The non-crossing rule

A perhaps overfanciful pictorial representation of this avoided crossing is portrayed in Fig. 3.8. If Iscr and 2sct were of different symmetry species, they 

Chapter 3. Diatomic Molecules and their Molecular Orbitals 

To solve this problem in detail let us limit ourselves to the simplest situation the two-state model (Appends D). Let us consider a diatomic molecule and such an intemuclear distance Rq that the two electronic adiabatic states r/rj (r Rq) and 4 2(r Rq)) correspond to the non-degenerate (but close on the energy scale) eigenvalues of the clamped nuclei Hamiltonian Ho(Ro)- 

The crossing of the energy curves at a given R means that + = E-, and from this it follows that the expression under the square root symbol has to equal zero. Since, however, the expression is the sum of two squares, the crossing needs two conditions to be satisfied simultaneously  

Two conditions, and a single changeable parameter R. If you adjust the parameter to fulfil the first condition, the second one is violated and vice versa. The crossing E+ = E- may occur only when, for some reason, e.g., because of the symmetry, the coupling constant is automatically equal to zero, Vi2 = 0, for all R. Then, we have only a single condition to be fulfilled, and it can be satisfied by changing the parameter R, i.e. crossing can occur. The condition Vi2 = 0 is equivalent to 

If two states of a diatomic molecule have the same q mmetry, then the corresponding potential energy curves cannot cross. 

Our goal now is to show, in an example, what happens to adiabatic states (eigenstates ofii(R)), if two diabatic energy curves (mean values of the Hamiltonian with the diabatic functions) do cross. Although we are not aiming at an accurate description of the NaCl molecule (we prefer simplicity and generality), we will try to construct a toy (a model) that mimics this particular stem. 

---

Katriel J, Davidson ER (1980) The non-crossing rule triply degenerate ground-state geometries of CH. Chem Phys Lett 76 259... 

There may be special difficulties in reactions where the ordering of orbitals centred on the metal changes along the actual reaction path, because of configuration interaction and the non-crossing rule for states. 

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. 

The established methods of proving the non-crossing rule which are in the literature (32,33) have recently been criticised by Naqvi and Byers-Brown (34), but their proof in turn has been criticized by Longuet-Higgins (35). All proofs attempt to show that crossing will only occur if conditions are satisfied for two functions of the internuclear distanced which are assumed to be independent. If we write this in the general form... 

Once the two sides of a correlation diagram have been established, the states of the same symmetry and multiplicity are connected by straight lines in such a way as to observe the non-crossing rule identical states cannot cross as the strength of the interaction is changed. When this is done we have completed the correlation diagram. 

Thus we leam three things 1) the non-crossing rule is not obeyed in the present picture of unstable resonance states, 2) complex resonances may appear on the real axis and 3) unphysical states may appear as solutions to the secular equation. Thus avoided crossings in standard molecular dynamics are accompanied by branch points in the complex plane corresponding to Jordan blocks in the classical canonical form of the associated matrix representation of the actual operator. 

Molecular Crystals as a Consequence of the Non-Crossing Rule (Level Anti-Crossing). 

Enhancement via Albrecht s 5-term derives from the non-Condon dependence of the electronic transition moment upon the vibrational coordinate. Unlike the A-term, the 6-term arises from the vibronic mixing of two excited states and it is non-zero for scattering due to both totally symmetric and non-totally symmetric fundamentals, provided that they are responsible for vibronic coupling of the states. The latter only takes place for a vibrational fundamental whose irreducible representation is contained in the direct product of the irreducible representations of the two states. Thus, 6-term activity for a totally symmetric mode requires that the latter must vibronically couple two states of the same symmetry. As a consequence of the non-crossing rule this holds only for few excited states which are lying very close together. 

Each reactant state correlates with some state of the products along the potential. Vibrations and rotations that are similar in the reactant and product (conserved modes), remain in the same quantum state throughout the channel, in the sense that their quantum numbers remain the same throughout. Other modes that change between reactants and products (transitional modes), are subject to correlation rules. Channels with the same angular momentum are not permitted to cross, similar to the non-crossing rule in diatomic molecules. 

Figure 9. Schematic adiabatic channel potentials for a dissociation reaction, illustrating the development of barriers due to convergence of vibrational levels and the non-crossing rule. At the energy indicated, two of the channels are open and two are closed.

The non-crossing rule requires that adiabatic potential energy curves which belong to electronic states of the same symmetry species (same A, S for nonrelar tivistic curves, same Cl for relativistic curves) do not cross. The Wigner-Witmer... 

We can reach the same conclusions by using orbital correlation arguments. If the symmetries of the bonds that are broken and the symmetries of the bonds that are made always match up in pairs, the non-crossing rule will then guarantee that none of the corresponding pairs of orbitals will cross. If they do not cross, leading to a h5 othetical excited state product the reaction is allowed by the orbital correlation procedure. 

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. 

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. 

Sometimes it is said that the barrier results from an avoided crossing (cf. Chapter 6) of two diabatic hypersurfaces that belong to the same irreducible representation of the symmetry group of the Hamiltonian (in short of the same q mmetry ). This, however, caimot be taken literally, because, as we know from Cliapter 6, the non-crossing rule is valid for diatomics only. The solution to this dilemma is the conical intersection described in Chapter 6 (cf. Fig. 6.15). Instead of diabatic we have two adiabatic hypersurfaces ("upper and lower ), each consisting of the diabatic part I and the diabatic part II. A thermic reaction takes place as a rule on the lower hypersurface and corresponds to crossing the border between I and n. 

---