1.8 Calculation Methods
1.8.1 Hartree-Fock Theory
The molecular Hartree-Fock wave function is written as an antisymmetrized product (Slater determinant) of spin-orbitals, each spin-orbital (x1,x2,......xN) being a product of a spatial orbital Φ1 and a spin function (either α(spin up) or β(spin down)). The Slater determinant is given as:
Here x1(1) indicates a function that depends on the space and spin coordinates of the spin orbital with electron labeled as ‘1’.The expression for the Hartree-Fock molecular electronic energy EHF is given by the variation theorem as
|
(1.20) |
where D is the Slater-determinant Hartree-Fock wave function and 
and VNN are given by
|
(1.21) |
|
(1.22) |
Where 
.The operator is the sum of one-electron operators 
and two-electron operators
|
(1.23) |
where 
and , written in atomic units, are given by following expressions:
|
(1.24) |
|
(1.25) |
In the expression for EHF bracket notation has been used. It is an abbreviated representation for the definite integral over all space of an operator sandwiched between two functions:
|
(1.26) |
denotes the conjugate of fm. Since VNN does not involve electronic coordinates and D is normalized, we have 
. The Hartree-Fock energy of a polyatomic molecule with only closed subshells is