Atoms finite-difference calculations

Due to the energy-dependence of the Hamiltonian the Wood-Boring approach leads to nonorthogonal orbitals and has been mainly used in atomic finite difference calculations as an alternative to the more involved Dirac-Hartree-Fock calculations. The relation... 

Tupitsyn, I. I. HFDB 2003. Program for atomic finite-difference four-component Dirac-Hartree-Fock-Breit calculations written on the base of the hfd code [110]. 

Tupitsyn, I. I. and Mosyagin, N. S. grecp/hfj 1995. Program for atomic finite-difference two-component Hartree-Fock calculations with the generalized RECP in the //-coupling scheme. 

Hunter I.C., Jones I.P., Numerical experiments on the effects of strong grid stretching in finite difference calculations, Tech. Rep. AERE R-10301, United Kingdom Atomic Energy Authority, Harwell (1981)... 

Table 4.1 Exchange-only ground-state energies from ROPM and RHF calculations for noble gas atoms Coulomb (C) and Coulomb-Breit (C + B) limit in comparison with complete transverse exchange (C + T) (Engel et al. 1998a). For the RHF approximation the energy difference with respect to the ROPM is given, AE = tot(RHF) — tot(ROPM), providing results from (a) finite-differences calculations (Dyall et al. 1989) and (b) a basis-set expansion (Ishikawa and Koc 1994). All energies in mHartree. uext and c as in Ishikawa and Koc (1994).

Finite-Difference Calculations for Atoms and Diatomic Molecules in Strong Magnetic and Static Electric Fields... 

Explicit inclusion of relativistic effects in valence-only calculations has been by far less frequently attempted. Datta, Ewig and van Wazer [135] used a Phillips-Kleinman PP in a study of PbO, whereas Ishikawa and Malli [136] tested PPs of semilocal form in four-component atomic DHF finite difference calculations. This work was extended by Dolg [137] to four-component molecular DHF calculations with a subsequent correlation treatment. In addition a complicated form of Vcv based on the Foldy-Wouthuysen transformation [138] was derived by Pyper [139] and applied in atomic calculations [140]. For all these approaches the computational effort is significantly higher than for the implicit treatment of relativity, and the gain of computational accuracy is not obvious at all. 

A vibrations calculation is the first step of a vibrational analysis. It involves the time consuming step of evaluating the Hessian matrix (the second derivatives of the energy with respect to atomic Cartesian coordinates) and diagonalizing it to determine normal modes and harmonic frequencies. For the SCFmethods the Hessian matrix is evaluated by finite difference of analytic gradients, so the time required quickly grows with system size. 

Six-dimensional, numerically accurate four-atom wave packet calculations were pioneered by Zhang and Zhang [17] and Neuhauser [18]. While numerous details of the present RWP implementation differ from these earlier approaches, it should be noted that many of the general ideas remain the same. In applications, finite-sized grids and basis sets are introduced to describe the wave packet, and... 

With the above-described heat transfer model and rapid solidification kinetic model, along with the related process parameters and thermophysical properties of atomization gases (Tables 2.6 and 2.7) and metals/alloys (Tables 2.8,2.9,2.10 and 2.11), the 2-D distributions of transient droplet temperatures, cooling rates, achievable undercoolings, and solid fractions in the spray can be calculated, once the initial droplet sizes, temperatures, and velocities are established by the modeling of the atomization stage, as discussed in the previous subsection. For the implementation of the heat transfer model and the rapid solidification kinetic model, finite difference methods or finite element methods may be used. To characterize the entire size distribution of droplets, some specific droplet sizes (forexample,.D0 16,Z>05, andZ)0 84) are to be considered in the calculations of the 2-D motion, cooling and solidification histories. 

A centered finite difference scheme is used to calculate the position of the ith atom, x, a short time At in the future ... 

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