Highly correlated single reference studies of the O3 potential surface. Dissociation and atomization energies

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CHEMICAL PHYSICS LETTERS

17 November 1989

HIGHLY CORRELATED SINGLE REFERENCE STUDIES OF THE O3 POTENTIAL SURFACE. DISSOCIATION AND ATOMIZATION ENERGIES John F. STANTON, Rodney J. BARTLETT Quantum Theory Project, Departments of Chemistry and Physics, Universityof Florida, Gainesville, FL 32611, USA

David H. MAGERS and William N. LIPSCOMB Gibbs Chemical Laboratory, Harvard Universify, Cambridge, MA 02138, USA Received 14 August 1989; in final form 25 August 1989

Highly carrelated many-body perturbation theory (MBPT) and coupled-cluster (CC) calculations based on a single determinant reference function are applied to computation of the dissociation and atomization energies of ozone in its ground electronic state. While results obtained in SCF calculations are in error by > 100 kcal/mol for both quantities, many-body methods which include the effeCts of triply substituted determinants (MBPT (4) and CCSDT-I ) yield values in relatively good agreement with experiment. The present results are found to be superior to those obtained in multireference configuration interaction studies (MRCI ). Comparable accuracy is obtained with the empirically corrected Cl method and also in MRCI calculations, provided a correction is applied to account for the effects of unlinked clusters.

1. Introduction Recently, a great deal of attention has been paid to the problem of electron correlation in the ground state ( ‘A, ) of the ozone molecule [ 1-6 1. Generalized valence bond calculations carried out more than a decade ago revealed that the ground state was well described as a biradical with two unpaired x electrons localized on the terminal atoms and weakly coupled into the observed singlet electronic state [ 71. This effect gives rise to extensive configuration mixing between the [core...]4b$6a: la: and [core...]4b:6a:2b: configurations, thereby placing considerable demand on theoretical electronic structure methods. Although best treated within the framework of a multireference (MR) approach which incorporates (at least) the two configurations above into the MCSCF reference state, sophisticated trcatments of electron correlation - particularly those based upon coupled-cluster (CC) #’theory - which +I’For a review of CC methods and their application to chemistry, see ref. [ 81.

are capable of “recovering” from a poor reference function may be applied with some success even if a single determinant (one configuration) reference function is used. In the first paper of this series [ 11, we applied the highly correlated methods of many-body perturbation theory (MBPT) [9] and various CC implementations to a study of the structure and harmonic force field of ozone. The results of that study demonstrate a striking influence of electron correlation on the computed properties, particularly those involving the asymmetric region of conformation space (C, geometries) where the most prominent effects involve two singly excited determinants in addition to the second configuration listed in the paragraph above. At the MBPT( 2) level of theory with a DZP basis set, the predicted harmonic frequency of the asymmetric stretching mode (0s) is overestimated by 117%, while CCSD reduces the error to 14%. However, methods which include only an approximate treatment of connected triple excitations (T3) give values which differ significantly from the experimental value of 1089 cm-’ (128i cm-’

0 009-2614/89/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )

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N2 and 680 cm-’ (CCSDT(CCSDtT(CCSD)) 1) ). The surprisingly poor performance of CCSDT1 was subsequently traced to contributions of nonlinear terms in the equation for the T3 amplitudes those which are omitted from the partial CCSDT equations defined by the CCSDT-1 model. The simplest CC scheme which includes the most important nonlinear term in the T3 equation is called CCSDT2. This method gives a value of 1182 cm-’ for w3 [ 61, in better agreement with experiment than any other published theoretical value. In contrast to u3, however, properties which depend only on symmetric (C,,) structures of O3 appear to be far less sensitive to the model used to treat correlation effects. For example, at the MBPT(4), CCSD+T(CCSD), CCSDT-1 and CCSDT-2 levels, predicted bond lengths, bond angles and totally symmetric harmonic frequencies differ from one another by only 0.014 A, l.O”, 82 cm-’ and 38 cm-‘, respectively, and are in good agreement with experiment. Furthermore, differences between values of these properties computed at the SDQ-MBPT( 4) level and its infinite order CC counterpart, CCSD, are much smaller than that found for wj. Hence, it appears that single reference MBPT and CC methods (particularly those which include triple excitation effects) are capable of providing fairly realistic predictions of the totally symmefric properties of ground state ozone. The apparent difficulty associated with asymmetric structures is due to the presence of four important configurations in the molecular wavefunction for only slightly distorted structures [ 11, in contrast to the existence of only two for symmetric forms. Consequently, estimation of properties associated with C, geometries places even greater demand on the reference function than does determination of those which are independent of this part of the potential surface. In this Letter, we apply single reference MBPT and CC calculations to the computation of the total binding energy of ozone as well as to the dissociation energy for the reaction 0,(z,4,)-+02(3~; ) tO(3P), and compare our results with those of previous calculations, particularly the MR configuration inter-

RzUsinga larger [ 5s3p2d] basis, the CCSDt T( CCSD) value for w, is 327 cm-‘.

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action (CI) study of Ramakrishna and Jordan d3. Results obtained with three different basis sets are reported, and the correlation and basis set dependences of the predicted properties are briefly discussed.

2. Methods All calculations reported in this paper were performed with the ACES program system, developed by Bartlett and coworkers [ II]. Three basis sets were used: the DZ basis of Dunning and Hay [ 121 augmented by six Cartesian d-type functions with an exponent of 1.211 [13] (DZP);a [5s3pld] set composed of Dunning’s [ %3p] basis [ 111and the same shell of polarization functions used in the DZP basis; and finally, a [ 5s3p2dlf] basis which includes an additional shell of d functions with exponent 0.4844 ( =0.4x 1.211) and a shell of f functions with exponent 1.4. The calculations presented in this paper are based on the experimental equilibrium geometriesofozone [15] andO, 1161. Reference functions used in the MBPT and CC calculations were of the unrestricted Hartree-Fock (UHF) form for both monatomic and diatomic oxygen, while the restricted Hartree-Fock (RHF) method was used to construct reference functions for 0s. Although it is known that a lower SCF energy for ozone can be obtained with the UHF method [ IO,17 1, results obtained in RHF-based correlated calculations involving ozone appear to be superior to those based on UHF reference functions [ 4, lo]. Since the emphasis of the present study is on the correlated results, we have chosen to use an RHF reference for all calculations on ozone. Correlation energies are presented at the following levels of theory for all species: MBPT( 2)) MBPT( 3), DMBPT(4), SDQ-MBPT(4), MBPT(4), and also with the CCSD and CCSDT-1 coupled-cluster methods. For detailed discussions of the individual MBPT and CC models used here, the reader is referred to extensive review articles [ 8,9,18]. In correlated calculations carried out with the largest [ 5s3p2dlf] ba13 The [ 4s3pld] and 1Ss3pZdlflbasissets used by these workers in ref. [lo] and those used here are not identical, having slightlydifferent polarization exponents.

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sis set, determinants corresponding to changes in the occupation number of the three lowest MOs and the three corresponding virtual functions (e < - 18 au and >48 au, respectively) were excluded from the configuration space.

3. Results and discussion Total electronic energies for 0, O2 and ozone are presented in table 1; dissociation and atomization energies of O3 are collected in table 2 along with other theoretical results and the experimental values. As anticipated from the poor description of ozone provided by the single determinant RHF reference function, the stability of this molecule is grossly under-

17 November1989

estimated at the SCF level of theory, with the decomposition fragments lying below ozone on the generalized OS potential surface. When electron correlation is included at the simplest MBPT( 2) level, the important las+2bT double excitation responsible for the biradical character of ozone first enters the wavefunction. As a result, the binding and dissociation energies increase by > 100 kcal/mol and assume the proper sign for all basis sets. However, as demonstrated previously by Ramakrishna and Jordan, the contribution of doubly substituted determinants to the wavefunction of ozone is substantially overestimated by the first-order correction !?(I) [lo]. This observation is consistent with the present results, where the contributions of double excitations to the next two orders of perturbation theory

Table 1 Total electronic energiesof 0,O2and OScalculatedat

variouslevelsof theorywith the DZP, [Ss3pld] and [%3p2dlf] basis sets. Calculationsperformedat the experimentalequilibriumgeometriesof O2 and O1 WP) DZP ‘) SCF MBPT(2) MBPT(3) D-MBPT(4) SDQ-MBPT (4) MBPT(4) CCSD CCSDT- I

- 74.918389 -74.931671 - 74.934081 -74.933453 -74.934462 -74.933903 -74.935178

- 149.654018 - 150.018294 - 150.016811 -150.018294 - 150.025506 - 150.036397 - 150.025027 - 150.034896

-224.303010 -224.974525 -224.935536 -224.964129 -224.953735 -224.996916 -224.95236X -224.990126

-74.806976 -74.942496 -74.955581 -14.957864 -74.957018 -74.958252 -74.957443 -74.959005

- 149.657558 - 150.069441 - 150.062I52 - 150.072295 - 150.073123 - 150.089461 -150.071306 - 150.085876

-224.310603 -225.052362 -225.004045 -225.035120 -225.026355 -225.080805 -225.022324 -225.069043

-74.808830 -74.946372 -74.961049 -74.963877 -74.962451 -74.964727 -74.962869 -74.965716

- 149.670053 - 150.094700 - 150.089778 - 150.100870 - 150.098929 -150.118797 - 150.097252 -150.115261

-224.330767 -225.097832 -225.050697 -225.083894 -225.069409 -225.130800 -225.065678 -225.117816

-74.805656

[5s3pld] ‘) SCF MBPT(2) MBPT(3) D-MBPT(4) SDQ-MBPT(4) MBPT(4) CCSD CCSDT- I [5s3p2dlf] b’ SCF MBPT(2) MBPT(3)

D-MBPT(4) SDQ-MBPT(4) MBFT(4) CCSD CCSDT- I

a) All electronscorrelated. b, Core MOs and correspondingvirtualfunctionsomitted.

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Volume 163, number 4,5 Table 2 Energies of dissociation

17 November 1989

(I) and binding energies (II) of ozone, calculated at various theoretical levels. All values in kcal/mol

[ 553~ Id] basis a) [ _%3p2dlf] basis ‘)

I

II

I

II

I

II

-96.6 25.4

-69.2 141.1

-92.9 35.6

-60.1 162.3

-0.1

105.1

7.6 9.9

84.8 -

SCF MBPT(2)

-9x.3 23.8

-71.5 137.7

II-CASSCF-CISD, [ %3p2dlf] basis e)

18.9

100.5

MBPT(3)

- 8.1

88.2

II-CASSCF-CISD t MRDC b1 [ Ss3p2dlf] basis ‘J) QCISD(T),6-311G**basis”~d’ G 1c.d.c)

25.3 12.4 30.7

118.3 148.6

D-MBPT(4) SDQ-MBPT(4) MBPT(4) CCSD CCSDT- I

7.4 -3.3 16.4 -4.1 12.6

101.6 96.2 121.4 94.5 115.8

experiment f,

26.1

145.5

II-CASSCF, II-CASSCF,

[ 5s3p2dlf]

[ 5s3pld]

DZP

II

I

-8.6 3.1 - 2.4 20.8 -4.0 15.2

86.2 101.4 97.5 129.3 94.1 120.5

12.0 5.1 29.7 3.5 23.1 26.1

120.6 114.3 148.5 111.1 138.5 145.5

‘) From ref. [lo]. Basis sets differ slightly from those usedin the presentwork. b, Includes multireference Davidson-like correction, eq. (1). c)Calculation includes empirical corrections. d, Values computed from data in refs. [ 5,221. ‘) Gl theory zero-point correction parameters have been removed to compare with vibrationless experimental value. ‘) Value for I from ref. [lo]. II computed from 1 and data in refs. [ 15,161.

are large and have alternating signs. It is likely that the full effect of doubles (D-MBPT(co) or, equivalently, LCCD [ 191) would predict dissociation energies intermediate between the MBPT(2) and MBPT (3) values. Of particular importance are the singly, triply and quadruply (via T$ ) excited determinants which enter MBPT at fourth order. The contribution of triples, which can be inferred from the difference between SDQ-MBPT (4 ) and MBPT (4) results, amounts to > 19 kcal/mol in all calculations. Unlike MBPT methods, which treat all classes of excitation to a certain order in perturbation theory, coupled-cluster approaches treat certain classes of excitation to all orders. For systems which are well described by the reference function, differences between fourth-order MBPT properties and their infinite-order CC counterparts (SDQ-MBPT( 4) versus CCSD and MBPT(4) versus CCSDT (here approximated by CCSDT-1) ) are typically small. In the present case, differences between CCSD and SDQ-MBPT(4) results are relatively modest (~2 kcal/mol), while those between CCSDT-1 and MBPT(4) are approximately three times larger attesting to the important effects of iterative inclusion of triple excitations. It is important to note that previous studies have demonstrated that the CCSDT-1 336

model appears to overestimate the Ts contribution to the correlation energy of ozone due to its omission of nonlinear terms in the T3 equation defined by the CCSDT model [ 1,6]. Hence, reaction energies for I and II predicted by the complete CCSDT model (and, in particular, the CCSDT-2 model which includes the most important nonlinear T: term) might probably be slightly smaller than the CCSDT-1 values and consequently further from the MBPT( 4) results. Evidently, the contribution of singly, doubly and particularly triply excited determinants to fifth and higher orders of perturbation theory is significant in ozone. Some assessment of the higher-order corrections not included in CCSDT-1 can be based upon the benchmark full MBPT (5 ) calculations on ozone carried out recently by Kucharski et al. [ 51. They find that the fifth-order energy for this molecule is nearly twice as large as any of fifteen other examples, and that the MBPT(5) contribution of single, double and triple excitations is a substantial 12.5 kcal/mol. Of this, the ETT term which would be included in the full CCSDT model accounts for - 1.4 kcal/mol. More significant, however, is that the initial contribution of T4 (EgQ) is - 9.0 kcal/mol. Hawever, this is nearly cancelled by the + 8.6 kcal/ mol contribution from the ETQ and EqT terms. The results of our best calculations (those carried

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CHEMICAL PHYSICS LETTERS

out with the [ 5s3p2dlf] basis at the CCSDT-1 level) are in relatively good agreement with the experimental values. Due to the sensitivity of the theoretical results to both basis set and correlation effects, it is impossible to attribute the error to a specific deficiency in either of these factors. Based on our work and the calculations

of Ramakrishna

and Jordan

[ 10 1, one can expect a larger basis to predict a greater stability for ozone, thereby improving agreement with experiment. The probable effect of higher-order correlation effects, however, is more difficult to ascerlain. As noted above, the full CCSDT model would likely yield somewhat smaller reaction energies, but the probable importance of T., in ozone [ 51 suggests that the CCSDT and full CI (exact within a given basis set) results still differ by a few kcal/mol. Comparison of the present calculations with previous theoretical studies reveals that our results are the most accurate ab initio (no empirical correclions) reported to date for this system. The [ 5s3p2d 1f] -CCSDT-1 values are considerably better than those obtained in the II-CASSCF-CISD calculations of ref. [ lo] and with the QClSD( T) “quadratic configuration-interaction” method, using the 6-3 1lG** ( [ 4s3p 1d] ) basis #4. The two sets of empirically corrected results presented in the table are comparable to those obtained in our ab initio calculations, but use of both methods can be objectively criticized in the present case. First, the simple (unnormalized) multireference generalization of the Davidson correction used in ref. [ lo],

~Gx=

l(

C e1C.W

lc;l*

(&SD-EMCSF)

(1)

)

to adjust for the lack of size extensivity (improper scaling with size) inherent in MRCI methods lacks sound theoretical justification, and is particularly suspect when strong non-dynamic correlation effects are present. A recent study by Paldus, Wormer and Benard [ 191 suggests that this correction severely overestimates the magnitude of the correlation energy in quasidegenerate systems (which would tend to exaggerate the stability ofozone), and that a mod114This value was obtained from data presented in refs. [4,22]. Geometries used were the minimum energy MBPT(Z)/63 I G* SCF structures for 0 and O,, and that determinedat the QCISD(T)/6-31 lG** level for 0s.

17 November 1989

ified scalar correction proposed by Meissner [2 1 ] is more accurate. Indeed, the viability of eq. ( 1) was investigated in the work of Ramakrishna and Jordan, in which the authors concluded that the empirical correction factor probably overestimated the unlinked cluster correction to AE for I by approximately 5 kcal/mol. Finally the empirical Gl scheme of Pople et al. [ 22 ] is based on an additivity approximation, in which a number of computed and ad hoc corrections are added to the QCISD(T) energy evaluated with the 6-3 1 lG** basis set in an effort to reproduce the exact energy u at the minimum energy MBPT (2)/6-3 lG* geometry. In the present case, the effects of adding multiple polarization functions and including infinite-order correlation corrections are of striking importance and the assumption of additivity of these effects must be regarded cautiously. To facilitate comparison with the experimental vibrationless energy quoted in ref. [ IO], zero-point corrections computed according to the Gl prescription have been removed from the values in the table. The present study attests to the capability of CC methods to provide reasonably accurate estimates of electron correlation effects for systems exhibiting significant non-dynamic correlation involving the quasidegeneracy of two electronic configurations. Similar behavior has been noted in studies of bondbreaking - a process which can be qualitatively described by two functions of the Heitler-London type - where the CCSDT- 1 model provides a realistic picture of the potential surface over a large range of internuclear distances [ 231. It should be stressed, however, that the single-reference MBPT and CC methods used here are less suited to the task of mapping out the global potential surface of 03. As discussed in section 1, asymmetric structures of ozone present a great challenge to theory due to the presence of a significant number of strongly contributing configurations. Consequently, the current single-reference methods are likely to prove inadequate in

Strictly speaking, energies computed with the G 1 scheme include the contribution of zero-point vibrational motion. To facilitate comparison with the present values, zero-point corrections obtained according to the recipe given in ref. [ 221 were subtracted from Gl energies.

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studies of the dissociation pathway and transition state for the Os+02 ( 3X; ) + 0( 3P) process, where configuration mixing is extensive. However, since CCSDT-2 resolves the problem with the asymmetric stretch mode, higher levels including the full CCSDT model [24] might well be applicable to the transition state. Furthermore, T4 has now also been included in CC methods [ 25 ] which should offer ever greater stability.

Acknowledgement This research was supported by the National Science Foundation (Grant No. CHE85-15347) and the Office of Naval Research. We also thank.the Pittsburgh Supercomputing Center for a generous allocation of time on the Cray YMP, on which most of our calculations were performed. We thank Professor Ken Jordan of the University of Pittsburgh for bringing the problem of ozone dissociation to our attention.

References

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