In the Hartree-Fock scheme, the many particle effects are approximated by the interaction of each electron with an average electric field produced by all the remaining electrons in the system. The error introduced by this SCF approximation is known as the correlation energy. The correlation energy is a small percentage of the total energy of the system, but is usually on the order of the strengths of chemical bonds or ionization energies. Thus, the correlation energy is not negligible. A number of different techniques are available to determine the correlation energy. The most widely used is the Configuration Interaction (CI) method. (Other methods are perturbation techniques or a cluster expansion of the system or direct solution of the integro-differential Hartree-Fock equations.)
CI; the basic idea. For any system, there exists an infinite number of orbitals in addition to the HF orbitals found by the procedure outlined above. These higher energy orbitals can be used to construct other configurations with coefficients variationally determined. The CI wave function has the form,
where
,
are an orthonormal set of N electron configurations.
Varying
leads to the CI equations,
where I is the identity matrix.
The matrix elements of the Hamiltonian matrix are given by,
If i and j are of different symmetry then,
.
If M configurations are used in the CI expansion then one obtains M eigenenergies and the lowest energy is an upperbound to the true energy. This implies the total system need not be solved, only the lowest (or lowest two or three) eigenvalues need to be determined. In the above discussion, configuration means a symmtery adapted linear combination of Slater Determinants (denoted by D(i) ). Symmetry adapted means the configuration posseses all the symmetry of the molecular state being described. This is written as,
The most common CI calculation is the Multi-Configuration
Self Consistent Field (MCSCF) calculation. In the
MCSCF, the configuration parameters
from Eq. (2.5), and the single particle expansion parameters
Ci
from Eq. (2.2) are varied simultaneously. This calculation is an iterative
process. The
iteration involves forming the vector,
One makes an initial guess to the
and
.
Then solve for the
.
Using the new
,
determine new parameters
and
and repeat the calculation until the parameters converge to a specified
tolerance. The complication in the MCSCF is the determination of the
matrix elements. Seighbahn uses a unitary group approach and a separation
of the orbitals to the occupied and valance orbitals for rapid convergence.
To determine a molecular configuration, only the electrons outside of the closed shells contribute. Within a set of orbitals, matrix elements between determinants which differ by three or more spin orbitals are identically zero, hence single and double excitations of the reference HF configuration need to be considered. This is not really much of a limitation, there are many (infinitely many) excited configurations to consider.
These are the most common properties determined by ab initio calculations.