Fock space energies/results

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]. 

Energy levels of heavy and super-heavy (Z>100) elements are calculated by the relativistic coupled cluster method. The method starts from the four-component solutions of the Dirac-Fock or Dirac-Fock-Breit equations, and correlates them by the coupled-cluster approach. Simultaneous inclusion of relativistic terms in the Hamiltonian (to order o , where a is the fine-structure constant) and correlation effects (all products smd powers of single and double virtual excitations) is achieved. The Fock-space coupled-cluster method yields directly transition energies (ionization potentials, excitation energies, electron affinities). Results are in good agreement (usually better than 0.1 eV) with known experimental values. Properties of superheavy atoms which are not known experimentally can be predicted. Examples include the nature of the ground states of elements 104 md 111. Molecular applications are also presented. 

A Fock space multireference coupled cluster method was described by Rittby and Bartlett <91TCA469>, applied to the calculation of ionization potentials and excitation energies of 1,2,4,5-tetrazines, and compared with conventional ab initio calculations and experimental results. 

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... 

What combination of Fock-space operators will produce, when acting on a closed-shell state vector o)> a resultant vector equivalent to the wavefunction used in Problem 8.1 Use the second-quantization form of the Hamiltonian (p. 82) and the anticommutation properties of the creation and annihilation operators (p. 81) to give an alternative derivation of the energy expression found in Problem 8.1. Hint Use operators a , etc. to destroy or create up-spin or down-spin electrons in orbital ipr- It is convenient to use indices i, j,... for orbitals in 0o and m, n,... for the virtual set used in the O.]... 

Representative applications of the NVPA Fock-space CC method to actinide systems are presented below. Many calculations have been carried out over the last 15 years, involving various heavy and superheavy atomic and molecular systems (not limited to actinides), with dozens of transition energies calculated per system. Most atomic results agreed with experiment within a few hundredths of an eV. Molecular applications of the RFSCC are less precise, due to the symmetry limitations on molecular basis sets. Still, our calculations of heavy molecular systems, including actinide compounds, yield state-of-art benchmark molecular parameters. A fuller description may be found in the original publications and in our recent reviews [6,7]. 

This can only be true if p2 (xj, Xj) = 0. In other words, this result tells us that the probability of finding two electrons with the same spin at the same point in space is exactly zero. Hence, electrons of like spin do not move independently from each other. It is important to realize that this kind of correlation is in no way connected to the charge of the electrons but is a direct consequence of the Pauli principle. It applies equally well to neutral fermions and - also this is very important to keep in mind - does not hold if the two electrons have different spin. This effect is known as exchange or Fermi correlation. As we will show below, this kind of correlation is included in the Hartree-Fock approach due to the antisymmetry of a Slater determinant and therefore has nothing to do with the correlation energy E discussed in the previous chapter. 

Results of similar accuracy as relativistic TFDW are found with a simple procedure based on near-nuclear correction which leave space for further improvements. For the reasons mentioned at the end of previous section the direct way to improve the present approach seems to be the refinement of the near nuclear corrections, a problem that we have just tackled with success in the non-relativistic framework [31,32]. The aim was to describe the near-nuclear region accurately by means of using the quantum mechanical exact asymptotic expression up to of the different ns eigenstates of Schodinger equation with a fit of the semiclassical potential at short distancies to the exact asymptotic behaviour (with four terms) of the potential near the nucleus. The result is that the density below Tq becomes very close to Hartree-Fock values and the improvement of the energy values is large (as an example, the energy of Cs+ is improved from the Ashby-Holzman result of-189.5 keV up to -205.6, very close to the HF value of -204.6 keV). This result makes us expect that a similar procedure in the relativistic framework may provide results comparable to Dirac-Fock ones. 

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