One-center-expansion

We will refer to this as the one center expansion. The components pRx are easily calculated in LMTO and are known on a radial mesh. In the interstitial region we choose to expand the charge density in the SSW s xp) and their first two energy derivatives. 

The first step is to use tp) and tp) to create a fitted charge density that has the same value and slope as the true charge density on the hard core spheres. This is thus a continuous, differentiable fit. As input we need the value and slope of the charge density on the a spheres. This is directly obtainable form the one center expansion. 

Typically we fit up to the / = 3 components of the one center expansion. This gives a maximum of 16 components (some may be zero from the crystal symmetry). For the lowest symmetry structures we thus have 48 basis functions per atom. For silicon this number reduces to 6 per atom. The number of random points required depends upon the volume of the interstitial region. On average we require a few tens of points for each missing empty sphere. In order to get well localised SSW s we use a negative energy. 

At the restoration stage, a one-center expansion in the spherical harmonics with numerical radial parts is most appropriate both for orbitals (spinors) and for the description of external interactions with respect to the core regions of a considered molecule. In the scope of the discussed two-step methods for the electronic structure calculation of a molecule, finite nucleus models and quantum electrodynamic terms including, in particular, two-electron Breit interaction may be taken into account without problems [67]. 

One-center expansion was first applied to whole molecules by Desclaux Pyykko in relativistic and nonrelativistic Hartree-Fock calculations for the series CH4 to PbH4 [81] and then in the Dirac-Fock calculations of CuH, AgH and AuH [82] and other molecules [83]. A large bond length contraction due to the relativistic effects was estimated. However, the accuracy of such calculations is limited in practice because the orbitals of the hydrogen atom are reexpanded on a heavy nucleus in the entire coordinate space. It is notable that the RFCP and one-center expansion approaches were considered earlier as alternatives to each other [84, 85]. 

All molecular spinors 4>p can be restored as one-center expansions in the cores using the nonvariational one-center restoration (NOCR) scheme [26, 27, 90, 92, 93, 94, 19, 97, 98] that consists of the following steps ... 

The results obtained with the one-center expansion of the molecular spinors in the T1 core in either s p, s p d or s p d f partial waves are collected in Table 4. The first point to notice is the difference between spin-averaged SCF values and RCC-S values the latter include spin-orbit interaction effects. These effects increase X by 9% and decrease M by 21%. The RCC-S function can be written as a single determinant, and results may therefore be compared with DF values, even though the RCC-S function is not variational. The GRECP/RCC-S values of M indeed differ only by 1-3% from the corresponding DF values [89, 127] (see Table 4). 

The scientific interests of Huzinaga are numerous. He initially worked in the area of solid-state theory. Soon, however, he became interested in the electronic structure of molecules. He studied the one-center expansion of the molecular wavefunction, developed a formalism for the evaluation of atomic and molecular electron repulsion integrals, expanded Roothaan s self-consistent field theory for open-shell systems, and, building on his own work on the separability of many-electron systems, designed a valence electron method for computational studies on large molecules. 

For covalently bonded atoms the overlap density is effectively projected into the terms of the one-center expansion. Any attempt to refine on an overlap population leads to large correlations between p>arameters, except when the overlap population is related to the one-center terms through an LCAO expansion as discussed in the last section of this article. When the overlap population is very small, the atomic multipole description reduces to the d-orbital product formalism. The relation becomes evident when the products of the spherical harmonic d-orbital functions are written as linear combinations of spherical harmonics ( ). 

Some time ago the one-center expansions (OCE) seemed to be prom-9... 

Figure 4.2. The relative error of the expansion of (a) the slope matrix using a one-center Taylor expansion and (b) using a two-center expansion.The error is defined in Eq. (4.35) and the expansion center is (0,0) for the one center expansion, and (0,0) and (-10,0) for the two-center expansion.

In the Bom-Oppenheimer approximation, the nuclei are fixed and have zero momenta. So the momentum density IT(p) is an intrinsically one-centered function whereas p(r) is a multi-centered function. Thus one-center expansions in spherical harmonics work well for one-electron momentum densities [51-53]. The leading term of such an expansion is the spherically averaged momenmm density IIq p) defined by... 

Molecular calculations. Molecular relativistic ab initio DF codes with electron correlation are still in development (see, for example. Refs. 86,87 and the corresponding chapters in this issue). Correlation effects are included there at the Cl [88], MBPT (the seccmd order Moller-Plesset, MP2 [89,90]), or the CCSD levels [91,92]. They are too computer time intensive and still not sufficiently economic to be applied to the heaviest element systems in a routine manner, especially to those studied experimentally. DF molecular codes, some without correlation, were recently used for small molecules of the heaviest elements. The main aim of those calculations was to study relativistic and correlation effects on some model systems like lllH, 117H, 113H, (113)2 or II4H4 [93-99]. Some pioneer calculations by PyykkO for Rfitt and SgHe using the one-center expansion DF method should also be mentioned here [100-102]. 

Vertical excitation energies of the singlet and triplet states and A, resulting from transitions of the 5ai electron to the orbitals 4s through 7s, 4p to 6p, 3d, and 4d, were calculated by a one-center expansion approximation [17]. 

Methods DHF, Dirac-Hartreer-Fock DHFS, Dirac-Hartree-Fock-Slater HF, Hartree-Fock OCE, one-center expansion MS, multiple-scattering DV, discrete-variational QR, quasirelativistic INDO, intermediate neglect of differential overlap WHT, Wolfsberg-Helmholz QR-EHT, quasirelativistic two-component extended Httckel EHT, extended Hllckel. 

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... 

In the case of atoms and molecules with central symmetry, in which case a one-center expansion is feasible, these equations are solved numerically. In the case of molecules with no central symmetry and in the case of crystals, this procedure is not possible and one must expand the molecular orbitals (MOs) or crystal orbitals (COs) as a linear combination of some basis functions (LCAO expansion). This was first conducted for molecules by Malli and Oreg< and applications can be found for diatomic or linear molecules, the only exception being the HjCO molecule, which was treated by Aoyama et... 

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