What is actually the difference between valence bond theory and molecular orbital theory?

Question

Recently I have read about both of the concepts in my book (Physical Chemistry by Atkins, Paula). It was literally a reading; though I could understand the language and superposition of orbitals, inversion symmetry, and so on, I never could really comprehend how these two theories differ (or agree) with each other. I have googled it but couldn't gain a lucid explanation between the differences in their approaches. So, now my question is:

How do these two theories differ from each other? What are the differences between their approaches?

Also, in VB theory, resonance plays a pivoting role; why isn't such concept needed in MO theory? Why it is told that "MO theory provides a global, delocalized perspective on chemical bonding"? This portion of the question was split to a separate question: Why is the resonance concept not required in molecular orbital theory?

Edit: After viewing some of the answers, I think IMHO, I had to clear what I was really asking. By saying "difference between MOT & VBT", I am not intending to mean that just write the different predictions the two theories can separately succeed in making. What I really mean is the difference between their approaches & their way of explanation. Of course, the case is not like "Hey, look, one theory is outdated; just follow the other one..."; they are both important and hence do want to know the difference between their pathways of analyzing the cases.

Answer 1

TL;DR

VB theory treats atomic orbitals (including hybridized orbitals) as providing a good mathematical/physical description of the true form of the molecular wavefunction. MO theory uses atomic orbitals (with Gaussian radial functions) as a tool of computational convenience in an effort to define a molecular wavefunction that in its final form often bears little resemblance to the original atomic orbitals, or to the VB theory description.

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Long Version

The key difference between the VB and MO theories lies in their respective assumptions of the underlying mathematical/physical form of the electronic wavefunction.

VB theory holds tightly to the mathematical structure of atomic orbitals (AOs), assuming that the electronic structure of a molecular system is described adequately by retaining the atomic orbitals (possibly hybridized) from each constituent atom, and overlapping them to form the needed description of bonding. VB calculations are computationally much more complex: the AO basis is generally non-orthogonal, and the linear algebra and other techniques required to cope with this non-orthogonality are much more involved.

More broadly, the need to invoke resonance (superposition of multiple valence-bond structures) and other concepts highlights that, in fact, often this adherence to atomic orbitals and their valence-bond overlaps is not adequate for a proper description, at least not without extensive use of methods analogous to the configuration interaction of MO theory. VB theory does hold certain advantages, however—for example, I believe the AO basis does sometimes allow for a more direct linkage to fundamental chemical concepts such as hybridization and Lewis electon-pair theory. To note, Sason Shaik is one of the strongest modern proponents of the theory I know of—he describes some of these strengths and their applications in his recent retrospective article (doi:10.1016/j.comptc.2017.02.011).

MO theory, at its core, casts away most assumptions regarding the mathematical structure of the electronic wavefunction, as long as properties like indistinguishability and the Pauli exclusion principle remain satisfied. Calculations within MO theory can use a wide variety of mathematical structures to try to represent the wavefunction, ranging from gridded numerical quadrature approaches (see here and here, and also orbital-free DFT) to Slater orbital computations to the Gaussian-orbital based calculations used most commonly today.

One potential point of confusion is that MO theory in its most common implementations makes direct use of atomic orbitals in composing the molecular wavefunction (see LCAO theory). The distinction to be made here is that instead of attributing significance to the atomic orbitals as an intrinsically good descriptor of electronic structure, they are used merely because they afford a convenient mathematical form for efficient computation, in trying to construct the MOs. Particular advantages are obtained from the use of atomic orbitals constructed from Gaussian functions, since the product of two Gaussians is another Gaussian (even in multiple dimensions) and numerical integration of n-dimensional Gaussian functions is straightforward.

It should be noted that, for as much of an improved description it affords, MO theory still is an approximation. For a given system, the electronic wavefunction just IS the highly-abstract mathematical object that satisfies the appropriate Schrodinger equation. MO theory is used extensively because it provides one of the most computationally tractable paths to getting in the neighborhood of the "real answer."

Practical Aspects of MO theory

The "simplest version" of MO theory for a time-independent system under the Born-Oppenheimer Approximation is the Hartree-Fock method, which provides a remarkably good description considering the significant simplification represented by its mean-field approximation of the electron-electron interactions. Hartree-Fock is in general not good enough for computations at the level of chemical accuracy (~$1\mathrm{\ kcal/mol}$), however, and numerous methods have been developed to refine it: the most often used techniques are Moller-Plesset perturbation theory, configuration interaction, and coupled cluster.

In-Depth Reading on MO Theory

For a deep dive into the underlying mathematics of MO computations, it's hard to beat the classic Szabo & Ostlund. As a more readable, lighter-weight introduction to MO theory and its implementations, I would recommend Frank Jensen's Introduction to Computational Chemistry.

Answer 2

Comparing modern valence bond and electronic structure theories one can argue that the generalized valence bond (GVB) wave function can be regarded as a special form of the multi-configurational self-consistent field (MCSCF) wave function.¹

Thus, for instance, for the hydrogen molecule, the GVB wave functions has the following form (ignoring normalization hereinafter for simplicity), $$ \Psi_\mathrm{GVB} = \Big( f(1)g(2) + g(1)f(2) \Big) \Big( α(1)β(2) - β(1)α(2) \Big) \, , $$ where $f = a + kb$ and $g = b + ka$ are non-orthogonal orbitals and $C$ is a normalization factor. This wave function is a linear combination $$ \Psi_\mathrm{GVB} = C_\mathrm{c} \Psi_\mathrm{c} + C_\mathrm{i} \Psi_\mathrm{i} $$ of the covalent and ionic structures described by $$ \Psi_\mathrm{c} = \Big( a(1)b(2) + b(1)a(2) \Big) \Big( α(1)β(2) - β(1)α(2) \Big) \, , \\ \Psi_\mathrm{i} = \Big( a(1)a(2) + b(1)b(2) \Big) \Big( α(1)β(2) - β(1)α(2) \Big) \, . $$ The GVB orbitals $f$ and $g$ are determined variationally: $f$ and $g$ are expanded over a basis and expansion coefficients are varied to minimize the energy of the wave function.

The simplest two configurational MCSCF wave function for the hydrogen molecule would also be a linear combination of two wave functions. But in MCSCF to avoid non-orthogonal functions (which complicates the matter) a bonding orbital $\phi_1$ and an anti-bonding orbital $\phi_2$ are introduced and the total wave function is written as a linear combination of two Slater determinants describing two different electron configurations, $\phi_1^2$ and $\phi_2^2$, i.e., $$ \Psi_\mathrm{MSSCF} = C_1 \Phi_1 + C_2 \Phi_2 \, , $$ where $$ \Phi_1 = \left| \begin{matrix} \phi_1 & \bar{\phi}_1 \end{matrix} \right| = \phi_1(1) \phi_1(2) \Big( α(1) β(2) - β(1) α(2) \Big) \, , \\ \Phi_2 = \left| \begin{matrix} \phi_2 & \bar{\phi}_2 \end{matrix} \right| = \phi_2(1) \phi_2(2) \Big( α(1) β(2) - β(1) α(2) \Big) \, . $$

However different $\Psi_\mathrm{GVB}$ and $\Psi_\mathrm{MSSCF}$ appear to be, if the same basis set is used to expand the orthogonal MCSCF orbitals $\psi_1$ and $\psi_2$ as well as the non-orthogonal GVB orbitals $f$ and $g$, it can be shown that these wave functions are identical. For details, see Exercise 4.9 in Ref. 1.

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1) Szabo, A.; Ostlund, N.S. Modern Quantum Chemistry: Introduction to Advanced Electronic Stucture Theory; Dover: Mineola, NY, 1989 (revised in 1996). pp. 258-261.

Answer 3

Here, I am referring Valence Bond Theory by VBT and Molecular Orbital Theory by MOT.

1.In VBT, atomic orbitals of the combining atoms retain a large amount of their individual character. (For example, see Orbital Hybridization). Whereas, in MOT atomic orbitals of the combining atoms lose their individual identity in the resulting Molecular Orbital.

2.Secondly, VBT fails to explain properties like para-magnetism of some compounds like O2, whereas MOT easily explains this.

3.Also, VBT fails to explain possibility and existence of ions like H2$^+$, whereas MOT is able to do this successful.

Why it is told that "MO theory provides a global, delocalized perspective on chemical bonding"?

Quoting from the Wikipedia page of MOT,

In MO theory, any electron in a molecule may be found anywhere in the molecule, since quantum conditions allow electrons to travel under the influence of an arbitrarily large number of nuclei, as long as they are in eigenstates permitted by certain quantum rules. Thus, when excited with the requisite amount of energy through high-frequency light or other means, electrons can transition to higher-energy molecular orbitals.

Why isn't such concept needed in MO theory?

See this question, which directly addresses this. However, I would recommend you to read How does molecular orbital theory deal with resonance?

Thank You!

Answer 4

In the words of a commentary in the Journal of Chemical Education:

At their theoretical limits, both VB and MO are equivalent; they simply approach that limit from different sides.

Source:https://doi.org/10.1021/acs.jchemed.1c00919

The authors argue that in introducing the VB method in General Chemistry, it is equated with Lewis structures, which does not fully represent modern VB theory.

In the article, they also describe the simplest forms of the two theories and how they differ:

As the names suggest, valence bond theory considers a molecule as a collection of bonds, whereas molecular orbital theory views the molecule as a whole. [...] VB is presented as a single configuration of electrons in hybrid orbitals, spin paired in bonds, while MO is presented as a single configuration of molecular orbitals arranged in order of increasing energy and filled with electrons in accordance with Hund’s rule.

One way of describing the different approaches is to look at the structure of methane. The symmetry of the atomic orbitals (the p-orbitals specifically) does not match the geometry of the hydrogen atom location. In VB, the carbon atom orbitals are mixed and matched (hybridized) to make localized bonds with hydrogen. In MO, the hydrogen s-orbitals are mixed and matched (linear combinations of AOs) to match the symmetry of the p-orbitals of carbon. At a higher level of theory, which recognizes that electrons are indistinguishable, you arrive at the same distribution of electrons in the molecule.