LCAO expansion

All m oleciilar orbitals are com biiiations of the same set of atom ic orbitals they differ only by their LCAO expansion coefficients. HyperC hem computes these coefficients, C p. and the molecular orbital energies by requiring that the ground-state electronic energy beat a minimum. That is, any change in the computed coefficients can only increase the energy. 

The LCAO expansion coefficients are found from the matrix eigenvalue problem... 

The Extended Hiickel model treats all valence electrons within the spirit of the TT-electron model. Each molecular orbital is written as an LCAO expansion of the valence orbitals, which can be thought of as being Slater-type orbitals (to look ahead to Chapter 9). Slater-type orbitals are very similar to hydrogenic ones except that they do not have radial nodes. Once again we can understand the model best by considering the HF-LCAO equations... 

The size-dependence of the intensity of single shake-up lines is dictated by the squares of the coupling amplitudes between the Ih and 2h-lp manifolds, which by definition (22) scale like bielectron integrals. Upon a development based on Bloch functions ((t>n(k)), a LCAO expansion over atomic primitives (y) and lattice summations over cell indices (p), these, in the limit of a stereoregular polymer chain consisting of a large number (Nq) of cells of length ao, take the form (31) ... 

I means the valence state ionization potential for the atomic orbital n, stands for the core charge, and Cj, and Xm are coefficients and atomic orbitals in the LCAO expansion... 

Hence, through the LCAO expansion we have translated the non-linear optimization problem, which required a set of difficult to tackle coupled integro-differential equations, into a linear one, which can be expressed in the language of standard linear algebra and can easily be coded into efficient computer programs. 

Mulliken s formula for Nk implies the half-and-half (50/50) partitioning of all overlap populations among the centers k,l,... involved. On one hand, this distribution is perhaps arbitrary, which invites alternative modes of handling overlap populations. On the other hand, Mayer s analysis [172,173] vindicates Mulliken s procedure. So we may suggest a nuance in the interpretation [44] departures from the usual halving of overlap terms could be regarded as ad hoc corrections for an imbalance of the basis sets used for different atoms. But one way or another, the outcome is the same. It is clear that the partitioning problem should not be discussed without explicit reference to the bases that are used in the LCAO expansions. 

Which properties are least well determined by the variational method The basis functions in the LCAO expansion are either Slater orbitals with an exponential factor e r or gaussians, e ar2 r appears explicitly only as a denominator in the SCF equations thus matrix elements are of the form < fc/r 0i> these have the largest values as r->0. Thus the parts of the wavefunction closest to the nuclei are the best determined, and the largest errors are in the outer regions. This corresponds to the physical observation that the inner-shell orbitals contribute most to the molecular energy. It is unfortunate in this respect that the bonding properties depend on the outer shells. 

Schmidtke et al. [30] presented a simplified model of the K integrals, which retains the Stevens delocalization coefficients featuring in the LCAO expansion of the metal d-orbitals over the ligands but does not allow for differences in the radial parts of the metal function. More-center and ligand-centered contributions are neglected. In this approximation the K integrals can be expressed as follows ... 

Here xb2 and xe represent the coefficients of the b2 and e-type 3d-functions in the appropriate LCAO expansion. B and C are the spherical Racah parameters of the d-only model. The point made by Schmidtke et al. [30] is that small deviations of the d-only limit may indeed give rise to sizeable splittings. As an example parameter values xb2 = 1, te = 0.992 can reproduce level spacings of 200 cm 1 in agreement with experiment. The model also explains why the splitting increases from the more ionic chloro to the more covalent iodo substituent. 

Substituting into the Hartree-Fock equations Fiji = v. j/ (5.44) the Roothaan-Hall linear combination of basis functions (LCAO) expansions i//, = csi4>s (5.52) for the MO s 1jj gave the Roothaan-Hall equations (Eqs. 5.56), which can be written compactly as FC = SCe (Eqs. 5.57). 

Summary of the steps in a single-point Hartree-Fock (SCF) calculation using the Roothaan-Hall LCAO expansion of the MO s... 

Using the same type of LCAO expansion as for molecules (see eq. (2.8)), the crystal orbitals are expanded as Bloch sums of the basis function centred at site fi in cell / ... 

N+/is the number of unit cells of the polymer chain. The LCAO expansion coefficient matrices C(k) and the corresponding energy eigenvalues, e(k), are obtained by solving the eigenvalue problem for each value of k-... 

In Equation (2.6), we know q and associated energy E and the coefficients cn and ci2 of its LCAO expansion. Incorporating Equation (2.6) in Equation (2.5) gives... 

Similarly, expanding the KS potential in an LCAO expansion makes molecular density-functional calculations practical [9]. For metals and similar crystalline solids, it is best to expand the Kohn-Sham potential in momentum space via Fourier coefficients. For molecular solids various real-space method are under investigation. For molecules studied with the big, well-chosen Gaussian basis sets of quantum chemistry, it is undoubtedly best to expand the KS potential in linear-combination-of-Gaussian-type-orbital (LCGTO) form [10]. 

LCAO expansion of the MOs [15]. In the DV-Xa MO method based on the Hartree-Fock-Slater approach, the exchange-correlation potential is approximated by the simple Slater form [16] Vxc(r) = —3a 3p(r)/47i 1/3, where the coefficient a is a scaling parameter (fixed at 0.7 in the present study) and p(r) is the local electron density at a position r. The basis functions for the MO calculation consisted of atomic orbital wave eigenfunctions obtained in numerical form, which included the ls-6s, ls-5s, ls-6p, ls-4p, and ls-2p orbitals for Ba, Sr, Pb, Ti, and O ions, respectively... 

Inserting the LCAO expansions for %x and of the form of (10) in (11), the matrix element of the angular momentum operators, and hence the matrix element of hso on the left hand side of (11) is easily seen to be a linear combination of terms of the form... 

Band-theoretical method using a LCAO expansion of the delocalized Bloch orbital (often with integral approximations)... 

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 ( ). 

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