Coulson-Fischer functions

The Weinbaum and Coulson-Fischer functions, though formally identical, have provided the prototypes for two somewhat divergent developments in... 

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. 4 Potential energy curves for the hydrogen molecule obtained from (a) the Hartree-Fock function [96], (b) the ODC multiconfiguration self-consistent field flmction, (c) the generalized Coulson-Fischer function [77] and (d) the extended James-CooUdge flmction of Krfos and...

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... 

The generalization of a Coulson-Fischer type wave function to the molecular case with an arbitrary size basis set is known as Spin Coupled Valence Bond (SCVB) theory. ... 

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 Coulson—Fischer orbitals 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... 

Since c1 and c2 are variationally optimized, expansion of v1,mo-ci should lead to exactly the same VB function as vB-fuii in Equation 3.4, leading to the equalities expressed in Equation 3.62 and to the equivalence of mo-ci and vB-fuii (see Exercise 3.1) The equivalence also includes the Coulson Fischer wave function Th h (Eq. 3.5) which, as we have seen, is equivalent to the VB-full description (see Exercise 3.2). 

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 ... 

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. 

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... 

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... 

Such a VB function is known as a Coulson-Fischer (CF) type. The c constant is fairly... 

Let us conclude this section by returning to the electron density from the Coulson-Fischer wave function. Writing a normalization factor K on the RHS of Eq. (24), one finds... 

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. 

The Coulson-Fischer wave function for H2 can be considered as the start of the Unrestricted Hartree-Fock (UHF) approach in quantum chemistry, which is the most general single determinant method. We shall not proceed further along this line, but instead ask ourselves if there is a way to correct the simation such that we obtain a wave function that dissociates correctly while preserving the spin and space symmetry of the wave function. The CF wave function gives acmally a hint. What happens if we simply skip the trouble-some triplet term in Eq. (22). This gives rise to a wave function that is a linear combination of two closed shell determinants ... 

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. 

Keywords Coulson-Fischer wave function Coulson-Fischer analysis Coulson-Fischer theory Modern valence bond theory Multireference correlation problem Collaborative virtual environment... 

In this chapter, we propose the creation of a new collaborative virtual environment for the development of the Coulson-Fischer method for molecular wave functions. It is proposed that this environment should be open. This chapter gives some background to the project. 

The purpose of this essay is to examine the Coulson-Fischer wave function [18] and the approach of Coulson and Fischer from the perspective of contemporary... 

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].)... 

Hence the approximation (1), which is a prototype of multistructure valence bond theory, is equivalent to approximation (4), a prototype of molecular orbital configuration interaction theory, and both (1) and (4) are equivalent to (13), the Coulson-Fischer wave function. 

Evidence for this renaissance is seen in the number of monographs [53-55], edited volumes [56-58] and review articles [52, 59, 60, 62, 63, 61, 64-69] on valence bond theory published in recent years. These works display a rich variety of theoretical machinery inspired by the valence bond picture of molecular structure. Some of the methodologies - particularly in the so-called modern valence bond theories introduced by Gerratt and Lipscomb [70,71] under the name spin-coupled wave functions and developed by Gerratt [72,73] and his collaborators [68,74-86] over the past 40 years - exploit the Coulson-Fischer ansatz. As we have seen in Section 3 and will consider further in Section 6, the Coulson-Fischer theory presents a third way of constructing approximate molecular wave functions which combine many of the advantages of both molecular orbital theory and valence bond theory. 

In a paper pubhshed in 1953 as part of a series under the general title The molecular orbital theory of chemical valency, Hurley, Lennard-Jones and Pople [87] presented A theory of paired electrons in polyatomic molecules. The pah-function model of Hurley et al. employed a Coulson-Fischer-type wave function to describe each pair of electrons in a polyatomic molecule. Orthogonality constraints were imposed between orbitals associated with different pairs of electrons in order to render the theory practical, i.e. computationally tractable. Hurley presented the corresponding orbital equations in a subsequent paper [88] which was published in 1956. 

The original Coulson-Fischer wave function for the H2 molecule can be generalized [77] by approximating each of the space orbitals by an expansion in terms of finite analytical basis functions xk, k = 1,2,..., . The algebraic approximation is implemented by means of the linear expansions... 

In order to develop a Coulson-Fischer approach for an A-electron systems, we assume that the spatial molecular wave function is approximated by a single product of N orbitals... 

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