Pseudo-orbital transformation

A further reduction of the computational effort in investigations of electronic structure can be achieved by the restriction of the actual quantum chemical calculations to the valence electron system and the implicit inclusion of the influence of the chemically inert atomic cores by means of suitable parametrized effective (core) potentials (ECPs) and, if necessary, effective core polarization potentials (CPPs). Initiated by the pioneering work of Hellmann and Gombas around 1935, the ECP approach developed into two successful branches, i.e. the model potential (MP) and the pseudopotential (PP) techniques. Whereas the former method attempts to maintain the correct radial nodal structure of the atomic valence orbitals, the latter is formally based on the so-called pseudo-orbital transformation and uses valence orbitals with a simplified radial nodal structure, i.e. pseudovalence orbitals. Besides the computational savings due to the elimination of the core electrons, the main interest in standard ECP techniques results from the fact that they offer an efficient and accurate, albeit approximate, way of including implicitly, i.e. via parametrization of the ECPs, the major relativistic effects in formally nonrelativistic valence-only calculations. A number of reviews on ECPs has been published and the reader is referred to them for details (Bala-subramanian 1998 Bardsley 1974 Chelikowsky and Cohen 1992 Christiansen et... 

The analytical forms of the modern PPs used today have little in common with the formulas we obtain by a strict derivation of the theory (Dolg 2000). Formally, the pseudo-orbital transformation leads to nodeless pseudovalence orbitals for the lowest atomic valence orbitals of a given angular quantum number l (one-component) or Ij (two-component). The simplest and historically the first choice is the local ansatz for A VCy in Equation (3.4). However, this ansatz turned out to be too inaccurate and therefore was soon replaced by a so-called semilocal form, which in two-component form may be written as... 

Similarly as in Section 1.2, one starts from atomic AE reference calculations at the independent-particle level (some kind of quasi-relativistic HF or fully relativistic DHF). The first step now in setting up pseudopotentials consists in a smoothing procedure for valence orbitals/spinors ( pseudo-orbital transformation ). In the DHF case, to be specific, the radial part ( )/ of the large component of the energetically lowest valence spinors for each //-combination is transformed according to... 

Mitroy et al. (1984) carried out an extensive configuration-interaction calculation of the structure amplitude (q/ 0) for correlated target and ion states. The long-dashed curve in fig. 11.7(a) shows their momentum distribution multiplied by 2. They found that the dominant contribution came from the pseudo-orbital 3d, calculated by the natural-orbital transformation. Pseudo-orbitals are localised to the same part of space as the occupied 3s and 3p Hartree—Fock orbitals and therefore contribute to the cross section at much higher momenta than the diffuse Hartree—Fock 3d and 4d orbitals. The measurements show that the 4d orbital has a larger weight than is calculated by Mitroy et al, who overestimate the 3d component. 

In order to reduce the size of the PW basis set pseudo potentials (PP) of the dual-space type [12,13] are used. The latest implementation of the GPW method [34] has been done within the CP2K program and the corresponding module is called Quickstep [32]. In this implementation the linear scaling calculation of the GPW KS matrix elements is combined with an optimizer based on orbital transformations [33]. This optimization algorithm scales linearly in the number of basis functions for a given system size and, in combination with parallel computers, it can be used for systems with several thousands of basis functions [33,34]. 

Reductions in the basis set used to represent the valence orbital can be only achieved if by admixture of core orbitals radial nodes are eliminated and the shape of the resulting pseudo ip) valence orbital core region (pseudo-valence orbital transformation)... 

It may be asked how accurate energy-consistent pseudopotentials will reproduce the shape of the valence orbitals/spinors and their energies. Often radial expectation values < r > are used as a convenient measure for the radial shape of orbitals/spinors. Due to the pseudo-valence orbital transformation and the simplified nodal structure it is clear that values n < 0 are not suitable, since the resulting operator samples the orbitals mainly in the core region. Table 2 lists orbital energies, < r > and < > expectation values for the Db [Rn] 5f 6d ... 

This set of equations can be solved directly, i.e., without proceeding by the pseudo-canonical transformation described in Appendix 3 for the more general case. The only quantities needed are the spherically averaged s and / associated with localized orbitals and the long-range dipole-dipole tensors. The update formula to get the nth approximation to the amplitude matrix element is... 

Such a feature of equivalent pseudo-orbitals in establishing the localization of the valence orbital and eigen-value consecrates the reality of the valence reality, on the one hand, and corresponds to those involving localization measures through imitary orthogonal transformations, described before, on the other hand. 

The equations may be simplified by choosing a unitary transformation (Chapter 13) which makes the matrix of Lagrange multipliers diagonal, i.e. Ay 0 and A This special set of molecular orbitals (f> ) are called canonical MOs, and they transform eq. (3.40) mto a set of pseudo-eigenvalue equations. 

There are two approaches to map crystal charge density from the measured structure factors by inverse Fourier transform or by the multipole method [32]. Direct Fourier transform of experimental structure factors was not useful due to the missing reflections in the collected data set, so a multipole refinement is a better approach to map charge density from the measured structure factors. In the multipole method, the crystal charge density is expanded as a sum of non-spherical pseudo-atomic densities. These consist of a spherical-atom (or ion) charge density obtained from multi-configuration Dirac-Fock (MCDF) calculations [33] with variable orbital occupation factors to allow for charge transfer, and a small non-spherical part in which local symmetry-adapted spherical harmonic functions were used. 

The CASSCF energy (but not RASSCF) is invariant to rotations among the inactive orbitals (compare SCF) and also to rotations among the active orbitals. This can be used, for example, to transform to localized orbitals, or to pseudo-natural orbitals,... 

We may again chose a unitary transformation which makes the matrix of the Lagrange multiplier diagonal, producing a set of canonical Kohn-Sham (KS) orbitals. The -resulting pseudo-eigenvalue equations are known as the Kohn-Sham equations. 

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