Secular determinant elements

The secular determinant can now be set up and results in a tridiagonal determinant since we only have nonzero matrix elements in the diagonal and the two neighboring elements. Hence,... 

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. 

As before, the overlap integrals are neglected and we can proceed directly to the secular determinant. After the substitutions are made and each element is divided by (3, the result can be shown as follows (where x = a — E)/ft) ... 

The integrals of the H and S matrices are generally referred to as matrix elements. The condition for a non-trivial solution is that the secular determinant should vanish,... 

The three-center orbitals (3.229a)-(3.229c) can also be considered to arise from the 3 x 3 secular determinant for interaction of the three hybrids, with Fock-matrix elements... 

All of the above conventions together permit the complete construction of the secular determinant. Using standard linear algebra methods, the MO energies and wave functions can be found from solution of the secular equation. Because the matrix elements do not depend on the final MOs in any way (unlike HF theory), the process is not iterative, so it is very fast, even for very large molecules (however, fire process does become iterative if VSIPs are adjusted as a function of partial atomic charge as described above, since the partial atomic charge depends on the occupied orbitals, as described in Chapter 9). 

In the secular determinant, overlap matiix elements are defined by... 

The only terms remaining to be defined in the assembly of the HF secular determinant are the one-electron terms for off-diagonal matrix elements. These are defined as... 

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... 

The characters x r f°rm a reducible representation of the molecular orbitals of this molecule that is, the orbitals still have finite interaction elements between them, and the secular determinant must be reduced further to diagonal form. The number of times af that a reducible representation occurs in an irreducible representation is given by (140)... 

It is now possible to solve the secular determinant for the energy levels of these 7t orbitals with the assumptions already listed (Sec. II.A). For simplicity, overlap is neglected, and if Q denotes the Coulomb term of an isolated carbon 2pn atomic orbital, we put W—Q — E. Then the determinant Hjj-SUE =0 may be evaluated using the above symmetry orbitals. For example, the matrix element //13 is obtained as follows ... 

It is a simple matter to calculate the orbital energies for the hypothetical complex C6H5X—Cr, since all the required matrix elements, Hu (coulomb terms) and //<7 (resonance integrals) of the secular determinant... 

When the secular determinant in (1.207) is in diagonal form (all off-diagonal elements equal to zero), it follows that the initially chosen unperturbed wave functions ipj0) of the degenerate level are the correct... 

Since the value of M does not affect E ° the unperturbed levels are (2y+l)-fold degenerate. Hence, before forging ahead, we must be sure that we have the correct zeroth-order wave functions for the perturbation (4.36). (See Section 1.10.) It was noted in Section 1.10 that when the secular determinant is diagonal, we have the right zeroth-order functions. We now show that the functions (4.38) give a diagonal secular determinant. An off-diagonal element has the form... 

List the SALCs so that all those belonging to a given representation occur together in the list. Use this list to label the rows and columns of the secular determinant. Only the elements of the secular determinant that lie at the intersection of a row and a column belonging to the same irreducible representation can be nonzero, and these nonzero elements will lie in blocks along the principal diagonal. The secular determinant will therefore be factored. 

The solution of the secular equation Fy —F5y = 0 requires the evaluation of the constituent matrix terms Fy. The Fy s are, however, themselves functions of the coefficients of the atomic orbitals amt through Pjel and therefore can only be evaluated by solving the secular equation. The Hartree-Fock procedure thus requires that a preliminary guess be made as to the values of the molecular population distribution terms Pici these values are then used to calculate the matrix elements Fy and thence solve the secular determinant. This, in turn, provides a better approximation to the wave function and an. .improved set of values of Pm. The above procedure is repeated with this first improved set and a second improved set evaluated. The process is repeated until no difference is found between successive improved wave functions. Finally, it may be shown that when such a calculation has been iterated to self-consistency the total electronic energy E of a closed shell molecule is given by... 

In some Hiickel packages, the input (the atoms and their connectivities) must be introduced as a secular determinant. The latter can be written merely by looking at the structural formula. Let if be the element in row i and column ) and set... 

It may hence be pertinent to draw the attention to two kinds of development of ligand field theory which have taken place since 1962. They are both related to the Wolfsberg-Helmholz model (27) where it is assumed that the non-diagonal elements between orbitals centered on afferent atoms are proportional to the product of their overlap integral and their average energy with a proportionality constant k usually assumed to have values between 1.6 and 2. Thus, the secular determinant for two interacting orbitals is ... 

We now apply the concepts developed in the preceding section to the system of just two nuclei, first considering the case in which there is no spin coupling between them. We digress from our usual notation to call the two spins A and B, rather than A and X, because we later wish to use some of the present results in treating the coupled AB system. The four product basis functions are given in Eq. 6.1. We now compute the matrix elements needed for the secular determinant. Because there are four basis functions, the determinant is 4 X 4 in size, with 16 matrix elements. Many of these will turn out to be zero. For <3CU we have, from... 

With all off-diagonal elements equal to zero, the secular determinant becomes... 

Before extending our treatment to two coupled nuclei, it is helpful to consider some general conditions that cause zero elements to appear in the secular equation. With this knowledge, we can avoid the effort of calculating many of the elements specifically for each case we study furthermore, the presence of zero elements usually results in the secular determinant being factored into several equations of much smaller order, the solution of which is simpler than that of a high order equation. 

The secular determinant can be further simplified by considering only interactions between first-nearest neighbors. In this case, aU the other matrix elements become equal to zero. Using the familiar Hiickel notation, Eq. 5.16 then looks hke ... 

The steps to be followed may be summarized. Secular determinants must be constructed for each of the doubly degenerate levels in both directions. First-order Zeeman coefficients must be evaluated for each direction. Matrix elements connecting the three secular determinants must be evaluated to yield second-order Zeeman coefficients. The first-and second-order Zeeman coefficients must be substituted into the Van Vleck equation to yield the anisotropic magnetic susceptibilities x and x - Generally, anisotropic magnetic properties are discussed in terms of /x and n since the variation of these anisotropic components are much more easily visuaUzed. 

The matrix elements in the secular determinant now rapidly increase in complexity because a single Hab value (now Fjj) can be composed of hundreds or thousands of coulomb... 

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