The Non-Crossing Rule

When computing multiple potential energy surfaces for a system, for instance, the ground state and electronic excited states, the possibility of the surfaces having the same value of potential energy, U(R), occurs. This might be, for example, where a minimum on the upper surface comes into the region of a maximum on the lower surface. What happens at this point is of great interest to chemists and is discussed in this section.

The non-crossing rule was quantitatively formulated in 1929 by von Neumann and Wigner,[28] proving the theorem put forward earlier by Hund.[29] It states that: potential energy curves corresponding to electronic states of the same symmetry cannot cross.

When two wavefunctions are mixed to form a better wavefunction, the new energies repel each other. This is best shown in the form of a molecular orbital (MO) diagram (Figure 11.1):

Figure 11.1 - Molecular orbital (MO) diagram for H2

The energies ε_A and ε_B are the energies of the original (atomic) wavefunctions, prior to mixing. The energies E₁ and E₂ represent the energies of the new (molecular) wavefunctions after mixing.
E₂ is higher in energy than either of the atomic wavefunctions, and E₁ is lower in energy than the original wavefunctions. This is typical of anti-bonding and bonding molecular orbitals, respectively.

Consider an analogous situation; that of an NaCl molecule. We can write two wavefunctions for this: an ionic (Na⁺Cl^–) wavefunction and a covalent (Na—Cl) wavefunction. At any fixed internuclear separation, the best wavefunction is a mixture of these two wavefunctions, determined by the variational principle (see section 7.1) – equation (11.1):

As the internuclear separation changes, the ionic and covalent wavefunctions will, at some point, cross (Figure 11.2, below):

Figure 11.2 - Crossing of ionic and covalent potential energy curves for NaCl

The curves are based on simple coulombic attraction between charged spheres (of which there is zero for the covalent model), and repulsion of nuclear charge and physical interaction between hard spheres.
To obtain the best wavefunctions and energies at any particular point, we must mix the ionic and covalent wavefunctions. The resulting energies repel each other. Therefore, at every internuclear separation, the energies of the original wavefunctions are “split apart.” At the crossing point, the new energies (which will be closer to the true energies than the original ionic and covalent wavefunction energies) split apart and the curves no longer cross. This point of nearest approach is called an avoided crossing point (Figure 11.3).

Figure 11.3 - Avoided crossing for the mixed wavefunctions of the ionic and covalent NaCl energies

Two potential energy curves can cross if their symmetries are sufficiently different. Hund wrote that if two potential energy curves cross, the electronic state must be degenerate at the point of crossing.[29]
von Neumann and Wigner gave the mathematical proof for Hund’s argument, given here in a contracted form:

The correct electronic wavefunctions for all energy levels except the two potential energy curves which the crossing of we wish to investigate are assumed to be known. Arbitrary wavefunctions, orthogonal to each other and the other ‘known’ wavefunctions, ψ₁ and ψ₂ are chosen. Each of these electronic wavefunctions can be represented as in equation (11.2):

(11.2)

Equation (11.2) can be represented as a Slater determinant in the form shown in equation (11.3):

(11.3)

For (11.3) to have degenerate solutions, it is necessary that:

(11.4)

In order to satisfy these conditions, at least two independently variable nuclear coordinates are required.
Clearly, for diatomics, the non-crossing rule will be enforced as there is only one parameter, the internuclear separation, and so degeneracy is not possible.
For polyatomics, however, there are sufficient parameters available to achieve degeneracy, so the non-crossing rule is apparently not enforced and states of similar symmetry may freely cross in polyatomic systems.
In general, three parameters are necessary, since H₁₂ may be complex.

However, avoided crossings (also called near intersections) are also seen in polyatomic systems. This caused much controversy, circa 1972, concerning the non-crossing rule when Naqvi[30] proposed that the non-crossing rule did indeed apply to polyatomic surfaces.
Naqvi observed that where H₁₁ = H₂₂, the possibility that H₁₂ = 0 can be excluded provided that the potential energy surfaces of the two states are not equal. They went on to conclude that the conditions in are not satisfied given that the energies are equal.
They propose two conditions for a crossing at R₀, thus:

(11.5)

…where F₁₂ is given by equation (11.6):

(11.6)

The non-crossing rule is then justified, Naqvi wrote, on the basis that there is no reason why these two conditions should be simultaneously satisfiable.

Naqvi’s work was quickly rebuffed by Longuet-Higgins,[31] where he argued that the conditions given in are automatically satisfied simultaneously when a crossing occurs. He further showed that intersections of more than two surfaces are possible, and that, for an n-fold degeneracy, ½n(n+1)–1 independent conditions are required. This means, Longuet-Higgins reasoned, that three-fold degeneracies (requiring five independent parameters) are possible (and quite likely) in molecules containing four or more atoms (where six independent variables are accessible).
Hatton, et al. also studied the situation of avoided crossings in polyatomic surfaces and suggested that such near intersections may be the effects of approximate calculations, since the surfaces are later found to cross in an exact solution of the electronic wave equation.[32]
This is, however, not to say that all avoided crossings in polyatomic surfaces are actually intersections.
The traditional viewpoint is now accepted that polyatomic potential energy surfaces can cross.