Heitler-London wavefunction

In the Kekule structure 3 there are four singly-occupied carbon and nitrogen 2p% and 2p7ty AOs. Labelling these orbitals as px(Q, Px(N), p (C) and py(N), the primary 5 = 0 Heitler-London wavefunction for the n electrons which occupy these orbitals is given by Eq.(8),... 

Numerical values of the interaction energies for these Heitler-London wavefunctions, taken from Magnasco (2008), are given in Tables 1.2 and 1.3. The energies are optimized variationally with respect to the values of the orbital exponents Co of the atomic Is STOs on A and B. 

One powerful approach to the problem of bond breaking is the valence bond method. To understand the basic idea of the valence bond picture, let us consider the electronic structure of the ground, singlet state of the hydrogen molecule. The form of the valence bond (or Heitler-London) wavefunction is... 

One widely used valence bond theory is the generalised valence bond (GVB) method of Goddard and co-workers [Bobrowicz and Goddard 1977]. In the simple Heitler-London treatment of the hydrogen molecule the two orbitals are the non-orthogonal atomic orbitals on the two hydrogen atoms. In the GVB theory the analogous wavefunction is written ... 

In the case of the hydrogen molecule-ion H2" ", we defined certain integrals Saa, Taa, Tab, Labra- The electronic part of the energy appropriate to the Heitler-London (singlet) ground-state wavefunction, after doing the integrations... 

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). 

When Heitler-London AO-type wavefunctions (i.e.. ..aabP +. ..baaP in which a and b are AOs) are used to represent electron-pair 7i c(CN) and 7i y(CN) bonds, it can be deduced [2,4,16, cf. also Eq.(ll) below] that VB structure 7 is equivalent to resonance between the Kekule Lewis structure 3 and the Dewar or "long-bond" Lewis structures 11-13. Only nearest-neighbour spin-pairing is indicated in increased-valence structures [2-5,10]. When the "long" or formal bonds are omitted from structures 11-13, these structures are designated as singlet diradical structures [2-4]. 

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. 

Initially, the MO and VB approaches were developed with different aims in view the primary task of MO theory was to explain the electronic spectra of molecules, while the VB method was concerned mainly with the problems of bonding and valency. This is directly reflected in the construction of the wavefunctions used in the most well-known examples of the two approaches, the Hartree-Fock (HF) method and classical (Heitler-London ) VB theory. 

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. 

In Section ILF it was shown that the total dimer Heitler-London energy for a dimer wavefunction where and are SCF wavefunc-... 

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