Dirac calculations

R. E. Stanton, S. Havriliak. Kinetic balance A partial solution to the problem of variational safety in Dirac calculations. J. Chan. Phys., 81(4) (1984) 1910-1918. 

R. S. Stanton and S. Havriliak,/. Ghent. Phys., 81,1910 (1984). Kinetic Balance A Partial Solution to the Problem of Variational Safety in Dirac Calculations. 

These coefficients give p in Mbar when T is in volts (i.e., in units of 11,605.6°K). The expression for b rj) is an analytic fit to values of b obtained from the Thomas-Fermi-Dirac zero-temperature pressure. The electronic temperature dependence is calculated from the zero-temperature pressure by assuming the latter to be the pressure of a degenerate Fermi gas, and then determining the temperature dependence of pressure of a Fermi gas at any density of interest. This procedure is equivalent to obtaining b from temperature dependent Thomas-Fermi-Dirac calculations. 

We also want to compare the amount of work in a spin-free modified Dirac calculation with that in a nonrelativistic calculation. One of the reasons given for the development of relativistic effective core potentials (RECPs) was the expense of four-component Dirac-Fock calculations. RECPs are often used in spin-free form, but the comparison is made with the spin-dependent unmodified Dirac approach. The comparison of the cost of a spin-free all-electron calculation with a nonrelativistic calculation will give a more realistic indication of the relative cost of the incorporation of spin-free relativistic effects. 

The Dirac equation can be readily adapted to the description of one electron in the held of the other electrons (Hartree-Fock theory). This is called a Dirac-Fock or Dirac-Hartree-Fock (DHF) calculation. 

There are several ways to include relativity in ah initio calculations more efficiently at the expense of a bit of accuracy. One popular technique is the Dirac-Hartree-Fock technique, which includes the one-electron relativistic terms. Another option is computing energy corrections to the nonrelativistic wave function without changing that wave function. 

Relativistic density functional theory can be used for all electron calculations. Relativistic DFT can be formulated using the Pauli formula or the zero-order regular approximation (ZORA). ZORA calculations include only the zero-order term in a power series expansion of the Dirac equation. ZORA is generally regarded as the superior method. The Pauli method is known to be unreliable for very heavy elements, such as actinides. 

DHF (Dirac -Hartree-Fock) relativistic ah initio method DHF (derivative Hartree-Fock) a means for calculating nonlinear optical properties... 

CALCULATION OF THE ELECTRONIC STRUCTURE OF ANTIFERROMAGNETIC CHROMIUM WITH A SINUSOIDAL SPIN DENSITY WAVE BY THE METHOD OF DIRAC FUNCTION LINEAR COMBINATION... 

The next example will illustrate the technique of calculating moments when the probability density function contains Dirac delta functions. The mean of the Poisson distribution, Eq. (3-29), is given by... 

The physical interpretation of the quantum mechanics and its generalization to include aperiodic phenomena have been the subject of papers by Dirac, Jordan, Heisenberg, and other authors. For our purpose, the calculation of the properties of molecules in stationary states and particularly in the normal state, the consideration of the Schrodinger wave equation alone suffices, and it will not be necessary to discuss the extended theory. 

When wave mechanical calculations are made according to the Schrodinger equation, the probability of finding the electron in a node is zero, but this treatment ignores relativistic considerations. When such considerations are applied, Dirac has shown that nodes do have a very small electron density Powell, R.E. J. Chem. Educ., 1968,45,558. See also Ellison, F.O. and Hollingsworth, C.A. J. Chem. Educ., 1976, 53, 767 McKelvey, D.R. J. Chem. Educ., 1983, 60, 112 Nelson, P.G. J. Chem. Educ., 1990, 67, 643. For a review of relativistic effects on chemical structures in general, see Pyykko, P. Chem. Rev., 1988, 88, 563. 

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]. 

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