Calculating the orthogonalizing matrix

The standard analytic procedure involves calculating the orthogonal transformation matrix T that diagonalizes the mass weighted Hessian approximation H = M 2HM 2, namely... 

The orthogonalizing matrix S 1/2 is calculated from S the integrals S are calculated and assembled into S, which is then diagonalized ... 

Having specified a Hartree-Fock calculation on singlet HHe+, with H—He = 0.800 A (1.5117 bohr), using an STO-1G basis set, the most straightforward way to proceed is to now calculate all the integrals, and the orthogonalizing matrix S 1/2... 

Calculate the integrals Trs, Vrs for each nucleus, and the two-electron integrals (ru ts) etc. needed for Grs, as well as the overlap integrals Srs for the orthogonalizing matrix derived from S (see step 3). Note in the direct SCF method (Section 5.3) the two-electron integrals are calculated as needed, rather than all at once. 

The /rth fragment density matrix P (ip) is obtained from an ab initio calculation for the parent molecule M. within a local coordinate system. Vector (p Kr) represents the set of AOs of the parent molecule M., with reference to the local coordinate system. In a local coordinate system with axes aligned with those of the macromolecular coordinate system, vector represents the same sequence of AOs at the same nuclear centers. These two representations are related by the orthogonal matrix transformation T ... 

The valence-bond approach plays a very important role in the qualitative discussion of chemical bonding. It provides the basis for the two most important semi-empirical methods of calculating potential energy surfaces (LEPS and DIM methods, see below), and is also the starting point for the semi-theoretical atoms-in-molecules method. This latter method attempts to use experimental atomic energies to correct for the known atomic errors in a molecular calculation. Despite its success as a qualitative theory the valence-bond method has been used only rarely in quantitative applications. The reason for this lies in the so-called non-orthogonality problem, which refers to the difficulty of calculating the Hamiltonian matrix elements between valence-bond structures. 

The overlap matrix S is calculated and used to calculate an orthogonalizing matrix asinEqs (4.107) and (4.108) ... 

As in section 3, the diabatic representations are obtained finding the orthogonal matrix T such that TT = P. Again, the diabatic representation is not unique because T is defined within an overall p-independent orthogonal transformation. In actual calculations, one has to manipulate the potential energy matrix V = Te(p)T, whose large dimensions are often the bottleneck in practice. Proper choice of T is therefore crucial. The practical implementation (see the final section) of hyperspherical harmonics as the proper diabatic set is of great perspective power. 

Use the overlap integrals 5 and an orthogonalization procedure to calculate the A matrix of coefficients that will produce orthonormal basis functions Xs t isXr... 

The choice of the basis set. One may perform a new S transform for the H conformation and identify the new orthogonalized AOs with that of the bicentric problem. This is the most direct solution but the orthogonaliz-ation tails will be different and the use in this new basis of the effective bielectronic Hamiltonian given in Eq. (136) for instance may result in uncontrolled effects. One may also express the bielectronic operator of Eq. (136) in the non-orthogonal basis set and calculate the Hamiltonian matrix of H in the basis of non-orthogonal determinants, antisymmetrized products of Is AOs. The problem to solve is then of H-ES) type and it faces a typical non-orthogon ity problem of VB methods, which has been a major drawback of these approaches. 

In modem quantum chemistry packages, one can obtain moleculai basis set at the optimized geometry, in which the wave functions of the molecular basis are expanded in terms of a set of orthogonal Gaussian basis set. Therefore, we need to derive efficient fomiulas for calculating the above-mentioned matrix elements, between Gaussian functions of the first and second derivatives of the Coulomb potential ternis, especially the second derivative term that is not available in quantum chemistry packages. Section TV is devoted to the evaluation of these matrix elements. 

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