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Modern valence bond theory
Modern valence bond theory is the quantum chemistry method that implements valence bond theory (VB) as the general bonding theory that governs the subsequent quantum chemical computations. The approach consists in making a linear combination of all the relevant structures, which are the quantum mechanical representation of for example resonance structures of the system to be described. Subsequently the coefficients of this linear combination are determined variationally using a computer program. Although the concept "modern valence bond theory" dates from the beginning of expressing valence bond theory in the wave function terminology, nowadays is also largely associated to the group of newly developed methods that made the popularity of valence bond theory in computational chemistry resurface; especially spin-coupled valence bond theory.

History
Modern valence bond theory traces back to 1927, when Heitler and London published their homonymous theory. For the first time, bonding properties of the hydrogen molecule were calculated based on quantum mechanical considerations.

Even though modern valence bond theory gained first a positive response due to the similarity with the experimentalist's bonding point of view, the complexity of the formulism (primarily due to its use of non-orthogonal orbitals) made MO-related methods take over quantum chemistry as they were faster. Thus, the early popularity of modern valence bond theory declined, being thought as an inefficient and obsolete technique. It is only recently with the development of the formulism and the general computational tools that the popularity of this method increased again. These developments are in part due to and described by Gerratt, Cooper, Karadakov and Raimondi (1997); Li and McWeeny (2002); Joop H. van Lenthe and co-workers (2002); Song, Mo, Zhang and Wu (2005); and Shaik and Hiberty (2004).

The VB wave function
The fundamental difference between MO based methods and modern VB methods is the way the wave function is constructed. In modern VB theory the wave function ($$ \Psi $$) is created by forming a linear combination of structures ($$\Phi_{i}$$):
 * $$ \Psi = \sum \limits_{i} c_{i} \Phi_{i} $$

Such a structure can for example represent a resonance structure of a molecule. The idea behind a structure is that it is chemically meaningful and combines all relevant orbitals to capture the targeted effects. In general a structure only has to be an eigenfunction of the $$\hat{S}_{z}$$ and $$\hat{S}^{2}$$  spin operators. For this reason the wave function is also referred to as spin-coupled wave function. To ensure these properties, it is usually written as a linear combination of determinants ($$ \Delta $$) with fixed spin-coupling coefficients:
 * $$ \Phi = \sum \limits_{j} s_{j} \Delta_{j} $$

Each determinant is the antisymmetrized product of atomic orbitals ($$ \phi $$) (i.e. all those which the user selected based on chemical knowledge). Usually these orbitals are non-orthogonal and expanded into a set of basis functions ($$ \chi $$) such that:
 * $$ \phi = \sum \limits_{k} s_{k} \chi_{k} $$

The total energy of the VB wave function is then given by:
 * $$ E_{VB} = \frac{\langle \Psi | \hat{H} | \Psi \rangle}{\langle \Psi | \Psi \rangle} =\frac{\langle \sum \limits_{i} c_{i} \Phi_{i} | \hat{H} | \sum \limits_{j} c_{j} \Phi_{j} \rangle}{\langle \Psi | \Psi \rangle} =\frac{\sum \limits_{i,j} C_{i} C_{j} \langle \Phi_{i} | \hat{H} | \Phi_{j} \rangle}{\sum \limits_{i,j} C_{i} C_{j}  \langle \Phi_{i} | \Phi_{j} \rangle} $$

In modern VB theory the coefficients are determined using the variational method. Thus they belong to the class of Ab initio methods. Depending on whether the orbitals are optimized alongside the different structure coefficients and whether there are any restrictions on the orbitals, different methods emerge from this approach.

Example: wave function of the hydrogen molecule
Let us consider the hydrogen molecule as an example. In classical VB theory or with a minimal basis set, two 1s atomic orbitals ( $$ \phi_{a}$$ and $$\phi_{b} $$), which are centered on the two hydrogen atoms, are coupled into a singlet bond. Using these spin-coupled orbitals one can construct a covalent structure:


 * $$ \Phi_{covalent} = | \phi_{a}^{\alpha}(1) \phi_{b}^{\beta}(2) | + | \phi_{b}^{\alpha}(1) \phi_{a}^{\beta}(2)| $$

The covalent structure represents the case when the electrons are shared between the two atoms. Alternatively one can also write an ionic structure, which represents the situation when both electrons are localized at either atom rather than in the center.


 * $$ \Phi_{ionic} = | \phi_{a}^{\alpha}(1) \phi_{a}^{\beta}(2) | + | \phi_{b}^{\alpha}(1) \phi_{b}^{\beta}(2)| $$

Both structures can be viewed as a way to represent the hydrogen molecule much like resonance structures. Although the later one might seem a bit more exotic it has it's relevance and is important for the dissociation properties of the hydrogen molecule. This illustrates one difficulty with VB. That is one has to come up with all relevant structures otherwise the result may not be accurate. The final wave function in this example is a linear combination of these two structures.


 * $$ \Psi_{VB} = \Phi_{ionic} + \Phi_{covalent} = C_{1} \bigg( | \phi_{a}^{\alpha}(1) \phi_{b}^{\beta}(2) | + | \phi_{b}^{\alpha}(1) \phi_{a}^{\beta}(2)| \bigg) + C_{2} \bigg( | \phi_{a}^{\alpha}(1) \phi_{a}^{\beta}(2) | + |  \phi_{b}^{\alpha}(1) \phi_{b}^{\beta}(2)|  \bigg)  $$

It can be shown that in the case where both coefficients are equal to 1 the expression equals the one obtained by MO based Hartree–Fock method within a minimal basis. Similarly the expression above is the same as obtained by MCSCF. This demonstrates that there is a connection between the VB and MO theory. While the above result is equally obtained by classical VB theory, the modern valence bond theory replaces the simple linear combination of the two atomic orbitals with a linear combination of more orbitals in a larger basis set. The two resulting valence bond orbitals look like an atomic orbital on one hydrogen atom slightly distorted towards the other hydrogen atom. Modern valence bond theory is thus an extension of this Coulson–Fischer method.

Ab Initio valance bond methods
As for MO based electronic structure methods also for VB methods there is a hierarchy of different methods, which offer a particular level of accuracy. These methods differ in the selection of coefficients, which are optimized, the restrictions applied to the orbitals and the numbers of configurations allowed. Most methods use n valence bond orbitals for n electrons and they are usually divided into those methods, which keep the orbitals localized and others which allow for delocalization. Usually the methods using localized orbitals yield results, which are easier to relate to the effect of different structures. Some examples form the spectrum of methods are provided below. For a more extensive list see Shaik and Hiberty, 2007.

Spin-coupled theory
If a single set of orbitals is combined with all linear independent combinations of the spin functions, we have spin-coupled valence bond theory. The spin-coupled theory (SC) developed by Gerratt and Pyper in 1977 is based on the single-configuration spin-coupled wave function, which gives a  great physical perspective of correlated electronic structure. In this method the total wave function is optimized using the variational method by varying the coefficients of the basis functions in the valence bond orbitals and the coefficients of the different spin functions. In other cases only a sub-set of all possible spin functions is used. Many valence bond methods use several sets of the valence bond orbitals. The SC method removes any orthogonality and perfect-pairing restrictions which allows the orbitals to overlap freely with each other. Due to these characteristics, the method has been commonly used as theoretical support of some concepts such as orbital hybridization, or resonance between Kekule structures. The SC method has the following chemical characteristics: in the SC method, the shape of the orbitals and the spin-coupling does not require any constraint since both are determined only by the variational principle; it takes into account the correlated electrons; the orbitals generated with SC method are unique.

Valance Bond SCF (VBSCF)
The valance bond self-consistenf field (VBSCF) method due to van Lenthe and Balint-Kurti is the VB equivalent to MO based  multi-configurational SCF  (MCSCF) method. It is also referred to as a non-orthogonal MCSCF method, since it is based on an extension of Brillouin's theorem to non-orthogonal orbitals. Similar to the spin-coupled VB method the wave function is build as a linear combination of structures. During the VBSCF optimization not only the coefficients of the structures are varied but also the orbital coefficients. However, in contrast to the SC method there are neither restriction on the number of configurations nor on their occupation. By this procedure the method is able to take into account correlation effects as does it's orthogonal counterpart. Since the orbitals are non-orthogonal this method is computationally more expensive than MO-based MCSCF. Thus various schemes for restricting orbitals have been developed. Often only a subset of orbitals is optimized, while keeping the rest fixed and orthogonal. However, this possibility to restrict orbitals also allows to localize them on specific atoms. With this possibility the effect of delocalization and resonance can be studied in a quantitative manner. Thus this method enables to investigate core ideas of chemical bonding. Furthermore using a subset of orbitals is often a reasonable simplification, since not all orbitals are always relevant for the chemical phenomenon under consideration. Such reasoning is the same as for CASSCF calculations. However, also similar to CASSCF calculations the accuracy of the results largely depends on the quality or relevance of the chosen structures and the number of orbitals, which are optimized. In principle by lifting all restrictions on the orbitals and including all possible structures the same result as in a full CI  is obtained albeit at a much higher computational effort.

Valance Configuration Interaction (VBCI)
The valence bond configuration interaction (VBCI) method was developed by Wu et al. and includes dynamic correlation using configuration interaction  (CI). VBCI is a post-VBSCF method that improve the energetics after a VBSCF calculation. It is the equivalent of the CI in the MO approach and the different levels of truncation are named in a similar way (VBCIS for single excitations, VBCISD for singles and doubles, etc.). Since this method has no other limitations than the level of truncation, it can be improved systematically with increasing computer power and software improvements. This method still expresses the wave function in terms of a minimal number of effective structures that explains the chemistry of the problem in a simply way.

The CI calculation defines virtual orbitals and excited structures, and uses the occupied orbitals and fundamental structures obtained from VBSCF. The virtual orbitals are localized on the same parts as the occupied orbitals and the excited structures are obtained by a restricted replacement of occupied orbitals by virtual orbitals. The excited structure and the fundamental one have the same electronic pairing pattern and charge distribution and describe the same classical VB structure. This excited structures serve to relax and give dynamical correlation to the fundamental structure. The accuracy of the method is comparable to CCSD and CCSD(T) methods.

Breathing Orbitals VB (BOVB)
The Breathing Orbital Valence Bond method was proposed by Hiberty et al. in 1994. This method basically allows the wave function to have different orbitals for the different VB structures, which represent the molecular system. This approach is easily understood in terms of Lewis structures and permits to characterize the elementary bonds. The BOVB method has two main objectives: yielding accurate dissociation energy curves as well as keeping wave function compact and clearly explained in terms of Lewis structures. These objectives can be achieved by following these two steps: (1) generate all relevant structures of a given electronic system, (2) obtain an optimal VB description by means of a proper orbital optimization. This guarantees that a variational combination of the VB structures accurately reproduces the energetics of this electronic system.

Applications
Charge-Shift Bonding: The Charge-Shift bond has been proposed through the results of VB theory calculations as a new class of chemical bond that sits along the well familiar families of covalent and ionic bonds where electrons are shared or transferred respectively. This bond derives its stability from the resonance of ionic forms rather than the covalent sharing of electrons, which are often depicted as having electron density between the bonded atoms. A notable characteristic of the charge shift bond model is that, the predicted electron density between the bonded atoms is very low compared to the description of the same bond using a covalent bond model, which agrees with the experimental data of atoms such as F2.

Another example where charge shift bonding has been used to explain the low electron density found experimentally is in the central bond between the inverted tetrahedral carbons in [1.1.1]propellanes. Thus the Charge-Shift bond model successfully explains the characteristics of certain atoms and molecules where the conventional bond model fails.