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18.2: The Single-Determinant Wavefunction

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  • 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:

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


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

    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).

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