Pseudo wave function

The basic idea of the pseudopotential theory is to replace the strong electron-ion potential by a much weaker potential - a pseudopotential that can describe the salient features of the valence electrons which determine most physical properties of molecules to a much greater extent than the core electrons do. Within the pseudopotential approximation, the core electrons are totally ignored and only the behaviour of the valence electrons outside the core region is considered as important and is described as accurately as possible [54]. Thus the core electrons and the strong ionic potential are replaced by a much weaker pseudopotential which acts on the associated valence pseudo wave functions rather than the real valence wave functions (p ). As... 

In the PP theory, the valence electron wave function is composed of two parts. The main part is the pseudo-wave function describing a relatively smooth-varying behavior of the electron. The second part describes a spatially rapid oscillation of the valence electron near the atomic core. This atomic-electron-like behavior is due to the fact that, passing the vicinity of an atom, the valence electron recalls its native outermost atomic orbitals under a relatively stronger atomic potential near the core. Quantum mechanically the situation corresponds to the fact that the valence electronic state should be orthogonal to the inner-core electronic states. The second part describes this CO. The CO terms explicitly contain the information of atomic position and atomic core orbitals. 

In the inner core region, the pseudospinors are smoothed, so that the electronic density with the pseudo-wave function is not correct. When operators describing properties of interest are heavily concentrated near or on nuclei, their mean values are strongly affected by the wave function in the inner region. The four-component molecular spinors must, therefore, be restored in the heavy-atom cores. 

Imagine that a valence-electron stale i/ (r)> could be written as a smooth pseudo wave function , corrected to be orthogonal to all core states c> ... 

Since both bulk and surface states are molecular in character, the wave functions of atoms in both types of position can be calculated by the same method. Appelbaum and Hamann [70] assume two-dimensional periodicity along the surface and make the same Fourier expansion of the pseudo-wave function as for the bulk, except that at each of a set of discrete surface normal co-ordinates a different set of expansion coefficients is used. These sets can be integrated from outside the surface into the bulk. Well inside the bulk, these wave functions are matched to bulk states of similar lateral symmetry and the matching condition determines energies and wave functions. 

There is one other point we should make concerning this formulation. The orthogonali-zation in Eq. (D-1) punches a hole in the pseudo wave function at each core. We see in fact that for each state, a fraction <(c (p>, called the orthogonalization hole, is... 

A pseudo-potential is an effective potential which effectively replaces the atomic all-electron potential such that core states are eliminated and the valence electrons are described by pseudo-wave functions with significantly fewer nodes. Only the chemically significant valence electrons are dealt with explicitly, while the... 

Figure 13. Sketch illustrating the plane wave pseudopotential concept, modified from Payne et al. (1992). The all electron wave function (v /v) is rapidly oscillating at small r due to the strong ionic potential of the core (Z/r). A veiy large number of plane waves would be required to mimic this cusped behavior. Instead, the Z/r potential of the core is replaced by a weaker pseudopotential (Vpseudo) that acts on a set of pseudo wave functions (vj/pseudo) up to the cutoff distance r. Vpseudo is chosen such that v /pseudo is smoothly varying inside the core ard best matches the behavior of v /v outside the core.

Despite these complications, what one has gained is a rather smooth pseudo wave function, which has a large degree of arbitrariness. Indeed, if one considers the pseudo hamiltonian, acting on a core state ... 

This arbitrariness has e.g. been exploited by Harrison to construct a pseudo wave function which is as smooth as possible [Ref. 9]. 

Several refinements and alternative formulations of the pseudopotential concept have been worked out. E.g. the arbitrariness in the pseudo wave function, can be transferred into an arbitrariness in the pseudopotential, to which a unique pseudo wave function belongs. This leads to the... 

For the special case /c >= En — Sc) c >i one has to do with the earlier pseu-dopotential (3.7), with an arbitrariness in the pseudo wave function as discussed... 

In the ab-initio pseudopotential theory, one essentially tries to take advantage of the arbitrariness in the pseudopotential or in the pseudo wave function, avoiding the introduction of adjustable parameters. For instance, Harrison developed an OPW-like scheme to make the pseudo wave function as smooth as possible [Ref. 9]. With this approach, detailed information is required on the core energies and core wave functions. 

The meaning of the concept of norm-conservation can be understood from the Phillips-Kleinman type of approach, discussed so far. From the relation between the wave function and the pseudo wave function, one obviously obtains ... 

Therefore, if the pseudo wave function is normalized, the true wave function is not. In the Phillips-Kleinman approach, the density is given by ... 

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 pseudo wave function, when normalized, becomes identical to the true valence wave function beyond some core radius Re-... 

One of the best known applications of pseudopotential theory, is in the theory of simple metals. The basis for its success lies in the large freedom in the form of the pseudo wave function. In simple metals, the electrons behave quasi free, and one might hope that the pseudo wave functions can be fairly well described by plane waves, upon which the pseudopotential acts as a perturbation. This possibility is a direct consequence of the smoothness of the pseudo wave function, in contrast to the true wave function, for which... 

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