Hamiltonian matrix effective

I h e preceding discussion mean s that tli e Matrix etjuatiori s already described are correct, except that the Fuck matrix, F. replaces the effective one-electron Hamiltonian matrix, and th at K depends on th e solution C ... 

The principal semi-empirical schemes usually involve one of two approaches. The first uses an effective one-electron Hamiltonian, where the Hamiltonian matrix elements are given empirical or semi-empirical values to try to correlate the results of calculations with experiment, but no specified and clear mathematical derivation of the explicit form of this one-electron Hamiltonian is available beyond that given above. The extended Hiickel calculations are of this type. 

As a consequence, the effective Hamiltonian matrix restricted to the ground state and the first excited state of the fast mode has a block structure. 

The eigenvalue equations of the two diagonal blocks of the effective Hamiltonian matrix is characterized by the equations... 

Note that because the effective Hamiltonian matrix is not Hermitian, the eigenvectors are not orthogonal. However, when ac is small, the orthogonality properties are satisfactorily verified. 

As such, the total effect of the Hamiltonian matrix may be obtained by summing partial [Pg.270]

The standard treatment employs equal Coulomb parameters a for all six carbon atoms, and equal resonance parameters /3 for all six CC bonds. In effect, a is the origin and /3 the unit in a scaled representation of the Hiickel hamiltonian matrix as... 

One wishes to calculate exactly the energy of A B — C (Eq. 3.50) relative to a situation where A and B — C are separated, in the effective VB Hamiltonian framework, as in the preceding exercise. Rewrite Equation 3.50 so that the two determinants exhibit maximum orbital and spin correspondence. Calculate the energies of the unnormalized determinants abc and acb, and the Hamiltonian matrix element ( afoc // acfo ). The following simplifications will be used... 

A difference between the qualitative VB theory, discussed in Chapter 3, and the spin-Hamiltonian VB theory is that the basic constituent of the latter theory is the AO-based determinant, without any a priori bias for a given electronic coupling into bond pairs like those used in the Rumer basis set of VB structures. The bond coupling results from the diagonalization of the Hamiltonian matrix in the space of the determinant basis set. The theory is restricted to determinants having one electron per AO. This restriction does not mean, however, that the ionic structures are neglected since their effect is effectively included in the parameters of the theory. Nevertheless, since ionicity is introduced only in an effective manner, the treatment does not yield electronic states that are ionic in nature, and excludes molecules bearing lone pairs. Another simplification is the zero-differential overlap approximation, between the AOs. 

Here, grs is a parameter that is quantified either from experimental data, or is calculated by an ab initio method as one-half of the singlet-triplet excitation energy gap of the r—s bond. In terms of the qualitative theory in Chapter 3, grs is therefore identical to the key quantity —2(3 5 - This empirical quantity incorporates the effect of the ionic components of the bond, albeit in an implicit way. (c) The Hamiltonian matrix element between two determinants differing by one spin permutation between orbitals r and s is equal to grs. Only close neighbor grs elements are taken into account all other off-diagonal matrix elements are set to zero. An example of a Hamiltonian matrix is illustrated in Scheme 8.1 for 1,3-butadiene. 

From Eq. (50), an overlap matrix element is exactly a PPD and can easily be evaluated from the routine for PPDs, while Hamiltonian matrix elements may be obtained by a similar routine to that for PPDs, where 2x2 sub-PPDs are replaced with effective sub-PPDs of one-electron and two-electron integrals. 

The procedure leading from the exact /V-electron Hamiltonian (2) to the Heisenberg Hamiltonian matrix (47) is very instructive, but it is rather lengthy. Much simpler is the use of effective Hamiltonians which in the space of N-electron eigenfunctions of S2 and S2 are represented by the same matrix. Furthermore, using the effective Hamiltonians may bring another insight into the nature of the interactions described by the model. The simplest effective Hamiltonian in the pure spin is... 

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