Fock space relativistic

Fsrcc is a multi-reference Fock-space relativistic coupled-cluster program by Eliav and Kaldor for correlated calculations on the ground and excited states of molecules. 

A pilot calculation on CdH using one-, two- and four-component Fock space relativistic coupled-cluster methods has been published by Eliav et al. (1998b). The calculated values obtained were in very good agreement with experiment. While the four-component method gives the best results, one- and two-component calculations include almost all the relativistic effects. 

If not otherwise stated the four-component Dirac method was used. The Hartree-Fock (HF) calculations are numerical and contain Breit and QED corrections (self-energy and vacuum polarization). For Au and Rg, the Fock-space coupled cluster (CC) results are taken from Kaldor and co-workers [4, 90], which contains the Breit term in the low-frequency limit. For Cu and Ag, Douglas-Kroll scalar relativistic CCSD(T) results are used from Sadlej and co-workers [6]. Experimental values are from Refs. [91, 92]. 

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. 

Successful model building is at the very heart of modern science. It has been most successful in physics but, with the advent of quantum mechanics, great inroads have been made in the modelling of various chemical properties and phenomena as well, even though it may be difficult, if not impossible, to provide a precise definition of certain qualitative chemical concepts, often very useful ones, such as electronegativity, aromaticity and the like. Nonetheless, all successful models are invariably based on the atomic hypothesis and quantum mechanics. The majority, be they of the ah initio or semiempirical type, is defined via an appropriate non-relativistic, Born-Oppenheimer electronic Hamiltonian on some finite-dimensional subspace of the pertinent Hilbert or Fock space. Consequently, they are most appropriately expressed in terms of the second quantization formalism, or even unitary group formalism (see, e.g. [33]). 

Because of its size-extensivity and faster convergence with respect to excitation level Coupled cluster theory has replaced Cl theory as the dominant approach in ab initio correlation calculations. Like MBPT the theory is still mainly applied in cases where the exact wave function is dominated by a single determinant, but multireference methods have been formulated and begin to enter mainstream quantum chemistry. Generalization of the algorithms to the relativistic no-pair level can again be achieved via the spinorbital formulation of the methods. I will first discuss the single reference method and then consider the Fock space method [40] that uses multi-reference wavefiinctions for ionized or excited states. 

Studies of rare earth or transition metal complexes often necessitate use of multireference wave functions. Among the Coupled Cluster type methods one can distinguish two main lines of approach to incorporate multireference character in the reference wave function. In the Hilbert space method one computes a single wave function for a particular state, while in the Fock space method one tries to obtain a manifold of states simultaneously. Since the latter method [40] has recently been implemented and applied in conjunction with the relativistic Hamiltonian [48-50] we will focus on this approach. 

There is only one subtle point with regard to the no-pair approximation that deserves some attention. In the non-relativistic case the Fock space formalism without truncation of the T operators gives just an alternative parametrization of the foil Cl wave function. In the relativistic case the situation is more complex because the states of interest may contain a different number of electrons than the reference state. This means that the no-pair approximation is less appropriate as it is based on a mean-field potential due to a different number of electrons. Formally this problem might be tackled by lifting the no-pair restriction but it will be very hard to turn the resulting complicated formalism into an efficient algorithm. The corrections would probably be small since the difference in potential mainly affects the valence region where the potential is small relative to the rest mass term anyway. 

Figure 1. Total nonrelativistic multi-configuration Hartree-Fock energy, relativistic corrections (estimated as the difference between the multi-configuration Dirac-Hartree-Fock and Hartree-Fock energies) and correlation contributions (estimated from correlation energy density functional calculations) for the group 4 elements. The multi-configuration treatments were carried out with the atomic structure code GRASP [78] and correspond to complete active space calculations with the open valence p shell as active space. The nonrelativistic results were obtained by multiplying the velocity of light with a factor of 10 .

Accurate Relativistic Fock-Space Calculations for Many-Electron Atoms... 

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... 

The correct frame of description of interacting relativistic electrons is quantum electrodynamics (QED) where the matter field is the four-component operator-valued electron-positron field acting in the Fock space and depending on space-time = (ct, r) (x = (ct, —r)). Electron-electron interaction takes place via a photon field which is described by an operatorvalued four-potential A x ). Additionally, the system is subject to a static external classical (Bose condensed, c-number) field F , given by the four-potential (distinguished by the missing hat)... 

The expression to the right of the dot is the model state P f >Rei = f o >Rei, which implies that the expression to the left is the relativistically covariant wave operator (also a Fock-space operator)... 

The momentum wave functions in various atomic models are calculated for arbitrary atomic orbitals. The nonrelativistic hydrogenic, the Hartree-Fock, the relativistic hydrogenic, and the Dirac-Fock models are considered. The momentum wave functions are obtained as a Fourier transform of the wave function in the position space. The Hartree-Fock and the Dirac-Fock wave functions in atoms are given in terms of Slater-type orbitals (STO s), i.e. the Hartree-Fock-Roothaan (HFR) method and the relativistic HFR (RHFR) method. All the wave functions in the momentum space can be expressed analytically in terms of hypergeometric functions. 

E. EUav, U. Kaldor, B. A. Hess. The relativistic Fock-space coupled-duster method for molecules CdH and its ions. 

The polarizability of T1 and element 113 has been calculated using the fully relativistic ab initio Dirac-Coulomb Fock-space coupled-cluster method and the finite field procedure. For Tl, the theoretical value is in good agreement with experiment. In group 13, the atomic polarizability increases from A1 to Ga, attains a maximum for In and then decreases towards Tl and furthermore towards element 113. So, element 113 presents the smallest polarizability, which results from the large relativistic contraction and stabilization of the 7pi/2 orbital. These values have then been used to estimate the adsorption enthalpies of Tl and element 113 on polyethylene and teflon surfaces and have shown that the difference of enthalpy attains 6 kJ/mol, which should be enough to separate and identify them. 

For systems with more than two open shells, it is in general necessary to resort to multireference methods. This section has dealt only with state-specific coupled-cluster methods, also known as state-universal methods or Hilbert space methods, for which a considerable amount of effort has been expended on nonrelativistic multireference methods. The alternative, which is much more suited to multireference problems, is the valence-universal or Fock space method, which has been developed for relativistic systems by Kaldor and coworkers (Eliav and Kaldor 1996, Eliav et al. 1994, 1998, Visscher et al. 2001). 

The ionization potentials and first few excitation energies of element 112 have been calculated by Eliav, Kaldor, and Ishikawa applying the Fock-space RCC method. " The calculations revealed that the relativistic stabilization of the 7s orbital is large and comparable to that of element 111. As a result, the ground state configurations of 112+ and 112 + are 6d 7s ( 5/2) 6d 7s J = 4). The equivalent electronic... 

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