Nondegenerate case

The problem of finding a vector is usually solved by representing the required vector as an expansion with respect to some natural set of basis vectors. Following this method one can expand the vector of the n-th order correction to the k-th unperturbed vector- 44 n terms of the solutions b p (eigenvectors) of the unperturbed problem eq. (1.51)  

By this, the expansion coefficients uffl are themselves of the 0-eth order in A. The restriction l / k indicates that the correction is orthogonal to the unperturbed vector. In order to get the corrections to the /c-th vector, we find the scalar product of the perturbed Schrodinger equation for it written with explicit powers of A with one of the eigenvectors of the unperturbed problem p (j k). For the first order in A we get  

Collecting the correction to the wave function on the left side and all the perturbation terms on the right side we get 

The first order correction to the fc-th eigenvector (wave function) then reads as follows  

This result indicates that the Rayleigh-Schrodinger PT is expected to be well applicable in those cases in which these fractions are small. Inserting eq. (1.62) into expressions for the energy corrections we get the well-known explicit expression for the second-order ones  

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For this case study, we again adopt the zero-overlap approximation but consider the nondegenerate case, assuming that sA > . The secular equation, from equation (3.15), becomes... 

Figure 3.5. Orbital interaction diagrams a) nondegenerate case (b) degenerate case.

This secular equation is an algebraic equation of degree n with n roots ,(1),. .., (,) that give the n first-order energy corrections. Substitution of each (,) in turn into (1.205) allows one to solve for the set of coefficients ctj (/=I,..., ) that go with the root, ,(1). Having found the n correct zeroth-order wave functions <)py(0) (j = 1,..., ), we can then proceed to find Ej<2 ipjl and so on the formulas for these corrections turn out to be essentially the same as for the nondegenerate case, provided that (pj0) is used in place of 0). 

The second-order correction to the energy of a ground state is always negative as all Ef] > eP for j / 0 (nondegenerate case). 

For the first power of A we have just the same equation as for the nondegenerate case ... 

The first equation as in the nondegenerate case, can be satisfied by taking the correction to be orthogonal to the unperturbed ground state vector which is satisfied by any linear combination of vectors from the manifold related to the /-th degenerate eigenvalue. Inserting the required expansion we get ... 

The numerator Pu(x) follows from the truncation of Eq. (5) at power xM. By this procedure, the Pade approximation problem is solved explicitly in the nondegenerate case (Baker and Graves-Morris, 1981). 

Alternatively, the sign of both exciton bands may be reversed. For the more general, nondegenerate case of coupled electric dipoles involving two different chromophoric groups, the rotatory strength R is given by... 

We see that the first order density change 8 cannot be zero if 8v is not a constant. In contrast to the nondegenerate case equation (164) does not imply that the function X is invertible, only the relation in equation (162) is invertible where the inverse only exists on the set of v-representable density variations within the set B. 

Using procedures similar to those for the nondegenerate case, one can now find the first-order corrections to the correct zeroth-order wave functions and the second-order energy corrections. For the results, see Bates, Volume I, pages 197-198 Hameka, pages 230-231. 

This last equation shows that the desired coefficients are proportional to the sums of the characters of the group operations which transfer the displacement Si to Si. Therefore the final result is the same as for the nondegenerate cases, namely,... 

In both degenerate and nondegenerate cases the resultant upper mdecular level is destabilized more than the lower one is stabilized. 

The coefficients c, and Cz, for / = 1,2 wllbe obtained for the degenerate and nondegenerate cases described above. 

Hie form of the solutions f of this equatitm will be examined for a degenerate case ( i - 2) and for the general nondegenerate case (e e ). 

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