Relativistic correlation energy

An interesting example of relativistic MC-SCF theory is the isoelectronic Be-like series. While the non-relativistic correlation energy is linear in Z (due to the degeneracy), the Z-dependence of the relativistic... 

Exclusion Principle. The energy associated with the filled vacuum is an unobservable constant which should be subtracted from a given physical model. Calculations which go beyond an independent particle model but are carried out using only the positive energy branch of the Dirac spectrum are said to be carried out within the no virtual pair approximation. Such calculations essentially follow the procedures adopted in non-relativistic studies. The relativistic and non-relativistic correlation energy calculations differ only in the model used to defined the reference independent particle model. 

Y. Ishikawa, K. Koc. Relativistic many-body perturbation theory based on the no-pair Dirac-Coulomb-Breit Hamiltonian Relativistic correlation energies for the noble-gas sequence through Rn (Z=86), the group-llB atoms through Hg, and the ions of Ne isoelectronic sequence. Phys. Rev. A, 50(6) (1994) 4733-4742. 

Table 3.7. Comparison of LDA [18], Cl (estimated from non-relativistic CI-calculations for the three innermost electrons and the experimental ionization potentials of all other electrons [25]) and MBPT2 [26] correlation energies for neutral atoms - non-relativistic correlation energy, AE - relativistic contribution...

The effect of core-electron correlation is small, as shown in Table 11.16. It should be noted that the valence and core correlation energy per electron pair is of the same magnitude, however, the core correlation is almost constant over the whole energy surface and consequently contributes very little to properties depending on relative energies, like vibrational frequencies. It should be noted that relativistic corrections for the frequencies are expected to be of the order of 1 cm" or less. ... 

In this review, we have mainly studied the correlation energy connected with the standard unrelativistic Hamiltonian (Eq. II.4). This Hamiltonian may, of course, be refined to include relativistic effects, nuclear motion, etc., which leads not only to improvements in the Hartree-Fock scheme, but also to new correlation effects. The relativistic correlation and the correlation connected with the nuclear motion are probably rather small but may one day become significant. 

Froman, A., Calculation of correlation energies and relativistic corrections of some He- and Ne-like systems/ ... 

For both the DZ and TZ sets a contracted function was included for the 6p orbital, but this was deleted in the QZ and 5Z sets due to near linear dependence. The contraction was also deleted from the 5Z set for the same reason. Figure 7 plots the correlation energies for both nonrelativistic and DK-relativistic CISD calculations. The CBS limits using a extrapolation of the QZ and 5Z correlation energies are -391.8 and -418.0 m /, for NR and DK, respectively. 

Figure 7. All-electron correlation energies from non-relativistic (NR) and DKrelativistic CISD calculations on the Hg atom using cc-pVnZ-NR and cc-pVnZDK basis sets, respectively.

For all results in this paper, spin-orbit coupling corrections have been added to open-shell calculations from a compendium given elsewhere I0) we note that this consistent treatment sometimes differs from the original methods employed by other workers, e.g., standard G3 calculations include spin-orbit contributions only for atoms. In the SAC and MCCM calculations presented here, core correlation energy and relativistic effects are not explicitly included but are implicit in the parameters (i.e., we use parameters called versions 2s and 3s in the notation of previous papers 11,16,18)). 

The Wlc total atomization energy at 0 K of aniline, 1468.7 kcal/mol, is in satisfying agreement with the value obtained from heats of formation in the NIST WebBook 39), 1467.7 0.7 kcal/mol. (Most of the uncertainty derives from the heat of vaporization of graphite.) The various contributions to this result are (in kcal/mol) SCF limit 1144.4, valence CCSD correlation energy limit 359.0, connected triple excitations 31.7, inner shell correlation 7.6, scalar relativistic effects -1.2, atomic spin-orbit coupling -0.5 kcal/mol. Extrapolations account for 0.6, 12.1, and 2.5 kcal/mol, respectively, out of the three first contributions. 

The plan of this paper is as follows - In section 2, the basic experimental data required in the re-evaluation of the empirical correlation energies of the N2 CO, BF and NO molecules are collected. The essential theoretical ingredients of our re-determination are given in section 3 including new fully relativistic calculations including the frequency independent Breit interaction and electron correlation effects described by second order diagrammatic perturbation theory for the Be-like ions B", C, O" ... 

Two key pieces of theoretical data are required to obtain an empirical estimate of the correlation energy from the experimental data collected in the preceding section the total molecular Hartree-Fock energy and the relativistic corrections . It is implicit in the definition of the correlation energy presented in Eq. (1) that the total electronic energy ofa given molecule, Ef, may be divided into three constituent parts,... 

In order to establish the plausibility of the argument that the calculation of relativistic corrections to the second-order correlation energy in Be-like ions may be used to estimate reliably the corresponding quantity in many-electron molecules, we have included a calculation, in Table 8, of the many-body corrections to Ne. The relativistic correction to the correlation derived from these calculations is 1.257 x 10 hartree, which should be compared with our previous calculations [38] of the same quantity in 1.277 x 10 hartree, and with the independent calculations by Drake... 

From this, we may deduce that the relativistic correction to the correlation energy is dominated by the contribution from the s electron pair, and that the total relativistic effect involving the exchange of a single transverse Breit photon is obtained to sufficient accuracy for our present purposes at second-order in many-body perturbation theory. 

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