IIT JEE , AIEEE Forum - Chemistry , Organic Chemistry

nitansh

In chemistry, a molecular orbital is a region in which an electron may be found in a molecule.^[1] In general, atomic orbitals combine to form molecular orbitals. The probability of finding an electron in a given region can be obtained by solving the Schrödinger wave equation.

Overview

In quantum chemistry, i.e. electronic structure theory, the molecular electronic states, which are the eigenstates of the electronic molecular Hamiltonian, are expanded (see configuration interaction expansion and basis) into linear combinations of anti-symmetrized products (Slater determinants) of one-electron functions. The spatial components of these one-electron functions are called molecular orbitals (MO). When one considers also their spin component, they are called molecular spin orbitals.

Most methods in computational chemistry today start by calculating the MOs of the system. A molecular orbital describes the behavior of one electron in the electric field generated by the nuclei and some average distribution of the other electrons. In the case of two electrons occupying the same orbital, the Pauli principle demands that they have opposite spin.

Qualitative discussion

For an imprecise, but qualitatively useful, discussion of the molecular structure, the molecular orbitals can be obtained from the "Linear combination of atomic orbitals molecular orbital method" ansatz (using eventually the concept of hybridized orbitals).

In this approach, the molecular orbitals are expressed as linear combinations of atomic orbitals, as if each atom were on its own.

The linear combination of atomic orbitals approximation for molecular orbitals was introduced in 1929 by Sir John Lennard-Jones. His ground-breaking paper showed how to derive the electronic structure of the fluorine and oxygen molecules from quantum principles. This qualitative approach to molecular orbital theory represents the dawn of modern quantum chemistry.

Some properties:

The number of molecular orbitals is equal to the number the atomic orbitals included in the linear expansion,
If the molecule has some symmetry, the degenerate atomic orbitals (with the same atomic energy) are grouped in linear combinations (called symmetry adapted atomic orbitals (SO)) which belong to the representation of the symmetry group, so the wave functions that describe the group is known as symmetry-adapted linear combinations (SALC).
The number of molecular orbitals belonging to one group representation is equal to the number of symmetry adapted atomic orbitals belonging to this representation,
Within a particular representation, the symmetry adapted atomic orbitals mix more if their atomic energy level are closer.

More quantitative approach

To obtain quantitative values for the molecular energy levels, one needs to have molecular orbitals which are such that the configuration interaction (CI) expansion converges fast towards the full CI limit. The most common method to obtain such functions is the Hartree-Fock method which expresses the molecular orbitals as eigenfunctions of the Fock operator. One usually solves this problem by expanding the molecular orbitals as linear combinations of gaussian functions centered on the atomic nuclei (see linear combination of atomic orbitals and basis set (chemistry)). The equation for the coefficients of these linear combinations is a generalized eigenvalue equation known as the Roothaan equations which are in fact a particular representation of the Hartree-Fock equation.

Simple accounts often suggest that experimental molecular orbital energies can be obtained by the methods of ultra-violet photoelectron spectroscopy for valence orbitals and X-ray photoelectron spectroscopy for core orbitals. This however is incorrect as these experiments measure the ionization energy, the difference in energy between the molecule and one of the ions resulting from the removal of one electron. Ionization energies are linked approximately to orbital energies by Koopmans' theorem. While the agreement between these two values can be close for some molecules, it can be very poor in other cases.

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Overview

Qualitative discussion

More quantitative approach