The computation of oscillator strengths and optical rotatory strengths from molecular wavefunctions. The electronic states of H2O, CO, HCN, H2O

m-Ch@k3lPi;ysics 0 North-Ho&d

25 (1977)409-424 Publishing Company

ArviRAUKandJose M.BARRIEL Departmentof CilemMIy,ihiversity of Calgary,Calgary,Alberta,Canadn Received 7 March 1977

Amethod is described for the determination of excited state wavefunctions from which transition moments over one electron operators can readily be determined correct to fust order in perturbation theory. These are used to calculate electric dipole oscillator strengths for transitions to all of the lower lying singlet electronic states of H20, CO, HCN. CH20, Hz02

and CzH4. In addition, optical rotatory strenghts are computed for Hz02 and a twisted form of&Ha. Provided that diffuse functions are incorporated in the basis set, the computed properties for both valence and Rydberg states agree reasonably well with experimentally determined values where these are available for comparison. It is suggested that for Rydberg states, an empirical correction equal to the difference between the experimentally determined vertical ionization potential and that estimated by Koopmans’ theorem, be added to the excitation energy obtained by configuration interaction where the space of configurations is restricted to single excitations from the ground state. Excitation energies adjusted in this way agree very well with experimentally determined vertical excitation energies and with values obtained by more rigorous methods.

Lhtroduction Assignment

of observed electronic

spectra on the

basis of quantum mechanical calculations falls into the range of problems which require going beyond the Hartree-Fock level ofapproximation. The construction of open shell excited state wavefUnctions from orbitals optiniized for the closed shell ground state is unsatisfactory, in general, tihile separate optimization of thk excited state described by a single configuration yields energieS which are subject to the same kind of correlation error as exists for the Hartree-Fock ground state. The correlation defects can not be assumed to cancel when one takes the difference in energies to obtain an excitation energy. ‘Fhe most commonly used and conceptually the most straightforward procedure beyond the Hartree-Fock leirel is configuration interaction (CI), which involves diagonaliiation of the hamiltonian matrix in the space of configurations constructed by replacements within the set of o&upied orbit& by members from the set of virtual orbitals; Experience has shown that if the set of configurations is selected from all single and double

replacements (excitations) with respect to a reference determinant (configuration), most of the correlation error in the energy of the state whose description is dominated by the reference configuration is accounted for. Unfortunately, if the basis set is reasonably large, construction and diagonalization of the hamiltonian matrix becomes very difficult if not impracticable. We will not attempt to review the many Important advances that have been made to maximize the efficiency of CI, given that a truncation of the configuration space is necessary. Notable among these are the many natural orbital techniques [l-5], methods which avoid construction of the hamiltonian matrix [6] and procedures which attempted to avoid the bottleneck imposed in most methods by the necessity to transform to molecular integrals [7-lo]. Use of perturbation theory to abate some of the difficulties involved in CI has long been advocated [6,9-141 and this is the approach we have adopted and describe below. In order to assign electronic spectra, it is desirable to have not only the predicted excitation energy, but also some information regarding oscillator strengths and, where applicable, optical rotatory strengths as’well. We

410

A. Ra& J.M. .&rriet/Computation of oscillarorsrreugthsand optical rotatoy strengths .

here describe a procedure for the computation ofelectronic oscillator strengths and optical rotatory strengths which can be applied with moderate computationaleffort to small and medium sized molecules. Experimentally, the area under the absorption band is the optical transition intensity, or oscillator strength, L namely f= 4.32 X 10-g~e(v)dv,

(1)

where E is the molar extinction coefficient and v is the frequency in cm-’ . By applying time dependent perturbation theory to the interaction of the molecule with the radiation field, one obtains the quantum mechanical expression for the dipole oscillator strength, fan, for a transition between two states, 0, and II, &, =(2m/3Fi”e2)(E,-~o)I~~OI~IJI,~12,

(2)

where P is the electric dipole moment operator, a sum of one electron position operators fk, p=er=eCr k

li’

(3)

.

ator has the consequence that the mixed product form off",should be less sensitive to correlation errors in the Kound state [15] or the excited state wavefunctions. If the particular molecule lacks anS, (n > 1)aXi.s of symmetry, the oscillator strengths for absorption of right and left circularly polarized light differ.rhe resulting circular dichroism (CD) can be measured experimentally, the fundamental quantity being the rotatory strength R which is related to the area under the CD band by R =0.696

X 1042j.[6(X)]/X

dA,

(7)

in cgs units, [0] is the molecular ellipticity here given as a function of wavelength, X, RW)l = 3300 Ce&)) - e,@)l.

(8)

The quantum mechanical expression for the rotatory strength, Ron, of a transition between the two states, 0 and 11,can be derived by time-dependent perturbation theo:y [17], and is expressed as R On = ImCc~~l~I~,)(rL,lmItL,>},

(9)

where “hu” means the imaginary part of, and m is the magnetic dipole operator.

and J$~ and 3/, arc the state wavefunctions for molecular states with energies E,, and En, respectively_ The commutator relationship 1% rl = -(fi21m)V,

(4)

allows one to rewrite the electric dipole transition moment as ~~,lrl~~~=~~2~~~E,~-EE0)1~~~l~J~~~-

(9

Eq. (5) is an identity for exact eigenfunctions of the molecular hamiltonian operator,%, and allows an independent calculation of the excitation energy (En Eo)_ The degree to which the transition moments
fan=~<~,lvl~,~~9,lrlJ/,>.

(6)

Use of expression (6) for fanhas advantages which have been previously discussed [l&I61 _Srkfiy stated, it can be shown that the antihermiticity of the V oper-

Use of expression (9) for the determination of rotational strength has the practical disadvantage that, since approximate wavefunctions must be used (and hence, eq. (5) is not obeyed), the result is not, in general, independent of the choice of origin. Application of rclationship (5) removes this difficuhy, namely

where the operators V and r X V are sums of sne-electron operators.

2. Perturbative configuration interaction In this section we discuss the application of perturbation theory to obtain CI wavefunctions for the ground and excited states, Which are suitable for the computation of transition moments ofone-electron operators.

A. Rauk.J.M. BarrieI@nputation of oscillatorstrengthsand opticalrotatoy strengths

Perturbation theory; correct top-order for the energy or (p - 1) order for the wavefunction, varies as N*P, where Nis the number of basis functions. Therefore, if the number of basis functions is not so large that the integral transformation constitutes a major problem (say, Ns 40), one could obtain, with little additional effort after the SCF MO calculation, the energies to second order and the wavefunctions to first order for ground and excited states. In the usual CI approach, the configurations which constitute the CI basis set are generated by successive replacement of occupied spin orbitals of the ground state HF wavefunction. When representing a state as a linear combination of configurations, there is usually a small set of configurations, A, which interact strongly, and the remainder, often a very Iarge set, B, of confIgurations which individually contribute very weakly but which collectively have a large contribution to the energy. If the sets A and B are regrouped, one can express the harniltonian or interaction matrix in partitioned form, ff AA H.4n H=

(12) HBA

HBB

HAA is usually small enough to be diagonalized exactly. Thus the zero-order hamiltonian matrix is chosen as HAA

Ho =

I

0

’ (13) %

I

where (AB)jj = (HI&.

411

in the eigenvalues, the off-diagonal elements ofNBB are not requi-ed and the number of cor&gu~ations to be taken into account is drastically reduced.

3. Transition moments In order to obtain oscillator strengths and optical rotatory strengths according to eqs. (6) and (1 l), one needs to svaluate integrals of the type C$,lSl~~l,>, where the operator is a simple sum of one-electron operators,

6=96,.

(16)

We proceed to give the expressions which result after making tfre following simplifying assumptions, which are valid for closed shell molecules without appreciable diradical character in their ground state. (1) The A group for the ground state consists only of the sirlgle determinantal solution of the HartreeFock calculation, $J~~. It is well known that $,, yields the one-electron density correct to first order (2) Only single excited configurations contribute to the A group for the excited states. With these assumptions the ground and excited state wavefun
~o=~,r:+‘Cao DE DE$ DE' and

si..

(14)

Having defied the zero-order problem, straightforward application of perturbation theory, using the EpsteinNesbet unperturbed hamiltonian [14,18,19] yields

_ whereBl is the true hamiltonian. IfE,. is taken as (@ lfl@, then one has the Rayleigh-Schriidinger (RS) perturbation expansion [141. Use of($,l%XlJI> forEi corresponds to the Brillouin-Wigner (BW) expanston [ 141. If one does not go beyond second order

+.m DE DE4 DE

(18)

where E is a parameter temporarily introduced to keep track of the zero and fust order contributions. The subscripts SE, DE, and TE represent singly, doubly and triply excited configurations. The coefficients X are given Implicitly in eq. (15). Substitution of eqs. (17) and (18) into <$,I 61 JI,>, keeping terms only to first order in E, yields

412

A. Rauk. J.M. Barn&Zbmputation

of osnllator strengths and optical rotatory strengths

arise from two excitations of the same spin (SS)‘or different spin fpsj.Thespecific expressions for the coefficients by RS perturbation theory are

+e

c

EgA

X”SE(J,HF 161&.,)

(W Several points can be made with respect to eq. (19). The configurations in the excited state description which are doubly or more highly excited do not contribute to the transition moment to first order. This is a consequence of assumption (1) above. Doubly excited configurations in the ground state description do contribute to first order. Thus, within the scope of the above reasonable assumptions, one can evaluate transition moments correct to first order by considering only singly excited configurations in the excited state description, and at most, onIy those doubly excited configurations of the ground state description which yield non-zero one electron matrix eIements with configurations of the excited state A group. Each of the configurational wavefunctions is constructed to be an eigenfunction of& and S’ with spin angular momentum zero i.e. a singlet. Eq. (19) may be written in terms of the molecular orbitals involved in the excitations (E set to l),

$Uyyv = (vu lji)lAl$u~&.

(23)

is the nth eigenvalue of HAA if n is less than the dimension of the zero order block, HAA, or is given by the equation below, with i=j and II = u, if It is greater. In the latter case the summation in equation (2 1) reduces to a single term with the component of the eigenequal to 1. The energy denomivector ofH nators are AA’ c”U’7

En

Eti =eu - ei- Jti -t-X.

(24)

Ul’

~~~“=$+%-~i-~j-J,+~,-J~+~jy

-Jb

- Jjt +Jv+Juu,

(25)

AkfuD; = AE$$s” + I$,, + Kli - Kii - Ku,.

+

c

c

(i,u)EA (i,u)+(i,u)

x&,, ’

(26)

Jij and Kii are the usual Coulomb and exchange integrals expressed as special cases of the general repulsion integral by Jii = (ii&>,

(27)

and Kv = (Qlji).

(28)

The general repulsion integral (uil I@ is defined by Subscripts i,j refer to the occupied manifold of ground state SCF orbitals while U, u refer to unoccupied orbitals. The couples (i, u), refer co the particular replacement of orbital i by orbital u in generating a configuration_ The limits for the sums are discussed below. The superscripts OSS and ODS indicate that the coefficients belong to terms in the ground state (0) expansion which

‘uilvj’ =~~~~(I) pi

3(l)

oiler

dr, dr2-

(29)

Lastly, ei is the expectation value of the Fock operator with respect to the molecular orbital, 3, i.e. the molecdar orbital energy.

A. Rauk, J.M BarnXfComputation of oscilIatorstrengths and optical rotatory strengths

4. Excitation energies Since the contribution of doubly and triply excited (relative to the closed shell grollnd state) configurations to the excited state description do not appear to fust order in the computed transition properties, a considerable saving in computational effort is achieved by neglecting these altogether. Thus, the sums required to calculate the energy to second order by RayleighSchrijdinger perturbation theory are restricted to run over singly excited configurations. This simplification has a considerable effect on the absolute value of the computed vertical excitation energy, since the majcr portion of orbital relaxation and correlation of the “unexcited” N-l electrons of the excited state can only be accomplished by double (and to a lesser extent higher) excitations. This is particularly true in the case of Rydberg states which are derived by excitation of an electron from a single occupied orbital and where the average distance of the excited electron from the nuclei can be many times the molecular dimensions. In such a case, the remaining electronic distribution will closely resemble that of the molecular ion, and in the present approximation, its energy will be estimated by Koopmans’ theorem [2OJ _ Although we expect configuration mixing among the singly excited configurations to ensure that the Rydberg states arising by excitation from a particular orbital are in the correct order and with approximately correct energy separations, it is apparent that all of the vaIues will be shifted relative to the observed spectral values by the amount of error (due to correlation error and lack of orbital relaxation) introduced by the neglect of the doubly and triply excited configurations. Workers using variations of the “frozen core approximation” in the study of molecular Rydberg states have compensated for this source of discrepancy by introducing the appropriate experimentally determined ionization potentials [ZI-251. Such an empirical correction is appropriate in the present method as well, where the difference between the experimentally determined ionization potentials and those estimated by Koopmans’ theorem may be taken as an approximate measure of the error introduced by lack of orbital relaxation and non-cancellation of correlation errors. Thus, the quantity A_@ + R), AE(i-,R)=AE

talc + (IP$V’ + Ei,

(30)

413

where (IP)?@ is the experimentally determined vertical ionizationbotential of.the ith occupied molecular orbital, and ej is the Hartree-Fock molecular orbital energy, should be a better approximation to the true excitation energy if the excited state is a Rydberg state describable by excitations from a single occupied orbital i. Where these conditions are satisfied, we have tabulated A@+ R) as well as AEcdc, in the tables. In cases where the excited state is a valence state, use of eq. (30) is not appropriate since strong interactions still exist between the excited electron and the remainder. In a number of cases, the excited state is described in terms of configurations derived from excitations out of more than just one of the occupied orbit&. This situation corresponds to some reorganization of the “unexcited core”. Where it occurs, the fractional excitation out of each occupied orbital has been indicated in the tables, as a percent and eq. (30) has been applied with reference to the orbital out of which most of the excitation occurs if the state is a Rydberg state.

5. Results and discussion The results of calculations on H,O, CO, HCN, H,O,, CH,O and C2H4 are shown in tables I-6 and are d$ cussed in separate subsections below. In all cases, the experimentally determined molecular geometry was used and kept constant in the description of the various excited states. The moderate sized 9sSp/4s basis set, contracted to 4s2p/2s [28] is common to all calculations and is referred to as the ‘valence basis”. This basis set is adequate to give a qualitative description of the molecuIar closed shell ground state, and for valence excited states, but is inappropriate for the description ofmany of the excited states with Rydberg, or intermediate character. Accordingly, for each of the molecules investigated, the valence basis was augmented by the addition of functions with suitably small exponents. As part of the present study involves exploring the basis set dependence of the method described in sections 24, various sets of diffuse functions were used in individual cases and these are described separately below. For the sake of brevity, we do not indulge in lengthy tabulations or discussions of results obtained with different basis sets, but restrict our comments to the salient features.

‘_rab&1 1’ ‘[_ ~;1 _

;

~Excitation energies an&oscillator strengths for the water molecule

State 1. _.x-

_-

.--=_

talc. a)

-_ -$,

Excitation energy (eV) eq. (30) b)

Oscillator &en&h expt.

--lb, - 3sq

‘-42 lbl + 3pbz IA1 3a1 - 3sal

8.59(9.95) 10.35

ref. [24] -_--

talc.

expt.

ref. 1271

0.041 g)

0.059

7.31

7.40d)

7.30

0.0464

9.07

9.1 c)

9.04

_

-

10.56(10.64j

9.70

9.70 d)

9.70

0.0972

0.05 g)

0.069

11.33(10.99)

10.05

10.00 d)

10.04

OX!093

-

0.012

‘Al

lbt -+ 3pal lb1 - 3pbI

11.45(8.69)

10.17

10.17 d)

10.16

-

0.013

% ‘AZ

lb, + 4sal lb, 4 3dbz

11.95 12.29

10.67 11.01

10.63 e,

10.64 10.87

‘fh

o.ob90 -0.0019

-

*B*

3al + 3pbl

12.33(14.63)

11.47

11.4 e)

11.46

0.0127

-

E

lh I&

lb* -t 3dal Zbl - 3da2

12.56(10.85) 12.59(10.28)

11.28 11.31

10.99 f) 11.07 a)

11.07

0.0156 0.0217

-

T

lh

Ibl + 3dbl

12.67(7.87)

11.39

11.12 f)

11.17

0.0096

-

‘RI

lb, - 3dal

12.67(7.99)

11.39

11.12 t-J

11.17

0.0080

-

a) b, ‘) g)

The numbers in parentheses are obtained by the ratio (JlolVl $n)/~$olrl I$,)_ For Ib,,e= -13.90, IP= 12.62,ref. [31];for 3al.E=-15.54, tP (~‘7)” 1468,ref. Ref. [32] _d, Ref. [33] _e, Assignedby Goddard and Hunt, ref. [24] _6 Ref. [34] _ Ref. [30] _

[31].

5.1. Water The space of configurations was generated by considering all possible single excitations from all of the The computed results for the water molecule with occupied orbitals except the atomic Is core orbitals, geometryOH=0.95 A, HOH= 105’ arecomparedwith into all of the virtual orbitals except the highest antiexperimental data and some previous calculations in bonding orbitals. The zero-order (A) group of configutable 1. All of the excited singlet states ofwater are rations for the excited state was chosen as the set of Rydberg states. The valence basis set has been augail configurations for which the value of ~(@~s(~$~)/ mented by the addition of a single set of s (as = O-025), (Q$$X~$~~ -
A. Rauk, J_M_-Bam~ei/Computation

of oscillator strengths and optical rotatoiy

strength

415

Table 2 Excitation energies and oscillator strengths for carbon monoxide Excitation energy (eV)

State

talc. a) X ‘l-l

n+lT*

i-

n+n*

‘z-

eq. (30) b,

Oscillator strength ref. [ZS]

exp.

8.62 (8.89)

8.39 c)

8.7

10.78 ‘)

12.0 12.8

0.209

0.195

0.24

-

_

0.041

0.017

0.060 0.084

P-+-IT *

n-+

11.95 (9.94)

c” ‘I;’

n-r

12.67C11.57)

11.53

11.40 c)

0.083

0.170

~‘II

n-t

12.63(10.73

IL.49

11.52 ‘1

0.058

0.10 1

‘c-

n-*

13.68(11.62)

12.54

12.37 @

0.014

*rI

l-l+

13.93U2.66)

12.79

c

10.81

ref. [ 291

‘2+

z

9.8 10.2

exp. =)

9.52 10.04

6’A

9.9 d) 10.5 d)

talc.

O.Oi8

‘) The numbers in parenthesesare obtained by the ratio ($olVl~n)/~~olrl $$. b~Forn,~=-15.l5,IP=14.01,~ef.]3l~;foora,e=-17.77,IP=16.91,ref.[31]. ‘) Ref. [35]. d, Ref. [36]. d Ref. [3i] __

state rather than as ‘B, (lb1 -+3dal), on the basis of the larger calculated oscillator strength of the ‘B, + IA, transition (0.0277 versus 0.0156). Exper@ent$ oscillator strengths for AZ-+2 ( ‘Bl + ‘A,) and B +X (IAl +- ‘Al) transitions have been determined [30] _The tabulated value for x +x falls in the range of the eper@_ental determinations. The calculated value fs B ,‘X ULOOIa_rgeby a factor of two. Our values for C 6 X and D + X agree reasonably well with those obtained by a considerably more extensive CI treatment [27]. 5.2. Carbon monoxide The computed results for carbon monoxide with CO = 1.128 t\ are shown in table 2, The valence basis set was augmented with two sets of diffuse s (0, (Y~= 0.010, 0.015; C, (Y~= 0.035,0.008) and p (0, up = 0.05,O.ol; C, IY = 0.025,0.005) functions [29]. l!h e first singlet excited state is found to be ‘n (n + K*). The computed excitation energy and oscillator strength are in good agreement with experimental values. The diffuse orbitals are not much used in the description of this state and equally good agreement with experiment is obtained if they are omitted from the basis set, AL?= 8.65 eV, f = 0.205. Inspection of the x* orbital reveals that the more diftkse p fknction of the valence basis set is an important component of the 8* orbital. The integral (lr*l?ln*), where r is the distance

from carbon in a direction perpendicular to the CO bond, has the value (2) = 3.12 (in units of bohr squared) indicating that the n + K* state is valence in character. The same is true for the ‘Z- and ‘A states which arise from the n + ‘II*excitations. For the remaining states, the diffuse orbitals are absolutely required. It is apparent from table 2, that the calculated energies of the first three states ’ II, ‘Z- and ‘A are in moderate agreement with experimental values while application of eq. (30) is essential to_obtai> good agreement for the rest. Except for the A 4 X transition, the calculated oscillator strengths agree only to within a factor of two or three with the experimentally determined values. 5.3. I&drocyanic acid The computed results for hydrocyanic acid with CN = 1.153 a and CH = 1.066 A are shown in table 3. The valence basis sets for C and N were augmented with two sets of s (C, (Y~= 0.035,0.008; N, or, = 0.050,0.011) and p (C, o[~ = 0.025,0.005; N, ap = 0.037,0.0075) functions, and a single set of diffuse s ((rs = 0.025) and p (arP ~0.020) [29] functions were added to the hydrogen atom. At first sight, the electronic states of HCN and CO should be very similar since they are isoelectronic molecules. There are, however, three essential differences. First, the orbital of HCN which corresponds to the highest occupied non-bonding u orbital of CO is a bond-

416

Table 3 Excitation energies and oscillator str&ths of HCN .

State -

Excitation energy (eV) talc. a)

x kci5 ‘A

27-E

7.21

IT-+**

7.74

c 53

50-+%T*

10.02(11.90)

P-+3S

10.40(10.71)

‘n

- _

eq.

(30) h)

exp.

Oscillator

strength

talc.

exp. ’

_

I ?-4 c) >729d) 8.14 e) 10.22

-

.-

0.0566

-

0.1564

-

0.1088 _

_

‘n

r-+3pa

lO.gO(9.64)

10.72

IA

;r+ 3pn

11.35

11.17

k1,+ _

r-+3piT

11.39

11.21

n+3pn

11.46(8.02)

11.28

0.0186

-

‘n

Ir+3d

12.16(10.42)

11.98

0.0386

-

-

-

.._ ~.

-

a) The numbers in parentheses are obtained by the ratio G$olVI tinX($clrl $J_ b, Forr, E =-13.79, IP= 13.61, refs. [31,38,39]; fork, E =-15.75, IP = 14.00 ref. [38]. c) The adiabatic transition enerw is 6.49 eV. The vertical excitation energy is estimated to OCCLU at Y” = 7 of the ‘A”%-+ x state I-

d) ~‘~~~~atic ‘) Ref. [40]

transition

energy [40,41] .

_

ing orbital (So) of the C--H bond and much lower in

energy. Secondly, HCN has the possibility of ready dissociation and some of its lower excited states may not be bound states. Thirdly, HCN has the possibility of becoming non-linear in its excited states. Indeed, there is ample evidence that the first three singlet states, which arise from rr + rr* excitations of the linear molecule, are significantIy non-linear. In table 3, the calculated and experimental excitation energies are for vertical transitions. Since the fast three states are components of the ‘Z-and ‘A states arising from a + rr* transitions in the linear molecule, transitions are electric dipole forbidden, but are observed as two very weak bands in the experimental spectrum. The a* orbital of both lZ_ and ‘A states is similar to the n* orbital of the n + r* state of CO. The expectation values, Gr*(?jrr**>,where, as before, r is the distance from carbon in a direction perpendicular to the molecular axis, are (?2) = 3.37 and (?>= 3.79, respectively. The next state, assigned to F-X, arises from the excitation of a 50 electron (C-H bonding orbital) to a d-like orbital. The calculated excitation energy is in reasonable agreement with the experimental value, if a large adjustment for correlation and relaxation errors

is made. However, application of eq. (30) is questionable since this state is not very diffuse in character (
417

A. Rauk, J.M. BarrielfConzputationof oscillator strengths and optical rotatory strengths T@le4 Excitation energies, oscillator strengths and optical rotatory strengths for Hz02 OscilIator

Excitation-energy (eV)

State

Optical rotatory

strength c,

strength talc. a)

eq. (30)

exp. b,

[RI’

.

2.516

[RIva

3.525

‘A

4b --r 3so*

6.24(38.13)

‘9

81% 5a _3so* 14% 4a 1

7.51(11.95)

‘9

4b+3so

9.07 (7.96)

6.87 d,

‘A

73% 5a --f 3so 16% 4b 1

9.72(LO.51)

7.81 e,

‘A

15% 5a -3sti* 12% 4b }

10.48(11.54)

8.28 d,

0.0131

21.95

9.31

71% 4b ‘3F? 20% 4a I

11.06 (9.62)

8.86 d,

0.1371

-0.05

0.40

5a+ 3pn*

11.25(10.92)

9.34 e)

0.0388

-23.73

4b-t

11.37(11.12)

9.17 d,

0.0172

11.j8C11.73)

‘B

4.04 d)

0.0013

5.60 e)

0.0207

-9.24

-7.68

i.0

0.0054

15.10

15.66

7.7

0.0078

0.0944

-23.99

0.025

12.59

-9.21

-10.10 0.0 10

6.78

a) The numbers in parentheses are obtained by the ratio (I$~~V~Q )/(&olrl I$,& b, Taken from a spectrum published in ref. 1481, see text for discussion. =) [RI’= 254 R (au) = 1.08 R (cgs x lo*‘); in [RIP AE, A& is in atomic units, 1 au = 27.21 eV. d)For4b,e=-13.71,1P= llSlre~[56]. e, For 5a, E = -14.47, IP = 12.56, ref. [56]. cited states are given in table 4. The absolyte config~.~ration for the molecule is designated “M” [44] and is

I H

rai H

“---$\ ,.

EH

w

(bl

Fig. 1. The absolute configurations of hydrogen peroxide (M) and ethylene twisted by loo (R). ’

shown in fig. 1. The lowest singlet state of H,O, reached upon vertical excitation is ‘A (4u+ 3su*) whic$arises from excitation of an electron from the non-bonding 4b orbital (one of the oxygen lone pairs) to a compact, and strongly antibonding combination (a*) of the 3s orbit& of the oxygen atoms. The second singlet state, ‘B, corresponds to excitation of one of the non-bonding Sa electrons to the same 3su* antibonding orbital. Both states are probably dissociative. The experimental UV spectrum of H202 below 6.7 eV is continuous with no indications of absorption bands [45] i. The first ‘A and ‘B excited states are predicted to be in this range if the large adi For a discussion, see ref. [46].

418

A. Rauk.JM BarrieljComputation o_foscillator strengths and optical rotatow strengfits

justment (z 2 eV.for correlation and relaxation errors is made via eq. (30). The photolytic decomposition of peroxides to give.onlyradicaIs is well known and has been discussed theoretically [47]_ On the basis of the higher calculated oscillator strength (table 4), we would conclude that the dominant decomposition pathway is via the second (‘B) state. Schiirgers and Welge [48] have published the absorption spectrum of H,O, in the 6.2 - 10 eV range. The absorption increases over this range except for a single broad maximum at 7.7 eV and the suggestion of a shoulder at about 7-O eV. The third and fourth singlet states, ‘B and ‘A, are calculated (after compensating for the rather large discrepancy between calculated and experimental ionization potentials) to fall in this region. Both arise from excitations to the symmetric combination of the 3s orbitals on the oxygen atoms (3s~) from the non-bonding orbitals. On this basis, both are probably bound states. The CI expansions for all of the states listed in table 4 are quite complex, being in some cases, combinations of configurations derived by substitution of more than one occupied orbital. We have indicated the relative importance of the various origins of excitation and an approximate description of the terminal orbitals where these descriptions are relatively simple. A more detailed description of these states would require a more extensive analysis than is warranted by the scope of this paper, and by the quality of the basis set used In this case. Little is known experimentally of the optical activity of H,O, or of peroxides in general. H,O, in the gaseous or liquid phases is not resolvable because of the low barrier [49] to interconversion of the enantiomeric conformations. H202 does apparently crystalliie in a single enantiomeric form [SO] but the absolute configuration has not been determined. Coupled HartreeFock-Roothaan perturbation theory has been used to c&ulate the rotatory power (p) as a function of energy below the first absorption [51]. fl was shown to be dominated by electric dipole radiation polarized along the O-O bond. We find that the polar&af.ion of the first ‘B + ‘A transition is along the O-O bond but it is not obvious from the present study if any single transition dominates the sign of 9. The circular dichroism spectra of molecules containing the related disulfide linkage have been studied theoreticaUy [52, 531 and experimentally [54] . The signs of rotation of

the first two transitions ‘A + ‘A, IB +- IA of HZ02 agree with the predictions and observations in the case of disulfides, and with the general rule for chromophores of C, symmetry enunciated by Wagni&re and Hug [SS]. 5.5. Formaldehyde The electronic states of formaldehyde have been very extensively studied, both theoretically and experimentally. We do not attempt to review these studies, but refer the reader to the references cited below and in table 5. Considerable discussion persists regarding the specific assignment of the observed Rydberg states, and the whereabouts of the s+ 6 state which is mysteriously absent from the experimental spectrum_ The calculated vertical excitation energies and osciliator strengths for’transitions to the lower lying excited states of Formaldehyde are presented in table 5 and compared with experimental data. The geometry is CO = 1.203 8, CH = 1.lOl, HCH = 116.5”. The valence basis set has been augmented with diffuse s and p functions on all atoms as described in sections 5.2 and 5.3. In addition, a single set of d (ad = 0.0118) functions has been added to the carbon atom. This is the largest basis set (54 functions) employed in the present study. The large number of diffuse functions is necessary to describe Rydberg s and p orbitals up to n = 4 and the Rydberg 3d functions. The experimental absorption spectrum is very well resolved, and well characterized with respect to transition energies and oscillator strengths. The lowest singlet state is ‘A2 (n + n*) for which the vertical excitation energy of the dipole forbidden transition is 3.49 eV. The calculated value, 4.29 eV, agrees poorly, and the value obtained is independent of basis set size. The ‘A2 (n + d) state is calculated to be a valence state, as are three other states derived from excitations out of lower occupied orbit& Into the n* orbital_ The calculated excitation energies for these valence states are: 21Bl (Sal + or*) 9.58 eV; 5lA (n + n*) 10.49 eV; 51A2 (1bI +7~*) 11.53 eV. The last two of these are not shown in table 5. The descriptiori of the 5lA, state as n + n* is somewhat arbitrary since 49% of its oneelectron density is due to configurations which arise from excitations out of the 5al (0 lone pair) orbital [57] . The oscillator strength calculated for the 5’4, + llAl transition is f= 0.0358. Previous calculations

A. Rauk, J.M_ Barriel/Campufation of oscillator strengtfrsand optical rotatory strengths Table 5 Excitation

ene@es and oscillator

strengths

State

419

for formaldehyde

Excitation talc.

Oscillator

energy (eV)

a)

eq. (30) b,

exp. ‘1

talc.

strength exp. c)

exp. d)

1

‘Aa

n-rx *

4.29

1

‘Ba

n+ 3so

8.67(5.61)

7.40

7.08

0.0 144

0.038

2

tB2

I1- 3pq

9.468.68)

8.19

7.96

0.0328

0.017

0.017

2

‘A,

n-+ 3pb2

9.58C6.89)

8.31

8.14

0.0382

0.038

0.032

2

‘Aa

n+3pbl

9.78

8.51

3

‘Al

{ ;:;rz3db2

0.0084

0.010

0.015

0.0164

0.012

0.017

==0.0002

3.49

10.36(6.05)

9.09

8.88

9.26 9.41 9.43 9.44 9.44

9.02

0.028

5

‘B2

n+4s

2

‘B,

5al+n

6

tB2

n+ 4pai

10.53(8.03) 10.68C9.471 10.70(9.1Oj 10.71 10.71(5.44) 9.5LV20.09) 10.89(9.82)

4

rA,

n+4pbz

10.91C7.48)

9.64

0.032

lAa

n-+4pb,

10.98

9.71

0.0140 -

0.028

4 7

‘B2

n-t5pal

11.40(9.09)

10.13

0.0 104

0.009

0.022

3

‘Ba

n+ 3dq

4

‘Bz

n+ 3dal

1

rB1

3

‘A,

n+3da2 n-L3db, *

0.0023 0.0005 5.0006 0.0033

9.62

0.0088 9.62

a) The numbers in parentheses are obtained by the ratio($olvl~,)l(~oIrl~~~. b, Forn, E= -12.15 eV.IP= 10.88 eV, ref. 1311. _ _ ‘j Ref. 1611. d, Ref. [62].

have qlso predicted that the s + n* state would fall

near the first ionization potential [58-601, amuch larger oscillator strength [58,59] . The 2lBl

(5al

+ n*) valence

state

is not

but with likely to

be assigned to a band in the experimental spectrum be-

cause of its low calculated oscillator strength and the fact that it occurs in the neighbourhood of the 3d and 4p Rydberg states. With the largest basis set, the Rydberg states of formaldehyde are readily identified and characterized. The calculated energies, after application of eq. (30), are within 0.2 eV of the experimental values with the exception of first state, 1 ‘B2 (n + 3so) where the deviation is an acceptable 0.32 eV. The calculated oscillator strengths are gene&y in good agreement with the experimental values, the worst cases being for the first two ‘B2 +- 1 lA, transitions where the discrepancy is a factor of two. We believe that an enlargement of the diffke basis set on oxygen would reduce this discre-

panty. The present calculations suggest a reversal (on the basis of excitation energy) of the assignments for the second and third Rydberg states previously made both theoretically [58-601 and experimentally [61, 621. Although the experimentally obtained oscillatnr strengths are quite different [6 1,621, the calculated oscillator strengths Iisted in table 5 are quite similar. Comparison of values obtained by the dipole length (f(r - r)), mixed (f(V - r), table 5), and dipole velocity cf(V-V)) forms of the oscillator strength expressions, Iends only tenuous, additional support to the present assignment. The values for the second and third transitions, respectively, are: f(r - r) 0.0309,0.0461; f(V -r) 0.0328,0.0382;f(v -V) 0.0348,0.03 17. There are no difficulties with the remaining assignments which can be made on the basis of calculated oscillator strengths, energies and quantum defects. Of the n -+ 3d transitions only two, n + 3db, and n + 3dal: are predicted to have significant oscillator strength, and

420

A.-Rauk. J.M. Bam~l~Campuration of oscilIatorstrengths and optical rotatoty strengths

Table 6 .Excitation

energies, oscillator strengths and absolute rotatory strengths of ethylene

State

Excitation energ calc.a1

eq. (30) b)

exp. c)

1

‘h.u nd3sa

7.21(7.52)

7.42

7.10

1

‘Big

7.80

8.01

7.45

7.93

8.14

7.25

g 1~’ 3@,2,

1 ‘Bzg ?i-3sbl,, I’B lU Vx+n

*

8.17(7.34)

1 2

‘Ag

y’

3@3,

‘B3u

n +

JFa,

1

‘A,

ir + 3pb;g

1 2

‘Bzu n- 3dblg ‘B IU b + 3pbzE (IT*)

2

$g

(n)

16’6a-+ 82% ,C-

1BI,

8.45 9.03

8.84

9.05

9.48

9.31(15.65)

952

9.50

9.71

999

10.20

w~_~+z*

[65]_

d, Ref.

0.269 -

0.34 -

8.25

0.009 1 _

0.0087 _

-

[79]:

e, See footnote

[RI’

[Rlv~

-23.62

-6.48

-2.96

-0.71 0.15

0.53

-126.96 -

-33.65

-4.67 _

-1.70 0.62

2.26 -28.39

-10.44

0.0938

196.45

63.67

_

0.0003

1.79

0.72

O.CiOOS 9.05

d,

(twisted)

-

0.0 126 _

0.34

096

0.0118

-5.46

-3.18

0.000 1

-0.05

-0.02

0.0128

10.27

a) The numbers in parentheses were obtained by the b, Fern, e= -lO.iO, IP= 10.51 eV, ref. [31].

c, Ref.

0.3 15 -

9.18

9.27

exp.

7.66

0.0117

n+4sag

‘B2g

-

0.0134

r-+4sblu

3

0.0020

-

0.0120

‘B zg

3dag

-

0.072

‘Bs,

r--c

0.04

0.0047

8.89

3

x+

0.0827

8.61

2

‘Au

0.0865

9.23

9.45

*B3g

(twisted)

9.34

9.24

1

Calc.

(planar)

9.02

3pb,, 14% OCH -

2

CaIC.

9.13(9.00) 9.36

81% OCM+“* 3

8.24 8.82(9.98)

Rotatory-strength

Oscillator strengtt-

(eV)

_

-

88.55

36.25

-

ratio ($a101$,,)/($olrl$,).

c of tab!e 4.

the values are in reasonable agreement with experiment. The n + 4s transition is predicted to have an energy similar to the remaining n +3d transitions, but is not observable because of its very low calculated oscillator strength. Two of the n * 4p transitions are expected to have measurable oscillator strengths and to be nearly degenerate. The combined oscillator strength, f= 0.0228, of the n + 4pal and n + 4pb, transitions is in reasonable agreement with the experimental values (table 5). 5.6. Ethylene The calculated excitation energies and oscillator strengths are displayed in table 6. The valence basis set was augmented by the addition of a single set of diffuse s (~ys= 0.015) and p (aF = 0.011) fimctions onto each carbon atom and a single diffuse s (01s= 0.025)

function into each hydrogen. Two d fimctions (ad = 0.059,0.0059) which do not contribute to the ground state wavefunction but allow the construction of ‘B and ‘BZu Rydberg states were also added to each car2 on atom. The experimentally determined geometry (CH = 1.080, CC = 1.332, HCH = 118”) was used. The orbitals and states are classified according to the irreducible representations of the point group D,,. Following Merer and Mulliken r64] , the configuration of the ground state is ‘Ag -. .(3a.J2(lb3 )2 (ib3u)‘. The symmetry designationoftheV(Ir+n j”j stateisBl,. It is particularly important to decide on the valence or Rydberg nature of the various states since adjustment for probable correlation and relaxation errors via eq. (30) would have severe consequences on the relative positions of some df the excited singlet states and a similar impact on the interpretation. Iri particular, Rydberg states which can be identified as arising from

A. Rauk, J.&I. Baniii/Camputation

of oscillator strengths and optical rotatory strengths

excitations out of the lb3 (uCH) and 3a (or& orbitalsare shifted by 1.28 e% and 1.40 c$ respectively, to lower energies relative to Rydberg states which arise from Ib3,, (7~)excitation. In order to determine whether application ofeq. (30) is appropriate, we have evaluated for each of the four states which involve excitations to n*-like orbitals (r is the distance from one of the carbon atoms in a direction perpendicular to the molecular plane). The values are: llBlu V (n + n*), (?, = 15.3; 2lBt, (rr + n*), (h = 58.3; 2tBr (ciC+ + n*), ($>= 3 -48; 2fB,, (occ + n*), (I *) = 5 .82. As a second criterion, we compare values for excitation energies and oscillator strengths for the same four states calculated with and without diffuse basis functions. The values are (without diffuse functions in parentheses): 1‘Blu V(n + ?T*),At? = 8.17(8.38) eV,f= 0.315 (0.363); 2rBln (n+n*), QE = 9.13 (20.2) eV,f= 0.72 (-); 2’Bt (u~_~ --, n*), AE = 9.36 (9.34) eV,f(twisted) = 0.09f8 (0.068); 2$a (UC-C +rr*), AI?= 10.27 (10.24) eV,f(twisted) = 0.0134 (0.000). By the second criterion, only the second ‘Blu state is clearly Rydberg in character since the diffuse Functions are required to describe its rr* orbital. The energies of the two gerade states are unaffected by omission of the diffuse basis functions. These states, by both criteria, must be regarded as valence states. We are left with the thorny issue of how to characterize the V state! By the first criterion, it may be regarded as a Rydberg state, by the second criterion, a valence state. Buenker and Peyerimhoff [63] have recently published a thorough analysis of the V state of ethylene, concentrating on tbe effect of various basis sets and various levels of sophistication in CI procedures on the description of the V state. Their results were similar to those cited above and they concluded that the V state is best characterized as “an essentially valence-shell species, with associated oscillator strength in the 0.25 to 0.30 range. Its orbital is nevertheless seen to possess a considerable amount of diffuseness”_ We conclude that application of eq. (30) to any of the ( + rr*) states except the second lBlu state is inappropriate. By a similar reasoning all of the remaining states are Rydberg states. The singlet states are listed in order of increasing energy in table 6 after application of eq. (30) to all except the three valence states, as discussed above. The excitationenergies for ethylene twisted 10’ in the sense shown in fig. 1 are not included in table 6 since they

421

were uniformly lower than the planar excitation energies by only 0.07 C 0.01 with three exceptions which reflect interactions between 8, states: the *Br,, V (rr + a*) state was lowered by 0.21 eV upon twisting; the ‘BIf (or-? + rr*) state was raised by 0.09 eV; the second Blu n+ 3ps*) state was lowered by 0.12 eV. The combined oscillator strengths for these states are very nearly the same, indicating a redistribution of absorption intensity. We return to this point in the discussion of the optical activity below. The electronic spectrum of ethylene has been the subject of an extensive review by Merer and Mulliken [64] and by Robin [65], to which the reader is referred for references to much of the earlier experimental and theoretical work. The first ultraviolet absorption of ethylene is assigned as the triplet T-N system and consists of a very weak progression of diffuse bands starting at 3.54 eV and peaking at 4.60 eV. Beginning at 4.67 eV the absorption curve rises steeply giving rise

to a long progression of broad bands with a maximum at 7.66 eV. This is the V-N ‘Blu (rr+ n*) system which we calculate to lie 8.17 eV above the ground state. The observed absorption envelope of the V state is apparently not suifted or broadened under increased pressure [66] suggesting that it is a valence state. However, the precise nature of this state has been the subject of considerable controversy among theoreticians owing to the almost universal failure to predict the vertical excitation energy to within 0.4 eV and the calculated large spatial extension [63] _As discussed above, we conclude that the V state can be called a valence state or perhaps a state with mixed character. At 7.10 eV, superimposed on the V-N bands, there begins a set of sharp bands which “disappear” (by broadening asymmetrically to the high frequency side) under high pressure of an inert gas [66] . These are assigned as the first Rydberg transition R + N ‘B3” (rr + 3sag), which we calculate to occur at 7.42 eV. The discrepancy between the calculated and experimentally estimated oa:illator strengths for this transition may in part be due to the difficulty of separating this complex set of bands from the much stronger underlying V-N system [63]. The theoretical and experimental values for the combined oscillator strengths of the two bands agree quite well. At higher frequencies the V-N system decreases steadily and one finds a sharp Rydberg transition at 8.89 eV followed by another, slightly broader one at

422

A. Rauk, JM. Barriel/Computation

ofosc&‘ator strengths and optical rotatory strengths

9.05 eV. We have assigned these to 2lB,” and 3 ‘B3,, , respectively. In all, four Rydberg series have been identified, all of which converge to an ionization potential of 10.50 eV [67] which corresponds closely to the value 10.51 eV obtained by photoelectron spectroscopy [3 I] for ionization from the T orbital. The very weak Rydberg bands at 8.25 eV and 8.6 1 eV [67] are assigned to the weak electric dipole allowed transitions to states Z1B3u and 11B7u, respectively. Thus all of the six bands observed 6 the UV spectrum of ethylene can be assigned to electric dipole allowed transitions, the average deviation between calculated and observed energies being 0.35 eV. In a study of ethylene by the trapped electron method, a distinct peak was detected at 9.18 eV [6X, 691 and Robin has suggested the possible uncovering of a forbidden valence shell transition [65]. According to the present calculations two Rydberg states, 31Ble G-r-++b& and qlB - zg (?r+4sblu) and one valencestate, 2B,, (uC_n + T ) are candidates for this position. We have assigned the 9.18 eV peak to the valence state 2lB,, (uCH + n*), transitions to which are calculated to%ave an energy of 9.36 eV. This valence state plays a central role in the optical activity of olefims as we shall see below. Some experimental evidence though not definitive for the existence of quadrupole allowed transitions to two states (of B, or Bz symmetry) near the V state has been obtained [70,71] _It has been argued convincmgly by Drake and Mason [72] and by others [73] that the interaction of at least one such state with the V state is necessary to account for the couplet of strong CD bands of opposite sign which are found in the region of the V-N transition of many chiral olefins. The BL state may be either of R * a* or of cc_H + 1~* character [72] _As seen above, we have calculated that the ucCH--f d state (2’B,,) falls 1.2 eV above the V state. There are, in fact, two diffuse states 1’B and 1‘B,, which are calculated to occur below the + state and we have tentatively assigned these to the experimentally observed bands at 7.45 eV [70] and 7.25 eV [7 l] , respectively. These assignments agree with the early predictions by Merer and Mulliken [64] and with more elaborate CI calculations [63], but not with a recent analysis by Mulliken [74]. In so far as the twisted ethylene molecule can serve as a model for chiral olefiis, the experimentally observed features [72] are reproduced. The Rydberg

‘B3u (n + 3sa ) transition which occurs at lower energy than the V-#transition has a small, but significant rotational strength. At higher energies one obtains the couplet of strong, oppositely_ signed, CD bands. The experimentally observed separation of these bands is approximately 1.6 eV [75], which is close to the separation calculated for twisted ethylene, 1.5 eV. The signs of the rotatory strengths agree with those previously calculated [76,77] and found (for rrans-cyclooctene) [78] _ The lowest state of B, symmetry, 1 ‘Big (r+3pb2a), interacts very little with the V state upon twisting of the molecule, in spite of its nearness in energy. The reason is presumably its diffuse nature. On twisting, the transition to this state gains little intensity and has a negligible rotational strength. The second, third and fourth B, states interact moderately under the small perturbation introduced by the IO’ twist. Transition to the third state, 2lB,,,, which arises from excitation of a rr electron into a diffuse n*-like, or molecular “d”, orbital is computed to have a significant .osciUator strength, 0.072, in the planar case but to have a very low oscillator strength upon twisting. On the other hand, the dipole forbidden transition to the valence 2lB,, state gains substantial intensity under the influen:e of the twist. The composition of these three states after twisting, as a percentage of excitation from the u (lb,,, magnetic dipole allowed) and rr (lb,,, electric &pole ahowed) orbitak is as follows: l’B,,,6% a,91% 7r;21B,,,, 11%0,89% n;2’Blg, 83% u, 16% P. Although the actual values of the percentage compositions are somewhat basis set dependent, the relatively small amount ofmixing of configurations to which transitions are electric dipole allowed (TT+ a*) with configurations to which transitions are magnetic dipole allowed (u+ rr*) points to the origin of the rotational strengths observed and calculated_ The large calculated values of rotational strengths for transitions to the two valence states are largely due to the presence of cross terms such as Cxrr*C~rr*
A. Rauk. J.M. Barrfel/Computation of oscillator strengths and optical rotatorj-’strengths

B, configuration (to which transitions are electric dipo Be forbidden, magnetic dipole allowed) to the CI expansion of the same derived state. One or the other of the coefficients may be quite small in absolute magnitude but the product of the two allowed transition moments dominates the magnitude of the term and the overall magnitude of the rotational strength. The signs of the coefficients determine the sign of the rotational strength.

6. Conclusions The procedure outlined in sections 2-4 is a relatively efficient way of obtaining transition moments over operators correct to first order in RayleighSchrBdinger perturbation theory. From the results obtained for the test cases discussed in detail in the previous section, the results are equally good for transitions to both valence and Rydberg states, provided that the appropriate diffuse functions are included in the basis set. Vertical excitation energies correct to second order in perturbation theory agree reasonably well for transitions to valence states. In the case of Rydberg states, an empirical estimate for the amount of error due to correlation effects and lack of orbital relaxation is suggested, based on the difference between experimentally determined ionization potentials and ionization potentials determined by Koopmans’ theorem_ In most cases, this adjustment to the excitation energgies leads to almost quantitative mentally determined values.

agreement

with experi-

Acknowledgement The calculations have been performed at University of Calgary Data Centre and at the University of London Computer Centre. The fmancial support of the National Research Council of Canada is gratefully acknowledged. We thank Tom Ziegler for many useful discussions, and Sylvia Jacyno and Jim Parker for technical and programming assistance.

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423

Phys. 50 (1969) 2161. [3] A.J. Coleman, Rev. Mod. Phys. 35 (1963) 668. [4] CF. Bender and E.R. Davidson, I. Phys. Chem. 70 (1966) 2615. [S] W.Meyer,lntern. J.Quant.Chem.S5 (1971)341; J. Chem. Phys. 58 (1973) 1017. [6] B. Roos,Chem. Phys. Letters 15 (1973) 153. . [7] W.hIeyer, J.Chem.Phys.64 (1976)2901. [8] C.E. Dykstra and H.F. Schaeffer III, J. Chem. Phys. 65 (1976) 2740. [9] R. Ahlrichs and F. Driessler, Theoret. Chim. Acta 36 (19751275. [lo] R.F. ILawman, S.D. Bloom and CF. Bender.Chem. Phys. Letters 32 (1975) 483. [ll] C. hlo!ler and M.S.Plesset, Phys. Rev.,46 (1934) 616. [12] K.A. Brueckner, Phys. Rev. 100 (1955) 36. [13] J. Goldstone, Proc. Roy. Sot. A239 (1957) 267. [14] N.S. Ostlund and M.F. Bowen, Theor. Chim. Acta 40 (1975) 175. [ 151 A.E. Hansen, hlol. Phys. 13 (1967) 425. [ 161 R.J. Adler, M. Trsic and W.G. Laidlaw, J. Chem. Phys. 64 (1976) 4802. 1171 L. Rosenfcld, Z.Physik. 52 (1928) 161. 1181 P.S. Epstein, Phys. Rev. 28 (1926) 695. [ 191 R.K . Nesbet, Proc. Roy. Sot. A230 (1955) 312,322. [20] J.C. Slater, Quantum chemistry of molecules and solids, Vol. 1 !McGraw-Hill, New York, 1963) p. 93. [21] H. Lefebvre-Brian, C.M. Moser and R.K. Nesbet, J. Mol. Spectry. 13 (1964) 418. [22] D. Demoulin and M. Jungen, Theoret. Chim. Acta 34 (1974) 1. [23] D. Demoulin,Chem. Phys. 17 (1976) 471. 1241 W.A. Goddard III and W.J. Hunt, Chem. Phys. Letters 24 (1974) 464. [25] W.J. Hunt and W.A. Goddard III, Chem. Phys. Letters 3 (1969)414. [26] N.W. Winter, \[‘.A. Goddard III and F.W. Bobrowicz, J. Chem. Phys. 62 (1975) 4325. [27] R.J. Buenker and S.D. Peyerimhoff, Chem. Phys. Letters 29 (1974) 253. [28] T.H. Dunning Jr., J.Chem.Phys. 53 (1970) 3385. [29] M.H.Wood, Chem: Phys. Letters 28 (1974) 477. 1301 K. Watanabe and hl. Zelikoff, J. Opt. Sot. Am. 43 (1953) 753. [3 I] D.W. Turner, C. Baker, A.D. Baker and C.R. Brundle. Molecular photoelectron spectroscopy, (Wiley, New York, 1970). 1321 A.Chutjian, R.I. Hall and S. Trajmar, J. Chem. Phys. 63 (1975) 892. 1331 S. Trajmar, W. Williams and A. Kuppermann, J. Chem. Phys. 54 (1971) 2274. 1341 WC. Price, J. Chem. Phys. 3 (1936) 147. [351 E.N. Lassettre and A. Skerbele, J. Chem. Phys. 54 (197 1) 1597. I361 G. Henberg, T. Hugo, S. Tilford and J. Simmons, Can. J. Phys. 48 (1970) 3004.

_-4.&j

--I

A. Rauk, J&f. Barriel/Computattin of oscillator strengths and optical rntaio;

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