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

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 Of course, the case is not like "Hey, look, one theory is outdated; just follow the other one..."; they are both important and hence do want to know the difference between their pathways of analyzing the cases.

• It is a good question and a lot of books only scratch the surface of the theories anyway. Understanding both on a conceptual basis is necessary to distinguish the difference and similarities. There are plenty of resources of comparison available online, maybe UC Davis helps, I have not read it, but the following might also help: chem.wisc.edu/courses/562/fall08/PDFfiles/Hoffmann.pdf Jul 12 '15 at 6:12
• UC Davis archive link. :-) Aug 8 '21 at 15:55

# 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.

# 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.

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.

• A lovely answer, but are you then saying that the only real difference is due to the choice of basis sets and that one (MO) is thus more adaptable (and computationally convenient) than the other? Feb 27 '17 at 12:43
• @porphyrin I wrote this answer before I'd learned much about the modern implementations of VB theory. I now know differently, and have plans at some point to revise this. That said, yes, I think both points of your question are correct. However, the VB analogues of correlated post-HF MO methods are able to provide quite good wavefunctions in many cases, so modern VB theory is actually much more useful than this answer makes it sound. IIUC, though, one has to make a deeper dive into VB as compared to MO before obtaining practically useful results. Feb 27 '17 at 14:28
• @porphyrin To note one key difference, whereas MO theory works intrinsically with orthogonal MOs, IIUC the "working orbitals" of VB theory are not constrained to be orthogonal. Feb 27 '17 at 14:29
• thank you, very clear. So I suppose the non-orthogonality must make diagonalising matrices very time consuming vs orthogonal ones for the same level of accuracy? Feb 27 '17 at 21:12
• @porphyrin I have to assume so -- I'm not too familiar with the mathematical particulars. I think it also makes the equation systems dramatically more complicated when you can't assume $\left<i|j\right>=0$ for $i\neq j$. Feb 27 '17 at 22:53

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.

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.

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"?