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Molecular PhysicsAn International Journal in the Field of ChemicalPhysicsPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713395160

Singlet-triplet transitions in three-atomic moleculesstudied by time-dependent MCSCF and densityfunctional theoryBoris Minaev a; Ingvar Tunell a; Pawel Salek a; Oleksandr Loboda a; Olav Vahtrasa; Hans Ågren aa Laboratory of Theoretical Chemistry, The Royal Institute of Technology, SE-10691,Sweden

Online Publication Date: 10 July 2004

To cite this Article: Minaev, Boris, Tunell, Ingvar, Salek, Pawel, Loboda, Oleksandr, Vahtras, Olav and Ågren, Hans(2004) 'Singlet-triplet transitions in three-atomic molecules studied by time-dependent MCSCF and density functionaltheory', Molecular Physics, 102:13, 1391 — 1406

To link to this article: DOI: 10.1080/00268970410001668435URL: http://dx.doi.org/10.1080/00268970410001668435

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Singlet–triplet transitions in three-atomic moleculesstudied by time-dependent MCSCF and density

functional theory

BORIS MINAEV, INGVAR TUNELL, PAWEL SALEK, OLEKSANDR LOBODA,OLAV VAHTRAS and HANS AGREN*

Laboratory of Theoretical Chemistry, The Royal Institute of Technology,SE-10691, Sweden

(Received 8 December 2003; accepted 15 December 2003)

Singlet–triplet transition moments and phosphorescence lifetimes have been calculated for thethree-atomic molecules HCN, O3, H2O, H2S, GeF2, GeCl2 and GeBr2 by time-dependentdensity functional theory (DFT) utilizing quadratic response functions in order to qualifyDFT which recently has become available for studies of this kind [TUNELL, I., RINKEVIVIUS, Z.,VAHTRAS, O., SALEK, P., HELGAKER, T., and AGREN, H., 2003, J. chem. phys., 119, 11024].Comparison with ab initio and experimental data indicates that DFT exhibit results of similarquality as explicitly correlated methods which indicates that it indeed is a viable approach forsinglet–triplet transitions. O3 provides an intriguing example in that a systematic investigationof the singlet–triplet transition moment of its Wulf band indicates a clear advantage of theDFT technique despite the multiconfigurational character of the electronic structure of thismolecule. The electronic spin–spin coupling and the hyperfine nuclear coupling constants havealso been calculated in order to further characterize the triplet state in the spectra of theinvestigated systems.

1. Introduction

Triplet excited states of chemically stable moleculesare very important not only for spectroscopic andphotochemical properties but also in chemical reactivityand catalysis [1–3]. Spin pairing is a well-knownattribute of a covalent chemical bond, leading to aground state that has a zero spin and a singlet spinmultiplicity (S0 state) of many stable covalently boundcompounds. Light absorption induces an electronexcitation, e.g. HOMO to LUMO, without spin flip,leading to an excited state that also is a singlet (S1). Thefirst excited triplet state (T1) is always lower in energythan the S1 state in such molecules, since the HOMO–LUMO exchange integral is positive [4]. The S1!T1

non-radiative transition between close lying excitedstates is often more probable than the radiativetransition S1!S0 (fluorescent fast emission) and alarge part of the excited molecules are accumulated inthe metastable triplet state. These have a relatively longlifetime because the T1!S0 transition is spin-forbidden,but sooner or later they have to spontaneously emit, thatis afterglow or phosphoresce. Terenin [5] and later on

Lewis and Kasha [6] thus interpreted the observedphosphorescence of many complicated organic com-pounds in solid solvents as due to T1!S0 transitions.

In small molecules (like H2O, HOCl, BrNO) thetriplet excited states are often dissociative [3, 7, 8]and the singlet–triplet (S–T) transition cross-section istherefore often an important characteristic of thephotodissociation process [9]. At the same time otherthree-atomic molecules have bound triplet excited states(O3, NO�2 , HCN, HSiCl, GeCl2 etc.) [10]. For simplepolyatomic molecules only a few optical transitionsto such states have been observed [8–13]. Instead, thereare many electron-impact measurements of the tripletexcited states of stable small molecules in the gas phase[14–16]. In such experiments the spin-forbidden char-acter of the S–T excitation is removed by interactionwith the spin of the incoming electron; thus S–S as wellas S–T transitions are observed in the energy-lossspectrum of impact electrons scattered at differentangles [14].

In optical absorption measurements the very weakintensity of the T1 S0 transition is a difficult subject forabsolute intensity calibration.Measurements of radiativetransitions have mostly been carried out under bulk con-ditions, being affected by self-absorption, collision- or

*Author for correspondence. e-mail: agren@theochem.kth.se

MOLECULAR PHYSICS, 10 JULY 2004, VOL. 102, NO. 13, 1391–1406

Molecular Physics ISSN 0026–8976 print/ISSN 1362–3028 online # 2004 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals

DOI: 10.1080/00268970410001668435

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lattice-induced non-radiative quenching, or by solventeffects. Furthermore, the main body of data forintensities of singlet–triplet transitions tabulated inthe literature [4] has been deduced from phosphores-cence measurements in solid solvents at 77K in whichthe emitter spin state distribution cannot be wellcharacterized.It follows from the survey above that information on

intensity for the S–T transitions is very scarce even forsmall molecules [10]. At the same time knowledge of thisprocess is important for many areas of applications [8,10, 14, 17–21] and also for a deeper understanding ofelectronic structure of molecular excited states [22–25].Triplet states have three spin sublevels which are spliteven in the absence of an external magnetic field andeach spin sublevel has its own radiative transitionprobability to the ground S0 state [26]. Vibronic patternsof the S–T transitions also provide rich informationabout spin-selective processes and about the dependenceof spin-orbit coupling (SOC) on vibrational movement[27, 28]. Thus much more important observables are ingeneral available for spectroscopy of S–T transitionsthan for the ordinary singlet states [28].So far, analysis of the complicated S–T absorption

and emission spectra of small three-atomic systemshas been performed at the ab initio level only for fewmolecules [8, 11, 29, 30]. The purpose of the presentpaper is to qualify the recently developed time-dependent density functional theory (TD-DFT) thatencompasses singlet–triplet transitions and phosphores-cence [31]. While the main motivation for TD–DFT isthe applicability toward large species for which explicitlycorrelated methods are too costly or where the standardHartree–Fock based method show singlet–triplet insta-bilities, we focus in this work on small molecules as theyconstitute testbeds with more abundant experimentaldata as well as ab initio data. The present paper providesS–T transition probabilities to separated spin-sublevelscalculated for two low-lying triplet states of hydrogencyanide, ozone and germanium dihalides by the quad-ratic response (QR) technique based on the CASSCF[32] and DFT [33] methods. A comparative study ofT1 S0 absorption in the H2O and H2S molecules isalso presented. The spin splittings (ZFS and spin-rotational coupling parameters) for the stable tripletstates are calculated by the CASSCF method [34–36] inorder to further characterize the triplet states.

2. Method of calculations

The effect of perturbations can be formulated in termsof response functions. Photon absorption is in lowestorder caused by the interaction between its electricfield and the dipole moment of the molecular charge

distribution, and a singlet–triplet transition is accord-ingly spin-forbidden. The presence of spin–orbit coup-ling destroys the spin symmetry and a non-vanishingtransition amplitude is obtained which may be calcu-lated as the residue of a quadratic response functioninvolving the combined perturbations of the electricdipole and the spin–orbit interactions [37]. Shortly, thesinglet–triplet transition intensities can be derived fromthe following three steps in calculating DFT responsefunctions (see [33] for details).

Given a model functional E½��; ��� and parametriza-tion of the Kohn–Sham (KS) determinant

j~00i ¼ e���j0i; ��rs ¼Xrs�

�rsayr�as� ð1Þ

the ground state densities are determined by solving theKS equations [38].

�E ¼ h0j½���; HH�j0i ¼ 0, ð2Þ

HH ¼X�

Zd�CC�ðrÞ

y �E���ðrÞ

���ðrÞ; ð3Þ

CC�ð ~rr Þ ¼Xk

�kðrÞak�: ð4Þ

The solutions, the KS orbitals �k, form a representationof the ground-state density of the real system

�� ¼ h0jCCðrÞy�CC�ðrÞj0i � h0j��ðrÞj0i: ð5Þ

In the presence of a perturbation VV , HH þ VV (cf.equation (3)) is now the Hamiltonian of the non-interacting system; it governs the time-evolution of theKS orbitals. The ground state is expanded in a powerseries in the perturbation strength,

�� ¼ �ð0Þ� þ �ð1Þ� þ �ð2Þ� . . . ð6Þ

and we assume that the zero-order ground-state �-and � densities are equal. Interpreting equation (1) as aparametrized time-evolution operator acting on theconverged KS determinant, we apply a time-dependentvariational principle (based on the Ehrenfest theoremfor an operator Q) to solve response equations for thenth-order parameters ��ðnÞ

�nh0j Q; e�� HH½��� þ V� i �hh@

@t

� �e���

� �j0i ¼ 0: ð7Þ

1392 B. Minaev et al.

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The nth order response functions, i.e. the nth-orderchanges of a property, may then be identified from theexpansion of a property A

A ¼ hAAi þ h0j½��ð1Þ; AA�j0i þ þh0j½��ð2Þ; AA�j0i

þ1

2h0j½��ð1Þ; ½��ð1Þ; AA��j0i . . . : ð8Þ

The quadratic response function in the frequencydomain is given by

hhA;V;Vii!1;!2¼h0j½��!1;!2 ; AA�j0iþP12h0j½��

!1 ;½��!2 ; AA��j0ið9Þ

and the singlet–triplet transition amplitude is evaluatedas the residue at a triplet state excitation energy with thedipole operator as the primary property and the spin–orbit operator as the perturber [37] (X is an arbitrarytriplet operator)

hSjrjTi ¼ lim!2!!T

ð!2 � !TÞhhr;Hso;Xii0;!2

hSjXjTi: ð10Þ

This formalism is implemented in the DALTONprogram [39] and utilized in the present work.The zero-field splitting (ZFS) parameters of the low-

lying triplet states of HCN, O3 and GeCl2 moleculeshave in this work been calculated as expectation valuesof spin–spin coupling (SSC) by the CASSCF techniquewhich has been discussed recently [40]. The zero-fieldsplitting in the excited triplet states of polyatomicmolecules is described by the tensor Dij, which is widelyused to analyse EPR and ODMR spectra. It defines theeffective spin Hamiltonian given by the scalar productsof the following terms [5]:

HS ¼Xij

Di;jSiSj; ð11Þ

where Si is the ith Cartesian component of the totalelectron spin operator. In a coordinate system x, y, zwith the ZFS eigenfunctions the effective spinHamiltonian can be written as

HS ¼ �XS2x � YS2y � ZS2z; ð12Þ

where Dx;x ¼ �X , Dy;y ¼ �Y and Dz;z ¼ �Z. Sincethe Di,j is a traceless and symmetric tensor (X þ YþZ ¼ 0) which is diagonal in its principal axes system it

can be described by only two independent parameters, Dand E [5]

HS ¼ D S2z �1

3S2

� �þ EðS2x � S2yÞ: ð13Þ

In equation (13) the choice of axes is such that thej Tzi eigenfunction describes the spin sublevel with thelargest splitting; that is, (j Z j > j X j, j Y j). In that casethe z axis is the main axis of the ZFS tensor, equation(13), where

D ¼ �3

2Z; E ¼

1

2ðY �XÞ: ð14Þ

The spin–spin contributions to the the splitting ofthe lowest triplet state has been calculated using theDALTON program [39], while the calculations of thespin–orbit contributions, obtained by diagonalizing thespin–orbit Hamiltonian, employed the MOLCAS [41]code where spin–orbit coupling integrals are estimatedby employing the atomic mean-field approximation [42].

For calculations of the HFC constants the unrestrictedBecke’s three-parameter hybrid-exchange-correlationfunctional (UB3LYP) functional [43] implemented inthe Gaussian 98 program [44] was used together with thedouble-zeta basis set EPR-II and the triple-zeta basis setEPR-III [45], which were specifically optimized for com-puting HFCs. The isotropic HFC constant, a, is relatedto the spin density at the corresponding nucleus by

a ¼ ð8p=3ÞgegN�e�NjC0j2, ð15Þ

where ge is the electron gyromagnetic ratio, �e is theelectron (Bohr) magneton, �N is the nuclear magneton,gN is the nuclear gyromagnetic ratio of the correspond-ing nucleus, and jC0j

2 is the unpaired electron spindensity at the nucleus.

The spin dipole coupling tensors are characterized byanisotropic interaction and can be described by thehyperfine tensor:

TijðNÞ¼gegN�e�N

XP��� h’jr

�5kNðr

2kN�ij�3rkNirkNjÞj’i,

ð16Þ

where P��� is an element of the spin density matrix.

The components of the total hyperfine tensor, A, areobtained as a sum of isotropic HFC constant andanisotropic electron-nuclear spin dipole coupling:Aii ¼ aþ Tii.

3. Results and discussion

We start by analysing the HCN molecule. Almostall physical properties of this species have been studiedby theory but still not the singlet–triplet spectra [25],

Singlet–triplet transitions in three-atomic molecules 1393

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despite that they serve as most intriguing predictions forsuch a benchmark species with importance for astron-omy and laser physics. Subsequently, we compareMCSCF QR and DFT QR results for singlet–triplettransition intensities in ozone, another benchmarkmolecule in atmospheric photochemistry and spectros-copy [18, 46]. The T1 S0 photoabsorption by waterand sulphur dihydride will be theoretically considerednext. Finally, the weak band system in the visible regionof GeF2, GeCl2 and GeBr2 emission [47] will beinterpreted by the DFT QR method as an examplewith a heavier element involved.

3.1. HCN moleculeHydrogen cyanide (HCN) is a poison; at the same

time it is a benchmark molecule in quantum chemistryand spectroscopy [10, 15, 19, 25, 48–53] with a hugenumber of studies of its ground-state properties beingpublished. For a review one can recommend, forexample, [54]. Compared with the electronic groundstate, only a few papers have been devoted to theelectronically excited states, but this area is growingvery fast [25, 52, 53, 55]. The first investigation ofthe absorption spectra of HCN refered to the vacuumultraviolet (UV) region [56]. Very intense diffuse bandswere found near 112 nm and sharp bands in the region155–170 nm; a much weaker band system in the range170–200 nm was studied in detail by Herzberg [10].A renewed interest in the excited states of the HCNmolecule was sparked by the central role of hydrogencyanide in the CN laser system [17]. Later on it wasfound that HCN is an important intermediate and finalproduct for molecular formation in dense interstellarclouds [15, 19, 48]. Low-lying electronic states of HCNcan provide channels for cooling down electrons in boththese situations; these states may be intermediates inpredissociation which produce electronically excited CNradicals [18]. The fluorescence spectrum of HCN(A1A00!X1�þ) has been reported [57], but phosphores-cence has never been observed [10]. Nevertheless, thelowest triplet state of hydrogen cyanide are predicted [49]to be physically stable. They can be populated in inter-stellar clouds via inverse predissociation [15] and a searchfor them should therefore be of astrophysical interest.The observed optical absorption spectra of HCN

starts near 200 nm and consists of a progression ofdiffuse peaks serving as a classical example of predis-sociation in polyatomic systems [10]. The first two S–Sabsorption bands (between 193 and 160 nm) arerelatively weak and were assigned by Herzberg [10]to A1A00 and B1A00 states. The assignment of thefirst band is now firmly established by numerousab initio calculations [24, 25, 49, 52, 53, 55, 58]. The

predissociation features strongly depend on rovibra-tional state quantum numbers and the particularities ofthis state have been the subject of detailed studies of thedynamics of the dissociated fragments [25, 52, 53, 55,58]. At the same time the low-lying triplet excited statesof HCN have not received much attention [10, 49]; asfar as we know, there exists no studies for these statesin optical spectroscopy. Considering the great interest inthe spectra of the HCN molecule as a benchmark speciesin spectroscopy of polyatomic molecules [10, 53] and itsimportance in astrophysical applications [48] we decidedto predict the S–T transition probability in order toprovide an easier spectroscopic search of these bands.

Ab initio calculations [24, 49, 53, 55, 58] illustrate thecomplexity of the low lying electronic excitations ofhydrogen cyanide and our results (tables 1–4) reproducethe main features of the spectrum. The molecule is linearin the ground state X1�þ with the main electronconfiguration . . .ð5sÞ2ð1pÞ2, where the ð5sÞ molecularorbital (MO) partly has some lone pair character as anitrogen 2pz atomic orbital (AO) and partly some C–Nbonding character. After some precalculations we endedup with the use of the following complete active spaces(CAS) in the MCSCF calculations of the HCN excitedstates: in the Cs point group we have 4 orbitals frozenand 4 orbitals active in a0 symmetry; then 2 orbitals ofthe a00 symmetry are active. Thus we have 6 activeelectrons in 6 MOs in this, the smallest, active space(CAS-1). A similar CAS includes one additional a0

active orbital (CAS-2). The largest CAS includes 8active electrons on 9 MOs: 3 frozen and 6 active MO’s inthe a00 symmetry and 3 orbitals in the a00 symmetry areactive (CAS-3). Comparison of MCSCF QR results inthese three types of CAS calculations are given in table2. Since the results are not very different we used CAS-2for the main body of calculations of the S–T transitionmoments at different geometries.

The first excited triplet state T1 correlates with the3�þ symmetry in the linear HCN structure, which isproduced mainly by a 1p!2p excitation—a typical pbonding to p* antibonding transition. The T1 state isstabilized by the bending mode and has a minimum atthe angle ffHCN¼ 124�. Because of the antibonding pp*character the C–N bond is weaker in the T1 triplet stateand has a longer distance rCN¼ 1.315 A. This bent statehas a3A0 symmetry. The !2 bending mode has muchhigher frequency (1510 cm�1) in the triplet a3A0 statethan in the ground state (756 cm�1). All results are hereobtained from our MCSCF geometry optimization inCAS-2 (Table 4).

The angle dependence of the three low lying states(X1A0; a3A0 and b3A00) were calculated by the MCSCF(figure 1) and the B3LYP DFT (figure 2) methods.Properties calculated byMCSCF and DFT are presented

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in tables 1, 3 and 4 together with results of othertheoretical and experimental studies. The angle depen-dences (figures 1–4) were obtained for fixed bondlengths: rCH¼ 1.093 A, rCN¼ 1.327 A, as optimized inthe 3A00 state by the MCSCF method with CAS-2. Thesebond lengths are quite similar for the T1 and T2 states,thus we used them for the study of angle dependenceof the total energy (figures 1 and 2) and of the S0–Tn

transition moments (figures 3 and 4). The DFT methodslightly underestimates the transition energy in theregion of the equlibria of the triplet states (figures 1

and 2) and the lowest triplet state a3A0 is more tightlybound with respect to the bending mode in the DFTmethod.

The DFT B3LYP calculations provide electric dipolemoments in the ground X1�þ state and in the a3A0 statethat are equal to 2.98 and 1.69D, respectively; in theCI method of [49] they are equal to 3.01 and 1.90D,

Table 1. Vertical transitions from the ground X1�þ state of the HCN molecule at the experimental geometry (rCN ¼ 1.156 A,rCH ¼ 1.064 A, angle HCN ¼ 180�); cc-pVTZ basis set. M is the transition moment (au), f is the oscillator strength and �p isthe phosphorescence radiative lifetime (s).

State Methodd �E/eV My;zðTz;yÞ f �p

a3�þ ( 3A0

) DFT 5.80 0.000181 9.31� 10�9 0.22

a3�þ ( 3A0

) MCSCF 6.24 0.000199 12.2� 10�9 0.15

a3A0

expera 5.46;5.6;5.83b –

b3�( 3A0 0

) DFT 6.61 0.000219 1.55� 10�8 0.10

b3�( 3A0 0

) MCSCF 7.48 0.000268 2.63� 10�8 0.05

b3A0 0

5.99;6.8;6.18b –

b3A0 0

experc 6.0 –

A1��( 1A0 0

) MCSCF 8.78 0 0 –

A1��( 1A0 0

) DFT 7.80 0 0 –

B1�( 1A0 0

) MCSCF 9.21 0 0 –

B1�( 1A0 0

) DFT 6.61 0 0 –3� MCSCF 8.52 0.000750 1.63� 10�7 –3� DFT 6.70 0.000781 2.11� 10�7 –1� MCSCF 9.59 0.364 0.031 –1� DFT 7.57 0.641 0.076 –1� MCSCF 11.47 0.075 0.001 –3� DFT 9.51 0.311 0.023 –3� MCSCF 13.14 0.780 0.199 –3� DFT 11.44 0.970 0.267 –

21�þ ( 1A0 0

) MCSCF 13.06 1.080 0.373 –

21�þ ( 1A0 0

) DFT 10.43 0.857 0.188 –

aElectron impact; [14].bUnassigned vibronic peaks; [14].cElectron impact; [15].dMCSCF results are obtained with CAS-2.

Table 3. Adiabatic S0–Tn transitions (between the groundand excited states equilibria) of HCN molecule.

State Method �E/eV M/au f

a3A0 CIa 4.40 –

a3A0 MCSCFa 4.95 0.000249 1.69� 10�8

a3A0 DFTb 4.68 0.000278 1.99� 10�8

b3A00 CIa 5.43 – –

b3A00 MCSCFb 6.25 0.000684 1.61� 10�7

b3A00 DFTc 6.01 0.000655 1.42� 10�7

aFrom [49].bTransition moment M is calculated at the upper state

geometry (without account of Franck-Condon factors).cCalculated at the upper state geometry optimised by

MCSCF method.

Table 2. MCSCF calculations of the S0–T2 transitionmoment (au) at the intermediate geometry of HCN;bond lengths correspond to the optimized geometry ofthe b3A00 state, the angle HCN¼ 180� corresponds to theground state.

CAS EX/au

�EX�b/

eV

My(Ty)¼Mz(T

z)/

(au) f

CAS-1 �92.9882 5.289 0.000189 9.26� 10�9

CAS-2 �92.9933 5.281 0.000194 9.74� 10�9

CAS-3 �93.0133 5.283 0.000203 1.03� 10�8

Singlet–triplet transitions in three-atomic molecules 1395

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respectively. Thus the decrease of the C–H bondpolarization in the triplet state is well reproduced bythe DFT method. The experimental dipole moment ofthe HCN molecule in the ground state is 2.986� 0.004D[49] being in excellent agreement with the DFT result. Inorder to further qualify the electronic structure methods,we have also calculated the nuclear quadrupole coupling(NQC) constants at the ground-state optimized struc-ture: for 14N and 2H nuclei the NQC constants along themolecular axis are equal �4.76 and 0.23MHz, respec-tively, in good agreement with the experimental valuesof �4.71 and 0.21MHz [60]. In the first excited tripletstate (a3A0) the NQC tensor for the 14N nucleus consistsof the following components: Vxx¼�2.76, Vyy¼�4.88andVzz¼ 7.64MHz. The C–N axis direction (x) does notcorrespond to the largest component of the NQC tensorat nitrogen anymore. Some rotation of the NQC tensor

upon the S0–T1 transition occurs also at the deuteriumnucleus: Vxx¼�0.08, Vyy¼�0.11 and Vzz¼ 0.19MHz.

The spin-projection selective S–T transition momentsare very much similar in MCSCF and DFT QRcalculations including their angle dependence (figures 3and 4). We first analyse the vertical transitions asthey provide the most intensive lines in the absorptionspectra. The molecule is linear in this case (table 1 andffHCN¼ 180� in figures 3 and 4). For the first T–Stransition, a3��O X1�þ, the O¼ j �þ�j ¼ 1 sublevelsare active; the ZFS parameter D¼�3/2X, where x isthe molecular axis, is negative D¼�1.896 cm�1; it isdetermined mostly by spin–spin coupling (E¼ 0 inlinear a3�� state). This means that the active spin-sublevels (Ty;z, O¼ 1) are lower than the dark spin-sublevel (Tx, O¼ 0). The a3��1 X1�þ transition ispolarized perpendicularly to the molecular axis and is

Table 4. Results of geometry optimization for three low lying states of the HCN molecule. is a dipole moment (Debye), !i isvibrational frequency (cm�1), in parenthesis—infrared transition intensity (kmmol�1).

State Method rCH rCN ff � /D !1 !2 !3

X1�þ CIa 1.068 1.165 180.0 3.01

X1�þ MCSCF 1.0663 1.1453 180.0 3.06 3582 (154) 756 (37) 2112 (7)

X1�þ DFT 1.0649 1.1463 180.0 2.98

X1�þ exper.b 1.066 1.153 180.0 2.98 3312 (53) 713.5 (58) 2097 (0.14)

a3A0 CIa 1.100 1.301 122.0 1.90

a3A0 MCSCF 1.0812 1.3147 124.2 1.86 3227 (286) 1510 (137) 1070 (28)

a3A0 DFT 1.1094 1.2796 120.7 1.69

b3A00 CIb 1.111 1.315 119.6 1.83

b3A00 MCSCF 1.090 1.337 117.7 1.74 3138 (274) 1371 (34) 988 (153)

aFrom [49].bFrom [10, 49, 59].

100 120 140 160 180HCN angle

-93

-92.95

-92.9

-92.85

-92.8

Ene

rgy,

a.u

.

X1Σ+

a3A′

b3A′′

Figure 1. Potential energy curves for the ground singlet stateand for two low-lying triplet states in the HCN moleculecalculated by the MCSCF method. Excited states areobtained from the response technique.

100 120 140 160 180HCN angle

-93.35

-93.3

-93.25

-93.2

Ene

rgy,

a.u

.

X1Σ+

a3A′

a3A′′

Figure 2. Potential energy curves for the ground singlet stateand for two low-lying triplet states in the HCN moleculecalculated by the DFT B3LYP method. Excited states arefrom the response technique.

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relatively intense (table 1, figure 4). The DFT andMCSCF methods provide similar transition momentsand only slightly different transition energies (table 1).Under symmetric stretch motion (prolongation of thebonds) the transition moment decreases (@M=@Q¼�0.000 36 au). In absorption, the a3A0 X1�þ bandwould be dominated by a bending progression due tothe linear to bent nature of the transition. As followsfrom our calculations, the oscillator strength f for thevertical T S0 excitation is relatively large ( f�10�8),thus one can hope that this transition could be observedin the near UV region (280–250 nm).

By decreasing the HCN angle (bending vibration) theMy;zðT

z;yÞ dipole transition moments decrease (figure 3)almost by a factor of two. At the same time the dark(upper) spin-sublevel Tx with zero spin projection onthe main top axis (close to the C–N bond) thengets radiative with a strong polarization perpendicularto the molecular plain. The MzðT

xÞ component ofthe electric dipole transition moment increases veryfast, as follows from both methods (figure 3). Thispeculiarity indicates that the bending progression inthe a3A0 X1�þ absorption spectrum does not followa simple Franck–Condon intensity distribution. A

100 120 140 160 180HCN angle

0

0.0001

0.0002

0.0003

0.0004(b)

Mb(S

0 - T

1a ), a

.u.

Mz(T

x)

Mx(T

z)

Mz(T

y)

My(T

z)

Figure 3. (a) The S0–T1 (X1A0–a3A0) transition moments inthe HCN molecule calculated by the MCSCF QRmethod. Transitions to Tz, Ty and Tx spin sublevels areshown by dotted, solid, and dashed lines, respectively. Mb

means light polarization along the b axis. (b) S0–T1 (X1A0–

a3A0) transition moments in the HCN molecule calculatedby the DFT B3LYP QR method. Transitions to Tz, Ty

and Tx spin sublevels are shown by dotted, solid, anddashed lines, respectively. Mb means light polarizationalong the b axis.

100 120 140 160 180HCN angle

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007(b)

Mb(S

0 - T

2a ), a

.u.

Mx(T

x)

My(T

x)

My(T

y)

Mz(T

z)

Mx(T

y)

Figure 4. (a) The S0–T2 (X1A0–b3A) transition moments inthe HCN molecule calculated by the MCSCF QRmethod. Transitions to Tz, Ty and Tx spin sublevels areshown by dotted, solid and dashed lines, respectively.Mb means light polarization along b axis. (b) The S0–T2

(X1A0–b3A) transition moments in the HCN moleculecalculated by DFT B3LYP QR method. Mb means lightpolarization along b axis.

100 120 140 160 180HCN angle

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007(a)

Mb(S

0 - T

2a ), a

.u.

Mx(T

x)

My(T

x)

Mx(T

y)

Mz(T

z)

My(T

y)

100 120 140 160 180HCN angle

0

5e-05

0.0001

0.00015

0.0002

0.00025

(a)M

b(S0 -

T1a ),

a.u

.

Mz(T

x)

Mx(T

z)

Mz(T

y)

My(T

z)

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complicated dependence of the T1–S0 transition momenton bending vibrations (figure 3) for different spinsublevels is a very important characteristic feature ofthe predicted HCN spectrum. For the second T2–S0transition the dependence ofMbðT

aÞ on the ffHCN angleis even more dramatic (figure 4). Selection rules [7] allowfive non-zero components for the b3A00 X1�þ transi-tion. The main top axis component of the electric dipoletransition moment MxðT

xÞ increases from zero to0.0006 ea0, the value being three times larger than M?at the linear structure (figure 4: M? ¼MzðT

zÞ ¼MyðTyÞ

at ffHCN¼ 180�). One has to note that the S–Ttransition moment is a result of a summation of ahuge amount of dipole allowed contributions weightedby SOC mixing; the fact that both methods, DFTand MCSCF, reproduce a very similar dependence oftransition moments on the bending angle and their spinselectivity indicates a high accuracy and reliability of theDFT method.The splittings of the electron spin sublevels (fine

structure) and of the nuclear spin sublevels (hyperfinestructure) have been studied for HCN in the lowesttriplet state a3A0 at different geometries. The isotropicFermi-contact coupling constants calculated at theequilibrium by the DFT method with the EPR-3 basisset of [61] and B3LYP functional are equal to:að14N)¼ 0.027 92 au¼ 4.51MHz, að13C)¼ 0.355 58 au¼199.87MHz, að1H)¼ 0.057 89 au¼ 129.38MHz. A rela-tively big isotropic hyperfine splitting at the 13C nucleusand at the proton is a characteristic feature of the EPRspectrum of the triplet-excited HCN species.The fine structure of the electron spin sublevels Ta

n

which is also called zero-field splitting was calculatedat the optimised equilibrium geometry by the MCSCFmethod with CAS-2. The ZFS in the a3A0 and b3A00

states is found equal to D¼�1.0700 cm�1, E¼�0.0485 cm�1 and D¼ 0.7085 cm�1, E¼ 0.0610 cm�1,respectively, where D ¼ � 3

2X , E¼ 1

2ðZ � YÞ in both

cases. The x axis is the main axis of the top which almostcoincides with the C–N bond direction, and the z axis isperpendicular to the molecular plain (figure 5). TheseZFS parameters are calculated as spin–spin couplingexpectation values. The SOC contribution is estimatedas very small for the lowest triplet state (less than0.01 cm�1 for the D value). A similar SOC estimation isanticipated for the second triplet state. We have, forcomparison purposes, also calculated the g-tensor forboth triplet states with the MCSCF method (usingCAS-2). In the lowest triplet state a3A0 it is equalto: gxx¼ 2.003 59, gyy¼ 2.002 31, gzz¼ 2.003 44. Theg-tensor in the b3A00 state is rather different:gxx¼ 1.994 22, gyy¼ 2.002 83, gzz¼ 2.003 24. The sameb3A00 optimized geometry was here used for the ZFSand g-tensor calculations for the two states. The angle

dependence was also studied for all tensors, indicatingonly small variations in the vicinity of the equilibrium.There is a large difference in the gxx componentsfor two low-lying triplet states of hydrogen cyanide.For the T1 state the deviation from the freeelectron value is positive (0.001 54), and for theT2 state it is negative (�0.008 13). This means thatthe spin-rotation coupling constant along the main topaxis (�xx¼�2A�gxx) would be different for the twotriplet states. The calculated rotational constants areequal for the a3A0 and b3A00 states; A¼ 25.195,B¼ 1.301, C¼ 1.237 cm�1 and A¼ 21.202, B¼ 1.284,C¼ 1.211 cm�1, respectively. Thus both fine structurefeatures, spin splitting, D ¼ � 3

2X , and spin-rotation

coupling constant along the main top axis, �xx, aredifferent by the sign and by the absolute value. Suchdifferences would be helpful in a rotational analysis ofthe two transitions, S0–T1 and S0–T2, as they are quitesimilar in intensity and as their band systems overlapeach other because of long progressions of the bendingmode.

3.2. OzoneThe S–T transitions in the ozone molecule have been

the subject of much interest during last decade for bothexperimentalists [18, 21, 46, 62–67] and theoreticians[9, 11, 12, 22, 23 68–70]. The triplet states of ozone areconnected with the kinetics of formation and dissocia-tion processes O2þOÐO3, and are therefore importantfor the photochemistry of the atmosphere [21, 62, 69,70]. The ground state of the ozone molecule is a singletwith a large dose of bi-radical character [71]. Thus it isnot surprising that the triplet states of ozone are verylow lying in energy [72]. The topicity [73] of the_OO�O� _OO bi-radical is very high [1, 69, 74], whichleads to a number of triplet excited bi-radicals: 3A2,

3B1

and 3B2. The former two are of sp nature, the latterone is a pp* analogue of the ground state. The verticalexcitation 3B2 X1A1 from the ground state requiresless energy than the vertical excitations to other triplet

H

C N x

y

y

z

(1) (2)

O

O O

Figure 5. Molecules and choice of axes.

1398 B. Minaev et al.

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states 3A2;3 B1 X1A1. On the grounds of this fact, the

3B2 X1A1 transitions have been considered [72] for apossible assignment of very weak absorption bands ofozone in the near-IR–visible region, known as the Wulfbands [10]. Calculations by the INDO CI method withaccount of SOC including vibronic perturbations forthe bending mode have provided the perpendicular(x) polarized transition with a relatively weak intensity(f � 10�8) [72]. Until 1993 [62] the Wulf band wasassigned to the singlet-singlet 1A2 X1A1 forbiddentransition [20, 75–77] on the grounds of the proposal in[71]. Since 1993 high-sensitivity absorption spectroscopyhas been used to study this very weak band [18, 62–65].Anderson et al. observed for the first time [62] arotational structure in the origin of the Wulf band.From analysis of this structure they found the firstspectroscopic evidence in favour of the 3A2 X1A1 [62]assignment earlier proposed in [23]. Bouvier et al.provided high-resolution absorption spectra for the range9100–10 500 cm�1 [18, 46, 64–66] and found rotationaland fine-structure constants for the 3A2 state. Ab initiocalculations of the S–T transition intensity [12, 78] andrecent experimental studies [18, 21, 46, 63–67] agree inthe assignment of the 3A2 X1A1 transition in the absorp-tion spectra of ozone. A very weak 3B2 X1A1 transitionhas also been identified among other rovibronic bandsin the Wulf region [64].In this paper we want to compare the MCSCF QR

results for the S–T transition intensity in ozoneabsorption [78] with the recently developed DFT QRtechnique. We used the cc-pVTZ [79], 6-311G** and6-311þþG(3df,3pd) basis sets [80] in both types ofcalculations of the S–T transition probability. For thea1, b1, b2, a2 irreducible representations of the C2v groupwe chose the inactive space as (4 0 2 0); this means 12active electrons in all CAS spaces. The simplest activespace (4 3 3 1) is called CAS-1, which provides 53 484determinants for the ground-state expansion. The nextCAS, CAS-2 (4 3 3 2), consists of 157 016 determinants.In this case inclusion of the a2 empty orbital does notchange the results very much, and the CAS-2 results aretherefore not discussed further. CAS-3 (4 3 4 1) andCAS-4 (4 3 4 2) provide 213 640 and 736 464 determi-nants, respectively; the largest CAS, CAS-5 is (5 3 4 2),includes about two million determinants. This big CASis necessary in the case of the 6-311þþG(3df,3pd) basisset which provides much lower empty orbitals, whileall other active spaces are compatible with smaller basissets. All methods provide qualitatively reasonableresults for those geometries of the O3 molecule whereall lowest roots are properly converged. We use theMCSCF QR calculations with the CAS-1 activespace on a large number of grid points which span awide range of bending and stretching variables. For

comparison the results of the big CASs are alsoconsidered for a limited number of geometry points.

The experimental ground state geometry (rO�O¼1.278 A, ffOOO¼ 116.8� [10]) is well reproduced in allmethods. The largest basis CAS-5 calculation provides1.2763 A and 116.9�, respectively. DFT B3LYP with cc-pVTZ basis set gives reasonable agreement: r¼ 1.256 A,ffOOO¼ 118.1�. The total energy of the DFT method(�225.499 071 au) is much lower than the more accurateCAS-5 calculation (�224.639 64 au). At the same timethe charge distribution and the dipole moment aresimilar in both methods and agree with published data[22, 68]. Three low-lying excited triplet states havegeometry and vibrational frequencies similar to thoseobtained in previous works [22, 68]. For example, theoptimized geometry of the lowest 3A2 triplet state in theDFT B3LYP method with the cc-pVTZ basis set isr¼ 1.331 A, ffOOO¼ 98.2�, which is close to the experi-mental finding (1.345 A and 98.9�, respectively [64]). Theadiabatic excitation energy (0.81 eV) is in reasonableagreement with a full CI treatment (0.86 eV [77]). Wehave used the QR method to calculate the S–T transitionmoments for more than a hundred grid points withdifferent CASs and basis sets, with the purpose tocompare MCSCF and DFT results for ozone S–Ttransition intensities. We remind one that this moleculehas a very peculiar predissociative character and anon-closed shell ground state. The single-determinantHartree–Fock method is non-applicable for ozone [71,72] while the MCSCF QR results strongly depend on thechosen CAS space and basis set. The DFT QR approachprovides here a definite advantage over the MCSCF QRmethod, which is quite ironic considering that ozoneoften is coined as a prime example for MCSCF.

Let us first compare results of the two methods(MCSCF and DFT) for the vertical Tn S0 transitions,calculated at the same experimental geometry (table 5).The most important state is the 3A2 triplet state as itdetermines the largest part of the Wulf band intensity.Thus the vertical 3A2 X1A1 transition intensity inozone would correspond to the maximum of thebending progression (220 band). This long progressionstarting with the 200 band at about 9500 cm�1 andproceeding until 15 000 cm�1 has been analysed byAnderson and co-workers [62, 63], the assignment ofwhich was confirmed by high-resolution spectroscopy[18, 46, 64–66]. From the measured cross-sections of theleast-broadened rotational lines the radiative lifetimeof 0.2� 01 s was inferred [64]. It should be noted thatall the upper rovibronic levels are more or less pre-dissociative and the effective lifetime of the 3A2 stateshould be much shorter. Although the Franck–Condonfactors for theWulf band are known [64], here we are notgoing to consider all details of the rather complicated

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rovibronic structure. The purpose of this study is merelyto compare MCSCF and DFT results for the S–Ttransition intensities and estimate the reliability of bothmethods.As one can see from table 5, the results of the MCSCF

method strongly depend on the CAS and basis set chosen.Even the transition energy changes from 1.1 to 1.9 eV.Other methods [64] provide �E in the range 1.6–2 eV.DFT B3LYP slightly underestimates the transitionenergy, which is a common aspect of the method [33].However, the relative intensities for the S–T transitionsto different spin-sublevels are consistent. The mostimportant is the My transition moment to the Ty spin-sublevel. This becomes close to 0.0039 au in DFTcalculations, while in CAS-1 and CAS-3 it differsdramatically; CAS-1 underestimates—and the CAS-3strongly overestimates—the value of the large CAScalculation (0.0036 au) [11] (we note that the spin-sublevels x and y of the 3A2 state are interchanged in[11]; there are also obvious misprints in the specifica-tion of the active spaces). In order to reproduce theabsorption cross section of the Wulf band, Braunsteinand Pack [23] fitted the dipole transition moment tothe value 0.0047 au, and we find the DFT values to beclose to this estimation, see table 5. One should notethat the measured cross sections of the least-broadenedrotational lines from high resolution spectroscopy [46,64] provide a slightly smaller transition moment thanwhat Braunstein and Pack [23] obtained. Thus the DFTresults seem very reasonable. Another shortcoming ofthe MCSCF QR calculations is a very strong MzðT

zÞ

component in the CAS-1 approach, table 5. The ratioMyðT

yÞ/MzðTzÞ is equal to 0.37 using the the cc-pVTZ

basis set, although an analysis of rotational lines [46, 64]indicates that this ratio is close to 10. Again the DFTratio MyðT

yÞ/MzðTzÞ ¼ 11.7 (6-311G*) is in a good

agreement with experimental data [46, 64].The 3B2 X1A1 transition is found to be very weak

in agreement with previous calculations [11, 12] and

experiments [46, 63, 64]. Its MxðTzÞ component is

stronger than the MzðTxÞ component, which is almost

negligible: MxðTzÞ ¼ 1.6�10�4 au for the vertical

3B2 X1A1 excitation in the CAS-4/cc-pVTZ method:MzðT

xÞ ¼ 2�10�6 au. Similar results are obtained byother methods. The 3B2 X1A1 transition is thus out-of-plain polarized with an oscillator strength of about10�9 which is close to the result of [11]. The dependenceon the asymmetric stretching motion of the transitionmoment indicates some weak vibronic activity of the3B2 X1A1 transition in agreement with the old semi-empirical prediction in [72]. The intensity ratio of the3A2 X1A1/

3B2 X1A1 transitions is not close to 100by no means, as follows from pure electronic calcula-tions [11, 12]. Franck-Condon factors for the 200 bandsare much more favourable for the 3B2 X1A1 transi-tion. An account of the asymmetric stretch providesadditional complications connected with conical inter-section and predissociation. From the present calcula-tions it is yet clear that the 3A2 X1A1 and

3B1 X1A1

absorption bands are predicted to be relatively intense.The Wulf absorption system is nearly exclusively dueto these two transitions [64]. The first one contributesmostly to the long wavelength range. Because of thelarge change in the equilibrium angle upon excitation(�18� for the 3A2 state and þ 7� for the 3B1 state [64,68]), we have studied the angle dependence of the S–Ttransition moments, see tables 6 and 7. In order tosimulate the real intensity in the progression of !2

bending vibrations (with frequency changing from705 cm�1 in the ground state to 530 cm�1 in the 3A2

state; the bond length increases from 1.278 A until1.345 A, respectively [62, 64, 68]), we consider theintermediate bond length rO�O¼ 1.3 A in a qualitativesense of the r-centroid approximation [10, 81]. As onecan see from these tables, the MCSCF results stronglydepend on the CAS and basis set; see for exampleMyðT

zÞ values for the angle ffOOO¼ 110� in table 7.While the value of MyðT

zÞ is very unstable with respect

Table 5. Vertical 3A2 X1A1 transition intensity in ozone calculated by different methods. �E is a transition energy (eV), Mb(Ta)

is a transition moment along b axis to the Ta spin-sublevel in au, kb is the Einstein coefficient (s�1) for spontaneous emissionpolarized along the b axis, �p is the simulated phosphorescence radiative lifetime (s) for the corresponding upper state. Theoscillator strengths f, kb and �p are calculated without account of the Franck–Condon factors.

Method Basis �E My(Ty) Mx(T

x) Mz(Tz) ky kx kz �p/s f (10�7)

CAS-1 6-311G* 1.43 0.00192 �0.000249 0.001353 11.50 0.19 5.69 0.06 1.96

CAS-1 cc-pVTZ 1.17 0.00138 �0.000287 0.003605 3.22 0.14 21.85 0.04 4.28

CAS-1 6-311þ þG(3df,3pd) 1.05 0.00230 �0.000249 0.001353 10.88 0.19 5.45 0.06 1.91

CAS-3 6-311G* 1.91 0.00684 �0.000051 0.000158 344.8 0.02 0.18 0.00 21.9

CAS-3 cc-pVTZ 1.92 0.01269 �0.000153 0.000088 1205 0.17 0.05 0.001 75.7

CAS-4 cc-pVTZ 1.89 0.00432 �0.000229 0.000110 133.7 0.37 0.09 0.007 8.67

DFT 6-311G* 1.19 0.00394 �0.000096 0.000337 28.21 0.01 0.20 0.03 4.59

DFT cc-pVTZ 1.24 0.00388 �0.000147 0.000243 30.62 0.04 0.12 0.03 4.62

1400 B. Minaev et al.

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to basis set in CASSCF QR calculations at many points,it is quite stable in the DFT QR method (table 7).Similar results are obtained for the important MyðT

yÞ

component of the 3A2 X1A1 transition; it changesdramatically in CASSCF QR calculations at manyangles (110�, 1150�, 120�) when we compare two basissets, while being well reproduced in the DFT calcula-tions. Again, the CASSCF ratio MyðT

yÞ/MzðTzÞ is

changed between the 6-311G* and cc-pVTZ basis sets;the latter basis provides definitely erroneous results asfollows from high-resolution rotational analysis [64, 66].The 6-311G* basis set gives a ratio of about 3 which israther far from the experimental ratio [65].

A rotational analysis of three vibronic bands 000; 211

and 210 [18, 64, 66] indicates that the MyðTyÞ/MzðT

zÞ

ratio increases with the increase of angle and all DFTvalues, see table 6, seem very reasonable in this respect:The MyðT

yÞ/MzðTzÞ ratio increases from 6.4 at 110� to

14.7 at 120� in the DFT QR calculations using the6-311G* basis set. A similar ratio is obtained in DFTwith the cc-pVTZ basis sets (table 6).

The complicated ozone molecule and its S–T spectra[18, 64, 66] are well reproduced in the DFT approach anddespite the fact that the DFT method is based on thesingle-determinant density, it gives quite good geometryof the ground state and first excited triplet state of

Table 7. Calculated 3B1–X1A1 transition intensity in ozone. The distance rO-O¼ 1.3 A is chosen fixed in analogy with the r-centroid

approximation. All notations are the same as in table 5.

Method Basis �E ff OOO Mz(Ty) My(T

z) kz ky �p/s f (10�7)

CAS-3 6-311G* 1.86 110� 0.000674 0.003293 3.09 73.72 0.01 5.14

CAS-1 6-311G* 1.78 110� 0.000760 0.001513 3.47 13.75 0.06 1.21

CAS-1 cc-pVTZ 1.80 110� 0.000679 0.000391 2.87 0.95 0.26 0.27

CAS-1 6-311G* 1.64 115� 0.000716 0.001239 2.40 7.2 0.10 0.82

CAS-1 cc-pVTZ 1.65 115� 0.000654 0.000533 2.07 1.37 0.29 0.29

CAS-1 6-311G* 1.48 120� 0.000660 0.000962 1.51 3.21 0.21 1.11

CAS-1 cc-pVTZ 1.49 120� 0.000620 0.000911 1.36 2.95 0.23 0.97

CAS-1 6-311G* 1.12 130� 0.000484 0.000433 0.35 0.28 1.59 0.26

CAS-1 cc-pVTZ 1.13 130� 0.000498 0.002137 0.37 6.91 0.14 1.33

DFT 6-311G* 1.10 110� 0.000756 0.002210 0.82 6.97 0.13 1.47

DFT cc-pVTZ 1.06 110� 0.000820 0.002315 0.85 6.77 0.13 1.29

DFT 6-311G* 0.86 115� 0.000788 0.001695 0.42 1.96 0.4 0.74

DFT cc-pVTZ 0.91 115� 0.000739 0.001628 0.43 2.13 0.39 0.71

DFT 6-311G* 0.65 120� 0.000705 0.001211 0.14 0.42 1.79 0.34

DFT cc-pVTZ 0.69 120� 0.000680 0.001172 0.16 0.49 1.51 0.31

Table 6. Calculated 3A2–X1A1 transition intensity in ozone at different angles. The distance rO-O¼ 1.3 A is chosen fixed as

analogous to the r-centroid approximation. All other notations are the same as in Table 5.

Method Basis �E ff OOO My(Ty) Mx(T

x) Mz(Tz) ky kx kz �p/s f (10�7)

CAS-1 6-311G* 1.07 110� 0.002107 �0.000266 0.000763 5.86 0.09 0.77 0.15 1.34

CAS-1 cc-pVTZ 0.96 110� 0.000655 �0.000262 0.001280 0.40 0.06 1.52 0.49 0.50

CAS-1 6-311G* 1.31 115� 0.002283 �0.000254 0.000786 12.42 0.15 1.47 0.07 1.89

CAS-1 cc-pVTZ 1.16 115� 0.000429 �0.000275 0.001531 0.31 0.13 3.91 0.23 0.74

CAS-1 6-311G* 1.50 120� 0.002430 �0.000212 0.000825 21.42 0.16 2.47 0.04 2.44

CAS-1 cc-pVTZ 1.32 120� 0.000111 �0.000284 0.001979 0.03 0.19 9.49 0.10 1.29

CAS-1 6-311G* 1.82 130� 0.002603 �0.000766 0.000845 43.52 3.76 4.59 0.02 8.11

CAS-1 cc-pVTZ 1.51 130� 0.002778 �0.000249 0.003596 28.49 0.23 47.72 0.10 7.67

DFT 6-311G* 0.57 110� 0.002943 �0.000131 0.000461 1.69 0.003 0.04 0.57 1.22

DFT cc-pVTZ 0.61 110� 0.002936 �0.000139 0.000348 2.09 0.005 0.03 0.47 1.35

DFT 6-311G* 0.92 115� 0.003751 �0.000118 0.000384 11.57 0.01 0.12 0.06 3.20

DFT cc-pVTZ 0.96 115� 0.003700 �0.000148 0.000282 12.93 0.02 0.07 0.07 2.95

DFT 6-311G* 1.19 120� 0.004417 �0.000034 0.000301 34.83 0.002 0.16 0.02 5.68

DFT cc-pVTZ 1.22 120� 0.004337 �0.000124 0.000213 37.62 0.03 0.09 0.02 5.71

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this bi-radical species. It also reproduces the permanentdipole moment z¼ 0.64Debye (0.53Debye [10]) inthe ground state in good agreement with experiment(0.53Debye [10]). The S–T transition intensities fromDFT also agree well with all recent experimental results.

3.3. Water and other dihydrides of halogensAbsorption spectra of stable dihydrides of halogens

H2O, H2S, H2Se and H2Te have been investigated inthe UV region (including far-UV) since 1936 [10].The first electronic transitions in these spectra occur atfairly short wavelengths, in H2O and H2S, for example,they produce a broad continuum, extending from 186to 145 nm and from 270 to 190 nm, respectively [10],and are connected with photodissociation of the typeHþAH.The photodissociation in the first absorption band in

water is a well studied process [9]. During long time itwas considered as a prototype of a fast and direct bondbreaking reaction on a single potential-energy surface(in the 1B1 state) [9, 82]. MCSCF QR calculations ofHOCl, HOBr and H2O molecules indicate that the long-wave tail of S–T absorption can also make a smallcontribution to the photodissociation cross-section [7, 8].Recent research has shown that photodissociationof HOD at 193 nm cannot be explained without thesuggestion that the T1 S0 transition contributes to thebond breaking [9]. The OD/OH ratio is surprisinglysmall during HOD photodissociation at 193 nm; itcontradicts results of dynamical calculations whichonly take into account the singlet-singlet absorption1B1 X1A1 [9]. Due to the lighter mass of the H atomin comparison with deuterium, the OH bond stretchessignificantly farther out into the dissociation channelthan the OD bond does. Because of that circumstancethe O–H photodissociation is more easily excited bylow-energy quanta (193 nm) than the O–D cleavage,thus from this argument the OD/OH ratio should bevery large. However, the the measured ratio is stillsurprisingly small [9]. The puzzle is explained by theadditional account of the T1 S0 transition [7–9]. Inorder to explain the measured ratio, the 3B1 X1A1

transition dipole moment of roughly 0.04Debye wasinferred from [9], while MCSCF QR calculations havepredicted 0.008Debye (0.0031 ea0) for the verticalexcitation [8]. Recent calculations by coupled-clusterresponse theory gives exactly the same value of thevertical 3B1 X1A1 transition dipole moment [29].In the following we discuss the Tn S0 transitions in

a series of halogen dihydrides on the basis of the DFTQR method in order to compare with previous result forthe water molecule [8, 29] and to make some predictionfor the H2S and H2Se molecules. We have employed thebasis which recently has been tested and employed in

studies of NMR spectra [83]. For hydrogen and oxygenit includes 9s3p1d and 14s10p3d1f uncontracted AObasis sets, respectively. The ground state geometry(r¼ 0.958 A, ffHOH¼ 104.5�) [10] is used for the verticalS–T transitions (ground-state energy is equal to76.435 2359 au). With B3LYP DFT response we getsinglet–triplet transition energies of 6.82, 8.84 and8.76 eV for the 3B1,

3A1 and 3A2 states, respectively.The transition moments to the lowest triplet state 3B1

have the largest component MzðTyÞ ¼ 0.002 62 au in

good agreement with the coupled-cluster response(0.002 64 au) [53] and MCSCF QR (0.002 65 au) [8]results. This S–T transition is polarized along the C2 (z)axis (figure 5) and populates the Ty spin-sublevel, whichis the lowest one according to our ZFS calculations.The next allowed S–T transition (when SOC isaccounted for) is polarized along the y axis; thetransition moment MyðT

zÞ ¼ 0.002 01 au is here slightlylower. The ratio MzðT

yÞ/MzðTyÞ should be important

for spin-selectivity of the fine structure population ofthe OH(X2�) product. In the present DFT calculationsthis ratio (1.3) is smaller than that obtained from theMCSCF method (1.48) [8]. The total 3B1 X1A1

transition moment is equal to 0.0033 au in a reasonableagreement with previous results (0.0031 au) [8, 53].

It has already been proposed that the photodissocia-tion of the H2S molecule in the first absorption bandis similar to photodissociation of water [10]. Indeed, theabsorption spectrum and the rovibrational state dis-tribution of the AH(X2�) products are very similar [82].However, detailed theoretical studies have proved thatthe photodissociation mechanisms are rather differentfor these two molecules [9, 82]. Cleavage of H2S entailstwo strongly coupled states, 1B1 and

1A2 in C2v startingsymmetry, which have conical intersections. Anotherdifference is connected with the fact that the firstabsorption band of the H2S molecule has a pronouncedtail at long wavelengths which MRCI calculations ofsinglet states fail to reproduce [9, 82]. Our DFT QRcalculation predicts a lowest triplet state of 3A2

symmetry close lying with the 3B1 state; the verticalexcitation energies are equal to 5.36 and 5.73 eV,respectively (the ground-state energy equal to�399.362241 au is here obtained for the experimentallyderived geometry, r¼ 1.336 A, ffHSH¼ 92.1� [82]).

The 3B1 X1A1 transition moment has the follow-ing components: MzðT

yÞ ¼ 0.007 56 au, MyðTzÞ ¼

�0.006 67 au. The formal spin-averaged radiative life-time of the 3B1 state is 147 ms. The transition momentfor the lowest-energy 3A2 X1A1 absorption bandconsists of: MyðT

yÞ ¼ 0.00572 au, MxðTxÞ ¼ 0.001 97 au

and MzðTzÞ ¼ 0.003 54 au. The phosphorescence

radiative lifetime of the H2S molecule is equal to107 ms in the high-temperature limit (this is a spin-

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averaged lifetime of the 3A2 state). We find thatthe first absorption band (3A2 X1A1 transition)is slightly weaker, f¼ 6.45�10�6, than the secondsinglet–triplet transition, 3B1 X1A1, with an oscillatorstrength of f¼ 1.43�10�5. Such intensity can beobserved as a weak tail in the long wavelength region.In comparison with the H2O molecule, for which

the very weak 3B1 X1A1 transition is overlapped bythe much more intense singlet-singlet band, the S–Tabsorption in the H2S molecule has a pronounced tail atlong wavelengths. This owes to the fact that the first S–Stransition (1A2 X1A1 transition with vertical energy5.9 eV) does not overlap any of the two T1;T2 S0

bands. Similar T1;T2 S0 absorption and photo-dissociation is predicted for the H2Se molecule butwith an order of magnitude higher oscillator strengthsand cross-sections.

3.4. Germanium dihalidesMuch attention has recently been paid to transient

species which contain germanium, because of their roleas intermediates in semiconductor fabrication [54].The most useful analytical method for the control ofsemiconductor growth is spectroscopy, thus quite a fewlaser-induced fluorescence (LIF) studies of germanium-containing molecules (HGeCl, HGeBr, GeCl2, GeF2)have been published [47, 84–86]. The LIF spectra ofgermanium dichloride have been obtained by pyrolysisof HGeCl3 in the supersonic jet [47]. Besides the strongfluorescence of LIF at 320 nm, a weak laser-inducedphosphorescence in the visible region (450–400 nm)has been detected and vibrationally analysed. Bothbands have been attributed to emissions from the 1B1

and 3B1 states, respectively, on the ground of UHFcalculations [47]. In the following we analyse our DFTQR calculations for the phosphorescence intensity3B1!X1A1 in light of the findings briefly reviewedabove. The electric dipole transition moment is pre-sented in figure 6 as a function of the Cl–Ge–Cl angle.The optimized geometry of the 3B1 state is r(Ge–Cl)¼ 2.187 A, ffCl–Ge–Cl¼ 118.6�, in good agreementwith the results of [47].The DFT prediction for the adiabatic excitation

energy 3B1 X1A1 is 2.55 eV compared to the experi-mental value of 2.76 eV [47]. Theory predicts a smallchange in the frequency of the symmetric stretchvibration (1) upon the T–S transition in agreementwith experiment: 1¼ 399 cm�1 in the ground state and1¼ 393 cm�1 in the triplet 3B1 state [47]. At the sametime the bending frequency changes remarkably, from2¼ 159 cm�1 in the ground state to 2¼ 118 cm�1 in thetriplet excited state [47]. Our calculated values are hereequal to 156 and 121 cm�1, respectively.

The observed triplet state lifetime is reported to be17.4 ms [47]. Our DFT QR calculation predicts theradiative lifetime equal to 51 ms (an account of amore realistic experimental energy for the phosphores-cence 3B1!X1A1 transition provides �p¼ 43 ms). This ismuch shorter then the corresponding radiative lifetimesin the other molecules discussed above and is a clearindication of the heavy atom effect of the Ge and Clatoms. The earlier reported data about weak emissionof GeCl2 in the 670–560 nm region [87] are not sup-ported by our calculations. We do instead support theconclusion of [47] that the 670–560 nm emission is notdue to GeCl2 or that the lower state is not the groundstate.

100 120BrGeBr Angle

-0.06

-0.04

-0.02

0

0.02

0.04

(b)

Mb(S

0 - T

1a ) (a

.u.)

Mz(T

y)

My(T

z)

Figure 6. The S0–T1 (X1A1–a3B1) transition moments in (a)

GeCl2 and in (b) GeBr2 molecules calculated by DFTB3LYP QR method with cc-pVTZ basis set.

90 100 110 120ClGeCl angle

-0.03

-0.02

-0.01

0

0.01

0.02

0.03(a)

Mb(S

0 - T

1a ) (a

.u.)

My(T

z)

Mz(T

y)

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Germanium difluoride spectra have been detectedin a limited number of studies. For a review we refer toa paper by Karolczak et al. [86]. Martin and Mererdiscovered a weak emission near 330 nm, which theytentatively assigned as the 3B1!X1A1 transition; thisassignment was confirmed by a laser-induced jetspectrum analysis and by ab initio CI calculations [86].Our DFT results for the 3B1 state energy and geometryoptimization are in a good agreement with data ofKarolczak et al. [86]. The bond angle is predicted toincrease by 15� on excitation in agreement with resultof Karolczak et al. (16�) and accounting for a longprogression of bending vibrations in the 3B1!X1A1

phosphorescence of GeF2 [86]. The radiative triplet statelifetime has not been measured so far; the calculated �pvalue (table 8) follows the expected trend for the theinternal heavy atom effect. For GeBr2 the phosphores-cence has not yet been observed, and we therefore hopethat our prediction can be used for the search of such asemission.

4. Conclusions

In the present paper we have employed a recentlydeveloped [33] time-dependent density functional theorymethod utilizing quadratic response functions to studysinglet–triplet transitions in three-atomic molecules.Comparative calculations have been carried out with thecomplete active space multiconfigurational SCF method.These species are chosen partly because of the interestin their spectral properties and partly because theyprovide grounds for qualifying density functional theoryfor singlet–triplet transitions and phosphorescence.In particular, DFT circumvents the instabilities fortriplet excitations encountered by the commonly applied

random phase approximation (response for Hartree-Fock reference wave functions), which rather unpredic-tably appear for different systems. DFT also overcomethe poor scaling of explicitly correlated wavefunctionmethods, e.g. MCSCF which overcome triplet instabil-ities with even small active spaces.

The main attention was paid to the HCN and O3

molecules, as predictions of singlet–triplet transitionintensity and fine structure are most intriguing for suchbenchmark species with importance for combustionand astrophysical research. The calculated S–T transi-tion moments in ozone for the three low-lying tripletstates 3A2;

3 B2 and 3B1 are found to be in goodagreement with recent experimental data and withprevious ab initio calculations, indicating a somewhatsurprising capacity of the DFT technique even for ozonewith its well-known multiconfigurational character.

The next studied group of three-atomic moleculeswere dihydrides of AB2 type. The photodissociation inthe first absorption band in water has been consideredfor a long time as a prototype of a fast and direct bondbreaking in the singlet 1B1 state. A similar behaviour isanticipated for the H2S molecule. Our DFT calculationssupport a recent suggestion that a tiny contribution ofthe T1 S0 transition explains the small OD/OH ratioin HOD photodissociation at 193 nm. More evidentcontributions of two T1;2 S0 transitions are here foundfor H2S photodissociation at 230 nm (3A2) and 216 nm(3B1). The HþHS cleavage entails these two stronglycoupled triplet states which have conical intersections.

We have confirmed the assignment of 3B1!X1A1

phosphorescence in GeF2 and GeCl2 [86] and havepredicted the phosphorescence lifetime for GeBr2.The earlier reported data about weak emission ofGeCl2 in the 670–560 nm region [87] are not supportedby our calculations. These and other results of thepresent work indicate that quadratic response time-dependent density functional theory shows capacity forprediction of singlet–triplet transition properties, whichwe also intend to exploit in coming studies of largerspecies.

This work has been supported by the Wenner-GrenFoundations (B. Minaev), by the Swedish ScienceResearch Council (OV) and the Swedish Foundationfor Strategic Research (PS).

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Table 8. The calculated T1–S0 transitions in germaniumdihalides (au).

3B1 GeF2 GeCl2 GeBr2

Adiabatic energy

DFT/B3LYP 0.1303 0.0934 0.0832

MRSDCIb 0.1243c 0.0960 0.0884

Exp. 0.1381 0.1016

Vertical energya

DFT 0.1182 0.0857 0.0755

�pa 3.6� 10�4 5.1� 10�5 1.3� 10�5

Transition momenta

My(Tz) �0.0130 �0.0239 �0.0524

Mz(Ty) 0.0080 0.0143 0.0346

aT1!S0 at the triplet state geometry; DFT/B3LYP QRwith cc-pVTZ basis set.

bFrom [47, 86].cCISDþQ/DZP: [86].

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