Molecular orbitals valence bond wavefunction

The virtual orbitals are then employed to construct the set of singly- and doubly-excited configurations which provide the final MO-VB wavefunction, where the SCF-MI determinant represents the zero order state. The final wavefunction has a general Molecular Orbital-Valence Bond form ... 

Wavefunction Vq refers to the situation where both electrons are on nucleus a or nucleus b, i.e., ionic structures. Now it is obvious that the valence bond wavefunction covalent structure, while the molecular orbital wavefunction t has an equal mixture of covalent and ionic contributions. Similarly, expanding the wavefunction in eq. (3.2.27) yields... 

The optimal c2/ci ratio is -0.73. More importantly, the improved valence bond wavefunction p + and the improved molecular orbital wavefunction t/fj are one and the same, thus showing these two approaches can lead to identical quantitative results. 

Figure 2.9. The qualitative behavior of the ground state energy of titanium oxide and nickel oxide calculated with a VB model that includes some ionic contribution (amount X). A pure valence bond wavefunction has = 0 and a molecular orbital state has A = 1. For titanium oxide the MO model predicts an energy closer to the real value than a VB calculation. For nickel oxide the valence bond model is more realistic.

The other limiting situation is the equilibrium separation of the protons in the H2 molecule. The molecular orbital SCF picture places the two electrons in one spatial orbital, a The valence bond wavefunction achieves this form if both orbitals a and b in... 

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... 

Both the early molecular orbital and the early valence bond approaches used wavefunctions (a) Molecular orbital, e.g. Pauling L (1928) Chem Rev 5 173. Lennard-Jones E (1929) Trans Faraday Soc 25 668. (b) Valence bond Heitler W, London F (1927) Z Phys 44 455... 

Ab initio modem valence bond theory, in its spin-coupled valence bond (SCVB) form, has proved very successful for accurate computations on ground and excited states of molecular systems. The compactness of the resulting wavefunctions allows direct and clear interpretation of correlated electronic structure. We concentrate in the present account on recent developments, typically involving the optimization of virtual orbitals via an approximate energy expression. These virtuals lead to higher accuracy for the final variational wavefunctions, but with even more compact functions. Particular attention is paid here to applications of the methodology to studies of intermolecular forces. 

In molecular orbital theory, molecular orbitals are formed by linear combinations of atomic orbitals, and the bonding molecular orbital is < >a + b.14 The molecular wave function is the product of the wave functions for the two electrons or [< >a(l) + < >b(l)][a(2) + <( b(2)]. When the energy is calculated using this wavefunction, it is found to be 80.0 kcal/mol with a length of 0.732A. In this case, the valence bond result is slightly more satisfactory than the molecular orbital result, but neither is really satisfactory. 

In certain Instances it may be appropriate to determine the ground state wavefunction using the generalized valence-bond (GVB) technique (21). This is especially true for closed shell anions where the pair of electrons in the highest occupied molecular orbital (HOMO) may be described by the wavefunction... 

Molecular orbitals (MOs) are derived mathematically by a linear combination of the wavefunctions for the atomic orbitals (AOs) of the individual atoms in a molecule. Usually, only the atomic orbitals of the valence electrons are considered, because these are the electrons involved in bonding. We can visualize the formation of MOs as proceeding from overlap of the AOs of the valence electrons. 

Although the use of strokes to represent bonds between atoms in molecules comes from the nineteenth century, the electron pair concept as necessary for the understanding of chemical bonding was introduced by G.N. Lewis (1875-1946) in 1916 (ref. 90) following Bohr s, then recently proposed, model of the atom. Indeed, the Lewis model still lies at the basis of much of present-day chemical thinking, although it was advanced before both the development of quantum mechanics and the introduction of the concept of electron spin. In a more quantitative way, it found a natural theoretical extension in the valence-bond approximation to the molecular wavefunction, as expressed in terms of the overlap of (pure or hybridized) atomic orbitals to describe the pairing of electrons, coupled with the concept of electron spin. 

This process of constructing functions for the various resonant formulae, followed by an adequate combination of them, is mathematically more complex than the mathematics of molecular orbital theory. It is therefore understandable that, after the initial preference of chemists for the v.b. bond theory which has a closer relation to Lewis structures - especially due to the contribution of Linus Pauling - m.o. theory became increasingly popular. In addition, m.o. theory leads directly, not only to fundamental states (through the occupied m.o.), but also to excited states (through vacant m.o.) of molecules. In recent years, however, a new form of valence-bond theory has been developed that is more amenable to computation (spin-coupled valence-bond theory) in which the molecular wavefunction is expressed as a linear combination of all the coupling schemes of the various electrons corresponding to the same resultant spin (ref. 97). 

Valence bond (VB) theory may be used as an alternative to molecular orbital (MO) theory for computational organotin studies. Most MO calculations of organotin systems use Gaussian, GAMESS, or Amsterdam density functional (ADE) program suites. A variety of VB methods exist, and although VB wavefunctions are more difficult to calculate, some VB methods can also be implemented in these programs. ... 

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