Coulson-Fischer theory

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... 

We submit that the Coulson-Fischer ansatz affords a third way in quantum chemistry that is distinct from the traditional valence bond and molecular orbital theories. In the next two sections, we very briefly survey the current state of the art in valence bond theory and in the multireference correlation problem based on the molecular orbital theory, before considering the Coulson-Fischer theory in more detail in Section 6. 

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. 

Here we are proposing the creation of an open collaborative virtual environment for the development of the Coulson-Fischer method for molecular wave functions. To this end we have created web pages at http //quantumsystems.googlepages.com/ cve theCoulson-Fischertheory, which is intended to form an element of a collaborative virtual environment for the further advancement of the Coulson-Fischer theory. In his recently published report, Karadakov [52] expresses his view that... 

The generalised valence bond (GVB) method, developed by Goddard in 1970, is one of the simplest and oldest valence bond methods that use flexible orbitals in a general way. The generalised Coulson-Fischer theory for the hydrogen molecule mentioned above is used to describe every electron pair in a molecule. The orbitals for each electron pair are expanded in terms of the full basis set and are non-orthogonal. Orbitals from different pairs are forced to be orthogonal. This condition simplifies the calculations but may lead to some difficulties [160,161],... 

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. ... 

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. 

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. 

He points to the relatively small number of researchers developing vb methodology, coupled with the lack of coordination of their efforts. The proposed open collaborative virtual environment should facilitate a new level of interaction and coordination but it envisaged that rather than focussing on vb theory, the emphasis should be on the Coulson-Fischer approach - a third way in quantum chemistry. 

Note that the rules and formulas that are expressed above in the framework of qualitative VB theory are independent of the type of orbitals that are used in the VB determinants purely localized AOs, fragment orbitals or Coulson-Fischer semilocalized orbitals. Depending on the kind of orbitals that are chosen, the h and S integrals take different values, but the formulas remain the same. 

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. 

Correlation in atoms introduces largely quantitative changes. As seen above, qualitative changes can come about in condensed phases. Here, we consider briefly the case of the stretched H2 molecule, going back to the work of Coulson and Fischer [48] (CF). Their work is a forerunner of the so called generalized valence bond theory. 

Sixty years ago, in 1949, Coulson and Fischer published a seminal paper [18] in the Philosophical Magazine, entitled Notes on the Molecular Orbital Treatment of the Hydrogen Molecule. In this note, they presented a wave function for the hydrogen molecule, which, whilst retaining a simple physical picture, combines the advantages of the two rival theories of molecular electronic structure, mo and VB theories. Let us briefly summarize the discussion given by Coulson and Fischer. 

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