molecules. It has nent arising from the 1⌬ electronic state of CH lies only been found that in addition to the nonrelativistic Renner- g 2 slightly above its ground electronic state at bent geometries, Teller coupling term, which is of second order in the bending as a result of strong Renner-Teller and pseudo-Renner-Teller amplitude ͑in 2⌸ electronic states͒, there exist SO-induced vibronic interactions. vibronic coupling terms which are of first order in the bend- The calculation of accurate rovibronic spectra of CH ing amplitude.1–3 In the present work, we proceed one step 2 poses a considerable theoretical challenge due to its large- further and develop a model for the description of the combined effects of Renner-Teller and SO coupling of linear tri- amplitude bending motion and the strong nonadiabatic cou- atomic molecules with a half-filled ␲ shell. This approach plings involved. Extensive calculations of the CH2 spectra have previously been performed, based on a systematic treat- employs the microscopic Breit-Pauli SO operator for two 22–24 active electrons. It may be applied for a wide range of linear ment of the Renner-Teller vibronic coupling. While the or quasilinear molecules ranging from simple carbenes to analysis of this effect has reached a high level of sophistica- 25 heavier molecular systems. tion for the low-lying electronic states, the treatment of the 4,5 Carbenes play an important role in combustion, SO coupling in CH2 is still largely phenomenological, as it is stratospheric6 and interstellar chemistry,7,8 as well as in or- based on an effective SO coupling parameter between the 9–12 1 3 ganic and organometallic reactions and processes that in- A1 and B1 electronic states at bent geometries. This param- volve thermal degradation of small organic molecules.13 The eter is taken either as an empirical constant or is considered ͑ ͒ 25 prototypical carbene methylene CH2 has been a favorite to be a function of the bending normal coordinate. target of spectroscopists14–16 and theoreticians17–20 for nearly Prototypical molecules in their own right are the haloge- ͑ ͒ half a century, due to its fundamental role as a radical inter- nated derivatives of CH2, i.e., CH D X and CX2 with X=F, mediate species in gas-phase chemical reactions.21 Moreover, Cl, Br. They can be considered as model systems for the it is sufficiently small to serve as a benchmark molecular study of the spectroscopy and dynamics on three coupled potential energy surfaces, providing insight into the reactiv- ͒ a Electronic mail: [email protected]. ity and electronic structure of carbenes as well as the inter- ͒ b Electronic mail: [email protected]. play between SO coupling and Renner-Teller vibronic inter-

Downloaded 01 Apr 2008 to 131.152.105.39. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 124318-2 Sioutis et al. J. Chem. Phys. 128, 124318 ͑2008͒ actions. These species also serve as benchmarks for both the two stretching vibrational modes on the vibronic cou- theory and experiment. The ground electronic state of the pling problem is not considered here. Their effect is to “tune” 3⌺− 1⌬ 1⌺+ dihalocarbenes, as well as of SiH2 and GeH2, corresponds to the energy separations among the , , and electronic the lower Renner-Teller-split component of the 1⌬ electronic states. The present formalism can be extended to include the 1 state which is of A1 spatial symmetry. The two lowest elec- effects of the totally symmetric stretching vibrational modes, tronic states of CH͑D͒X are ˜X 1AЈ and Ã 1AЉ at bent geom- see, e.g., Refs. 27–29. etries. These arise from the Renner-Teller splitting of the The spin-vibronic molecular Hamiltonian Hˆ can be ex- electronically degenerate 1⌬ ground electronic state at the pressed as linear configuration. The triplet electronic state with 3AЉ spa- 3 − Hˆ Tˆ Hˆ Tˆ Hˆ Hˆ ͑ ͒ tial symmetry arises from the ⌺ electronic state and lies in = N + el = N + es + SO, 1 the vicinity of these Renner-Teller-split electronic states. The 3 Tˆ Hˆ ã AЉ state plays a double role, as it is involved in SO cou- where N is the nuclear kinetic-energy operator and el is pling with the ˜X 1AЈ state and contributes to the perturbations the electronic Hamiltonian which includes the kinetic energy of the Ã 1AЉ electronic state of CH͑D͒X at the linear geom- of the electrons, the electron-nuclear and electron-electron ͑ Hˆ ͒ etry. interaction terms represented by es , as well as the SO ͑ Hˆ ͒ The goal of the present work is the theoretical study of interaction given by SO . If we express the dimensionless the interplay of the Renner-Teller and SO interactions in lin- bending normal mode in the polar-coordinate representation ear triatomic molecular systems with a half-filled ␲ shell. ͑␳, ␾͒ and use atomic units, the nuclear kinetic-energy op- While the combined effects of ⌺-⌸ and SO vibronic cou- erator is given by plings on the electronic structure of linear molecules have 2ץ 1 ץ ץ ␻ 1 previously been investigated employing the microscopic Tˆ ͫ ͩ␳ ͪ ͬ ͑ ͒ N =− + . 2 ␾2ץ ␳ ␳2ץ ␳ץ Breit-Pauli͒ spin-orbit coupling operator,26,27 no correspond- 2 ␳͑ ing information about the combined ⌺-⌬ and SO interactions has been elaborated thus far. We develop the theoretical for- The electrostatic part of the Hamiltonian is written as malism which is required for a systematic and comprehen- 2 3 Q 1 ͒ ͑ ͪ ١2 n sive description of combined Renner-Teller and SO vibronic Hˆ ͩ 1 es = ͚ − 2 m − ͚ + , 3 coupling effects in quasilinear molecules with a half-filled ␲ m=1 n=1 rmn r12 orbital. We illustrate the effects of particular coupling terms by variational energy-level calculations for selected models. and the Breit-Pauli SO operator for two electrons is The carbenes as well as heavier linear triatomic species will 2 be the systems of choice for the application of these meth- Hˆ Hˆ ͑m͒ Hˆ ͑12͒ ͑ ͒ SO = ͚ SO + SO , 4 ods. The results of the present analysis will facilitate a better m=1 understanding of the complicated perturbation mechanisms involved in these species as well as an unambiguous inter- where pretation of the experimentally observed irregular patterns of 3 Q ͒ ͑ ͒͒ ١ ͑ their spin-vibronic energy levels which still defy a compre- Hˆ ͑m͒ 1 ␤2 ˆ n ͑ ϫ SO = 2 g eSm͚ 3 rmn − i m 5 hensive spectroscopic analysis. n=1 rmn The remainder of this paper is arranged as follows. The derivation of the Hamiltonian and the analysis of its symme- represents its one-electron part ͑m=1,2͒ and try properties are reviewed. The effect of the Renner-Teller 2 ͑ ͒ ig␤ ͔͒ −2١ ١͑ and other electrostatic vibronic-coupling parameters on the Hˆ 12 = e ͓Sˆ ͓r ϫ 3 − 1 1 + SO 3 1 12 1 2 ⌺ , ⌬, and ⌺ potential energy surfaces for a model mo- 2r12 lecular system is analyzed. The SO-induced vibronic cou- 6͒͑ ͔͔͒ −2١ ١͑ Sˆ ͓r ϫ + pling constants of the Hamiltonian are explicitly related to 2 21 2 1 topographic features of these potential energy surfaces. Their is the corresponding two-electron contribution.30 Here, g impact on the 3⌺−, 1⌬, and 1⌺+ spin-vibronic energy levels is Ӎ ␤ 2.0023 refers to the g factor of the free electron and e discussed. =1/2c designates the Bohr magneton. In Eq. ͑2͒, ␻ refers to the harmonic vibrational fre- II. THEORY quency of the bending vibrational mode. The normal coordi- A. Spin-vibronic Hamiltonian in the diabatic nates of the bending vibrational mode QϮ are given by the representation relationship Consider the two electrons of a half-filled ␲ shell mov- Ϯ ␳ Ϯi␾ ͑ ͒ QϮ = Qx iQy = e , 7 ing in the field of three linearly arranged nuclei along the z ͑ ͒ axis with effective nuclear charges Q1, Q2, and Q3. The elec- where Qx ,Qy are the dimensionless bending normal coor- ͑ ͒ ͑ ͒ tronic states that arise from this electronic configuration are dinates. In Eqs. 5 and 6 , r1 and r2 refer to the radius 3 − 1 1 + ͑ ͒ ⌺ , ⌬, and ⌺ . In this work, their mutual interaction and vectors of electrons 1 and 2, respectively, Rn n=1,2,3 cor- ͑ coupling due to the degenerate bending vibrational mode of respond to the nuclear radius vectors, rmn=rm −Rn rmn ␲ ͉ ͉͒ ͑ ͉ ͉͒ symmetry are examined. To simplify things, the effect of = rmn , r12=r1 −r2 =−r21 r12= r12 , and

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␴͑m͒ ␴͑m͒ ␴͑m͒ ͑ ͒ The two-electron diabatic electronic basis functions ␺⌳, Sm = i ˆ x + j ˆ y + k ˆ z . 8 ⍀ which are orthonormal, antisymmetric with respect to elec- ͑m͒ ͑m͒ ͑m͒ In Eq. ͑8͒, ␴ˆ , ␴ˆ , and ␴ˆ denote the Pauli spin matrices ˆ 2 ˆ ˆ ˆ x y z tron exchange, and eigenfunctions of S , Sz, Lz, and Jz, are and i, j, and k symbolize the unit vectors of the three- given by dimensional space. 1 ͑␪ ␪ ͒ The symmetry properties of the electronic Hamiltonian ␺+2 ͑ ͒ ͑ ͒ i 1+ 2 ͑␣ ␤ ␣ ␤ ͒ ͑ ͒ = P 1 P 2 e 1 2 − 2 1 , 14 Hˆ +2 ͱ2 el are represented by the commutators

͓Hˆ Jˆ ͔ ͑ ͒ 1 ͑␪ ␪ ͒ ͑␪ ␪ ͒ el, z =0 9 ␺0 ͑ ͒ ͑ ͓͒ i 1− 2 −i 1− 2 ͔͑␣ ␣ ͒ ͑ ͒ = P 1 P 2 e − e 1 2 , 15 +1 ͱ2 and 1 ͑␪ ␪ ͒ ͑␪ ␪ ͒ ␺0 P͑ ͒P͑ ͓͒ei 1− 2 e−i 1− 2 ͔͑␣ ␤ ␣ ␤ ͒ ͑ ͒ ͓Hˆ Tˆ͔ ͑ ͒ 0 = 2 1 2 − 1 2 + 2 1 , 16 el, =0, 10 1 ͑␪ ␪ ͒ ͑␪ ␪ ͒ ␾0 ͑ ͒ ͑ ͓͒ i 1− 2 −i 1− 2 ͔͑␣ ␤ ␣ ␤ ͒ ͑ ͒ where 0 = 2 P 1 P 2 e + e 1 2 − 2 1 , 17 ץ ץ 1 ͑1͒ 1 ͑2͒ 1 ͑␪ ␪ ͒ ͑␪ ␪ ͒ Jˆ = Lˆ + Sˆ =−i + ␴ˆ − i + ␴ˆ ͑11͒ ␺0 ͑ ͒ ͑ ͓͒ i 1− 2 −i 1− 2 ͔͑␤ ␤ ͒ ͑ ͒ ␪ 2 z −1 = ͱ P 1 P 2 e − e 1 2 , 18ץ ␪ 2 zץ z z z 1 2 2 is the z component of the electronic ͑orbital+electron spin͒ 31 1 ͑␪ ␪ ͒ ␺−2 ͑ ͒ ͑ ͒ −i 1+ 2 ͑␣ ␤ ␣ ␤ ͒ ͑ ͒ angular momentum operator and = P 1 P 2 e 1 2 − 2 1 , 19 −2 ͱ2 ͑1͒ ͑2͒ Tˆ = ␴ˆ ␴ˆ c.c. ͑12͒ ͑ ͒ ͑ ͒ y y where P m = P rm ,zm , m denotes the label of the electron, and ͑␣,␤͒ correspond to the two possible electron spin stands for the time-reversal symmetry operator.31,32 Here, ␪ ͑ ͒ eigenfunctions. The wave functions defined in Eqs. ͑15͒, represents the electronic azimuthal angular coordinate, ␴ˆ m z ͑16͒, and ͑18͒ are antisymmetric with respect to reflection corresponds to the z projection of the spin angular momen- through the molecular plane of the Dϱ or Cϱ␷ point group, tum of the mth electron, and c.c. denotes the operation of h while the one defined in Eq. ͑17͒ is symmetric and is distin- Tˆ 31,32 complex conjugation. is an antiunitary operator and guished by the label ␾. The above set of diabatic electronic obeys the relation Tˆ 2 =1 for molecular systems with an even basis functions is defined in the absence of interelectronic number of electrons ͑as is the case here͒. interactions. ͑ ͒ The full molecular Hamiltonian defined in Eq. 1 also The time-reversal operator Tˆ has the following effects possesses time-reversal symmetry and satisfies the axial on the six diabatic electronic basis functions: symmetry property, i.e., it commutes with the z component ͑ ͒ Tˆ͑␺Ϯ2͒ ␺ϯ2 ͑ ͒ of the total electronic+nuclear angular momentum opera- Ϯ2 =− ϯ2c.c. 20 tor, which is defined by Tˆ͑␺0 ͒ ␺0 ͑ ͒ Ϯ1 = ϯ1c.c. 21 ץ Jˆ Ј = Jˆ − i . ͑13͒ ␾ץ z z Tˆ͑␺0͒ ␺0 ͑ ͒ 0 =− 0c.c. 22 Jˆ Ј The eigenvalues of z are denoted by the spin-vibronic angular momentum quantum numbers ␮ and are integers Tˆ͑␾0͒ ␾0 ͑ ͒ 0 =− 0c.c.. 23 ͑␮=0,Ϯ1,Ϯ2,...͒. In the absence of SO coupling interactions, the electronic orbital ͑⌳͒ and electron spin ͑⌺͒ projec- Hence, the time-reversal symmetry operator Tˆ in the diabatic tion quantum numbers are good quantum numbers. In the representation is given by presence of SO coupling, ⍀=⌳+⌺ is a good quantum num- ␺+2 000 00−1 ␺+2 ber in the fixed-nuclei approximation. When the nuclear mo- +2 +2 ␺0 ␺0 tion is included, only ␮ remains a good quantum number. +1 000 010 +1 ␲ ͑ ͒ Ϯi␪ ␺0 ␺0 We introduce -type molecular orbitals P r,z e 0 00−1000 0 ͑ ͒ Tˆ = c.c.. which are solutions of the one-electron Hartree-Fock ␾0 000−100 ␾0 Schrödinger equation at the linear configuration, where r, z, 0 0 0 ΂ ΃ 0 ΂␺ ΃ ΂␺ ΃ 010 000 and ␪ represent cylindrical electronic coordinates. We define −1 −1 33–35 ␺−2 −100000 ␺−2 a two-electron diabatic spin-orbital electronic basis set −2 −2 ͑ ͒ diabatic spin orbitals for a linear triatomic molecular ͑24͒ system.1–3 The basis set consists of the two electronic components of the 1⌬ electronic state ͑with electronic orbital Following the methodology of Refs. 1, 3, and 26,a6 angular momentum quantum numbers ⌳= Ϯ2͒ and the elec- ϫ6 spin-vibronic Hamiltonian matrix can be derived in the tronic states corresponding to the 3⌺− and 1⌺+ spatially non- six-dimensional spin-orbital electronic Hilbert space of the degenerate electronic states ͑⌳=0͒. Their spin angular mo- diabatic electronic basis functions defined in Eqs. ͑14͒–͑19͒. ⌺ Ϯ Hˆ mentum quantum numbers are =0, 1. The matrix elements of el are expanded in a Taylor series in

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TABLE I. The electrostatic ͑a and c͒ and SO-induced ͑g, d, dЈ, and b͒ vibronic coupling coefﬁcients of the bending vibrational mode that parametrize the model electronic Hamiltonian, deﬁned in the diabatic electronic basis set, and their relationships to the terms of the Taylor series expansion of the potential.

Symbol Description Deﬁnition

4ץ 1 a Quartic Renner-Teller coupling coefficient ͗␺Ϯ2͉Hˆ ͉␺ϯ2͘ ͒ ␳4 Ϯ2 el ϯ2ץ͑ 4! 0 2 ץ 1 1 + 1 c Quadratic ⌺ − ⌬ coupling coefficient ͗␺Ϯ2͉Hˆ ͉␾0͘ ͒ ␳2 Ϯ2 el 0ץ͑ 2! 0 3⌺− 1⌺+ ͑͗␺0͉Hˆ ͉␾0͒͘ g Zeroth-order 0 − 0 coupling coefficient 0 SO 0 0

Ϯ ץ 1 − 3 d Linear ⌺Ϯ − ⌬Ϯ coupling coefﬁcient ͗␺ 2͉Hˆ ͉␺0 ͘ ͒ ␳ Ϯ2 SO Ϯ1ץ͑ 2 1 0

ץ + 1 − 3 dЈ Linear ⌺Ϯ − ⌺ coupling coefﬁcient ͗␺0 ͉Hˆ ͉␾0͘ ͒ ␳ Ϯ1 SO 0ץ͑ 0 1 0 3ץ 3⌺− 1⌬ 1 Ϯ2 0 b Cubic ϯ1 − Ϯ2 coupling coefﬁcient ͗␺ ͉Hˆ ͉␺ ͘ ͒ ␳3 Ϯ2 SO ϯ1ץ͑ 3! 0

the bending amplitude ␳, keeping expansion terms up to calculation of the individual coefﬁcients, performing the in- ␳ ␪ ␪ fourth order in . Angular momentum conservation and time- tegration over the angular variables 1 and 2 as well as over reversal symmetry determine the nonvanishing matrix ele- the discrete spin variables ͑see, e.g., Ref. 3͒. More details are ments. A further simpliﬁcation is achieved by the explicit given in Appendix A. The result of this derivation is

␾ ␾ ␾ ␾ E1⌬ d␳ei c␳2e2i b␳3e3i a␳4e4i +2 0 ␾ ␾ ␾ d␳e−i E3⌺− dЈ␳ei b␳3e3i +1 0 0− 1 1 0 0 E3⌺− g 0 0 Hˆ Tˆ Hˆ ͩTˆ ␻␳2 ␭␳4ͪ 0 ͑ ͒ = N + el = N + + 1 + ␾ ␾ ␾ ␾ , 25 2 4! c␳2e−2i dЈ␳e−i g E1⌺+ dЈ␳ei c␳2e2i 0 − ␾ ␾ ␾ ΄ 3 −3i −i 3 − i ΅ b␳ e dЈ␳e E ⌺ d␳e 0 0 − −1 − ␾ ␾ ␾ ␾ a␳4e−4i b␳3e−3i c␳2e−2i d␳e−i E1⌬ − 0 − −2

where 1 represents the 6ϫ6 unit matrix and ␭ denotes the interstate vibronic coupling parameters are summarized in quartic anharmonic-oscillator parameter. Table I. Note that most of the SO-induced coupling param- ͑E1⌬ ;E1⌬ ͒, ͑E3⌺− ;E3⌺− ;E3⌺− ͒, and E1⌺+ correspond eters would be absent if a phenomenological expression of +2 −2 +1 0 −1 0 to the energies of the 1⌬, 3⌺−, and 1⌺+ SO-coupled electronic the SO coupling operator36,37 were used instead. Equation states at the linear geometry ͑␳=0͒, respectively. The energy ͑25͒ reveals, in particular, that the Breit-Pauli SO operator is separations between the 3⌺− and 1⌬ potential energy surfaces not diagonal in the diabatic electronic basis. as well as between the 1⌬ and 1⌺+ states at ␳=0 are desig- Among all vibronic coupling matrix elements, the pa- E E nated as 1 and 2, respectively, in the following. While the rameter g is the only one which has an effect on the potential vibronic parameters a ͑quartic Renner-Teller coupling be- energy surfaces at the linear geometry. Specifically, it tween the two electronic components of 1⌬͒ and c ͑quadratic couples the 3⌺− state with the 1⌺+ state. As a result of this 1⌺+ − 1⌬ coupling͒ are of nonrelativistic ͑electrostatic͒ char- coupling, the spin degeneracy of the 3⌺− state is lifted ͑see ͑ 3⌺− 1⌺+ ͒ ͒ acter, the parameters g zeroth-order 0 − 0 coupling , Sec. III B for details . It has previously been shown that ͑ 3⌺− 1⌬ ͒ Ј ͑ 3⌺− 1⌺+ ⌺ d linear Ϯ1 − Ϯ2 coupling , d linear Ϯ1 − 0 cou- nonzero SO matrix elements exist between states with dif- ͒ ͑ 3⌺− 1⌬ ͒ ͑⌬ ͒ pling , and b cubic Ϯ1 − ϯ2 coupling are of SO origin. ferent multiplicities S=1 when these behave differently 30 The effects of the zeroth-, first-, second-, third-, and fourth- under the ␴␷ operation. The SO coupling does not lift the order vibronic coupling parameters on the topography of the degeneracy of the 1⌬ state at ␳=0 in conformity with group potential energy surfaces will be discussed below. The defi- theoretical considerations. The SO-coupled electronic Hamil- nitions of the SO-induced, Renner-Teller, and 1⌺+ − 1⌬ tonian matrix of Eq. ͑25͒ can be diagonalized for ␳=0 by a

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constant ͑␳- and ␾-independent͒ unitary transformation. The III. RESULTS AND DISCUSSION vibronic matrix presumably cannot be transformed to a The concept of the electronic adiabatic potential energy ␳0. block-diagonal form for surface ͑PES͒ is well-defined when the Born-Oppenheimer approximation is valid, particularly if SO coupling is not relevant for the problem. The Renner-Teller effect is an ex- B. Vibronic basis set and vibronic eigenfunctions emplary case of a breakdown of the Born-Oppenheimer approximation. However, it is still useful to use the concept of The basis set used for the variational calculation of vi- the adiabatic PESs for Renner-Teller active molecules in the bronic energy levels corresponds to direct products of elec- presence of SO coupling. In the following, the topographies tronic and vibrational wave functions ͑spin-vibronic basis of two types of adiabatic PESs will be described. The first set͒ and is represented by Hˆ ͑ ͒ type refers to the eigenvalues of es Sec. III A , while the ͉⌳ ⌺ ͘ ͉⌳ ⌺͉͘ ͘ ͑ ͒ Hˆ Hˆ ͑ , ,n,l = , n,l , 26 second corresponds to the eigenvalues of es+ SO Secs. III B and III C͒. The PESs are obtained as parametric func- ⌳ Ϯ ⌺ Ϯ with electronic quantum numbers =0, 2, =0, 1, and tions of the nuclear positions by numerical diagonalization of vibrational quantum numbers n=0,1,2,..., l=n, n−2,..., the 6ϫ6 Hamiltonian matrix ͑we did not succeed in obtain- ͉⌳ ⌺͘ −n+2, −n. The electronic wave functions , refer to the ing analytical expressions for the eigenvalues even with the spin-orbital diabatic electronic basis set defined in Eqs. aid of symbolic-mathematical tools͒. ͑ ͒ ͑ ͒ ͉ ͘ 14 – 19 , while the n,l are the eigenfunctions of the two- We also investigate the effects of SO-induced vibronic 38 dimensional isotropic harmonic oscillator. coupling interactions ͑Sec. III B͒ as well as the combined ͓Hˆ Jˆ Ј͔ 1 + 1 Since , z =0, the vibronic eigenfunctions can be effects of Renner-Teller, ⌺ − ⌬, and SO-induced coupling classified by the spin-vibronic quantum number ␮, ͑Sec. III C͒ on the spin-vibronic energy levels of the 3⌺−, 1⌬, and 1⌺+ electronic states for representative parameter values. Jˆ Ј␺ ␮␺ ͑ ͒ 1⌺+ 1⌬ z ␮ = ␮. 27 Since the Renner-Teller and − electrostatic couplings have a negligible impact on the energy-level diagrams for The spin-vibronic basis set of Eq. ͑26͒ is constructed by first reasonable parameter values, they will not be considered any specifying the electronic angular momentum quantum num- further in the present study. bers ⍀ and then selecting suitable values of l of the ͉n,l͘ vibrational wave functions for a given value of ␮. A. Hamiltonian without SO coupling The spin-vibronic eigenfunctions with total angular mo- The effects of the quartic 1⌬ Renner-Teller and quadratic mentum quantum number ␮, ␺␮, satisfy the Schrödinger 1⌺+ − 1⌬ electrostatic vibronic coupling mechanisms on the equation adiabatic PESs of the 3⌺−, 1⌬, and 1⌺+ electronic states of a linear triatomic molecule are illustrated in Fig. 1. The figure Hˆ ␺␮ = E␺␮ ͑28͒ displays cross sections of the PESs ͑for any azimuthal angle ␾͒ along the dimensionless bending normal coordinate ␳ for for Hˆ of Eq. ͑1͒. They are expressed as linear combinations a system with the parameters E =E =5␻ and ␭/␻=0.2 in of the basis functions ͑26͒ for a given quantum number ␮.A 1 2 the limit of vanishing SO coupling interactions. The param- second quantum number n␮ is used to label the eigenvalues eters were expressly chosen to highlight the effect of the ͑ordered by increasing energy͒ for a given ␮. We, thus, de- involved coupling mechanisms on the adiabatic PESs; that is, note eigenkets as ͉␮,n␮͘. comparatively large values of the coupling parameters were Based on the above, the spin-vibronic eigenfunctions chosen. Note that the ordinate scale of these diagrams is in ͉␮,n␮͘ may be written as 3 − units of ␻ and that the energy of the ⌺ state, i.e., E3⌺−,is0 ␳ ͉␮ n ͘ ͚ Cn␮͉⌳͉͘⌺͉͘n l͘ ͑ ͒ for =0. , ␮ = k , , 29 ͑ ␳ ͒ k The lowest curve near =0 represents a slice across the 3⌺− PES. The next-lowest function gives the 1⌬ PES, while n␮ 1⌺+ where the index k of the expansion coefficients Ck repre- the uppermost curve depicts the PES. The energy order- sents all symmetry-allowed combinations of the basis-set ing of these potential energy curves is based on Hund’s rule. quantum numbers ⌳, ⌺, n, and l. To simplify the notation, The 3⌺− potential is threefold spin degenerate. The 1⌺+ PES we shall hereafter drop the Ϯ signs of the ␮ quantum num- is nondegenerate, while the 1⌬ PES retains its twofold elec- bers. tronic degeneracy only in the limit of vanishing Renner- Equation ͑26͒ defines a complete basis set. The basis set Teller coupling and 1⌺+ − 1⌬ quadratic vibronic coupling. must be truncated to some manageable size for the numerical The Renner-Teller theory predicts39,40 that a 1⌬ electronic diagonalization of the Hamiltonian matrix. The calculated state of a linear molecule will interact with the ␲ vibration in eigenvalues are tested for convergence by systematically in- fourth order, leading to two PESs which touch at the linear creasing the number of vibrational basis functions until ad- configuration. ditional basis functions have a negligible effect on the eigen- The PESs of Fig. 1͑a͒ were constructed considering only values. The final matrices used were of the order of 103. The the quartic Renner-Teller vibronic coupling effect, assuming nonvanishing matrix elements of the spin-vibronic Hamil- a/␻=0.01. Upon displacement from the linear geometry ͑␳ tonian are given in Appendix B. =0͒ along the bending coordinate, the 3⌺−, 1⌬, and 1⌺+ elec-

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3⌺− and 1⌺+ functions at ␳Ϸ Ϯ4.5 in Fig. 1͑a͒. The Renner- Teller splitting of the electronically degenerate 1⌬ PESs, being a fourth-order effect, is expected to take place at larger displacements from the linear geometry than for ⌸-type Renner-Teller-split PESs. The slice across the lowest sheet of the 1⌬ PESs exhibits a double-well topography if the parameter a/␻ exceeds a certain critical value. Figure 1͑b͒ exhibits the ramiﬁcations of the 1⌺+ − 1⌬ quadratic electrostatic interaction on the PESs for the example case of c/␻=0.1. Only the shapes of the 1⌺+ and 1⌬ PESs are expected to change, as this interaction does not involve vibronic coupling with the 3⌺− electronic state. The 1⌺+ − 1⌬ interaction removes the degeneracy of the 1⌬ components.

B. Hamiltonian with SO coupling, excluding electrostatic vibronic coupling 1. Potential energy surfaces Figure 2 is constructed based on the same general principles which have been applied for Fig. 1, but now the pure SO-induced vibronic coupling effects are considered. Spe- cifically, the diagrams ͑a͒–͑d͒ of Fig. 2 examine the effects of the zeroth-order ͑g/␻=6͒, first-order ͑d/␻=2͒, ͑dЈ/␻=2͒, and third-order ͑b/␻=0.1͒ SO-induced vibronic coupling interactions on the 3⌺−, 1⌬, and 1⌺+ PESs, respectively ͑the parameter values are given in the caption͒. The parameter values were chosen with the express purpose of illustrating and enhancing the SO-induced vibronic coupling effects. Let us first examine the implications of the zeroth-order 3⌺− 1⌺+ 0 − 0 SO vibronic coupling parameter g for these PESs. If we assume a linear geometry ͑␳=0͒, then the 6ϫ6 electronic Hamiltonian matrix of Eq. ͑25͒ decouples into a diagonal matrix and a 2ϫ2 submatrix which is responsible for 3⌺− 1⌺+ the mixing of the 0 and 0 electronic states. The elec- 1⌬ 3⌺− tronic states Ϯ2 and Ϯ1 retain their twofold degeneracies and their potentials remain unaffected. The linear ͑first order in ␳͒ SO-induced vibronic coupling parameters d and dЈ are expected to have, apart from the zeroth-order parameter g, the largest impact on the topography of the 3⌺−, 1⌬, and 1⌺+ PESs. We first investigate the effect of the parameter d which refers to the coupling 3⌺− 1⌬ FIG. 1. Potential energy as a function of the bending coordinate for selected between Ϯ1 and Ϯ2. If we assume that all parameters cases of Renner-Teller and 1⌺+ − 1⌬ electrostatic vibronic couplings for ⌺ except d are zero, then the 6ϫ6 electronic Hamiltonian ma- and ⌬ electronic states of quasilinear triatomic molecules with a half-filled trix of Eq. ͑25͒ reduces to two 2ϫ2 submatrices which mix ␲ shell. The ordinate scale is in units of ␻, while the abscissa represents the ͑ ␳ ͒ 1⌬ 3⌺− 3⌺− ␳ for 0 the Ϯ2 and Ϯ1 electronic states. The 0 and dimensionless bending normal coordinate . The curves shown here have 1⌺+ been constructed for E =E =5␻ and ␭/␻=0.2. The energy E3⌺− is defined 0 electronic states remain unaffected. While the twofold 1 2 3⌺− 1⌬ as 0 for ␳=0. ͑a͒ Renner-Teller vibronic coupling of the two electronic degeneracies of the Ϯ1 and Ϯ2 PESs are retained, their 1⌬ ͑ /␻ ␭/␻ /␻ ͒ ͑ ͒ 3 − components of a =0.01; =0.2; c =0 . b Quadratic vibronic curvatures change. The ⌺Ϯ PES develops a double- 1 + 1 1 coupling of ⌺ and ⌬ ͑c/␻=0.1; ␭/␻=0.2; a/␻=0͒. minimum shape along the bending normal coordinate, while 1⌬ ␳ the Ϯ2 function becomes steeper as a function of upon 3 ͑1 1 ͒ 1 /␻ ͓ ͑ ͔͒ Ј 3⌺− tronic states reduce to B1, A1 , B1 and A1 in C2␷ geom- increase of d see Fig. 2 b .Asd involves the Ϯ1 and ͑ 1⌺+ etry, respectively. Note that an in-plane y axis is assumed in 0 electronic states, the vibronic coupling interaction is ͒ 3⌺− 1⌺+ this work. As the Renner-Teller effect becomes stronger, the now manifested between the Ϯ1 and PESs. If we as- interaction between the two 1⌬ electronic components in- sume that all parameters except dЈ are zero, then the 6ϫ6 creases; the splitting between them becomes more pro- electronic Hamiltonian matrix of Eq. ͑25͒ reduces to a 4 ϫ ͑ ␳ ͒ 3⌺− 1⌺+ nounced and starts at smaller displacements from the refer- 4 submatrix which mixes for 0 the Ϯ1 and 0 ͑ ͒ 1⌺+ 3⌺− ence geometry not shown . Note the crossing between the electronic states. The well is compressed and the Ϯ1 1⌬ Renner-Teller-split potential energy functions with the PES exhibits a flattening, while retaining its double degen-

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eracy upon displacement from the linear geometry ͓see Fig. ͑ ͔͒ 1⌬ 3⌺− 2 c . The potential remains unaltered. The Ϯ1 PES exhibits similar topographical characteristics in Figs. 2͑c͒ and 2͑b͒ since the parameter values of dЈ/␻ and d/␻ are the same in both cases. Of fundamental importance for kinetics and spectroscopic studies is the accurate determination of the barrier to linearity of the lowest PES which is defined as the energy difference between the bent equilibrium geometry and the “transition-state” linear geometry. While the effect of the cubic SO-induced vibronic coupling parameter b on the 3⌺− and 1⌬ PESs is less important than that of the zeroth-order and first-order contributions, it introduces interesting changes to the topographies of these PESs. This parameter represents third-order vibronic cou- 3⌺− 1⌬ pling between Ϯ1 and ϯ2. When all coupling parameters except b are zero, the 6ϫ6 electronic Hamiltonian matrix of Eq. ͑25͒ reduces to a 4ϫ4 submatrix which mixes ͑for ␳ ͒ 1⌬ 3⌺− 0 the Ϯ2 and ϯ1 diabatic electronic states. The quali- tative characteristics of this interaction are depicted in Fig. 2͑d͒, where one can notice that one component of the 3⌺− 1⌬ PES becomes flatter upon bending, while the Ϯ2 PES be- 3⌺− 1⌺+ comes steeper. The potentials of the 0 and 0 states remain unperturbed.

2. Spin-vibronic energy levels Central to this work is the impact that the linear SO- induced vibronic coupling parameters d and dЈ have on the 3⌺−, 1⌬, and 1⌺+ spin-vibronic energy levels. Figures 3͑A͒ and 3͑B͒ show the energy-level diagrams of the spin- vibronic energy levels with calculated energies lower than 5␻. The molecular parameters d/␻ in ͑A͒ and dЈ/␻ in ͑B͒ vary between 0 and 0.5. The energy levels were computed E ␻ E ␻ ␭/␻ assuming 1 =1.5 , 2 =1.8 , and =0. The ordinate scales of the diagrams of ͑A͒ and ͑B͒ are in units of ␻, while their abscissae refer to the dimensionless parameters d/␻ and dЈ/␻, respectively. The solid, long-dashed, dash-dotted, dotted and medium-dashed lines correspond to energy levels with ␮=0, 1, 2, 3, and 4, respectively. The symmetry labels of the states were determined in the same way as in Ref. 36. Note the correlation of the spin-vibronic energy levels with the harmonic-oscillator vibronic energy levels, as well as the electronic states from which the latter levels arise in panels ͑A͒ and ͑B͒ of Fig. 3. As the linear SO-induced vibronic interaction parameter d involves coupling between the electronic states 3⌺− and 1⌬, we expect energy-level perturbations to take place among their energy levels satisfying ⌬␷=1. The degeneracy of the lowest two ␮=0, 1 ͑3⌺− ,␷=0͒ spin-vibronic energy levels in ͑A͒ is lifted since the latter one interacts with the FIG. 2. Potential energy as a function of the bending coordinate for selected ␮ ͑1⌬ ␷ ͒ ⌺ ⌬ =1 , =1 energy level. The former energy level re- cases of SO-induced vibronic coupling in and electronic states of qua- 1 silinear triatomic molecules with a half-ﬁlled ␲ shell. The ordinate scale is mains unperturbed as there is no ͑ ⌬,␷=1͒ energy level with in units of ␻, while the abscissa represents the dimensionless bending nor- ␮=0 to interact with. Similarly, the ␮=2 ͑1⌬,␷=0͒ energy ␳ E mal coordinate . The curves shown here have been constructed for 1 ␮ ͑3⌺− ␷ ͒ ͓ E ␻ ␭/␻ ␳ level interacts with the =2 , =1 energy level. Note = 2 =5 and =0.2. The energy E3⌺− is deﬁned as 0 for =0. The dia- 1⌬ ␮ 3⌺− ͑ ͒ ͑ ͒ ͑ ͒ that this level can cross with the =1,3 levels as grams a – d represent “large” vibronic interactions: a zeroth-order vi- 1 3⌺− 1⌺+ ͑ /␻ ␭/␻ /␻ Ј/␻ shown in Fig. 3͑A͔͒. The ͑ ⌬,␷=1͒ energy levels with ␮ bronic coupling of 0 and 0 g =6; =0.2; d =0; d =0; /␻ ͒ ͑ ͒ 3⌺− 1⌬ ͑ /␻ ␭/␻ 3 − b =0 , b linear vibronic coupling of Ϯ1 and Ϯ2 d =2; =3,1 interact with the same-␮ quantum-number ͑ ⌺ ,␷=2͒ g/␻ dЈ/␻ b/␻ ͒ ͑ ͒ 3⌺− 1 =0.2; =0; =0; =0 , c linear vibronic coupling of Ϯ1 and levels. Finally, the ␮=4, 2, and 0 ͑ ⌬,␷=2͒ energy levels 1⌺+ ͑ Ј/␻ ␭/␻ /␻ /␻ /␻ ͒ ͑ ͒ 0 d =2; =0.2; g =0; d =0; b =0 , and d cubic vibronic 3 − 3⌺− 1⌬ ͑ /␻ ␭/␻ /␻ /␻ interact with the ␮=4, 2, and 0 ͑ ⌺ , ␷=3, 3, and 1, 3͒ levels, coupling of Ϯ1 and ϯ2 b =0.1; =0.2; g =0; d =0; dЈ/␻=0͒. respectively. The same principles apply in Fig. 3͑B͒, where

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͑ ͒ ␻ 3⌺− 1⌬ 1⌺+ ͑ ͒ 3⌺− FIG. 3. Color online Calculated spin-vibronic energy levels with energies lower than 5 of the , , and electronic states for A linear Ϯ1 1⌬ ͑ ͒ ͑ ͒ 3⌺− 1⌺+ ͑ Ј͒ /␻ Ј/␻ − Ϯ2 SO vibronic coupling d and B linear Ϯ1 − 0 SO vibronic coupling d , as a function of d and d , respectively. The ordinate scale is in units of ␻, while the abscissae in ͑A͒ and ͑B͒ refer to the dimensionless quantities d/␻ and dЈ/␻, respectively. The solid, long-dashed, dash-dotted, dotted, ␮ E ␻ E ␻ and medium-dashed lines represent energy levels with =0, 1, 2, 3, and 4, respectively. The energy levels were calculated for 1 =1.5 , 2 =1.8 ,and ␭/␻=0. the SO-induced vibronic coupling takes place between the energy level interacts with the ␮=2 ͑3⌺− ,␷=1,3͒ energy 3⌺− and 1⌺+ electronic states. For example, the lowest ␮ levels, and the variation of their energies reflects the coun- =1 ͑3⌺− ,␷=0͒ energy level interacts with the ␮=1 ͑1⌺+ ,␷ terbalance of these interactions. =1͒ energy level, while the ␮=0 ͑3⌺− ,␷=0͒ energy level The interaction pattern of the spin-vibronic energy levels remains unaffected, as there is no ␮=0 ͑1⌺+ ,␷=1͒ energy is relatively trivial when the g parameter is varied. An im- level to interact with. What is new in Fig. 3͑B͒ is that only portant observation is that the energy-level perturbations be- one of the two degenerate ␮=0 3⌺− ͑shown for ␷=1 and 3͒ come relevant already for small values of g/␻. Since g is of energy-level components may interact with suitable 1⌺+ en- zeroth order in ␳ ͑that is, purely electronic͒, it can become ergy levels due to the dЈ spin-vibronic coupling. significantly larger than the bending vibrational frequency ␻ Figure 4 illustrates the effects of third-order ͑parameter in molecules with heavy atoms. b͒ SO vibronic coupling on the 3⌺−, 1⌬, and 1⌺+ spin- vibronic energy levels. This figure was constructed based on C. Hamiltonian with SO coupling, Renner-Teller ͑ ͒ ͑ ͒ the same principles as applied for Figs. 3 A and 3 B . All coupling, and 1⌺+ − 1⌬ coupling spin-vibronic energy levels with calculated energies lower than 5␻ are retained in the diagram. The parameter b/␻ 1. Potential energy surfaces varies in the range of ͑0,0.10͒. Figure 4 displays interesting Figure 5 illustrates the combined effects of Renner- higher-order spin-vibronic coupling effects. The increase of Teller, 1⌺+ − 1⌬ electrostatic, and SO-induced vibronic cou- the parameter b/␻ causes degenerate groups of energy levels pling interactions on the 3⌺−, 1⌬, and 1⌺+ PESs of a tri- to fully resolve into their individual components. What is atomic linear molecule for selected parameter values. The unique here is the early onset of extensive interstate energy- cross sections along the bending normal coordinate cover the level interactions. For example, the lowest ␮=2 ͑1⌬,␷=0͒ most important regions of the 3⌺−, 1⌬, and 1⌺+ PESs, such

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fects of the d and dЈ linear couplings translate into a con- 3⌺− 1⌬ 1⌺+ comitant interaction between the Ϯ1 and the Ϯ2 and 0 PESs. 3⌺− 1⌬ The interaction between the Ϯ1 and Ϯ2 PESs is further enhanced by the inclusion of the cubic coupling term ͑b͒, as shown in Fig. 5͑c͒. There appear two different sets of bent-geometry stationary points, those that correlate with the 3⌺− ␳Ϸ Ϯ lowest of the two Ϯ1 PESs, which are located at 6, 3⌺− and the stationary points of the 0 PES, which are posi- tioned at ␳Ϸ Ϯ2. Any combination of the linear and cubic vibronic coupling interactions, or the combined effects of the linear coupling interactions alone, leads to SO splitting of the degen- 3⌺− 1⌬ erate components of the Ϯ1 and Ϯ2 PESs. Consider the exemplary case of combined d and dЈ interactions. The 1⌬ 3⌺− former concerns the Ϯ2 − Ϯ1 coupling, whereas the latter 1⌺+ 3⌺− Ј refers to the 0 − Ϯ1 coupling. The d and d effects are 3⌺− “competing” for the Ϯ1 energies. Therefore, the d vibronic coupling quenches the dЈ vibronic coupling and vice versa. The joint action of Renner-Teller, 1⌺+ − 1⌬ electrostatic, and SO-induced vibronic coupling effects on the 3⌺−, 1⌬, and 1⌺+ PESs is illustrated in Fig. 5͑d͒ which combines the effects of Fig. 5͑c͒ plus the Renner-Teller and 1⌺+ − 1⌬ electrostatic vibronic coupling ͑a/␻=0.002; c/␻=0.04; g/␻=4; dЈ/␻=2; d/␻=2; b/␻=0.1͒. Note the small, but non- negligible, repositioning of the local energy minima of the 3⌺− ͑ ͒ ͑ ␳Ϸ Ϯ ͒ lowest Ϯ1 PES in Fig. 5 d at 7 . Overall, one may conclude that the Renner-Teller and 1⌺+ − 1⌬ electrostatic effects play only a moderate role for the shape of the PESs, while the SO-induced vibronic coupling effects have a deter- mining impact on these. FIG. 4. ͑Color online͒ Calculated spin-vibronic energy levels with energies ␻ 3⌺− 1⌬ 1⌺+ 3⌺− lower than 5 of the , , and electronic states for cubic Ϯ1 1⌬ ͑ ͒ /␻ 2. Spin-vibronic energy levels − ϯ2 SO vibronic coupling b as a function of b . The ordinate scale is in units of ␻, while the abscissa refers to the dimensionless quantity b/␻. The solid, long-dashed, dash-dotted, dotted, and medium-dashed lines rep- The purpose of Fig. 6 is to illustrate, through the results resent energy levels with ␮=0, 1, 2, 3, and 4, respectively. The energy levels of numerical computations for a model system, how the E ␻ E ␻ ␭/␻ were calculated assuming 1 =1.5 , 2 =1.8 , and =0. combined effects of the SO-induced vibronic coupling terms of the vibronic Hamiltonian translate into energy-level interactions and patterns for the 3⌺− and 1⌬ electronic states. The as the stationary points of bent-geometry energy minima as E ␻ spin-vibronic energy levels were computed for 1 =1.5 , well as linear-geometry local minima or local maxima. E =1.6␻, and ␭/␻=0. Only those spin-vibronic energy lev- ͑ ͒ 2 Figure 5 a was computed considering the zeroth-order els that are associated with the ␯=0 and 1 harmonic- 3⌺− 1⌺+ ͑ /␻ ͒ 0 − 0 SO coupling effect g =4 plus the linear vi- oscillator energy levels are included in this figure. The spin- 3⌺− 1⌺+ ͑ /␻ Ј/␻ ͒ bronic coupling between Ϯ1 and 0 g =4; d =2 . vibronic energy levels have been organized in two distinct Figure 5͑b͒ incorporates the same interactions as Fig. 5͑a͒ groups, each one of which is linked to the 3⌺− and 1⌬ elec- 3⌺− 1⌬ plus the linear Ϯ1 − Ϯ2 vibronic coupling interaction tronic states. These two groups of energy levels are differen- ͑g/␻=4; dЈ/␻=2; d/␻=2͒. Figure 5͑c͒, finally, combines tiated by distinct indentations of their corresponding ␯=0 the topographic potential-energy-surface characteristics of and 1 harmonic-oscillator labels. The far left-hand column the 3⌺−, 1⌬, and 1⌺+ electronic states of Fig. 5͑b͒ with those presents the results of energy-level calculations for the unperturbed harmonic-oscillator case ͓column ͑a͔͒. The energy introduced by cubic SO vibronic coupling between 3⌺− and Ϯ1 levels contained in columns ͑b͒–͑d͒ were calculated in the 1⌬ ͑g/␻=4; dЈ/␻=2; d/␻=2; b/␻=0.1͒. ϯ2 presence of pure SO coupling, representing the incremental ͑ ͒ Once the linear SO coupling interaction parameter d is addition of the g ͑g/␻=2.0͒, dЈ and d ͑dЈ/␻=d/␻=0.6͒, and ͑ ͒ included in Fig. 5 b , a number of interesting effects become b ͑b/␻=0.04͒ SO coupling parameters, respectively. visible. What distinguishes Fig. 5͑b͒ from Fig. 5͑a͒ is that in The parameter g lifts the threefold degeneracy of the 3⌺− 3 − 3 − the former figure, the twofold degeneracies of the Ϯ1 and ͑ ⌺ , ␷=0͒ energy levels of ⌺ vibronic symmetry into a 1⌬ ͑3⌺−͒ Ϯ2 PESs are fully resolved, which would not have taken twofold degenerate level 1 and a nondegenerate energy Ј ͑3⌺− ͓͒ ͑ ͔͒ place if the d and d linear coupling effects were considered level 0+ see Fig. 6 b . The sixfold vibronic degeneracy independently, as was done in Sec. III B. The combined ef- of the ͑3⌺−, ␷=1͒ energy level of 3⌸ vibronic symmetry is

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͑3⌸ also lifted, producing a fourfold degenerate level 0Ϯ 3⌸ ͒ ͑3⌸ ͒ + 2 and a twofold degenerate level 1 . The vibronic degeneracy of the ͑1⌬, ␷=1͒ energy level of 1⌸+ 1⌽ vibronic symmetry is maintained in column ͑b͒. Note that the ͑3⌺−, ␷ ͒ 3⌺− 3⌸ =0,1 spin-vibronic energy levels 1 and 1 are tuned into accidental quasiresonance in column ͑b͒ of Fig. 6. The inclusion of the dЈ and d linear coupling effects in column ͑c͒ lifts the aforementioned degeneracies and quasidegen- eracy of column ͑b͓͒see column ͑c͔͒. The ͑1⌬, ␷=1͒ energy 1⌸ 1⌽ levels 1 and 3 are tuned into accidental quasidegen- eracy in column ͑d͒. The value of the cubic coupling parameter b which is included in the energy-level calculation of column ͑d͒ is too small to cause signiﬁcant changes of the energy-level pattern of column ͑c͒. The effects of SO vibronic coupling would be more pronounced if the 1⌬ PE ␳ function had a minimum at 0, as it happens in CH2, 1⌬ 3⌺− CHX, CX2, SiH2, GeH2, etc. Then the lowest and vibronic levels would be quasidegenerate.

IV. CONCLUSIONS The combined effects of Renner-Teller, 1⌺+ − 1⌬, as well as SO-induced vibronic coupling interactions on the PESs of the 3⌺−, 1⌬, and 1⌺+ states arising from a half-filled ␲ orbital of a linear triatomic molecule have been described, employing the microscopic ͑Breit-Pauli͒ expression of the SO operator. The microscopic theory involves six SO-coupled electronic wave functions with spin-orbital angular momentum projections ⍀= Ϯ2, Ϯ1, 0Ϯ. The spin-vibronic Hamiltonian matrix has been constructed in the diabatic representation, considering terms up to fourth order in the bending displacement. The symmetry properties of the SO-induced as well as electrostatic vibronic coupling effects have been described in detail. We have provided the first systematic and complete analysis of the SO vibronic coupling effects in the manifold of electronic states which arise from a half-filled ␲ orbital in a quasilinear molecule. The Taylor expansion of the vibronic matrix and the numerical calculation of the vibronic energy levels can straightforwardly be extended to higher orders if this should be necessary. The present analysis provides the foundation for a comprehensive analysis and computational prediction of the vibronic spectra of CH2 as well as of heavier linear triatomic halogenated carbenes of the R1CR2 type, which are subject to increasing SO interactions, such as CHX with X=F, Cl, Br, I, their deuterated analogs, CX2,or various molecular species such as SiH2, GeH2, etc.

ACKNOWLEDGMENTS The authors thankfully acknowledge financial support by the Deutsche Forschungsgemeinschaft via a research grant as FIG. 5. Potential energy as a function of the bending coordinate for selected well as a visitor grant for L. V. P. This work has also been cases of Renner-Teller, 1⌺+ − 1⌬ electrostatic, and SO-induced vibronic cou- supported by the Fonds der Chemischen Industrie. plings in ⌺ and ⌬ electronic states of quasilinear triatomic molecules with a half-filled ␲ shell. The ordinate scale is in units of ␻, while the abscissa ␳ 3⌺− 1⌬ 1⌺+ represents the dimensionless bending normal coordinate . The curves APPENDIX A: DERIVATION OF THE „ , , … SO E E ␻ ␭/␻ shown here have been constructed for 1 = 2 =5 and =0.2. The energy VIBRONIC COUPLING HAMILTONIAN E3⌺− is defined as 0 for ␳=0. ͑a͒͑g/␻=4; dЈ/␻=2; ␭/␻=0.2; a/␻=0; c/␻=0; d/␻=0; b/␻=0͒, ͑b͒͑g/␻=4; dЈ/␻=2; d/␻=2; ␭/␻=0.2; a/␻ Consider the matrix elements of the electronic Hamil- /␻ /␻ ͒ ͑ ͒͑ /␻ Ј/␻ /␻ /␻ ␭/␻ =0; c =0; b =0 , c g =4; d =2; d =2; b =0.1; =0.2; tonian Hˆ +Hˆ with the electronic basis functions defined a/␻=0; c/␻=0͒, and ͑d͒͑a/␻=0.002; c/␻=0.04; g/␻=4; dЈ/␻=2; d/␻ es SO /␻ ␭/␻ ͒ ͑ ͒ ͑ ͒ Hˆ Hˆ =2; b =0.1; =0.2 . in Eqs. 14 – 19 . We expand es+ SO in powers of the

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FIG. 6. ͑Color online͒ Energy-level correlation diagram of the spin-vibronic energy levels of the 3⌺− and 1⌬ electronic states of a Renner-Teller active mode of ␲ symmetry, calculated for selected cases of SO-induced vibronic coupling for the following parameter values: ͑a͒͑a/␻=0; c/␻=0; ␭/␻=0; g/␻=0; dЈ/␻=0; d/␻=0; b/␻=0͒, ͑b͒͑a/␻=0; c/␻=0; ␭/␻=0; g/␻=2.0; dЈ/␻=0; d/␻=0; b/␻=0͒, ͑c͒͑a/␻=0; c/␻=0; ␭/␻=0; g/␻=2.0; dЈ/␻=0.6; d/␻ /␻ ͒ ͑ ͒͑ /␻ /␻ ␭/␻ /␻ /␻ /␻ /␻ ͒ E ␻ =0.6; b =0 ,and d a =0; c =0; =0; g =2.0; dЈ =0.6; d =0.6; b =0.04 . The calculation of the energy levels assumes that 1 =1.5 and E ␻ ␯ 3⌺− 1⌬ 2 =1.6 . Only the spin-vibronic energy levels arising from the =0 and 1 harmonic-oscillator levels of the and electronic states are presented in the ﬁgure. The solid, long-dash dotted, short-dash dotted, and short-short dotted lines represent energy levels with ␮=0, 1, 2, and 3, respectively.

Ϯi␾ bending normal mode QϮ =␳e . The requirement that the Jˆ Ј angular-momentum operator z leads to the following struc- Hamiltonian must be totally symmetric with respect to the ture of the Hamiltonian matrix:

␺+2 ␺0 ␺0 ␾0 ␺0 ␺−2 +2 +1 0 0 −1 −2 ␾ ␾ ␾ ␾ ␾ ␺+2 E1⌬ ␳ i ␳2 2i ␳2 2i ␳3 3i ␳4 4i +2 d1 e h e c e b e a e Ã ␾ ␾ ␾ ␾ ␾ ␺0 ␳ −i E3⌺− ␳ i ␳ i ␳2 2i Ј␳3 3i +1 d1 e d2 e d3 e k e b e Ã ␾ Ã ␾ ␾ ␾ H ␺0 ␳2 −2i ␳ −i E3⌺− Ј␳ i Ј␳2 2i ͑ ͒ el = 0 h e d2 e g d3 e h e . A1 Ã ␾ Ã ␾ Ã ␾ ␾ ␾0 ␳2 −2i ␳ −i E1⌺+ Ј␳ i ␳2 2i 0 c e d3 e g d2 e c e Ã ␾ Ã ␾ Ã ␾ Ã ␾ ␾ ␺0 ␳3 −3i ␳2 −2i Ј ␳ −i Ј ␳ −i E3⌺− Ј␳ i −1 b e k e d3 e d2 e d1 e Ã ␾ Ã ␾ Ã ␾ Ã ␾ Ã ␾ ␺−2 ␳4 −4i Ј ␳3 −3i Ј ␳2 −2i ␳2 −2i Ј ␳ −i E1⌬ −2 a e b e h e c e d1 e

The diagonal elements of the matrix ͑A1͒ and the ele- bending displacement. The parameter a represents the ments c␳2e2i␾ and a␳4e4i␾ arise from the electrostatic Hamil- fourth-order Renner-Teller coupling within the 1⌬ state. All Hˆ other matrix elements of Eq. ͑A1͒ arise from the SO tonian es. The parameter c represents the vibronic coupling of the 1⌬ state and the 1⌺+ state in second order of the operator.

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͑0͒ 1 2 1 4 We include only the lowest-order nonvanishing contribu- 1 + ␻␳ ␭␳ ͑ ͒ E ⌺ = E1⌺+ + 2 + 24 + ¯ , A4 tion for each matrix element and ignore SO corrections to matrix elements of the electrostatic Hamiltonian. It can be shown by explicit calculation of matrix elements with the basis functions ͑14͒–͑19͒ that the coupling parameters where we have assumed, for simplicity, that the bending vi- ␻ ␭ a,b,c,d,... are real. The diagonal matrix elements have the brational frequency and the anharmonicity parameter are ␲2 expansion the same in the three electronic states arising from the conﬁguration. ͑0͒ 1 2 1 4 1 ␻␳ ␭␳ ͑ ͒ E ⌬ = E1⌬ + 2 + 24 + ¯ , A2 Requiring the vanishing of the commutator of the Hamil- tonian matrix ͑A1͒ with the time-reversal operator ͑24͒,we ͑ ͒ Ј Ј Ј Ј 0 1 2 1 4 obtain the relations b =−b, d1 =−d1, d2 =−d3, d3 =−d2, E3⌺− = E3 − + ␻␳ + ␭␳ + , ͑A3͒ ⌺ 2 24 ¯ hЈ=h, and thus,

͑ ͒ 0 ␳ i␾ ␳2 2i␾ ␳2 2i␾ ␳3 3i␾ ␳4 4i␾ E1⌬ d1 e h e c e b e a e ͑ ͒ ␳ −i␾ 0 ␳ i␾ ␳ i␾ ␳2 2i␾ ␳3 3i␾ d1 e E3⌺− d2 e d3 e k e − b e ͑ ͒ ␳2 −2i␾ ␳ −i␾ 0 ␳ i␾ ␳2 2i␾ h e d2 e E3⌺− g − d2 e h e H 1 ␻␳2 1 ␭␳4 ͑ ͒ el = ͑ + ͒1+ ͑ ͒ . A5 2 24 ␳2 −2i␾ ␳ −i␾ 0 ␳ i␾ ␳2 2i␾ c e d3 e g E1⌺+ − d3 e c e ͑ ͒ ΄ ␳3 −3i␾ ␳2 −2i␾ ␳ −i␾ ␳ −i␾ 0 ␳ i␾ ΅ b e k e − d2 e − d3 e E3⌺− − d1 e ͑ ͒ ␳4 −4i␾ ␳3 −3i␾ ␳2 −2i␾ ␳2 −2i␾ ␳ −i␾ 0 a e − b e h e c e − d1 e E1⌬

͑ ͒ ͑ ͒ The matrix A5 can be further simpliﬁed by the explicit Hˆ 1 = ␬͑S · L + S · L ͒, calculation of matrix elements of the SO operator of Eqs. SO 1 1 2 2 ͑4͒–͑6͒ with the electronic basis functions ͑14͒–͑19͒.Itis Hˆ Hˆ ͑2͒ ␬͑ Q Q ͒ ͑ ͒ convenient to write SO as an expansion in the bending am- SO = S1 · 1 + S2 · 2 , A9 plitude ␳,

Hˆ ͑3͒ ␬͑ R R ͒ SO = S1 · 1 + S2 · 2 . Hˆ Hˆ ͑0͒ Hˆ ͑1͒ Hˆ ͑2͒ ͑ ͒ SO = SO + SO + SO + ¯ . A6 L Q R The operators k, k, and k, k=1,2, are differential operators in electronic coordinate space and can be obtained The leading term can be written as K from the k by somewhat lengthy calculations. It is obvious that the parameter k in Eq. ͑A5͒ vanishes by ͑ ͒ Hˆ 0 = ␬͑S · K + S · K ͒, ͑A7͒ spin integration, since the basis functions differ in two spin SO 1 1 2 2 Hˆ orbitals, while SO is a single-particle operator in spin space. It is straightforward to show that the matrix element with ␳2 2i␾ ͗␺0͉Hˆ ͑2͉͒␺−2͑͒͘ h e = 0 SO −2 A10 ␬ 1 ␤2 = 2 ig e , vanishes by spin integration. For the matrix element

͑ ͒ ␾ Q 1 ͗␺0 ͉Hˆ 1 ͉␺0͘ = d ␳ei , +1 SO 0 2 ͒ ͑ ͔͒ ١ ١͑ ϫ ͓ ͒ ١ K n ͑ ϫ 1 =−͚ 3 r1n 1 + 3 r12 1 −2 2 , A8 n r r 1n 12 it can be shown that

͗␺0͉Hˆ ͑1͉͒␺0 ͘Ã ͗␺0 ͉Hˆ ͑1͉͒␺0͘ Q 1 0 SO +1 =− +1 SO 0 . ͔͒ ١ ١͑ ϫ ͓ ͒ ١ K n ͑ ϫ 2 =−͚ 3 r2n 2 + 3 r21 2 −2 1 . n r2n r21 Hˆ Since el must be Hermitian, this implies d2 =0. With the simpliﬁed notation d1 =d, d3 =dЈ, we have the Hamiltonian of The higher-order terms can analogously be written as Eq. ͑25͒.

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