One-electron secular equation

From the one-electron secular equation, it follows that the gerade solution (in momentum space) has the form ... 

Koga and others [10] have shown that the loss of accuracy resulting from tmncation of the basis set can be reduced by replacing the one-electron secular equation, (34), with an equation based on a second iteration of the integral equation (31). The second-iterated integral equation has the form ... 

Molecular Orbitals Based on Sturmians 5.1 The One-Electron Secular Equation... 

There seems to be a certain complementarity between the degree of difficulty in evaluating HKL for various one-electron sets y)k and the order of the secular equation needed to obtain a certain accuracy in the result. The work carried out in getting extensive tables of molecular integrals has also been of essential value for facilitating the calculation of the matrix elements HKL. 

Table X gives an idea of the strength of the various expansion methods, and it shows that, by using the principal term only, one can hardly expect to reach even the above-mentioned chemical margin, even if the wave function W gO(D) is actually very close in the helium case. This means that one has to rely on expansions in complete sets, and the construction of the modern electronic computers has fortunately greatly facilitated the numerical solution of secular equations of high order and the calculation of the matrix elements involved. For atoms, the development will probably go very fast, but, for small molecules one has first to program the conventional Hartree-Fock scheme in a fully self-consistent way for the computers, before the next step can be taken. For large molecules and crystals, the entire situation is much more complicated, and it will hence probably take a rather long time before one can hope to get a detailed understanding of the correlation phenomena in these systems.

For each value of j there is a set of equations, but the different sets are all identical except for the replacement of the subscript j by some other, say j. The condition for the solubility of this set of equations is that their determinants vanish. The l roots of this secular equation, W = Wj (j = 1,2,...,/) give the energies of the l molecular orbitals. We then assign two electrons (one with positive and one with negative spin) to the lowest... 

The problem of evaluating the effect of the perturbation created by the ligands thus reduces to the solution of the secular determinant with matrix elements of the type rp[ lICT (pk, where rpj) and cpk) identify the eigenfunctions of the free ion. Since cpt) and cpk) are spherically symmetric, and can be expressed in terms of spherical harmonics, the potential is expanded in terms of spherical harmonics to fully exploit the symmetry of the system in evaluating these matrix elements. In detail, two different formalisms have been developed in the past to deal with the calculation of matrix elements of Equation 1.13 [2, 3]. Since t/CF is the sum of one-electron operators, while cpi) and cpk) are many-electron functions, both the formalisms require decomposition of free ion terms in linear combinations of monoelectronic functions. 

The function F(l,2) is in fact the space part of the total wave function, since a non-relativistic two-electron wave function can always be represented by a product of the spin and space parts, both having opposite symmetries with respect to the electrons permutations. Thus, one may skip the spin function and use only the space part of the wave function. The only trace that spin leaves is the definite per-mutational symmetry and sign in Eq.(14) refers to singlet as "+" and to triplet as Xi and yi denote cartesian coordinates of the ith electron. A is commonly known angular projection quantum number and A is equal to 0, 1, and 2 for L, II and A symmetry of the electronic state respectively. The linear variational coefficients c, are found by solving the secular equations. The basis functions i(l,2) which possess 2 symmetry are expressed in elliptic coordinates as ... 

Thus the many-center one-electron problem is easily solved, provided that the integrals shown in equation (65) can be evaluated. The reciprocals of the parameters can then be identified with the roots of the secular equation (63). 

The first set of equations govern the Cj amplitudes and are called the CI- secular equations. The second set determine the LCAO-MO coefficients of the spin-orbitals (f>j and are called the Fock equations. The Fock operator F is given in terms of the one- and two-electron operators in H itself as well as the so-called one- and two-electron density matrices yij and Tyj i which are defined below. These density matrices reflect the averaged occupancies of the various spin orbitals in the CSFs of VP. The resultant expression for F is ... 

Prior to considering semiempirical methods designed on the basis of HF theory, it is instructive to revisit one-electron effective Hamiltonian methods like the Huckel model described in Section 4.4. Such models tend to involve the most drastic approximations, but as a result their rationale is tied closely to experimental concepts and they tend to be inmitive. One such model that continues to see extensive use today is the so-called extended Huckel theory (EHT). Recall that the key step in finding the MOs for an effective Hamiltonian is the formation of the secular determinant for the secular equation... 

In his valuable paper Molecular Energy Levels and Valence Bonds Slater developed a method of formulating approximate wave functions for molecules and constructing the corresponding secular equations.1 Let a,b, repreamt atomic orbitals, each occupied by one valence electron, and a and 0 represent the electron spin functions for spin orientation -f i and — J, respectively. Slater showed that the following function corresponds to a valence-bond structure with bonds a-----b, c---d, and so forth ... 

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