One centre expansion

Central Field Model of Tetrahedral and Octahedral Molecules.—The idea is very simple, and has long been exploited in the sense of one-centre expansions of molecular wave functions in a molecule like CH4. However, to exemplify the way the density description can afford answers to questions (i)—(iii) above, we take the model literally in which, in methane for example, we smear the protons uniformly over the surface of a sphere of radius R, equal, in the methane example to the C—H bond length. 

To conclude this short description of the Dirac-Fock one-centre expansion method (a more extensive presentation can be found in Ref 45) we list in the table below most of the model molecular systems computed with that method and the main conclusions drawn from these calculations (see Table 7.3 in Ref 2 for a full list of references to the results summarized here). 

One-centre expansions Pyykkb and Desclaux performed ab initio one-centre expansions of the molecular spinors of small molecular species using finite difference methods. These studies included diatomic hydrides [191,192] and systematic investigations of chemical trends in the properties of hydrides which placed the heavy central atom in tetrahedral [193] and octahedral [194] environments. These calculations were amongst the first to enable the invesiga-tion of the likely chemical properties of some of the superheavy elements at a plausible level of theory. 

By means of the one-centre expansion (2.4) of the Bloch sum of MTO tails, the required tail cancellation is seen to occur if... 

Fig.5.2. The convergence area for the expansion theorem (5.14) (hatched area), and the one-centre expansion (5.29) (cross-hatched area)...

Below we show that as an alternative to the multi-centre expansion (5.28), the wave function may be written in terms of a one-centre expansion of the form... 

The LCMTO secular matrix is now simply obtained by inserting the one-centre expansion (5.29) into the matrix (5.37). We find... 

The one-centre expansion (6.15) is specialised to the case where R = 0, and is valid inside the atomic sphere centred at the origin. It may be used to derive the LMTO equations and with the normalisation implied by (6.11) it is consistent with the secular matrices (5.46,47) in the ASA. In linear methods in band theory [6.2] Andersen presented the one-centre expansion in the form (6.15) and derived the LMTO formalism from that assumption. His LMTO formalism is equivalent to that presented here apart from the normalising factor [/S/2 (- -1)] 1 appearing in the definition (6.11) of the energy-dependent muffin-tin orbital. 

The Bloch sum of muffin-tin orbitals x (r) is most conveniently expressed by the one-centre expansion (6.15) valid inside the sphere at the origin... 

Contrasting with the one-centre expansion for the inter-particle distance which always terminates when used with AOs because of the orthogonality properties of the spherical harmonics of the first kind. 

Turning to molecules of the form XH, also considered by Allan et al., where X ranges from Be to F, then the one-centre expansion model of the writer (March, 1952) guides the scaling, to yield... 

In principle, any molecular wavefunction can be expanded in terms of a complete set of functions centred at any convenient point in space. This approach, which is referred to as the one-centre expansion method, the central-field approximation or the united-atom method, goes back to the earliest days of atomic and molecular physics. 

The vast majority of contemporary molecular calculations overcome the convergence problems of the one-centre expansion method, which are associated with the description of off-centre nuclei, by employing basis functions located on a number of centres usually coinciding with the nuclei in the molecule. Rather accurate calculations for molecules containing more than one non-hydrogenic nucleus can be performed by using a multi-centre basis set. The use of such basis sets does, however, give rise to new problems. 

The one-centre expansion method has already been discussed in Section IV. A. Here the use of the one-centre expansion method in calculations for large molecules which take account of electron correlation effects will be briefly discussed. ... 

Pyykko (1979b) used the Dirac-Hartree-Fock one-centre expansion method for the monohydrides to calculate relativistic values for the lanthanide and actinide contraction, i.e. 0.209 A for LaH to LuH and 0.330A for AcH to LrH. The corresponding nonrelativistic value derived from Hartree-Fock one-center expansions for LaH and LuH is 0.191 A, i.e., for this case 9.4% of the lanthanide contraction is due to relativistic effects. The experimental value of 0.179 A would suggest a correlation contribution of-14.4% to the lanthanide contraction if one assumes that the relativistic theoretical values are close to the Dirac-Hartree-Fock limit, which is certainly not true for the absolute values of the bond lengths themselves. Moreover, it is well known that for heavy elements relativistic and correlation contributions are not exactly additive. Corresponding nonrelativistic calculations for AcH and LrH have not been performed and experimental data are not available to determine relativistic and electron correlation effects for the actinide contraction. Table 8 summarizes values for the lanthanide and actinide contraction derived from theoretical or experimental molecular bond lengths. It is evident from Ihese results... 

Desclaux, J.R, 1983b, Diiac-Fock one-centre expansion method, in Relativistic Effects in Atoms, Molecules and Solids, NATO ASI Series, Series B Physics, Vol. 87, ed. G.L. Malli (Plenum, New York) p. 213. 

The one-centre expansion of the MEP is not adequate, however, for the large number of chemical applications we mentioned in Sect. 4. In fact the expansion theorem holds for points r lying outside a sphere containing all the elements of the charge distribution. In molecules, this condition is never formally satisfied, because Q r) has an exponential decay. The difference between V (r) and the exact multipole expansion of K (r) is generally called the penetration term ... 

The description of the photoionization process by means of a method based on the Density Functional Theory (DFT) is reviewed. The present approach is based on a basis set expansion in B-spline functions, which are particularly suited to deal with the boundary conditions of the continuum states. Both Kohn-Sham (KS) and its extension to the Time Dependent (TD-DFT) formalism are considered. The computational aspects of the method are described the implementations for atoms, for molecules in One Centre Expansion (OCE) and for molecules with the Linear Combination of Atomic Orbital (LCAO) scheme. The applications of the method are discussed, from atoms to large fullerenes, with comparison with available experimental data. 

In this chapter we describe the implementations of the method, underlying the most important and original features. In particular we will try to describe how it has been possible, starting from atomic calculation, to extend the method to a molecular One-Centre Expansion (OCE) and to a more elaborate LCAO multicentre formulation particularly suited for molecular systems of moderate size. 

Apart from the slow convergence of the one-centre expansion there are regions of overlap where it is a poor representation of the interaction. Ng et al. discuss the effect of charge overlap on the quadrupole-quadrupole interaction [50]. Using their parameterization I have drawn the ISM + 00 interaction for two CO2 molecules in the T-orientation, (0. = ir/2, 0. = 0). Curve A represents, J ... 

This formula will be called the "one-centre expansion of r" in terms of Gegenbauer poljmomials. It is a generalization of the so-called Neumami s expansion used in electrostatics. 

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