The solution of secular equations

Once the G and F matrices are obtained, the next step is to solve the matrix secular equation  

The F and G matrix elements of abentXY2 molecule are given in Eqs. 1.140 and 1.141, respectively. The secular equation for the Aj species is quadratic  

GiiFn + G12F21 — X G F 2 + G12F22 G2 F + G22F21 G21F12 + G22F22 X 

If this is expanded into an algebraic equation, the following result is obtained  

Although the bond distance is involved in both the G and F matrices, it is canceled during multiphcation of the G and F matrix elements. Therefore, any unit can be used for the bond distance. 

This assumpiion docs not cause serious error in final results, since F is small in most cases. 

If the force constants in terms of the generalized valence force field are selected as 

Using these values, we find that Eq. 14.3 becomes 

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This is an eigenvalue problem of the form of Eq. III.45 referring to the truncated basis only, and the influence of the remainder set is seen by the additional term in the energy matrix. The relation III.48 corresponds to a solution of the secular equation by means of a modified perturbation theory,19 and the problem is complicated by the fact that the extra term in Eq. III.48 contains the energy parameter E, which leads to an iteration procedure. So far no one has investigated the remainder problem in detail, but Eq. III.48 certainly provides a good starting point. 

Since P depends on the solution of the secular equation, which in turn depends on P, it is clear that we must solve iteratively for the molecular orbitals. In general, we will consider only the first few iterations and start the first iteration with = ZM, where is the effective charge of the nuclear core of the pth orbital (for more than one orbital per atom we have ZA = EM(y4) Zfi). The potential surface of the system is then approximated by... 

This suggests that as a simple variation treatment of the system for smaller values of tAb we make use of the same wave functions uUa and uUb, forming the linear combinations given by solution of the secular equation as discussed in Section 26d. The secular equation is... 

A represents the lack of orthogonality of uUjL and uXSb- Because of the equivalence of the two functions, the relations Baa = Bbb and Bab = Bba hold. The solutions of the secular equation are... 

Let us consider a case in which an ion (donor, D ) and a solvent (acceptor. A) form a CT complex. The ground state energy (see Fig. 5) can be obtained as a solution of the secular equation ... 

In the example considered above, Arj - A/s is the only symmetry coordinate of species B2. Thus, it results in a factor of degree one in the completely reduced secular determinant It is therefore a normal coordinate. On the other hand, the two normal coordinates of species Ai are linear combinations of the symmetry coordinates Acr and Arj + Ar2. They can only be found by solution of the secular equations. 

The change x may represent the effect of a substituent or heteroatom at the wth position. In this case the v electron energy levels, charges g free valences Fg and bond orders pgt can be obtained by direct solution of the secular equations (8) using... 

Methods are introduced for generating many-electron Sturmian basis sets using the actual external potential experienced by an N-electron system, i.e. the attractive potential of the nuclei. When such basis sets are employed, very few basis functions are needed for an accurate representation of the system the kinetic energy term disappears from the secular equation solution of the secular equation provides automatically an optimal basis set and a solution to the many-electron problem is found directly, including electron correlation, and without the self-consistent field approximation. In the case of molecules, the momentum-space hyperspherical harmonic methods of Fock, Shibuya and Wulfman are shown to be very well suited to the construction of many-electron Sturmian basis functions. 

The value of po for a particular state of an atom appears as a root of the secular equation, and is therefore determined by diagonalization of the matrix Ti/, /, equation (19), a matrix which is independent of po- Thus the values of the orbital exponents k are completely determined by solution of the secular equation. In the lowest approximation, we can represent the ground-state of... 

Solution of the secular equation for benzene gives the following energy levels and eigenfunctions ... 

Applying the variational principle to obtain the optimum energy requires a solution of the secular equations which may be written in determinant form as (61). //, and H22 are the energies of diagonal matrix element (64). 

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). 

If the basis set used is finite and incomplete, solution of the secular equation yields approximate, rather than exact, eigenvalues. An example is the linear variation method note that (2.78) and (1.190) have the same form, except that (1.190) uses an incomplete basis set. An important application of the linear variation method is the Hartree-Fock-Roothaan secular equation (1.298) here, basis AOs centered on different nuclei are nonorthogonal. Ab initio and semiempirical SCF methods use matrix-diagonalization procedures to solve the Roothaan equations. 

The As values are obtained from the solution of the secular equation... 

Solution of the secular equation amounts to finding the roots of an iVth order equation in E. The N roots are the energies of the N molecular orbitals the forms of the orbitals in terms of the basis atomic orbitals 9are found by substituting each value of E, in turn, back into Equations A2.13 and solving for the c s using the additional condition that each MO tf)t is to be normalized,... 

The determination of the normal modes and their frequencies, however, depends upon solving the secular equation, a 3N X3N determinant. This rapidly becomes nontrivial as N increases. Methods do exist which somewhat simplify the computational problem. Thus, if the molecule has symmetry, the 3Ar X 3N determinant can be resolved into sub-determinants of lower order, each of which involves only normal frequencies of a given symmetry class. These determinants are of course easier to solve. (We will return shortly to the subject of symmetry considerations since they not only aid in the solution of the secular equation, but they permit the determination — without any other information about the molecule — of many characteristics of the normal modes, such as their number, activity in the infrared and Raman spectra, possibilities of interaction, and so on.) In addition, special techniques have been developed for facilitating the setting up and solving of the secular equation [Wilson, Decius, and Cross (245)]. Even these, however, become prohibitive for the large N encountered in complex molecules such as high polymers. 

The expansion coefficients and energy eigenvalues are found from solutions of the secular equation Hnn — SnnE = 0. The nonzero matrix elements Hnn can be expressed in terms of the coefficients of the potential... 

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... 

For a given set of atomic orbitals Xp, the one- and two-electron integrals (Equation 6.26 and Equation 6.28) can be calculated, and all terms of the Fock matrix in the AO basis, I jlv (Equation 6.25), would be known except for the density matrix, Ppo, which depends on the LCAO-MO coefficients as solutions of the secular equation (Equation 6.23) and thus on F. Accordingly, the density matrix depends on the evaluation of the Fock matrix, which depends on the elements of the density matrix. 

In summary, the solution of the secular equations, Eqs. (32)-(34), corresponding to non-zero mass particles, may be written in analogy with Eqs. (2)—(7) as... 

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