The pseudopotential concept

Heine V 1970. The Pseudopotential Concept. Solid State Physics 24 1-36. 

The first explanation and use of such a pseudopotential is due to Heilman5 (1935) who used it in atomic calculations. More recently the pseudopotential concept was reformulated by Phillips and Kleinman7 who were interested in its application to the solid state.8-10 Research in both solid- and liquid-state physics with pseudopotentials was reviewed by Ziman,11 and work in the fields of atomic spectroscopy and scattering has been discussed by Bardsley.12 For an earlier review on applications to the molecular environment the reader is referred to Weeks et a/.13 In this article we shall concentrate on molecular calculations, specifically those of an ab initio nature. Our objective in Section 2 has been to outline the theoretical origins of the pseudopotential approximation, and in Section 3 we have described some of the techniques which have been used in actual calculations. Section 4 attempts to present results from a representative sample of pseudopotential calculations, and our emphasis has been to concentrate on particular molecules which have been the subjects of investigation by the various approaches, rather than to catalogue every available calculation. Finally, in Section 5, we have drawn some conclusions on the relative merits of the different methods and implementations of pseudopotentials. Some of the possible future developments are outlined in the context of the likely progress in quantum chemistry. 

The pseudopotential concept was advanced a long ago [1] and is based on the natural energetic and spatial separation of core and valence electrons. The concept allows a significant reduction in computational efforts without missing the essential physics of phenomena provided the interaction of core and valence electrons is well described by some effective (model) Hamiltonian. Traditionally, pseudopotentials are widely used in the band structure calculations [2], because they allow convenient expansions of the wavefunctions in terms of plane waves suited to describing periodical systems. For molecular and/or nonperiodical systems, the main advantage of pseudopotentials is a... 

The most straightforward and formally sound route to account for non-electrostatic effects is provided by the use of the pseudopotential concept. As shown by Phillips and Kleinman63, the orbitals in a selected subsystem (pseudoorbitals - 4>iK ) can be obtained from the following equation involving the pseudo-Hamiltonian (HPK) ... 

Hansen JP, McDonald IR (1986) Theory of Simple Liquids. Academic Press, London Heine V (1970) The pseudopotential concept. Solid State Phys 24 1-37 Heilman H (1937) Einfuhrung in die (Jrrantrrmchemie. Deuticke, Leipzig... 

The pseudopotential concept has strongly evolved together with its applications, and so many models have been developed, that it is not always easy to recover the basic ideas and assumptions. 

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... 

As discussed in the chapter on the pseudopotential concept, an extension of the Heine-Animalu pseudopotentials was provided by Bachelet, Hamann and Schliiter, who calculated norm-conserving ionic pseudopotentials from atomic calculations for all the elements, including relativistic effects, and fitted their results with a relatively small set of analytic functions, the parameters of which were tabulated. The resulting matrix elements - - ... 

Implementations have been realized using Gaussian functions (GTO s) ([38, 39] and Slater-type orbitals (STO s) [5, 40, 41], and numerical basis sets [42, 43, 44]. The auxiliary basis may be avoided by the use of a purely numerical representation of the potential on a grid (usually called DVM - Discrete Variational Method [45, 5]), by certain approximations for the potential (Multiple Scattering concept within the so-called mufl5n-tin approximation - [46]), the linear combination of muffin-tin orbitals [47, 3], and in connection with the pseudopotential concept the application of plane-wave basis expansions - see, e.g.. Ref. [112]. 

V. Heine, The pseudopotential concept. Solid State Phys. 24, 1-36 (1970). 

The traditional valence-only MO schemes are Extended Hiickel and CNDO with its subsequent modifications. Present-day computing facilities make it possible to move one step further, to the ab initio treatment of valence electrons through the use of pseudopotential (PP) methods. The essentials of such methods will be illustrated in the following, through a description of the NOCOR formulation3, which will then be used for extensive calculations on sulphoxide and sulphone systems. The general concepts exposed in the foregoing sections will be illustrated by many examples. 

The EPM required some measured data to determine the Fourier coefficients of the pseudopotential. However, the most modem approaches follow the Fermi [5] concept of developing a pseudopotential to yield a wave function without nodes that coincides with the all-electron atomic wave function outside the core and is still normalized. Several methods were developed [16-19] in the 1970s and 1980s, and new methods for constructing useful pseudopotentials continue to appear in the literature. The applications discussed here are mostly based on the pseudopotentials developed using the approach described in Ref. [19]. The important point to empha-... 

Explain the concept of a pseudopotential. Aluminium is fee with a lattice constant of a = 7.7 au. It is well described by an Ashcroft empty core pseudopotential of core radius 1.1 au. Show that the lattice must be expanded by 14% for the 2n/a(200) Fourier component of the pseudopotential to vanish. 

Subsequent cellular methods, on which there is an enormous literature, will not be described here. We shall, however, need to introduce- certain ideas, particularly that of the pseudopotential. We begin by introducing the concept of the muffin-tin potential due to Ziman (1964a). This is illustrated in Fig. 1.9. The tight-binding approximation is appropriate for states with energies below the muffin-tin zero ( bound bands in Ziman s notation). If the energy is above the... 

Energy bands and a number of properties have been described here in terms of L( A() theory and matrix elements given by formulae such as Eq. (20-6). We turn now to the origin of those formulae and to a description of the electronic structure that proves useful for other properties. The formulae for the matrix elements will in fact be obtained from transition-metal pseudopotential theory, but the principal results can be obtained from the theory of Miiflin-Tin Orbitals, which we discu.ss first. Moreover, one of the central concepts of Muffin-Tin Orbital theory is necessary for using transition metal pseudopotential theory to obtain the formulae for the interatomic matrix elements. The analysis in this section and the next is somewhat analogous to the use of free-clectron theory to obtain the form and estimates of the magnitudes of the matrix elements used in the LCAO theory, and here the consequences arc just as rich. 

The concept of atomic or ionic size is one that has been debated for many years. The structure map of Figure 1 used the crystal radii of Shannon and Prewitt and these are generally used today in place of Pauling s radii. Shannon and Prewitt s values come from examination of a large database of interatomic distances, assuming that intemuclear separations are given simply by the sum of anion and cation radii. Whereas this is reasonably frue for oxides and fluorides, it is much more difficult to generate a self-consistent set of radii for sulfides, for example. A set of radii independent of experimental input would be better. The pseudopotential radius is one such estimate of atomic or orbital size. 

The first attempts for using pseudopotential concepts within the FSGO model have been made by Barthelat and Durand " between 1972 and 1976 they took... 

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.

The concept of quasi-free conduction electrons implies that their scattering by the ion core potential in the solid is rather weak. From here the modem theory of the pseudopotential has been developed. This theory shows that it is possible to reproduce the scattering of electron waves by replacing the deep potential at each site of the ionic core by a very much weaker effective potential, the pseudopotential. Thus the total pseudopotential in the metal or the semiconductor, which the conduction electrons feel, is fairly uniform, and the replacement of the real potential by the pseudopotential is a perfectly rigorous procedure. Furthermore, the fact that the total pseudopotential is fairly flat means that one can apply perturbation theory in order to calculate electron energies, cohesion, optical properties, etc. [20]. 

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 ... 

As an illustration of the usefulness of the pseudopotentiai concept in terms of perturbation theory, let us perform this first order calculation. The pseudopotential contains the electron potential V, and a repulsive term from the core states. The electron potential V, is split up in two contributions one term due to the ions with their core electrons, and a second term due to the other valence electrons, the distribution of which remains to be calculated ... 

It is explained elsewhere in this Volume that the self-consistent charge density and ionic potential corresponding to a given configuration of atoms not only determine the system s total energy - but also provide the fo qe gacting on all atoms through the theorem of Hellmann-Feynman (HF). The details of application in the context of DF, and u jJ.n the concept of the pseudopotential, are discussed elsewhere it will be examined... 

These inconveniences may be partly overcome by using another representation of atom Y, for instance by introducing a pseudohalogen in a semi-empirical computation. In this case the properties of the pseudohalogen may be adjusted to mimic the actual Y atom. This approach is close to the use of a pseudopotential on atom Y, which has been proposed as an efficient solution to this problem. Nevertheless, the extraction of the pseudopotential may not be very familiar to a nonspecialist and it may be difficult to reparametrize this pseudopotential if one wants to adjust the representation of the classical atom to a nonstandard local situation. It is the reason why we looked for another solution which does not require special knowledge other than the standard concepts of quantum chemistry. 

An early method of describing electrons in crystals was the method of nearly free electrons we shall refer to it as the NFE model. In this the potential energy V(x, y, z) in (6) is treated as small compared with the electron s total energy . This is, of course, never the case in real crystals the potential energy near the atomic core is always large enough to produce major deviations from the free-electron form. Therefore, until the introduction of the concept of a pseudopotential , it was thought that the NFE model was not relevant to real crystalline solids. 

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