Expansion of the Secular Determinant

For simplicity, die following development of this example will be limited to die special case in which m 1 mi m and k = ki = k. Then, the expansion of the secular determinant of Eq. (86) yields... 

So there is a direct correlation between the order of a perturbation type expansion of the secular determinant problem and the index of the lowest moment which can influence a structural problem. The energetic importance of these higher order terms clearly decreases as 2 n increases via the denominator of the perturbation expansion. This relationship between the moments approach and more traditional viewpoints is therefore quite a strong and interesting one. 

Expansion of the secular determinant results in an algebraic equation with six roots. These are ... 

Solution of the polynomial equation that results from expansion of the secular determinant equation 1.30 provides m orbital energies e, (/= I, 2,. . . , m) which, according to the variational theorem, are a set of upper bounds to the true orbital energies. Written in matrix notation, equation 1.30 becomes... 

As an example of a symbolic calculation. Fig. 5 displays a Mathcad solution of one 4X4 block of the secular determinant of a Hiickel molecular orbital calculation done in Exp. 41 for orf/ro-benzosemiquinone (compare with Table 41-2). Mathcad is a software package for numerical analysis but also makes use of a subset of the symbohc routines of Maple. The algebraic expansion of the determinant is generated and solved with two... 

Equation (3-111) is the secular equation for this problem. [Pauling and Wilson discuss the meaning of the term secular in this context.] Expansion of the determinant gives a polynomial in X that is called the characteristic equation ... 

The expansion of the corresponding secular determinant leads to the relation... 

First, it is possible to simplify the secular equation (2) by means of symmetry. It can be shown by group theory (140) that, in general, the integrals Hi and Si are nonzero only if the orbitals < , and j have the same transformation properties under all the symmetry elements of the molecule. As a simple example, the interaction between an s and a pn orbital which have different properties with respect to the nodal plane of the pn orbital is clearly zero. Interaction above the symmetry plane is cancelled exactly by interaction below the plane (Fig. 13). It is thus possible to split the secular determinant into a set of diagonal blocks with all integrals outside these blocks identically zero. Expansion of the determinant is then simply the product of those lower-order determinants, and so the magnitude of the... 

It is now preferable to solve the secular determinant directly with the aid of a computer but expansion is clearer for the reader. 

In this connection a more detailed account will only be given of a method that permits calculation of HMO orbital energies of a model of a heterocycle without expanding the secular determinant from the knowledge of molecular orbital energies and expansion coefficients of the parent hydrocarbon.11-15 Now that automatic computing machines are commonly used for quantum-chemical calculations we see the chief merit of the method in that it permits one to study the effect of empirical parameters on energy characteristics in a clear-cut and concise manner. 

Expansion of the Hiickel orbital (HMO) secular determinant for a PAH graph gives the characteristic polynomial P(G X) = det X1 — A where I is the identity matrix and A is the adjacency matrix for the corresponding graph [11]. The characteristic polynomial of a N carbon atom system has the following form... 

After expansion of the determinant the secular equation reduces to an algebraic equation of nth order with respect to e. It has, in general, n roots. The ground-state energy is equal to the minimum root... 

The form of the secular equation [Eq. (1), Sec. 4-4] is convenient to use in showing how to get the secular equation in expanded form, that is, in the form of an ordinary algebraic equation in the unknown quantity X. A determinant is expanded by forming the sum of all products, with proper signs, constructed by selecting one and only one element from each row and column, One such product is obviously made up of the elements on the principal diagonal. All other terms in the expansion are obtained by substituting for two or more of these s. Consider the terms in One of them is obtained... 

As a preliminary step in finding the MOs of a molecule, it is helpful (but not essential) to construct linear combinations of the original basis AOs such that each linear combination does transform according to one of the molecular symmetry species. Such Unear combinations are called symmetry orbitals or symmetry-adapted basis functions. The symmetry orbitals are used as the basis functions Xs in the expansions = 2sCj,a s [Eq. (14.33)] of the MOs The use of basis functions that transform according to the molecular symmetry species simplifies the calculation by putting the secular determinant in block-diagonal form. This will be illustrated below. 

The applications of this model is mostly restricted to hydrogen and helium like systems. Winkler [25] has calculated the detachment energies of H embedded in a variety of Debye plasmas. He used a correlated description of the two-particle wave functions introducing interparticle coordinate in the expansion of the basis set. The linear variational parameters are determined by solving the generalized secular equation... 

Expansion of this determinant gives a single equation containing the unknown E. Any value of E satisfying this equation is associated with a nontrivial set of coefficients. The lowest of these values of E is the minimum average energy achievable by variation of the coefficients. Substitution of this value of E back into Eqs. (7 6) produces n simultaneous equations for the n coefficients. Equation (7-47) is referred to as the secular equation, and the determinant on the left-hand side is called the secular determinant. 

The above considerations suggest that fairly good approximate wave-functions, when written in terms of determinants based on an intelligent choice of orbitals, may yield expansions of the form (3.1.4) containing only a small number of terms—though an exact function would require an infinite number. We consider the theory of such expansions in detail in later sections and here simply recall that the coefficients that yield best approximations to the electronic states and their energies may always be determined, in principle, by solving the secular problem of Section 2.3, namely... 

The expansion of this determinant is a polynomial of degree n in A, giving the characteristic or secular equation... 

A. Partial Derivatives and Polarizability Coefficients Expansion of (8) yields a polynomial, the characteristic or secular polynomial, whose roots are determined by the values of the parameters , vw- The ground state energy (12) is likewise a function of the (a,j3) parameter values, as are all quantities such as AO coefficients in the MO s, charges q bond orders p t, etc. It is possible, therefore, to specify the h partial derivative with respect to any or at an arbitrary point defined by a set of values (a,j8) in the parameter space, and to make expansions such as... 

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