Secular determinant matrix

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,... 

This condition on the so-called secular determinant is the basis of the vibrational problem. The roots of Eq. (59), X, are the eigenvalues of the matrix product GF, while the columns of L, the eigenvectors, determine the forms of the normal modes of vibration. These relatively abstract relations become more evident with the consideration of an example. 

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 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... 

The sum is called the partially Fourier transformed dynamical matrix, which depends only on Q, and For each wave vector Q the normal mode frequencies of the crystal can be found by setting the secular determinant equal to zero ... 

The system comprised of equations 3.62 and 3.63 has solutions only when the determinant of coefficients and 2 (secular determinant) is zero. The coefficient matrix, or dynamic matrix, is... 

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). 

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... 

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... 

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

The energies of the (/-orbitals for the system , , i = 1-5, are then obtained by diagonalization of the real symmetrical matrix Zfy, i = dzi...dy2. The real (/-orbital linear combinations which correspond to these energies are then obtained by substituting the solutions, into the sets of simultaneous equations derived from the secular determinant. 

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... 

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... 

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... 

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 secular determinant for the remaining three-by-threc matrix may be obtained immediately and solved to give... 

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. 

As a result, the secular determinant for this function becomes a 10 by 10 matrix, the solution of which is not trivial however, the value of the lowest energy term, E, gives a more accurate value which is in accord with the variational principle. The use of still larger basis sets will push the value even lower. 

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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