Pseudovalence orbital

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... 

Having a nodeless and smooth pseudovalence orbital pjj and the corresponding orbital energy v,/y at hand, the corresponding radial Fbck equation... 

The scalar-relativistic effects can be easily absorbed into the effective potential by taking the all-electron (AE) calculation results of the same order of relativistic approximation as the references to parametrize the potentials. Taking the two-component (or even four-component) form of the pseudo-valence orbitals, the spin-orbit coupling effect can also be absorbed into the ECR Because the pseudovalence orbitals are energetically the lowest-eigenvalue eigenvectors of the Fock... 

In pseudopotential calculations one has a different nodal structure of the pseudovalence orbitals in comparison to the all-electron valence orbitals. (di WdR dr) and its expectation values are neither vanishing nor small. On the contrary, it is assumed and has numerically been shown for Au2 that the main contribution to the relativistic bond-length changes stem from relativistic corrections to the Hellmann-Feynman force resulting from the pseudopotential ... 

There are two main lines of ECPs, i.e., the model potential (MP) technique, which utilizes valence orbitals with a nodal stmcture corresponding exactly to those of the AE valence orbitals, and the PP scheme, which uses valence orbitals exhibiting a simplified nodal structure with respect to the AE valence orbitals, i.e., the so-called pseudovalence orbitals. This chapter will only focus on the PP approach, while the chapter from Barandi an and Seijo will deal with MPs. 

The pseudopotential approximation was originally introduced by Hellmann already in 1935 for a semiempirical treatment of the valence electron of potassium [25], However, it took until 1959 for Phillips and Kleinman from the solid state community to provide a rigorous theoretical foundation of PPs for single valence electron systems [26]. Another decade later in 1968 Weeks and Rice extended this method to many valence electron systems [27,28], Although the modern PPs do not have much in common with the PPs developed in 1959 and 1968, respectively, these theories prove that one can get the same answer as from an AE calculation by using a suitable effective valence-only model Hamiltonian and pseudovalence orbitals with a simplified nodal structure [19],... 

Since there are no core functions in equation 6.4, the pseudovalence orbitals belonging to the lowest Hartree-Fock (HF) or Kohn-Sham solutions for each angular momentum / are thus nodeless [6]. 

For subsequent usage in molecular valence-only calculations compact valence basis sets were generated, i.e., the pseudovalence orbitals were fitted by using a nonlinear least-squares procedure, similar to the one for fitting the potentials, to a linear combination of Gaussian functions... 

A closer look at equation 6.20 will tell us about the singularity problem in the PP for a radial node of the pseudovalence orbital [Pg.159]

For the accuracy of PP calculations the applied valence basis sets are as important as the PPs themselves. Therefore, these basis sets should be optimized very carefiilly. Furthermore, only a valence basis set corresponding to the PP under consideration will provide a reliable description of the pseudovalence orbitals, which exhibit different radial shapes in the spatial core region for different PPs [19]. Thus, even valence basis sets corresponding to PPs for the same element and with the same core size are not transferable. If one needs a more extended basis set than provided, the original basis set can just be augmented by adding diffuse functions. 

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