2.2: Origins of MO Theory - The “Impossible” Problem

The “Impossible” Problem

One of the principal goals for this course is to apply quantum mechanical principles to understand the structures and properties of molecules. Most molecules of interest to modern chemists are polyatomic, but the strategies used to solve the Schrödinger equation for atoms and diatomic molecules can also be applied to polyatomics. For simplicity, this discussion will begin with the simplest diatomic molecule, \(\ce{H_2^+}\), which will serve as a useful illustration of the challenges and strategies that also apply to more complicated systems. The non-relativistic, spin-free, time-independent Hamiltonian of in atomic units (using the notation in Figure 1.1) is:

\[\hat{H} = -\frac{1}{2}\left(\nabla_1^2 + \nabla_2^2 + \frac{\nabla_A^2}{m_p} + \frac{\nabla_B^2}{m_p}\right) - \frac{1}{r_{1A}} - \frac{1}{r_{1B}} - \frac{1}{r_{2A}} - \frac{1}{r_{2B}} + \frac{1}{r_{12}} + \frac{1}{R}. \label{chap:mot:eq:h}\]

Figure 2.1: Coordinate system used in the Hamiltonian for \(\ce{H_2^+}\).

Breaking this down into its physical components, the individual terms mean:

• Kinetic energies of electrons and protons (proton mass is \(\sim\)1836\(m_e\) in atomic units). \[-\frac{1}{2}\left(\nabla_1^2+\nabla_2^2+\frac{\nabla_A^2}{m_p}+\frac{\nabla_B^2}{m_p}\right)\]
• Electron-proton attraction terms for each pair. The numerators are \(-1\) because the protons and electrons have charges of \(+e\) and \(-e\) respectively (\(e \equiv 1\) in atomic units). \[\frac{-1}{r_{1 A}}, \frac{-1}{r_{1 B}}, \frac{-1}{r_{2 A}}, \frac{-1}{r_{2 B}}\]
• Electron-electron repulsion.\[\frac{1}{r_{12}}\]
• Proton-proton repulsion. \[\frac{1}{R}\]

The goal is to find the set of wavefunctions \(\ket{\psi}\) that satisfy the Schrödinger equation \(\hat{H}\ket{\psi_n} = E_n\ket{\psi_n}\). In other words, we seek a set of mathematical functions involving the coordinates of the nuclei and electrons that, after taking various second derivatives and multiplying by a number of functions of the coordinates, returns the exact same set of functions, each multiplied by a constant which corresponds to its energy (i.e., the eigenfunctions and eigenvalues of the Hamiltonian operator). Unfortunately, this equation is not at all trivial. In particular the presence of the \(r_{12}\) electron correlation term makes analytical solution of this equation extremely difficult– so much so that right now there is no known exact solution to Equation \ref{chap:mot:eq:h}!

In one sense, the entire field of chemistry is a solved problem: we know exactly what equation governs the nature of molecules. The trouble is that we don’t know how to solve it exactly! All is not lost; however, as we can use a variety of strategies (i.e., perturbation theory, the variational principle, or even just ignoring terms) to approximately solve the Schrödinger equation. The main theory we will use to describe the electronic structure of molecules is Molecular Orbital (MO) Theory, in which we allow electrons to occupy “orbitals” that may span the entire molecule rather than being constrained to a single atom or bond. This is in contrast to Valence Bond Theory (and its hybridized variants) which associate electrons with particular chemical bonds and localized lone pairs. As we will see, both of these theories involve thinking about electrons as if they can be treated more or less independently of one another by approximating the problematic \(r_{12}\) term.

The Born-Oppenheimer Approximation

Molecular orbital theory is based on two main approximations that involve ignoring terms. Our first approximation is the Born-Oppenheimer approximation, which argues that nuclear motion is very slow compared to the motion of the electrons and can be neglected when considering the behavior of the electrons. The practical effect of this approximation is to separate the Hamiltonian into an electronic part, in which the nuclear kinetic energy terms are set to 0 and their positions (i.e., all distances \(R\) that involve nuclear positions) are constant. The Schrödinger equation is solved for the electrons first, yielding the energy levels for the electrons. The electronic wavefunctions are functions only of the coordinates of the electrons, and do not directly involve any nuclear coordinates because their positions are held constant.

This calculation can be repeated at many nuclear positions to determine the electronic energy as a function of geometry, and then later the Schrödinger equation can be solved for the nuclei, where the nuclear kinetic energy is added to the electronic potential energy from the electronic Hamiltonian as a function of the nuclear coordinates. As it turns out, the energies of the nuclear states are generally small compared to the electronic energies, and therefore the electronic wavefunction and energies are primarily responsible for determining a molecule’s geometry, bonding, and chemical reactivity. For this reason, it is sometimes said that the field of chemistry is all about the energy associated with electrons. We will return to the nuclear Hamiltonian later in the course; for now, the rest of this chapter and the following chapter will focus only on the electronic Hamiltonian under the Born-Oppenheimer approximation.

After making the Born-Oppenheimer approximation, the Hamiltonian for becomes:

\[\hat{H} = -\frac{1}{2}\left(\nabla_1^2 + \nabla_2^2 \right) - \frac{1}{r_{1A}} - \frac{1}{r_{1B}} - \frac{1}{r_{2A}} - \frac{1}{r_{2B}} + \frac{1}{r_{12}} + \frac{1}{R}, \label{chap:mot:eq:hbo}\]

where \(R\) is constant since the nuclear coordinates are fixed. Thus, the \(1/R\) term is just the nuclear repulsion energy, which we will write as \(E_N(R)\) to remind ourselves that although the nuclear energy is independent of the coordinates of the electron, it does depend on the internuclear separation coordinate \(R\). If we change the nuclear geometry, the value of \(E_N\) will be different, but still independent of the coordinates of the electrons. Although this is a major simplification (the wavefunction now depends only on 6 coordinates: the \(x\), \(y\), and \(z\) coordinates of each of the two electrons), the problematic \(r_{12}\) electron correlation term remains, which makes this still a difficult– and perhaps mathematically impossible– problem that forces us to use additional approximations. However, we will choose approximations that are systematically improvable: they will us a useful chemical model that we can use to rationalize and predict molecular behavior, and that model converges toward the “exact” answer as more computational power is applied to the problem. Together, these approximations form the basis of MO theory. To see how MO theory works, though, it will be easiest to first take a step back to a simpler problem, and come back to armed with some initial results.