Use of Pseudopotentials

When core electrons are eliminated, the Hamiltonian for the valence electrons of an atom becomes 

The effective potentials normally used in analytic variational calculations are nonlocal potentials that involve angular projection operators that cannot be simply transferred into QMC calculations. In the earliest QMC calculations to use effective potentials, Hurley and Christiansen and Hammond et a 7 avoided this difficulty with the use of local potentials defined in terms of trial wavefunctions. The use of effective potentials is, by its nature, not exact and 

The results of calculations using effective core potentials of the several types may be compared with experimental measurements, but more useful comparisons can be made with all-electron calculations for the same systems. For example, in studying the use of effective core potentials in QMC calculations, Lao and Christiansen calculated the valence correlation energy for Ne and found excellent agreement with previous full-CI benchmark calculations. They recovered 98-100% of the valence correlation energy and could detect no significant error due to the effective potential approximation. 

The advantages of using pseudopotentials are dramatically illustrated by DQMC calculations for the Fe atom carried out by Mitas for all electrons, for a neon core pseudopotential, and for an argon core pseudopotential the relative computational effort for a fixed statistical uncertainty was in the order 6250, 60, and 1, respectively. Thus, the appeal of pseudopotentials is strong. Of course, the additional (systematic) uncertainty introduced with the use of pseudopotentials is a disadvantage. Additional work will undoubtedly resolve the relative advantages and disadvantages. 

A sampling of studies using effective potentials, model potentials, effective Hamiltonians, and related devices is given in Table The en- 

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The main problem related to the use of pseudopotentials in studies of solids under pressure is to make sure that the overlap of ionic cores does not increase significantly when interatomic distances decrease. The present study is certainly not affected by this potential pitfall since Ti-O distances typically change by no more than 0.1 A over the pressure range investigated. However, theoretical studies of fluorite and related phases at pressures of around 100 GPa should be performed with added caution. 

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. 

Local-density potentials greatly simplify the computational problems associated with defect calculations. In practice, however, such calculations still are very computer-intensive, especially when repeated cycles for different atomic positions are treated. In most cases the cores are eliminated from the calculation by the use of pseudopotentials, and considerable effort has gone into the development of suitable pseudopotentials for atoms of interest (see Hamann et al., 1979). 

A further simplication often used in density-functional calculations is the use of pseudopotentials. Most properties of molecules and solids are indeed determined by the valence electrons, i.e., those electrons in outer shells that take part in the bonding between atoms. The core electrons can be removed from the problem by representing the ionic core (i.e., nucleus plus inner shells of electrons) by a pseudopotential. State-of-the-art calculations employ nonlocal, norm-conserving pseudopotentials that are generated from atomic calculations and do not contain any fitting to experiment (Hamann et al., 1979). Such calculations can therefore be called ab initio, or first-principles. ... 

In practice, use of pseudopotentials typically does not lead to significant improvement in computational speed. This is because most integrals involving core basis functions are vanishingly small and can be eliminated prior to actual calculation. What pseudopotentials accomplish, however, is extension of the range of methods for elements for which all-electron basis sets are not available. 

For most recent developments in the use of pseudopotentials, useful papers are Hamann et al (1979) and Cohen (1982). Cottrell (1988) has given a very readable outhne of the present position. 

A proposal for the proper use of pseudopotentials in molecular orbital cluster model studies of chemisorption , J.Chem.Phys. 81, 3594... 

The Use of Pseudopotentials in Molecular Calculations arbitrary non-local potential ... 

Hinchliffe and Bounds review in detail the calculation of the electric and magnetic properties of molecules. Finally, the use of pseudopotentials in molecular calculations is extending the range of ab initio calculation to molecules containing heavy atoms, and Dixon and Robertson survey this rapidly growing field. 

Further uses of pseudopotentials are numerous. The most obvious (and rather widely known ones) are to continue with the PP or MP Hamiltonians for a widely understood combination of the core and valence shells and to apply standard ab initio techniques to electrons in the valence subspace only. We do no elaborate further on this as the hybrid nature of the pseudopotential methods is rather obvious from the above and its more specific applications in a narrower QM/MM hybrid context will be described later. 

Relativistic effects in heavy atoms are most important for inner-shell electrons. In ab initio and DFT calculations these electrons are often treated through relativistic effective core potentials (RECP), also known as pseudopotentials. This approach is sometimes called quasirelativistic, because it accounts for relativity effects in a rather simplified scalar way. The use of pseudopotentials not only takes into account a significant part of the relativistic corrections, but also diminishes the computational cost. 

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. Pseudopotenlial perturbation theory is an expansion in which the ratio W/Ey of the pseudopotential to the kinetic energy is treated as small, whereas for covalent solids just the reverse quantity, Ey/W, should be treated as small. The distinction becomes /wimportant 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 KIcinman (1959) nor in the more recent application of the Empirical Pseudopotenlial Method u.scd 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. 

This rule centers on the much-used concept of ionic size, a concept that has been refined, or rather been better defined in recent years by the use of pseudopotential radii (Zunger and Cohen, 1978). Thus, the use of pseudopotential radii (r,) to define atomic size more closely, and the iden-... 

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