Fock space approach

FULL CLUSTER EXPANSION THEORIES IN FOCK SPACE 7.1 Preliminaries for a Fock Space Approach... 

In this section, we shall motivate towards the need for a Fock-space approach to generate core-valence extensive cluster expansion theories (i.e., of type (a3)), introduced in Sec.2. Since we have to maintain size-extensivity of the energy... 

The Fock space approach has the potential advantage of exploiting the fact that operators written in the occupation number representation are independent of electron number, so that all the manipulations involving H and can be performed at the operator level first, which is somewhat simpler and more transparent than working with the matrix-elements involving functions > and. ... 

Furthermore, since the same operators appear in all the n-valence sectors of the Fock-space, the projections onto the wave-functions can be performed at the very end, and it is possible to treat systems with varying n (i.e., different degrees of ionization) on the same footing, and in an explicitly size-extensive manner with respect to the valence electrons. As it has turned out, only the Fock—space approach has the potentiality to furnish explicitly connected size-extensive theories for a general incomplete model space/91-96/, which shows its flexibility and generality. 

Multireference coupled cluster methods, which started development more recently, are generally divided into two types. Hilbert space CC methods use multiple reference functions to obtain a description of a few states, including the reference state (for a review see (4)). Fock space methods (for a review see (5)), on the other hand, provide direct state-to-state energy differences, relative to some common reference state. The Fock space approach is particularly well-suited to the calculation of ionization potentials (IPs), electron affinities (EAs), and excitation energies (EEs). For principal IPs and EAs, FSCC is equivalent (6, 7) to the EOM-IP and EOM-EA CC methods (1, 2, 7, 8). In this paper, we will focus primarily on the IP problem. 

The Fock space approach assumes that the reference wave functions for the states of interest can be written in terms of determinants that share a common closed shell core. This closed shell core is then considered as the zero order wave function from which the energy of states of interest can be calculated as an electron affinity or ionization energy. In the calculation of these quantities one may incorporate the necessary multireference character by allowing mixing between different reference wave functions. The first step of the procedure is to define the reference or model space that spans a number of sectors of the Fock... 

Brillouin-Wigner coupled cluster theory Fock-space approach Journal of Chemical Physics 117, 9580 (2002)... 

The relativistic coupled cluster method starts from the four-component solutions of the Drrac-Fock or Dirac-Fock-Breit equations, and correlates them by the coupled-cluster approach. The Fock-space coupled-cluster method yields atomic transition energies in good agreement (usually better than 0.1 eV) with known experimental values. This is demonstrated here by the electron affinities of group-13 atoms. Properties of superheavy atoms which are not known experimentally can be predicted. Here we show that the rare gas eka-radon (element 118) will have a positive electron affinity. One-, two-, and four-components methods are described and applied to several states of CdH and its ions. Methods for calculating properties other than energy are discussed, and the electric field gradients of Cl, Br, and I, required to extract nuclear quadrupoles from experimental data, are calculated. 

By adopting the no-pair approximation, a natural and straightforward extension of the nonrelativistic open-shell CC theory emerges. The multireference valence-universal Fock-space coupled-cluster approach is employed [25], which defines and calculates an effective Hamiltonian in a low-dimensional model (or P) space, with eigenvalues approximating some desirable eigenvalues of the physical Hamiltonian. The effective Hamiltonian has the form [26]... 

In summary, Erdahl s treatment is more general and allows a more concise formulation because he works in Fock space, conserving only the parity of the number of particles however, he finds it necessary to restrict the coefficients to be real. We work at fixed particle number and have no reason for the restriction to real coefficients. If the Hamiltonian should be general Hermitian, in which case the RDM must likewise be assumed to be general Hermitian, then our approach leads to Hermitian semidefinite conditions. 

Let us consider the 5s, 5p, 5d orbitals of lead and Is orbital of oxygen as the outercore and the ai, a2, os, tti, tt2 orbitals of PbO (consisting mainly of 6s, 6p orbitals of Pb and 2s, 2p orbitals of O) as valence. Although in the Cl calculations we take into account only the correlation between valence electrons, the accuracy attained in the Cl calculation of Ay is much better than in the RCC-SD calculation. The main problem with the RCC calculation was that the Fock-space RCC-SD version used there was not optimal in accounting for nondynamic correlations (see [136] for details of RCC-SD and Cl calculations of the Pb atom). Nevertheless, the potential of the RCC approach for electronic structure calculations is very high, especially in the framework of the intermediate Hamiltonian formulation [102, 131]. 

A. Landau, E. Eliav, Y. Ishikawa, U. Kaldor, Mixed-sector intermediate Hamiltonian Fock-space coupled cluster approach, J. Chem. Phys. 121(14) (2004) 6634. 

Each spin orbital is a product of a space function fa and a spin function a. or ft. In the closed-shell case the space function or molecular orbitals each appear twice, combined first with the a. spin function and then with the y spin function. For open-shell cases two approaches are possible. In the restricted Hartree-Fock (RHF) approach, as many electrons as possible are placed in molecular orbitals in the same fashion as in the closed-shell case and the remainder are associated with different molecular orbitals. We thus have both doubly occupied and singly occupied orbitals. The alternative approach, the unrestricted Hartree-Fock (UHF) method, uses different sets of molecular orbitals to combine with a and ft spin functions. The UHF function gives a better description of the wavefunction but is not an eigenfunction of the spin operator S.2 The three cases are illustrated by the examples below. 

Results given by the EOMEA and EOMIP methods are equivalent to those of certain variants of the Fock-space coupled-cluster (FSCC) method. For a discussion of this correspondence, as well as an overview of and references to the general FSCC approach, see Ref. 266. 

This application of FSCC methods is a recent development, but along with its IP-EOM-CC twin, there are few such applications in the literature (8, 33), and they frequendy deal with symmetry-breaking problems. Comparisons are possible with single reference calculations, though as we shall see, the comparison of high-level SRCC calculations with Fock-space results does not always lead to an unambiguous determination of which approach is more reliable. Nevertheless, FSCC is very well suited to such applications and will undoubtedly... 

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