# 18.2: The Single-Determinant Wavefunction

The simplest trial function of the form given above is the single Slater determinant function:

$| \Psi \rangle = \big| \phi_1\phi_2\phi_3 ... \phi_N \big|.$

For such a function, the CI part of the energy minimization is absent (the classic papers in which the SCF equations for closed- and open-shell systems are treated are C. C. J. Roothaan, Rev. Mod. Phys. 23 , 69 (1951); 32 , 179 (1960)) and the density matrices simplify greatly because only one spin-orbital occupancy is operative. In this case, the orbital optimization conditions reduce to:

$\hat{F} \phi_i = \sum\limits_j \epsilon_{i,j} \phi_j ,$

where the so-called Fock operator $$\hat{F}$$ is given by

$\hat{F} \phi_i = h \phi_i + \sum \limits_{j(occupied)}\left[ \hat{J}_j - \hat{K}_j \right]\phi_i .$

The coulomb ($$\hat{J}_j$$) and exchange ($$\hat{K}_j$$) operators are defined by the relations:

$\hat{J}_j \phi_i = \int\phi^{\text{*}}_j(r')\phi_j(r') \dfrac{1}{\big| r-r' \big|}d\tau ' \phi_i(r)$

and

$\hat{K}_j \phi_i = \int\phi^{\text{*}}_j(r')\phi_i(r') \dfrac{1}{\big| r-r' \big|}d\tau ' \phi_j(r) .$

Again, the integration implies integration over the spin variables associated with the $$\phi_j$$ (and, for the exchange operator, $$\phi_i$$ ), as a result of which the exchange integral vanishes unless the spin function of $$\phi_j$$ is the same as that of $$\phi_i$$ ; the coulomb integral is non-vanishing no matter what the spin functions of $$\phi_j \text{ and } \phi_i$$.

The sum over coulomb and exchange interactions in the Fock operator runs only over those spin-orbitals that are occupied in the trial $$\Psi$$. Because a unitary transformation among the orbitals that appear in $$| \Psi \rangle$$ leaves the determinant unchanged (this is a property of determinants- det (UA) = det (U) det (A) = 1 det (A), if U is a unitary matrix), it is possible to choose such a unitary transformation to make the $$\epsilon_{i,j}$$ matrix diagonal. Upon so doing, one is left with the so-called canonical Hartree-Fock equations:

$\hat{F} \phi_i = \epsilon_i\phi_j ,$

where $$\epsilon_i$$ is the diagonal value of the $$\epsilon_{i,j}$$ matrix after the unitary transformation has been applied; that is, $$\epsilon_i$$ is an eigenvalue of the $$\epsilon_{i,j}$$ matrix. These equations are of the eigenvalue-eigenfunction form with the Fock operator playing the role of an effective one-electron Hamiltonian and the $$\phi_i$$ playing the role of the one-electron eigenfunctions.

It should be noted that the Hartree-Fock equations $$\hat{F} \phi_i = \epsilon_i \phi_j$$ possess solutions for the spin-orbitals which appear in $$\Psi$$ (the so-called occupied spin-orbitals) as well as for orbitals which are not occupied in $$\Psi$$ ( the so-called virtual spin-orbitals). In fact, the F operator is hermitian, so it possesses a complete set of orthonormal eigenfunctions; only those which appear in $$\Psi$$ appear in the coulomb and exchange potentials of the Fock operator. The physical meaning of the occupied and virtual orbitals will be clarified later in this Chapter (Section VII.A).