Electronic correlation

From Freepedia

Electronic correlation refers to the interaction between electrons in a quantum system whose electronic structure is being considered. The term correlation stems from mathematical statistics and means that two distribution functions, f(x) and g(y), are not independent of each other.

Introduction

Within the Hartree-Fock method of quantum chemistry, the antisymmetric wave function is approximated by a single Slater determinant. Exact wave functions, however, cannot generally be expressed as single determinants. The single-determinant approximation does not take into account the correlation between electrons with opposite spin, leading to a total electronic energy different from the exact solution of the non-relativistic Schrödinger equation within the Born-Oppenheimer approximation. Therefore the Hartree-Fock limit is always above this exact energy. The difference is called the correlation energy, a term coined by Löwdin.

A certain amount of electron correlation is already considered within the HF approximation, found in the electron exchange term describing the correlation between electrons with parallel spin. This basic correlation prevents two parallel-spin electrons from being found at the same point in space and is often called Fermi correlation. Coulomb correlation, on the other hand, describes the correlation between the spatial position of electrons with opposite spin due to their Coulomb repulsion. There is also a correlation related to the overall symmetry or total spin of the considered system.

Mathematical viewpoint

For two independent electrons a and b,

<math>\rho(\mathbf{r}_a,\mathbf{r}_b) \sim \rho(\mathbf{r}_a)\rho(\mathbf{r}_b)</math>,

where ρ(r_a,r_b) represents the joint electronic density, or the probability of finding electron a at r_a and electron b at r_b. Within this notation, ρ(r_a,r_b)dr_adr_b represents the probability of finding the two electron in the respective volume elements dr_a and dr_b.

If these two electrons are correlated, then the probability of finding electron a at a certain position in space depends on the position of electron b, and vice versa. In other words, the product of their independent density functions does not adequately describe the real situation. At small distances, the uncorrelated pair density is too large, and too small at large distances - the electrons tend to "avoid each other".

See also

• Configuration interaction
• Coupled cluster
• Hartree-Fock
• Moller-Plesset
• Post-Hartree-Fock
• Strongly correlated material