Resonance theory - hydrogen molecule

Slater determinants are usually eonstrueted from molecular spinorbitals. If, instead, we use atomic spinorbitals and the Ritz variational method (Slater determinants as the expansion functions) we would get the most general formulation of the valence bond (VB) method. The beginning of VB theory goes back to papers by Heisenberg. The first application was made by Heitler and London, and later theory was generalized by Hurley, Lennard-Jones and Pople.  

The essence of the VB method can be explained by an example. Let us take the hydrogen molecule with atomic spinorbitals of type Ha a and si /3 denoted shortly as a a and fc/3 centred at two nuclei. Let us construct from them several (non-normalized) Slater determinants, for instance  

The functions 3, i/ 4 and the normalized difference tf i — ip2 ( hl is a normal- Hemer-London ization factor) function 

The VB method relies on optimization of the e q ansion coefficients c in front of these stwctures in the Ritzprocedure (p. 202) coialent 

The covalent structure itself, hl, was one great success of Walter Heitler and Fritz London. For the first time the correct description of the chemical bond was 

---

According to resonance theory, a molecule of naphthalene can be considered to be a hybrid of three Kekule structures. One of these Kekule structures, the most important one, is shown in Fig. 14.15. There are two carbon atoms in naphthalene (C4a and C8a) that are common to both rings. These two atoms are said to be at the points of ringfusion. They direct all of their bonds toward other carbon atoms and do not bear hydrogen atoms. 

Resonance theory can also account for the stability of the allyl radical. For example, to form an ethylene radical from ethylene requites a bond dissociation energy of 410 kj/mol (98 kcal/mol), whereas the bond dissociation energy to form an allyl radical from propylene requites 368 kj/mol (88 kcal/mol). This difference results entirely from resonance stabilization. The electron spin resonance spectmm of the allyl radical shows three, not four, types of hydrogen signals. The infrared spectmm shows one type, not two, of carbon—carbon bonds. These data imply the existence, at least on the time scale probed, of a symmetric molecule. The two equivalent resonance stmctures for the allyl radical are as follows ... 

Curve 1 represents the total energy of the hydrogen molecule-ion as calculated by the first-order perturbation theory curve 2, the naive potential function obtained on neglecting the resonance phenomenon curve 3, the potential function for the antisymmetric eigenfunction, leading to elastic collision. 

We have applied Pauling s theory to the molecular hydrogen cluster as follows the nonmetallic cluster is well described by the usual Kekule structure (1-2 3-4) (Fig.2), where orbitals 1 and 2 are at one hydrogen molecule and 3 and 4 are at the other one. The synchronized resonance is the mechanism in which the system alternates between structures (1-2 3-4) and (1-4 2-3) (anti-Kekule) (Fig.2), breaking simultaneously the two original covalent bonds and forming two new ones. 

Abstract In this chapter we discuss the influence of ir-electron delocalization on the properties of H-bonds. Hence the so-called resonance-assisted hydrogen bonds (RAHBs) are characterized since such systems are mainly classified in the literature as those where TT-electron delocalization plays a very important role. Both the intramolecular and intermolecular RAHBs are described. RAHBs are often indicated as very strong interactions thus, their possible covalent nature is also discussed. Examples of the representative crystal structures as well as the results of the ab initio and DFT calculations are presented. Additionally the RAHB systems, and the other complexes where rr-electron delocalization effects are detectable, are characterized with the use of the QTAIM (Quantum Theory Atoms in Molecules ) method. The decomposition scheme of the interaction energy is applied to expand the knowledge of the nature of the RAHBs. 

Wave function (1) is the standard covalent-ionic resonance of vb theory. The parameter k can take values from k = 0, which corresponds to the pure covalent description of the hydrogen molecule ground state (this is the wave function first considered by Wang [47] when he introduced a variable screening constant into the Heitler-London wave function), through to k — oo, which corresponds to a pure ionic description. The best wave function of the form (1), determined by invoking the variation theorem to determine the optimal screening constant, was first reported by Weinbaum [48] in 1933. 

The first quantitative theory of chemical bonding was developed for the hydrogen molecule by Heitler and London in 1927, and was based on the Lewis theory of valence in which two atoms shared electrons in such a way that each achieved a noble gas structure. The theory was later extended to other, more complex molecules, and became known as valence bond theory. In this approach, the overlap of atomic orbitals on neighbouring atoms is considered to lead to the formation of localized bonds, each of which can accommodate two electrons with paired spins. The theory has been responsible for introducing such important concepts as hybridization and resonance into the theory of the chemical bond, but applications of the theory have been limited by difficulties in generating computer programs that can deal efficiently with anything other than the simplest of molecules. 

Application of valence bond theory to more complex molecules usually proceeds by writing as many plausible Lewis structures as possible which correspond to the correct molecular connectivity. Valence bond theory assumes that the actual molecule is a hybrid of these canonical forms. A mathematical description of the molecule, the molecular wave function, is given by the sum of the products of the individual wave functions and weighting factors proportional to the contribution of the canonical forms to the overall structure. As a simple example, the hydrogen chloride molecule would be considered to be a hybrid of the limiting canonical forms H—Cl, H Cr, and H C1. The mathematical treatment of molecular structure in terms of valence bond theory can be expanded to encompass more complex molecules. However, as the number of atoms and electrons increases, the mathematical expression of the structure, the wave function, rapidly becomes complex. For this reason, qualitative concepts which arise from the valence bond treatment of simple molecules have been applied to larger molecules. The key ideas that are used to adapt the concepts of valence bond theory to complex molecules are hybridization and resonance. In this qualitative form, valence bond theory describes molecules in terms of orbitals which are mainly localized between two atoms. The shapes of these orbitals are assumed to be similar to those of orbitals described by more quantitative treatment of simpler molecules. 

This discovery was to be the beginning of the use of exchange terms in the quantum mechanics of atoms and molecules. It became the key factor that shortly afterward allowed Walter Heitler and Fritz London to obtain the first successful quantum mechanical calculation of the covalent bond in the simplest case of a diatomic hydrogen molecule. Exchange terms would also pave the way for the notion of quantum mechanical resonance and the development of the quantum mechanical theories of bonding by Linus Pauling and many others. ... 

---