Matrix diagonalization perturbation method

The RF interference associated with an infinite number of effective RF fields is a typical problem addressed by the Floqute theory.28 32 To solve the problem, one may, however, face the diagonalization of a matrix with infinite dimensions, which is often unlikely to be done analytically. Certain approximations, such as the perturbation method, may be used.32... 

Then the (w -f 1) x (m -f 1) eigenvalue matrix can be constructed from (9) and can be solved exactly by direct diagonalization or with the aid of perturbation methods for the eigenvalues with the external magnetic field along any principal axis, (m -f 1) Energy values and therefore m Am, = 1... 

Simulations of the X-band spectra by approximate (2nd order perturbation theory) and exact (matrix diagonalization) methods differ slightly in this case, see Fig. 3.24 in Section 3.4.I.7. Some spectral features caused by the fact that the high-field approximation does not strictly apply are worth noting ... 

The traditional approach, in which the whole matrix H is computed and eigenvalues and eigenvectors are found by standard diagonalization techniques, is obviously unable to deal with this situation (a matrix with 10 rows and columns would not even fit in the computer ). In the last chapter such difficulties were avoided by perturbation methods, but other difficulties remained in particular, the expansion is infinite, convergence is uncertain, only selected terms can be summed, and for open-shell and excited states there are further problems. 

Yarkoni [108] developed a computational method based on a perturbative approach [109,110], He showed that in the near vicinity of a conical intersection, the Hamiltonian operator may be written as the sum a nonperturbed Hamiltonian Hq and a linear perturbative temr. The expansion is made around a nuclear configuration Q, at which an intersection between two electronic wave functions takes place. The task is to find out under what conditions there can be a crossing at a neighboring nuclear configuration Qy. The diagonal Hamiltonian matrix elements at Qy may be written as... 

Due to the size of the variational problem, a large Cl is usually not a practicable method for recovering dynamic correlation. Instead, one usually resorts to some form of treatment based on many-body perturbation theory where an explicit calculation of all off-diagonal Cl matrix elements (and the diagonalization of the matrix) are avoided. For a detailed description of such methods, which is beyond the scope of this review, the reader is referred to appropriate textbooks295. For the present purpose, it suffices to mention two important aspects. 

Several methods exist for calculating g values. The use of crystal field wave functions and the standard second order perturbation expressions (22) gives g = 3.665, g = 2.220 and g = 2.116 in contrast to the experimentaf values (at C-band resolution) of g = 2.226 and g 2.053. One possible reason for the d screpancy if the use of jperfXirbation theory where the lowest excited state is only 5000 cm aboye the ground state and the spin-orbit coupling constant is -828 cm. A complete calculation which simultaneously diagonalizes spin orbit and crystal field matrix elements corrects for this source of error, but still gives g 3.473, g = 2.195 and g = 2.125. Clearly, covalent delocalization must also be taken into account. 

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