Excitation Energies with Cost-Reduced Variant of the Active-Space EOMCCSDT Method: The EOMCCSDt-3̅ Approach Han-Shi Hu and Karol Kowalski* William R. Wiley Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, K8-91, P.O.Box 999, Richland, Washington 99352, United States ABSTRACT: In this paper, we discuss the performance of several simplified variants of equation-of-motion coupled cluster method (EOMCC) with iterative inclusion of singles, doubles, and active-space triples (EOMCCSDt). In particular, we explore simplified EOMCCSDt approaches that enable one to generate the triply excited amplitudes in an on-the-fly manner. The original EOMCCSDt formulation has already demonstrated great success in encapsulating the most important excited-state correlation effects due to triples. In analogy to the original EOMCCSDT-3 formulation, the proposed approach can bypass the typical bottlenecks associated with the need for storing triply excited amplitudes. In this paper, we illustrate the performance of several approximate EOMCCSDt methods, named EOMCCSDt-3̅ and EOMCCSDt-3̅, on typical benchmark systems including C 2 ,N 2 , ozone, ethene, and E-butadiene molecules. These new methods yield excitation energies close to the EOMCCSDt ones. The extrapolation of excitation energies for basis sets ranging from cc-pVDZ to cc-pV6Z for N 2 and C 2 shows very good convergence to the experimental results for states dominated by single excitations. The performance of the EOMCCSDt-3̅x approach is also compared with the results obtained with popular CCSDR(3) and CC3 approaches. ■ INTRODUCTION The equation-of-motion coupled cluster (EOMCC) meth- ods 1-6 or closely related linear response CC, 7-12 symmetry adapted cluster configuration interaction formalisms, 13 and spin-flip EOMCC methods 14 have evolved into one of the major tools for calculating vertical excitation energies (VEE) and for the characterization of excited-state potential energy surfaces. Over the last few decades, there has been significant effort toward establishing the hierarchy of various EOMCC approximations, which has enabled accurate descriptions of a wide spectrum of excited states characterized by various configurational structures of corresponding wave functions. For example, for relatively small systems, excited states dominated by single excitations can be very accurately described by the EOMCC approximation involving single and double excitations (EOMCCSD approach 1-3 ). For more complicated states containing non-negligible contributions from doubly excited (with respect to the reference determi- nant) configurations, the inclusion of collective three-body excitations in the EOMCC formalism is required for accurate results. However, full inclusion of triply excited amplitudes in the EOMCC formalism (resulting in the EOMCCSDT formalism 15 ) comes at a high numerical price, which is proportional to N 8 , where N symbolically designates the system size. Several noniterative methods have been introduced with the purpose of mimicking the effect of triply excited amplitudes in calculating vertical excitation energies such as the EOMCCSD(T) 16 and EOMCCSD(T ̃ ) 17 formulations, several noniterative methods originating from various formalisms including linear response CC, 18 methods of moments of the CC equations (MMCC), 19-21 spin-flip formalism, 14,22,23 or perturbative techniques for similarity transformed Hamilto- nian. 24-27 Unfortunately, the conclusions about the EOMCC accuracies inferred from the benchmark calculations for small molecular systems cannot be directly extrapolated toward larger molecular assemblies. As shown in the series of papers discussing EOMCC calculations for singly excited states in molecules composed of several tens of light atoms, 28-31 the EOMCCSD approach for these systems no longer matches the level of accuracy of the EOMCCSD method for small systems. These studies clearly indicate a growing role of higher-rank excitations (three-body effects) in describing singly excited states of large systems. The importance of this higher-rank excitations was shown in refs 32-34. which have recently been reiterated in refs 27, 35, and 36 in studies of DNA bases. Although the noniterative EOMCC methods accounting for triple excitations in most cases improve the EOMCCSD results, the recent studies of (TiO 2 ) n clusters 37 pointed out possible problems of these noniterative formulations in describing singly excited states. In order to achieve agreement with experimentally inferred data, the iterative inclusion of the triple excitation was necessary. It was also demonstrated that active-space EOMCCSDT formulations (or EOMCCSDt for short) 15 provide an efficient way of attaining near-EOMCCSDT Received: June 12, 2013 Published: September 27, 2013 Article pubs.acs.org/JCTC © 2013 American Chemical Society 4761 dx.doi.org/10.1021/ct400501z | J. Chem. Theory Comput. 2013, 9, 4761-4768