Four-component coupled cluster method

If effects from electron correlation on parity violating potentials shall be accounted for in a four-component framework, the situation becomes more complicated than in the Dirac Hartree-Fock case. This is related to the fact, that in four-component many body perturbation theory (MBPT) or in a four-component coupled cluster (CC) scheme the reduced density matrices on the respective computational level are required in order to determine the parity violating potentials. Since these densities were not available in analytic form, Thyssen, Laerdahl and Schwerdtfeger [153] used a finite field approach to compute parity violating potentials in a four-component framework on a correlated level. This amounts to adding the parity violating operator with different scaling factors A to the 

---

E. Eliav and U. Kaldor, The Relativistic four-component coupled-cluster method for molecules Spectroscopic constants of SnH4, Chem. Phys. Lett. 248, 405 (1996). 

In the following subsections I will outline the methodologies used for the calculation of P-odd effects in chiral molecules, namely the methodology of the pioneering four-component study by Barra, Robert and Wiesenfeld, the Dirac Hartree-Fock method and the four-component coupled cluster approach. 

For more details on configuration interaction methods that include spin-orbit coupling we refer to the reviews by Marian [796,797] and by Hess, Marian and Peyerimhoff [767]. Finally, we also mention that the four-component coupled-cluster approaches discussed in section 8.9 have two-component relatives (see Refs. [798,799] for examples). 

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. 

For atomic structure calculations, the four-component MCSCF approach was the method of choice for a long time. The implementation of methods for treating very large CSF spaces, particularly Davidson-type diagonalization techniques, produced a tool to compete with highly precise experimental measurements. The coupled-cluster method—or MBPT approach as it is usually called in the physics community—turned out to be a valuable, alternative, size-consistent method. 

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. 

E. EUav, U. Kaldor. Relativistic four-component multireference coupled cluster methods Towards a covariant approach. In P. Carsky et al., Ed., Recent Progress... 

L. K. Sorensen, T. Fleig, J. Olsen. A Relativistic Four- and Two-component Generalized-active-space Coupled Cluster Method. Z. Phys. Chem., 224 (2010) 671-680. 

L. K. Sorensen, J. Olsen, T. Fleig. Two- and four-component relativistic generalized-active-space coupled cluster method Implementation and application to BiH. /. Chem. Phys., 134 (2011) 214102. 

Several groups have developed four-component methods for the calculation of polyatomic molecules in the last decade. While several codes for DHF calculations exist by now, " programs for calculations on polyatomic molecules including correlation by means of ab initio techniques appeared only very recently. They feature relativistic multi-reference configuration interaction, second-order M0ller-Plesset coupled-cluster methods. ... 

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 primary objective of most applications carried out so far was to assess the performance of the PPP-VB method for diverse alternant and nonaltemant -electron systems of aromatic, nonaromatic or antiaromatic character, both electrically neutral and charged. The main emphasis was on ground states of different spin multiplicity, even though some preliminary calculations were also carried out for excited states. The PPP-VB codes were also employed to provide the approximate three- and four-body connected cluster components for the so-called VB-corrected coupled cluster (CC) approach [71]. In the following, we briefly point out the most important aspects of the PPP-VB method and illustrate them with a few typical results. 

Accounting for relativistic effects in computational organotin studies becomes complicated, because Hartree-Fock (HF), density functional theory (DFT), and post-HF methods such as n-th order Mpller-Plesset perturbation (MPn), coupled cluster (CC), and quadratic configuration interaction (QCI) methods are non-relativistic. Relativistic effects can be incorporated in quantum chemical methods with Dirac-Hartree-Fock theory, which is based on the four-component Dirac equation. " Unformnately the four-component Flamiltonian in the all-electron relativistic Dirac-Fock method makes calculations time consuming, with calculations becoming 100 times more expensive. The four-component Dirac equation can be approximated by a two-component form, as seen in the Douglas-Kroll (DK) Hamiltonian or by the zero-order regular approximation To address the electron cor-... 

---