A multiconfiguration selfconsistentfield response study of one and twophoton dipoletransitions between the X 1Σ+ and A 1Π states of CODage Sundholm, Jeppe Olsen, and Poul Jo/rgensen Citation: The Journal of Chemical Physics 102, 4143 (1995); doi: 10.1063/1.468542 View online: http://dx.doi.org/10.1063/1.468542 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/102/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Multiconfiguration selfconsistent field quadratic response calculations of the twophoton transition probability rateconstants for argon J. Chem. Phys. 101, 4931 (1994); 10.1063/1.467415 Firstorder nonadiabatic coupling matrix elements from multiconfigurational selfconsistentfield response theory J. Chem. Phys. 97, 7573 (1992); 10.1063/1.463477 Multiconfiguration selfconsistentfield calculation of the dipole moment function and potential curve of NO(X 2Π) J. Chem. Phys. 62, 864 (1975); 10.1063/1.430537 Multiconfiguration selfconsistentfield calculation of the dipole moment function of CO(X 1Σ+) J. Chem. Phys. 60, 4130 (1974); 10.1063/1.1680880 Atomic Multiconfiguration SelfConsistentField Wavefunctions J. Chem. Phys. 50, 684 (1969); 10.1063/1.1671117

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A multiconfiguration self-consistent-field response study of one-and two-photon dipole transitions between the X 1S1 and A 1P statesof CO

Dage Sundholma)

Institut fur Physikalische Chemie, Lehrstuhl fu¨r Theoretische Chemie, Universita¨t Karlsruhe, D-76128Karlsruhe, Germany

Jeppe OlsenTheoretical Chemistry, Chemical Centre, University of Lund, P.O. Box 124, S-22100 Lund, Sweden

Poul Jo”rgensenDepartment of Chemistry, Aarhus University, DK-8000 Aarhus C, Denmark

~Received 2 September 1994; accepted 28 November 1994!

The one- and two-photon dipole transitions between theX 1S1 and theA 1P states of CO havebeen studied by means of multiconfiguration self-consistent-field linear and quadratic responsemethods. The vibrationally averaged oscillator strength for the 0–0 one-photon dipole transitionbetween theX 1S1 andA 1P states obtained using the linear response method is 1.3131022 ascompared to the experimental results of 0.96~14!31022, 1.08~7!31022, and 1.1131022. Thetwo-photon transition probability rate constant, obtained using the quadratic response method, forthe 0–1 vibrational band of theX–A transition of 7310259 cm4 s is more than six orders ofmagnitude smaller than the experimental result of 3.5310252 cm4 s. We suggest that the experimentshould be reconsidered. The dipole moment of theA 1P state obtained from quadratic responsecalculations on the ground state atR52.332 a.u. is20.0441 a.u. suggesting an anomalous polarityalso for theA 1P state. The experimental value is60.059~20! a.u. © 1995 American Institute ofPhysics.

I. INTRODUCTION

The ~X–A! transition between thep4s2 X 1S1 groundstate and thep4sp* A 1P state of carbon monoxide was firststudied more than a century ago.1 This transition has sincethen been the subject of many experimental2–9 andtheoretical10–15 studies. Hesser2 and Carlsonet al.3 reportedlifetime measurements of theA 1P state and derived oscil-lator strengths and dipole transition moments from the life-times and emission intensities. Lee and Guest5 deduced theoscillator strength for the 0–0 vibrational bands of low-lyingtransitions of CO from measured peak absorption cross sec-tions. Fieldet al.6 determined the dipole transition momentfor the X–A transition from lifetimes ofA 1P vibrationallevels. Le Flochet al.8 reported recently very accurate mea-surements of the~n920! bands of theX–A transition.DeLeon9 deduced the dipole transition moment function fortheX–A transition for the range 1.9,R,3.4 a.u. from laserinduced fluorescence measurements on highly vibrationallyexcited CO molecules. TheX–A transition was also one ofthe first transitions for which a two-photon transition prob-ability rate constant was reported.16–19 Cooperet al.12,14,15

calculated spectroscopical constants and dipole momentfunctions for the first few lowest1S and 1P states, and re-ported oscillator strengths between vibrational levels of theX–A transition. The oscillator strengths were deduced fromthe one-photon dipole transition moment functions obtainedin configuration interaction~CI! calculations.

In this work we study theX 1S1 and theA 1P states ofCO using the multiconfiguration self-consistent field

~MCSCF! method for the ground state wave function, andthe one-photon dipole transition properties are obtained frommulticonfiguration linear response~MCLR! functions. Thedipole moments for the excited states, the transition momentsbetween excited states, and the two-photon dipole transitionmoments between theX 1S1 andA 1P states are obtainedfrom multiconfiguration quadratic response~MCQR! func-tions. The linear response method is able to provide a gooddescription of excited states which are dominated by singleexcitations. In a coupled cluster singles and doubles linearresponse calculation~CCSDLR!,20 the percentage of thet1amplitudes was 93.8%, and a relatively good description ofthe A 1P state is expected. The MCSCF linear responsefunction was originally derived by Yeager and Jo”rgensen21

and by Dalgaard.22 The MCSCF quadratic response func-tions were derived by Olsen and Jo”rgensen.23,24 A modernimplementation of the MCLR method is reported byJo”rgensen, Jensen, and Olsen.25 The implementation of theMCQR method has recently been described by Hettemaetal.26 for singlet operators and by Vahtraset al.27 for tripletoperators.

II. BASIS SET AND CONFIGURATION SPACE

The Dunning’s correlation consistent aug-cc-p VTZ ba-sis set28,29 is the starting point for the basis set used in thiswork. The aug-cc-p VTZ basis set is further augmented by aset of diffuse functions. The exponents of the added basisfunctions ~one s, p, d, and f function for C and O! wereobtained by dividing the smallest exponents of the originalaug-cc-p VTZ basis by 3. Only the true spherical harmoniccomponents of thed and f functions are included, yielding148 primitive functions contracted to 124 basis functions.

a!On leave from the Department of Chemistry, P.O. Box 19~Et. Hesperiank.4!, FIN-00014 University of Helsinki, Finland.

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All calculations are performed inC2v symmetry, and thecomplete active space calculations~CAS! are labeled4220~8!, 6331~8!, and 7441~8!. This notation means that, forexample, in the smallest CAS calculation the number of ac-tive orbitals of each irreducible representation is 4, 2, 2, and0, respectively. The number of active electrons is givenwithin parentheses. This corresponds to four actives andfour activep orbitals, and to three inactives orbitals, i.e.,the 1sC, 1sO, and 2sO orbitals. The active spaces were se-lected based on second-order Mo” ller–Plesset~MP2! naturaloccupation numbers.30 The Hartree–Fock energy calculatedat R52.132 a.u. using the present basis set is2112.781 68as compared to the HF limit of2112.790 95 a.u.31 The4220~8! and the 7441~8! CAS energies are2112.935 12 and2113.033 65 a.u., respectively.

The one- and two-electron integrals over generally con-tracted Gaussian-type orbitals were obtained using theHERMIT integral program.33 The calculations were carried outwith the SIRIUS-ABACUS-RESPONSEsuite of codes for multi-configuration self-consistent-field ~MCSCF! wavefunctions,32,34 for multiconfiguration linear response func-tions ~MCLR!,25,35 and for multiconfiguration quadratic re-sponse functions~MCQR!.26,27,35Calculations were also car-ried out with a single reference state giving random phaseapproximation~RPA! results. The spectroscopical propertiesand the transition moments between vibrational levels wereobtained from the MCSCF, MCLR, and MCQR data usingthe program for vibration–rotation spectrum of diatomicmolecules in theMOLCAS program package.36

III. SPECTROSCOPIC PROPERTIES

The potential energy curve for theX 1S1 ground stateof CO was obtained in 4220~8! CAS SCF calculations forinternuclear distances between 1.532 and 3.332 a.u. The ver-tical excitation energy to the lowestA 1P state was calcu-lated for each geometry using the linear response method. Byadding the excitation energy to the ground state energy, the

total energy of theA 1P excited state and the correspondingpotential energy curve are obtained. In Table I, the total en-ergy of theX 1S1 andA 1P states is given as a function ofthe internuclear distance.

The spectroscopical properties such as the equilibriumbond lengthRe , the energy minimum, the harmonic vibra-tional frequency~ve!, the rotational constant (Be), andae

were for the two states deduced from the energy data inTable I using theMOLCAS program package.36 This involvesthe standard procedure of generating vibrational wave func-tions and energies from the potential curves and then obtainthe constants by fitting the energy expansion in terms ofnand J to the calculated energies. The anharmonicites arerather sensitive to the number of vibrational states includedin the fitting and they will not be discussed further. Thespectroscopic constants of the ground state of12C16O are inTable II compared to previously calculated values and ex-perimental results. A similar comparison for theA 1P stateis given in Table III. The term energy~Te! is the energydifference between the minima of the two potential energycurves.

The spectroscopic properties of theX 1S1 obtained inthe 4220~8! CAS calculations are all in fairly good agree-ment with previously calculated12,14,37–40and experimental4

results. The spectroscopic constants of theA 1P state areslightly more inaccurate than the corresponding propertiesfor the ground state. However, the calculated bond length isonly 0.013 a.u. longer than the experiment~0.006 a.u. forX 1S1!, the harmonic vibrational frequency is 18 cm21 toolarge, and the term energy is 0.116 eV too large. In general,the spectroscopic properties of the excited state agree with

TABLE I. The total energies for CO~X 1S1! and CO~A 1P! obtained in4220~8! complete active space calculations as a function of the internucleardistance~in a.u.!.

R E(X 1S1) E(A 1P)a

1.532 2112.408 642 2112.018 0771.732 2112.759 022 2112.371 8421.932 2112.900 683 2112.543 1772.032 2112.927 153 2112.588 5382.082 2112.933 061 2112.604 0772.132 2112.935 119 2112.615 7522.182 2112.933 964 2112.624 1412.232 2112.930 142 2112.629 7532.332 2112.916 296 2112.634 3502.532 2112.873 147 2112.626 1042.732 2112.821 151 2112.606 0982.832 2112.794 550 2112.594 3282.932 2112.768 396 2112.582 2023.132 2112.718 871 2112.558 3403.332 2112.674 385 2112.536 560

aTheE(A 1P) is obtained by adding the linear response excitation energy toE~X 1S1!.

TABLE II. The spectroscopic properties of theX 1S1 state of12C16O ascompared to previously calculated and experimental values.

MethodRe

~a.u.!ve

~cm21!Be

~cm21!ae

~cm21! Ref.

CASSCF 2.138 2202 1.92 0.0171 PWa

MCSCF 2.130 2206 1.94 0.0170 38CI 2.141 2184 12CI 2.153 2152 1.89 0.0177 14QCI-SD~T! 2.144 2147 0.0176 39MP4SDQ 2.138 2163 0.0182 37CASSCF 2.142 2162 0.0169 37MRD-CI 2.16 2210 40Exp. 2.1322 2170 1.93 0.0175 4

aPresent work.

TABLE III. The spectroscopic properties of theA 1P state of12C16O ascompared to previously calculated and experimental values.

MethodRe

~a.u.!ve

~cm21!Be

~cm21!ae

~cm21!Te

~eV! Ref.

CAS-LRa 2.347 1536 1.59 0.0200 8.184 PWCI 2.330 1504 8.21 12CI 2.362 1477 1.60 0.0216 8.276 14MRD-CI 2.34 1487 8.315 40Exp. 2.3344 1518 1.61 0.0232 8.068 4

aObtained from linear response calculations.

4144 Sundholm, Olsen, and Jo”rgensen: Transitions between the X 1S1 and A 1P states of CO

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previously calculated12,14,40 and experimental4 results. ThepresentTe of 8.184 eV is significantly closer to the experi-mental value than the most recent configuration interaction~CI!14 and multireference configuration interaction~MRD-CI! values of 8.276 and 8.315 eV, respectively. Judged fromthe degree of accuracy for the spectroscopic constants wemay conclude that theX 1S1 state is relatively well de-scribed in the present 4220~8! CAS calculations, and that it ispossible to provide accurate potential energy curves for theA 1P state by adding the vertical excitation energies to thetotal energy of the ground state. The vertical excitation ener-gies were obtained in 4220~8! complete active space linearresponse~CAS-LR! calculations.

IV. ONE-PHOTON TRANSITIONS

By using the multiconfiguration linear responsemethod,23 several excitation energies and the correspondingone-electron transition moments can be obtained simulta-neously with relative computational ease and quite high ac-curacy for small and medium sized molecules. Knuts,Vahtras, and Ågren41 reported recently transition energiesand oscillator strengths for azabenzenes obtained using theMCLR method. The main advantage with the response ap-proach is that once the correlation space of the referencestate~the ground state! is known, the excitation spectra andthe transition moments can be obtained without any furtherconsiderations of the nonorthogonality and the interactionsbetween the excited states. In CI calculations, the orthogo-nality condition for the states is also automatically fulfilled.However, to find an orbital basis in which all states are bal-ancedly treated is not an uncomplicated task.14,15 In MCSCFstudies of transition moments, orthogonality restrictions onthe excited states have often been introduced leading to aless accurate description of the excited state than of theground state. Malmqvist42,43 has recently shown that it isindeed possible to compute the transition moments effi-ciently between nonorthogonal MCSCF states also for mol-ecules such as pyrazine.44

In Table IV, the calculated transition moment function„^(x1 iy)/A2&… is compared with previously calculated andexperimental results. The present transition moment function

agrees well with the transition moment function reported byKirby and Cooper,15 while the calculated and experimentaltransition moment functions differ. The analytical expressionfor the experimental transition moment function~in Debye!by Field et al.6 is 7.4860.34 @120.68320.008

10.006~R/Å!#, and theexpression for the transition moment function~in Debye! byDeLeon9 as corrected in Ref. 15 is 14.3960.41@1–1.17760.022~R/Å!10.35060.013~R/Å!2#. The error barsfor the transition moment function by Field6 atR52.132 a.u.is 60.05 a.u. and for the transition moment function givenby DeLeon9 the uncertainty atR52.132 a.u. is60.25 a.u.The calculated transition moment functions are probablymore accurate than the measured ones.

The vibrationally resolved oscillator strength is obtainedfrom the vibrationally averaged transition moment (Dn9n8)as

f n9n85~2/3!gDEn9n8~Dn9n8!2 ~1!

g is the degeneracy factor which is 2 for theX→A absorp-tion process. The oscillator strength for then850, n950 bandof the X→A transition is given in Table V. The HF-LR~RPA! and CAS-LR oscillator strengths are obtained fromequation 1 using the transition moment functions in TableIV, the potential energy curves given in Table I, and theexperimentalDE of 8.068 eV. The HF value is about eighttimes smaller than the oscillator strength obtained in the4220~8! CAS calculation. The oscillator strength obtained byKirby and Cooper15 from CI calculations and using the ex-perimental Rydberg–Klein–Rees~RKR! potential energycurves is significantly larger than the present CAS result.However, the transition moment function and the spectro-scopic constants of the two states agree quite well with thepresent CAS calculations and with experimental results, in-dicating that the difference in the oscillator strengths appearsdue to the use of different methods to deduce the oscillatorstrength from the transition moment and energy data. Usingthe energy and transition moment data of Kirby and Cooper15

and the program for vibration–rotation spectrum of diatomicmolecules in theMOLCAS package,36 the oscillator strengthbecomes 1.2231022 which agrees better with the presentCAS result and with experiment. By using the experimental

TABLE IV. The one-photon dipole transition moment function for theX 1S1 to A 1P transition in CO obtained from CAS-LR calculations ascompared to previously calculated and experimental results~in a.u.!.

R PW Calc.a Exp.b Exp.c

1.532 0.8354 1.562 1.3141.732 0.9176 0.860 1.219 1.1011.932 0.7787 0.801 0.920 0.8882.132 0.6288 0.636 0.666 0.6752.332 0.5076 0.504 0.456 0.4632.532 0.4072 0.396 0.291 0.2502.732 0.3212 0.305 0.1692.932 0.2525 0.224 0.093

aInterpolated data from Ref. 15.bReference 9. A new fit to the experimental data quoted in Kirbyet al. ~Ref.15!.cReference 6.

TABLE V. The oscillator strength for the 0–0 vibrational band of theX 1S1

to A 1P transition as compared to literature values.

Method Oscillator strength Reference

HF-LRa 0.16031022 PWb

CAS-LRa 1.3131022 PWb

CI 1.5531022 15CI 1.2231022 15b,c

Exp. 0.96~14!31022 5Exp. 1.08~7!31022 3Exp. 1.1131022 2Exp. 1.5631022 6Exp. 1.1931022 6d

aA linear response calculation.bThe experimental energy difference of 8.068 eV is used.cRecalculated in this work using the data of Ref. 15.dRecalculated in this work using the energy data of Ref. 15 and the experi-mental transition moment function of Ref. 6.

4145Sundholm, Olsen, and Jo”rgensen: Transitions between the X 1S1 and A 1P states of CO

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transition moment function of Fieldet al.6 and the potentialenergy curves of Kirby and Cooper,15 we obtain an oscillatorstrength of 1.1931022 which is now in closer agreementwith the other experimental results. The transition momentsbetween vibrational levels are often estimated by multiplyingthe electronic transition moment by the overlap between thevibrational functions. The overlap between the 0–0 vibra-tional functions obtained in CAS-LR and CI15 calculationsare 0.323 and 0.318, respectively, yielding oscillatorstrengths of 1.6331022 and 1.6231022 from the CAS-LRand CI data, respectively. This shows that it is necessary tohave an accurate treatment of the vibrational averaging toobtain oscillator strengths of high accuracy for comparisonwith experiment. The present CAS-LR oscillator strength is1.3131022 as compared to the three experimental values of0.96~14!31022,5 1.1131022,2 and 1.08~7!31022.3

V. TWO-PHOTON TRANSITIONS

In the dipole approximation, using atomic units and thelength form of the interaction operator, the expression for thetwo-photon transition probability rate constantKi2 f is

Ki2 f5a04t0~2p!3a2v1v2d

TP. ~2!

In Eq. ~2!, a0 and t0 are conversion factors to atomic units~a050.529 177 2631028 cm/a.u., t052.418 884 4310217

s/a.u.!, a is the fine structure constant~a57.297 353 031023!, v1 andv2 are the photon energies~in a.u.!, anddTP is a rotationally averaged two-photon tran-sition moment~in a.u.!. Equation ~2! implicitly assume adelta function line shape associated with each~monomode!radiation field. Actual experiments involve fields with finitewidth line shape functions.45,46To be able to compare calcu-lated and measured two-photon transition probability rateconstants, this laser dependency must be removed. The ex-pression for the two-photon transition probability rateWif

~units of s21! can be written as

Wif5T2G~1,2!I 1I 2 , ~3!

where a function of the laser shape of the radiation fields,G(1,2) ~units of s!, was explicitly introduced, and whereI 1and I 2 are radiation fluxes of the two lasers. The notation ofRef. 45 was adopted for the two-photon cross sectionT2 ~incm4!

T25Ki2 f /t0 . ~4!

With the definitions of Eqs.~3! and ~4! it becomes evidentthat the laser independent two-photon cross sectionT2 ~ortheKi2 f rate constant!, introduced by McCannet al.,19,45 isthe rate constant deduced from the experiments that may becompared to the calculated rate constant@Eq. ~2!#. For detailssee also Ref. 47.

The rotationally averaged two-photon transition momentin Eq. ~2! is obtained as48

dTP5FdF1GdG1HdH , ~5!

dF5(a,b

SaaSbb* /30, ~6!

dG5(a,b

SabSab* /30, ~7!

dH5(a,b

SabSba* /30. ~8!

The coefficientsF, G, H depend on the polarization ofthe incident light beams. For linearly polarized laser beamswith parallel polarization planes we haveF5G5H52.48 Amore general expression for the polarization dependency oftheF, G, andH constants has been derived by Moccia andRizzo.49 The Sab factors ~a,b5x,y,z! are the two-photontransition moments in Cartesian coordinates and may be ob-tained from quantum mechanical calculations. The two-photon transition momentsSab are as the one-photon transi-tion moments strongly dependent on the vibrational motionsof both the initial and the final states, and must be vibra-tionally averaged for every specific vibrational level of theupper and lower state before comparison with experiment.

The expression for the two-photon transition momentbetween the initial state~often the ground state! and the finalstate derived using time-dependent lowest order perturbationtheory ~LOPT! in the dipole approximation is50

Sab5(j

^ i uau j &^ j u~b2^ i ubu i &!u f &~v j2v f1v1!

1^ i ubu j &^ j u~a2^ i uau i &!u f &

~v j2v1!. ~9!

In Eq. ~9!, ui & is the initial state~the reference!, u f & is thefinal state, andu j & are the intermediate states.a andb aredipole operators,v j andv f are the energies of the interme-diate and final states, respectively,v1 is the photon energy. Itis assumed thatv11v25v f . It should be emphasized thatthe LOPT maintains its validity where the photon energiesare far from the intermediate state resonance condition, i.e.,as long as the denominators in Eq.~9! do not become toosmall.

In the quadratic response formulation of Olsen andJo”rgensen23,24 as implemented by Hetemaet al.,26 the two-photon transition moment may be expressed in terms of thetwo linear response vectorsNa(v f2v1) andN

b(2v1) andthe eigenvectorsXf of the generalized linear response eigen-value equation@see Eqs.~10!–~13!#

Sab5E@3#1B@2#C1A@2#B1A@2#C, ~10!

E@3#52Nja~v f2v1!~

eEjkl@3#1eEjlk

@3#1v1eSjlk

@3#

2v feSjkl

@3#!Njb~2v1!Xkf , ~11!

B@2#C52Nja~v f2v1!

eb jk@2#Xkf , ~12!

A@2#B1A@2#C52Njb~2v1!~

ea jk@2#1eak j

@2#!Xkf . ~13!

We refer to Refs. 23, 24, and 26 where the details can befound about the definitions of the individual matrix elements.When the eigenvectorXf and the eigenvaluev f are calcu-lated the two sets of linear equations forNa(v f2v1) andNb(2v1) are determined and the two-photon transition ma-

4146 Sundholm, Olsen, and Jo”rgensen: Transitions between the X 1S1 and A 1P states of CO

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trix elementsSa,b are obtained. The solution of the two lin-ear equationsNa(v f2v1) andN

b(2v1) has the effect ofreplacing the explicit summation over the complete set ofintermediate states in Eq.~9!.

The two-photon transition probability rate constant fortheX 1S1 to A 1P transition has been measured by Ferrellet al.18 They report a value of 5310211 cm4W22 s21 for therate constantk. The rate constantk is laser dependent andrelated to the laser independent rate constantKi2 f by

Ki2 f5khn1hn2t0 /G~1,2! . ~14!

In Eq. ~14!, G(1,2) is a laser shape function of 1.5310212 s,t0 is the conversion factor for time~in s! to atomic units, andhn1 andhn2 are the photon energies. In the CO experimentonly one laser was used for the excitation, i.e.,hn15hn256.5813310219 J, which corresponds to al of301.83 nm. Ferrellet al.18 do not give any value for the lasershape functionG(1,2). However, McCannet al.

19 claimedthat the laser bandwidth was the same as for the lasers usedin a two-photon experiment on O2. The present value for

G(1,2) is deduced from the O2 data of Ref. 19. By usingk55310211 cm4W22 s21, G(1,2)51.5310212 andhn15hn256.5813310219 J, the experimental laser inde-pendent transition probability rate constant,Ki2 f , becomes3.5310252 cm4 s. The experimentalKi2 f value18 for theX–A transition becomes then about 104 times larger than thereported experimental rate constants~Ki2 f5T2t0! for O2.

19

The reason is not understood.The calculated two-photon transition moments obtained

in Hartree–Fock quadratic response~HF-QR! and 4220~8!complete active space quadratic response~CAS-QR! calcu-lations are given in Table VI as a function of the internucleardistance. The two nonvanishing two-photon transition mo-ments obtained in the CAS-QR calculations decrease mo-notonously with increasing internuclear distance until the en-ergy difference between the two states becomes close to thelaser frequency. A comparison with the two-photon transitionmoments obtained in HF-QR calculations shows that the cor-relation contribution varies between20.2 a.u. atR51.532a.u. and 0.5 a.u. atR5Re52.132 a.u. For largerR values theHF-QR values for the transition moments become unreliablebecause the energy splitting approaches the laser frequency.The correlation contribution is not very large, but it is ofsame order of magnitude as the transition moment in thevicinity of the equilibrium bond length resulting in signifi-cant changes in the transition probability rate constants. Theremaining correlation contribution of the 4220~8! CAS-QRcalculations was estimated by performing a 7441~8!CAS-QR calculation atR52.132 a.u. In Table VII, the tran-sition probability rate constantsKi2 f obtained in the HF-QR,the 4220~8! CAS-QR, and the 7441~8! CAS-QR calculationsat R52.132 a.u. are compared. TheKi2 f of 2310255 cm4 sobtained in the HF-QR calculation is about 1 order of mag-nitude larger than theKi2 f of the 4220~8! CAS-QR calcula-tion, while the 7441~8! CAS-QR value forKi2 f of 5310256

cm4 s is 2.5 times larger than the 4220~8! CAS-QR value.The qualitative agreement between theKi2 f values obtainedin the HF-QR and the two CAS-QR calculations shows that

TABLE VI. The two-photon dipole transition moment function for theX 1S1 to A 1P transition in CO obtained in the HF and CAS calculations~in a.u.!.

R Sx,z~HF! Sz,x~HF! Sx,z~CAS! Sz,x~CAS!

1.532 4.1988 4.5643 4.0106 4.38291.732 3.1112 3.7295 3.0493 3.58611.932 0.3566 1.0981 0.8161 1.42822.032 0.2615 0.73432.082 0.0915 0.47542.132 20.5525 20.2164 20.0228 0.27052.182 20.0980 0.10512.232 20.1430 20.02332.332 20.6333 20.6868 20.1708 20.19962.532 20.4842 20.6009 20.1325 20.33062.732 20.3268 0.4441 20.0912 20.30332.832 20.0857 20.26342.932 20.0921 20.22383.132 20.2341 20.2951

TABLE VII. The rotationally and vibrationally averaged two-photon transition moments~dTP in a.u.! and thecorresponding two-photon transition probability rate constants~Ki2 f in cm

4 s! for X 1S1 to A 1P transitions inCO as compared to experimental results.

Methoda dTP b Ki2 f Ref.

HF-QRc 3.9431022 2.02310255 PWCAS-QR„4220~8!…c 4.0931023 2.09310256 PWCAS-QR„7441~8!…c 9.9731023 5.10310256 PWCAS-QR„4220~8!…~n950, n850! 1.2431024 6.32310258 PWCAS-QR„4220~8!…~n950, n851! 1.4131025 7.19310259 PWCAS-QR„4220~8!…~n950, n852! 1.4131024 7.19310258 PWCAS-QR„4220~8!…~n951, n850! 1.3631023 6.97310257 PWCAS-QR„4220~8!…~n951, n851! 7.1131024 3.64310257 PWCAS-QR„4220~8!…~n951, n852! 1.5231024 7.79310258 PWCAS-QR„4220~8!…~n952, n850! 2.8631023 1.46310256 PWCAS-QR„4220~8!…~n952, n851! 2.0031024 1.02310257 PWCAS-QR„4220~8!…~n952, n852! 1.8331024 9.36310258 PWExperiment~n950, n851! 3.5310252 17,18,19

aDouble prime denotes the vibration level of the1S state.bv15v250.142 85 a.u. is used.cSingle point value calculated atR52.132 a.u.

4147Sundholm, Olsen, and Jo”rgensen: Transitions between the X 1S1 and A 1P states of CO

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the transition probability rate constants obtained in the4220~8! CAS-QR calculations are of the correct order ofmagnitude, and reasonable estimates for the transition prob-ability rate constants between vibrational levels can be de-duced from the 4220~8! CAS-QR data. In Table VII, thevibrationally averaged rate constants are given for the threelowest vibrational levels of theA 1P and theX 1S1 states.The transition probability rate constant for the@1S1~n950!,1P~n851!# transition calculated on the 4220~8! CAS level isabout 53106 times smaller than the experimental result. Thetwo-photon transition moment functions change sign in thevicinity of the equilibrium bond distance of the ground state.The vibrationally averaged transition moments of then950transitions become therefore very sensitive to the shape ofthe potential energy curves. By using the potential energydata of Kirby and Cooper15 and the present two-photon tran-sition moment functions, the two-photon transition probabil-ity rate constant for the~n950, n851! transition became5.2310258 cm4 s as compared to the value of 7.2310259

cm4 s obtained using the CAS and CAS-LR potential energydata and the same transition moment functions. However, therate constants of the other transitions reported in Table VIIare less sensitive to the shape of the potential energy curves.For example, the present value for the rate constant for the~n951, n850! transition is 7.0310257 cm4 s, while a value of5.8310257 cm4 s is obtained using the Kirby and Cooperpotential energy curves and the present two-photon transitionmoment functions.

The relative uncertainty of the single point CAS-QRtransition probability rate constant is estimated to be about2.5, i.e., the relative difference between theKi2 f values ofthe two CAS calculations. The uncertainty of the~n950,n851! transition probability rate constant introduced by thevibrationally averaging is about a factor of 10. However, theerror bars of the calculated rate constant are 5 to 6 orders ofmagnitude smaller than the discrepancy between the mea-sured and calculated values for the transition probability rateconstants. We conclude that the measured rate constant isincorrect and that the two-photon experiment of theX–Atransition in CO should be reconsidered. Preliminary calcu-lations on O2 also show that the experimental rate constantsfor the O2 transitions reported by McCannet al.19 are moreaccurate than the experimental CO rate constants.18

VI. DIPOLE MOMENTS FROM QUADRATIC RESPONSEFUNCTIONS

The dipole moment of excited states and the one-photondipole transition moment between excited states can also bedetermined by using the quadratic response method. In thetime-dependent LOPT approach, the two-photon transitionmoment is formally obtained as the sum of the dipole tran-sition moment between the initial state and the intermediatestates multiplied by the dipole transition moment betweenthe intermediate state and the final state, divided by an en-ergy difference, see Eq.~9!. The solution of the two sets oflinear equationsNa(v f2v1) andN

b(2v1) gives a compu-tationally simple way of determining the contributions fromall the intermediate states. However, the individual contribu-tion from specific intermediate states may also be calculated

from response theory. Linear response calculations providethe excitation energies and the one-photon dipole transitionmoments between the ground state and the intermediatestates in question, and from a quadratic response calculationthe one-photon dipole transition moments between the inter-mediate states and the final state can be calculated. For thecasej5 f in Eq. ~9!, the contribution is proportional to thetransition moment between the initial and final state multi-plied by the difference between the dipole moments of thefinal and initial states. Once the dipole transition momentand the ground state dipole moment are known, the dipolemoment of the final state can be deduced from expression~9!.

The calculated dipole moments of theX 1S1

@m~X 1S1!# and theA 1P @m~A 1P!# states are in Table VIIIcompared to previously calculated and experimental values.The ground state dipole moment obtained in the 4220~8! and7441~8! CAS calculations differ by only 0.005 a.u. The4220~8! CAS value form~X 1S1! calculated atR52.132 a.u.is 20.0770 a.u. as compared to the experimental value of20.048 a.u.51 The dipole moment of theA 1P state is alsoclose to zero.52 The measurement ofm~A 1P! was based onthe linear Stark effect, and the sign ofm~A 1P! could there-fore not be deduced from the experimental data. The presentm~A 1P! obtained from the 4220~8! CAS quadratic responsecalculation atR52.332 a.u. is20.0817 a.u. and a 6331~8!CAS-QR calculation yields a value of20.0441 a.u. form~A 1P!. The equilibrium bond length for theA 1P state is2.334 a.u. The CAS-QR value form~A 1P! is about as largeas the experimental value but with the opposite sign. Accord-ing to the present CAS-QR calculations on CO, theA 1Pstate has also the anomalous polarity C2O1. All previouslycalculated values form~A 1P! are positive.12,14Them~A 1P!obtained by Cooper and Kirby14 is also very small but posi-tive ~0.073 a.u.!. They considered the good agreement withexperiment perhaps fortuitous because the wave function didnot include external correlation effects which have beenshown to be important in obtaining accurate dipolemoments.53

TABLE VIII. The dipole moments of theX 1S1 andA 1P states of CO ascompared to previously calculated and experimental values~in a.u.!.a

Method R m~X 1S1! R m~A 1P! Ref.

HF1QRb 2.132 0.1041 2.132 20.1605 PW4220~8!CASSCF1QRb 2.132 20.0770 2.132 20.2816 PW7441~8!CASSCF 2.132 20.0720 PW4220~8!CASSCF1QRb 2.332 0.0559 2.332 20.0817 PW6331~8!CASSCF1QRb 2.332 0.0794 2.332 20.0441 PWMCSCF 2.132 20.0931 38CI 2.132 20.049 2.335 0.23 12MCRPAc 2.132 20.130 2.132 0.212 13CI 2.132 20.126 2.35 0.073 14MP4SDQ 2.138 20.047 37CASSCF 2.142 20.13 37Exp. 2.132 20.048 2.35 60.059~20! 51.52

aThe positive sign means C1O2. 1 a.u.52.541 75 D.bm~X 1S1! is obtained from a CAS calculation andm~A 1P! is obtainedfrom a quadratic response calculation.cA multiconfiguration random phase approximation calculation.

4148 Sundholm, Olsen, and Jo”rgensen: Transitions between the X 1S1 and A 1P states of CO

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The HF-QR and 4220~8! CAS-QR values for the dipoletransition moments between the first few1S1 and1P statesare given in Table IX. The transition moments are calculatedat R52.132 a.u. using the linear and quadratic responsemethods. The correlation contributions are for most of themsmall. However, the transition moment for theB 1S11F 1S1 is halfed by electron correlation effects. Aswe saw previously, though the electron correlation effects onthe transition moments are small in the single point calcula-tions, the correlation effects on the oscillator strengths of thetransitions between vibrational levels may become large. Forthe X 1S1–A 1P transition the transition moment atR52.132 a.u. is hardly at all affected by electron correlationeffects, but the HF-LR and 4220~8! CAS-LR values for theoscillator strengths of the 0–0 transition differ by a factor of8 ~see Table V!. Most of the CAS-LR values for the transi-tion moments given in Table IV agree well with the CI resultby Kirby and Cooper.15 The largest discrepancy is found fortheX 1S1–B 1S1 transition for which the CAS-LR transi-tion moment is about six times larger than the CI result. Thepresent HF-QR~or RPA! values for the transition momentsbetween the excited states given in Table IX also agree wellwith the RPA transition moments calculated by Sengelo”v andOddershede54 using a smaller basis set.

VII. CONCLUSIONS

TheX–A transition in CO has been studied by means ofmulticonfiguration linear and quadratic response methods.The spectroscopic constants of the ground and excited stateare in a good agreement with previously calculated and ex-perimental results. The potential energy curve of theA 1Pstate is obtained by adding the vertical excitation energy tothe ground state energy. The excitation energy is obtainedusing the linear response method. The one-photon dipole

transition moment function for theX–A transition has alsobeen obtained using the linear response method. The transi-tion moment function agrees with the transition momentfunction calculated by Kirby and Cooper15 using a CImethod. The calculated transition moment functions areprobably more accurate than the transition moments func-tions deduced from experimental data.6,9 The oscillatorstrength for the 0–0 vibration band of theX–A transitionwas deduced from the transition moment function and thepotential energy curves. The calculated oscillator strength of1.3131022 is slightly larger than the experimental oscillatorstrengths of 1.1131022, 1.08~7!31022, and 0.96~14!31022

measured by Hesser,2 Carlsonet al.,3 and Lee and Geust,5

respectively. The previously best theoretical value for theoscillator strength was calculated by Kirby and Cooper15

who reported an oscillator strength of 1.5531022. They de-duced it from a CI transition moment function and the ex-perimental RKR potential energy curves.

The two-photon dipole transition between theX 1S1

andA 1P states has been studied using the multiconfigura-tion quadratic response method. The calculated two-photontransition dipole moments are vibrationally and rotationallyaveraged. The two-photon transition probability rate con-stants for the few lowest vibrational levels are given. Thecalculated two-photon transition probability rate constant forthe ~n950, n851! band of theX–A transition is 7.19310259

cm4 s while the corresponding experimental laser-independent rate constant is 3.5310252 cm4 s. The calculatedand measured values for the rate constants differ by a factorof about 106. We recommend that the measurement should bereconsidered.

The dipole moment of theA 1P state is deduced fromquadratic response functions. Them~A 1P! obtained in a6331~8! CAS-QR calculation is found to be small and nega-tive ~20.0441 a.u.! i.e., suggesting that the CO molecule intheA 1P state also has an anomalous polarity C2O1.

ACKNOWLEDGMENTS

The research reported in this article has been supportedby grants from The Academy of Finland, the Swedish Natu-ral Science Research Council, the Nordic Council of Minis-ters, and the Alexander von Humboldt Stiftung.

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TABLE IX. The transition momentsa between the first few1S1 and 1Pstates of CO calculated atR52.132 a.u. as compared to previously calcu-lated values~in a.u.!.

Transition HFb CASb RPA ~Ref. 54! CI~Ref. 15!

X 1S1–B 1S1 0.4461 0.3986 0.452 0.061X 1S1–C 1S1 0.5140 0.5895 0.502 0.462X 1S1–F 1S1 0.4455 0.3777 0.418X 1S1–A 1P 0.6282 0.6288 0.627c 0.636X 1S1–E 1P 0.3470 0.3690 0.344 0.308B 1S1–C 1S1 2.9892 3.1341 2.5000 2.819B 1S1–F 1S1 0.3762 0.1853B 1S1–A 1P 0.5497 0.6173 0.5561 0.875B 1S1–E 1P 3.6304 3.7011 4.821C 1S1–F 1S1 3.6855 3.8618C 1S1–A 1P 0.0733 0.1165 0.1930 0.347C 1S1–E 1P 1.1602 0.8579 0.585F 1S1–A 1P 0.2595 0.3129F 1S1–E 1P 1.2126 1.0986A 1P–E 1P 0.4314 0.4291 0.207

aThe sign of the transition moments is redundant. All transition moments aregiven with positive sign.bLinear response calculations for transitions between the ground and excitedstates and quadratic response calculations for transitions between the ex-cited states.cThe multiconfiguration RPA value of Ref. 13 is 0.636 a.u.

4149Sundholm, Olsen, and Jo”rgensen: Transitions between the X 1S1 and A 1P states of CO

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