Determinantal equation

In summary, to solve an eigenvalue equation, we first solve the determinantal equation ... 

There are several ways to find the solutions of these simultaneous equations. One approach is to find the two roots of the determinantal equation known as the secular equation ... 

Instead of using just energy conservation, Moody (1975) derived a revised model that takes into account all the conservation laws. He found that critical flow rate is given by a determinantal equation that gives G as a function of p, X, and S. 

As before, the total energy, E, can be minimized by optimizing the mixing coefficients c so that dE/dci is zero. In the case of a system comprising two configurations, this leads to the same sort of determinantal equation that was obtained when we minimized the en-... 

When this is compared with Eq. (1.15) we see that we have solved our problem, if is the column of V and is the diagonal element of Thus the diagonal elements of are the roots of the determinantal equation Eq.(1.16). 

Now consider the variation problem with + 1 functions where we have added another of the basis functions to the set. We now have the matrices 7/( +0 and and the new determinantal equation... 

Physically meaningful solutions of the system of equations (2) are obtained only for certain values of molecular orbital energies, Eit which follow from the determinantal equation... 

Solving a determinantal equation of the type (6) (orbital energies) and systems of equations of the type (2) (expansion coefficients) gives the results shown in the following tabulation. 

The purpose of this book is to show how the consideration of molecular symmetry can cut short a lot of the work involved in the quantum mechanical treatment of molecules. Of course, all the problems we will be concerned with could be solved by brute force but the use of symmetry is both more expeditious and more elegant. For example, when we come to consider Huckel molecular orbital theory for the trivinylmethyl radical, we will find that if we take account of the molecule s symmetry, we can reduce the problem of solving a 7 x 7 determinantal equation to the much easier one of solving one 3x3 and two 2x2 determinantal equations and this leads to having one cubic and two quadratic equations rather than one seventh-order equation to solve. Symmetry will also allow us immediately to obtain useful qualitative information about the properties of molecules from which their structure can be predicted for example, we will be able to predict the differences in the infra-red and Baman spectra of methane and monodeuteromethane and thereby distinguish between them. 

Clearly it is much easier to solve one 3x3 and two 2x2 determinantal equations than the 7x7 determinantal equation which occurs when no use is made of symmetry. 

The determinant of A is nonzero if A is nonsingular, so the solutions to the two detenninantal equations must be the same. B A is the inverse of A 1B, so its characteristic roots must be the reciprocals of those of A" B. There might seem to be a problem here since these two matrices need not be symmetric, so the roots could be complex. But, for the application noted, both A and B are symmetric and positive definite. As such, it can be shown (see Section 16.5.2d) that the solution is the same as that of a third determinantal equation involving a symmetric matrix. 

As a very persuasive illustration of the effectiveness of symmetry factorization in reducing a computational task that would be entirely impractical without a digital computer to one that is a straightforward pencil-and-paper operation, we shall again consider the naphthalene molecule. It has been shown in Section 7.1 that the secular equation for the n MOs is the 10 x 10 determinantal equation, 7.1-15, if the set of 10pn orbitals is used directly for constructing LCAO-MOs. 

It follows from the properties of determinants that, if the entire determinant is to have the value zero, each block factor separately must equal zero. Thus the 10 x 10 determinantal equation has been reduced to two 2x2 and two 3x3 secular equations. For example, the energies of the two MOs of Au symmetry are given by the simple secular equation... 

Thus, there are two Au MOs, two B2g MOs, three Blu MOs, and three B2g MOs. By constructing SALCs corresponding to these representations, the well-nigh hopeless problem of solving a 10 x 10 determinantal equation is reduced to the tractable task of solving two quadratic and two cubic equations. This has already been illustrated in Section 7.2. 

Obviously, this leads to two different energy values, ° Hn, instead of the same one, EH, for both states. The entire situation is formally analogous to the occurrence of a 2 x 2 determinantal equation for the energies of two individual orbitals of the same symmetry. In practice, however, it is a little different, since there is no simple way to estimate the magnitude of the interaction energy. This is caused by interelectronic repulsion and is difficult to compute accurately. 

This determinantal equation gives the extremum energy W of the best linear combination of and B for the system. Expanding the equation gives... 

The ground state is Aj (ajj3 )2 (bf3 )2 (ax )2 (b2 )2. Notice that the 3x3 determinantal equation for the aj orbital gives three roots one strongly bonding, one nearly nonbonding, and one strongly antibond-... 

Consider the propenyl system. In the secular determinant the i,i-type interactions will be represented by x, adjacent i,. /-type interactions by 1 and nonadjacent i, /-type interactions by 0. For the determinantal equation we can write (Fig. 4.25)... 

When the molecular orbitals are taken as a linear combination of the atomic pz orbitals only, the form taken by the equations amounts to solving a determinantal equation ... 

The eigenequation (71) can formally be rewritten as a problem of finding zeros of the matrix equation (ul — U )Qk = 0. This matrix equation can operationally be solved by conversion to its corresponding scalar counterpart, that is, a determinantal equation known as the secular equation [2] ... 

Expanding the determinantal equation (235) by the Cauchy rule, we readily obtain the following continued fraction ... 

Hirata S, Nooijen M, Bartlett RJ (2000) High-order determinantal equation-of-motion coupled-cluster calculations for electronic excited states. Chem Phys Lett 326 255—262. 

But the solution of this determinantal equation leads to an algebraic equation of the sth degree in X. The s roots of this equation give the values of Xi,. . ., Xs and so uniquely determine X. 

We have dropped the subscripts j on X because every value of j leads to the same determinantal equation. 

The secular determinantal equation is set up in the usual manner, the wavefunctions corrected for the crystal-field interaction are used in the perturbation treatment, energies are generated, and these are used in conjunction with the secular equations to generate new wavefunctions that have now been corrected for spin-orbit coupling. These corrected wavefunctions are used for the calculation of the Zeeman effect. 

A great simplification occurs when the disturbance of the impurity is confined to a single site and to a sin e orbital (or to a single apprcq>riately symmetrized combination of orbitals). The Fredholm determinantal equation (5.9) is one-dimensional and becomes... 

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