Molecular orbital

In quantum chemistry, a molecular orbital (MO) is a one-electron function that is "smeared out" over a whole molecule. Usually an MO is a linear combination of atomic orbitals (an LCAO), which is a weighted sum of (almost) all atomic orbitals in the molecule. Molecular symmetry plays an important role in molecular orbital theory. In general MOs are adapted to symmetry, which has the consequence that atomic orbitals of a symmetry different than that of an MO do not contribute to the MO (have weight factor zero).

The expansion coefficients (weight factors) of an MO can be determined by semi-empirical MO theory, which may or may not be of the self-consistent field type, or by ab initio methods. The most common ab initio method for determining MO coefficients is the self-consistent field method of Hartree and Fock. Lately, local density methods for computing MOs have found increasing application.

Definition of molecular orbital

A molecular orbital (MO) depends on several position vectors of one and the same electron, labeled 1: r_A1, r_B1, r_C1, ... where A, B, C, ... are different points in space (usually nuclear positions). The oldest example of an MO (without use of the name MO yet) is in the work of Burrau (1927) on the single-electron ion H₂⁺. Burrau applied spheroidal coordinates (a bipolar coordinate system) to describe the wave functions of the electron of H₂⁺.

Lennard-Jones introduced the following linear combination of atomic orbitals (LCAO) way of writing an MO φ:

$\phi (\mathbf {r} _{1})=\sum _{A=1}^{N_{\mathrm {nuc} }}\sum _{i=1}^{n_{A}}c_{Ai}\chi _{i}(\mathbf {r} _{A1}),\qquad c_{Ai}\in \mathbb {C} ,\qquad \qquad \qquad \qquad (1)$

where A runs over N_nuc different points in space (usually A runs over all the nuclei of a molecule, hence the name molecular orbital), and i runs over the n_A different AOs centered at A. The complex coefficients c _Ai can be calculated by any of the existing effective-one-electron quantum chemical methods. Examples of such methods are the Hückel method—a non-self-consistent field method— and the Hartree-Fock method—a self-consistent field method.

Computation of molecular orbitals

In the hands of Friedrich Hund, Robert S. Mulliken, John Lennard-Jones, and others, molecular orbital theory was established firmly in the 1930s as a (mostly qualitative) theory explaining much of chemical bonding, especially the bonding in diatomic molecules.

MO energy level diagram for H₂

To give the flavor of the theory we look at the simplest molecule: H₂. The two hydrogen atoms, labeled A and B, each have one electron (a red arrow in the figure) in an 1s atomic orbital (AO). These AOs are labeled in the figure 1s_A and 1s_B. When the atoms start to interact the AOs combine linearly to two molecular orbitals:

$\sigma _{g}=N_{g}(1s_{A}+1s_{B})\quad {\hbox{and}}\quad \sigma _{u}=N_{u}(1s_{A}-1s_{B}).$

In equation (1) above a general expression for an LCAO-MO is given. In the present example of the σ_g MO of H₂, we have N_nuc = 2, n_A = n_B = 1, χ_A1 = 1s_A, χ_B1 = 1s_B, c_A1 = c_B1 = N_g. With regard to the form of the MOs the following: Because both hydrogen atoms are identical, the molecule has reflection symmetry with respect to a mirror plane halfway the H—H bond and perpendicular to it. It is one of the basic assumptions in quantum mechanics that wave functions show the symmetry of the system. [Technically: solutions of the Schrödinger equation belong to a subspace of Hilbert (function) space that is irreducible under the symmetry group of the system]. Clearly σ_g is symmetric (is even, stays the same) under

$\mathrm {\scriptstyle Reflection:} \quad 1s_{A}\longleftrightarrow 1s_{B},$

while σ_u changes sign (is antisymmetric, is odd). The Greek letter σ indicates invariance under rotation around the bond axis. The subscripts g and u stand for the German words gerade (even) and ungerade (odd). So, because of the high symmetry of the molecule we can immediately write down two molecular orbitals, which evidently are linear combinations of atomic orbitals.

The normalization factors N_g and N_u and the orbital energies are still to be computed. The normalization constants follow from requiring the MOs to be normalized to unity. In the computation we will use the bra-ket notation for the integral over x_A1, y_A1, and z_A1 (or over x_B1, y_B1, and z_B1 when that is easier).

$1=\langle \sigma _{g}|\sigma _{g}\rangle =N_{g}^{2}\left(\langle 1s_{A}|1s_{A}\rangle +\langle 1s_{B}|1s_{B}\rangle +\langle 1s_{A}|1s_{B}\rangle +\langle 1s_{B}|1s_{A}\rangle \right)=N_{g}^{2}(2+2S).$

We used here that the AOs are normalized to unity and that the overlap integral is real

$\langle 1s_{A}|1s_{B}\rangle =\langle 1s_{B}|1s_{A}\rangle \equiv S.$

Applying the same procedure for N_u, we find for the two normalization factors

$N_{g}=\left[2+2S\right]^{-1/2}\quad {\hbox{and}}\quad N_{u}=\left[2-2S\right]^{-1/2}.$

At this point one often assumes that S ≈ 0, so that both normalization factors are equal to ½√2.

In order to calculate the energy of an orbital we introduce an effective-one-electron energy operator (Hamiltonian) h,

${\begin{aligned}E_{\sigma _{g}}&=\langle \sigma _{g}|h|\sigma _{g}\rangle \\&={\frac {1}{2}}\left(\langle 1s_{A}|h|1s_{A}\rangle +\langle 1s_{B}|h|1s_{B}\rangle +\langle 1s_{A}|h|1s_{B}\rangle +\langle 1s_{B}|h|1s_{A}\rangle \right)\\&\equiv {\frac {1}{2}}(q_{A}+q_{B}+\beta _{AB}+\beta _{BA})=q+\beta .\end{aligned}}$

Since A and B are identical atoms equipped with identical AOs, we were allowed to use

$\langle 1s_{A}|h|1s_{A}\rangle =q_{A}=q_{B}=q=E_{1s}\quad {\hbox{and}}\quad \langle 1s_{A}|h|1s_{B}\rangle =\beta _{AB}=\beta _{BA}=\beta .$

Likewise

$E_{\sigma _{u}}=\langle \sigma _{u}|h|\sigma _{u}\rangle =q-\beta .$

The energy term β is negative (causes the attraction between the atoms). One may tempted to assume that q = −½ hartree (the energy of the 1s orbital in the free atom). This is not the case, however, because h contains the attraction with both nuclei, so that q is distance dependent.

If we now look at the figure in which energies are increasing in vertical direction, we see that the two AOs are at the same energy level q, and that the energy of σ_g is a distance |β| below this level, while the energy of σ_u is a distance |β| above this level. In the present simple-minded effective-one-electron model we can simply add the orbital energies and find that the bonding energy in H₂ is 2β (two electrons in the bonding MO σ_g, ignoring the distance dependence of q). This model predicts that the bonding energy in the one-electron ion H₂⁺ is half that of H₂, which is correct within a 20% margin.

If we apply the model to He₂, which has four electrons, we find that we must place two electrons in the bonding orbital and two electrons in the antibonding MO σ_u, with the total energy being 2β − 2β = 0. So, this simple application of molecular orbital theory predicts that H₂ is bound and that He₂ is not, which is in agreement with the observed facts.