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Variational Calculation of the Vibronic Spectrum in the X2пu

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Variational Calculation of the Vibronic Spectrum in the X2пu J. Serb. Chem. Soc. 83 (4) 439–448 (2018) UDC 544.14:539.216:519.6+537.5+54.02 JSCS–5087 Original scientific paper 2 Variational calculation of the vibronic spectrum in the X Πu – electronic state of C6 MARKO MITIĆ1*#, MILAN MILOVANOVIĆ1#, RADOMIR RANKOVIĆ1#, STANKA JEROSIMIĆ1# and MILJENKO PERIĆ1,2 1Faculty of Physical Chemistry, University of Belgrade, Studentski trg 12, P. O. Box 47, PAK 105305, 11158 Belgrade, Serbia and 2Serbian Academy of Sciences and Arts, Knez Mihailova 35, 11158 Belgrade, Serbia (Received 29 November, revised and accepted 4 December 2017) Abstract: A variational approach for ab initio handling of the Renner–Teller effect in six-atomic molecules with linear equilibrium geometry is elaborated. A very simple model Hamiltonian suitable for the description of small-amp- litude bending vibrations in Π electronic states of arbitrary spin multiplicity was employed. The computer program developed within the framework of the 2 – present study was tested on the example of the X Πu state of C6 . The results are compared with those generated in corresponding perturbative calculations. – Keywords: Renner–Teller effect; ab initio calculations; variational approach; C6 . INTRODUCTION For a long time, the Renner–Teller (R–T) effect1 was investigated only in triatomic molecules. The first theoretical study of this kind of vibronic coupling in Π electronic states of molecules with linear equilibrium geometry was per- formed by Petelin and Kiselev.2 A simple model for ab initio handling of the R–T effect, accompanied by spin–orbit coupling in Π and Δ electronic species has been developed and tested on a number of examples.3–13 In these studies, the vibronic Schrödinger equation was solved using both variational and perturbative approaches. Perturbative calculations of the vibronic spectra in five-14 and six- -atomic15,16 molecules were also realized. A method for handling the R–T effect in molecules with an arbitrary number of nuclei was sketched and perturbative formulae for several particular coupling schemes were derived.17 For an exhaust- ive literature survey of this topic, the reader is referred to the papers cited above. In two recent studies,13,18 it was explicitly shown that the vibronic coupling in molecules with more than three nuclei could be understood as a combination * Corresponding author. E-mail: [email protected] # Serbian Chemical Society member. https://doi.org/10.2298/JSC171129001M 439 ________________________________________________________________________________________________________________________Available on line at www.shd.org.rs/JSCS/ (CC) 2018 SCS. 440 MITIĆ et al. of the classical R–T effect, arising as a consequence of the interaction between two components of an electronic state spatially degenerate at the linear geometry and split upon bending, and unavoidable (avoided) crossings of the correspond- ing potential surfaces at non-linear nuclear arrangements, similar to those in the classical Jahn–Teller effect.19 In the second of these papers,18 a variational method for calculating the vibronic spectrum in five-atomic molecules was elab- 2 orated and applied to calculate the low-energy vibronic spectrum in the X Πu – state of C5 . In the present study, for the first time variational ab initio calcul- 2 ations of vibronic levels in a six-atomic molecule, namely in the X Πu state of – C6 , were realized. This system was handled in a previous study using the perturbative formulae derived therein. One of the aims of the present study is to check the reliability of this second-order perturbative treatment and to cover the coupling cases for which the perturbative formulae are not available. THE MODEL HAMILTONIAN The model Hamiltonian used herein for ab initio handling of the Renner–Teller effect and spin–orbit coupling in Π electronic states of six-atomic molecules is subject to the follow- ing restrictions:18 the equilibrium molecular geometry is linear; the harmonic approximation is applied; the coupling between the bending and stretching vibrational modes is neglected; the end-over-end rotations are not considered; the spin–orbit operator is assumed in the pheno- menological form and the asymptotic (linear limit) electronic wave functions are used in mat- rix representation of the kinetic energy operator. Thus, the model Hamiltonian has the form: ˆˆ=++ ˆˆ H HTHeb SO (1) where Ĥe is the electronic Hamiltonian, T̂ b is the kinetic energy operator for bending motions of the nuclei, and ĤSO is the electronic spin–orbit operator. A six-atomic molecule with linear equilibrium geometry, whose nuclei have the masses m1, m2,…, m6, is considered in general. The equilibrium distances between the nuclei are denoted by r1, r2,…, r5. The z-axis was chosen to coincide with the molecular axis at the equilibrium nuclear arrangement (see Fig. 1 of Ref. 17). Infinitesimal bending vibrations were considered herein. To describe the four doubly degenerate bending modes, the Cartesian bend- 15,16 ing coordinates ρ1x, ρ2x, ρ3x, ρ4x and ρ1y, ρ2y, ρ3y, ρ4y were introduced. In terms of the derivatives of these coordinates, the kinetic energy operator (in atomic units) for infinitesimal bending vibrations is represented by a sum of x- and y- contributions: ˆˆ=+ ˆ TTxTybb() b () (2) where 224 mm++++2 m m+∂ Txˆ ()=− ii112+ + i i − 222∂ρ 2 i=1mmii+++++11112 rii m i rr ii m i m i ri+1 x 3 2 11mm+++∂ −++2 ii12 + (3) 2 ∂∂ρρ i=1mrriii++11 mm i ++ 12 i ri+1 m i +++ 212 rr ii ixix + 1, 2 ∂2 + 1 ∂∂ρρ i=1 mrriii+++212 ixix + 2, ________________________________________________________________________________________________________________________Available on line at www.shd.org.rs/JSCS/ (CC) 2018 SCS. THE RENNER–TELLER EFFECT IN SIX-ATOMIC MOLECULES – VARIATIONAL APPROACH 441 ˆ and with a completely analogous form for Tyb (). For the handling of symmetric six-atomic = = = molecule of the type ABCCBA, when mm43, mm52, mm61, it is convenient to intro- ρ ρ ρ ρ ρ duce the symmetrized Cartesian bending coordinates C1x , C2x , T1x , T2x and C1y , ρ ρ ρ C2y , T1y , T2y , defined as: ρρρρρρ=+11 =+ C1x ()4xx 1,, C2 x()3 x 2 x 22 (4) ρρρρρρ=−11 =− T1x ()4xx 1, T2 x()3 x 2 x 22 and analogously for the y-coordinates. The new coordinates describe two collective cis- and two trans-bending vibrations in the planes xz and yz. A straightforward transformation of exp- ression (3), as well as of its y-counterpart, results in: 11∂∂22 ∂ 2 ∂ 2 TTTˆˆˆ=+=− + − + + bCT μρ∂∂22 ρ μ ∂ ρ 2 ∂ ρ 2 22C1 C1xy C1 C2 C2 x C2 y 11∂∂22 ∂∂ 22 ++−+− (5) μρρρρμρρ∂∂ ∂∂22 22C12 C1xx C2 C1 yy C2 T1 T1xy T1 11∂∂22 ∂2 ∂ 2 −+++ μρ∂∂22 ρ μ ∂ ρρρ ∂ρ 22T2 T2xy T2 T12 T1xx T2 T1T2yy where 22 2 μμ==mm123 mr12 r m23 mr2 C1 ,,C2 22+++()2 mm+ mm12 r12 mm 131 r r 2 m 23 mr 23 222 μμ==mmrr23121mmmrr123 2 C12 ,,T1 (6) mr++ m() r r 22+++()2 21 3 1 2 mm12 r12 mm 131 r r 2 m 23 mr mmrr22 mmrrr2 μμ==2323 , 23132 T2 22T12 ++ + 4/2/2mr()()++ r mr 2/2mr21() r 2 r 3 mr 3 ()rr12 22 3 33 3 The kinetic energy operator is equivalent to that derived in Ref. 15; the only difference is that now the factors 1/ 2 were used in Eqs. (4) instead of 1/2, having been employed before. When the coupling of the two components of the Π electronic state is neglected, the potential energy part of the bending Hamiltonian in the harmonic approximation has the form: =+=11ρρ22 + + ρ 2 + ρ 2 + VVCT V k C1()k C2() 22C1xy C1 C2 x C2 y +++++ρρ ρρ1 ρ22 ρ kkC12() C1xx C2 C1 yy C2 T1 () (7) 2 T1xy T1 ++++1 ρρ22 ρρρρ kkT2 ()T12() CTxx T2 T1 yy T2 2 T2xy T2 Note that also the potential energy operator is separated (besides into a cis- and a trans- -part) into x- and y-contributions. ________________________________________________________________________________________________________________________Available on line at www.shd.org.rs/JSCS/ (CC) 2018 SCS. 442 MITIĆ et al. In the next step, the bending normal coordinates are introduced in a way presented in the literature,15 and finally, the kinetic and potential energy are expressed in terms of the polar φ components ()qii, of the dimensionless normal coordinates. In this way, relations (5) and (7) are transformed into: 1114 ∂∂∂22 Tˆ =− + + ω (8) b ∂∂∂222φ i 2 i=1qqqqiiiii ω where i are the bending vibrational frequencies. The potential energy part of the Hamiltonian becomes: 4 = 1 ω 2 Vq i i (9) 2 i=1 The spin–orbit (S–O) operator in the phenomenological form is: ˆˆ= ˆ HSOALS SO zz (10) ˆ ˆ where ASO is the spin–orbit constant, and Lz and Sz are the components of the electronic orbital and spin angular momentum along the molecular axis. The model Hamiltonian (1) is represented in the two-dimensional electronic space span- ned by the “diabatic” electronic eigenfunctions ψ Λ and ψ −Λ defined as: 11 ψψψψψψΛτ=+eiii()'",−− Λ = e τ() '" − i (11) 22 where ψ ' and ψ " are the eigenfunctions of the electronic operator Ĥe obtained within the framework of the Born–Oppenheimer approximation (“adiabatic” electronic functions) for the two components of the Π state, and τ is an angle defined as:4 44 εωω φφ+ ij i jqq i jsin() i j 1 ij==11 τ = arctan (12) 2 44 εωωqq cos() φφ+ ij i j i j i j ij==11 with the quantities εij determined by the energy difference of the adiabatic potentials at planar geometries. The sense of τ is that it defines the plane with the property that one of the adiabatic electronic wave functions (ψ′) is invariant when reflected on it, and the other (ψ″) changes its sign upon the reflection.
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