Pseudopotential

In quantum Chemistry, the method of the pseudopotential (or pseudopotential ) is an attempt to replace the Coulomb potential of interaction of the core and the effects of the electrons of heart, strongly bound, by an effective potential interacting only with the electrons of valence. The approximation of the pseudopotential is of great interest in the theoretical calculation of the electronic structure of the matter because it makes it possible to treat explicitly only the electrons of valence; thus allowing an important profit in the computer's resources necessary to calculations. A pseudopotential can be generated for a Atome in a electronic Configuration of reference chosen arbitrarily. It is then used in calculations of the properties of the electrons of valence for other systems (molecules, solids…) ; the description of the states of heart remaining unchanged whatever the studied system. The diagram which makes it possible to generate a pseudopotential is not single what explains the development of several classes of pseudopotentials in particular the pseudopotentials called “to preserved standard” and “ultrasoft” (called sometimes pseudopotentials of Vanderbilt).

Approach of the pseudopotential

The approach of the pseudopotential is based on the fact that a large majority of the physical properties and chemical of materials depend only on the behavior of the electrons of valence. Indeed, in a Atome only the electrons the pérophériques ones (of small number) contribute to the formation of the chemical bonds whereas the electrons of heart (in great number) are them strongly related to the atomic nucleus and thus not very sensitive to the environment close to the atom. The distribution of the electrons of heart can thus be regarded as practically unchanged even when the atom is placed in a chemical medium different from that of the isolated atom. It is on this property that the approximation known as is based of the cold heart ( frozen-core approximation ) which consists in calculating, for the isolated atom, the electronic Configuration of the Ion. The advantage of this approximation is that the number of appearing electron in an explicit way in calculations (and thus the number of equation to be solved) is some strongly reduced; only the electrons of valence are taken into account (as example, for a crystal of 100 atoms of Silicium one passes from 1400 to 400 electrons).
If the approach of the cold heart consistue an important advance in the optics of a profit in the computer's resources its application as higher stated is not desirable. Indeed, the electrons of heart always appear in an implicit way. However, in quantum Mécanique all the functions of wave describing the electronic states (i.e. electrons) must be orthogonals between them (condition of generalized orthogonality). This condition forces the function of wave of valence to be orthogonal with all the functions of wave describing the states of heart. This constraint makes that the function of wave of valence has a nodal structure can favorable from a numerical point of view. It is thus more judicious to replace the true ion of heart by a Potentiel ionic manpower with which a function with softened wave is associated (i.e. without node). The use of an effective potential is justified by the quantum nature of the electrons which is such as the repulsive potential generated by the electrons of heart is compensated by the gravitational potential of the core ( annulment theorem ). It results an effective ionic potential from it relatively soft which acts only on the electrons of valence; the pseudopotential.

Empirical pseudopotential

The concept of pseudopotential was introduced into the years 1930 by Fermi. Thereafter, Hellmann uses this concept for calculation of the energy alkaline metal levels. These first pseudopotentials are qualified the empirical ones; what means that they are not obtained by calculation but are not parameterized as well as possible to reproduce experimental résultars of reference. The use of such a type of pseudopotential is based on two observations. First of all, if it were certain at that time that one could obtain in an exact way and by calculation of the pseudopotentials that passed by the resolution of complex calculations (implying the atomic function of wave) impossible to solve without computer's resources. The use of an empirical method much simpler thus went from oneself. Then, one can note that a certain number of elements can be described by pseudopotentials parametrized while providing an acceptable approximation of the interaction electron of valence - ion of heart. That was true in particular for alkaline metals, metals " simples" such as aluminum as well as the semiconductors. The use of these pseudopotentials goes permettrent in ten year to increase the field of knowledge in the field of the solid state before being replaced by more effective ab.initio pseudopotentials.

Method of the orthogonal plane waves

The method of the orthogonal plane waves (plane OPW for orthogonalized waves ) was introduced by Convers Herring into the years 1940. The method made it possible to better include/understand the nature of structure of material semiconductors band such as silicon and germanium and was the first to explain in a theoretical way why silicon is an indirect material with gap. The development of this method deserves to be introduced because this one is the direct ancestor of the concept of pseudopotentiel.

Mathematical formalism

In a concrete way, method OPW is a general approach which aims at building functions of bases for the description of the states of valence. These functions are in the following way defined:

\ chi^ {OPW} _q (R) = \ frac {1} {\ Omega} \ sum_j u_j (R)

The functions uj are arbitrary but require to be localized around the cores. Preceding definition, it then that chiqOPW is quite orthogonal with all the functions uj i.e. for all uj:

=0

If the functions uj are correctly chosen, the expression (X) can then be seen as being the sum of two contributions; a softened part (software), i.e. not comprising nodes, and a localized part. The softened part can be represented easily by a combination of plane waves what was the objective of Herring like it specifies it itself. Antoncik, in an independent way, publishes the same year a similar approach. The method of the pseudopotential of Philips-Kleinman-Antoncik (PKA) is the first to show that the condition of orthogonality in the area of heart between the states of heart and valence acts as a repulsive potential which tends to oppoer with the gravitational nuclear potential felt by the electrons of valence. Generally, these two effects combine to form a potential slightly repulsive, the pseudopotential.

Mathematical development

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Advantages and disadvantages

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Model pseudopotential

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Pseudopotential to preserved standard

The introduction of the principle of pseudopotential to preserved standard represents the most significant opening in the treatment of the electrons of heart. The approach was developed by Hamann, Schlüter and Chiang and was follows little time afterwards by a very similar but established method in an independent way by Kerker.

Methodology

The first stage in the generation of a pseudopotential to preserved standard consists in carrying out an atomic calculation ab.initio all-electrons. The electronic configuration of the atom can be arbitrarily selected. Generally it is about the atom in a neutral state. Calculation is done at the beginning of the equation of Khon-Sham written in its radial form:

rR_ {nl} (R) = \ epsilon_ {nl} rR_ {nl} (R)

The function of real wave is then replaced by a pseudofonction of wave with which a potential is associated models (the pseudopotential) selected to correctly reproduce the state-owned properties of valence. The mathematical diagram which makes it possible to generate the pseudofonction of wave is not single. There exists from the mathematical point of view a certain freedom in the choice of the method as testify some the many publications which present different manner of making.

Conditions on the pseudofonction

To obtain a pseudopotential to the most effective possible preserved standard, the pseudofonction of wave must answer a list of precise criterion

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