Pseudopotential theory

Abell G C 1985. Empirical Chemical Pseudopotential Theory of Molecular and Metallic Bonding Physical Review B31 6184-6196. 

L. Szasz, Pseudopotential Theory of Atoms and Molecules John Wiley Sons, New ork (1985). 

The basic idea of the pseudopotential theory is to replace the strong electron-ion potential by a much weaker potential - a pseudopotential that can describe the salient features of the valence electrons which determine most physical properties of molecules to a much greater extent than the core electrons do. Within the pseudopotential approximation, the core electrons are totally ignored and only the behaviour of the valence electrons outside the core region is considered as important and is described as accurately as possible [54]. Thus the core electrons and the strong ionic potential are replaced by a much weaker pseudopotential which acts on the associated valence pseudo wave functions rather than the real valence wave functions (p ). As... 

Szasz L (1985) Pseudopotential theory of atoms and molecules, Wiley, New York... 

See also Detonation, Longuet-Higgins Theory and Detonation, Pseudopotential Theories... 

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... 

Szadz,L. and Brown,L. (1976), New formulation of the pseudopotential theory for atoms with two valence electrons , J.Chem.Phys. 65, 1393... 

How is the molecular geometry of the liquid affected by the presence of an excess electron Two extreme cases may be immediately distinguished the free electron case and the localized electron case. The free electron may be adequately described by using a plane wave state. The physical significance of such a simple description is not trivial and will be clarified after the discussion of the pseudopotential theory. Here we... 

Up to this point we have used rather crude terminology to describe the free electron state in simple liquids, and we are now faced with the following dilemma is it possible to describe a free excess electron in a liquid in terms of a plane wave, neglecting the effect of the core electrons Such a simple description is justified by the pseudopotential theory introduced by Phillips and Kleinman (37) and by Cohen and Heine (5). 

Clearly any attempt to base FeK on such molecularly defined cores defeats the aims of pseudopotential theory. However, the approximate invariance of atomic cores to molecule formation implies that, of the total of Na electrons which could be associated with the centre A in an atomic calculation, nx are core electrons and n K will contribute to the molecular valence set. Thus we can define a one-centred Fock operator ... 

In this simple case there is no advantage to the pseudopotential calculation (the 3-21G( ) geometry is actually better ), but more challenging calculations on very-heavy-atom molecules, particularly transition metal molecules, rely heavily on ab initio or DFT (Chapter 7) calculations with pseudopotentials. Nevertheless, ordinary nonrelativistic all-electron basis sets sometimes give good results with quite heavy atoms [64]. A concise description of pseudopotential theory and specific relativistic effects on molecules, with several references, is given by Levine [65]. Reviews oriented toward transition metal molecules [66a,b,c] and the lanthanides [66d] have appeared, as well as detailed reviews of the more technical aspects of the theory [67]. See too Section 8.3. 

Although historically pseudopotentials appear in numerous disguises on which we do not dwell here, the modem derivation of the pseudopotential theory [36] is based... 

We conclude here that the matrix elements of d/dx are constant. This conclusion will follow also from the pseudopotential theory of covalent bonding in Chapter 18, and was found to Idc true of the matrix elements in the nonlocal pscudopotential calculations of Chclikowsky and Cohen (quoted by Phillips,... 

We might also expect the matrix elements of djdx to scale inversely with d among the homopolar semiconductors (and correspondingly, for the matrix elements of X to scale with d) and this is in fact predicted by the pseudopotential theory of Chapter 18. However, that does not describe the trends in Xi(0) well, and in Section 4-C, we shall allow the proportionality constant to vary from row to row in the Periodic Table. 

It is possible in principle to calculate all of these modes from the theory of the electronic structure, which is equivalent to the calculation of all the force constants. Indeed we will see that this is possible in practice for the simple metals by using pseudopotential theory. In covalent solids, even within the Bond Orbital Approximation, this proves extremely difficult because of the need to rotate and to optimize the hybrids, and it has not been attempted. The other alternative is to make a model of the interactions, which reduces the number of parameters. The most direct approach of this kind is to reduce the force constants to as few as possible by symmetry, and then to include only interactions with as many sets of neighbors as one has data to fit- for example, interactions with nearest and next-nearest neighbors. This is the Born-von Karman expansion, and it has somewhat surprisingly proved to be very poorly convergent. This simply means that in all systems there arc rather long-ranged forces. 

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... 

This matrix clement is the Fourier component of the pseudopotential with wave number equal to the difference q. In the more complete pseudopotential theory of Appendix D, the pseudopotential becomes an operator, so that IT (r) and c cannot be interchanged in the final step of Eq. (16-2), and the matrix element depends upon k also. This is not a major complication, but we shall utilize the simpler form given in the last step of Eq. (16-2), called the local approximation to the pseudopotential. 

This completes the specification of the pseudopotential as a perturbation in a perfect crystal. We have obtained all of the matrix elements between the plane-wave states, which arc the electronic states of zero order in the pseudopotcntial. We have found that they vanish unless the difference in wave number between the two coupled states is a lattice wave number, and in that case they are given by the pseudopotential form factor for that wave number difference by Eq. (16-7), assuming that there is only one ion per primitive cell, as in the face-centered and body-centered cubic structures. We discuss only cases with more than one ion per primitive cell when we apply pseudopotential theory to semiconductors in Chapter 18. Tlicn the matrix element will be given by a structure factor, Eq. (16-17),... 

In the covalent solids, the Jones Zone gap should be identified with the principal optical absorption peak previously identified with LCAO interatomic matrix elements. Thus it allows a direct relation between the parameters associated with the LCAO and with the pseudopotential theories. It is best, however, to simplify the pseudopotential analysis still further before making that identification. 

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