Nodeless pseudo-orbitals

Here 1/ is the effective potential and a>i i is a nodeless pseudo-orbital that can be derived from Xi, in several different ways. For first-row atoms, Christiansen, Lee and Pitzer (1979) suggest... 

C-H bond is replaced by calculating the difference in correlation energy between a are then replaced by a set of nodeless pseudo-orbitals. The regular valence orbitals will... 

The in (172) are still exact energies of the original Fock operator and the pseudo-orbital is now determined by - ... 

Fig. 11. Radial amplitude P(r) = R(r)-r of a valence Hartree-Fock orbital

Essentially two lines are followed for the derivation of pseudopotentials or effective-core potentials creating nodeless pseudo-orbitals. One approach uses as reference data the shape of the valence orbitals or spinors in the spatial valence region and their corresponding one-particle energies. In the spatial core region the one-particle functions are smoothed according to special prescriptions and the radial Fock equation is solved for the potential, which is then usually fitted by linear combinations of Gaussians times powers of r. 

An ab initio effective core potential method derived from the relativistic all-electron Dirac-Fock solution of the atom, which we call the relativistic effective core potential (RECP) method, has been widely used by several investigators to study the electronic structure of polyatomics including the lanthanide- and actinide-containing molecules. This RECP method was formulated by Christiansen et al. (1979). It differs from the conventional Phillips-Kleinman method in the representation of the nodeless pseudo-orbital in the inner region. The one-electron valence equation in an effective potential of the core electron can be written as... 

The action of the projection operator (-e +A) 0 ><0 is to raise the eigenvalue of the core orbital % to the value A. A new lower bound for the eigenvalue for the pseudo-orbital xl can be shown to be the lower of A and ej. In practice the core eigenvalues are usually shifted so as to be degenerate with the lowest valence eigenvalues of the same symmetry. The coefficients a in equation (37) can now assume values which allow the pseudo-orbital Xt to be nodeless and thus capable of representation by a smaller basis set expansion. 

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... 

Figure 10. Radial part of orbital products 2s 2p for gC entering in the 2s-2p exchange integral. Whereas the product formed with nodeless pseudo-valence orbitals generated by a 4-valence electron ([jHe] core) pseudopotential (PP) is always positive, the one formed with valence orbitals from all-electron (AE) calculations has a negative contribution in the core region due to the 2s radial node.

The value of the ECP depends not only on an electron s coordinates, but also on the projection of the wave function of one electron (holding the other electron coordinates constant) onto the spherical harmonics, T/m. The local term, WL+i(r), depends only on the distance of the electron from the nucleus. The angular potentials wi(r) are determined so that, beyond some cutoff distance, the pseudo-orbitals obtained from an ECP calculation match those of an all-electron calculation, but are nodeless and smoothly go to zero within the cutoff radius [133], The w are then fit to a Gaussian expansion [134] so that the potential can be rapidly integrated over Gaussian basis functions. 

While plane waves are a good representation of delocalized Kohn-Sham orbitals in metals, a huge number of them would be required in the expansion (O Eq. 7.67) to obtain a good approximation of atomic orbitals, in particular near the nucleus where they oscillate rapidly. Therefore, in order to reduce the size of the basis set, only the valence electrons are treated explicitly, while the core electrons (i.e., the inner shells) are taken into account implicitly through pseudopotentials combining their effect on the valence electrons with the nuclear Coulomb potential. This frozen core approximation is justified as typically only the valence electrons participate in chemical interactions. To minimize the number of basis functions the pseudopotentials are constructed in such a way as to produce nodeless atomic valence wavefunctions. Beyond a specified cutoff distance from the nucleus, Ecut the nodeless pseudo-wavefunctions are required to be identical to the reference all-electron wavefunctions. 

Having a nodeless and smooth pseudo valence orbital (p. and the corresponding orbital energy e ij at hand, the corresponding radial rock equation... 

In the effective core potential (ECP) approximation, is represented by a semi-local potential [74]. Unlike in the MCP methods, there are no core functions and the pseudo-valence orbitals are nodeless for the radial part, which is an essential approximation. The semi-local ansatz gives rise to rather complicated integrals over the Gaussian functions compared to the MCP methods, though efficient algorithms were developed for their solution. Relativistic and SO effects are treated by relativistic one-electron PPs (RPP) [76]... 

An ultrasoft type of pseudopotential was introduced by Vanderbilt (1990) and Laasonen et al. [1993] to deal with nodeless valence states which are strongly localized in the core region. In this scheme the normconserving condition is lifted and only a small portion of the electron density inside the cutoff radius is recovered by the pseudo-wavefunction, the remainder is added in the form of so-called augmentation charges. Complications arising from this scheme are the nonorthogonality of Kohn-Sham orbitals, the density dependence of the nonlocal pseudopotential, and need to evaluate additional terms in atomic force calculations. 

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