Extended valence bond method

Valence-bond descriptions of multiple bonds are also similar to the simple LCAOMO description. A double bond consists of a sigma bond and api bond, and a triple bond consists of a sigma bond and two pi bonds. Each bonding factor replaces one LCAOMO, but nonbonding orbitals are the same in both methods. There are no analogues to antibonding orbitals in the simple valence-bond method, so molecules with unpaired electrons such as O2 also cannot be well described in the simple valence-bond method. The extended valence-bond methods mentioned in Chapter 20 are more versatile. 

Resonance theory [15] contains essentially three assumptions beyond those of the valence bond method. Perhaps the most serious assumption is the contention that only unexcited canonical forms, non-polar valence bond structures or classical structures need be considered. Less serious, but no more than intuitive, is the proposition that the molecular geometry will take on that expected for the average of the classical structures. This is extended to the measurement of stability being greater the greater the number of classical structures. These concepts are still widely used in chemistry in very qualitative ways. 

Hiickel s application of this approach to the aromatic compounds gave new confidence to those physicists and chemists following up on the Hund-Mulliken analysis. It was regarded by many people as the simplest of the quantum mechanical valence-bond methods based on the Schrodinger equation. 66 Hiickel s was part of a series of applications of the method of linear combination of atom wave functions (atomic orbitals), a method that Felix Bloch had extended from H2+ to metals in 1928 and that Fowler s student, Lennard-Jones, had further developed for diatomic molecules in 1929. Now Hiickel extended the method to polyatomic molecules.67... 

Curves that go beyond the Flartree-Fock method have been calculated for certain systems, and the list grows monthly. Of special interest is the series on ground [62-64] and excited states [63, 64] of CO, on NaLi and NaLi+ using an extended Flartree-Fock method with optimized double-valence configurations [65], HeLi by a valence-bond method [66], and so on. A quite complete listing of all nonempirical potential-energy curves calculated through 1967 is included in the NBS report [33]. 

The cluster approach is, like the extended-surface approach, characterized by many different calculational schemes. A recent review stresses electronic aspects of bonding. In this review we have chosen to concentrate on geometric aspects. We shall discuss a number of major techniques in order of increasing computational complexity the extended H uckel theory, self-consistent Xa scattered wave calculations, and self-consistent ab initio Hartree-Fock and valence bond methods. In that order these techniques allow increasing accuracy. However, the cluster size must decrease simultaneously due to calculational complexity, ultimately reducing the degree of analogy with surfaces. 

A wide range of theoretical methods has been applied to the study of the structure of small metal clusters. The extremes are represented on the one hand by semi-empirical molecular orbital (Extended Huckel) (8 ) and valence bond methods (Diatomics-In-Molecules) ( ) and on the other hand by rigorous initio calculations with large basis sets and extensive configuration interaction (Cl) (10). A number of approaches lying between these two extremes have been employed Including the X-a method (11), approximate molecular orbital methods such as CNDO (12) and PRDDO (13) and Hartree-Fock initio molecular orbital theory with moderate Cl. 

Another fairly new method, using the electrostatic molecular potential, will not be discussed here since it is the subject of another contribution to this volume 50>. I will now consider methods that have had the widest application in the theoretical study of chemical reactivity, in order of increasing complexity a) molecular mechanics b) extended Htickel method c), d) empirical self-consistent field methods such as CNDO and MINDO e) the simplest ab initio approach f) the different S.C.F. methods, possibly including configuration interaction g) valence bond methods, and h) the dynamical approach, including the calculation of trajectories 61>. 

The first quantum-mechanical treatment of the hydrogen molecule was by Heitler and London in 1927. Their ideas have been extended to give a general theory of chemical bonding, known as the valence-bond (VB) theory. The valence-bond method is more closely related to the chemist s idea of molecules as consisting of atoms held together by localized bonds than is the molecular-orbital method. The VB method views molecules as composed of atomic cores (nuclei plus inner-shell electrons) and bonding valence electrons. For H2, both electrons are valence electrons. 

If we try to extend the unmodified valence bond method of Section 11-2 to a greater number of molecules, we are quickly disappointed. In most cases, our descriptions of molecular geometry based on the simple overlap of unmodified atomic orbitals do not conform to observed measurements. For example, based on the ground-state electron configuration of the valence shell of carbon... 

For this reason, there has been much work on empirical potentials suitable for use on a wide range of systems. These take a sensible functional form with parameters fitted to reproduce available data. Many different potentials, known as molecular mechanics (MM) potentials, have been developed for ground-state organic and biochemical systems [58-60], They have the advantages of simplicity, and are transferable between systems, but do suffer firom inaccuracies and rigidity—no reactions are possible. Schemes have been developed to correct for these deficiencies. The empirical valence bond (EVB) method of Warshel [61,62], and the molecular mechanics-valence bond (MMVB) of Bemardi et al. [63,64] try to extend MM to include excited-state effects and reactions. The MMVB Hamiltonian is parameterized against CASSCF calculations, and is thus particularly suited to photochemistry. 

Values of Bond Energies for Multiple Bonds.—In Section 3-5 there is given a table of values of bond energies for single bonds. In the construction of this table care was taken to make use of data for only those molecules to each of which an unambiguous assignment of a valence-bond formula could be made. This consideration of bond energies is extended in Table 6-1, which contains values for some multiple bonds, obtained by methods similar to those described in Section 3-5. 

A simple graphical method of formula ting the independent valence-bond structures for a molecule was discovered by Rumer.1 This method has been extended to permit the secular equation for a set of resonating valence-bond structures to be written without difficulty. Quantum-mechanical treatments of aromatic and conjugated molecules have been carried out by many investigators. The subject of molecular quantum mechanics is too extensive to be reviewed in this book. 

The distances found between platinum centers in these molecules have been correlated with the resonating valence bond theory of metals introduced by Pauling. The experimentally characterized partially oxidized one-dimensional platinum complexes fit a correlation of bond number vs. metal-metal distances, and evidence is presented that Pt—Pt bond formation in the one-dimensional chains is resonance stabilized to produce equivalent Pt—Pt distances.297 The band structure of the Pt(CN)2- chain has also been studied by the extended Huckel method. From the band structure and the density of states it is possible to derive an expression for the total energy per unit cell as a function of partial oxidation of the polymer. The equilibrium Pt-Pt separation estimated from this calculation decreases to less than 3 A for a loss of 0.3 electrons per platinum.298... 

Other common methods for representing the three-dimensional structures of molecules include Newman projections for showing conformational relationships and sawhorse figures. Newman projections look down a carbon-carbon bond so that the front carbon, designated by a circle, obscures the carbon directly behind it. Valences (bonds) to the front carbon extend to the center of the circle, while bonds to the rear carbon stop at the circle. Sawhorse projections have the carbon-carbon bond at oblique angles, which attempts to represent a perspective drawing of the molecule. Thus for 2-chloro butane, if one chooses to examine the 2,3 bond, then the sawhorse and Newman projections would be... 

Because the n-networks of benzenoid hydrocarbons are the classical prototypical example of delocalized bonding, they provide a crucial test for chemical-bonding theories. Here there is revealed a systematic organization for valence-bond views to describe such n-bonding. With an initiation near the ab initio realm a sequence of semiempirical valence-bond models is identified and characterized. The refinement from one model to the next proceeds via either a (perturbative) restriction to a smaller model space or orthogonalization of a suitable natural basis for the model space. The known properties of the models are indicated, and possible methods of solution are mentioned. A great diversity of work is outlined, related, systematized and extended. New research is suggested. 

Therefore, the dependence on the coefficients does not enter the gradient expression not for fixed orbitals, which is the classical Valence Bond approach and not for optimised orbitals, irrespective of whether they are completely optimised or if they are restricted to extend only over the atomic orbitals of one atom. If the wavefimction used in the orbital optimisation differs, additional work is required. This would apply to a multi-reference singles and doubles VB (cf. [20,21]). Then we would require a yet unimplemented coupled-VBSCF procedure. Note that the option to fix the orbitals is not available in orthogonal (MO) methods, due to the orthonormality restriction. 

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