A Comparison of Variational and Coupled-Cluster Calculations of

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J. Phys. Chem. 1996, 100, 6325-6331

6325

A Comparison of Variational and Coupled-Cluster Calculations of Molecular Properties: The Polarizabilities of BeO, 1Σg+, and C2, 1Σg+, 3Πu, and 3ΣgVudhichai Parasuk,†,§ Pavel Neogra´ dy,‡ Hans Lischka,*,† and Miroslav Urban‡ Institute for Theoretical Chemistry and Radiation Chemistry, UniVersity of Vienna, Wa¨ hringerstrasse 17, A-1090 Vienna, Austria, and Department of Physical Chemistry, Faculty of Science, Comenius UniVersity, BratislaVa, Mlynska´ dolina, 842 15 BratislaVa, SloVak Republic ReceiVed: October 9, 1995; In Final Form: December 30, 1995X

The parallel and perpendicular components of the static dipole polarizabilities of the title molecules in different spectroscopic states were calculated using a large variety of high-level electron correlation methods. Excitation energies between different states for C2 are presented as well. We used both single-reference coupled-cluster methods at different level of sophistication and complete active space SCF, multireference configuration interaction, and averaged quadratic coupled-cluster methods with various active spaces. The reliability of calculated properties for BeO and C2, known as difficult to calculate accurately, was deduced from the pattern of results observed by a systematic improvement of the level of sophistication and from a comparison of coupled-cluster and multireference variational methods.

1. Introduction The aim of this paper is to further extend our knowledge about the reliability of well-established methods of quantum chemistry aimed at accurate predictions of molecular properties. Obviously, in developing a new method the first step in testing its reliability is comparison with some reference data, either experimental or accurate theoretical. By “accurate theoretical” we mean full configuration interaction (FCI) benchmark data. Unfortunately, FCI is only possible with limited basis sets. However, the evaluation of the importance especially of higher excitations within the FCI will depend strongly on the quality and size of the basis set. A comparison with experiment also requires calculations with extended basis sets if various compensation effects, like the deficiency of the theoretical method overshadowed by the unsaturation of the basis set, should be kept at a minimum. Another, more generally applicable approach is to follow a pattern of results obtained with systematically improved theoretical methods. In fact, this is a typical procedure in ab initio quantum chemistry1 born in 1956. This systematic approach forms the basis of the present work. It is loosely related to our previous investigations2,3 aimed at setting error bars for a variety of coupled cluster (CC) methods.4-7 In the present work we extended the methodology by a group of variational methods, namely the multireference single and double excitation configuration interaction (MRCISD), the averaged coupled pair functional (ACPF),8 and the averaged quadratic coupled-cluster approach (AQCC).9 Both groups of methods, CC and variational, allow a systematic improvement which will help in the assessment of the reliability of results. Improvement in CC methods with a single determinant reference used in this work may be quite transparently followed by gradually extending the completeness of the exponential wave function expansion (in terms of the wave function perturbation expansion) in the iterative procedures in which the amplitudes †

University of Vienna. Comenius University. § Permanent address: Department of Chemistry, Faculty of Science, Chulalongkorn University, Phyathai Road, Patumwan, Bangkok 10330, Thailand. X Abstract published in AdVance ACS Abstracts, March 1, 1996. ‡

0022-3654/96/20100-6325$12.00/0

of the T1, T2, T3, etc. (single, double, triple, etc. excitation) operators are evaluated. Eventually one extends the perturbative sequence in noniterative procedures. The improvements within the group of variational methods is obtained by the systematic extension of the number of reference configurations used for the singles and doubles CI and of the molecular orbitals defining the active space. The crucial parameter is the number of configuration state functions (CSFs) in the reference wave function which depends on the number of active electrons (n) distributed among active orbitals (a) according to certain selection principles (see later). The selection of a set of active orbitals is often not straightforward. That means the “numerical experiments” with different numbers of active electrons and active orbitals have to reveal the quality of the calculation. The high quality of all methods used in this study was demonstrated in many previous papers (see e.g. refs 6, 7, 1012). However, there are notoriously difficult cases which were selected as our test molecules here. First is the highly ionic BeO molecule in which the near degeneracy of the beryllium 2s and 2p orbitals causes problems in some single reference methods.13 For example, Watts et al.3 have demonstrated that one of the approximations to CC with singles and doubles (CCSD), denoted as quadratic CISD (QCISD), completely fails for the calculation of the dipole moment and the dipole polarizability of this molecule. This holds even if triples are included in a noniterative waysQCISD(T). The reason is that the exp (T1 + T2) expansion which defines CCSD is incomplete in QCISD, and this does not guarantee the proper coupling of the two operators T1 and T2, especially in a situation when the amplitudes of T1 are large, as with BeO. Electrical properties are especially sensitive to this coupling. This follows from the fact that amplitudes of the T1 operator are of the first order in the presence of the external electric field, in contrast to the situation without an external field, in which case they are of the second order. This is actually the reason why we consider electrical properties of some molecules as a “hard” test of our quantum chemistry methods. Another “difficult” case is the C2 molecule, especially in its ground state, 1Σg+ for which it has been shown that a strong mixing between two configurations ...(2σg)2(2σu)2(1πu)4 © 1996 American Chemical Society

6326 J. Phys. Chem., Vol. 100, No. 15, 1996 and ...(2σg)2(3σg)2(1πu)4 occurs (see ref 14 and other references therein). It was also demonstrated by Watts and Bartlett15 that the T2 amplitudes, corresponding to the 2σu2 f 3σg2 excitations are very large (see also ref 3). Still, Watts and Bartlett15 were successful in obtaining reasonably accurate spectroscopic constants for C2 by CC methods. Martin et al.16,17 came to the same conclusion. So we were attracted by the idea to investigate the electrical properties, namely dipole polarizabilities, of C2 in its electronic ground state in a series of more demanding calculations. Also, we decided to calculate polarizabilities of the first and second excited states of C2, which were supposed to provide much less quasidegeneracy. These properties may also be interesting as such in view of a permanent interest in Cn clusters. 2. Methods, Basis Sets, and Other Computational Details 2.1. Methods. The “simplest” CC method used in the present work is CCSD.18 Triples are accounted for in the exponential expansion either accurately in the CCSDT method19 or approximately in the CCSDT-3 approximation.20,21 In this last method, some higher order wave function projections in the iterative evaluation of amplitudes are omitted. Also noniterative approaches, CCSD + T(CCSD)20 and its extension, CCSD(T),22,23 were applied. They differ by a single term (the fifthorder term when canonical orbitals are used as a reference), [5] of singles with triples, which is included the “interaction” EST in the CCSD(T) energy. The T(CCSD) term may be denoted as E[4] TT, i.e. it corresponds to the regular triples in the fourth order of the perturbation expansion with triple amplitudes obtained from the converged double excitation amplitudes from CCSD. Details may be found in the literature (see e.g. refs 24-26, with a specific emphasis to open-shell systems relevant in the present work). Crucial for the reliable accuracy of CCSD[5] is (T), presently the most often used CC method, is that EST small when the energy or any property related to it (e.g. energy derivatives) is calculated. In situations when this term is large one can suspect that other fifth-order contributions arising from singles, doubles, triples, and connected quadruples may be large as well. We demonstrated their importance at least for the closed-shell ground state of C2, which shows the largest multireference character. We should note that these terms are computationally quite demanding and thus not suitable for routine calculations. We also should add that the discussion of individual terms in noniterative procedures for triples is a bit more complicated when general noncanonical orbitals are used as a reference (see e.g. refs 24-26). Besides of the sequence of higher order contributions to the property of interest also some other diagnostics are described in the literature. The most often used is the T1 diagnostic27 which takes the Euclidean norm of the T1 amplitudes subject to Hartree-Fock orbitals and normalizes them. However, as was shown in a recent paper by Watts et al.,3 in some difficult cases the amplitudes of the T2 operator are large and cause problems, even if the T1 diagnostic is not extremely large. In the sequence of multireference calculations the reference configurations are constructed in the following way (see also Table 1): starting from a single-reference function denoted “SR” all single excitations into a subset of additional active orbitals are constructed giving the reference set “S”. Next all single and double excitations are taken resulting in reference set “SD”. Finally, all possible CSFs within the given active space (complete active space, “CAS(n/a)”) are chosen. These reference sets are used to construct all single and double excitations. The ACPF and AQCC methods use exactly the same expansion sets as the MRCISD does.

Parasuk et al. TABLE 1: Notation for Various Reference Spaces:a The Ground State of BeO, 1Σg+, Ground State of C2, 1Σg+, and Two Excited States of C2, 3Πu and 3ΣgSR S SD CAS(6/6) CAS(6/7)

BeO 1Σ+ (1σ)2(2σ)2(3σ)2(4σ)2(1πx)2(1πy)2 ...{(4σ)2(1πx)2(1πy)2;(5σ)(2πx)(2πy)(6σ)}S ...{(4σ)2(1πx)2(1πy)2;(5σ)(2πx)(2πy)(6σ)}SD ...{(4σ)(1πx)(1πy)(5σ)(2πx)(2πy)} ...{(4σ)(1πx)(1πy)(5σ)(2πx)(2πy)(6σ)}

SR S SD CAS(6/6) CAS(6/7)

C2 1Σg+ (1σg)2(1σu)2(2σg)2(2σu)2(1πux)2(1πuy)2 ...{(2σu)2(1πux)2(1πuy)2;(3σg)(1πgx)(1πgy)(3σu)}S ...{(2σu)2(1πux)2(1πuy)2;(3σg)(1πgx)(1πgy)(3σu)}SD ...{(2σu)(1πux)(1πuy)(3σg)(1πgx)(1πgy) ...{(2σu)(1πux)(1πuy)(3σg)(1πgx)(1πgy)(3σu)}

SR CAS(6/6) CAS(6/7)

C2 3Πu (1σg)2(1σu)2(2σg)2(2σu)2(1πux)1(1πuy)2(3σg)1 ...{(2σu)(1πux)(1πuy)(3σg)(1πgx)(1πgy)} ...{(2σu)(1πux)(1πuy)(3σg)(1πgx)(1πgy)(3σu)}

SR CAS(6/7) CAS(8/8)

C2 3Σg(1σg)2(1σu)2(2σg)2(2σu)2(1πux)1(1πuy)1(3σg)2 ...{(2σu)(1πux)(1πuy)(3σg)(1πgx)(1πgy)(3σu)} ...{(2σg)(2σu)(1πux)(1πuy)(3σg)(1πgx)(1πgy)(3σu)}

a SR stands for single reference, S for single excitations from the reference configuration indicated, SD for single and double excitations.

2.2. Basis Sets. Contracted Gaussian orbital basis sets used in the present study have been presented and tested in previous papers. We used the [5s3p2d] polarized basis set (POL) of Sadlej,28 which is economical enough and still specifically suitable to calculations of electrical properties. More extensive is the averaged ANO basis (Atomic Natural Orbitals) developed by Widmark et al.29 It is a generally contracted basis. We used the contraction [5s4p3d2f]. This basis set was extended by g-functions30 in some calculations. 2.3. Computational Details. Dipole moments and dipole polarizabilities have been computed by a finite-field technique using external electric fields of the strength 0.001 au. In the calculation of the parallel component of the polarizability the electric field was oriented in the z direction. In calculations on the perpendicular component the field has been oriented in the y direction which is important for the calculation of the 3Πu state (see the specification of the reference space in Table 1). First and second derivatives with respect to the electric field were calculated numerically from the energies. Unless stated otherwise, all occupied orbitals are correlated except for the K shell orbitals which are frozen. The calculations were carried out using D2h symmetry or one of its subgroups. In UHF CC calculations we exploied the ACESII program system.31 In calculations with the restricted open-shell HartreeFock (ROHF) reference with spin adaptation26,32 we used our computer program combined with the MOLCAS II program.33 Fifth-order contributions were calculated also by the Comenius program.34 The variational calculations were performed by the Vienna installation of the COLUMBUS program system.35-37 To this COLUMBUS version the HERMIT integral generator38-40 has been connected and was used for the finite field calculations. 3. Results and Discussion The description of electronic configurations of the ground state of the BeO and C2 molecules, and the two excited states of C2, together with a variety of reference configurations, is presented in Table 1. Total energies of all species with zero electric field are summarized in Table 2.

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J. Phys. Chem., Vol. 100, No. 15, 1996 6327

TABLE 2: A Selection of Total Energiesa with Zero External Field for BeO and C2 in the Ground and Two Excited States at the Experimental Geometryb

TABLE 4: T1 Diagnostics, Largest CCSD Amplitudes, and CI Expansion Coefficients of the Leading CSFs for the Ground State of BeO and the Ground and Excited States of C2a

molecular/state basis set/method

BeO

(1Σg+)

C2 (1Σg+)

C2(3Πu)

C2(3Σg-)

SCF MCSCF CAS(6/7) MCSCF CAS(8/8) CI SR CI CAS(6/7) AQCC SR AQCC CAS(6/7) CCSD CCSD(T) CCSDT

POL[5s3p2d] -89.429 143 -75.391 742 -89.554 408 -75.607 419 -89.658 439 -75.672 525 -89.682 052 -75.735 010 -89.670 692 -75.707 776 -89.685 718 -75.738 993 -89.703 870 -75.708 794 -89.714 159 -75.736 831 -89.715 237 -75.735 419

-75.593 455 -75.707 173 -75.736 664 -75.723 982 -75.742 389 -75.724 321 -75.737 831 -75.739 052

-75.504 733 -75.585 338 -75.697 724 -75.713 937 -75.708 899 -75.720 872 -75.709 256 -75.716 493 -75.717 986

SCF MCSCF CAS(6/7) MCSCF CAS(8/8) CI CAS(6/7) AQCC CAS(6/7) CCSD CCSD(T)

ANO[5s4p3d2f] -89.451 143 -75.405 172 -89.578 569 -75.620 512 -89.763 423 -75.788 796 -89.768 098 -75.794 450 -89.778 404 -75.759 124 -89.795 540 -75.793 651

-75.604 897 -75.786 485 -75.794 042 -75.772 380 -75.790 746

-75.514 243 -75.595 337 -75.760 444 -75.769 216 -75.754 988 -75.766 057

T1 largest diagnostics amplitudes

0.041

0.039

0.034

a Energies are given in Hartrees. b R(BeO): 1.331 Å. R(C ): 1Σ + 2 g 1.243 Å, 3Πu 1.312 Å, 3Σg- 1.369 Å (see ref 44).

TABLE 3: Dipole Moments and Dipole Polarizabilitiesa BeO Calculated at the Experimental Geometryb Using Various Methods and Basis Setsc method

ref space

µ

R|

R⊥

2.95 2.95 2.61 2.50 2.50 2.65 2.55 2.46 2.43 2.43 2.11 2.37 2.36 2.50 2.45 2.40 2.40 2.39 2.56 2.26 2.43 2.29 2.40

19.7 19.7 24.6 26.1 26.1 25.6 29.0 31.8 33.2 33.6 85.4 38.6 39.7 33.7 32.9 37.1 35.9 36.6 29.8 45.9 34.4 44.9 36.0

22.4 22.4 29.8 31.8 32.1 28.9 34.1 34.2 34.8 35.0 d 35.0 33.0 37.0 34.9 35.0 35.1 31.3 38.7 35.1 35.0 34.3

20.4 27.0 33.1 33.1 36.1 28.3 42.8 34.4

21.9 30.2 32.5 32.5 32.5 28.9 34.7 32.6

POL[5s3p2d] SCF MCSCF

CI

ACPF AQCC

CCSD CCSD+T(CCSD) CCSD(T) CCSDT-3 CCSDT

S SD CAS(6/6) CAS(6/7) SR S SD CAS(6/6) CAS(6/7) SR CAS(6/6) CAS(6/7) SR S SD CAS(6/6) CAS(6/7)

ANO basis set [5s4p3d2f] 2.96 CAS(6/6) 2.54 CAS(6/6) 2.47 CAS(6/7) 2.47 AQCC CAS(6/6) 2.42 CCSD 2.60 CCSD+T(CCSD) 2.33 CCSD(T) 2.46 SCF MCSCF CI

-0.089 -0.030 -0.040

BeO, 1Σg+ (ref 3) ANO [5s4p3d2f] ...{(4σ)2(1πy)2(1πy)2} T1(4σ-5σ) T1(4σ-6σ) T2(1π2-2π2)

0.083 -0.294 -0.047 0.080

C2 1Σg+ POL[5s3p2d] ...{(2σu)2(1πux)2(1πuy)2} T1(2σg-3σg) T2(2σu-3σg2) T1(2σu-3σg) T2(1πu2-3σg2)

0.046 -0.035 -0.093

C2, 3Πu POL[5s3p2d] ...{(2σu)2(1πux)1(1πuy)2(3σg)1} T1ββ(2σg-3σg) T1ββ(2σu-3σu) T2RβRβ(1πux2σu-1πgx3σg)

0.023 -0.059 0.047

C2, 3ΣgPOL[5s3p2d] ...(2σu)2(1πux)1(1πuy)1(3σg)2 T1ββ(2σu-3σu) T2RRRR(1πux1πuy-1πgx1πgy) T2RβRβ(2σu2σu-1πgy2πgy)

of 0.012

a

In au. b See Table 2. c CC results are taken from ref 3. d Result not meaningful.

3.1. BeO, 1Σg+. Both coupled cluster and variational calculations of electrical properties of BeO are summarized in Table 3. Single-reference CC results are taken from ref 3. It was mentioned above that for this molecule the proper coupling of T1 and T2 amplitudes is crucial. The incomplete coupling in QCISD results in a dramatic failure of this method.3 Both

excitation

CI expansion coefficientsb

0.943

0.859 0.320

0.910

0.933

a All values are calculated at the experimental geometry. b Taken from the CI CAS(6/7) calculation.

T1 and T2 amplitudes are relatively large, but the T1 diagnostic, 0.041 with the ANO [5s4p3d2f] basis set, is relatively small. Largest amplitudes are presented in Table 4. Very instructive is the comparison of CC with CI and AQCC results. It is seen that calculations with a single reference configuration are generally very sensitive to the selected method. This contrasts to virtually all methods which use a multireference wave function. The exception is MCSCF where the inaccuracy of the dipole moment and polarizability is obviously caused by the missing dynamical correlation effects. Both CASSCF (6/ 6) and (6/7) are not capable of recovering this deficiency. On the other hand the MRCI, ACPF, and AQCC methods with the same set of references mutually agree very well (only ACPF is a little overeshooting for R|). Results with CAS(6/6) and CAS(6/7) references are almost identical. The explanation lies in the fact that in both these references the most important orbitals, i.e. 4σ, 5σ,. 2πx, and 2πy are already included in the CAS(6/6) space. The disagreement of CCSD+T(CCSD) with CCSD(T) has already been discussed.3 It demonstrates the importance of the [5] which couples singles and triples. CCSDfifth-order term EST (T) agrees very well with complete iterative CCSDT which in turn differs very little from AQCC CAS(6/7) results. This gives a high confidence to both these results as well as to the much more practical and applicable CCSD(T) method which, even if triples are included noniteratively, performs quite well. The CCSD(T) dipole moment is higher by only 0.04 au than the AQCC CAS(6/6) value and the parallel polarizability is lower by less than 2 au. The perpendicular components are almost identical with the two methods in the ANO [5s4p3d2f] basis set. The high accuracy of CCSD(T) results for electrical properties is a bit surprising if one realizes that just a single

6328 J. Phys. Chem., Vol. 100, No. 15, 1996

Parasuk et al.

TABLE 5: Dipole Polarizabilitiesa for the Ground State 1Σ + of the C Molecule, Calculated at the Experimental g 2 Geometryb Using Various Methods and Basis Sets method

ref space

R|

R⊥

44.4 25.6 24.3 24.5 58.8 25.8 25.7 25.8 68.2 26.0 25.9 26.0 24.4 44.7 31.9 26.8 26.1

-1.2 21.3 18.1 18.4 -13.0 18.2 18.6 19.9 99.9 21.3 21.6 20.2 20.3 61.9 33.6 21.3 20.0

44.6 24.4 24.7 25.9 25.9 26.1 26.1 25.2 44.4 31.8

-0.2 18.7 19.0 20.5 20.6 20.8 21.0 21.1 65.4 36.3

44.6 24.4 25.9 26.1 25.3 44.3 31.7

-0.3 18.7 20.5 20.8 21.2 66.0 36.6

POL [5s3p2d] SCF MCSCF CI

AQCC

SD CAS(6/6) CAS(6/7) SR SD CAS(6/6) CAS(6/7) SR SD CAS(6/6) CAS(6/7)

CCSD CCSD+T(CCSD) CCSD(T) CCSDT-3 CCSDT ANO [5s4p3d2f] SCF MCSCF CI AQCC

CAS(6/6) CAS(6/7) CAS(6/6) CAS(6/7) CAS(6/6) CAS(6/7)

CCSD CCSD+T(CCSD) CCSD(T) ANO [5s4p3d2f1g] SCF MCSCF CI AQCC CCSD CCSD+T(CCSD) CCSD(T) a

CAS(6/6) CAS(6/6) CAS(6/6)

In au. b See Table 2.

[5] fifth-order term, EST , is considered. This leads us to the conclusion that other fifth-order terms cancel successfully. The credibility of our results is further supported by a relatively small basis set dependence. For the same methods both basis sets used in this paper, POL [5s3p2d] and ANO [5s4p3d2f], give very similar results. 3.2. C2, 1Σg+. Accurate calculations of the electrical properties of the C2 molecule is an even more challenging task than that for BeO. The multireference character of its ground state is well known, which implies that single reference-based approaches are supposed to have only little chance to provide accurate results. Still, Watts and Bartlett15 obtained a quite satisfactory equilibrium distance, harmonic frequency, and dissociation energy of this molecule. Martin et al.17 are even more optimistic in their conclusion concerning the performance of CCSD(T) in calculations of vibrational spectra, relative stabilities, and excitation energies of various Cn (n ) 1-8) compounds. Extended MRCI and FCI calculations on spectroscopic properties have been reported by Kraemer and Roos,14 Bauschlicher and Langhoff,41 Pradhan et al.,42 and Peterson.43 In the latter two, careful investigations of basis set convergence have been carried out. Electrical properties, however, are expected to be even more demanding, as our results in Table 5 confirm. SRCI as well as SR-AQCC fail to predict reasonable polarizabilities. CC results also demonstrate that the problem is really difficult. Both parallel and perpendicular components differ with CCSD+T

TABLE 6: Individual Contributions to the Correlation Energy, the Parallel and Perpendicular Polarizabilities of the 1Σ + Ground State of C a,b g 2 contributionc

energy contributiond

R|

R⊥

CCSD T(CCSD) E[5]TT E[5]DT E[5]QT E[5]QQ E[5]ST CCSD+TQ[CCSD]

-0.317 052 75 -0.320 763 2 -0.002 910 34 +0.004 068 93 +0.006 145 52 -0.007 156 51 +0.004 039 90 -0.344 941 57

-19.91 +20.29 +1.08 -3.27 -5.78 +1.08 -12.78 -19.31

+21.51 +41.55 +6.24 -7.74 -9.40 +10.33 -28.26 +34.22

a Calculated at the experimental geometry; see Table 2. b The POL basis was used. c E[5]TT in the notation means the fifth-order contribution which arises from the interaction of triples in the noniterative energy expression, i.e. the term 〈|T3WT3|0〉 where W is the two electron correlation perturbation, S, D, T, and Q means singles, doubles, triples, and connected quadruples; see ref 23. d At zero external field.

(CCSD) and its fifth-order extension, CCSD(T), even more than we found in BeO. With all three basis sets the CCSD+T (CCSD) value of R| is higher by about 30% than its CCSD(T) counterpart. The situation is even worse for R⊥: it is about two times higher with CCSD+T(CCSD) than with CCSD(T). This is an indication that C2, 1Σg+, is really a problem for singlereference CC. Once again, all three basis sets show approximately the same pattern with all methods applied. So, let us discuss the POL [5s3p2d] calculations, for which we have the largest variety of results, more carefully. First we note that once we use the multireference wave functions, either CASSCF(6/6) or (6/7), the results differ relatively little for CI as well as for AQCC. CASSCF itself behaves in the same way. It is interesting that even though dynamical correlation is important, CASSCF works quite well. We suppose that this is to some extent fortuitous. The good agreement of results obtained with CAS(6/6) and CAS(6/7) active spaces is explained, as with BeO, by the fact, that the dominant orbitals are already included in the (6/6) active spacesthe additional 3σu orbital does not contribute heavily to dominant excitations. Concerning single-determinant CC methods, the dramatic difference between the two noniterative methods, CCSD+T(CCSD) and CCSD(T), suggests that no noniterative method is reliable. In fact, CCSD(T) overestimates R| in comparison to full CCSDT by 5.8 au, i.e. about by 22%. R⊥ is even worse; the overestimation of 13.6 au represents 68% of the CCSDT value. CCSD(T) relies just on well-balanced [5] fourth-order contributions with EST , a single fifth-order term, which is large. This holds for both R| and R⊥. We then expect that other fifth-order contributions could be large as well, which is demonstrated in Table 6. Even if all E[5] terms were considered, the inaccuracy, especially in R⊥, would not be removed, because simply the T1 and T2 amplitudes, used in all noniterative methods following CCSD, are not accurate enough and moreover are large. This holds even if the T1 diagnostic for C2, 1Σg+, is not larger than that for BeO. Also, the T1 amplitudes are comparable in these two isoelectronic systems (both being as large as 0.08-0.09), but the highest T2 amplitude is almost 0.3 for C2, in comparison to 0.04 for BeO. It corresponds to the 2σu2 f 3σg2 excitation and shows that C2 1Σ + really needs a multideterminant reference, at least with g CCSD and noniterative triples. The low weight of the leading CSF (Table 4) is in accord with this finding. In contrast to noniterative methods both iterative procedures, CCSDT-3 and full CCSDT, lead to very similar results and are capable of recovering the deficiency of the single reference. Both agree excellently with AQCC-CAS(6/7) and CI-CAS(6/7).

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J. Phys. Chem., Vol. 100, No. 15, 1996 6329

TABLE 7: Dipole Polarizabilitiesa for the First Excited State 3Πu of the C2 Molecule, Calculated at Experimental Geometryb Using Various Methods and Basis Sets method

ref space

TABLE 8: Dipole Polarizabilitiesa for the Second Excited State 3Σg- of the C2 Molecule, Calculated at Experimental Geometryb Using Various Methods and Basis Sets

R|

R⊥

20.1 43.0 62.7 58.5 40.1 35.2 39.9 32.1 35.4 39.5 35.3 37.1 39.4

9.0 22.1 23.7 29.2 24.4 24.7 24.8 25.2 22.8 30.2 25.3 24.3 24.4

SCF MCSCF CI

19.6 65.4 31.0 27.7 34.2 39.4 34.9

9.4 24.2 24.9 25.4 22.8 31.3 26.2

AQCC

method

POL [5s3p2d] SCF MCSCF CI AQCC

CAS(6/6) CAS(6/7) SR CAS(6/6) CAS(6/7) CAS(6/6) CAS(6/7)

CCSD CCSD+T(CCSD) CCSD(T) CCSDT-3 CCSDT

CAS(6/7) CAS(6/7) CAS(6/7)

AQCC

a

CAS(6/7) CAS(6/7) CAS(6/7)

19.6 65.4 31.0 27.9 34.2 39.5 35.0

3.3. The 3Πu and 3Σg- Excited States of C2. Results for the polarizability for the first excited state are presented in Table 7, those for the second excited state in Table 8. These highspin open-shell systems were calculated mostly with the UHF reference for CC methods. The influence of the spin contamination and the spin adaptation was verified by a comparison with results using the restricted open-shell HF (ROHF) reference. A. The 3Πu State. In contrast to BeO and C2, 1Σg+, for both the CI and the AQCC methods the parallel polarizabilities differ quite significantly between CAS(6/6) and CAS(6/7) reference spaces. The perpendicular component, on the other side, is not so sensitive to the choice of reference configurations. The reason for such a behavior is the importance of the 3σu orbital as an excitation orbital, not present in the (6/6) active space. Another indication of the deficiency of the active space is the large difference between CASSCF (6/6) and (6/7) results for R| (31% of the CASSCF (6/7) value). The R⊥ values agree quite well. Even larger reference spaces would be needed to confirm the CAS(6/7) results for R|. Various approximations to the single-determinant CCSDT results provide smaller differencies for both components of the polarizability than CI and AQCC. Even if the T1 diagnostic is only slightly lower than that for the ground state of C2, the values of the largest amplitudes (ordered in absolute value) are “only” 0.046 for T1 and -0.093 for T2. This is an even more important indication than the T1 diagnostics. Interestingly enough, R| computed with CCSD+T(CCSD) agrees this time with full CCSDT better than CCSD(T). This may be a consequence of lower T1 amplitudes than for other cases. However, we should not overemphasize this finding. B. The 3Σg- State. This is a case of a good single-reference wave function as one can see from the weight of the leading

28.2 27.6 27.9 28.1 28.4 28.3 28.5 28.6 28.5

ANO [5s4p3d2f] SCF MCSCF CI

CAS(8/8) SR CAS(6/7) SR CAS(6/7)

CCSD CCSD+T(CCSD) CCSD(T)

28.0 27.7 27.4 27.5 27.8 27.7 27.9 28.0 27.9

ANO [5s4p3d2f1g] SCF MCSCF CI

9.4 24.2 24.8 25.4 22.8 31.3 26.2

In au. b See Table 2.

CAS(8/8) SR CAS(6/7) SR CAS(6/7)

CCSD CCSD+T(CCSD) CCSD(T)

ANO [5s4p3d2f1g] SCF MCSCF CI AQCC CCSD CCSD+T(CCSD) CCSD(T)

R|

POL [5s3p2d]

ANO [5s4p3d2f] SCF MCSCF CI AQCC CCSD CCSD+T(CCSD) CCSD(T)

ref space

AQCC CCSD CCSD+T(CCSD) CCSD(T) a

CAS(8/8) SR CAS(6/7) SR CAS(6/7)

28.1 27.7 27.3 27.4 27.7 27.6 27.9 27.9 27.8

In au. b See Table 2.

configuration listed in Table 4. More support for this conclusion can also be found by comparing the polarizability values computed with single-reference and multireference CI and AQCC methods as well as by the good agreement between the CC, MRCI, and MR-AQCC methods (see Table 8). In Table 8 only results for the parallel component of the polarizability are given since we encountered difficulties in calculating the perpendicular component of the polarizability. This is because under the reduced symmetry of C2V introduced by the addition of the perpendicular external field, the in-plane component of the 3Πu state and the 3Σg state belong to the same irreducible representations, the 3Πu state being the lower one. Under this constraint, the regular single determinant-based calculations could not be carried out. MCSCF and MRCI calculations on the second root could be done. However, with our present choices of reference spaces it was difficult to obtain a balanced description of the molecule with and without external field. Much larger reference spaces and CI expansions, going beyond the scope of this work, would be necessary in order to obtain sufficiently converged results. For the 3Σg- state we have also examined the influence of eventual spin contamination which may arise when the UHF reference is used. Using our approach26,32 for CCSD(T) calculations with the ROHF reference and with a spin adaptation of the most important part of the double excitation amplitudes of the T2 amplitudes, we calculated the polarizabilities of the 3Σ - state. The denominator was constructed from the diagonal g part of the Fock operator. Also we used the approach based on semicanonical orbitals as implemented in ACESII. We found differences of both spin-adapted and nonadapted ROHF results with the UHF reference less than 0.1 au, which is completely negligible.

6330 J. Phys. Chem., Vol. 100, No. 15, 1996

Parasuk et al.

TABLE 9: Excitation Energiesa 1Σg+ f 3Πu b and 3Πu f 3Σ -,c with a Variety of Methods and Basis Sets g method

ref space

1Σ + g

f 3Πu

3Π

u

f 3Σg-

ANO [5s4p3d2f] ROHF-SCF MCSCF CI AQCC CCSD(T)d ROHF-CCSD(T)sd,e ROHF-SA1-CCSD(T)dd,f CCSD CCSD+T(CCSD) CCSD(T) MCSCF CI AQCC CCSD CCSD+T(CCSD) CCSD(T)

CAS(6/7) CAS(6/7) CAS(6/7)

ANO [5s4p3d2f1g] CAS(6/7) CAS(6/7) CAS(6/7)

19119 -3427 -507 -90 -890 -894 -976 2909 -1365 -636

4819 4300 5449 3817 5611 5418

-2644 -592 -243 2841 -1456 -726

5743 5476 3828 5641 5446

Selected Data from the Literature -613 MRCI [8s6p3d]g -654 MRCI [5s4p3d2f1g]h -142 MRCI+Q [5s4p3d2f1g]h -981 CMRCI estim. basis set limiti -443 CMRCI+Q estim. basis set limiti -772 CCSD(T),cc-pVQZi -885 CCSD(T), pVQZk -716 experimentl

5593 5648

5615 5718

a All values in cm-1; 1 au } 219 474.6 cm-1. b The energy difference E(1Σg+) - E(3Πu). c The energy difference E(3Σg-) - E(3Πu). d Including the correlation of the two inner-shell orbitals. e Restricted openshell HF with semicanonical orbitals (see ref 24). f ROHF reference; spin-adapted T2 amplitudes corresponding to excitations from doubly occupied orbitals to virtual orbitals, with diagonal Fock operator used in denominator in noniterative triples (see ref 26). g Reference 14. h Reference 41. i Reference 43. j Reference 16. k Reference 15. l Reference 44.

3.5. The 1Σg+ f 3Πu and 3Πu f 3Σg- Excitation Energies of C2. The calculation of the excitation energies shown in Table 9 is a byproduct extracted from our zero field energy values. The first excitation energy is as small as -716 cm-1 44 and requires high accuracy obtained for both ground and excited states. They differ, however, quite significantly in their multireference character and, as previously noticed by Watts and Bartlett15 provide another hard test for quantum chemical methods. The splitting between the two triplet states is less demandingsboth are much less quasidegenerate than the ground state. Let us mention that the POL 5s3p2d basis, specifically designated primarily to electrical properties, leads to the incorrect stabilities of the two states (it prefers the 3Πu state with almost all methods used). The ANO 5s4p3d2f basis set is already not too far from results with its g extension counterpart. This latter basis set is identical in size and comparable in quality to the cc-pVQZ basis set used by Peterson43 who showed that this basis set gave results close to the basis set limit for the spectroscopic constants of C2. MCSCF leads to a 1Σg+/3Πu splitting which is much too large. Addition of at least 4σg and 4σu orbitals is needed to achieve a balanced description of both states at the CAS level.14 The CI CAS(6/7) results are still not converged with respect to the AO basis set but give the most reasonable results in the group of variational methods. The AQCC methods leads to sizeconsistency corrections which reduce the excitation energy in absolute value considerably. The MRCI results of Bauschlicher and Langhoff41 also give singlet/triplet splittings which are too small in absolute value. Their quadruples corrections show

similar trends as our AQCC results. Somewhat larger singlet/ triplet splittings are obtained by Peterson43 in his CMRCI calculations. This is probably due to the larger CAS(8/8) reference space in his work. The inclusion of quadruples corrections leads also to an underestimation of the excitation energy in absolute value with all basis sets used. Due to the large difference between CCSD+T(CCSD) and CCSD(T) excitation energies the latter value is not considered to be too reliable as well, even though it is closest to the experimental ones. Triples are extremely important: the CCSD value predicts the 3Πu state much more stable than the ground state. The lower accuracy for excitation energies with single reference CC methods was noticed earlier by Watts and Bartlett.15 In addition, we found from our CC calculations that correlating the four inner-shell electrons increases the excitation energy (in absolute value) by about 264 cm-1. At the first sight, this change seems to be large, but the effect of core electrons of about 0.03 eV is quite common. We should note, however, that our basis is not specifically designed for inner-shell correlation effects. The UHF CCSD(T) result is very close to that with ROHF semicanonical orbitals.24 The spin adaptation26 brought a change of about 82 cm-1, which is small but not negligible. Our best value, -726 with CCSD(T) and the ANO [5s4p3d2f1g] basis set is very close to the experimental value, but further increases (all in absolute value) due to the abovementioned inner-shell electron correlation and due to a small contribution arising from the spin adaptation is to be expected. Thus, we can estimate that our best CCSD(T) value may be as large as -1000 cm-1 after all corrections. The energy separation between the two triplet states is much less sensitive to the method and results are much more consistent, as we could expect on the basis of the previous discussion. AQCC CAS(6/7) and CCSD(T) values agree very well, CCSD+T(CCSD) and CCSD(T) results differ by less than 200 cm-1. The deviation from experiment is less than 300 cm-1. 4. Conclusions In this work, the assessment of the reliability of the calculations is based on trends which are observed when the basis set and the level of sophistication of the theoretical methods is gradually improved. Among the many indicators well known in the literature we concentrated especially on the following ones: (1) the sensitivity to extensions and/or changes of reference spaces in MRCI calculations; (2) the size of the coefficients of the leading CSFs in the variational methods; (3) changes when the perturbative order of the energy or the wave function in CC is increased (large differences between CCSD+T(CCSD) and its extension, CCSD(T), usually indicates possible problems of the single-reference noniterative CC methods); and (4) large values of the T1 diagnostics, or, even more important, large excitation amplitudes in CC. These indicators have not been applied so much to extrapolate results to basis set and excitation level limits but to use them as diagnostics which tell more generally about the reliability of the methods used. One important aspect of our work was that indicators derived from MRCI and CC methods have been used at the same time and compared to each other. Very similar results are obtained for the electrical properties of BeO using MR CI, MR AQCC, and the single reference CCSD(T) methods and thus are not too sensitive to further improvement of the theoretical level. Therefore, these values

Variational and Coupled-Cluster Calculations can be considered as reliable, despite the previous sceptiscism concerning single reference CC results. The recommended value of the dipole moment of BeO is 2.42-2.46 au, the parallel polarizability is 34.4-36.1 au, and the perpendicular polarizability is 32.5-32.6 au, confirming previous CC results.3 Due to the strong multireference character of the 1Σg+ state of C2, polarizabilities are predicted to be less accurate for this state than for BeO. The excellent agreement of AQCC CAS(6/7) with MRCI CAS(6/7) and with the full iterative CCSDT method with the POL basis and the insensitivity to the extension of the active space gives good credibility to the AQCC CAS(6/7) values. The suggested value for the parallel component is 26 au and for the perpendicular component 21 au. The extremely large difference between CCSD+T(CCSD) and CCSD(T) results for the perpendicular component reveals a possibility of problems with CCSD(T). CCSD(T) predicts that the parallel component is smaller than the perpendicular one, in disagreement with full CCSDT and AQCC CAS(6/7) results. A different situation is found for the 3Πu state. Results (especially for the parallel component) are sensitive to the selection of the active space, MR-CI and AQCC differ significantly for different CAS choices, while CC results are relatively more stable. On the basis of a comparison with the full CCSDT results with a smaller basis we recommended the values of 35-39 and 25-26 au for the parallel and perpendicular components of the polarizability, respectively. The R| value for the 3Σg- state is calculated to lie between 27 and 28 au with all methods. It is quite insensitive to the basis set and is thus considered as very reliable. The very small excitation energy between the 1Σg+ and 3Πu states is extremely difficult to calculate accurately due to the multireference character of the 1Σg+ state. Even if our best CCSD(T) value of -726 cm-1 agrees with the experimental value, -716 cm-1 excellently, we cannot claim that this value is reliable enough. The MRCI CAS(6/7) calculations gives a 1Σ +/3Π excitation energy which is too small in absolute value g u compared to the results given by Peterson.43 It is further decreased by size-consistency corrections at the AQCC level. These corrections are rather large and seem to overshoot. However, considering still existing deficiencies in the reference wave function and in the basis set and missing electron correlation contributions of the core orbitals as discussed above the present AQCC value could be consistent after applying further corrections. Calculated values for the triplet-triplet splitting are more reliable (between 5446 and 5743 cm-1 with CCSD(T), AQCC CAS(6/7)) and in good agreement with the experimental value of 5718 cm-1. Acknowledgment. This work was carried out with the support of the Austrian “Fonds zur Fo¨rderung der wissenschaftlichen Forschung”, Project No. P9032-CHE and by the COST action D3. We also are grateful for support by the exchange programs between the University of Vienna and the Comenius University in Bratislava. One of us (V.P.) is thankful for a fellowship by the Austrian Academic Exchange Service. We are grateful to Dr. T. U. Helgaker for giving us the HERMIT program package. We thank the Slovak grant agency for support of this work, under contract No. 1/1455/1994.

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