Some pseudopotential approaches

It is precisely the repulsive term in the core, which is responsible for the fact that the pseudopotential is an integral operator. Since such operators 

Some confusion might arise from the term valence energy, which means the energy associated with the valence electrons of the atom but in a solid the energies of these valence electrons constitute the conduction bands and/or the higher valence bands. 

One of the simplest early approximations is the so-called empty-core 

However, in the following sections we will not discuss this model ap- 

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. 

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Briefly summarizing the results presented in Tables 1-5, one could conclude that the pseudopotential approach is promising for simulations of large-scale systems. A special advantage appears to be presented by a combined approach when some region of a large system is considered all-electronically, whereas the rest is replaced with pseudoatoms. The latter will lead to a drastic reduction in computational efforts without the loss of accuracy. 

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. 

A lot of theoretical work has been done in order to explain the size dependent properties of semiconducting nanocrystals. These methods are primarily based on the effective mass approximation, pseudopotential approaches or the tight binding scheme. Each of these methods has certain advantages and disadvantages. We shall explore these methods in some detail in Section 11.5. 

This set of lectures reviews some of the many recent theoretical accomplishments in this area. Experimental results are discussed only in so far as they bear on a theoretical result. In addition, in order to limit the scope of this review, emphasis will be placed on a few theoretical techniques—primarily the pseudopotential approach. Specific prototype systems are considered to illustrate the accomplishments of the theory for semiconductors, insulators, and transition metals. Some details of the calculations and results will be given, but the reader should go to the original papers for more specifics. 

The pseudopotential approach was used in [28] and was also applied in the density-functional formalism [29]. For chemical problems, ab initio effective core potentials including some relativistic effects may be of practical use [30]. 

In a second approach of the reactivity, one fragment A is represented by its electronic density and the other, B, by some reactivity probe of A. In the usual approach, which permits to define chemical hardness, softness, Fukui functions, etc., the probe is simply a change in the total number of electrons of A. [5,6,8] More realistic probes are an electrostatic potential cf>, a pseudopotential (as in Equation 24.102), or an electric field E. For instance, let us consider a homogeneous electric field E applied to a fragment A. How does this field modify the intermolecular forces in A Again, the Hellman-Feynman theorem [22,23] tells us that for an instantaneous nuclear configuration, the force on each atom changes by... 

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-... 

W. Fickett in "Detonation Properties of Condensed Explosives Calculated with an Equation of State Based on Intermole-cular Potentials , LosAlamosScientific-LabRept LA-2712(1962), pp 38-42, reports that pseudopotential theories are obtd by an approach completely different from perturbation theories. The problem of defining a system of detonation products consisting of both solid carbon in some form and a fluid mixt of the remaining product species has been formally rearranged to a single fictitious substance with an extremely complicated compn- temp-dependent potential function , called the pseudopotential. The fictitious substance corresponding to this potential is clearly non-conformal with the components of the mixt... 

One interesting scheme based on density functional theory (DFT) is particularly appealing, because with the current power of the available computational facilities it enables the study of reasonably extended systems. DFT has been applied with a variety of basis sets (atomic orbitals or plane-waves) and potential formulations (all-electron or pseudopotentials) to complex nu-cleobase assemblies, including model systems [90-92] and realistic structures [58, 93-95]. DFT [96-98] is in principle an ab initio approach, as well as MP2//HF. However, its implementation in manageable software requires some approximations. The most drastic of all the approximations concerns the exchange-correlation (xc) contribution to the total DFT functional. 

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 delocalized (right-hand) side of Fig. 1.1 involves some form of calculation on the full lattice such as a band-theory calculation. Again, the Hartree-Fock wave function may be employed in an ab initio method or some approximate method such as Huckel band theory, or the local-exchange approximations employed leading to augmented-plane-wave or ab initio pseudopotential (PP) methods. As an alternative to band theory, the development of the ionic approach using pair potentials or modified electron-gas (MEG) theory has proved effective for certain crystalline species. 

Numerical integration schemes allow an opportunity to test the numerical nonempirical pseudopotentials without their fit by analytical functions, which can lead to a considerable reduction in computational efforts. Employment of atomic pseudopotentials only at some selected atoms of a system while treating the rest all-electronically makes an impression of the consistency and reliability of such a combined approach. The results obtained for the MgO clusters embedded in some effective pseudopotential surroundings demonstrate a promise of the approach for compensation of broken bond effects. Specifically, the approach offers a tool for a substantial reduction of the artificially introduced nonequivalence of partial densities and the effective charges for atoms equivalent in the lattice. It is worth to mention that our approach can be modified further in many ways because numerical integration schemes can be easily applied/adapted even in those cases where the analytical methods become too complicated. 

Most plane wave calculations use ultrasoft pseudopotentials (USPP),15,28 which describe the core electrons of atoms in a mathematically efficient form that greatly reduces the computational cost associated with heavy atoms. An increasing number of calculations used the projector augmented wave (PAW) approach instead.28 In most circumstances where both approaches can be used, the differences between USPP and PAW calculations are minor. Some exceptions to this observation include transition metals with large magnetic moments (e.g., Fe) and alkali metals.28... 

The unknown functions, Cij r),riij r) in (57) and (58) need to be parameterized in some way. In a first attempt we have chosen gaussians with variance and amplitude as new variational parameters [16]. This form was shown to be suitable for homogeneous electron gas [13]. Approximate analytical forms for ij r) and r]ij r), as well as for the two-body pseudopotential, have been obtained later in the framework of the Bohm-Pines collective coordinates approach [14]. This form is particularly suitable for the CEIMC because there are no parameters to be optimized. This trial function is faster than the pair product trial function with the LDA orbitals, has no problems when protons move around and its nodal structure has the same quality as the corresponding one for the LDA Slater determinant [14]. We have extensively used this form of the trial wave function for CEIMC calculations of metallic atomic hydrogen. 

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