Degenerate case

Let us consider the situation when the eigenvalues of the unperturbed Hamiltonian II 0) are respectively gi, -fold degenerate. In this case the unperturbed Schrodinger equation reads  

Equating as previously coefficients at the different powers of A on the two sides we get for 0-eth power of A 

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

In the non-degenerate case it was an inhomogeneous equation with the nonvanishing right hand part, which could be used to determine the first order energy /A1) and the expansion coefficients of the first order wave function It is not like this in the 

The point is that the vectors k 4 satisfying the unperturbed Schrodinger equation, if used to expand 44 make the right hand side disappear and the equation becomes a uniform one. The only thing we can do is to use it to determine the proper expansion coefficients of the zeroth order wave function b p in terms of the degenerate subspace as well as the first order energy. (The first order wave function is usually not calculated/considered in the degenerate case.) 

---

Whenever a fiinction can be written as a product of two or more fiinctions, each of which belongs to one of the synnnetry classes, the symmetry of the product fiinction is the direct product of the syimnetries of its constituents. This direct product is obtained in non-degenerate cases by taking the product of the characters for each symmetry operation. For example, the fiinction xy will have a symmetry given by the direct product of the syimnetries of v and ofy this direct product is obtained by taking the product of the characters for each synnnetry operation. In this example it may be seen that, for each operation, the product of the characters for Bj and B2 irreducible representations gives the character of the representation, so xy transfonns as A2. 

This shows that the set of functions (the Gp in this degenerate case) that are eigenfunctions of S can also be eigenfunctions of R. 

The popular and well-studied primitive model is a degenerate case of the SPM with = 0, shown schematically in Figure (c). The restricted primitive model (RPM) refers to the case when the ions are of equal diameter. This model can realistically represent the packing of a molten salt in which no solvent is present. For an aqueous electrolyte, the primitive model does not treat the solvent molecules exphcitly and the number density of the electrolyte is umealistically low. For modeling nano-surface interactions, short-range interactions are important and the primitive model is expected not to give adequate account of confinement effects. For its simphcity, however, many theories [18-22] and simulation studies [23-25] have been made based on the primitive model for the bulk electrolyte. Ap-phcations to electrolyte interfaces have also been widely reported [26-30]. 

Finally, it should be added that the conventional problem statement and pointwise solution format can be interpreted as a particular degenerate case of our more general formulations. As the minimum acceptable size for zones in the decision space decreases, the different performance criteria converge to each other and X gets closer and closer to x. Both approaches become exactly identical in the extreme limiting case where Ax = 0, m = 1,..., M, which is the particular degenerate case adopted in traditional formulations. 

Left-multiplying (32) by i occupied in a non degenerate case, using (19),... 

Note that since X — 1, Eq. (95) leads to the degenerate case where bgn — 0, and thus the sum of the correction terms over all species is null. Also, if one of the mass fractions is null3 (say Yy — 0) then Xy — Xy i and thus the correction term for such mass fractions will be null due to Eq. (96). 

However, perturbation-theoretic expressions such as Eqs. (1.24) and (2.7) are problematic in the degenerate case when donor and acceptor orbitals have equal energies.7 In this case we can directly formulate the interaction of orbitals a and b (with equal energies e A = b = e. and / ab = <a. F b ) in terms of a limiting variational model with 2x2 secular determinant... 

The procedure of the previous section provides a means of going from one basic feasible solution to one whose/is at least equal to the previous/(as can occur, in the degenerate case) or lower, if there is no degeneracy. This procedure is repeated until (1) the optimality test of relations (7.15) is passed or (2) information is provided that the solution is unbounded, leading to the main convergence result. 

Degenerate cases such as those above are not frequently encountered. More often, A = 0. Let... 

A localized operation is a degenerate case of a joint action with a distinct receiver, in which nothing is known or stated about the initiator s identity or attributes. All relevant aspects of the initiator are abstracted into the input and output parameters of the operation. The following is a fully localized operation that cannot refer to the initiator at all. 

The idea of a joint action as an abstraction of many possible interaction protocols was inspired by the work of Disco [Kurki-Suonio90], We treat localized actions as a degenerate case of joint actions. Clarifying the meaning of parameters (and the corresponding rales and documentation for refinement) in the context of objects, and the treatment of use cases, we believe to be unique to our approach. Traditional use cases are described by Jacobsen in [Jacobsen92],... 

Now, we turn to the degenerate case. Consider the solutions of the Schrodinger equation... 

Basically, two types of approaches are developed here iterative (optimization-based) approaches like the one by Sippl et al. [101] and direct approaches like the one by Kabsch [102, 103], based on Lagrange multipliers. Unfortunately, the much expedient direct methods may fail to produce a sufficiently accurate solution on some degenerate cases. Redington [104] suggested a hybrid method with an improved version of the iterative approach, which requires the computation of only two 3x3 matrix multiplications in the inner loop of the optimization. 

An estimate of the longest relaxation time can be obtained by applying the perturbation theory for linear operators to the degenerated case of the zero eigenvalue of the matrix K. We have. ..,Ay i)-I-fcy Q-I-o(fc,),... 

Equations (3.33), (3.34), (3.38), and (3.39) are valid under the condition that the perturbation be weak compared to energy differences, that is, hab/ ea — / ) 1. The case 1 and case 2 studies revealed the situation in the degenerate case. We now interpolate the intermediate situation, remembering that overlap will not be zero except when dictated to be so by local symmetry. However, overlap will always be small when dealing with interactions between molecules. Refer to Figures 3.1-3.4 for the defined quantities. 

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

Thun, under the general conditions stated above, p.m. is valid to the second approximation ivith respect to the eigenvalues and to the first approximation with respect to the eigenvectors in the degenerate case too, provided =... 

Note that eigenvector orthonormality can also be safely assumed for the degenerate case, cii = ap without loss of generality.)... 

Thus, in the degenerate case also,, (0) is invariant to all symmetry operations. 

This variant is especially valuable in near-degenerate cases and when several roots are required. 

For example, in recent years Macosko and Miller (MM)37-40 have developed an attractively simple method which at first sight appears to be basically new. However, a closer inspection reveals the MM approach as being a degenerate case of the more general cascade theory. The simplicity is unfortunately gained at the expense of generality, and up-to-date conformation properties are not derivable by the MM-technique. 

It might be thought that the preceding discussion of a trivial special case was uninteresting, but this is not so, for the structure of the system in the degenerate case 71 = 72 = 0 contains the clue to the structure of the general case. Writing the equations for this case as ... 

---