Ab initio study of the singlet-triplet transitions in hypobromous acid

Ab initio study of the singlet–triplet transitions inhypobromous acid

B.F. Minaev* , H. Agren

Department of Physics and Measurement Technology, Linko¨ping University, S-58183, Linko¨ping, Sweden

Received 20 January 1999; received in revised form 18 February 1999; accepted 1 March 1999

Abstract

The absorption spectra of the HOBr molecule in the near-ultraviolet and in the visible regions produced by excitation to thedissociative triplet and singlet states have been studied by the multi-configuration self-consistent field method with linear andquadratic response techniques. A large basis set and the complete Breit–Pauli form of the spin–orbit coupling operator are used.It is shown that the singlet–triplet (S–T) transition to the lowest triplet statea3A00 ← X1A0 is responsible for a weak absorptionin the visible region (440–650 nm) recorded recently by Barnes et al. (J. Phys. Chem. 100 (1996) 453). The photodissociation ofthe HOBr molecule by visible light near the peak of the solar actinic flux through the S–T transition indicates the importance ofSOC effects in atmospheric photochemistry of heavy halogens. The transition is polarized along the O–Br bond in agreementwith the experimental measurements of OH Doppler line profiles (Barnes et al., J. Phys. Chem. 100 (1996) 453). The second S–T transitionb3A0 ← X1A0 enters into the region of the first singlet–singlet (S–S)A1A00 ← X1A0 absorption and produces anappreciable contribution to the total cross section at wavelengthsl < 340–360 nm in addition to the S–S absorption. The S–Tabsorption decreases the concentration and the lifetime of tropospheric hypobromous acid and influences the ozone depletionproblem to a great extent.q 1999 Elsevier Science B.V. All rights reserved.

Keywords:Ab initio study; Hypobromous acid; Absorption spectral studies

1. Introduction

Hypobromous acid, HOBr, plays an important rolein stratospheric ozone depletion processes [2–4]. It isproduced in the atmosphere by the following reac-tions: by gas-phase disproportionation of two radicals

HO2 1 BrO! HOBr 1 O2 �1�and by heterogeneous reactions involving the hydro-lysis of bromine nitrate, BrONO2, on aerosol particlesin the night time stratosphere [2]

H2O 1 BrONO2 ! HOBr 1 HNO3: �2�

Important loss processes of HOBr are reaction withatomic oxygen [4] and photolysis [2,3]. The role ofthe HOBr molecule in photochemistry of the upperatmosphere is highly dependent on its photolysis inthe long wavelength region (l $ 300 nm). The photo-dissociation of HOBr

HOBr 1 hn! HO�2P�1 Br�2P� �3�can lead to an effective ozone depletion processesthrough the chain [8]

OH 1 O3 ! HO2 1 O2; �4�

Br 1 O3 ! BrO 1 O2 �5�and a recombination of the products of the reactions

Journal of Molecular Structure (Theochem) 492 (1999) 53–66

0166-1280/99/$ - see front matterq 1999 Elsevier Science B.V. All rights reserved.PII: S0166-1280(99)00084-6

* Corresponding author. Fax:1 46-13-13-75-68.E-mail address:[email protected] (B.F. Minaev)

www.elsevier.nl/locate/theochem

(4) and (5) reproduces hypobromous acid again (Eq.(1)). Thus the net process 2O3 ! 3O2 occurs, whichdetermines the role of hypobromous acid in the cata-lytic cycle for ozone destruction [8]. Recent measure-ments indicate that the ozone depletion potential forbromine is even greater than that of chlorine [2,8]. Animportant sequence of these reaction schemes is thathypobromous acid can couple ozone destructivecycles involving bromine with those involvingchlorine and OH radicals [1,3]. As reaction (3) isthe rate-determining step of the chain, an accuratedetermination of the UV absorption cross section ofhypobromous acid is necessary for assessing the roleof this molecule in the photochemical reaction cyclesin the stratosphere [1–8].

Spectroscopic data on hypobromous acid aresparse, although recent studies by Orlando andBurkholder [9] and by Barnes et al. [1] have providedaccurate measurements of its absorption in the near-UV and visible regions. A big difficulty in recordingthe electronic absorption spectra of HOBr moleculesis caused by interference from strong absorptionbands of Br2O and Br2; these species are typicallypresent in large concentrations as parts of the sourcechemistry required to generate such a chemicallyunstable molecule as HOBr; it is usually generatedfrom water vapor and Br2 [1,9]. Orlando andBurkholder [9] were able to minimize the problemof HOBr absorption spectra contamination by carefulbackground subtraction; they obtained an accurateabsorption cross section for the spectral region 240–400 nm even though there is an overlap with the Br2Oand Br2 absorption. Barnes et al. [1] have used analternative method of recording an absorption-likespectrum of the photodissociating molecule. Theymonitored the yield of the OH photofragment whilescanning the wavelength of the incident light. A newabsorption band (l t 400–650 nm) of hypobromousacid centered at 440 nm was detected by this methodand tentatively assigned to a triplet state of HOBr [1].

We have recently attempted [10] to support thishypothesis by ab initio calculations of the HOBr spec-tra with an account of spin–orbit coupling in theframework of a response theory approach. We havealso performed similar calculations for the HOClmolecule [10] and our prediction of a very weakabsorption in the range 380–410 nm determined bythe first singlet–triplet (S–T)transition of hypochlorous

acid was confirmed recently by the measurements ofBarnes et al. [11].

In this paper we shall present the results for hypo-bromous acid in detail.

2. Method of calculations

The S–T transition probability may be described interms of a response function that corresponds to themolecular response to spin–orbit coupling which istreated as a perturbation [12]. A big advantage ofthis theory is that the sum-over-states expression forthe S–T transition dipole moment [13,14] is replacedby solution of sets of linear equations, which may besolved by iterative techniques. The multi-configura-tional self-consistent field (MCSCF) method with acomplete active space (CAS) is used for the responsefunctions calculations. For details we refer to Ref.[12]. The SOC operator is implemented here in thecomplete Breit–Pauli form [12].

The singlet–singlet (S–S) transitions are calculatedby the linear response (LR) method [15]; the poles ofthe response function give the excitation energies andthe residues at the excitation frequency give the S–Stransition moments to the corresponding excitedstates. The S–T transitions are forbidden in the non-relativistic approximation; spin–orbit couplingmixes the spin multiplicity and gives rise to a non-vanishing transition dipole moment which can becalculated as the residue of a quadratic response(QR) function at the excitation frequency of theS–T transition [12]

Ma�Tb� � kSuX

i

eai uTbl: �6�

The MCSCF response method employs the opera-tors associated with orbital and configuration relaxa-tion to describe the effect of an external perturbation[12]. As the orbitals are allowed to relax in response tothe external field, the MCSCF response method ismore efficient than conventional configuration inter-action (CI) approaches. Both orbital excitation opera-tors and configuration excitation operators describecorrelation. Conventional CI methods use only thelatter, and call for more elaborate calculations toobtain an equal correlation level.

Calculations of the S–T transition moment (6) by

B.F. Minaev, H. A˚gren / Journal of Molecular Structure (Theochem) 492 (1999) 53–6654

the sum-over-states expression in conventional CImethods usually lead to a slow convergence [14]. Trun-cation of the sum cannot guarantee even a semi-quantitative accuracy in ab initio methods [12]. The

restricted CI method also leads to size-consistencyerrors when the S–T transition dipole moment isstudied as a function of vibrational displacements oras a function of reaction coordinate.

We employ Cartesian triplet componentsTb �b�x; y; z�which are eigenfunctions of the zero-field-split-ting Hamiltonian [13,14]. These are related to thespherical components (high external magnetic fieldspin-functions) by an irreducible tensor transforma-tion [13]

Tx � T21 2 T1��2p ; Ty � i

T21 1 T1��2p ; Tz � T0

: �7�

As a result of high anisotropy of SOC in molecules,each spin sublevel of the triplet stateTb is associatedwith a specific S–T transition probability. The selec-tion rules for the S–T transitions in molecules with Cs

symmetry can be found, e.g. in Ref. [10].In this work calculations of spectral properties of

B.F. Minaev, H. A˚gren / Journal of Molecular Structure (Theochem) 492 (1999) 53–66 55

Fig. 1. Choice of axes for the HOBr molecule (a); the biradicalstructure of the equilibrium lowest triple state (b).

Table 1Geometry optimization for the lowest singlet and triplet states of the HOBr molecule.r denotes the internuclear distances (A˚ ), andn thevibrational harmonic frequency (cm21). IR intensities (km/mol) are given in parentheses,En is the total energy (hartree) of then-th state

Method rO–H (A) rO–Br (A) /HOBrk n1, O–H n2, bend n3, Br–O En (hartree)

Ground state, 11A0

3-21Ga 1.008 1.9410 101.92 3334.6 (8.3) 1226.1 (41.4) 534.5 (3.4) 2 2635.11549Ahlrichs VDZa 1.000 1.9530 103.56 3434.4 (52.8) 1185.5 (33.4) 506.3 (2.2) 2 2647.47392Sadlejb 0.964 1.853 102.96 2 2647.88932Sadlej, MP2c 0.96 1.850 102.3 2 2648.56084MP2d 0.966 1.865 100.9 3959 1138 616 2 2648.27875CISDe 0.955 1.842 102.8 3991 1233 653 2 2648.01040MRCISDe 0.966 1.897 100.1 2 2647.91641CCSD (T)f 0.964 1.884 100.9 3837 1180 573 2 2648.06638CCSD (T)g 0.964 1.853 102.3 3807 1197 608 2 2648.22065Experimentalh 0.961 1.834 102.3 3614.9 1162.6 620.2First excited triple state, 13A00

3-21Ga 1.017 8.22i 0 3170.9(32.9) – – 2 2635.03880Ahlrichs VDZa 1.007 4.106j 0 3358.2 138.2 52.3 2 2647.42068

a This work. MCSCF in (6a0, 3a00) CAS.b This work. MCSCF in (6a0, 3a00) CAS. Partial geometry optimization.c The second order Møller–Plesset perturbation theory [18]. Calculation at fixed geometry.d Ref. [4]. Basis set 9s6p2d of Binning and Curtiss [22] for bromine and 6-311G(d,p) for oxygen and hydrogen. The second order Møller–

Plesset perturbation theory [18].e Ref. [4]. The same basis set as in footnote a, CI for single and double excitations. Multireference CISD on MCSCF orbitals.f Ref. [4]. Coupled-cluster method with singles, doubles and partly triple excitations in the same basis set as in footnote d.g Ref. [23]. Coupled-cluster method with singles, doubles and partly triple excitations in TZ2P basis.h From Ref. [4].i This structure corresponds to the dissociation Br1 H–O.j This structure corresponds to the linear Br–H–O complex withrBr–H � 3.099 A.k Angles in degrees.

HOBr molecule have been performed by the MCSCFQR method using the basis set of Sadlej [16]. Thiscontains [15s12p9d–9s7p4d] functions for brominewith a total number of contracted orbitals in MOexpansion of 83. Calculations with smaller bases(VDZ set of Ahlrichs et al. [17] and 3-21G basis setof Pople et al. [18]) are used for comparison. TheVDZ basis set includes [14s10p5d–5s4p2d] functions

for bromine with smaller sets for H and O atoms. Thetotal number of AOs is 38. The 3-21G basis set is alsomoderate (the [12s9p3d–5s4p1d] contraction forbromine) and consists totally of 33 AOs. The choiceof axes for HOBr molecule in the ground state (a) andin the first excited triplet state with optimized geome-try (b) is given in Fig. 1. We used a modified versionof the Dalton code [20]. The modification concerns


Table 2Vertical excitation energies (eV) and oscillator strengths (in parentheses) for the S–S and S–T transitions in the HOBr molecule at theexperimental ground state geometry (with one exceptionc). The first two columns correspond to this work; Sadlej basis set results are obtainedin CAS (6,3)

State Ahlrichs VDZa Sadlej CISb MRCIb,c MRCIb Experimentald

13A0 0 2.98 (8.12× 1025) 2.64 (7.28× 1028) 2.95 2.48 2.84 2.82e

13A0 0 3.82 (7.19× 1024) 3.43 (2.01× 1024) 3.72 3.44 3.80 , 3.5f

11A0 0 3.96 (7.86× 1024) 3.78 (3.58× 1024) 4.16 (6× 1024) 3.29 (1.8× 1026) 3.66 (1.00× 1025) 3.54g

21A0 5.00 (5.59× 1023) 4.70 (1.71× 1023) 5.08 (5.50× 1023) 4.24 (1.00× 1023) 4.69 (1.60× 1023) 4.43h

21A0 0 6.41 (0) 7.02 (0) 6.77 7.1631A0 0 6.94 (7.93× 1023) 7.05 (3.74× 1023) 7.89 (8.90× 1023) 5.57 (3.70× 1023) 6.38 (5.90× 1023)31A0 8.77 (0) 8.54 (6.72× 1022)41A0 9.09 (3.08× 1021) 9.06 (2.40× 1021)

a Extended CAS (7a0, 3a0 0; 14 electrons in 10 MOs).b Ref. [4].c Optimized geometry.d Ref. [1].e Weak continuous band with a maximum at 440 nm.f This absorption overlaps with a band atlmax� 350 nm.g A band atlmax� 350 nm.h A strong band atlmax� 280 nm.

Table 3The T1 ← S0 transition HOBr molecule. MCSCF QR calculations of the vertical transition momentMa�Tb� � k11A0ue

Pi ai u1

3A00�Tb�l (in1026 a.u.) near the experimental ground state geometry;DES,T is the transition energy,fa is the oscillator strength for theT1 ← S0 absorptionpolarized along thea axis

Ma (Tb) 3-21Ga 3-21Gb AhlrichsVDZb AhlrichsVDZb,c Sadleja Sadlejb

Mx(Tx) 706 715 620 707 64 53

My(Ty) 2517 2605 2122 1843 2952 2904

Mz(Ty) 4609 4137 6499 6866 1977 1690

My(Tz) 23651 23477 3077 31452 31354 31813

Mz(Tz) 9867 9888 11756 11888 10089 10099

Transition energy (eV) and oscillator strengthDES,T (eV) 3.17 3.07 2.92 2.90 2.72 2.64fx 3.87× 1028 3.84× 1028 2.74× 1028 3.55× 1028 2.73× 10210 1.82× 10210

fy 4.39× 1025 4.19× 1025 6.8 × 1025 7.06× 1025 6.61× 1025 6.6 × 1025

fz 9.21× 1026 6.63× 1026 1.29× 1025 1.34× 1025 7.04× 1026 6.78× 1026

ftotal 5.31× 1025 4.86× 1025 8.09× 1025 8.4 × 1025 7.32× 1025 7.28× 1025

a The experimental ground state geometry;rO–Br� 1.834 A, rOH � 0.964 A, angle HOBr� 102.38.b The fixed ground state geometry;rO–Br� 1.85 A, rOH � 0.96 A, angle HOBr� 102.38.c Extended CAS (7a0, 3a0; 14 electrons in 10 MOs).

the total S–T transition probability for molecules withsymmetry lower than C2v.

3. Results and discussion

At the equilibrium ground state geometry theHartree–Fock electronic configuration of the HOBrmolecule is

11A0 � �core�30�12a0�2�13a0�2�14a0�2�5a00�2�15a0�2

� �16a0�2�6a00�2:�8�

The core orbitals plus the lowest occupied valenceorbital, 12a0 MO with low energy (21.39 a.u.), whichdescribes mostly the 2s-AO of oxygen, were inactivein the MCSCF CAS calculations. All other occupiedMOs (4, 2) and three empty MOs (2, 1) have beenincluded in the MCSCF active space (12 electrons in 9orbitals) which corresponds to 3560 determinants ofthe1A0 symmetry for the reference ground state. Here,and in the following, the notation (n, m) refers to thenumber of orbitals of (a0, a00) symmetry in the CAS.Hence we shall use the CAS (6, 3) throughout until


Fig. 2. Dissociation potential (the O–Br bond length dependence ofthe total energies) for the ground and few lowest excited states in theHOBr molecule calculated by the MCSCF LR method: (a) theground state 11A0 and the two lowest excited triplet states; (b)extension of the previous figure with addition of the singlet excitedstates. Then1A0 states are denoted by solid lines, then1A00 states aredenoted by dashed lines, the triplet state are denoted by dot-dashedlines.

Fig. 3. Bending potential (the HOBr angle dependence of the totalenergies) for the ground singlet state and for the two excited tripletstates in the HOBr molecule calculated by the MCSCF LR method.

otherwise mentioned in few illustrative examples(Tables 2 and 3). This rather small active space isquite reasonable. Extension of occupied orbitals(inclusion of the MO 12a0) almost does not changethe transition energies and transition moments for thesix lowest excited states [19]. Extension of the emptyMOs is also unreasonable; the three lowest emptyorbitals, included in this CAS, are quite well separatedin energy from a large group of higher MOs. The sametrend is seen from the natural orbital occupationnumbers obtained by MP2 calculations. The emptyorbitals, which were included in the CAS, have occu-pation numbers 0.06, 0.027 and 0.026, whereas for the

rest of the empty MOs the occupation numbers do notexceed 0.01. Further attempts to increase CAS by theextension of a number of empty orbitals lead tounreasonable S–T transition moments (about 1 a.u.)because of a wrong balance from different quasi-degenerate state contributions. Vertical transitionenergies to the six lowest excited states calculatedby the MCSCF LR method are quite close in all thetested CAS spaces and are similar to the results ofrecent calculations by Francisco et al. [4] with largenumber of configurations in the MRCI approach. Inthe following we start with the results of the geometryoptimization of HOBr molecule in two bound states.


Fig. 4. The O–Br bond length dependence of the S–S transition dipole moments calculated by the MCSCF LR method: (a) then1A00 ← 11A0

transition dipole moments (alongx axis). The 11A00 ← 11A0 transition moment is denoted by a solid line,n� 2 is denoted by dotted line,n� 3 bydashed line; (b) then1A 0 ← 11A 0 transition dipole moments; solid line forn� 2, dotted line forn� 3, dashed line forn� 4; (c) the 21A0 ← 11A0

transition dipole moments (alongy andz axes) in a big scale.

Then we discuss the potential energy curves and theS–S and S–T transition moments as functions of theO–Br distance for fixed experimental ground stateparameters. The potential energy curves of the tripletstates and the S–T transition moments will be studiedas functions of the/HOBr angle at the fixedexperimental bond lengths in order to complete theinformation about the S–T absorption intensityobtained in the framework of the Born–Oppenheimerapproximation.

3.1. Dissociation and bending potentials of the groundand excited states

Results of geometry optimization of the HOBrmolecule in the ground state and in the first excitedtriplet state by the MCSCF CAS (6, 3) method withfew different basis sets are presented in Table 1. Thecalculated ground state geometry and the infra-redspectrum are in reasonable agreement with the experi-mental data.

Fig. 2(a) shows the dissociation potentials of theground singlet and of the two triplet states at fixedO–H bond length (0.964 A˚ ) and angle/HOBr �102.968. The two lowest excited triplet states 13A00

and 13A0 of hypobromous acid are strongly repulsivein the vicinity of the vertical excitation from theground 11A0 state. The lowest triplet 13A00 state poten-tial gets flat at long O–Br distances (rO–Br $ 2.2 A) incontrast to the second triplet (13A0) state, which is still

a descending function in this region (Fig. 2(a)). Theflat part of the 13A00 state potential has a weak mini-mum atrO–Br� 2.34 Awith a dissociation energyDe�1.31 kJ/mol. This minimum is more deep and occursat a shorter O–Br distance than the correspondingminimum in the HOCl molecule [10]. However, thisis not a real minimum on the triplet state potentialenergy hypersurface as the angle and the O–H bondwere fixed. The absolute minimum for the lowesttriplet state has been obtained by MCSCF geometryoptimization; it corresponds to the rotation of the O–Br bond until the angle/HOBr becomes zero. Simul-taneously, a cleavage of the O–Br bond occurs, so thebiradical structure Br…HO (Fig. 1(b)) is produced.Such a biradical dissociates in 3-21G basis set duringthe MCSCF geometry optimization, but it is stable inAhlrichs VDZ basis set calculation. The bendingvibration frequencyn2 � 138 cm21 is quite large forthe weak intermolecular complex, whilen3 �52 cm21 is typical for van-der-Waals forces. Thebending vibration is predicted to be very intensivein the infra-red spectrum of the biradical (Table 1).Similar results have been obtained for the HOClsystem [10].

The dissociative behavior of the eight lowestexcited states obtained by MCSCF LR calculationsin Sadlej basis set for HOBr molecule is presentedin Fig. 2(b) at the same fixed O–H bond length(0.964 A) and angle/HOBr � 102.968. The twofirst excited singlet states 11A00 and 21A0 have


Fig. 4. (continued)

repulsive potentials which are quite similar to thecorresponding potentials of their triplet counterparts.Excitation to these states will lead to a fast, direct,dissociation. The next two singlet states have avoidedcrossings with Rydberg excitations at short O–Brdistances. The upper states shown in Fig. 2(b) arecharacterized by many avoided crossings. For exam-ple, the first avoided crossing between the upper statesof the1A00 symmetry (the second and the fourth states)is of the valence-Rydberg type (the upper branch ofthe avoided crossing is not shown). Then the avoidedcrossing between valence excited states occurs,namelysO–Br* ← pO–Br* andsO–Br* ← pO–Br excita-tions localized on the O–Br bond with subsequenttransformation to the OH and Br orbitals. As bothradicals have few degenerate states (in the absenceof SOC) a new avoided crossing can be predicted.Similar behavior is seen for the excited1A0 statecurves (Fig. 2(b)).

The bending potentials of the ground state and ofthe two lowest excited triplet states, 13A00 and 13A0, ofhypobromous acid at fixed bond distances (rO–Br �1.834 A, rO–H � 0.964 A) calculated with the basisset of Sadlej are shown in Fig. 3. All states are bentin the Frank–Condon region; the upper triplet state,13A0, gets linear at shorter (# 1.7 A) O–Br distances.At the equilibrium bond length it has two shallowminima at the angles/HOBr� 1198 and/HOBr�1768. The first excited triplet state, 13A00, is almostinsensitive to the angle variation in the Frank–Condon region; it has a minimum at/HOBr �109.88 but the bending potential is very flat to theleft from this point (Fig. 3). The bending vibrationscould be easily excited upon the 13A00 ← 11A0 absorp-tion and lead finally to a particular rotational distribu-tion of the OH product.

We have not presented here the bending poten-tials of the excited singlet states, but they areavailable on request. The first singlet excitedstate, 11A00, has a bending potential very similarto that of the triplet state. Calculations by gridpoints with the MCSCF LR method indicatesthat the 21A0 state surface in linear geometries has aconical intersection with the ground 11A0 state atrO–

Br t 2.5 A. Hence the linear structure has a barrier fordissociation and recombination reactions H–O–BrOOH 1 Br. The rigidity factor for the recombinationprocess is quite high; only angles in the vicinity

/HOBr t 1008 correspond to the reaction path forboth processes.

The predicted BrHO triplet structure near thedissociation limit (Table 1) has no importance forthe photo-dissociation process as the Br and OHfragments pass through the region of long O–Brdistances (r t 3–4 A) with high velocity. But thistriplet state of the radical-pair-like structure (Fig.1(b)) could probably be important for the recombina-tion reaction Br(2P) 1 OH and for the processdescribed by Eq. (1).

In comparison with the singlet state encounters, thetriplet state collisions of two radicals occur more oftenbecause of the statistical weight ratio (3:1). Theweakly bound triplet radical pair (Fig. 1(b)) formedat slow collision conditions in the cold stratosphere isa metastable spin trap. It has a few quasi-degeneratestates with very large spin–orbit coupling matrixelements between the singlet and triplet counterparts.It means that the Tw S intersystem crossing can occurleading to the total enhancement of the recombinationrate and yield.

The disproportionation reaction (1) must proceedon the triplet state potential energy surface as theproducts HOBr1 O2 are in the total triplet state.The separated radical pair HO2 1 BrO could be eitherin the singlet or in the triplet state. The triplet entrancechannel can easily provide a hydrogen abstractionfrom the peroxide radical. At the same time the hydro-gen insertion into the O–Br bond can occur on thesinglet and triplet surfaces; the intermediate productconsisting of two triplet molecules BrHO and O2 canacquire singlet, triplet and quintet multiplicities aswell. When the oxygen molecule departs and takesaway two non-paired electrons, the triplet speciesBrHO produced by this separation can play the roleof a spin trap. Again, as SOC in the BrHO species isvery strong, the Tw S transition can occur inside theBr–HO biradical. After that an easy migration of thebromine towards the oxygen atom on the singlet statepotential energy surface will complete the reaction(1). Such a complicated scenarium seems to be quiteprobable for slow collisions in a cold stratosphere.

Some peculiar spin effects should be considered forother processes (3)–(5) [10,21]. The triplet states ofozone are definitely involved in reactions (4) and (5)in order to provide the necessary spin uncoupling forthese thermal processes.


3.2. The singlet–singlet transition probabilities

Fig. 4 presents the S–S transition moments calcu-lated by the MCSCF LR method with the basis set ofSadlej as functions of the same dissociation reactioncoordinate. All transitions have the initial singletground state, 11A0, denoted in the following as S0.The S–S transitions to then1A00 states (Fig. 4(a)) arepolarized along thex axis (perpendicular to the mole-cular plane). Transitions to the first excited singletstate (11A00) is rather weak; the maximum value ofthe transition dipole moment is equal 0.092 a.u. atrO–Br� 2.35 A. At the same time this function extendsfar away to the dissociation limit. As a result of such along stretch of the transition moment function we canunderstand the wide diffuse absorption band of theHOBr molecule in the range 320–400 nm [4,9] asattributed to the transition 11A00 ← 11A0. This is asO–Br* ← pO–Br* excitation localized on the O–Brbond. Both atoms almost equally contribute to theseantibonding orbitals at the equilibrium ground stategeometry; at slightly longer O–Br distances($1.85 A) the pO–Br* MO (6a00) quickly transformsto the Br orbital. ThesO–Br* MO has a small but non-zero admixture of s-AO character on the oxygen atomand this determines a low intensity of the transition.The bromine contribution is negligible as the s-AOcharacter on the Br atom is very small. The maximumof this band (lmax t 350 nm [9]) is in a good agree-ment with our calculated excitation energy (3.68 eV)

at the equilibrium ground state geometry (rO–Br �1.85 A) which corresponds tolmax � 337 nm. Theoscillator strength for this vertical excitation (f �0.00037) is in qualitative agreement with the observedintensity [9]. A steep descent of the dissociativepotential of the 11A00 state (the lowest dashed line inFig. 2(b)) provides a diffuse nature of the 11A00 ← 11A0

absorption band. We shall discuss this band in the lastsection.

The vertical transitions to the upper states of thesame symmetry, 21A00 and 31A00, are in the UV regiont6:8 eV (Table 2). They do not contribute much tothe absorption cross section (Fig. 4(a)).

Corresponding transitions to then1A0 states aremuch more intensive (Table 2, Fig. 4(b) and (c)).Transition moments for the first 21A0 ← 11A0 excita-tion of this type are shown separately in Fig. 4(c) dueto the particular importance of the correspondingabsorption band (lmax � 280 nm). The transitioncould be interpreted as asO–Br* ← nBr excitation.At the ground state equilibrium geometry both theses-MOs have appreciable coefficients on the hydrogenatom; finally the hydrogen and oxygen contributions(including s–p polarization) are important for theintensity of this transition (the Br atom contributesvery little despite of the high localization of the tran-sition density on bromine). At longer O–Br distances,the nBr orbital (16a0-MO) gets more localized; thehydrogen and oxygen contributions diminish and thetransition moment decreases (Fig. 4(c)).


Table 4The T2 ← S0 transition in the HOBr molecule. MCSCF QR calculation of the vertical transition dipole momentMa�Tb� �k11A0ue

Pi ai u1

3A0�Tb�l (in 1026 a.u.) near the experimental ground state geometry;DES,T is the transition energy,fa is the oscillator strengthfor theT2 ← S0 absorption polarized along thea axis

Ma(Tb) 3-21Ga 3-21Gb AhlrichsVDZb AhlrichsVDZb,c Sadleja Sadlejb

My(Tx) 34194 34317 40892 40677 39669 40395

Mz(Tx 7687 8001 8666 8646 10805 10977

Mx(Ty 50552 43932 – 72558 23500 24173

Mx(Tz) 26237 23349 – 32871 7293 7638

Transition energy (eV) and oscillator strengthDES,T (eV) 4.27 4.16 3.81 3.73 3.52 3.43fx 3.40× 1024 2.52× 1024 – 5.8× 1024 5.22× 1025 5.4 × 1025

fy 1.24× 1024 1.20× 1024 1.56× 1024 1.51× 1024 1.36× 1024 1.37× 1024

fz 6.19× 1026 6.53× 1026 7.02× 1026 6.84× 1026 1.00× 1025 1.01× 1025

ftotal 4.68× 1024 3.78× 1024 – 7.38× 1024 1.98× 1024 2.01× 1024

a The experimental ground state geometry;rO–Br� 1.834 A, rOH � 0.964 A, angle HOBr� 102.38.b The fixed ground state geometry;rO–Br � 1.85 A, rOH � 0.96 A, angle HOBr� 102.38.c Extended CAS (7a0, 3a00; 14 electrons in 10 MOs).

The peculiar contributions of the s–p polarizationdetermine a strange behavior of theMy/Mz compo-nents ratio for this transition moment as a functionof the O–Br distance (Fig. 4(c)). TheMy componentof the transition moment has maximum value just at

the Frank–Condon region (Fig. 4(c)). The calculatedoscillator strength for the vertical 21A0 ← 11A0 transi-tion (f t 0.002, Table 2) is in qualitative agreementwith the observed intensity [9]. Vertical excitations tothe upper states (n � 3, 4) are in the far-UV region( t 8.6 eV); their intensity is very high (seeMy compo-nent of the 41A0 ← 11A0 transition, dashed line in Fig.4(b)). At first glance they seem to be of no importancefor atmospheric photochemistry of the HOBr mole-cule, since the UV sun flux is absorbed by the Schu-mann–Runge continuum of the oxygen molecule. Butin the framework of perturbation theory the S–T tran-sitions (which occur in very important atmospherictransparent visible and near-UV regions) can borrowintensity from the 41A0 ← 11A0 UV transition. Such aredistribution of the absorption intensity, which isaccounted here by QR theory (and not accounted innon-relativistic quantum chemistry), is very efficientfor the HOBr molecule and leads to important conse-quences for the atmospheric photochemistry [10].

In the following we shall consider the absorptionspectrum of the HOBr molecule with particular atten-tion to the S–T absorption by analysis of the transitionmoments as functions of the O–Br stretching andbending coordinates presented above. We shall usethe QR method for calculations of the 13A00 ← 11A0

and 13A0 ← 11A0 transitions dipole moments in thevicinity of the Frank–Condon region. Results arepresented in Tables 3 and 4, respectively. A summaryfor the total spectrum calculations is given in Table 2.

3.3. The 13A00 ← 11A0 transition (T1 ← S0)

The orbital structure of the transition is determinedmostly by the 17a0 ← 6a00 (s*O–Br ← p*O–Br) excita-tion (the same as for the singlet counterpart). We donot completely agree with the qualitative interpreta-tion of this transition given in Ref. [4, Fig. 1]. This isnot thes* ← nBr excitation, wherenBr is the out-of-plane non-bonding orbital on bromine, but rathers* ← p* excitation, wherep*-MO has a large contri-bution from the oxygen atom. The MCSCF QR resultsof the 13A00 ← 11A0 vertical transition moment calcu-lations in HOBr molecule are presented in Table 3.The T–S transitions to the three spin sublevels of thetriplet stateTb have different intensities and polariza-tions, which follow simple selection rules [10]. Eachspin sublevelTb (7) has a zero spin projection on theb


Fig. 5. TheT1 ← S0 (3A00 ← 11A0) transition dipole momentscalculated by the MCSCF QR method: (a) The O–Br bond lengthdependence of theT1 ← S0 transition dipole moments; (b) the HOBrangle dependence of theT1 ← S0 transition dipole moments.

axis. Fig. 5(a) presents all the transition moments forthe T1 ← S0 absorption in HOBr calculated with thebasis set of Sadlej as the functions of the O–Brdistance.

Transitions13A00 ← 11A0 to the y and z spin sub-levelseach have their particular in-plane polarization.The transition to thez spin sublevel is the most inten-sive one (Fig. 5(a)). Following our notation (6), thistransition dipole moment is given as

Ma�Tz� � kS0uX

i

ai uTzl; �9�

wherea [ y, z are in-plane axes. They component ofthis vector is the largest one (Fig. 5(a)). The ratioMy /Mz is slightly changed along the dissociation coordi-nate (Fig. 5(a)), but the vectorM (Tz) is mostlyoriented along the O–Br axis. The transition to theyspin sublevel is much weaker (Fig. 5(a)). The orienta-tion of theM (Ty) vector changes a few times as thesystem moves along the dissociation coordinate.

Transition13A00 ← 11A0 to the x spin sublevelis xpolarized (x-axis is perpendicular to the molecularplane) and is practically negligible: see dotted linein Fig. 5(a). The choice of axes (Fig. 1(a)) is notarbitrary; it is very close to the directions of themagnetic axes determined by the zero-field splitting(ZFS) Hamiltonian, which accounts for intramolecu-lar magnetic interactions—spin–spin coupling andSOC (in the second order) [14]. They-directionalmost coincides with the main axis of the ZFS tensorat the equilibrium geometry. The orientation of theyandzZFS-axes is slightly changed along the dissocia-tion reaction coordinate but it is still close to thechoice shown in Fig. 1(a). This is important for analy-sis of the spin alignment in the dissociation products.

The Tb1 ← S0 excitations to bothy andz spin sub-

levels can borrow intensity from spin-allowedn1A0–11A0 transitions because of the SOC mixing

kn1A0uHbsou1

3A00�Tb�l: �10�In Eq. (10)b [ y, z. The most intensive S–S absorp-tion is determined by the 41A0–11A0 transition whichhas polarization close to the O–Br axis (Fig. 4(b)).The SOC integral (10) is large for theb � z compo-nent. This simple analysis explains some qualitativefeatures of theTb

1 ← S0 transition intensity in HOBr.Vertical excitation corresponds almost to the peak

on the My(Tz) transition moment curve (Fig. 5(a)).

This result determines a comparative importance ofthe T1 ← S0 absorption in the visible-near UV regionfor HOBr photolysis. From Fig. 5(a) it follows that theMy(T

z) transition moment is large even at long O–Brdistances: atr � 2.6 A its value is 0.02 a.u. This is acomparatively large value (compare with theS1 ← S0

transition moment,n� 1 in Fig. 4(a)). Accounting forthe overlap of the ground state vibrational wave func-tion and theT1 state nuclear continuum we can quali-tatively predict a long tail of theT1 ← S0 absorption ofHOBr; the valueuMy�Tz�u2 is large enough even atlarge O–Br distances (2–2.6 A˚ ) where the S–T tran-sition energy is small and nuclear wave functions stillhave non-zero overlap. The spectrum of hypobromousacid recorded by Barnes et al. [1] exhibits a very longweak tail of theT1 ← S0 absorption in the visibleregion until 650 nm. Our results support the tentativeassignment [1] and make it clear that the recentmodels of bromine chemistry in the Arctic tropo-sphere [2] should be reconsidered with account oftheT1 ← S0 absorption of hypobromous acid in visibleregion.

Angle dependence of the T1 ← S0 transition momentis presented in Fig. 5(b). The total 13A00 ← 11A0 tran-sition moment is almost parallel to the O–Br bondwhen the bond angle varies in the Frank–Condonregion (100–1058) but it deviates from this directionat higher angles/HOBr (Fig. 5(b)). In general a rota-tion of the total transition moment in they, z planefollows the O–Br bond rotation, but not exactly. Theyandz component of the most intensiveTz

1 ← S0 tran-sition get equal at/HOBr� 1328 (they andzprojec-tions of the other Ty

1 ← S0 transition are alsoequalized near this point,/HOBr � 1338). Clearly,the O–Br bond projections ony andzaxes are equal at/HOBr � 1358. The out-of-plane componentMx(T

x)is an increasing function of the angle until/HOBr�1758, but it is still negligible in the middle range of theO–Br bond rotation. At the linear structure, theT1 andT2 states get degenerate (this is the 13P state). TheT1 ← S0 transition moment along the molecular axis(O–Br direction),Mz(T

y) in Fig. 5(b), is much higherthan the perpendicular componentMy(T

z). The ratioMz(T

y)/My(Tz) � 3.47 corresponds to an angle of the

transition moment deviation from the molecular axisequal to 168. These results may be used for analysis ofthe rotational state distribution of the OH radical andfor spin alignment in both products.


The triplet state of the HOBr molecule could bestabilized in a rare gas matrix; in that case a phosphor-escence should be observed. Probably, the angle of theT state could be large and our results (Fig. 5(b))

should be useful for analysis of the EPR and phos-phorescence measurements.

3.4. The 13A0 ← 11A0 transition (T2 ← S0)

The second triplet state (13A0) potential energycurve is very close to the first singlet excited statepotential (the first dashed line in Fig. 2(b)). So theT2 ← S0 transition enters to the region of the firstS1 ← S0 absorption and seems to be overwhelmedby the 11A00 ← 11A0 band. In general, the S–S transi-tions are usually much more intensive than the S–Ttransitions. But this is not true for theT2 ← S0 (13A0 ←11A0) andS1 ← S0 (11A00 ← 11A0) transitions in HOBr[10]. In the following, we shall show that this is not anartifact of perturbation theory (and of the QR)approach.

Let us start to analyzethe T2 ← S0 (13A0 ← 11A0)transitions to the y and z spin sublevels. According tothe selection rules [10], these transitions have out-of-plane (x) polarization and borrow intensity from then1A00 ← 11A0 andn3A0 ← 13A00 transitions. Both theS–S and T–T transitions withn� 1, 2 are of very lowintensity.

In the framework of perturbation theory, the contri-bution of the firstS1 ← S0 (11A00 ← 11A0) transition tothe T2 ← S0 transition dipole moment

Mx�Tb� � k11A0ueX

i

xi u13A0�Tb�l

� k11A00uHbsou1

3A0�Tb�lE�13A0�2 E�11A00� k11A0je

Xi

xi j11A00l

�11�(whereb [ y, zare in-plane axes of spin quantization)could be overestimated since the energy gapE(13A0)–E(11A00) is not much higher than the SOC matrixelement. In calculations of HOCl the perturbationtreatment was doubtful at some O–Cl distances (thecc-pVDZ basis set calculation) [10]: nearrO–Cl �1.97 A the T2 (13A0) and S1 (11A00) states were soclose in energy (the energy gap was only 0.017 eV)so the vicinity of this point was excluded from thetotal intensity summation [10].

In HOBr the situation is much more safe (exceptAhlrichs VDZ basis set results, Table 4). Though theenergies of theT2 (13A0) andS1 (11A00) states are ratherclose in the Frank–Condon region (Fig. 2(b)) the


Fig. 6. TheT2 ← S0 (3A0 ← 11A0) transition dipole moments calcu-lated by the MCSCF QR method: (a) the O–Br bond length depen-dence of theT2 ← S0 transition dipole moments; (b) the HOBr angledependence of theT2 ← S0 transition dipole moments.

perturbation theory is still applicable for almost thewhole region of the O–Br bond lengths. The energygap in the Frank–Condon region, 0.27 eV, is muchhigher than the SOC matrix elements

k11A00uHbsou1

3A0�Tb�l �12�which are equal to 334.2 cm21 for thez-spin subleveland 786 cm21 for they-spin sublevel. The ratio of theSOC integral to the energy gap is 0.15 and 0.36,respectively. The square of the last value providesonly 13% of theS1 ← S0 intensity borrowing so theperturbation theory is applicable. Besides that, thisSOC mixing is only important for theMx componentsof the Ty;z

2 2 S0 transitions, which are less intensivethan the other transition,Tx

2 2 S0 (Fig. 6(a) and (b)).1

In HOBr the in-plane polarization of theT2 ← S0

absorption prevails over the out-of-plane polarizationin a wide region of the studied O–Br distances (Fig.6(a)) and angle variations (Fig. 6(b)). The total transi-tion dipole moment vector has almost constant orien-tation (at least in the Frank–Condon region); it pointsout-of-plane, but has the largest projection along theO–Br axis. This follows from an analysis ofthe T2 ←S0 (13A0 ← 11A0) transitions to the x spin sublevel.According to the selection rules discussed in Ref.[10], these transitions are in-plane (y,z) polarized.Calculations along the dissociation reaction coordi-nate show (Fig. 6(a)) that this absorption has predo-minantly y polarization (along the O–Br axes). Thevertical transition momentMy(T

x) for the T2 ← S0

absorption (Fig. 6(a)) is about 0.04 a.u in all basissets (Table 2), which is comparable with the spin-allowed S1 ← S0 transition (Fig. 4(a), solid line).The transition momentMy(T

x) has its maximum nearthe Frank–Condon region (Fig. 6(a)), so it produces agreat contribution to theT2 ← S0 absorption crosssection. Thez-polarization (O–H bond direction) isless intensive (Fig. 6(a), dotted line). Similar behaviorhas been obtained before for HOCl [10]. This transi-tion borrows intensity mostly from the very intensive41A0 ← 11A0 transition (and also from othern1A0 ←

11A0 excitations). The energy gapsE(13A0)–E(n1A0)all are quite large, so results of perturbation treatmentare reliable.

The angle dependence of the T2 ← S0 (13A0 ← 11A0)transition momentis shown in Fig. 6(b). The perpen-dicular component rapidly diminishes with theincreasing bending angle. So the intensity borrowingfrom the 11A00 ← 11A0 transition becomes unimportantwhen the angle increases. At the same time, the in-plane polarized transition intensity does not changemuch. There is only a rotation of the in-plane transi-tion dipole moment vector, i.e. a redistributionbetweenMy(T

x) andMz(Tx) components. They andz

projections of theTx2 ← S0 transition moment are

equalized just at/HOBr � 1358, where y and zprojections of the O–Br bond vector are also equal.The transition momentMa(T

x) is almost parallel to theO–Br bond direction in the region 115–1658. Thelargest deviation occur at/HOBr � 1758 and thenat/HOBr� 1808 the vectorM (Tx) coincides with theO–Br bond direction again. The total transitionmoment for theT2 ← S0 excitation in the linear struc-ture is the same as for theT1 ← S0 transition (compareFigs. 6(b) and 5(b)); the parallel to the perpendicularcomponent ratio,Mz(T

x)/Mx(Tz) � 3.47, corresponds

to the same deviation angle from the molecular axis(16.18), which was obtained for theMz(T

y)/My(Tz)

components of theT1 ← S0 transition. TheT1, T2 ←S0 excitations become degenerate components of the3P ← 1S1 transition in the linear molecule.

As it was mentioned before, the considered secondT2 ← S0 transition enters the region of the firstS1 ← S0

absorption at the equilibrium ground state geometry.The calculated oscillator strength for theT2 ← S0

vertical transition in HOBr is quite high (f � 2 ×1024, Table 4, Sadlej basis set) with 70% of intensitypolarized along the O–Br axis. Though the in-plane/out-of-plane polarization ratio strongly depends onthe basis set used (Table 4), a strong in-plane compo-nent (the O–Br axis) is definitely present in theT2 ←S0 absorption. This is in a contrast to theS1 ← S0

absorption which in entirely out-of-plane polarized.So the first intensive near UV absorption band,lmax� 350 nm, in HOBr [1,9] should consist of twotransitions,T2 ← S0 andS1 ← S0, with different polar-ization.

TheS1 ← S0 transition moment (Mx) increases withlarger O–Br separations (from 0.06 a.u. at the


1 We should remind that this discussion refers to calculationswith the basis set of Sadlej. Calculation of theMz transition moment(11) in CAS (6,3) with Ahlrichs VDZ basis set (Table 4) failed at thepoint rO–Br� 1.85 A, since the energy gapE(13A0)–E(11A00) was toosmall (0.045 eV). The transition moment (11) acquired an unreason-able value (0.2 a.u.) since the perturbation treatment cannot beapplied at this point.

equilibrium point up to 0.09 a.u. atrO–Br� 2.3 A, Fig.4(a)). This behavior determines a diffuse absorptionband in the HOBr spectrum withlmax� 350 nm. Thisis a precursor of the stronger peak atlmax� 280 nm(theS2 ← S0 transition) [9]. A diffuse absorption bandin the HOBr spectrum (lmax� 350 nm) determined bythis S1 ← S0 transition has a significant contributionfrom theT2 ← S0 excitation. Both transition momentsare increasing functions of the O–Br distance in theFrank–Condon region. The functionMy(T

x) increasesmore strongly (Fig. 6(a)) and has a maximum atshorter distancerO–Br � 2.06 A, which is closer tothe Frank–Condon region. Though theS1 ← S0 transi-tion is more intensive (Table 2, Sadlej basis set), bothtransitions should be qualitatively important for theintensity of the absorption band at 350 nm in theHOBr molecule.

4. Conclusions

QR calculations with the MCSCF ground statewave function provide a reliable interpretation ofthe absorption spectrum of hypobromous acid. Thecalculated oscillator strength for the firstT1 ← S0

transition (13A00 ← 11A0) is about 8× 1025, so it iseasily observed in the absorption spectrum at 440 nm[1]. The nextT2 ← S0 transition is more intensive andenters the region of the firstS1 ← S0 absorption (lmaxt350 nm) of hypobromous acid [4,9]. The most reliableMCSCF QR calculations with basis set of Sadlejpredict f � 2 × 1024 which is comparable with theoscillator strength of theS1 ← S0 transition itself (f �3.8 × 1024). These findings qualitatively agree withintensity measurements of the long-wave absorptionin the HOBr molecule [1,9].

Absorption by the first and second excited tripletstates of hypobromous acids makes an importantcontribution to the overall atmospheric photochemicalactivity of this molecule. The model of sudden ozonedepletion in the Arctic troposphere [2] includesuncertainties in the HOBr photolysis rates; thishampers the assessment of the bromine chemistrycontribution to the Arctic ozone loss. An additional

important contribution of the S–T long wavelengthabsorption to HOBr photolysis rate may thus haveserious implications for the ozone depletion model[1].

References

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Ab initio study of the singlet-triplet transitions in hypobromous acid

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Transcript of Ab initio study of the singlet-triplet transitions in hypobromous acid