Determinants secular

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

This ean be cleared of fraetions by multiplying by 1260 to obtain the secular determinant... 

Fill out the secular determinant for the linear combination in Problem 4. 

Hi2 is the resonance integral, usually symbolized by p. In a homonuclear diatomic molecule Hi I = H22 = a, which is known as the Coulomb integral, and the secular determinant becomes... 

The Hiickel MO method is based on the LCAO method for diatomic molecules discussed in Section 7.2.1. Extension of the LCAO method to polyatomic molecules gives a secular determinant of the general type... 

In the case of benzene, Hiickel treatment of the six 2p orbitals on the carbon atoms and perpendicular to the plane of the ring leads to the secular determinant... 

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,... 

For simplicity, die following development of this example will be limited to die special case in which m 1 mi m and k = ki = k. Then, the expansion of the secular determinant of Eq. (86) yields... 

Coordinates such as these, which have the symmetry properties of the point group are known as symmetry coordinates. As they transform in the same manner as the IRs when used as basis coordinates, they factor the secular determinant into block-diagonal form. Thus, while normal coordinates most be found to diagonalize the secular determinant, the factorization resulting horn the use of symmetry coordinates often provides considerable simplification of the vibrational problem. Furthermore, symmetry coordinates can be chosen a priori by a simple analysis of the molecular structure. 

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. 

Their difference, when substituted in Eq. (58) leads to the secular determinant in its more usual form, viz. 

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... 

Set up the secular determinant for the rr-system of naphthalene and factor it a explained in the last paragraph of this chapter. 

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. 

As a simple illustration of the development of the secular determinant, consider the water molecule. A reasonable set of internal coordinates consists of the changes in lengths of the two bonds and the variation in the bond angle. Thus, from Eq. (45) and Fig. 2,... 

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. 

T ie determinant in Eq. (59) is of course a secular determinant, a description that refers to its application to the temporal evolution of a mechanical system, historically in astronomy. It will re-appear later in this chapter in the development of the variation method. 

The secular determinant as presented above involves the first-order perturbations of the Hamiltonian and the energy. More generally, it is formulated in terms of the Hamiltonian and the total energies of the perturbed system. From Eqs. (12) and (16),... 

The expansion of the corresponding secular determinant leads to the relation... 

For a homonuclear diatomic system in the Hilrikel approximation the integrals given by Eqs. (128)—(131) take the simple forms Haa = Hyy = or, Hab = Hta = P and 5 = 0. The atomic orbitals involved, Xa and xtb are of coarse the px orbitals of carbon atoms a and b, respectively. The resulting secular determinant is then simply... 

Butadiene exists in two equilibrium structural isomers. They are represented in Fig. 7. However, with the usual Hflckel approximation these two structures cannot be distinguished, as interactions between nonadjaeent atoms have been neglected. Thus for either isomer, or even a hypothetical structure in which the carbon skeleton is linear, the secular determinant is the same, namelv. 

Consider the trans isomer of butadiene. Both the symmetry operations that define the group < 2h and the characters of the representation r are given in Table 3. The reduction of this representation leads to Tn =2Bg 2Aa. Thus, two linear combinations of the atomic orbitals can be constructed of symmetry Bg and two others of symmetry A. Their use will factor the secular determinant into two 2x2 blocks, as described in the following paragraph. 

As a final exercise for the reader, consider the naphthalene module (symmetry 02h) as shown in Fig. 10. Application of the HUcKel method leads to a lOx 10 secular determinant (see problem 30). However, with the application... 

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