Coulson/Fischer orbitals

The results of a valence bond treatment of the rotational barrier in ethane lie between the extremes of the NBO and EDA analyses and seem to reconcile this dispute by suggesting that both Pauli repulsion and hyperconjugation are important. This is probably closest to the truth (remember that Pauli repulsion dominates in the higher alkanes) but the VB approach is still imperfect and also is mostly a very powerful expert method [43]. VB methods construct the total wave function from linear combinations of covalent resonance and an array of ionic structures as the covalent structure is typically much lower in energy, the ionic contributions are included by using highly delocalised (and polarisable) so-called Coulson-Fischer orbitals. Needless to say, this is not error free and the brief description of this rather old but valuable approach indicates the expert nature of this type of analysis. 

Here, a and b are purely localized AOs, while energy minimization, are generally not very delocalized (e < 1), and as such they can be viewed as distorted orbitals that remain atomic-like in nature. However, minor as this may look, the slight delocalization renders the Coulson—Fischer wave function equivalent to the VB-full wave function (Eq. 3.4a) with the three classical structures. A straightforward expansion of the Coulson—Fischer wave function leads to the linear combination of the classical structures in Equation 3.6. 

In any one of the above cases, improvement of the wave function can be achieved by using Coulson—Fischer orbitals that take into account ionic... 

Note that if instead of using purely localized AOs for a and b, we use semilocalized Coulson-Fischer orbitals, Equation 3.37 will no more be the... 

An important feature of the BOVB method is that the active orbitals are chosen to be strictly localized on a single atom or fragment, without any delocalization tails. If this were not the case, a so-called "covalent" structure, defined with more or less delocalized orbitals like, e.g., Coulson-Fischer orbitals, would implicitly contain some ionic contributions, which would make the interpretation of the wave function questionable [27]. The use of pure AOs is therefore a way to ensure an unambiguous correspondence between the concept of Lewis structural scheme and its mathematical formulation. Another reason for the choice of local orbitals is that the breathing orbital effect is... 

We can elaborate this VB formulation for the cycloaddition by replacing the nearest-neighbour active-space AOs in VB structures 50 and 51 with Coulson-Fischer orbitals [34(b)]. Thus if a and b are now the singly-occupied carbon and oxygen AOs of HCNO, and c and d are the singly-occupied carbon AOs of HCCH, the c and d AOs in structure 50 can be replaced by the Coulson-Fischer MOs c + k d and d + k"c. In structure 51, a + Ad, b + Ac, c + K b and d + K"a can replace the a, b c and d AOs. Use of these orbitals permits additional canonical Lewis VB structures to be included in the equivalent Lewis structure resonance scheme. The mechanism can then accommodate some charge transfer between the HCNO and HCCH reactants. The more-flexible wavefunction of Eq.(13),... 

The calculations reported so far are based on unconstrained mixing of all valence functions as a result, the optimized orbitals differ greatly from those pictured by Pauling, which - although usually hybrids - were strictly monocentric in character. The optimized forms resemble more closely the Coulson-Fischer orbitals of Sect.2, being distorted AOs which result in considerably increased overlap in the bond regions. In this general context, such AOs have been referred to as overlap-enhanced... 

Fig. 1 Amplitude of the generalized Coulson-Fischer orbitals for the methane molecule. See text for details (Taken from [79].)...

Fig. 3 Amplitude of the second Coulson-Fischer orbital for the ground state of the hydrogen molecule at the equilibrium nuclear separation. This orbital is obtained from the orbital shown in Fig. 1 by reflection in the plane perpendicular to the intemuclear axis and passing through its mid-point (Taken from [79].)...

Here, p is the reduced resonance integral that we have just defined and S is the overlap between orbitals a and b. Note that if instead of using purely localized AOs for a and b, we use semilocalized Coulson-Fischer orbitals, Eq. [41] will not be the simple HL-bond energy but would represent the bonding energy of the real A B bond that includes its optimized covalent and ionic components. In this case, the origins of the energy would still correspond to the QC determinant with the localized orbitals. Unless otherwise specified, we will always use qualitative VB theory with this latter convention. 

Such a VB function is known as a Coulson-Fischer (CF) type. The c constant is fairly small (for H2 c 0.04), but by allowing the VB orbitals to adopt the optimum shape, the need for ionic VB stractures is strongly reduced. Note that the two VB orbitals in eq. (7.8) are not orthogonal, the overlap is given by... 

Mueller, C. R., and Eyring, H., J. Chan. Phys. 19, 1495, Semi-localized orbitals. I. The hydrogen molecule. Combination of Inui (1941) and Coulson-Fischer (1949). 

The wave function of H2 is expressed below as a formally covalent VB structure F< > using Coulson—Fischer (CF) orbitals [Pg.71]

All the VB methods that deal with semilocalized orbitals use a generalization of the Coulson—Fischer idea (12), whereby a bond is described as a singlet coupling between two electrons in nonorthogonal orbitals that possess small delocalization tails resulting from the variational orbital optimization. Albeit formally covalent, this description implicitly involves some optimal contributions of ionic terms, as a decomposition of the wave function in terms of pure AO determinants would show (see Eqs. 3.5 and 3.6). For a polyatomic... 

The generalized valence bond (GVB) method was the earliest important generalization of the Coulson—Fischer idea to polyatomic molecules (13,14). The method uses OEOs that are free to delocalize over the whole molecule during orbital optimization. Despite its general formulation, the GVB method is usually used in its restricted form, referred to as GVB SOPP, which introduces two simplifications. The first one is the perfect-pairing (PP) approximation, in which only one VB structure is generated in the calculation. The wave function may then be expressed in the simple form of Equation 9.1, as a product of so-called geminal two-electron functions ... 

There is still freedom in the choice of atomic orbitals used in Eq. (50). For instance, one can use fixed atomic orbitals, which eliminates the (sometimes costly) orbital optimisation. One can also use fully optimised, potentially delocalised orbitals in the spin-coupled / Coulson-Fischer sense. Finally, one can use real atomic orbitals by limiting each orbital to its own atom. This often gives a clearer physical picture of chemical bonding. It generates for instance optimal hybrids [9,10]. 

Here each orbital, (p/or (pr, is mainly localized on a single center but involves a small tail on the other center, so that the expansion of the Coulson-Fischer wave function vFcf (eq 7) in AO determinants is in fact equivalent to Fci in eq 5, provided the coefficient e is properly optimized. 

Fig. 3. VB mechanisms for electron conduction in lithium with (a) 2pa AOs and (b)

The Coulson-Fischer function for H2 serves as a simple example of these arguments, the orbitals molecular point group Dmh- By contrast, in the case of HF there is no subgroup of C for which the expansion (69) is finite, so that the valence orbitals are both invariant under the full group. 

Fig. 2. (a) The modulus squared of the overlap between the spin-up and spin-down orbitals in a spin-unrestricted Xa calculation on H2 the Coulson-Fischer point (b) The spin-polarized triplet, spin-restricted (R.Jfa), spin-unrestricted UA"a), energy-projected singlet (EQl)and self-consistently exchange-projected singlet (E02) Xa solutions for as a function of internuclear separation. 

Abstract The wave function of Coulson and Fischer is examined within the context of recent developments in quantum chemistry. It is argued that the Coulson-Fischer ansatz establishes a third way in quantum chemistry, which should not be confused with the traditional molecular orbital and valence bond formalisms. The Coulson-Fischer theory is compared with modern valence bond approaches and also modern multireference correlation methods. Because of the non-orthogonality problem which arises when wave functions are constructed from arbitrary orbital products, the application of the Coulson-Fischer method to larger molecules necessitates the introduction of approximation schemes. It is shown that the use of hierarchical orthogonality restrictions has advantages, combining a picture of molecular electronic structure which is an accord with simple, but nevertheless empirical, ideas and concepts, with a level of computational complexity which renders praetieal applications to larger molecules tractable. An open collaborative virtual environment is proposed to foster further development. 

Wave function (4) is the standard configuration interaction expansion in mo theory. The parameter can take values from -1 to -1-1. Putting /r = 0 gives the pure molecular orbital description first considered by Coulson [49] in 1937. Table 1 summarizes the behaviour of the approximate wave functions as a function of the parameters k and /r. in the Coulson-Fischer analysis. (This table is taken from the work of Coulson and Luz [50].)... 

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