Eigenvector

The exact ground-state eigenvalue and corresponding eigenvector... 

Nevertheless, equation (A 1.1.145) fonns the basis for the approximate diagonalization procedure provided by perturbation theory. To proceed, the exact ground-state eigenvalue and correspondmg eigenvector are written as the sums... 

In practice, the matrix (iL+R+K) is diagonalized first, with a matrix of eigenvectors, U, as in equation (B2.4.15)), to give a diagonal matrix. A, with the eigenvalues, X, of L down the diagonal. 

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). 

For slow exchange, a convenient matrix of eigenvectors is given by equation (B2.4.23). 

After coalescence, a possible set of eigenvectors is given in equation (B2.4.24). If these are substituted into (B2.4.22), the results are pure real, reflecting the fact that iP - 8 is now positive. 

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. 

Note that the Liouville matrix, iL+R+K may not be Hennitian, but it can still be diagonalized. Its eigenvalues and eigenvectors are not necessarily real, however, and the inverse of U may not be its complex-conjugate transpose. If complex numbers are allowed in it, equation (B2.4.33) is a general result. Since A is a diagonal matrix it can be expanded in tenns of the individual eigenvalues, X. . The inverse matrix can be applied... 

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... 

E.( ), possesses a complete set of eigenfiinctions, the matrix F whose dimension Mis equal to the number of atomic basis orbitals, has M eigenvalues e. and M eigenvectors whose elements are the. . Thus, there are... 

CISD Yes/No A/ transformed integrals n N to solve for one Cl energy and eigenvector... 

Aq becomes asymptotically a g/ g, i.e., the steepest descent fomuila with a step length 1/a. The augmented Hessian method is closely related to eigenvector (mode) following, discussed in section B3.5.5.2. The main difference between rational fiinction and tmst radius optimizations is that, in the latter, the level shift is applied only if the calculated step exceeds a threshold, while in the fonuer it is imposed smoothly and is automatically reduced to zero as convergence is approached. 

There is no simple genei al form for the adiabatic eigenvectors, except in the limits, k = 0 and i = 0, wlien, for example. 

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,... 

As noticed from Figure 15, the two surfaces u and 2 ate conelike potential energy surfaces with a common apex. The corresponding eigenvectors are... 

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