Correlation energy

      The Hartree-Fock method has important limitations.  Even the Hartree-Fock limiting energy is above the true energy of a molecule.  This is because correlation energy is not included.  The correlation energy can be defined as:

Ecorrelation = Etrue – EHartree-Fock              

 

·     This error arises because the interaction of one electron with another is treated as the interaction with a smoothed out, averaged electron density. 

 

·     In fact, the position of one electron affects the position of the other electron, because they repel one another. 

 

·     If one is on one side of the molecule, the other electron is likely to be on the other side. 

 

·     Their positions are correlated, an effect not included in a Hartree-Fock calculation. 

 

·     While the absolute energies calculated by the Hartree-Fock method are too high, relative energies may still be useful. 

 

·     If the correlation energy does not change during a process, the energy calculated for the process with the Hartree-Fock method will be close to the real value. 

 

·     The correlation energy does not change very much if the bonding in the molecule does not change much. 

 

·     So, for example, calculations of conformational properties can be done accurately at the Hartree-Fock level (e.g. calculating the rotational barrier in ethane). 

 

When the bonding in a molecule changes, though, the correlation energy changes significantly.  This means that the Hartree-Fock method performs very poorly for calculations on breaking chemical bonds.  The dissociation energy predicted is too high, and it may predict the wrong products to be formed (e.g. predicting the formation of ions when in fact neutral atoms should be formed by breaking a bond). 

 

For treating chemical reactions, we need to use methods which include electron correlation.  Many methods of this type use a Hartree-Fock calculation as a starting point.  An example is the use of perturbation theory, developed by Møller and Plesset and available in many ab initio molecular orbital programs.  The term MP2 is used for second-order perturbation theory.  A more sophisticated (and computationally demanding) approach is called configuration interaction (CI).  In this technique, different electronic configurations are allowed to interact, to give a better overall wavefunction.  The improved, configuration interaction, molecular wavefunction is written as a linear combination:

Yimproved, CI = aY0 + bY1 + cY2 + ….  (31)

·     Y0 is the wavefunction found first (e.g. from a Hartree-Fock calculation). 

 

·     Y1, Y2 etc. are wavefunctions based on the original Y0 in which some electrons have been moved into higher energy molecular orbitals which are unoccupied in Y0. 

 

·     This gives a better overall wavefunction, Yimproved, CI.  CI calculations tend to be extremely demanding of computer resources, because many excited electronic configurations (Y1, Y2…) are possible.  Usually only a limited number can be included. 

 

Recently, techniques based on density-functional theory have become popular.  The energy of the molecule is calculated as a function (strictly a functional) of the electron density. 

 

·     Density-functional theory calculations on molecules usually include a Hartree-Fock calculation. 

 

·     They are popular because they include electron correlation, but are not as demanding of computer resources as methods such as MP2 and CI. 

 

Semiempirical MO methods