Coulson contour

V. K. Motlitt and C. A. Coulson. Phil. Mag. [7], 3S. 634 (1937 . These authors also give instructive contour diagrams of carlwn atom He, 2Ir, and 2di hybrid AO s, using orthogonalized Stater 25 AO s. 

Based on this conclusion one can introduce the concept of the partial electron density and draw its contour map in the plane just above and below, say one Bohr radius, the molecular plane [24, 29, 34]. Again for XVI and XII the results of g3 are given in Fig. 5, which is the contribution of the highest three occupied Huckel MO s. Note that due to the pairing theorem proved by Coulson and Rushbrooke [35] the n-electron densities on all the component carbon 2pn orbitals are the same and the contour map of the conventional electron density cannot differentiate any of the local aromaticity of alternant hydrocarbon molecules. 

Electron density contour diagrams have been derived " (Figure 2) and potential energy curves and surfaces calculated In addition to this impressive quantity of accurate work on the parent compound a number of simplified schemes were proposed by Forster, Coulson and Moffit " and by Walsh these schemes have been widely employed for the interpretation of the photoelectron spectra and other properties of the derivatives of cyclopropane. 

Poles and residues of the electron propagator Gv, ( j provide the spin orbital energies and the molecular orbital amplitudes. The sum of the energies of the occupied spin orbitals is used as a measure of the total energy of the 7t-orbital system and Coulson observed that this could be expressed as a contour integral in the complex energy plane... 

The contour diagrams, in the first row, emphasize that the contours of mathematical functions are being represented. The orbital shapes confirm the symmetries of these contours and include their phases as the signs on the lobes of the individual pictures. But, phase is not determined, since only the square of the wave function is of physical significance, as the probability density. And so, for example, the sign is not given for the s-orbital. [After Coulson s Valence, R. McWeeny, OUP 1979 Roy McWeeny 1979]. 

The contour C consists of the one given in Fig. 3.1 plus an infinite semicircle in the upper half of the complex F plane or in fact any other contour obtainable from it by a deformation that does not bring it through any of the singularities. A particular contour used by Coulson is shown in Fig. 3.2... 

Figure 3.2 Contour used by Coulson (see text). The total energy can be obtained as...

Using the asymptotic form of D E) for large E-values and the contour integral expression in Eq. (3.38), one can derive the Coulson s energy formula... 

A convenient contour (apparently first used in quantum chemistry by Coulson, 1937) is the one shown in Fig. 13.1. The required result is then... 

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