Open-shell coupled-cluster method: Direct calculation of excitation energies

Volume128, number 1

CHEMICALPHYSICSLElTERS

OPEN-SHELL COUPLED-CLUSTER METHOD: DIRECT CALCULATION OF EXCITATION ENERGIES

11 July 1986

*

Uzi KALDOR and Azizul HAQUE ’ School of Chemrstry Received

2 March

Tel-Avrv University, 69978 Tel Avrv, Israel 1986; in final form 1 April 1986

An open-shell coupled-cluster method for the direct calculation of excitation energies is presented. As a first test, applications to atomic Be and Ne are carried out, with exact inclusion of T, and T2 operators and lowest-order inclusion of T3. Quasicomplete model spaces are used. Two ionization potentials and 16 excitation energies are calculated. They all agree with experiment to 0.15 eV.

1. Introduction The exp S or coupled-cluster method (CCM) [ 1,2] has been used widely in recent years for ab initio electronic structure calculations. Most applications involved non-degenerate systems (see ref. [3] for a recent review), although the single-reference method seems applicable even in quasi-degenerate situations [4]. A variety of multireference, open-shell (OSCCM) formulations [S--18] has also been described. Recently we reported the application of OSCCMto the direct calculation of electron affinities [ 19,201 and ionization potentials [21]. Single and double excitations were included to all orders (the CCSD approximation). Triple excitations, with one or more electrons excited out of the valence shell, turned out to be important in many cases. The lowest-order inclusion of triple excitations, calculated from the converged CCSD amplitudes while ignoring their own contributions to other excitations (the CCSD t T approximation), gave good electron affinities [20] and ionization potentials [21] for all the systems studied. The purpose of the present work is to apply a similar method to the direct calculation of excitation

energies. This extension of the method presents additional difficulties with regard to the construction of the model space, as discussed below. Pilot applications to several atomic excitations are reported.

2. Method The basic operator in OSCCM is the wave operator S2,which acts upon the Slater determinants in the model (or P) space, $8, and transforms them into the exact functions W , w

= i22\k”o (a = 1, ... . d) .

P is the projector upon the model space, \k;=Pw

0 009-2614/86/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(2)

The energies of the system are given by the eigenvalues of the effective Hamiltonian Heff = PHi2P,

(3)

where H is the physical Hamiltonian of the system. Following Lindgren [lo], we use the normal-ordered

OSCCM , S2={expT).

* Supported in part by the US-Israel Binational Science Foundation. ’ Present address: Department of Chemistry, University of Rochester, Rochester, NY 14627, USA.

(1)

(4)

The nature of the model space and of the excitation operator T depends upon the system under investigation. If the system is best described as having N elec45

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trons outside a clsoed shell, the model space will consist of determinants with N valence orbitals and T will include terms with 0, 1,2, .... N valence particles. The eigenvalues of Heff will then describe the energy of adding N electrons to the closed shell. Ionization potentials from a closed shell, on the other hand, may be obtained by applying OSCCMin a model space characterized by valence holes. In both cases, Haque and Mukhejee have shown [ 171 that the excitation operator may be partitioned according to the number of valence orbitals involved, T= T(o) t T(1) + T(2) + ... ,

(5)

with partial decoupling of the OSCCM equations, so that the equation for T(“l involves only T(“I elements with m < n. This strategy has been followed in our previous work [ 19-211. The treatment of excitations from a closed shell requires the simultaneous inclusion of valence holes and particles in the formalism, New types of diagrams

I

j

k

I

Fig. 1. Excitations included in the calculations. Tn(I,& de_ scribes excitations of n orbiti, of which I are valence particlesandm arevalence holes. Diagram (a) is Tf”lo), (b) Tz(o*O), (c) T(l*‘), (d) TilpO), (e) Tpp’), (f) T,$@‘),(g, h) $sr), (i) $~“~, (j) Z$Ovl),(k, 2) T3( *I). Downgoing lines are holes, upgoing lines are particles. Double arrows denote valence lines. Diagrams (a)-(h) are included in the CCSD scheme; the inclusion of diagrams (O-(Q) to the lowest order gives the CCSD + T method.

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11 July 1986

and T elevents are added (see fig. l), which can easily and systematically be derived from ones used previously. Eq. (5) is also easy to generalize, 7-z T(u,o) + T&u) + J@,l) + T&l) + ... ,

(6)

where the two superscripts denote the number of valence particles and holes, respectively, and the partial decoupling of equations still holds. A severe problem arises however with the structure of the model space. OSCCMis easiest to apply in a complete model space, including all possible combinations of valence orbitals. Jeziorski and Monkhorst derived a method capable of handling general model spaces [ 141, similar in spirit to a many-body perturbation theory (MBPT) presented earlier by Hose and Kaldor [22]. The method is rather difficult and has so far been applied in the linear approximation only [23]. The use of complete model spaces, while feasible if valence particles or holes exist, is highly problematic when both types have to be taken into account. Multiple excitations involving all valence holes and particles must then be included in the space, the dimensionality of which will be very large, requiring a great computational effort. Even more important, the broad energy span of the space will give rise to intruder states, leading to divergences in most cases of interest. Lindgren [24] has described a more general class of model spaces, called quasicomplete. These are spaces where an excitation from a model determinant can always be classified as internal (leading to another model space determinant) or external, the classiflcation being independent of the determinant on which the excitation acts. Complete model spaces have this property by definition, including as they do all possible combinations of the valence orbitals, but not all incomplete spaces share it. The effective Hamiltonian in a quasicomplete model space does include some disconnected diagrams; these are however few in number and are expected to be small [24]. The states investigated here are single excitations from a closed-shell system. A natural choice for a model space would therefore include all determinants with one valence hole and one valence particle, with valence orbitals selected properly. Such a space is quasicomplete, since a transition is internal if it does not change the number of valence holes and particles and external otherwise. A space of this type can be kept within reasonable size and energy span, and the

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corresponding excitation operator includes only the four terms shown explicitly in eq. (6). The T excitations were classified above by the number of valence holes and particles involved. A more common classification is according to the total number of excited electrons, T=T1+T2+T3+....

(7)

Most coupled-cluster calculations, for closed- or openshell systems, include only the double excitation T2 (CCD). The single excitation T, is often included as well, giving the so-called CCSD approximation. It should be noted that the CCSD scheme includes all disconnected triple and quadruple excitations, left out by the CISD (configuration interaction, singles and doubles) method. These account for the major part of the triple and quadruple excitations, as pointed out by Sinanoglu [25]. Lee and Bartlett recently calculated the connected triple excitations T3 for the closed-shell Be, system [26]. We found it necessary to include T3 in our OSCCMto obtain good predictions for electron affinities [20] and ionization potentials [21]. An exact calculation of T3 requires great computational effort, and its contribution was therefore obtained to lowest order only, from the converged CCSD excitation amplitudes (see ref. [20] for details). The same level of approximation is employed here. All the T elements used in this work are shown in fig. 1.

3. Calculations and discussion Excitations from the closed-shell atoms beryllium and neon were chosen for a first test of the method. The basis sets used were 6s5pld, based on the 6-3 1lG* sets of Krishnan et al. [27] with two diffuse s and p Gaussians added to describe the excited orbitals. Their exponents were 0.207 and 0.069 for Be, 0.132 and 0.043 for Ne. The Is orbital was kept frozen for both atoms (a Be calculation including excitations out of the 1s did not change the results significantly). The CCSD equations (T = T1 + T2) were solved for T(O*u),T( WI, ~K41) , and T(l*l). Quasicomplete model spaces [24] were used, comprising all determinants with one valence hole and one valence particle. For beryllium, the 2s was defmed as the valence hole, and the two lowest unoccupied s and three lowest p orbitals served as valence particles. The 2p orbital was

11 July 1986

Table 1 Excitation and ionization energies of Be (ev)

2s ionization 2s+ 2p sP ‘P 2s+3s 3S ‘S 2s+ 3p 3P ‘P 2s+4s 3S ‘S

Exp. (28)

CCSD

CCSD + T

9.32 2.72 5.28 6.46 6.78 7.29 7.46 8.00 8.09

9.28 2.84 5.61 6.48 6.84 7.34 7.46 8.08 8.28

9.28 2.82 5.44 6.46 6.76 7.30 7.39 8.06 8.23

the valence hole in the neon calculations, with the lowest unoccupied s and p orbitals the valence particles. Using the conver ed CCSD amplitudes, the lowest-order T$lso), Z$8,l) and @*l) were calculated [20], and their contributions to the transition energies added to the CCSD values. The core term Z”$‘*“’ was ignored, as it contributes only to the core correlation and not to the transition energies. The ionization potentials and excitation energies are shown in tables 1 and 2, and compared with the experimental results taken from Moore’s tables [28]. Most CCDS results for the beryllium atom agree well with experiment (within mO.1 eV), the major exception being the 2s-t 2p IP transition, which is 0.3 eV off. It is exactly this transition on which the inclusion of T3 has the largest effect (0.17 ev). All CCSD + T transition energies agree with experiment to within 0.16 eV, the average error being 0.07 eV. The CCSD is not very satisfactory for Ne, with the ionization potential too low by 0.5 eV and excitation energies 0.2-0.3 eV off. The inclusion of T3 is crucial here, Table 2 Excitation and ionization energies of Ne (ev)

2p ionization 2p-,3saP ‘P 2p+3p3S sD ‘D 3P ‘P ‘S

Exp. [28]

CCSD

CCSD + T

21.56 16.67 16.85 18.38 18.57 18.64 18.71 18.69 18.97

21.06 16.41 16.59 18.13 18.38 18.47 18.51 18.51 18.84

21.69 1q.71 16.85 18.49 18.73 18.78 18.84 18.84 19.12

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and it brings all transition energies to within 0.15 eV of experiment.

4. Conclusion The open-shell coupledcluster method has been shown capable of direct calculation of electron excitation energies from closed shells. The CCSD t T method, which accounts for single and double excitations to all orders and includes lowest-order connected triples, gives accurate results with reasonable computational effort. The advantage of calculating transition energies directly, whether they are ionization potentials, electron affinities or excitation energies, is that errors in the two states may be kept comparable, and some corrections which affect both equally need not be computed at all (e.g. T$‘$‘) terms in the present formulation). The method described here is now being applied to molecular systems.

Acknowledgement

We are grateful to professor I. Lindgren for sending us his work prior to publication.

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