Method of Moments of Coupled-Cluster Equations: A New Theoretical

Chapter 3

Downloaded by PENNSYLVANIA STATE UNIV on September 14, 2012 | http://pubs.acs.org Publication Date: August 14, 2002 | doi: 10.1021/bk-2002-0828.ch003

Method of Moments of Coupled-Cluster Equations: A New Theoretical Framework for Designing "Black-Box" Approaches for Molecular Potential Energy Surfaces 1

1

1

Piotr Piecuch , K a r o l Kowalski , Ian S. O . Pimienta , and Stanislaw A. Kucharski 2

1

Department of Chemistry, Michigan State University, East Lansing, MI 48824 Institute of Chemistry, Silesian University, Szkolna 9, 40-006 Katowice, Poland

2

The recently proposed method of moments of coupled-cluster equations ( M M C C ) is reviewed. The ground-state MMCC formalism and its excited-state extension via the equation-of-motion coupled-cluster ( E O M C C ) ap proach are discussed. The main idea of all MMCC meth ods is that of the noniterative energy corrections which, when added to the ground- and excited-state energies ob tained in approximate CC calculations, recover the exact energies. Approximate M M C C methods, including the renormalized C C S D ( T ) , C C S D ( T Q ) , and C C S D T ( Q ) ap proaches and the E O M C C - r e l a t e d M M C C ( 2 , 3 ) method, are described and examples of applications of these new approaches are given. It is demonstrated that the M M C C formalism provides a new framework for designing "black-box" approaches that give excellent description of entire potential energy surfaces at the small fraction of the effort associated with multireference calculations.

© 2002 American Chemical Society

In Low-Lying Potential Energy Surfaces; Hoffmann, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.

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Introduction One of the most important problems in coupled-cluster (CC) theory (1 -7) is extension of the existing single-reference C C (SRCC) methods to quasidegenerate and excited states. Potential energy surfaces (PESs) of ground and excited states represent a particularly challenging problem. The standard C C "black-boxes" for ground electronic states, such as the CCSD (CC singles and doubles) approach, and their perturbative triples and quadruples extensions, including CCSD[T] (8-10), CCSD(T) (11), CCSD(TQf) (It), and CCSDT(Qf) (12), fail to describe bond dissociation (cf., e.g., refe 6, 7, 13-20). The response C C methods (21-26) and their equation-of-motion C C (EOMCC) analogs (27-30) are capable of providing very good results for excited states dominated by singles (cf., e.g., refe 27-29, 31-37), but accurate calculations of excited states of quasidegenerate systems (particularly, excited states having large biexcited components) and of entire PESs of excited states with the standard response C C and EOMCC approximations, including the E O M C C S D (27-29), EOMCCSD(T) (51), EOMCCSD(t) (32), EOMCCSD(T') (32), EOMCCSDT-n (31,32), CCSDR(3) (36,37), and CC3 (5^-57) methods, are not possible (cf., e.g., refe 38-41). The genuine multi-reference C C (MRCC) approaches of the state-universal type (4,6,42-56) (the SUMR C C methods) have showed some promise in studies of molecular PESs (cf., e.g., refe 45, 47, 48, 51-53, 56), but the SUMRCC calculations are plagued by intruder states (51,52) and by multiple intruder solutions (51,54)- The applicability of the valence-universal M R C C methods (4,6,57-60), which also suffer from intruder states and unphysical multiple solutions (61,62), is limited to vertical excitation energies of atoms and molecules at their equilibrium geometries. Thus, in spite of tremendous progress in C C theory, which is nowadays routinely used in accurate calculations of various equilibrium properties of closed-shell and simple open-shell molecular systems, there is a need for new ideas that would extend the applicability of C C methods to entire molecular PESs. Several attempts have been made to remove the pervasive failing of the perturbative C C approximations at large internuclear separations. The representative examples include the externally-corrected SRCC methods (6,10, 63-74), the active-space SRCC approaches (14,19, 75-86), the orbital-optimized SRCC methods (39,87,88), and the perturbative C C approaches based on the partitioning of the similarity-transformed Hamiltonian (89,90) (see ref 91 for the original idea). Of all these approaches, the reduced MRCCSD (RMRCCSD) method (68-74), which uses the multi-reference configuration interaction (MRCI) wave functions to extract information about triply and quadruply excited clusters, and the active-space CCSDt and CCSDtq methods (19,85,86) and their earlier state-selective (SS) CCSD(T) and CCSD(TQ) analogs (14, 75-84) are particularly promising. The RMRCCSD approach can be used to success-



33 fully describe bond breaking and ro-vibrational term values in the ground state (6,68-70,72-74), and to calculate the lowest-energy excited state of a given symmetry (71,73). Unfortunately, it is not possible to apply the RMRCCSD method to several electronic states of the same symmetry and the cost of the MRCI calculations that are used to extract the triply and quadruply excited clusters may significantly increase the cost of the RMRCCSD calculations. The CCSDt and CCSDtq methods and their SSCCSD(T) and SSCCSD(TQ) analogs, in which higher-than-doubly excited components of the cluster operator are selected through active orbitals, have fewer Umitations. The CCSDt or SSCCSD(T) and CCSDtq or SSCCSD(TQ) approaches are less expensive than the MRCI methods and can easily be extended to excited states of the same or different symmetries via the E O M C C formalism The active-space C C approaches proved to be successful in describing quasidegenerate ground states (79,80,85), ground-state PESs involving bond breaking (14,19,81,84-86), highly ex cited vibrational states (86), and ground-state property functions (83) at the fraction of the computer cost associated with the parent C C S D T (CC singles, doubles, and triples) (92,93) and C C S D T Q (CC singles, doubles, triples, and quadruples) (79,94-96) approaches. The E O M C C extension of the CCSDt method, termed EOMCCSDt in which relatively small subsets of triexcited components of cluster operator Τ and E O M C C excitation operator R are selected through active orbitals, proved to be capable of providing excellent description of excited states domi nated by doubles and states having large triexcited components (includ ing excited states of molecules whose ground states are quasidegenerate). The EOMCCSDt approach proved to be successful in describing entire excited-state PESs at the fraction of the computer cost associated with MRCI and full EOMCCSDT (EOMCC singles, doubles, and triples) (41) calculations 97). The active-space SRCC methods and their E O M C C extensions are very promising and we will continue to develop them. They are relatively easy to use, although, in analogy to multireference approaches, they require choos ing active orbitals, which in some cases may be a difficult thing to do. From this point of view, the active-space C C methods are not as easy-to-use as the noniterative perturbative methods, such as CCSD(T) or CCSD(TQf), or their response C C or E O M C C extensions. Undoubtedly, it would be de sirable to have an approach that combines the simplicity of the noniterative C C schemes with the effectiveness with which the iterative active-space C C and E O M C C methods, such as CCSDt and EOMCCSDt, describe groundand excited-state PESs. It has recently been demonstrated that the applicability of the groundstate SRCC approaches, including the popular noniterative approximations, such as CCSD(T), can be extended to bond breaking and quasidegenerate states, if we switch to a new type of the SRCC theory, termed the method of moments of CC equations (MMCC) (7,16-18). It has further been demon-

(40,41)-

(40,41),

(40,41,



34 strated that the M M C C theory can be extended to exeited-state PESs, if we combine the M M C C and E O M C C (98) or M M C C and SUMRCC (18) for malisms. The main idea of the ground-state M M C C formalism (7,16-18) is that of the noniterative energy correction which, when added to the en ergy obtained in approximate SRCC calculations, such as CCSD or CCSDT, recovers the exact (full CI) energy. The noniterative energy corrections defining the EOMCC-based M M C C theory (98), added to the energies ob tained in approximate E O M C C (e.g., EOMCCSD) calculations, recover the full CI energies of excited states. It has been demonstrated that the M M C C formalism allows us to renormalize the existing noniterative SRCC approx imations, such as CCSD[T], CCSD(T), CCSD(TQ ), and CCSDT(Q ), so that they can correctly describe entire ground-state PESs (7,16-20). It has also been demonstrated that the excited-state M M C C theory, based on the E O M C C method, allows us to introduce a new hierarchy of simple noniterative C C approximations that remove the pervasive failing of the EOMCCSD and perturbative EOMCCSDT approximations in describing entire excited-state PESs (98). In our view, the M M C C theory represents an interesting development in the area of new C C methods for molecular PESs. The MMCC-based renormalized CCSD(T), CCSD(TQ), and CCSDT(Q) methods and the noniter ative MMCC approaches to excited states provide highly accurate results for ground and excited-state PESs, while preserving the simplicity and the "black-box" character of the noniterative perturbative C C schemes. In this chapter, we review the M M C C theory and new C C approximations that result from it and show the examples of the M M C C and renormalized C C calculations for ground and excited state PESs of several benchmark molecules, including HF, F 2 , N , and C H . The review of the previously published numerical results (7,16-20) is combined with the presentation of new results for the C 2 , N , and H 2 O molecules. f

f

+

2

2

The Method of Moments of Coupled-Cluster Equations: The General Formalism We begin our review of the M M C C theory with the ground-state for malism. The extension of the M M C C formalism to the E O M C C case is discussed in the next subsection.

The Ground-State Theory In the SRCC theory, we represent the ground-state wave function of an iV-electron system, described by the Hamiltonian if, in the following way: τ

|Φο)=β |Φ),


(1)

35 where Τ is the cluster operator and |Φ) is an independent-particle-model ( I P M ) reference configuration (usually, the Hartree-Fock determinant). I n the exact theory, Γ is a sum of all of its many-body components, including the iV-body one. In the standard S R C C approximations, the many-body expansion of Τ is truncated at some excitation level. Let us consider the standard S R C C approximation (hereafter referred to as method A), i n which Τ is approximated as follows:


niA T

w

T

(A)

=

Y^T ,

(2)

n

n=l where T , η = 1 , . . . , m A, are the many-body components of Τ included i n the calculations and ΤΠΑ < Ν (m A = 2 defines the C C S D method, m A = 3 defines the C C S D T method, etc.). The system of equations for the cluster amplitudes defining the T components has the following form: n

n

Λ

φ(*>#< >|Φ)=0,

(3)

where TW

TW

TW

HW=e- He

=(He )

(4)

c

is the similarity-transformed Hamiltonian of the C C theory, subscript C designates the connected part of the corresponding operator expression, and Q( ) is the projection operator onto the subspace of all excited config urations described by T^ \ i.e., A

A

τη A Λ ()

Method of Moments of Coupled-Cluster Equations: A New Theoretical

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