Phillips-Kleinman Pseudopotential

Non-uniqueness of the Pseudopotential.—The Phillips-Kleinman pseudopotential contains operators 2( v- k) k)<.p which shifts the eigenvalues of the core func-... 

We underline also the fundamental difference between the orbital-free embedding route (Eq. 53) and that offered by Phillips-Kleinman pseudopotentials. In the calculations based on Eqs. 31-32, the embedded orbitals of two different subsystems are not subject to the orthogonality condition which is explicitly imposed in Eq. 55. 

Using the so-called generalized Phillips-Kleinman pseudopotential [3]... 

The concept of the orthogonalisation hole, appearing in the pseudopotentials of the Phillips-Kleinman type, thus results from the fact that the pseudo wave function overestimates the electron charge inside the core region. The norm-conserving Bachelet-Hamann-Schliiter (BHS) pseudopotentials differ from the Phillips-Kleinman pseudopotentials in at least two important aspects ... 

The first reliable energy band theories were based on a powerfiil approximation, call the pseudopotential approximation. Within this approximation, the all-electron potential corresponding to interaction of a valence electron with the iimer, core electrons and the nucleus is replaced by a pseudopotential. The pseudopotential reproduces only the properties of the outer electrons. There are rigorous theorems such as the Phillips-Kleinman cancellation theorem that can be used to justify the pseudopotential model [2, 3, 26]. The Phillips-Kleimnan cancellation theorem states that the orthogonality requirement of the valence states to the core states can be described by an effective repulsive... 

Note that this is just the same condition under which the generalized pseudopotential of Weeks, Hazi, and Rice13 reduced to the Phillips-Kleinman form.7)... 

In the Reporters pseudopotential calculations31 a different approach is taken by replacing the Phillips-Kleinman term with an operator,... 

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... 

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. 

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. 

Up to this point we have used rather crude terminology to describe the free electron state in simple liquids, and we are now faced with the following dilemma is it possible to describe a free excess electron in a liquid in terms of a plane wave, neglecting the effect of the core electrons Such a simple description is justified by the pseudopotential theory introduced by Phillips and Kleinman (37) and by Cohen and Heine (5). 

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. 

Early work based on a statistical treatment of the core electrons was published by Hellmaim (1935) and Gombas (1935) for molecular and solid-state physics, respectively. Quantum-mechanical justifications in the framework of the Hartree and Hartree-Fock theories were given by Fenyes (1943) and Szepfalusy (1955, 1956), respectively. The first derivation of the pseudopotential approach within the Hartree-Fock formalism which came to general attention is due to Phillips and Kleinman (1959) and was later generalized by Hazi and Rice (1968). The accuracy and limitations of the pseudopotential approach... 

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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