True potential approximation

Presently, only the molecular dynamics approach suffers from a computational bottleneck [58-60]. This stems from the inclusion of thousands of solvent molecules in simulation. By using implicit solvation potentials, in which solvent degrees of freedom are averaged out, the computational problem is eliminated. It is presently an open question whether a potential without explicit solvent can approximate the true potential sufficiently well to qualify as a sound protein folding theory [61]. A toy model study claims that it cannot [62], but like many other negative results, it is of relatively little use as it is based on numerous assumptions, none of which are true in all-atom representations. 

The improvement came in the form of the coherent-potential approximation (CPA) (Soven 1967, Taylor 1967, Velicky et al 1968), which remedied the lack of self-consistency exhibited by the ATA. The crux of this approach is that each lattice site has associated with it a complex self-consistent potential, called a coherent potential (CP). The CP gives rise to an effective medium with the important property that removing that part of the medium belonging to a particular site, and replacing it by the true potential, produces, on average, no further scattering. Because the CPA is used for our discussion of chemisorption on DBA s, its mathematical formulation is given below. 

The propagation of the wavepacket is thereby reduced to the solution of coupled first-order differential equations for the parameters representing the Gaussian wavepacket, with the true potential being expanded about the instantaneous center of the wavepacket [i2(<),f(<)]. This propagation scheme is very appealing and efficient provided the basic assumptions are fulfilled. The essential prerequisite is that the locally quadratic approximation of the PES is valid over the spread of the wavepacket. This rules out bifurcation of the wavepacket, resonance effects, or strong an-harmonicities. 

Fig. 2.2 Actual molecules do not sit still at the bottom of the potential energy curve, but instead occupy vibrational levels. Also, only near qe, the equilibrium bond length, does the quadratic curve approximate the true potential energy curve...

Here R denotes the distance between the ion centres. The important condition is that the two spheres do not overlap. Equations (1.152)—(1.154) are approximate because of the implicit assumption that uniform potential (1.151) represents the true potential actually existing in the vicinity of the ion. In fact, this expression is perturbed by matching conditions on the boundary, which are neglected. 

No matter what degree of optimization can actually be achieved via this route, it must be remembered that vaginal delivery is only applicable to approximately 50% of the population. Thus it may be that the true potential of this route lies in the treatment of female-specific conditions, such as in the treatment of climacteric symptoms of the menopause etc., rather than more general applications such as insulin/peptide delivery. 

The ratio v /vq differs slightly from this harmonic ratio due to deviation of the true potential function from a quadratic form, as depicted in Fig. 1. A closer approximation to the solid curve can be had by adding cubic and higher anharmonic terms to U r), viz.,... 

This repulsive term tends to cancel the true potential of the core itself, a feature noted very early in the development of the pseudopotential theory that we have described. Ashcroft (1966) took advantage of this feature in proposing the empty-cote model of the pscudopotential. In this model, the repulsive term of Eq. (15-12) is combined with the Coulomb potential of a point ion and the core potential i , of Eq. (15-8) to give a potential due to each ion, which is approximated by... 

The free-clectron approximation described in Chapter 15 is so successful that it is natural to expect that any effects of the pseudopotential can be treated as small perturbations, and this turns out to be true for the simple metals. This is only possible, however, if it is the pscudopotential, not the true potential, which is treated as the perturbation. If we were to start with a frcc-electron gas and slowly introduce the true potential, states of negative energy would occur, becoming finally the tightly bound core states these arc drastic modifications of the electron gas. If, however, we start with the valence-electron gas and introduce the pscudopotential, the core states arc already there, and full, and the effects of the pseudopotential arc small, as would be suggested by the small magnitude of the empty-core pseudopotential shown in Fig. 15-3. 

Since Eq. (15) does not accurately represent the true potential-especially not if the trial model is relatively far away from the true minimum energy geometry — and because of the approximate nature of the transformation of Ag to AX, the calculated shifts of the atomic coordinates will not, in general, minimize the potential energy. Therefore the new model is used as input for another cycle of calculations, until the AXf are less than a prespecified value. It should be noted that the accuracy of Eq. (16) can be improved by including higher terms this would probably be outweighted by the increased amount of computer time needed. In the final cycle each Af is very nearly zero and only C-terms remain. The final potential can then be written as ... 

Still another way of using DFT, which does not depend directly on approximate solution of Kohn-Sham equations, is the quantification and clarification of traditional chemical concepts, such as electronegativity [6], hardness, softness, Fukui functions, and other reactivity indices [6, 175], or aromaticity [176]. The true potential of DFT for this kind of investigation is only beginning to be explored, but holds much promise. 

The only interaction in this model is a link-link repulsion it is short-range and of the order of a lattice edge. Actually, this approximation is a rather imperfect representation of reality. The true interaction contains simultaneously, a short-range repulsive interaction, or hard core, and an attractive part whose range is a little longer and which results from van der Waals forces. Experimentally, the fact that the mixing of the polymer with the solvent is endothermic is a manifestation of these attractive forces. The shape of the true potential is indicated in Fig. 4.5. 

The true potential function for a diatomic molecule departs from harmonicity, especially for large amplitude vibrations. It is useful to approximate an anharmonic potential using a Morse function (Figure 3)... 

It is known from quasi-relativistic calculations that this bare-potential approximation provides reliable results for valence electron properties. This is particularly true for the scalar-relativistic variants discussed in the next section [see, e.g.. Refs. [656,752]], but may be different for the truly two-component method including spin-orbit splitting [655]. For core-shell properties and for the spin-orbit splitting of high-angular-momentum orbitals this approximation will not be sufficient [643,659]. 

Figure 3.7 Harmonic oscillator approximation (dashed curve) to true potential energy curve e(/ ) in a diatomic molecule.

In ref. 151 the author studies the piecewise perturbation methods to solve the Schrodinger equation and the two form of this approach, i.e. the LP and CP methods. On each stepsize the potential is numerically approximated by a constant (in the case of CP) or by a linear function (in the case of LP). After that the deviation of the true potential from this numerical approximation is obtained by the perturbation theory. The main idea of the author is that an LP algorithm can be made computationally more efficient if expressed in a CP-like form. The author produces a version of order 12 whose the two main parts are a new set of formulae for the computation of the zeroth-order solution which replaces the use of the Airy functions, and an efficient way of obtained the formulae for the perturbation corrections. The main remark for this paper is that from our experience for these methods the computational cost is considerably higher than for the finite difference methods. 

Taking matrix elements of the hamiltonian between such states creates a secular equation which can be solved to produce the desired eigenvalues. Since the potential cannot be truly spherical throughout the WS cell, it is reasonable to consider it to be spherical within a sphere which lies entirely within the WS, and to be zero outside that sphere. This gives rise to a potential that looks like a muffin-tin, hence the name of the method Linearized Muffin-Tin Orbitals (LMTO). This method is in use for calculations of the band structure of complex solids. The basic assumption of the method is that a spherical potential around the nuclei is a reasonable approximation to the true potential experienced by the electrons in the solid. 

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