Fock-Space Coupled Cluster Approach

The coupled cluster (CC) approach is the most powerful and accurate of generally applicable electron correlation methods. This has been shown in many benchmark applications of 4-component relativistic CC methods to atoms [11-18] and molecules [19-31]. The CC method is an all-order, size-extensive, and systematic many-body approach. Multireference variants of relativistic 4-component CC methods capable of handling quasidegeneracies, which are important for open-shell heavy atomic and molecular systems, have been developed in recent years [15,17-19,21,31]. In particular, the multireference FSCC scheme [32,33] is applicable to systems with a variable number of particles, and is an ideal candidate for merging with QED theory to create an infinite-order size-extensive covariant many-body method applicable to systems with variable numbers of fermions and bosons [6,7]. 

The development of a general multiroot multireference scheme for treating electron correlation effects usually starts from consideration of the Schrodinger equation for a number (d) of target states. 

The physical Hamiltonian is divided into two parts, H = Ho+ V, so that V is a small perturbation to the zero-order Hamiltonian Hq, which has known eigenvalues and eigenvectors, Ho p)=E p). 

Q is the normal-ordered wave operator, mapping the eigenfunctions of the effective Hamiltonian onto the exact ones, = 1, It satisfies intermediate normalization, 

PQP = P. The effective Hamiltonian and wave operator are connected by the generalized Bloch equation, which for a complete model space P may be written in the compact linked form [35] 

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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]... 

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

A particular variant of the coupled cluster method, called Fock-space or valence-universal [49,50], gave remarkable agreement with experiment for many transition energies of heavy atoms [51]. This success makes the scheme a useful tool for reliable prediction of the structure and spectrum of superheavy elements, which are difficult to access experimentally. A brief description of the method is given below. A more flexible scheme with higher accuracy and extended applicability, the intermediate Hamiltonian Fock-space coupled cluster approach, is shown in the next section. 

Up to this point we have tailored the second-quantization formalism in close connection to the independent-particle picture introduced before. However, the formalism can be generalized in an even more abstract fashion. For this we introduce so-called occupation number vectors, which are state vectors in Fock space. Fock space is a mathematical concept that allows us to treat variable particle numbers (although this is hardly exploited in quantum chemistry see for an exception the Fock-space coupled-cluster approach mentioned in section 8.9). Accordingly, it represents loosely speaking all Hilbert spaces for different but fixed particle numbers and can therefore be formally written as a direct sum of N-electron Hilbert spaces. 

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. 

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. 

The basic relativistic equations are described in Sec. 2, and the Fock-space coupled cluster method is discussed in Sec. 3. The recently developed intermediate Hamiltonian approach is described and illustrated by several... 

A major advantage of the intermediate Hamiltonian approach is the flexibility in selecting the model space. This has been a major problem in applying the Fock-space scheme, as described at the beginning of this section. While in the Fock-space coupled cluster method one may feel lucky to find any partitioning of the function space into P and Q with convergent CC iterations, the intermediate Hamiltonian method makes it possible for the first time to vary the model space systematically and study the effect upon calculated properties. An example is given in Table 3, which shows the dependence of the calculated electron affinity of Cs on the model spaces Pm and Pi [55]. 

Excitation energies of atomic barium and radium were calculated in 1996 using the Fock-space coupled cluster method [57]. The model space in the 2-electron sector included all states with two electrons in the 5d, 6s and 6p orbitals, except the 6p states inclusion of the latter led to intruder states and divergence, so that incomplete model spaces had to be employed. In the intermediate Hamiltonian approach all these states (including 6p ) were in Pm, Pi was defined by adding states with occupied 7s-10s, 7p-10p, 6d-... 

The intermediate Hamiltonian Fock-space coupled-cluster calculations of Infante et al. [1146] were performed in an all-electron approach explicitly correlating 26 electrons and using large de[Pg.625]

Fock-space coupled cluster (FSCC) approach that has been mostly applied to actinides. A more extensive discussion can be found in Chapter 2 by Eliav and Kaidor. 

In this section we will introduce some wavefunction-based methods to calculate photoabsorption spectra. The Hartree-Fock method itself is a wavefunction-based approach to solve the static Schrodinger equation. For excited states one has to account for time-dependent phenomena as in the density-based approaches. Therefore, we will start with a short review of time-dependent Hartree-Fock. Several more advanced methods are available as well, e.g. configuration interaction (Cl), multireference configuration interaction (MRCI), multireference Moller-Plesset (MRMP), or complete active space self-consistent field (CASSCF), to name only a few. Also flavours of the coupled-cluster approach (equations-of-motion CC and linear-response CQ are used to calculate excited states. However, all these methods are applicable only to fairly small molecules due to their high computational costs. These approaches are therefore discussed only in a more phenomenological way here, and many post-Hartree-Fock methods are explicitly not included. 

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. 

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