Atomic orbital orthonormal

We also know that exact atomic orbitals are orthonormal, that is, Sa = 1 and = 0 for an LCAO. If we assume that orthonormality is canied from an LCAO into the molecular orbital, then S = I and, from Eq. (7-17),... 

Looking back, 1 seem to have made two contradictory statements about the basis fiinctions Xt used in the PPP model. On the one hand, I appealed to your chemical Intuition and prior knowledge by suggesting that the basis functions should be j garded as ordinary atomic orbitals of the correct symmetry (i.e. 2p orbitals). On the other hand, 1 told you that the basis functions used in such calculations are taken to be orthonormal and so... 

The variational problem is to minimize the energy of a single Slater determinant by choosing suitable values for the MO coefficients, under the constraint that the MOs remain orthonormal. With cj) being an MO written as a linear combination of the basis functions (atomic orbitals) /, this leads to a set of secular equations, F being the Fock matrix, S the overlap matrix and C containing the MO coefficients (Section 3.5). 

An X-ray atomic orbital (XAO) [77] method has also been adopted to refine electronic states directly. The method is applicable mainly to analyse the electron-density distribution in ionic solids of transition or rare earth metals, given that it is based on an atomic orbital assumption, neglecting molecular orbitals. The expansion coefficients of each atomic orbital are calculated with a perturbation theory and the coefficients of each orbital are refined to fit the observed structure factors keeping the orthonormal relationships among them. This model is somewhat similar to the valence orbital model (VOM), earlier introduced by Figgis et al. [78] to study transition metal complexes, within the Ligand field theory approach. The VOM could be applied in such complexes, within the assumption that the metal and the... 

If the constituent atomic orbitals %p have been orthonormalized as discussed earlier in this chapter, the overlap integrals <%vlXg> reduce to S v. 

Suppose that, instead of writing the secular determinant from an n x n array of atomic orbitals, we use an n x n array of n orthonormal, linear combinations of the basis set orbitals. Suppose, furthermore—and this is the key—we require these linear combinations to be SALCs, that is, each one is required to be a function which forms a basis for an irreducible representation of the point group of the molecule. Then, as shown in Chapter 5, all integrals of the types... 

Therefore, the dependence on the coefficients does not enter the gradient expression not for fixed orbitals, which is the classical Valence Bond approach and not for optimised orbitals, irrespective of whether they are completely optimised or if they are restricted to extend only over the atomic orbitals of one atom. If the wavefimction used in the orbital optimisation differs, additional work is required. This would apply to a multi-reference singles and doubles VB (cf. [20,21]). Then we would require a yet unimplemented coupled-VBSCF procedure. Note that the option to fix the orbitals is not available in orthogonal (MO) methods, due to the orthonormality restriction. 

In the CNDO/2 approach, only valence electrons are treated explicitly by choosing a minimum set of basis functions (atomic orbitals) for the molecule in its ground state. The main approximations are (1) Basis functions n are treated as an orthonormal set i.e., the overlap integrals are put to zero unless < u =r, in which case they are unity. (2) All two electron integrals which involve overlapping charge densities between... 

We have outlined above the procedure for the construction of orthonormal molecular-orbital and atomic-orbital Gel fand states and for the conversion of the latter to the non-orthogonal valence bond states. We require, in addition, a freeon Hamiltonian to compute the spectra of the several polyene systems. For this we employ the freeon, reduced Hiickel-Hubbard Hamiltonian which has the following form ... 

The integrals are calculated in terms of the atomic orbitals (AOs) and are subsequently transformed to the orthonormal basis. In some cases it may be more efficient to evaluate the expressions in the nonorthogonal AO basis. We return to this problem when we consider the calculation of the individual geometry derivatives. For the time being we assume that the Hamiltonian is expressed in the orthonormal molecular orbital (MO) basis. The second-quantized Hamiltonian [Eq. (8)] is a projection of the full Hamiltonian onto the space spanned by the molecular orbitals p, i.e., the space in which calculations are carried out. 

Multiconfiguration SCF (MC-SCF-LCAO-MO). If 4>lt , but for the best O then the variational principle, used simultaneously on both the a (the Cl coefficients) and the % (the atomic orbitals), will ensure that the will overlap as much as possible. 

The variational principle can be used to estimate the energy. If only the first-nearest neighbor interactions and an orthonormal (Eq. 5.15) set of atomic orbitals are considered, substitution of Eqs. 5.32 and 5.33 into Eq. 5.5 yields ... 

The general spherical harmonics are familiar, in low order, as the mutually orthonormal angular components of valence atomic orbitals. Now, the sufficient number of these functions to provide basis functions for the regular representations of the molecular point groups, in... 

These functions have the appropriate transformation properties for applications in calculation, as illustrated on the CDROM in the BonusPack , where sets of Hiickel theory calculations can be found for the energy spectra of regular orbits of I and Ih point symmetry decorated with His atomic orbitals. It is to be noted that, while the continuous functions are mutually orthonormal on the unit sphere, this property is not maintained in their discrete samplings on the 60 and 120-vertex orbit cages, and so a further orthogonalization transformation is required to restore orthogonality. 

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