Pair bond wavefunction

As indicated already in Chapter 2, unless stated otherwise (see for example Chapter 23), the equivalent Lewis structure resonance theory assumes that electron-pair bond wavefunctions are of the Heitler-London atomic orbital type - for example y(l)a(2) + a(l)y(2) a.nAy l)b(2) + b )a 2) for structures (6) and (7). Atomic formal charges are not indicated in the generalized valence bond structures that involve the Y, A, B, C and D atoms. 

Figure 3.2 Overlap of atomic ls-wavefunctions of H-atoms to form an H2 molecule with an electron-pair bond.

As an example, we take the electron pair bonds of NC—CN and CN— NC in ct symmetry between the 5a singly occupied highest occupied orbitals (SOMOs) of the two CN radicals, two systems treated in the next section. Below the 5ct orbitals, there are fully occupied orbitals, the most important one being the 4a N lone pair orbital (the aHOMo orbital, see next section). The wavefunction VF° is written in this case as follows ... 

We remind the reader that, with one hybrid or non-hybrid atomic orbital (AO) per atomic centre, the most-general singlet spin (S = 0) wavefunction of the Heitler-London type for the electron-pair bond A" B or A B is given by Eq.(l). 

Because structures 11-13 do not involve C-N triple bonds, Eq.(ll) shows that in Eq.(9), the Px(C)-px(N) and py(C)-py(N) spin-pairings form fractional 7ix(CN) and Jty(CN) electron-pair n-bonds via Heitler-London AO formulations of the bond wavefunctions. 

Shifted ST0-6G electronic energies ( = -E - 36.0 a.u.), and VB structural weights for resonance between VB structures I-IX. Structures VII-IX involve (2s)1(2p)1, (2s)1(2p)1 and (2p)2 configurations for the LiW. The wavefunctions for the electron-pair bonds involve Heitler-London AO formulations. 

To introduce the concepts, we must of course start from the classic 1927 paper of Heitler and London [1], referred to by Pauling himself [2] as the greatest single contribution to the clarification of the chemist s conception of valence. .. since G.N.Lewis s suggestion that the chemical bond between two atoms consists of a pair of electrons held jointly by the two . The electron-pair bond in the hydrogen molecule was described using a wavefunction of the form... 

The basic idea of the Heitler-London model for the hydrogen molecule can be extended to chemical bonds between any two atoms. The orbital function (10.8) must be associated with the singlet spin function cro,o(l > 2) in order that the overall wavefunction be antisymmetric [cf. Eq (8.14)]. This is a quantum-mechanical realization of the concept of an electron-pair bond, first proposed by G. N. Lewis in 1916. It is also now explained why the electron spins must be paired, i.e., antiparallel. It is also permissible to combine an antisymmetric orbital function with a triplet spin function, but this will, in most cases, give a repulsive state, such as the one shown in red in Fig. 10.2. 

The electron-pair bond plays a central role in the qualitative understanding of molecular structure. The pair-function or geminal approach attempts to put this electron-pair concept into a more quantitative form. In this approach the total wavefunction is assumed to be of the form of an antisymmetrized product of pair-functions ... 

To some extent the pair function methods seem to fall between two stools. They attempt to utilize chemical intuition in solving the complex quantum mechanical problem, and they do indeed possess an appealing simplicity of interpretation in terms of electron-pair bonds, especially at the GVB or separated pair level. The price that must be paid for a wavefunction of such an easily interpretable form is that such methods are unlikely, in general, to yield chemically accurate potential energy surfaces. 

The simple Heitler-London valence bond model for an electron-pair bond formed by the combination of two (hybrid atomic) orbitals has the wavefunction... 

It is one of the most basic assumptions of chemistry that a complex molecular electronic structure can be thought of as a series of environment-insensitive substructures with a large degree of autonomy. In particular for many molecules we think of the total structure as composed of pairs of electrons (inner shells, lone pairs, bond pairs) and this idea can be translated into a quantum-mechanical model in which each separate pair (or group) has its own wavefunction. If this is so, then most of the analysis which we have used for the total electronic structure can be taken over unchanged in a description of the separate pairs. 

The perfect-pairing formula (7.3.7) has been widely employed in qualitative discussions of the interactions determining the shape and stability of polyatomic molecules and in the interpretation of empirical additivity rules etc., which apply in many instances and appear to support the validity of a wavefunction representing a single well-defined set of localized electron pair bonds. It must be remembered, however, that the derivation rests upon an orthogonality assumption that intro-... 

The covalent-ionic resonance wavefunction for the A B electron-pair bond can be expressed according to Eqs. (27) and (28). 

For each of the increased-valence structures that we have described in this chapter, there is an apparent violation of the Lewis-Langmuir octet mle for some of the first-row atoms. However, because only the 2s and 2p orbitals are required for a minimal basis set description of the bonding, no real octet violation occurs in the molecular wave-function. Further, (with either Heitler-London atomic-orbital or 2-centre bond-orbital wavefunctions for nearest-neighbour electron-pair bonds, cf Chapter 23), these stmctures summarize resonance between standard and long-bond Lewis stmctures, each of which obeys the octet rule. This latter result has been demonstrated in Section 11-9 for the N2O4 increased-valence stractuie (16), whose component octet stmctures are displayed in Fig. 11-5. Further comments on apparent octet violation for atoms of first-row and higher-row elements are provided in Refs. 14, 15,22, 23 and 24. 

In this chapter, we illustrate how to obtain the optimal natural Lewis stmcture (NLS) formulation of the wavefunction in terms of optimal NBOs for shared pairs (bonds) and lone pairs of the conventional Lewis stmctural dot diagram. We also describe how to assess the accuracy of the NLS representation, comparing it with alternative Lewis stmctural formulations (alternative resonance stmctures ) that might be suggested. In Sections 4.1. 2, we first consider the relatively simple closed-shell molecules such as HF, CH3OH, or H2NCHO that conform to the octet mle. The... 

The Lewis-type (L) contribution is considered the easy part of chemical wavefunction analysis, because it corresponds closely to the elementary Lewis structure model of freshman chemistry. Nevertheless, controversy often arises over the magnitude of steric or electrostatic effects that are associated with the Lewis model itself [i.e., distinct from the resonance-type effects contained in (NL)]. The NBO program offers useful tools for quantifying both steric and electrostatic interactions in terms of the space-filling (size and shape) and dielectric properties (charge, dipole moment, etc.) of the electron pair bonds and lone pairs that comprise the Lewis structure model. This chapter discusses the physical nature and numerical quantitation of these important chemical effects, which are often invoked in a hand-waving manner that reflects (and promotes) significant misconceptions. 

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