Similarity-transformed Hamiltonian Research Articles

Renormalized coupled-cluster methods exploiting left eigenstates of the similarity-transformed Hamiltonian

Completely renormalized (CR) coupled-cluster (CC) approaches, such as CR-CCSD(T), in which one corrects the standard CC singles and doubles (CCSD) energy for the effects of triply (T) and other higher-than-doubly excited clusters [K. Kowalski and P. Piecuch, J. Chem. Phys. 113, 18 (2000)], are reformulated in terms of the left eigenstates Phimid R:L of the similarity-transformed Hamiltonian of CC theory. The resulting CR-CCSD(T)(L) or CR-CC(2,3) and other CR-CC(L) methods are derived from the new biorthogonal form of the method of moments of CC equations (MMCC) in which, in analogy to the original MMCC theory, one focuses on the noniterative corrections to standard CC energies that recover the exact, full configuration-interaction energies. One of the advantages of the biorthogonal MMCC theory, which will be further analyzed and extended to excited states in a separate paper, is a rigorous size extensivity of the basic ground-state CR-CC(L) approximations that result from it, which was slightly violated by the original CR-CCSD(T) and CR-CCSD(TQ) approaches. This includes the CR-CCSD(T)(L) or CR-CC(2,3) method discussed in this paper, in which one corrects the CCSD energy by the relatively inexpensive noniterative correction due to triples. Test calculations for bond breaking in HF, F(2), and H(2)O indicate that the noniterative CR-CCSD(T)(L) or CR-CC(2,3) approximation is very competitive with the standard CCSD(T) theory for nondegenerate closed-shell states, while being practically as accurate as the full CC approach with singles, doubles, and triples in the bond-breaking region. Calculations of the activation enthalpy for the thermal isomerizations of cyclopropane involving the trimethylene biradical as a transition state show that the noniterative CR-CCSD(T)(L) approximation is capable of providing activation enthalpies which perfectly agree with experiment.

Transcorrelated method for electronic systems coupled with variational Monte Carlo calculation

A Jastrow–Slater-type wave function is often used as a trial function for precise calculations of the total energy of electronic systems, where the correlation effect is taken into account by the Jastrow factor that directly depends on the distance between electrons. Since many-body integrals are inevitable there, the calculation totally depends on Monte Carlo sampling, and so, except for very simple cases, it is very difficult to optimize one-body wave functions in the Slater determinant which determine the nodal surfaces of the total wave function. Here we propose and demonstrate that the total wave function is efficiently optimized by coupling an ordinary variational Monte Carlo (VMC) technique with the transcorrelated method, in which the one-body wave functions are definitely obtained by solving Hartree–Fock-type self-consistent-field (SCF) equations derived from the similarity-transformed Hamiltonian. It is shown that the present method reproduces about 90% of the correlation energy for helium-like two-electron systems (H−, He, Li+, and Be2+) and gives much better results than the conventional VMC method using the Hartree–Fock orbitals for a Li atom, a Be atom, and a H2 molecule. It is also shown that the orbital energy appearing in the SCF equations gives a good approximation to the ionization potential.

New type of noniterative energy corrections for excited electronic states: Extension of the method of moments of coupled-cluster equations to the equation-of-motion coupled-cluster formalism

The recently proposed method of moments of coupled-cluster equations (MMCC) is extended to excited states via the equation-of-motion coupled-cluster (EOMCC) formalism. The main idea of the new MMCC theory is that of the noniterative energy corrections which, when added to the excited-state energies obtained in standard approximate EOMCC calculations, recover the exact energies. The MMCC corrections are expressed in terms of the generalized moments of the EOMCC equations. Approximate variants of the excited-state MMCC formalism, including the MMCC(2,3) approach, are introduced. In the MMCC(2,3) method, very simple energy corrections, expressed in terms of matrix elements of the triples-reference, triples-singles, and triples-doubles blocks of the EOMCCSD (EOMCC singles and doubles) similarity-transformed Hamiltonian, are added to the excited-state energies obtained in EOMCCSD calculations. The performance of the MMCC(2,3) approach is illustrated by the results of pilot calculations for the potential energy curves of ground and excited states of CH+.

Second-order perturbation corrections to singles and doubles coupled-cluster methods: General theory and application to the valence optimized doubles model

We present a general perturbative method for correcting a singles and doubles coupled-cluster energy. The coupled-cluster wave function is used to define a similarity-transformed Hamiltonian, which is partitioned into a zeroth-order part that the reference problem solves exactly plus a first-order perturbation. Standard perturbation theory through second-order provides the leading correction. Applied to the valence optimized doubles (VOD) approximation to the full-valence complete active space self-consistent field method, the second-order correction, which we call (2), captures dynamical correlation effects through external single, double, and semi-internal triple and quadruple substitutions. A factorization approximation reduces the cost of the quadruple substitutions to only sixth order in the size of the molecule. A series of numerical tests are presented showing that VOD(2) is stable and well-behaved provided that the VOD reference is also stable. The second-order correction is also general to standard unwindowed coupled-cluster energies such as the coupled-cluster singles and doubles (CCSD) method itself, and the equations presented here fully define the corresponding CCSD(2) energy.

A second-order correction to singles and doubles coupled-cluster methods based on a perturbative expansion of a similarity-transformed Hamiltonian

We present a new ansätz for deriving perturbative corrections to single-reference coupled-cluster based methods. The methods are based on a perturbative expansion of the Hamiltonian formed by a similarity transformation of the normal Hamiltonian. These new methods naturally contain triple and quadruple excitations at second order. However, because of a factorization approximation, the cost of the method only scales as n 7. Results are presented for a second-order correction to the optimized-orbital coupled-cluster doubles method [OD(2)]. For cases where the standard CCSD(T) method performs poorly, OD(2) gives significantly improved results.

Transformation of the Hamiltonian in excitation energy calculations: Comparison between Fock-space multireference coupled-cluster and equation-of-motion coupled-cluster methods

A use of the transformed form of the Hamiltonian in excitation energy calculations is discussed and a comparison between methods using this form is performed. It is shown that use of a similarity-transformed Hamiltonian can lead to separation of the ground state and excited state eigenvalue problems. Then standard approaches for determination of excited state energies can be used. Since these energies are usually quasidegenerate, it is convenient to use the effective Hamiltonian scheme. For different approximate methods, various advantages and disadvantages referring to desirable features of a ‘‘theoretical model chemistry’’ are discussed.