Huckel coefficient matrix

The tight-binding band structure calculations were based upon the effective one-electron Hamiltonian of the extended Huckel method. [5] The off-diagonal matrix elements of the Hamiltonian were calculated acording to the modified Wolfsberg-Helmholtz formula. All valence electrons were explicitly taken into account in the calculations and the basis set consisted of double- Slater-type orbitals for C, O and S and a single- Slater-type orbitals for H. The exponents, contraction coefficients and atomic parameters were taken from previous work [6],... 

The application of Cl using the SCF solutions is straightforward, being carried out in the way mentioned above. A Huckel Cl treatment requires, however, a different formulation. In this case, after the Huckel treatment has been carried out, a matrix with elements similar to those defined by Eqs. (8) is formed5 the expansion coefficients appearing in those equations are those determined in the simple Htickel approximation. Diagonalization of this matrix yields the new eigenvectors, that can now be used to construct the excited state functions for use in the Cl treatment. In fact this approximation represents a simplified SCF Cl procedure. 

The w parameter determines the weight of the charge on the diagonal elements. Since Qa is calculated from the results (MO coefficients, eq. (3.90)), but enters the Huckel matrix which produces the results (by diagonalization), such schemes become iterative. Methods where the matrix elements are modified by the calculated charge are often called charge iteration or self-consistent (Huckel) methods. 

The obvious and logical extension to the discussion so far outlined in this chapter is to ask whether, in the case of, say, a general, non-alternant hydrocarbon, it is possible and legitimate to perform an iterative calculation in which both Coulomb integrals ( 7.2 and 7.3) and resonance integrals ( 7.4) are varied simultaneously. In such a scheme, calculations based on relations (7-6) and (7-8) would be carried out in an iterative fashion such that the one-electron Hamiltonian-matrix [ar, 0rJ furnished qr and p identical with those which had served to calculate the particular ar - and / elements in question such LCAO-MO coefficients and energy-levels as were derived from this process would then in principle be truly self-consistent , in the sense implied. This approach has been called1122 the self-consistent Huckel-method or the / a/co" method . 

H was the matrix-component of the Hiickel effective-Hamiltonian operator, effective between two basis atomic-orbitals, 4>r and 4>s, Srs was the overlap integral between 4>r and s, and H was set equal to a, H to / . This is how we developed the simple HMO-approach in Chapter Two. What Roothaan did was to show that a formally similar determinant is obtained in a full treatment of the re-electrons, but that it involves a somewhat more complicated expression for the matrix-elements, H . Furthermore, he showed that this more-complicated expression somehow had to take into account interactions between any one re-electron and all the other re-electrons. We do not go into the details of this here, except to say that, in order to find the LCAO-MO coefficients for one molecular orbital, it is necessary to know all the others, because all the others appear in the expressions for the equivalent terms, Hrs. This is a very familiar situation which mathematicians have long known how to deal with and which we encountered during our discussion of the self-consistent" Huckel-methods in 7.2—7.5 it is necessary to use an iterative scheme. An initial guess is made of all the orbitals except one and these are used to calculate the H -terms for the one orbital which has not yet... 

However, the solution given by Eq. (4.277) is based on the form of effective independent-electron Hamiltonians that can be quite empirically constructed - as in Extended Huckel Theory (Hoffmann, 1963) such arbitrariness can be nevertheless avoided by the so-called self-consistentfield (SCF) in which the one-electron effective Hamiltonian is considered such that to depend by the solution of Eq. (4.266) itself, i.e., by the matrix of coefficients (C) this way we identify the resulted Hamiltonian as the Fock operator, while the associated eigen-problem rewrites the Hartree-Fock equation (4.267) under the mono-electronic wave-function representation ... 

The one-electron density matrix P in AO basis is calculated self-consistently after the initial set of coefficients C is known. The simple one-electron Hamiltonian of Huckel type is often used as an intial approximation to the Hartree-Fock Hamiltonian. It is important to remember that the one-electron approximation is made when the many-electron wavefunction is written as the Slater determinant or their linear combination. For bound states, precise solution of Schrodinger equation can be expressed as a (in general infinite) linear combination of Slater determinants. 

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