Equation Secular determinant

Using the solutions for X in the set of secular equations which corresponds to the secular determinant (equation 3) yields the linear combinations of the (/-orbitals which belong to the two types in the crystal field. Using the coordinate system of Figure 2, where the origin is at a cation in the simple cubic lattice and the Cartesian axes are the cell axes, one obtains the well known result that... 

The numbers involving the integrations over the d-orbitals with this potential can be absorbed into a constant q. Then the solutions to the secular determinant (equation 5) take the form... 

Besides the trivial solution, m, ° = 0 that corresponds to no vibration, there is a specific set of displacement values for which the solutions are non-trivial, the eigenvectors. These values are those where the eigenvalues, X, satisfy the secular determinant equation ... 

Let us look at a simple case to illustrate the predictions of first- and second-order perturbation and compare them with the full HMO calculation for the perturbed system. We choose ethylene as the parent system and look at inductive perturbation induced by introduction of a substituent at atom 1 or by replacing atom 1 by a heteroatom such N or O. The exact solution is obtained by solving the 2x2 secular determinant (Equation 4.17). 

The zero secular-determinant equation is then (c/ equation (2-28))... 

We have agreed that t , is such that when in the secular determinant (equation (2-28)) is assigned the value e, the determinant vanishes —this was by arrangement, for e, is a root of (2-29). If the determinant of the coefficients of the c/rj in the n secular-equations (2-32) is zero, then from the theory of determinants this means that one row of the determinant is a linear combination of the others—in other words, it means that one of the n equations in (2-32) is redundant. Contrary to initial appearances, therefore, there are only ( — 1) independent, simultaneous equations in (2.32), but there are n unknowns, ( /,, r = 1,2.n. Consequently, all that we can find by elimina-... 

Nj X Ny(Ny xN(j) matrices, where is, in principle, infinite. No approximation is inherent in Equations 8-10. Solution of the secular determinant equation (Equation 10) yields eigenvalues of Equation 7 with eigenvectors expressed as (finite) combinations of the sub-basis = 4>t. The representation given by Equations 8-10... 

The molecular orbital energies in this two orbital case, e, (/ = 1,2). are obtained by solving the secular determinant (equation 1.30) shown in equation 2.5 for this particular example... 

Fortunately, we do not have to go back to equation 4.1, make substitutions, take partial derivatives, and so on. Instead, we proceed directly to the secular determinant (equation 4.18). Thus, the determinant for allyl is shown in equation 4.35. 

With this change of basis, the secular determinant equation B—iE =0 becomes replaced by... 

To obtain a secular determinant equation analogous to (6.22) in the internal coordinate basis, both 2Tand 2V must be expressed in terms of S. Since 2T is readily given in terms of mass-weighted coordinates i, we need a transformation of the form... 

According to the rules of matrix multiplication, the product D D is a 3N — 6) X (3N — 6) square matrix. With 2T and 2V now consistently expressed in the S basis, the secular determinant equation B — iM M = 0 in the generalized coordinate basis becomes... 

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... 

For the equation set to be linearly dependent, the secular determinant must be zero... 

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. 

Equation (125) applies for all values of the index k — 1,2,..., m. It is a set of m simultaneous, homogeneous, linear equations for the unknown values of the coefficients c . Following Cramer s rule (Section 7.8), a nontrivial solution exists only if the determinant of the coefficients vanishes. Thus, the secular determinant takes the form... 

As both F and G are partitioned by the use of symmetry coordinates, the secular determinant is factored accordingly. The problem of calculating the vibrational frequencies is thus divided into two parts solution of a linear equation for the single frequency of species B2 and of a quadratic equation for the pair of frequencies of species Aj. 

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 parameters are defined in equations 45-48. The parameters A, B and T are negative quantities if the orientation of the basis orbitals is defined as shown in diagram 43, and the same is true for AA if we assume that the energy of the cr-orbital lies below that of the two jr-orbitals as indicated in presentation 44. The orbital energies j are obtained by solving the secular determinant given by equation 49, which yields the solutions given in equations 50. 

Figure 5. Mean band energy (E) vs. total reorganization energy [Eq. (18)]. The dashed curve is the equation, E = Et. The solid curves are calculated from an analytical expression (9). The solid dots are calculated from a diagonalization of the two-mode model secular determinant with c = —6.0, Ac, = 1.1, and vc, =...

Integration is along the y axis of the complex plane between — oo and + oo and (iy) is the argument in all secular determinants or minors the same limits of integration apply to all formulae in this section, and will therefore be omitted. j is the determinant obtained by crossing out the ath row and 6th column of A in equation (8) and A is the derivative of A with respect to the argument (iy). 

The properties of the minors of the secular determinant of an alternant hydrocarbon may again be used to show that the integrals for which the index is even in (44) and odd in (45) and (46) are zero. It follows that the finite change Aq is an odd function, of Sa, while AFg and Apgt are even. Any inequalities between values of any index for two different positions u), as defined in equations (31) to (34) which arise as first terms of the corresponding infinite series in (44) to (46), persist term-by-term in the expression for the exact finite changes (Baba, 1957). In consequence, the broad agreement with experiment found earlier in the description of ionic and radical reactions by the approximate method carries over to the exact form. 

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... 

Form die secular determinant, and determine the N roots Ej of the secular equation. 

Prior to considering semiempirical methods designed on the basis of HF theory, it is instructive to revisit one-electron effective Hamiltonian methods like the Huckel model described in Section 4.4. Such models tend to involve the most drastic approximations, but as a result their rationale is tied closely to experimental concepts and they tend to be inmitive. One such model that continues to see extensive use today is the so-called extended Huckel theory (EHT). Recall that the key step in finding the MOs for an effective Hamiltonian is the formation of the secular determinant for the secular equation... 

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). 

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... 

Insertion of each of these roots in turn into the covering equations of the secular determinant gives a set of five simultaneous equations in the five coefficients of the above symmetry orbitals in the corresponding molecular orbital. For example, the root = 2.047 gives the equations... 

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