Article pubs.acs.org/JCTC
Cite This: J. Chem. Theory Comput. 2018, 14, 72−91
Exploring the Accuracy of a Low Scaling Similarity Transformed Equation of Motion Method for Vertical Excitation Energies Achintya Kumar Dutta,† Marcel Nooijen,*,‡ Frank Neese,*,† and Róbert Izsák*,† †
Max-Planck-Institut für Chemische Energiekonversion, Stiftstr. 34-36, 45470 Mülheim an der Ruhr, Germany Department of Chemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
‡
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S Supporting Information *
ABSTRACT: The newly developed back transformed pair natural orbital based similarity transformed equation of motion (bt-STEOM) method at the coupled cluster singles and doubles level (CCSD) is combined with an appropriate modification of our earlier active space selection scheme for STEOM. The resulting method is benchmarked for valence, Rydberg, and charge transfer excited states of Thiel’s test set and other test systems. The bt-PNO-STEOM-CCSD method gives very similar results to canonical STEOM-CCSD for both singlet and triplet excited states. It performs in a balanced manner for all these types of excited states, while the EOM-CCSD method performs especially well for Rydberg excited states and the CC2 method excels at obtaining accurate results for valence excited states. Both EOM-CCSD and CC2 perform worse than bt-PNO-STEOMCCSD for charge transfer states for the test cases studied.
1. INTRODUCTION The accurate calculation of excitation energies is an important area of modern day quantum chemistry. Among the many methods available, coupled cluster (CC) theory has emerged as an accurate and the most systematically improvable way of calculating excitation energies.1 Single reference coupled cluster theory may be generalized to excited states2 in the framework of the so-called equation of motion (EOM) approach.3 The coupled cluster linear response (CCLR) formalism of Jørgensen and co-workers4 leads to identical results for excitation energies, although it follows a different logic. The SAC−CI approach of Nakatsuji is also very closed related5,6 to EOM-CC. The equation of motion coupled cluster method7−9 is generally used at the singles and doubles truncation level (EOM-CCSD) and predicts excitation energies for states dominated by single excitations with sufficient accuracy. However, the N6 scaling and the associated storage requirements of EOM-CCSD prohibit its use for larger molecules. The various approaches described in the literature to reduce the scaling of EOM-CCSD can be divided into three broad categories. The first one is based on the perturbative truncation of the similarity transformed Hamiltonian10,11 and the EOMCC vector.12 In recent times, Pal and co-workers have explored various flavors of perturbative truncation schemes within EOMCCSD.13−16 The second way is to approximate the two electron repulsion integrals (ERIs) using density fitting techniques17−29 and seminumerical approximations.30−33 The use of density fitting to accelerate Møller−Plesset and CC calculations has a long history,34−39 and more recently Krylov © 2017 American Chemical Society
and co-workers have proposed a resolution of identity (RI) implementation for EOM-CCSD.40 However, density fitting and resolution of identity techniques can only reduce the formal scaling of the so-called “Coulomb type” terms; the formal scaling of “exchange type” terms remains unaffected apart from the prefactor. This can be a problem since many of the expensive terms in EOM-CCSD, including the most expensive term involving the integral with four external indices, are of the exchange type. One needs to use special seminumerical techniques like tensor hyper contraction (THC)41 or chain of sphere approximation (COSX)42 to reduce the formal scaling of exchange type terms in EOMCCSD. The third approach involves the use of local or natural orbitals or both to arrive at a lower scaling implementation of EOM-CCSD. It started with the pioneering work of Crowford and King43 and Korona and Werner.44 Schütz and co-workers extended the local correlation approaches to excitation energies,45 properties,46 and geometry optimization47 in the context of the second-order approximate coupled cluster (CC2) method. Hättig and co-workers made use of pair natural orbitals (PNO) calculated from the CIS(D) method48 to obtain a lower scaling implementation of the CC249,50 and the ADC(2)-x51 method. Mata and Stoll on the other hand, advocated the use of natural transition orbitals to reduce the scaling of the EOM-CCSD method.52 In recent times, their Received: July 27, 2017 Published: December 5, 2017 72
DOI: 10.1021/acs.jctc.7b00802 J. Chem. Theory Comput. 2018, 14, 72−91
Article
Journal of Chemical Theory and Computation
compute the excitation energy ωk = Ek − E0 from the commutator form of the above equation
idea has been pursued by many groups to arrive at a lower scaling implementation of CC253−55 and EOM-CCSD.56,57 Nooijen and co-workers58,59 have proposed an alternative of approach of using a second similarity transformation to reduce the scaling of the excited state calculations. In the similarity transformed equation of motion (STEOM) approach, a similarity transformation is parametrized using ionization potential (IP) and electron affinity (EA) solutions of the EOM approach. This is then applied to the CC similarity transformed Hamiltonian to decouple the singles block from the doubles. Thus, the final STEOM equations need only be solved in the configuration interaction singles (CIS) space, as it is done in the case of time dependent density functional theory (TDDFT).60,61 Unlike TDDFT however, STEOM does take care of double excitations. However, STEOM requires the definition of an active space, which until recently had to be selected manually. In one of our previous studies,62 an automatic active space selection scheme was proposed to eliminate this issue. In another study, we have described a hybrid approach based on pair natural orbitals and similarity transformation to obtain a lower scaling coupled cluster based method for excited states, which can be applied to medium sized molecules.63 The resulting bt-PNO-STEOM-CCSD is a lower scaling successor to the STEOM-CCSD method of Nooijen and co-workers,58,59,64−66 since the ground state is calculated in a linear scaling fashion, eliminating the highest scaling ingredient of STEOM-CCSD. When combined with our automatic active space selection scheme, the bt-PNO-STEOMCCSD can be applied to a variety of interesting chemical problems, even if such a scheme does not solve all the existing problems of STEOM-CCSD. However, it is absolutely necessary to benchmark the accuracy of the bt-PNOSTEOM-CCSD carefully for excited states of various kinds before the method may be applied to solve chemical problems. The aim of this paper is to characterize the errors made by the bt-PNO-STEOM-CCSD method for valence, Rydberg, and charge transfer states and to provide slight improvements in the active scale selection scheme.
[H̅ , R̂ k]|Φ0⟩ = ωkR̂ k|Φ0⟩
For the calculation of excitation energies (EE), R̂ is a particle conserving operator, which has the following form EE
i,a
IP
R̂ k =
∑ r ii + ∑ rbjib†ji + ... (7)
b,j>i
and for electron affinities (EA) EA
R̂ k
=
∑ raa† + ∑ a
rbajb†ja† + ... (8)
j,a>b
The EOM-CC equations are generally solved using a modified Davidson iterative diagonalization procedure in which the EOM part scales as N6 for the EE problem, and N5 for IP and EA problem using singles and doubles truncation. 2.2. The STEOM Formalism. Following Nooijen and coworkers,58,59 one can perform a second similarity transformation ̂
̂
G = {e S}−1H̅ {e S}
(9)
where the operators Ŝ are normal ordered with respect to the reference function Φ0 and are defined as IP
EA
Ŝ = Ŝ + Ŝ
(10) Î P
In the singles and doubles approximation, S is represented as IP
S ̂ = Smi′{m†i′} +
1 ij † † Smb{m ib j} 2
(11)
1 ej † † Sab{a eb j} 2
(12)
and SÊ A has the form EA
Ŝ
= Sae′{a′† e} +
The index m and e denote subsets of the occupied and virtual orbitals, respectively. They are referred to as active orbitals, while the remaining inactive occupied or virtual orbitals are denoted using primed labels (i′, a′). It should be noted that the meaning of active space in STEOM-CC is very different from that in the complete active space (CAS) method. The active space in STEOM-CC has nothing to do with nondynamic correlation; it can be interpreted as the dynamic correlation of the quasi-particles. The aim of the second similarity transformation is to remove the dominant terms in the Hamiltonian G coupling double and single excitations,
̂
(2) H̅ = e−T HeT where T̂ contains the cluster amplitudes obtained by solving the ground state coupled cluster equations (3)
⟨Φi ′|G|Φm⟩ = ⟨Φbji|G|Φm⟩ = 0
2
In the EOM formalism, the ground state CC equation is extended to excited states by the action of a linear operator R̂ on the similarity transformed Hamiltonian HR ̅ ̂ k|Φ0⟩ = Ek R̂ k|Φ0⟩
(6)
i
