The vibrational contributions to second-order transition properties are considered in the framework of the Born—Oppenheimer approximation. It is shown that the usual formula for vibrational second-order transition matrix elements is incomplete and needs to be supplemented by a term of purely vibrational origin. This pure vibrational contribution is calculated for vibrational transition polarizabilities in LiH and BeF and found to be quite
significant. Its inclusion in theoretically calculated data for the Raman intensities appears to be necessary

Pluta, Tadeusz

Sadlej, A. J.

Acta Physica Polonica A

English

The vibrational contributions to second-order transition properties are considered in the framework of the Born-Oppenheimer approximation. It is shown that the usual formula for vibrational second-order transition matrix elements is incomplete and needs to be supplemented by a term of purely vibrational origin. This pure vibrational contribution is calculated for vibrational transition polarizabilities in LiH and BeF and found to be quite significant. Its inclusion in theoretically calculated data for the Raman intensities appears to be necessary

Pluta, T.

Biblioteka Nauki - repozytorium artykuÅÃ³w

Vibrational Corrections to Transition Properties

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: Vibrational corrections to transition properties  Author: Tadeusz Pluta, A. J. Sadlej  Citation style: Pluta Tadeusz, Sadlej A. J. (2000). Vibrational corrections to transition properties. "Acta Physica Polonica A" (Vol. 98, nr 4 (2000), s. 363-371.  Vol. 98 (2000)	 ACTA PHYSICA POLONICA A 	 No. 4VIBRATIONAL CORRECTIONS TO TRANSITIONPROPERTIEST. PLUTAInstitute of Chemistry, Silesian University, Szkolna 9, 40-006 Katowice, PolandAND A.J. SADLEJDepartment of Quantum Chemistry, Institute of ChemistryNicolaus Copernicus University, Gagarina 7, 87-100 Toruń, Poland(Received August 17, 2000)The vibrational contributions to second-order transition properties areconsidered in the framework of the Born—Oppenheimer approximation. It isshown that the usual formula for vibrational second-order transition matrixelements is incomplete and needs to be supplemented by a term of purelyvibrational origin. This pure vibrational contribution is calculated for vi-brational transition polarizabilities in LiH and BeF and found to be quitesignificant. Its inclusion in theoretically calculated data for the Raman in-tensities appears to be necessary.PACS numbers: 33.20.Fó1. IntroductionOver the past years a great deal of interest [1-3] has been given to vibra-tional corrections to different molecular properties [4]. Bishop, in his review of thesubject [5], stressed that accounting for the influence of molecular vibrations (androtations) on the calculated molecular properties is much more than just a simpleaveraging with respect to the given rovibrational (v, J) state. For properties ofthe second- and higher-orders the usual rovibrational correction δPv,J [6], whichfollows from averaging the so-called electronic contribution, Pel, to the total prop-erty value P in the given electronic-vibrational-rotational state of the molecule,needs to be supplemented by additional terms, δPrv, of rovibrational origin. Theseterms arise from the fact that in the presence of some external perturbation boththe electronic and nuclear motions are affected. Moreover, in several cases [1-3, 7]these, usually neglected δPΓv terms, have been found to make either large or evendominant contribution to the totαl, i.e., the ones which follow from physical mea-surements, values of properties.(363)364 T. Pluta, A.J. SadlejIt appears that so far most of attention has been focused on vibrational cor-rections to expectation values of different, mostly electric, properties of molecules.However, a similar mechanism should also add some vibrational and rotationalcontributions to transition properties. In the present paper we develop a theoryof the corresponding corrections to second-order vibrational transition properties.Such a property can be exemplified by the transition dipole polarizability betweentwo vibrational states. This is a quantity which determines the intensity in the vi-brational Raman scattering [8]. The magnitude of this effect is first estimated onthe basis of spectroscopic parameters. Its magnitude depends on the value of thevibrational transition dipole moment between the two states under considerationand on the difference between vibrational contributions to the dipole moment ofthese two states. Additional numerical illustration is given by direct calculationsof the transition polarizabilities in LiH and BeF.2. TheoryIn this paper we shall focus our attention on a single electronic state derivedunder the assumption of the validity of the Born—Oppenheimer approximation.Moreover, we shall assume that the overall rotation of the molecule has beenseparated out and the system is in its rotationless (J = 0) state. Transitionsbetween different states of the system will occur between vibrational levels ofthe same electronic state. For an external perturbation represented in the totalelectronic-nuclear Hamiltonian H = H(λ) by the perturbation term )AV, whose"strength" is governed by the parameter λ, the Born—Oppenheimer approximationleads to the notion of the electronic energy E = E(Q; λ) which is an eigenvalue ofthe electronic Schrodinger equationwhere r denotes the electronic coordinates and Q stands for the set of nucleargeometry parameters. The h-dependent Hamiltonian H(7.; Q, λ) isThe eigenvalue of (1) depends on both the nuclear coordinates and on the strengthof the perturbation V and defines the λ-dependent potential in what is called thenuclear Hamiltonian ħ= ħ(Q; λ) of the electronic state under considerationwhere TQ is the nuclear kinetic energy operator and E(Q; )A) includes the nuclearrepulsion terms. The nuclear eigenvalue problemgives the .\-dependent solutions which determine the total electronic-vibrationalfunction Ψv (r, Q; λ) of the electronic state 0(r; Q, λ):If the perturbation V enters the molecular Hamiltonian with some time depen-dent factor, its presence causes transitions between different electronic-vibrationalVibrational Corrections to Transition Properties	 365states. The single-quantum vibrational transitions are known to be governed bythe (first-order) transition matrix elementOwing to the assumption of the Born—Oppenheimer approximation one can definethe pure electronic contribution to the h-dependent first-order transition prop-erty (6). In the present case this electronic contribution is simply the expectationvalue of the operator V in the state 0(r; Q, λ):Thus, upon taking the λ → 0 limit of (6) one obtains the usual vibrational transi-tion elementwhose determination requires solely the λ-independent solutions of the electronicand nuclear Schrodinger equations. Because of the Hellmann—Feynman theoremthe only vibrational contributions to P(1);el v,vare due to .\-independent vibrationalwave functions of the two states involved in the transition. For V being the dipolemoment operator this theory gives the usual formula for the vibrational transitiondipole matrix element which determines the intensity of the IR bands [9].In the case of two-quantum transitions caused by the time dependent factorat the perturbation V their intensity is approximately governed [8] by the second--order transition matrix elementBy taking the λ → 0 limit of the derivative in (9) one finds thatwhereandMoreover,is the electronic contribution to the second-order property arising from the per-turbation V.In Eq. (10) the first term, P(2),elv,vgives the usual definition (11) of thesecond-order transition property for transitions between two vibrational statesdefined by the quantum numbers v and v'. However, if one takes into account that366 T. Pluta, A.J. Sadlejthe potential used in the nuclear equation also depends on the external pertur-bation, there is additional contribution P( 2 ) , v 1ó of purely vibrational origin. Thisadditional term contributing to the total second-order transition property (10)involves the first derivatives of vibrational functions determined at λ = 0. Thesederivatives are simply the first-order perturbed solutions, X(1)(Q; 0) arising fromthe perturbation expansion of the λ-dependent nuclear eigenvalue problem (4).Through the first-order in λ one findsandwhere the superscript "0" indicates the operators and solutions in the absence ofthe external perturbation, e.g., X,v(0)(Q) = Xv(Q; 0), E^0) = ev(0). Similarly,In terms of the complete set of solutions {X, 0 >} of the .\-unperturbed nuclearproblem the first-order vibrational functions can be written asand upon substitution into Eq. (13) give explicit sum-over-states formula for thepurely vibrational correction to second-order vibrational transition propertiesObviously, the usefulness of the sum-over-states expansion of P(2)vib v,v,'in its com-plete form (18) is rather limited. However, in the case of the mechanical andelectric harmonic approximations one finds that for each selected mode v' = v ± 1and each of the two sums in Eq. (18) is reduced to a single term. Within theseapproximations the estimate of (18) will bewhere ωe denotes the harmonic frequency associated with the given mode. Thisapproximate expression for P^ (2)vibbv,v+1may be helpful in estimating the magnitudeof the P(2),vib contribution to vibrational second-order transition properties. TheVibrational Corrections to Transition Properties 367evaluation of (19) requires only some lower-order data, i.e., the first-order vibra-tional transition property and vibrational averages of P( 1 ) , et. All these data canbe expressed in terms of spectroscopic constants and the first-order derivative ofp(1),el with respect to the normal coordinate under consideration [5, 6]. In thecase of polyatomic molecules the use of normal coordinates would require the as-sumption of the small amplitude vibrations. For most of the stable molecules inlow rovibrational states this assumption is quite well satisfied.3. Applications: transition dipole polarizability3.1. Vibrational transition dipole polarizabiltyThe formalism presented so far is essentially valid for any second-order vi-brational transition property. Its particular application discussed in this sectionrefers to intensities in vibrational Raman spectra [8]. Then, the perturbation op-erator V can be identified with the negative of some component, say al p , the dipolemoment operatorand the strength parameter λ acquires the meaning of the Fp (= F) component ofthe external electric field. Then,P ( 1 ) el(Q; 0) = µe ^ (Q; 0) becomes the electronicpart of the dipole moment surface. Similarly, P(2)el(Q; 0) = αel(Q; 0) is the elec-tronic part of the diagonal pp-component of the dipole polarizability surface. Tosimplify the notation we shall avoid explicit use of the tensor indices. Analogousidentification of symbols holds also for vibrational averages and vibrational tran-sition properties, i.e.,andIt is also useful to note that the pure vibrational contribution αvibv,v, to the transitionpolarizability αv,v can he, according to (13), expressed as the following derivative:while αelv,v, is simply a matrix element of αel(Q; 0) between the two vibrationalstates involved in the Raman transition.The origin of the additional vibrational contribution to the transition dipolepolarizability can be also elucidated in the framework of Placzek's theory of theRaman intensities [8, 10]. In this case one considers the induced electronic dipolemoment δ, el(Q; F) which arises from the presence of the oscillatory electric fieldF. In the linear approximation one has368	T. Pluta, A.J. Sadlejwith the total dipole moment derivative approximated by the electronic dipolepolarizability. i.e..If one takes into account that the presence of the electromagnetic field influencesalso the Born—Oppenheimer potential, then the correct expression for the totaldipole moment derivative in (25) readsThe additional term which accounts for the dependence of normal coordinates onthe external field gives rise to the correction α vib v,v' derived in this paper for thestatic limit of the Raman intensity formulae [8, 10].3.2. Vibrational contributions to the transition dipole polarizabiltyin LiH and BeFThe two molecules, LiH and BeF, have been selected to exemplify the theoryof vibrational contributions to transition dipole polarizabilities. We have calculatedpotential energy, dipole moment (z-component along the bond), and dipole polar-izability (zz-component) curves by using the complete active space self-consistentfield (CASSCF) method and the polarized basis sets Poili, PolLi, PolBe, andPolF [11]. The CASSCF calculations [12] have been carried out with active spacescomprising all valence orbitals and the is core orbital of the metal atom with 4 and11 electrons for LiH and BeF respectively. These calculations are essentially simi-lar to those carried out earlier by Fowler and one of the present authors [13]. Allelectric properties needed for the evaluation of the transition dipole polarizabilityhave been obtained by using the finite field perturbation method [14]. The fieldstrength values along the z-axis used to obtain finite difference approximations toenergy derivatives were F = 0.0, ±0.0025, and ±0.0050 a.u.The present calculations are not aimed at obtaining highly accurate datafor any of the studied properties. The main purpose of this numerical illustrationis to show the importance of purely vibrational contribution to transition dipolepolarizabilities. Since the evaluation of the transition matrix elements requiresthe energy and property curves to be determined over a rather wide range ofinteratomic distances, the use of the CASSCF method appears to be quite natural.With the present choice of the active orbital subspaces this method gives a properdissociation limit for each of the studied molecules.All parameters needed to estimate the pure vibrational contribution to thetransition dipole polarizability according to the approximate formula (19) are sum-marized in Table I. They are obtained from calculations of the electronic energyand dipole moment curves followed by the solution of the rovibrational problemwith the aid of the Numerov—Cooley method [12, 15] and calculation of the vi-brational dipole moment expectation values and transition dipoles. The electronicVibrational Corrections to Transition Properties	 369dipole polarizability curves, αel(Q; 0) were used in a similar way to obtain the α el v,v'values presented in Table II.Table II shows pure vibrational corrections estimated from the data of Table Iby using Eq. (19). The results referred to as directly calculated follow from thenumerical evaluation of the derivative given by Eq. (24). The matrix elementsentering this equation can be easily evaluated from the known dipole moment curveobtained in the absence of the external field and vibrational functions determinedin the presence of the external field.The examples presented in Table II show that the pure vibrational con-tribution to transition dipole polarizability is by no means negligible. For LiHit contributes about ten per cent of the total value of αv, v + 1 for v equal to0 and 1. In the case of the BeF molecule about 40 per cent of the total tran-sition dipole polarizability arises from the purely vibrational term. One has tostress that this term has been so far systematically neglected in evaluations ofthe Raman intensities. Since the calculation of αv ib v,v'for polyatomic molecules mayrepresent quite a task, its simple estimates become quite valuable. The estimationmethod proposed in this paper is found to give quite reasonable values of αvibv,v'It is of importance that this estimate involves the vibrational transition moment370	 T. Pluta, A. J. Sadlejand zero-point vibrational corrections [6, 16] to the electronic dipole moment invibrational states linked by the Raman transition, i.e., the quantities which areaccessible either experimentally or by computations. The first of these quantitiescan be either calculated theoretically or determined from infrared band intensi-ties. The calculation of zero-point vibrational corrections to molecular propertiesis becoming routine [2, 16, 17]. The last parameter, ω e , which enters Eq. (19) isusually known.4. ConclusionsIt has been shown that the usual expression for vibrational second-ordertransition properties needs to be supplemented by additional term of purely vi-brational origin. This term arises from the dependence of vibrational functions onthe external perturbation and is absent in the case of the vibrational first-orderproperties. As an example of the theory of vibrational effects on second-order tran-sition properties we have considered vibrational transition dipole polarizabilities.Estimates of the pure vibrational contributions and their directly calculated valuesshow that the so far neglected vibrational effect is by no means negligible and mayconsiderably affect the calculated theoretical data for the Raman intensities.The present treatment assumes that the transitions under consideration donot involve changes in the rotational state of the molecule. Moreover, it is assumedthat zeroth-order perturbation problem is well represented by the harmonic po-tential. This may to some extent limit the conclusions of the paper making themvalid only for relatively low rotational states and restricted to molecules for whichanharmonic terms are sufficiently small. The latter restrictions can be easily liftedby using the higher-order perturbation expansions with respect to anharmonicterms of the Born—Oppenheimer potential.AcknowledgmentsThis project was supported by the Committee for Scientific Research underthe grant No. 3 T09A 108 15.References[1] C.E. Dykstra, S.-Y. Liu, D.J. Malik, Adv. Chem. Phys. 75, 37 (1990) and refer-ences therein.[2] M.J. Cohen, A. Willetts, R.D. Amos, N.C. Handy, J. Chem. Phys. 100, 4467(1994) and references therein.[3] M.G. Papadopoulos, A. Willetts, N.C. Handy, A.E. Underhill, Mol. Phys. 88,1063 (1996).[4] A.D. Buckingham, Adv. Chem. Phys. 12, 107 (1967).[5] D.M. Bishop, Rev. Mod. Phys. 62, 343 (1990) and references therein.[6] A.D. Buckingham, J. Chem. Phys. 36, 3096 (1962); A.D. Buckingham, W. Urland,Chem. Rev. 75, 113 (1975).Vibrational Corrections to Transition Properties 371[7] M. Eckert, G. Zundel, J. Phys. Chem. 91, 5170 (1987); D.M. Bishop, J. Pipin,J.N. Silverman, Mol. Phys. 59, 165 (1986); D.P. Shelton, L. Ulivi, J. Chem. Phys.89, 149 (1988); L. Adamowicz, R.J. Bartlett, J. Chem. Phys. 84, 4988 (1986);L. Adamowicz, J. Chem. Phys. 89, 6305 (1988).[8] J. Brandmuller, H. Moser, Einfuhrung in die Ramanspektroskopie, DietrichSteinkopff Verlag, Darmstadt 1963, Ch. 24; D.P. Craig, T. Thirunamachandran,Molecular Quantum Electrodynamics, Academic Press, New York 1984.[9] K. Nakamoto, Infrared Spectra of Inorganic and Coordination Compounds, Wiley,New York 1963, Ch. 13; S. Califano, Vibrational States, Wiley, New York 1976.[10] G. Placzek, iu: Ilandbuch der Radiologie, Ed. E. Marx, Akademische Verlag,Leipzig 1934, p. 205.[11] A.J. Sadlej, Coll. Czech. Chem. Commun. 53, 1995 (1988); A.J. Sadlej, Theor.Chim. Acta 79, 123 (1991); A.J. Sadlej, M. Urban, J. Mol. Struct. (Theochem.)234, 147 (1991).[12] K. Andersson, M.R.A. Blomberg, M. Fulscher, V. Kellö, R. Lindh,P.-Å. Malmqvist, J. Noga, J. Olsen, B.O. Roos, A.J. Sadlej, P.E.M. Siegbahn,M. Urban, P.-O. Widmark, Molcas System of Quantum Chemistry Programs,Release 3. Theoretical Chemistry, Uuiversity of Lund, Lund, Sweden and IBMSweden, 1994.[13] P.W. Fowler, A.J. Sadlej, Mol. Phys. 73, 43 (1991).[14] A.D. McLean, M. Yoshimine, J. Chem. Phys. 46, 3682 (1967); A.D. McLean,M. Yoshimine, J. Chem. Phys. 47, 3256 (1967).[15] J.W. Cooley, Math. Comp. 15, 363 (1963); C.E. Dykstra, D.J. Malik, J. Chem.Phys. 87, 2806 (1987).[16] A.J. Russell, M.A. Spackman, Mol. Phys. 90, 251 (1997).[17] A.J. Russell, M.A. Spackman, Mol. Phys. 88, 1109 (1996).

Vibrational corrections to transition properties

https://rebus.us.edu.pl/bitstream/20.500.12128/8485/1/Pluta_Vibrational_corrections_to_transition_properties.pdf

Vibrational corrections to transition properties

Abstract

Similar works

Full text

Available Versions

Biblioteka Nauki - repozytorium artykuÅÃ³w

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

Vibrational corrections to transition properties

Abstract

Similar works

Full text

Available Versions

Biblioteka Nauki - repozytorium artykuÅÃ³w

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

Biblioteka Nauki - repozytorium artykuÅÃ³w