Valence bond, approximation theory

From the conceptual point of view, there are two general approaches to the molecular structure problem the molecular orbital (MO) and the valence bond (VB) theories. Technical difficulties in the computational implementation of the VB approach have favoured the development and the popularization of MO theory in opposition to VB. In a recent review [3], some related issues are raised and clarified. However, there still persist some conceptual pitfalls and misinterpretations in specialized literature of MO and VB theories. In this paper, we attempt to contribute to a more profound understanding of the VB and MO methods and concepts. We briefly present the physico-chemical basis of MO and VB approaches and their intimate relationship. The VB concept of resonance is reformulated in a physically meaningful way and its point group symmetry foundations are laid. Finally it is shown that the Generalized Multistructural (GMS) wave function encompasses all variational wave functions, VB or MO based, in the same framework, providing an unified view for the theoretical quantum molecular structure problem. Throughout this paper, unless otherwise stated, we utilize the non-relativistic (spin independent) hamiltonian under the Bom-Oppenheimer adiabatic approximation. We will see that even when some of these restrictions are removed, the GMS wave function is still applicable. 

At the present stage of theoretical development it is hardly possible to go far enough, in terms of either band theory or the Pauling valence bond approximation, with a quantitative treatment of crystals involving free electrons and a relatively large electronegativity difference between the components. Nevertheless, the matter merits more attention. 

Soon after the development of the quantum mechanical model of the atom, physicists such as John H. van Vleck (1928) began to investigate a wave-mechanical concept of the chemical bond. The electronic theories of valency, polarity, quantum numbers, and electron distributions in atoms were described, and the valence bond approximation, which depicts covalent bonding in molecules, was built upon these principles. In 1939, Linus Pauling s Nature of the Chemical Bond offered valence bond theory (VBT) as a plausible explanation for bonding in transition metal complexes. His application of VBT to transition metal complexes was supported by Bjerrum s work on stability that suggested electrostatics alone could not account for all bonding characteristics. 

An alternative to the MO method for the quantum mechanical treatment of molecular systems is the so-called Valence-Bond (VB) theory where molecular wavef unctions Eire obtained as linear combinations of covalent and ionic structures. It was shown long ago 181> that for distances larger than equilibrium distances, VB approximate wave functions should be better than MO functions of the same level, and hence VB theory should find its most profitable application in the evaluation of potential surfaces and reaction paths. Although true in principle, this statement has little influence in practice this is mostly because VB theory has only recently been formulated in a nonempirical form 182-184) so that applications are only just beginning to appear. 

Another empirical method that has been extensively used by Warshel and co-workers is the empirical valence bond (EVB) theory. jn this approach, it is assumed that a reaction can be described by some VB resonance structures. The analytical form of these VB functions can be approximated by appropriate molecular mechanics potentials, and the parameters of these MM potentials are calibrated to reproduce experimental or ab initio MO data in the gas phase as well as in the condensed phase. The combined EVB/MM method and its unique calibration procedure have been recently reviewed. > 2 it should be noted that Kim and Hynes presented a similar method, yielding a nonlinear Schrodinger equation. However, the solvent was treated as a dielectric continuum in the Kim-Hynes theory. Nevertheless, an interesting feature in the latter method is a consideration of nonequilibrium coupling between the solute and solvent. ... 

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Flamiltonian in the CSF basis. This contrasts to standard FIF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond sti uctures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

T orbital for benzene obtained from spin-coupled valence bond theory. (Figure redrawn from Gerratt ], D L oer, P B Karadakov and M Raimondi 1997. Modem valence bond theory. Chemical Society Reviews 87 100.) figure also shows the two Kekule and three Dewar benzene forms which contribute to the overall wavefunction Kekuleform contributes approximately 40.5% and each Dewar form approximately 6.4%. 

The VB and MO theories are both procedures for constructing approximations to the wavefunctions of electrons, but they construct these approximations in different ways. The language of valence-bond theory, in which the focus is on bonds between pairs of atoms, pervades the whole of organic chemistry, where chemists speak of o- and TT-bonds between particular pairs of atoms, hybridization, and resonance. However, molecular orbital theory, in which the focus is on electrons that spread throughout the nuclear framework and bind the entire collection of atoms together, has been developed far more extensively than valence-bond... 

Cooper et al. [30] were successful in rationalizing the striking variation in the S-S equihbrium bond lengths of FSSF (189.0 pm), ClSSCl (195.0 pm) and HSSH (205.5 pm) using the spin coupled (modern valence bond) theory. In the disulfur dihalides, but not for HSSH, incipient hypercoordinate character is observed at sulfur, with two partial t-like interactions in approximately perpendicular planes, and some antibonding character in the S-X (X=F or Cl) bonds. In other words, it is the form of t-like orbitals that is most rele-... 

In order to apply the theory, one first draws a valence bond formula with the correct constitution, including all lone electron pairs. This formula shows how many valence electron pairs are to be considered at an atom. Every electron pair is taken as one unit (orbital). The electron pairs are being attracted by the corresponding atomic nucleus, but they exercise a mutual repulsion. A function proportional to 1 /rn can be used to approximate the... 

The theory of resonance should not be identified with the valence-bond method of making approximate quantum-mechanical calculations of molecular wave functions and properties. The theory of resonance is essentially a chemical theory (an empirical theory, obtained largely by induction from the results of chemical experiments). Classical structure theory was developed purely from chemical facts, without any help from physics. The theory of resonance was also well on... 

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