No crossing rule

This rule is also known as the no-crossing rule. For a discussion, see Albright, T. A. Burdett, J. K. Whangbo, M.-H. Orbital Interactions in Chemistry John Wiley Sons New York, 1985 pp. 52-53. 

When the states in the weak-field limit are connected with like states in the strong-field limit using the no-crossing rule, the result is a correlation diagram, where there is a one-to-one correspondence between the term symbols from... 

A conical intersection needs at least two nuclear degrees of freedom to form. In a ID system states of different symmetry will cross as Wy = 0 for i j and so when Wu = 0 the surfaces are degenerate. There is, however, no coupling between the states. States of the same symmetry in contrast cannot cross, as both Wij and Wu are nonzero and so the square root in Eq. (68) is always nonzero. This is the basis of the well-known non-crossing rule. 

As an example we can take the excited states of NO. It has been shown that there are two excited states of the same symmetry ( 11) whose vibrational levels are best interpreted on the basis of diabatic curves which cross as in Fig. 1 (75-7 7). One of these states (B) arises from the electron excitation to an antibonding valence molecular orbital and the other (C) from excitation to a Rydberg orbital. The Born-Oppenheimer adiabatic curves cannot cross (by virtue of the non-crossing rule which is to be discussed in a later section) and must fullow the dashed curves shown in the figure. 

Thus the scalar product of vectors A and B in two-dimensional space is equal to the sum of the products of their components with no cross terms (e.g., AxBy). This result is actually only a special case of the general rule in p-dimensional space ... 

In this language, the implications of the W-H rules are that allowed reactions will proceed in a concerted fashion and that forbidden reactions will not, the argument in the latter case being that some other, non-concerted mechanism is likely to provide a more stable pathway. It is now fairly clear that there are exceptions to both of these generalizations. In this connection the concept of orbital isomerism is helpful. This was introduced by Dewar, Kirschner, and Kollmar,39 who postulated that a group of isomeric molecules can be divided into sets in such a way that members of any one set can be interconverted by paths which involve no crossing between the highest... 

The above conclusion applies as well to non-activated chemisorption, fn chemisorption, however, the chemical specificity of the reaction is such that factors like the crossing of electronic curves, the existence of steric constraints, and the coupling with the heat reservoir are expected to play a fundamental role, so that no general rule can be specified. 

A number of hydrated inorganic salts have also been studied by the inelastic neutron scattering (INS) lechnique. Since the proton scattering cross section is quite large, the INS spectrum reflects mainly the motion of the protons in the crystal. Furthermore, INS spectroscopy has no selection rules involving dipole moments or polarizabilities. Thus it serves as a complementary tool to vibrational spectroscopy in studying the hydrogen vibrations of hydrated salts. 

In solids, the selection rules can seldom he considered as absolute rules. The situation is reminiscent of towns with low traffic morals when the traffic light is green everyone crosses (allowed transition, no selection rules), at red there are still a few people who cross against the rules (forbidden transition, but selection rule slightly relaxed). The relaxation of selection rules is connected to wavefunction admixtures into the original, unperturbed wave functions. This can be due to several physical phenomena, like spin-orbit coupling, electron-vi brat ion coupling, uneven crystal-field terms, etc. Their treatment lies outside the scope of this book. The reader is referred to Refs. [11 and [3]. 

The secondary branches have no crossings for comb structures such as the one shown in Fig. 6.3, and rule (iii) does not apply. For the sake of generality, we consider the case that when the walker is at a site on the backbone it can jump to another site on the backbone with probability a or move onto the secondary branch with probability I - a. This generalization allows us to analyze, for example, structures as those shown in Fig. 6.4, treating them as equivalent to that in Fig. 6.3 by choosing an appropriate value for a. 

The non-crossing rule for a diatomic molecule was based on Eq. (6.47). To achieve the crossing, we had to make vanish two independent terms with only one parameter (the intemuclear distance R) to vary. It is important to note that in the case of a polyatomic molecule, the formula would be the same, but the number of parameters would be larger 3M — 6 in a molecule with M nuclei. For M = 3, therefore, one has already three such parameters. No doubt even for a three-atomic molecule, we would be able to make the two terms equal to zero and. therefore, achieve E+ = E- i.e., the crossing of the two diabatie hypersurfaces would occur. 

There are additional rules for assigning methylenes. Firstly, a methylene connecting only one methine peak is either one of mmm, rmi mrm, and rrr secondly, methylenes showing no cross-peak in the nietliylene-methylene corss-peak region is mrm or rrr and thirdly, a methylene connecting two methines is mmr or mrr. 

Figure 2 is a reproduction of the original figure. Note that there is no scale for the energy values. The most important result from this section, which may surprise the reader, is that HL concluded that the energy curves of E and E cannot cross likewise, the curves for Ep and E p do not cross. Thus, the paper of HL is not only the first quantum chemical study which explains the nature of the covalent bond it also contains the non-crossing rule. The authors clearly say that the two solutions a and a (likewise P and P ) can be combined linearily however, the exact form of the combination would not be predictable without further investigation. 

In (B) of Fig. 6.1 the py <— cr correspondence is induced by the second-order perturbation [02 0 61] 62 . It has no energetic consequences in the present example, because the orbitals involved are unoccupied. Moreover, successive or simultaneous displacement along coordinates belonging to any two non-totally symmetric irreps of desymmetrizes the system all the way down to Cl, in which - subject to the non-crossing rule - any pair of orbitals can be brought into correlation. The energetic consequences of composite displacements will be explored in subsequent examples, in which they are called upon to induce correspondence between occupied orbitals and where the initial symmetry is sufficiently high that they lead to only partial desymmetrization. 

Cross-linked Polymers. As the name indicates, these have two or more separate polymeric chains connected hy a cross-link it is implied that the crosslink is generally (1) relatively short (and not itself polymeric) and (2) at irregular intervals along the chains, hut there seem to he no firm rules on these points. Ladder polymers comprising two linear strands cross-linked at regular intervals are discussed under Douhle-Strand (Ladder) Polymers. 

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