The Eigenvector

The eigenvectors of the twelve phonons in the naphthalene crystal are sketched in Fig. 5.4 for the special case K = 0, that is at the F point of the 1st BriUouin zone (1st BZ). The modes 1, 2, 3, 5, 7 and 10 are pure translational vibrations. The vectors are roughly parallel to the crystal axes a, b and (/, o is perpendicular to the a-b plane. Each of these 3 directions corresponds to a mode in which the two molecules in the unit cell oscillate in phase (1, 2 and 3) and to a mode in which 

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It is convenient, for simple systems, to have explicit expressions for equation (B2.4.17). Since the original matrix is non-Hemiitian, the matrix fomied by the eigenvectors will not be unitary, and will have four independent complex elements. Let them be a, b, c and d, so that U is given by equation (B2.4.20). 

As with the uncoupled case, one solution involves diagonalizing the Liouville matrix, iL+R+K. If U is the matrix with the eigenvectors as cohmms, and A is the diagonal matrix with the eigenvalues down the diagonal, then (B2.4.32) can be written as (B2.4.33). This is similar to other eigenvalue problems in quantum mechanics, such as the transfonnation to nonnal co-ordinates in vibrational spectroscopy. 

The quantities (U and (UF ) in B2.4.34 are projections of the eigenvector J along 1. From the above equations, this can be interpreted as follows. The temi (UF ) is the amount that the transition J received from... 

At this stage, we are ready to prove that Kramers theorem holds also for the total angular momentum F. We will do it by reductio ad absurdum. Then, let Itte) be the eigenvector of H with eigenvalue E,... 

Step 3 The eigenvectors of C define 3N - 6 collective coordinates (quasi... 

The occurrence of the argument pj2 shows that these eigenvectors are defined up to a sign only. For a unique representation we have to cut the plane along a half-axis. By this, vector fields uniquely defined on the cut plane. They cannot, however, be continued over the cut, but change their roles there instead. Thus, we have the situation of a crossing at which the eigenvector field is discontinuous and Assumption (A) of Thm. 3 is hurt. 

A square matrix has the eigenvalue A if there is a vector x fulfilling the equation Ax = Ax. The result of this equation is that indefinite numbers of vectors could be multiplied with any constants. Anyway, to calculate the eigenvalues and the eigenvectors of a matrix, the characteristic polynomial can be used. Therefore (A - AE)x = 0 characterizes the determinant (A - AE) with the identity matrix E (i.e., the X matrix). Solutions can be obtained when this determinant is set to zero. 

TlypcrC hcin uses the eigenvector I ollowing inelhod described in Bnker, J, An Algorllhin for the Lociillon of Tran sition SLiiics, ... 

HyperChem uses the eigenvector following method described in Hakcr, J., 1 Compul. Chern.. 7, 385-395 (1986), wdierc the details of the procedures can be found. 

D is the diagonal matrix containing the eigenvalues A, of S, and U contains the eigenvectors of S. is the transpose of the matrix U. (This expression is often written U SU = D since for real basis functions = U. ) Then the matrix X is given by X = where... 

It turns out that the htppropriate X matrix" of the eigenvectors of A rotates the axes 7t/4 so that they coincide with the principle axes of the ellipse. The ellipse itself is unchanged, but in the new coordinate system the equation no longer has a mixed term. The matrix A has been diagonalized. Choice of the coordinate system has no influence on the physics of the siLuatiun. so wc choose the simple coordinate system in preference to the complicated one. 

Alternative procedure Mathcad. Follow the procedure above except that where QMOBAS is indicated, use Mathcad instead. Enter the Huckel molecular orbital matrix, modified by subtracting xl, with some letter name. For example, call the modified matrix A. Type the command eigenvals(A) = with the name of the modified HMO matrix in parentheses. Mathcad prints the eigenvalues. The command eigenvecs(A) yields the eigenvectors, which are useful in ordering the energy spectrum. 

The optimization procedure is canied out to find the set of coefficients of the eigenvector that minimizes the energy. These are the best coefficients for the chosen linear combination of basis functions, best in the sense that the linear combination of arbitrarily chosen basis functions with optimized coefficients best approximates the molecular orbital (eigenvector) sought. Usually, some members of the basis set of funetions bear a eloser resemblanee to the true moleeular orbital than others. If basis function a +i. 

In this case Mathcad has already arranged the eigenvectors in their proper order as can be verified by the following queries ... 

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