Effective-potential approximation

Although the goal here is the extension of molecular electronic structure techniques into the realm of heavy elements, it is important that as much as possible of the reliability of light-element work be retained. The accuracy of the effective potential approximation can most readily be determined by careful comparisons of molecular EP results with those obtained from allelectron calculations. At the present time this can be done easily only for the nonrelativistic case. Although comparisons can be made with experimentally determined properties, it should be kept in mind that in general highly accurate valence wave functions are required. 

Other complications are associated with the partitioning of the core and valence space, which is a fundamental assumption of effective potential approximations. For instance, for the transition elements, in addition to the outermost s and d subshells, the next inner s and p subshells must also be included in the valence space in order to accurately compute certain properties (54). A related problem occurs in the alkali and alkaline earth elements, involving the outer s and next inner s and p subshells. In this case, however, the difficulties are related to core-valence correlation. Muller et al. (55) have developed semiempirical core polarization treatments for dealing with intershell correlation. Similar techniques have been used in pseudopotential calculations (56). These approaches assume that intershell correlation can be represented by a simple polarization of one shell (core) relative to the electrons in another (valence) and, therefore, the correlation energy adjustment will be... 

In the effective potential approximation Mg is isoelectronic with Be. But, as can be seen in Figure 2, the Mg REP is composed of three terms (s, p and d) with the s and p both repulsive. As a result, even though the correlation correction is almost as large as in Be the multi-determinant correction resulting from Equation 3 is only a tenth as big (see Table II). The discrepancy between values from references 2) and ( ) is due to large statistical or extrapolation error. Note that unlike Be one cannot make comparisons with experimental results without first taking core-valence... 

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. 

To rationalize the two-centre approximation , the effective potential is written as... 

The result of this approximation is that each mode is subject to an effective average potential created by all the expectation values of the other modes. Usually the modes are propagated self-consistently. The effective potentials governing die evolution of the mean-field modes will change in time as the system evolves. The advantage of this method is that a multi-dimensional problem is reduced to several one-dimensional problems. 

With the above definitions, there is no additional overall phase factor to be included in (27). Eqs. (24)-(27) are the CSP approximation.Like TDSCF, CSP is a separable approximation, using a time-dependent mean potential for each degree of freedom. However, the effective potentials in CSP... 

We have used the basis set of the Linear-Muffin-Tin-Orbital (LMTO) method in the atomic sphere approximation (ASA). The LMTO-ASA is based on the work of Andersen and co-workers and the combined technique allows us to treat all phases on equal footing. To treat itinerant magnetism we have employed the Vosko-Wilk-Nusair parametrization for the exchange-correlation energy density and potential. In conjunction with this we have treated the alloying effects for random and partially ordered phases with a multisublattice generalization of the coherent potential approximation (CPA). 

The applicability of the Born-Oppenheimer approximation for complex molecular systems is basic to all classical simulation methods. It enables the formulation of an effective potential field for nuclei on the basis of the SchrdJdinger equation. In practice this is not simple, since the number of electrons is usually large and the extent of configuration space is too vast to allow accurate initio determination of the effective fields. One has to resort to simplifications and semi-empirical or empirical adjustments of potential fields, thus introducing interdependence of parameters that tend to obscure the pure significance of each term. This applies in... 

A disadvantage of this technique is that isotopic labeling can cause unwanted perturbations to the competition between pathways through kinetic isotope effects. Whereas the Born-Oppenheimer potential energy surfaces are not affected by isotopic substitution, rotational and vibrational levels become more closely spaced with substitution of heavier isotopes. Consequently, the rate of reaction in competing pathways will be modified somewhat compared to the unlabeled reaction. This effect scales approximately as the square root of the ratio of the isotopic masses, and will be most pronounced for deuterium or... 

Mehler EL, Guarnieri F (1999) A self-consistent, microenvironment modulated screened coulomb potential approximation to calculate pH-dependent electrostatic effects in proteins. Biophys J 75 3-22. 

Although the pseudopotential is, from its definition, a nonlocal operator, it is often represented approximately as a multiplicative potential. Parameters in some chosen functional form for this potential are chosen so that calculations of some physical properties, using this potential, give results agreeing with experiment. It is often the case that many properties can be calculated correctly with the same potential.43 One of the simplest forms for an atomic model effective potential is that of Ashcroft44 r l0(r — Rc), where the parameter is the core radius Rc and 6 is a step-function. 

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