
The NonCrossing 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 noncrossing 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):
The energies ε_{A} and ε_{B}
are the energies of the original (atomic) wavefunctions, prior to mixing.
The energies E_{}_{1} and
E_{2} represent the energies of the
new (molecular) wavefunctions after mixing. 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):
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.
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] 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, ψ_{1} and ψ_{2} are chosen. Each of these electronic wavefunctions can be represented as in equation (11.2):
Equation (11.2) can be represented as a Slater determinant in the form shown in equation (11.3):
For (11.3) to have degenerate solutions, it is necessary that:
In order to satisfy these conditions, at least two independently variable
nuclear coordinates are required. However, avoided crossings (also called near
intersections) are also seen in polyatomic systems. This caused
much controversy, circa 1972, concerning the noncrossing rule when Naqvi[30]
proposed that the noncrossing rule did indeed apply to polyatomic
surfaces.
…where F_{12} is given by equation (11.6):
The noncrossing 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 LonguetHiggins,[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 nfold
degeneracy, ½n(n+1)–1
independent conditions are required. This means, LonguetHiggins reasoned,
that threefold degeneracies (requiring five independent parameters) are
possible (and quite likely) in molecules containing four or more atoms
(where six independent variables are accessible). 

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[28] J. von Neumann and E.P. Wigner, Z. Physik, 30, 467, 1929 [29] F. Hund, Z. Physik, 40, 742, 1927 [30] K. Razi Naqvi, Chem. Phys. Lett., 15, 634, 1972 [31] H.C. LonguetHiggins, “The Intersection of Potential Energy Surfaces in Polyatomic Molecules”, Proc. R. Soc. Lond. Ser. A., 344, 147156, 1975 [32] G.J. Hatton, W.L. Lichten and N. Ostrove, “Nonnoncrossings in Molecular Potential Energy Curves”, Chem. Phys. Lett., 40, 437, 1976 
Potential Energy Surfaces and Conical Intersections • June 2002 • Ian Grant 