Coulson formulae

More generally, one has to consider the weights of VB structures, which are quantitatively related to physical properties like electron densities, net charges, and so on. According to the popular Chirgwin—Coulson formula (5), the weight of a given structure, d>x, is defined as the square of the coefficient plus one-half of the overlap population terms with all the other structures ... 

This formula is the VB analogue of the Mulliken population in MO-based calculations. The VB weights sum to unity if the wave function 1P, in Equation 3.54, is normalized. However, Equation 3.55 can be used even if the VB structures are not normalized, or even if represents an AO-based determinant rather than a VB structure. In such a case, it is useful to note that with the definition of the weights as in the Chirgwin—Coulson formula, the weight of a VB structure is equal to the sum of the weights of its constituting VB determinants. 

Then, the weight of 2123 is calculated by means of the Chirgwin—Coulson formula Eq. 3.55 ... 

The decomposition of the As coupling into isotropic and anisotropic constants leads to arsenic 4s and 4p spin densities equal to 0.06 and 0.75, respectively. From the resulting hybridization ratio X, close to 10.8, the out-of-plane angle can be calculated by using the Coulson formula and by assuming a symmetry (see Figure 2). The value of C (11°)... 

Coulson formula cos 0 = -1 /. If the hybrid orbitals are assumed to be oriented along the... 

Coulson concluded that the most important contribution to H-bonding is ionic resonance (5.29a). However, generations of empirical modelers have found it convenient to employ simple pairwise-additive Coulombic formulas with empirically fitted point charges to model H-bonds, and such empirical models have tended to encourage uncritical belief in the adequacy of a classical electrostatic picture of H-bonding. 

We select without proof, from the theory of Coulson and Longuet-Higgins (1947a,b), those formulae which relate to quantities used in the description of chemical reactivity, as follows ... 

Coulson (reference If) also gives formulas corresjwmding to the majority of the. V integrals listed here. However, his formulas for 0 are in terms of w, and A, in a form not adapted to the present computation and tabulation. His formulas for /-- 0 are identical with our Ivjs. (otj-(63j except that they are not normalized like ours to give. V values directly. 

Alternatively, they may be fixed from the outset. To the structures, weights can be attributed, which add up to one, using a formula given by Chirgwin and Coulson [7]. 

In this section we report the Coulson-type integral formulas for total re-electron energy and their modifications applicable for TRE, ef and efa. We consider here only the special cases when the respective conjugated systems are benzenoid hydrocarbons. 

The integral formulas for total rr-electron energy were invented by Coulson [2, 3, 6], Later, they were extensively elaborated (see [13], pp. 139-147). Their real usefulness in structure-property analysis became evident only after they were combined with the results of graph spectral theory [13, 18, 65, 66]. 

Slight modifications of Eq. (14) lead us to Coulson-type integral representations of TRE, ef and efa, as defined by Eqs. (9), (10) and (13), respectively. These integral formulas read ... 

Coulson derived a new method for computing P(t) that gives the Landau-Zener formula as a special case. He considered three cases (1) transition between two discrete states [approximate internal conversion when the molecule passes from one discrete (bound) state to another], (2) transition from a bound state to the continuum (predissociation), and (3) transitions between two states of the continuum (corresponding to scattering problems). Coulson omitted in his article the computational details for P(t) for case (1). For case (2) he gave... 

Coulson s analyses yielded different results for cases (2) and (3), in conflict with all previous work on this problem. Prior to this work it had been usual to apply the Landau-Zener formula both to predissociation and to scattering. 

The two concepts have on occasion been brought together Coulson and Duncanson[4] gave an explicit formula for sp-orbitals based on Slater type orbitals (STO s). Rozendaal and Baerends used hybrids to describe chemical bonding in a momentum representation [5], and more recently, Cooper considered the shape of sp hybrids in momentum space, and their impact on momentum densities [6], We would like to have a closer look at them, in terms of their functional behavior, their nodal structure and their topology. We will do... 

Starting the schematical historic outline now, one can suppose ETO integral calculation starts with the pioneering work of Hylleraas [3], Kemble and Zener [4], Bartlett [5], Rosen [6], Hirschfelder [7], Coulson [8] and Ldwdin [9], solidifying as a quantum chemical discipline with the publication of overlap formulae and tables by MuUiken et al. [10]. 

The definitions (13), (14), and (15) of bond orders and charges at atoms were proposed by the British chemist and physicist C. Coulson. Different definitions of the same quantities were suggested at various times by other scientists (R. Mulliken, K. Ruedenberg, L. Pauling and others). We shall not cite or analyse any of those definitions here. It is sufficient to note that after some additional simplifying assumptions all of them reduce to Coulson s formulas. 

The theoretical values of Table 3.9 were calculated in this way, and the agreement with experiment is impressive, in view of the criticism of the simple Landau-Zener formula by Bates and by Coulson. ... 

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... 

Hiickel improved his treatment of aromaticity in a second paper, through a method (referred to by Hiickel himself as method II) that still bears his name the HMO method (Hiickel s molecular orbital method) (Hiickel 1932 Pullman and Pullman 1952 Dewar 1969 Coulson, O Leary, and Mallion 1978). The end result was a quantum mechanical treatment of aromaticity, through the 4k+2 (where k = 0, 1, 2,. ..) rule as the criterion for aromaticity. The formula referred to the number of k electrons in a given organic cyclic compound, which would be classified as aromatic if the number was 2, 6, or 10. Furthermore, this explained the stability of the aromatic molecules with respect to reactions. One important result of his approach was that for aromatic and unsaturated compounds, the number of possible valence structures is not the same as the number of different states of determined energy, nor is a state of determined energy necessarily identified with a specific valence structure. 

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