Quantitative MO theory

The chemist's qualitative picture of molecular orbitals can be rigorously derived from a quantum chemical approach. In this section we give a very brief outline of the theory.

Schrödinger equation
In 1925 Erwin Schrödinger and Werner Heisenberg independently formulated a general quantum theory. Although the two formulations are mathematically equivalent, Schrödinger presented his theory in terms of partial differential equations and, within this framework, the energy of an isolated molecule can be obtained by the solution of the Schrödinger equation. In its time-independent form, this can be written as:


 * $$\hat{H} \Psi = E \Psi $$

where $$\hat{H}$$ is the Hamiltonian operator, $$\Psi$$ is the wavefunction, and E is the energy of the system relative to the state in which the nuclei and electrons are infinitely separated and at rest.

The masses of the nuclei are much larger and their velocities much smaller than those of the electrons, and it is possible to simplify the solution of the Schrödinger equation by separating it into two parts, one describing the motions of the electrons in a field of fixed nuclei and the other describing the motions of the nuclei. This is known as the adiabatic or Born-Oppenheimer approximation. Molecular orbital theory is concerned with finding approximate solutions to the first part, that is, the electronic Schrödinger equation:
 * $$\hat{H}^e \Psi^e = E^e \Psi^e $$

Each quantity is implicitly a function of the nuclear co-ordinates.

The orbital approximation
The orbital approximation simplifies the above equation by assuming that each electron is associated with a separate one-electron wavefunction or spin orbital, [chi]. Thus, Hartree proposed that the wavefunction could be expressed simply as a product of spin orbitals, one for each electron:


 * $$ \psi = \chi_1 (1) \chi_2 (2) ... \chi_n (n)\,\! $$

The LCAO approximation
Each spin orbital is actually a product of a spatial function, $$\chi_i (x,y,z)\,\!$$, and a spin function, α or β. The spatial molecular orbitals, $$\Phi_i\,\!$$, are usually expressed as linear combinations of a finite set of known one-electron functions. In the simplest case where these functions take the form of the atomic orbitals of the constituent atoms, this expansion is called a linear combination of atomic orbitals (LCAO):


 * $$\Phi_i = c_{1i}\chi_1 + c_{2i}\chi_2 + ... c_{Ni}\chi_N\,\!$$

Qualitatively, this is like saying that the two molecular orbitals in H2 are linear combinations of the 1s atomic orbitals:

&Phi;2 = 0.5&chi;a - 0.5&chi;b &Phi;1 = 0.5&chi;a + 0.5&chi;b

One-electron approximations
The simplest way to solve the Schrödinger equation is to treat each electron in isolation from the rest. This "one-electron approximation" leads to Hückel theory and Extended Hückel theory.