Phillips-Kleinman method

If one defines the core projector and the pseudo-orbital in the same way as done in the Phillips-Kleinman method, for relativistic spinor wavefunctions one obtains relativistic pseudo-orbitals and relativistic effective potentials... 

These early calculations used effective potentials obtained by the Phillips-Kleinman method (see the review of Krauss and Stevens ) wherein the pseudo-orbitals are taken to be linear combinations of the atomic orbitals of the same / and j. This method tends to underestimate the repulsive region of the potential energy curves. There is no reason that the pseudo-orbital must be linear combinations of core and valence orbitals. Christiansen, Lee and Pitzer have proposed a method for constructing pseudo-orbitals in which the pseudo-orbital is represented as... 

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 adaptation of the Phillips-Kleinman method to relativistic spinors yields relativistic pseudo-orbitals and relativistic ECPs given by... 

Figure 2 compares the pseudo-orbitals obtained from the shape-consistent Christiansen-Lee-Pitzer method with the Phillips-Kleinman, and the all-electron techniques for the chlorine 3s orbital. As seen from Figure 2, the maximum of the pseudo orbital obtained from the Phillips-Kleinman method, occurs at a shorter distance, compared to both the Christiansen-Lee-Pitzer orbital and the all-electron 3s Hartree-Fock orbital of the chlorine atom, which is the origin of under-estimation of the repulsive wall of the potential surface in the Phillips-Kleinman method. Furthermore, the shorter maximum in the Phillips-Kleinman method leads to shorter equilibrium bond distances using the Phillips-Kleinman potentials. 

Two pseudopotential schemes were developed in the framework of the DV-Xq method. One of them [42], [43] is based on the explicit inclusion of core atomic orbitals in the valence pseudoorbitals (the Phillips-Kleinman ansatz)... 

This method is probably as accurate as some other simple pseudopotential approaches. However, there appear to be some difficulties in improving it to the standard of some of the recent pseudopotential calculations. Attempts to use larger than minimal basis sets required the inclusion of a Phillips-Kleinman term in addition to the orthogonality procedure in order to prevent collapse of the valence orbitals into the core space. Thus in calculations on AlaQ, Vincait had to include not only the A1 3s and Cl 3j and 3p shells but also the A12p and Cl 2s and 2p shells explicitly in the valence-electron basis in order to obtain good results. Consequently this calculation was not substantially less expensive in computing time than an equivalent all-electron calculation. 

The above equation, although provides an exact relation could be useful in practice only if the pseudopotential can be reasonably well approximated without knowing the all-electron orbitals and eigenvalues e,. Such approximations are available to separate core and valence electrons. The Phillips-Kleinman formal route can also be used to separate electrons in different molecules64,65. This route is, however, not orbital-free. The environment needs to be described at the orbital-level. Therefore, this group of methods will not be discussed further here. 

Effective Core Potential methods are classified in two families, according to their basic grounds. On the one hand, the Pseudopotential methods (PP) rely on an orbital transformation called the pseudoorbital transformation and they are ultimately related to the Phillips-Kleinman equation [2]. On the other hand, the Model Potential methods (MP) do not rely on any pseudoorbital transformation and they are ultimately related to the Huzinaga-Cantu equation [3,4]. The Ab Initio Model Potential method (AIMP) belongs to the latter family and it has as a... 

Extensive introductions to the effective core potential method may be found in Ref. [8-19]. The theoretical foundation of ECP is the so-called Phillips-Kleinman transformation proposed in 1959 [20] and later generalized by Weeks and Rice [21]. In this method, for each valence orbital (pv there is a pseudo-valence orbital Xv that contains components from the core orbitals and the strong orthogonality constraint is realized by applying the projection operator on both the valence hamiltonian and pseudo-valence wave function (pseudo-valence orbitals). In the generalized Phillips-Kleinman formalism [21], the effect of the projection operator can be absorbed in the valence Pock operator and the core-valence interaction (Coulomb and exchange) plus the effect of the projection operator forms the core potential in ECP method. 

Prior to 1975 or so, the words ab initio did not exist in the scientific vocabulary for methods describing the electronic structure of the solid state. At that time, a number of very powerful and successful methods had been developed to describe the electronic structure of solids, but these methods did not pretend to be first principles or ab initio methods. The foremost example of electronic structure methods at that time was the empirical pseudopotential method (EPM) [1]. The EPM was based on the Phillips-Kleinman cancelation theorem [2], which justified the replacement of the strong, allelectron potential with a weak pseudopotential [1]. The pseudopotential replicated only the chemically active valence electron states. Physically, the cancelation theorem is based on the orthogonality requirement of the valence states to the core states [1]. This requirement results in a repulsive part of the pseudopotential which cancels the strongly attractive part of the core potential and excludes the valence states from the core region. Because of this property, simple bases such as plane waves can be used efficiently with pseudopotentials. 

Many of the effective potentials (relativistic or non-relativistic) are generated using the Phillips-Kleinman transformation. In this method, the explicit core-valence orthogonality constraints are replaced by a modified valence Hamiltonian. If one replaces the potential generated by core electrons by a potential Fj, then one can write the one-electron valence wave equation as... 

Fig. 2. A comparison of dJoiine 3s pseudo-orbitals generated with the Christiansen-Lee-Pitzer method and the Phillips-Kleinman pseudo-orbital and the all-electron Hartree-Fodc orbital.

The ECP method dates back to 1960, when Phillips and Kleinman suggested an approximation scheme for discarding core orbitals in band calculations [1]. They replaced the full Fock-operator with the following operator ... 

The generalization of the pseudopotential method to molecules was done by Boni-facic and Huzinaga[3] and by Goddard, Melius and Kahn[4] some ten years after Phillips and Kleinman s original proposal. In the molecular pseudopotential or Effective Core Potential (ECP) method all core-valence interactions are approximated with l dependent projection operators, and a totally symmetric screening type potential. The new operators, which are parametrized such that the ECP operator should reproduce atomic all electron results, are added to the Hamiltonian and the one electron ECP equations axe obtained variationally in the same way as the usual Hartree Fock equations. Since the total energy is calculated with respect to this approximative Hamiltonian the separability problem becomes obsolete. 

The remarkable conclusion of this argument is that though pseudopotentials can be used to describe semiconductors as well as metals, the pseudopotential perturbation theory which is the essence of the theory of metals is completely inappropriate in semiconductors. Pseudopotential perturbation theory is an expansion in which the ratio W/Ep of the pseudopotential to the kinetic energy is treated as small, whereas for covalent solids just the reverse quantity, EpiW, should be treated as small. The distinction becomes unimportant if we diagonalize the Hamiltonian matrix to obtain the bands since, for that, we do not need to know which terms are large. Thus the distinction was not essential to the first use of pseudopotentials in solids by Phillips and Kleinman (1959) nor in the more recent application of the Empirical Pseudopotential Method used by M. L. Cohen and co-workers. Only in approximate theories, which are the principal subject of this text, must one put terms in the proper order. 

Until the late 70 s the method employed to construct a pseudo-potential was based on the Phillips and Kleinman cancellation idea. A model analytic potential was constructed and its parameters were fitted to experimental data. However, these models did not obey condition (6.43). 

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],... 

Historically speaking, the orthogonalized plane wave (OPW) method should have been discussed before the pseudopotential method (Section 3.6). The work of Phillips and Kleinman, which is... 

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