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Fock space multi-reference coupled-cluster study of excitation energies and dipole oscillator strengths of ozone
Volume 193, number
CHEMICAL
5
PHYSICS
LETTERS
5 June 1992
Fock space multi-reference coupled-cluster study of excitation energies and dipole oscillator strengths of ozone Maria
Barysz
I, Magnus
Rittby
’ and Rodney
J. Bartlett
Quantum Theory Project, University of Florida, Gainesville, FL 3261 l-2085, USA Received
2 January
I992
The singlet and triplet excitation energies are investigated by the Fock space multi-reference coupled-cluster method. A procedure for evaluating dipole transition moments and oscillator strengths is also presented and applied. Good agreement is achieved for excitation energies and transition moments where experimental data are known.
1. Introduction The electronic spectrum of ozone has been the subject of numerous investigations over a long period of time, primarily because of the importance of the atmospheric ozone layer for the absorption of solar radiation. While studies of the absorption bands of ozone date from the work of Chappuis in 1880 [ 11, present experimental knowledge of its excited states is still largely derived from the three prominent spectral bands - the intense Hartley band in the ultraviolet (peak at 4.85 eV) and the weaker Chappuis band in the visible (peak at 2.13 eV) and Huggins bands ( z 4 eV). The very low energy region ( l-2 eV) was also studied by electron impact work showing a number of peaks which could correspond either to separate (triplet or optically forbidden) electronic states or to vibrational features of the same state according to the experimental evidence. The ozone spectrum in the far-ultraviolet region (from 5 to 10 eV ) is characterized by various broad maxima with a number of superimposed peaks at higher energy. The analysis is difficult because of the Correspondence to: R.J. Bartlett, Quantum Theory Project, University of Florida, Gainesville, FL 3261 I-2085, USA. ’ Permanent address: Institute of Chemistry, Silesian University, Szkolna 9,40-006 Katowice, Poland. ’ Texas Christian University, Department of Physics, Box 329 15, Fort Worth, TX 76 129, USA. 0009-2614/92/$
05.00 0 1992 Elsevier Science Publishers
lower resolution attained for this part of the spectrum and is furthermore complicated by the fact that the usual O2 contamination gives rise to an underlying intensity with a high absorption coefficient beginning around 7 eV. No assignment for the main structure in this region has been put forth yet, and only a tentative assignment for some of the peaks above 9 eV has been given. Great interest from both the experimental and theoretical sides has been directed toward the question of identifying the first few closely spaced ionization potentials which are found in photoelectron spectroscopy to lie at (vertical) 12.75, 13.03 and 13.57 eV respectively. Agreement has been reached only on the assignment of the first IP as being of a ‘A, state, but controversy still exists concerning the identification of the second and third IPs that correspond to the ‘A2 and 2B2 states. On the basis of experimental observation alone it is not possible to make this assignment with certainty. It is widely appreciated today that the traditional independent-particle model of electronic structure provides an inadequate description of ozone (see refs. [ 2- 12 ] and references therein), which has much biradical character. Besides its complex structure, the photoelectron spectrum of OS contains a number of unusually strong “shake-up” bands [ 13- 17 ] (which are due to electron correlation effects #’ ), and ioni#’ For a detailed photoelectron
B.V. All rights reserved.
discussion of electron spectra, see ref. [ 18 1.
correlation
effects in
373
Volume 193, number
5
CHEMICAL
PHYSICS
zation potentials predicted by Koopmans’ theorem are even qualitatively in error. This breakdown of the standard molecular orbital picture is largely due to strong configuration mixing between the lowest and highest energy determinants and the need for highly correlated models of electron correlation with a multi-reference configuration space. A survey of the available experimental data and computational results on the spectroscopy and kinetics has been published [ 7,19-2 11. The O3 molecule has been of interest and studied by one of us for a long time. Several results have been published which, until now, have mainly focused on the single reference coupled-cluster (CC) [ 22,231 studies of the O3 potential surface, dissociation and atomization energies and of the structure and harmonic vibrational frequencies of this molecule [ 2426]. In this paper the Fock space multi-reference coupled-cluster (FSMRCC) [ 27,281 method has been applied to the photoelectron spectra of O3 together with the lowest ionization potentials. Now, together with the computation of oscillator strengths, the method seems to be a very good tool for the complete description of spectra.
LETTERS
5 June 1992
cupied space of @,. The relevant reference spaces for the GS (0, 0), IP (0, l), EA (1, 0) and EE (1, 1) cases are as follows: %s=%
@iIp=@]@I,
%A = {a+Po,
@PEE = {~+~}~cl,
(3)
where the brackets { } indicate normal order and all active hole operators ,u, and particle operator a! are considered. Then, we can write the solutions to the Schrijdinger equation as yco,o, =s)y/oGSY
yy’o.l’=-J~
Y(l,O),SZylOEA,
Y(‘J)=QY&.
IP T
(4)
In this notation the GS is found in the solution of the (0,O) problem; ionization potentials (IP) in the (0, 1) problem; electron affinities (EA) in the ( 1, 0) problem and excitation energies (EE) in the ( 1, 1) problem. In the FSMRCC Q is the valence universal wave operator. This means that the same Q solves all eigenvalue problems: O={exp( T co.o,+~‘O.“+~‘l’O)+~(I,~)+...)}
) (5)
where
2. Theory In the Fock space multi-reference coupled-cluster theory we solve a problem of the form
(6)
W, =EIw,
(1)
for several states of I. The solution
a/i run over all unoccupied/occupied (2)
is built from the model or multi-reference function 9 that is represented in the space of configurations {@k}. The space of functions is constructed from some set of particles (a, /I, ...)/holes (p, Y, ...) creation/annihilation operators acting on the ground state (GS) wavefunction @,. From this point of view every state Y, is characterized by its multi-reference (model) function, q, expanded on the @space, and can be referred to by notation Yj ~3~~) where m, refers to the number of electrons created in the virtual space and mh refers to the number of electrons removed from the oc374
orbitals in oo,
T(O,‘)= z (plT(“-‘)li){p+i} +t C (,ublTJ”~‘) Iij),{p+b+ji}+... , r,i p,b
(7)
T(‘,O)= ,c, (c]rl’~o) ]cr){c+a} +t C (~d~T~‘~~)IaY)~{c+d+i~}+..., r,d 1.01
(8)
I-(‘,‘)= ac, (pIz-“J’la){p+a} + 3 (P4T &a
(‘~*)I(~i)~{~+d+ia}+... ,
(9)
the indices ~,a are defined by the active holes and particles. From eq. ( 1) and the definition of the model space, inserting into the Schrijdinger equation, we have rR-‘Hl2C@,=E,C@,,
(10)
(@,~i2-‘H~~C@,)=C,E,.
(11)
If we then define Heff=p(k,!)(a-
(13)
(14)
C,‘HeffCI=EI,
(15)
where EI is a diagonal matrix and C, is an eigenvector which, along with 52, defines the wavefunction Y, or Yk,l). If we now define the orthogonal space projectors Qck.‘) for k particles and 1 holes,
I&>(@jI Y
F
(16)
(17)
@,# Nk”’ > 3
that is, the projection operator over all function space not spanned by the model determinants, then projection by the Q-space Bloch equation of the form
will give equations
)
(21)
Af=(Y,‘II C)“2(
(22)
YJ;I YJ)“*
is for Y,= YJ the expectation value of the operator a in the state Y1;,and for Y,# YJ the transition moment of a. It can be shown that even if the operator &?C,& from eq. (2) is not a unitary one, after some manipulations it can be shown that the following expressions for the expectation value and the square of the transition moment absolute value are valid:
A:=(@,1 Gm-‘mGPI,>
,
(23)
IA:I’= I (@,I vw-‘mGP.d
12.
(24)
If one defines the operator @I=Q--’ PQ
(25)
and matrix element
where
Hap’k,”
A(n) =A+ti
(12)
where
1 _p(k.O=
culate dipole transition moments in the generalized coupled-cluster method has been applied to FSMRCC [ 28 1. If A^is a Hermitian operator representing a certain perturbation to our system described by Hamiltonian fi,
then the quantity
lHQ)P(k,‘) 9
(H”ff),,=(~,IP’k,‘)(52-1HS2)P(k,’)I~p),
Q(kO=
5 June 1992
CHEMICAL PHYSICS LETTERS
Volume 193, number 5
_- QH,_$(k.l) us the
(18)
multi-reference
Q(k*‘)(Hi2-SLH,fl)P(k,i)-0 -
~:=(@,lQ-‘m@JJ> y,= yco.0,,
(27)
yJ= yl(1.l)
(28)
with the normal-ordered
perturbation
coupled-cluster
(19)
or (@ck.‘) IHL2-1;2H,,(@‘k~” p >--0 3 I
(26)
and chooses
a,= ,
3
1 (ilrlj){i+j} 1.j
(29)
as a one-electron dipole moment operator, then the transition moment between the ground state and excited states can be calculated from the expression
(20)
where @j# {@Lk.”}. The form of the Q-space set {$j} will vary depending on the sector of interest. The Fock space multi-reference method also allows for a direct calculation of first- and second-order static properties as well as response functions. Here the Stolarczyk and Monkhorst [ 29 ] idea to cal-
IAJIZ= I (OolQ-‘A,(S2C,)@,) In a matrix representation, Jth row of the matrix,
12. where CJ indicates
I v-J/L WJC,(CJ) _‘WJ, the matrix elements
(30) the
(31)
WJ and WJ are 375
Volume
193, number
5
CHEMICAL
WJ=(@cisI PJI Q%E,O > 2
(32)
wJ=(@E,Oi
>
(33)
@J&S)
PHYSICS
(34)
In this application the higher particle operators l@,, have not been considered. The second-order quantity I VJI * has been used to calculate dipole length transition moment VJ and the dipole length oscillator strength f”=;
I vJ~2(~J-EG,)
(35)
>
where all quantities
are in hartree atomic units.
3. Results and discussion Calculations reported in this paper were carried out with the FSMRCC program developed by Rittby, Pal and Bartlett [ 3 11. A single contracted Cartesian Gaussian DZP basis set has been used in this work. This comprises Dunning’s 4s2p contraction [ 321 of Huzinaga 9s5p primitive set [ 331, augmented with a polarization exponent of 1.2 1 [ 341. Using single determinant self-consistent field (SCF) reference functions formed from these basis sets, correlation corrections were computed by means of Fock space multi-reference coupled cluster theory. The Czv geometry optimized in the CCSD method [22] hasbeenused(R,=1.263&8,=117.4”).The Table 1 Calculated
vertical transition
energies (in eV) from the 0s ground state to the electronic states below 5 eV and comparison
No.
State
Dominant excitation
POL-CI =)
MRD-CI
la,+2b, 6a,+2b, 4bz+2b, 4b2+2b,
1.60 2.01 2.09 2.34
1.20 1.59 1.44 1.72
I
‘B2
2 3 4
‘RI 3Az ‘A*
a) Ref. [7].
b, Ref.
376
[ 191.
5 June 1992
SCF configuration for ozone obtained with the DZP basis is [core] 3a: 2b:4a: 5a: 3b: 1b: 4b: 6a: 1a:2by7ay5b:. The results have been compared with data from the best calculations previously available: the polarization configuration interaction (POL-CI) technique [ 7 ] and large-scale configuration interaction calculations (including energy extrapolation) MRD-CI [ 191. In both cases a DZP quality basis set has been used. The comparison with recently published multi-configurational linear response MCLR results has also been made [ 2 11. These calculations employ a larger POL 1 basis [ 35 1. The l-2 eV energy region is primarily characterized by states which result from a single excitation out of one of the highest (energetically closely spaced) MOs of the ground state (4b:, 6a: and 1a$ ) into the lowest unoccupied type orbital 2b:. The calculated vertical transition energies together with experimental data and other computational results are given in table 1. The 3B2 state is found to be the lowest-lying in all methods, but the calculated spacing among all four states is relatively small. MRDCI gives the 3A2 below 3B, state, opposite to the POLCI and FSMRCC results. The FSMRCC and MCLR results are very close to each other and give the same ordering of all energy levels. The comparison with experiment shows that the obtained energies are in the experimental region. In the 2-5 eV and above 8 eV region (table 2) the first two singlet states ‘Bi and ‘B2 (known as Chappuis and Huggins bands) are still characterized mainly by single excitations into the 2b, MO’ while the spectrum beyond 8.0 eV is characterized by single excitation states populating the second-lowest unoccupied MO, the orbital 7a:.
where W”( WJ) represents the connected part of (Q-r l&)c and can be divided into zero-, one-, twoparticle operators &@~+@,+ti*+....
LETTERS
‘) Ref. [21].
b,
with experiment
MCLR ‘)
FSMRCC
Exp.
1.14 1.66 1.81 2.14
1.37 I .62 1.95 2.17
peaks at 1.29 1.43 1.55 1.67 1.80 1.92
5 June 1992
CHEMICAL PHYSICS LETTERS
Volume 193, number 5
Table 2 Calculated vertical transition energies (in eV) from the ground state to the various states in 2 to 5 eV and 8 eV No.
State
Dominant excitation
POL-CI 8)
6a,+2b, la2-r2b, 4bz-+7a, 6a,-+7a, 6a,+7a, la2A7a, 4b2-+7a,
2.41 6.12
5
‘B,
6 7 8 9 10 11
‘R* 3R* 3A, ‘A, )Az ‘R2
‘) Ref. [7].
b, Ref. [ 191. ‘) Ref. [21].
MRD-CI b,
1.95 4.97 8.45 8.44 9.29 9.30 10.05
In tables 1 and 2 one can see that all the states are dominated by single excited configurations. Our model space in the ( 1, 1) sector of the FSMRCC method has also been created by single excited configurations. The accurate results for ‘B2, 3BZand ‘Bz states support the known fact that good results can be achieved if the model space includes the dominant configurations. It should also be noted that the FSMRCC method is not a single-state oriented method like most multi-reference methods. The main idea of this method is to obtain many states at the same time. As a result, some accuracy for the single state can be lost, but the objective is to reproduce the whole spectrum. This conclusion is supported by the results presented in table 3, where the dipole transition moment and oscillator strength have been shown. One can see that the dipole oscillator strength, the quantity measured by experiment, is reproduced well. Some questions may arise if we look at the tran-
MCLR =I
FSMRCC
2.17 5.10
2.13 5.52 8.07 8.17 9.29 9.92 9.80
8.86 9.38
Exp.
2.13 4.86 8.55 8.45 9.32 10.2
sition moment. For the states ‘B2 and ‘Al (states 5 and 9 in table 3) it seems to be too small in comparison to other methods. Approximations including a relatively small basis are made in its evaluation, but it is di~cult to say much about the error since the final values of the oscillator strength are appropriate. One can also see that oscillator strengths and transition moments are better for the iB2 and ‘A, states (states 6 and 10 in table 3) than for the other one. Looking at the eigenvectors of the effective Hamiltonian He” shows that these states have more multi-reference character than the others. One might expect that the multi-reference coupled-cluster method should work better for those types of states. In table 3 the newest values of dipole transition moments and oscillator strengths in the length representation are shown. Again the comparison among POL-CI, MRD-CI, FSMRCC and expe~ment~ data has been made.
Table 3 Transition moment and oscillator strength for OJ dipole allowed transitions No.
State
transition moment ‘R* 5 ‘R* 6 ‘A* 9 ‘Rz 10 oscillator strength 5 ‘RI 6 ‘R* 9 ‘A, 11 ‘RZ *) Ref. [7].
“Ref.
POLCI a’
MRD-CI b,
MCLR ”
FSMRCC
0.028 1.228
0.017 1.202 0.011 0.648
0.012 0.863 0.034
0.007 0.983 0.003 0.687
4.71 x 10-S 2.3x 10-l
1.5x 10-S 1.76x IO-’ 0.24x 1O-2 1.05x 10-l
7.3X IO-6 0.93x 10-l 0.26X 10-f
2.85X 10-e 1.30x 10-i 0.25x IO-” 1.13x10-
Exp.
2x 10-5 0.88x IO-’
[ 191. “‘Ref. [21].
377
CHEMICAL PHYSICS LETTERS
Volume 193, number 5
5 June 1992
Table 4 Calculated vertical ionization potentials (in eV) for ozone obtained from various treatments Koopmans
State
*Al ZB2 2Az
15.31 15.64 13.27
‘) Ref. (191.
“Ref.
MRD-CI a)
12.44 12.40 12.79 [36].
MBPT b’
12.90 13.28 13.15
‘) Basis I: DZP (48 functions).
4. Summary We have presented FSMRCC calculations for free ozone. Results for properties (dipole transition moment and oscillator strength) and spectra (energy) have been given. The calculations have been done for the ground state open CIVfor ‘A,. We found that the theory can be successfully applied for obtaining spectra and its intensity. One should also emphasize that a small (three holes and three particles) active space has been used. The transition moment amplitudes were fast and well converged. Once the energy is given the oscillator strength can be achieved.
Acknowledgement
This work has been supported by the US Office of Naval Research ONR-NO00 14-9 1-J- 1282.
References [ I] J. Chappuis, Compt. Rend. Acad. Sci. (Paris) 91 ( 1880)
378
basis I c,
basis II d,
12.37 12.54 13.19
12.69 12.82 13.43
12.75 13.03 13.57
(‘A?) (*Bz?)
d, Basis II: [ 14s7p3d/Ss4p2d] (87 functions).
Ionization potentials of O+ The first three ionization potentials of ozone have also been treated in the present work and the results are compared in table 4, with those of previous theoretical studies, as well as with the pertinent experimental data. The present calculations (even if a better basis set is used) consistently give the ‘A,, ‘B2 and 2A2 ordering of states which supports earlier many-body perturbation theory (MBPT) [ 361 results but it conflicts with the conclusion of MRD-CI results.
985; N. Huggins, Proc. Roy. Sot. 48 ( 1890) 216.
Exp.
FSMRCC
[2] T.J. Lee, W.D. Allen and H.F. Schaefer III, J. Chem. Phys. 87 (1987) 7063; [ 3) M.V. Rama Krishna and K.D. Jordan, Chem. Phys. 115 (1987) 423. [4] Y. Yamaguchi, M.J. Frisch, T.J. Lee, H.F. Schaefer III and J.S. Binkley, Theoret. Chim. Acta 69 ( 1986) 337. [ 5 ] W.D. Laidig and H.F. Schaefer III, J. Chem. Phys. 74 ( 1981) 3411. [ 61 G. Karlstrom, S. Engstriims and B. Jiinsson, Chem. Phys. Letters 57 (1978) 390. [7] P.J. Hay and T.H. Dunning Jr., J. Chem. Phys. 67 (1977) 2290. [S] P.J. Hay, T.H. Dunning Jr. and W.A. Goddard III, J. Chem. Phys. 62 (1975) 3912. [9] W.A. Goddard, W.J. Hunt and P.J. Hay, Accounts Chem. Res. 6 (1973) 398. [IO] R.J. Bint andM.D. Newton, J. Chem. Phys. 59 ( 1973) 6220. [ 111 P.J. Hay, T.H. Dunning and H.F. Schaefer III, Chem. Phys. Letters 23 (1973) 457. [ 121 S. Rothenberg and H.F. Schaefer III, Mol. Phys. 2 1 ( 197 1) 317. [ 131 T. Cvita, L. Klasinc and B. Kovac, Intern. J. Quantum Chem. 29 (1986) 657. [ 141 J.M. Dyke, L. Golob, N. Jonathan, A. Morris and M. Okuda, J. Chem. Sot. Faraday Trans. 2 ( 1974) 1828. [ 151 C.R. Brundle, Chem. Phys. Letters 26 (1974) 25. [ 161 D.C. Frost, S.T. Lee and C.A. McDowell, Chem. Phys. Letters 24 (1974) 149. [ 171 T.N. Radwan and D.W. Turner, J. Chem. Sot. A ( 1966) 85. [ 181 G. Vendlin, Structure and bonding, Vol. 45 (Springer, Berlin, 1981) pp. I-117. [ 191 K.H. Thunemann, S.D. Peyerimhoff and R.J. Buenker, J. Mol. Spectry. 70 (1978) 432. [20] J.I. Steinfeld, S.M. Adler-Golden and J.W. Gallagher, J. Phys.Chem.Ref.Data16 (1987)911. [21] D. Nordfors, H. Agren and H.J.Aa. Jensen, Intern. J. QuantumChem.40(1991)475. [22] R.J. Bartlett, C.E. Dykstra and J. Paldus, in: Advanced theories and computational approaches to the electronic structure of molecules, ed. C.E. Dykstra (Reidel, Dordrecht, 1987) p. 127. [23] R.J. Bartlett, J. Phys. Chem. 93 (1989) 1697; Ann. Rev. Phys. Chem. 32 ( 198 1) 359.
Volume 193, number 5
CHEMICAL PHYSICS LETTERS
[ 241 J.F. Stanton, W.N. Lipscomb, D.H. Magers and R.J. Bartlett, J. Chem. Phys. 90 (1989) 1077. [ 251 J.F. Stanton, R.J. Bartlett, D.H. Magers and W.N. Lipscomb, Chem. Phys. Letters 163 (1989) 333. [26] J.D. Watts, J.F. Stanton and R.J. Bartlett, Chem. Phys. Letters 178 (1991) 471. [27] D. Mukherjee and S. Pal, Advan. Quantum Chem. 20 (1989) 291. [ 281 S. Pal, M. Rittby, R.J. Bartlett, D. Sinha and D. Mukherjee, J. Chem. Phys. 88 ( 1988) 4357, and references therein. [ 291 L.Z. Stolarczyk and H.J. Monkhorst, Phys. Rev. A 32 ( 1985) 725,743; 37 (1988) 1908,1926. [ 301 M. Barysz and R.J. Bartlett, in preparation.
5 June 1992
[31] R.J. Bartlett, G.D. Purvis III, G.B. Fitzgerald, R.J. Harrison, Y.S. Lee, W.D. Laidig, S.J. Cole, D.H. Magers, E.A. Salter, G.W. Trucks, C. Sosa, M. Rittby, S. Pal and M. Barysz, ACES I Program System; M. Rittby, S. Pal and R.J. Bartlett, J. Chem. Phys. 90 ( 1989) 3214. [32] T.H. Dunning Jr., J. Chem. Phys. 53 ( 1970) 2823. [33] S. Huzinaga, J. Chem. Phys. 42 (1965) 1293. [34] L.T. Redmon, G.D. Purvis III and R.J. Bartlett, J. Am. Chem. Sot. 101 (1979) 2856. [35] A. Sadlej, Coll. Czech. Chem. Commun. 53 (1988) 1995. [ 361 L.S. Cederbaum, W. Domcke, W. von Niessen and W.P. Kramer, Mol. Phys. 34 (1977) 381.
379
CHEMICAL
5
PHYSICS
LETTERS
5 June 1992
Fock space multi-reference coupled-cluster study of excitation energies and dipole oscillator strengths of ozone Maria
Barysz
I, Magnus
Rittby
’ and Rodney
J. Bartlett
Quantum Theory Project, University of Florida, Gainesville, FL 3261 l-2085, USA Received
2 January
I992
The singlet and triplet excitation energies are investigated by the Fock space multi-reference coupled-cluster method. A procedure for evaluating dipole transition moments and oscillator strengths is also presented and applied. Good agreement is achieved for excitation energies and transition moments where experimental data are known.
1. Introduction The electronic spectrum of ozone has been the subject of numerous investigations over a long period of time, primarily because of the importance of the atmospheric ozone layer for the absorption of solar radiation. While studies of the absorption bands of ozone date from the work of Chappuis in 1880 [ 11, present experimental knowledge of its excited states is still largely derived from the three prominent spectral bands - the intense Hartley band in the ultraviolet (peak at 4.85 eV) and the weaker Chappuis band in the visible (peak at 2.13 eV) and Huggins bands ( z 4 eV). The very low energy region ( l-2 eV) was also studied by electron impact work showing a number of peaks which could correspond either to separate (triplet or optically forbidden) electronic states or to vibrational features of the same state according to the experimental evidence. The ozone spectrum in the far-ultraviolet region (from 5 to 10 eV ) is characterized by various broad maxima with a number of superimposed peaks at higher energy. The analysis is difficult because of the Correspondence to: R.J. Bartlett, Quantum Theory Project, University of Florida, Gainesville, FL 3261 I-2085, USA. ’ Permanent address: Institute of Chemistry, Silesian University, Szkolna 9,40-006 Katowice, Poland. ’ Texas Christian University, Department of Physics, Box 329 15, Fort Worth, TX 76 129, USA. 0009-2614/92/$
05.00 0 1992 Elsevier Science Publishers
lower resolution attained for this part of the spectrum and is furthermore complicated by the fact that the usual O2 contamination gives rise to an underlying intensity with a high absorption coefficient beginning around 7 eV. No assignment for the main structure in this region has been put forth yet, and only a tentative assignment for some of the peaks above 9 eV has been given. Great interest from both the experimental and theoretical sides has been directed toward the question of identifying the first few closely spaced ionization potentials which are found in photoelectron spectroscopy to lie at (vertical) 12.75, 13.03 and 13.57 eV respectively. Agreement has been reached only on the assignment of the first IP as being of a ‘A, state, but controversy still exists concerning the identification of the second and third IPs that correspond to the ‘A2 and 2B2 states. On the basis of experimental observation alone it is not possible to make this assignment with certainty. It is widely appreciated today that the traditional independent-particle model of electronic structure provides an inadequate description of ozone (see refs. [ 2- 12 ] and references therein), which has much biradical character. Besides its complex structure, the photoelectron spectrum of OS contains a number of unusually strong “shake-up” bands [ 13- 17 ] (which are due to electron correlation effects #’ ), and ioni#’ For a detailed photoelectron
B.V. All rights reserved.
discussion of electron spectra, see ref. [ 18 1.
correlation
effects in
373
Volume 193, number
5
CHEMICAL
PHYSICS
zation potentials predicted by Koopmans’ theorem are even qualitatively in error. This breakdown of the standard molecular orbital picture is largely due to strong configuration mixing between the lowest and highest energy determinants and the need for highly correlated models of electron correlation with a multi-reference configuration space. A survey of the available experimental data and computational results on the spectroscopy and kinetics has been published [ 7,19-2 11. The O3 molecule has been of interest and studied by one of us for a long time. Several results have been published which, until now, have mainly focused on the single reference coupled-cluster (CC) [ 22,231 studies of the O3 potential surface, dissociation and atomization energies and of the structure and harmonic vibrational frequencies of this molecule [ 2426]. In this paper the Fock space multi-reference coupled-cluster (FSMRCC) [ 27,281 method has been applied to the photoelectron spectra of O3 together with the lowest ionization potentials. Now, together with the computation of oscillator strengths, the method seems to be a very good tool for the complete description of spectra.
LETTERS
5 June 1992
cupied space of @,. The relevant reference spaces for the GS (0, 0), IP (0, l), EA (1, 0) and EE (1, 1) cases are as follows: %s=%
@iIp=@]@I,
%A = {a+Po,
@PEE = {~+~}~cl,
(3)
where the brackets { } indicate normal order and all active hole operators ,u, and particle operator a! are considered. Then, we can write the solutions to the Schrijdinger equation as yco,o, =s)y/oGSY
yy’o.l’=-J~
Y(l,O),SZylOEA,
Y(‘J)=QY&.
IP T
(4)
In this notation the GS is found in the solution of the (0,O) problem; ionization potentials (IP) in the (0, 1) problem; electron affinities (EA) in the ( 1, 0) problem and excitation energies (EE) in the ( 1, 1) problem. In the FSMRCC Q is the valence universal wave operator. This means that the same Q solves all eigenvalue problems: O={exp( T co.o,+~‘O.“+~‘l’O)+~(I,~)+...)}
) (5)
where
2. Theory In the Fock space multi-reference coupled-cluster theory we solve a problem of the form
(6)
W, =EIw,
(1)
for several states of I. The solution
a/i run over all unoccupied/occupied (2)
is built from the model or multi-reference function 9 that is represented in the space of configurations {@k}. The space of functions is constructed from some set of particles (a, /I, ...)/holes (p, Y, ...) creation/annihilation operators acting on the ground state (GS) wavefunction @,. From this point of view every state Y, is characterized by its multi-reference (model) function, q, expanded on the @space, and can be referred to by notation Yj ~3~~) where m, refers to the number of electrons created in the virtual space and mh refers to the number of electrons removed from the oc374
orbitals in oo,
T(O,‘)= z (plT(“-‘)li){p+i} +t C (,ublTJ”~‘) Iij),{p+b+ji}+... , r,i p,b
(7)
T(‘,O)= ,c, (c]rl’~o) ]cr){c+a} +t C (~d~T~‘~~)IaY)~{c+d+i~}+..., r,d 1.01
(8)
I-(‘,‘)= ac, (pIz-“J’la){p+a} + 3 (P4T &a
(‘~*)I(~i)~{~+d+ia}+... ,
(9)
the indices ~,a are defined by the active holes and particles. From eq. ( 1) and the definition of the model space, inserting into the Schrijdinger equation, we have rR-‘Hl2C@,=E,C@,,
(10)
(@,~i2-‘H~~C@,)=C,E,.
(11)
If we then define Heff=p(k,!)(a-
(13)
(14)
C,‘HeffCI=EI,
(15)
where EI is a diagonal matrix and C, is an eigenvector which, along with 52, defines the wavefunction Y, or Yk,l). If we now define the orthogonal space projectors Qck.‘) for k particles and 1 holes,
I&>(@jI Y
F
(16)
(17)
@,# Nk”’ > 3
that is, the projection operator over all function space not spanned by the model determinants, then projection by the Q-space Bloch equation of the form
will give equations
)
(21)
Af=(Y,‘II C)“2(
(22)
YJ;I YJ)“*
is for Y,= YJ the expectation value of the operator a in the state Y1;,and for Y,# YJ the transition moment of a. It can be shown that even if the operator &?C,& from eq. (2) is not a unitary one, after some manipulations it can be shown that the following expressions for the expectation value and the square of the transition moment absolute value are valid:
A:=(@,1 Gm-‘mGPI,>
,
(23)
IA:I’= I (@,I vw-‘mGP.d
12.
(24)
If one defines the operator @I=Q--’ PQ
(25)
and matrix element
where
Hap’k,”
A(n) =A+ti
(12)
where
1 _p(k.O=
culate dipole transition moments in the generalized coupled-cluster method has been applied to FSMRCC [ 28 1. If A^is a Hermitian operator representing a certain perturbation to our system described by Hamiltonian fi,
then the quantity
lHQ)P(k,‘) 9
(H”ff),,=(~,IP’k,‘)(52-1HS2)P(k,’)I~p),
Q(kO=
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CHEMICAL PHYSICS LETTERS
Volume 193, number 5
_- QH,_$(k.l) us the
(18)
multi-reference
Q(k*‘)(Hi2-SLH,fl)P(k,i)-0 -
~:=(@,lQ-‘m@JJ> y,= yco.0,,
(27)
yJ= yl(1.l)
(28)
with the normal-ordered
perturbation
coupled-cluster
(19)
or (@ck.‘) IHL2-1;2H,,(@‘k~” p >--0 3 I
(26)
and chooses
a,= ,
3
1 (ilrlj){i+j} 1.j
(29)
as a one-electron dipole moment operator, then the transition moment between the ground state and excited states can be calculated from the expression
(20)
where @j# {@Lk.”}. The form of the Q-space set {$j} will vary depending on the sector of interest. The Fock space multi-reference method also allows for a direct calculation of first- and second-order static properties as well as response functions. Here the Stolarczyk and Monkhorst [ 29 ] idea to cal-
IAJIZ= I (OolQ-‘A,(S2C,)@,) In a matrix representation, Jth row of the matrix,
12. where CJ indicates
I v-J/L WJC,(CJ) _‘WJ, the matrix elements
(30) the
(31)
WJ and WJ are 375
Volume
193, number
5
CHEMICAL
WJ=(@cisI PJI Q%E,O > 2
(32)
wJ=(@E,Oi
>
(33)
@J&S)
PHYSICS
(34)
In this application the higher particle operators l@,, have not been considered. The second-order quantity I VJI * has been used to calculate dipole length transition moment VJ and the dipole length oscillator strength f”=;
I vJ~2(~J-EG,)
(35)
>
where all quantities
are in hartree atomic units.
3. Results and discussion Calculations reported in this paper were carried out with the FSMRCC program developed by Rittby, Pal and Bartlett [ 3 11. A single contracted Cartesian Gaussian DZP basis set has been used in this work. This comprises Dunning’s 4s2p contraction [ 321 of Huzinaga 9s5p primitive set [ 331, augmented with a polarization exponent of 1.2 1 [ 341. Using single determinant self-consistent field (SCF) reference functions formed from these basis sets, correlation corrections were computed by means of Fock space multi-reference coupled cluster theory. The Czv geometry optimized in the CCSD method [22] hasbeenused(R,=1.263&8,=117.4”).The Table 1 Calculated
vertical transition
energies (in eV) from the 0s ground state to the electronic states below 5 eV and comparison
No.
State
Dominant excitation
POL-CI =)
MRD-CI
la,+2b, 6a,+2b, 4bz+2b, 4b2+2b,
1.60 2.01 2.09 2.34
1.20 1.59 1.44 1.72
I
‘B2
2 3 4
‘RI 3Az ‘A*
a) Ref. [7].
b, Ref.
376
[ 191.
5 June 1992
SCF configuration for ozone obtained with the DZP basis is [core] 3a: 2b:4a: 5a: 3b: 1b: 4b: 6a: 1a:2by7ay5b:. The results have been compared with data from the best calculations previously available: the polarization configuration interaction (POL-CI) technique [ 7 ] and large-scale configuration interaction calculations (including energy extrapolation) MRD-CI [ 191. In both cases a DZP quality basis set has been used. The comparison with recently published multi-configurational linear response MCLR results has also been made [ 2 11. These calculations employ a larger POL 1 basis [ 35 1. The l-2 eV energy region is primarily characterized by states which result from a single excitation out of one of the highest (energetically closely spaced) MOs of the ground state (4b:, 6a: and 1a$ ) into the lowest unoccupied type orbital 2b:. The calculated vertical transition energies together with experimental data and other computational results are given in table 1. The 3B2 state is found to be the lowest-lying in all methods, but the calculated spacing among all four states is relatively small. MRDCI gives the 3A2 below 3B, state, opposite to the POLCI and FSMRCC results. The FSMRCC and MCLR results are very close to each other and give the same ordering of all energy levels. The comparison with experiment shows that the obtained energies are in the experimental region. In the 2-5 eV and above 8 eV region (table 2) the first two singlet states ‘Bi and ‘B2 (known as Chappuis and Huggins bands) are still characterized mainly by single excitations into the 2b, MO’ while the spectrum beyond 8.0 eV is characterized by single excitation states populating the second-lowest unoccupied MO, the orbital 7a:.
where W”( WJ) represents the connected part of (Q-r l&)c and can be divided into zero-, one-, twoparticle operators &@~+@,+ti*+....
LETTERS
‘) Ref. [21].
b,
with experiment
MCLR ‘)
FSMRCC
Exp.
1.14 1.66 1.81 2.14
1.37 I .62 1.95 2.17
peaks at 1.29 1.43 1.55 1.67 1.80 1.92
5 June 1992
CHEMICAL PHYSICS LETTERS
Volume 193, number 5
Table 2 Calculated vertical transition energies (in eV) from the ground state to the various states in 2 to 5 eV and 8 eV No.
State
Dominant excitation
POL-CI 8)
6a,+2b, la2-r2b, 4bz-+7a, 6a,-+7a, 6a,+7a, la2A7a, 4b2-+7a,
2.41 6.12
5
‘B,
6 7 8 9 10 11
‘R* 3R* 3A, ‘A, )Az ‘R2
‘) Ref. [7].
b, Ref. [ 191. ‘) Ref. [21].
MRD-CI b,
1.95 4.97 8.45 8.44 9.29 9.30 10.05
In tables 1 and 2 one can see that all the states are dominated by single excited configurations. Our model space in the ( 1, 1) sector of the FSMRCC method has also been created by single excited configurations. The accurate results for ‘B2, 3BZand ‘Bz states support the known fact that good results can be achieved if the model space includes the dominant configurations. It should also be noted that the FSMRCC method is not a single-state oriented method like most multi-reference methods. The main idea of this method is to obtain many states at the same time. As a result, some accuracy for the single state can be lost, but the objective is to reproduce the whole spectrum. This conclusion is supported by the results presented in table 3, where the dipole transition moment and oscillator strength have been shown. One can see that the dipole oscillator strength, the quantity measured by experiment, is reproduced well. Some questions may arise if we look at the tran-
MCLR =I
FSMRCC
2.17 5.10
2.13 5.52 8.07 8.17 9.29 9.92 9.80
8.86 9.38
Exp.
2.13 4.86 8.55 8.45 9.32 10.2
sition moment. For the states ‘B2 and ‘Al (states 5 and 9 in table 3) it seems to be too small in comparison to other methods. Approximations including a relatively small basis are made in its evaluation, but it is di~cult to say much about the error since the final values of the oscillator strength are appropriate. One can also see that oscillator strengths and transition moments are better for the iB2 and ‘A, states (states 6 and 10 in table 3) than for the other one. Looking at the eigenvectors of the effective Hamiltonian He” shows that these states have more multi-reference character than the others. One might expect that the multi-reference coupled-cluster method should work better for those types of states. In table 3 the newest values of dipole transition moments and oscillator strengths in the length representation are shown. Again the comparison among POL-CI, MRD-CI, FSMRCC and expe~ment~ data has been made.
Table 3 Transition moment and oscillator strength for OJ dipole allowed transitions No.
State
transition moment ‘R* 5 ‘R* 6 ‘A* 9 ‘Rz 10 oscillator strength 5 ‘RI 6 ‘R* 9 ‘A, 11 ‘RZ *) Ref. [7].
“Ref.
POLCI a’
MRD-CI b,
MCLR ”
FSMRCC
0.028 1.228
0.017 1.202 0.011 0.648
0.012 0.863 0.034
0.007 0.983 0.003 0.687
4.71 x 10-S 2.3x 10-l
1.5x 10-S 1.76x IO-’ 0.24x 1O-2 1.05x 10-l
7.3X IO-6 0.93x 10-l 0.26X 10-f
2.85X 10-e 1.30x 10-i 0.25x IO-” 1.13x10-
Exp.
2x 10-5 0.88x IO-’
[ 191. “‘Ref. [21].
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CHEMICAL PHYSICS LETTERS
Volume 193, number 5
5 June 1992
Table 4 Calculated vertical ionization potentials (in eV) for ozone obtained from various treatments Koopmans
State
*Al ZB2 2Az
15.31 15.64 13.27
‘) Ref. (191.
“Ref.
MRD-CI a)
12.44 12.40 12.79 [36].
MBPT b’
12.90 13.28 13.15
‘) Basis I: DZP (48 functions).
4. Summary We have presented FSMRCC calculations for free ozone. Results for properties (dipole transition moment and oscillator strength) and spectra (energy) have been given. The calculations have been done for the ground state open CIVfor ‘A,. We found that the theory can be successfully applied for obtaining spectra and its intensity. One should also emphasize that a small (three holes and three particles) active space has been used. The transition moment amplitudes were fast and well converged. Once the energy is given the oscillator strength can be achieved.
Acknowledgement
This work has been supported by the US Office of Naval Research ONR-NO00 14-9 1-J- 1282.
References [ I] J. Chappuis, Compt. Rend. Acad. Sci. (Paris) 91 ( 1880)
378
basis I c,
basis II d,
12.37 12.54 13.19
12.69 12.82 13.43
12.75 13.03 13.57
(‘A?) (*Bz?)
d, Basis II: [ 14s7p3d/Ss4p2d] (87 functions).
Ionization potentials of O+ The first three ionization potentials of ozone have also been treated in the present work and the results are compared in table 4, with those of previous theoretical studies, as well as with the pertinent experimental data. The present calculations (even if a better basis set is used) consistently give the ‘A,, ‘B2 and 2A2 ordering of states which supports earlier many-body perturbation theory (MBPT) [ 361 results but it conflicts with the conclusion of MRD-CI results.
985; N. Huggins, Proc. Roy. Sot. 48 ( 1890) 216.
Exp.
FSMRCC
[2] T.J. Lee, W.D. Allen and H.F. Schaefer III, J. Chem. Phys. 87 (1987) 7063; [ 3) M.V. Rama Krishna and K.D. Jordan, Chem. Phys. 115 (1987) 423. [4] Y. Yamaguchi, M.J. Frisch, T.J. Lee, H.F. Schaefer III and J.S. Binkley, Theoret. Chim. Acta 69 ( 1986) 337. [ 5 ] W.D. Laidig and H.F. Schaefer III, J. Chem. Phys. 74 ( 1981) 3411. [ 61 G. Karlstrom, S. Engstriims and B. Jiinsson, Chem. Phys. Letters 57 (1978) 390. [7] P.J. Hay and T.H. Dunning Jr., J. Chem. Phys. 67 (1977) 2290. [S] P.J. Hay, T.H. Dunning Jr. and W.A. Goddard III, J. Chem. Phys. 62 (1975) 3912. [9] W.A. Goddard, W.J. Hunt and P.J. Hay, Accounts Chem. Res. 6 (1973) 398. [IO] R.J. Bint andM.D. Newton, J. Chem. Phys. 59 ( 1973) 6220. [ 111 P.J. Hay, T.H. Dunning and H.F. Schaefer III, Chem. Phys. Letters 23 (1973) 457. [ 121 S. Rothenberg and H.F. Schaefer III, Mol. Phys. 2 1 ( 197 1) 317. [ 131 T. Cvita, L. Klasinc and B. Kovac, Intern. J. Quantum Chem. 29 (1986) 657. [ 141 J.M. Dyke, L. Golob, N. Jonathan, A. Morris and M. Okuda, J. Chem. Sot. Faraday Trans. 2 ( 1974) 1828. [ 151 C.R. Brundle, Chem. Phys. Letters 26 (1974) 25. [ 161 D.C. Frost, S.T. Lee and C.A. McDowell, Chem. Phys. Letters 24 (1974) 149. [ 171 T.N. Radwan and D.W. Turner, J. Chem. Sot. A ( 1966) 85. [ 181 G. Vendlin, Structure and bonding, Vol. 45 (Springer, Berlin, 1981) pp. I-117. [ 191 K.H. Thunemann, S.D. Peyerimhoff and R.J. Buenker, J. Mol. Spectry. 70 (1978) 432. [20] J.I. Steinfeld, S.M. Adler-Golden and J.W. Gallagher, J. Phys.Chem.Ref.Data16 (1987)911. [21] D. Nordfors, H. Agren and H.J.Aa. Jensen, Intern. J. QuantumChem.40(1991)475. [22] R.J. Bartlett, C.E. Dykstra and J. Paldus, in: Advanced theories and computational approaches to the electronic structure of molecules, ed. C.E. Dykstra (Reidel, Dordrecht, 1987) p. 127. [23] R.J. Bartlett, J. Phys. Chem. 93 (1989) 1697; Ann. Rev. Phys. Chem. 32 ( 198 1) 359.
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[31] R.J. Bartlett, G.D. Purvis III, G.B. Fitzgerald, R.J. Harrison, Y.S. Lee, W.D. Laidig, S.J. Cole, D.H. Magers, E.A. Salter, G.W. Trucks, C. Sosa, M. Rittby, S. Pal and M. Barysz, ACES I Program System; M. Rittby, S. Pal and R.J. Bartlett, J. Chem. Phys. 90 ( 1989) 3214. [32] T.H. Dunning Jr., J. Chem. Phys. 53 ( 1970) 2823. [33] S. Huzinaga, J. Chem. Phys. 42 (1965) 1293. [34] L.T. Redmon, G.D. Purvis III and R.J. Bartlett, J. Am. Chem. Sot. 101 (1979) 2856. [35] A. Sadlej, Coll. Czech. Chem. Commun. 53 (1988) 1995. [ 361 L.S. Cederbaum, W. Domcke, W. von Niessen and W.P. Kramer, Mol. Phys. 34 (1977) 381.
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