Brian C. Hoffman: SCF Theory

Basic Principles and
Hartree-Fock Theory

Notation Conventions

The notation used in quantum chemistry varies widely from one research group to the next; therefore, the notational conventions used in this dissertation will be provided within this section for easy reference. Many of the conventions adopted herein are those used in the book by Szabo and Ostlund. [19]

x_i The spin and spatial coordinates of the ith electron.

r_i The spatial coordinates of the ith electron.

$\Psi$ The exact wave function for the system.

$\Phi$ The trial wave function used in a variational method.

$\chi(x_i)$ A spin orbital.

$\psi(r_i)$ A spatial molecular orbital.

$\phi(r)$ An AO or SO basis function.

n The number of basis functions and MOs.

N The total number of electrons.

N $_\alpha$ The number of $\alpha$ spin electrons.

N $_\beta$ The number of $\beta$ spin electrons.

u, v, $\rho$ , $\sigma$ Indecies used to denote basis functions.

a, b, c, $\cdots$ Indecies typically used to denote occupied orbitals.

i, j, k, $\cdots$ Indecies typically used to denote virtual orbitals.

$\vert \chi_i(x_a) \cdots \chi_k(x_N) \rangle$ Shorthand for a Slater determinant listing only diagonal elements.

$\vert ab \cdots c \rangle$ Shorthand for a Slater determinant listing only thenumber of each diagonal element.

$\vert \Phi_i \rangle$ The ith N-electron basis function--a single Slater determinant or CSF.

$\vert \Phi_a^r \rangle$ N-electron basis function which differs by some reference function $\mid \Phi_0 >$ by replacement of spin-orbital a by spin orbital r.

$\hat{h}(i)$ One-electron Hamiltonian for electron i.

$\hat{H}$ The non-relativistic, electronic Hamiltonian.

H The Hamiltonian matrix.

H_ij The i, j-th element of the Hamiltonian in a basis set

of CSFs or determinants

$\hat{f}$ The Fock operator.

F The Fock matrix.

F_uv The u, v-th element of the Fock matrix in the AO basis.

S The overlap integral matrix.

S_uv Overlap of the u-th and v-th AO basis function.

$\langle a \vert h \vert b \rangle$ Dirac notation for the one-electron integral $\int \chi_a(x_1) \hat{h}(1) \chi_b(x_1) \; dx_1$

$[a \mid h \mid b]$ Chemists' notation for the one-electron integral $\int \chi_a(x_1) \hat{h}(1) \chi_b(x_1) \; dx_1$ .

$\langle ab \vert cd \rangle$ Dirac notation for the two-electron integral $\int \chi_a(x_1) \chi_b(x_2) \frac{1}{r_{12}} \chi_c(x_1) \chi_d(x_2) \; dx_1 \; dx_2$ .

$\langle ab \vert \vert cd \rangle$ Shorthand notation for $\langle ab \vert cd \rangle - \langle ab \vert dc \rangle$ .

$[ab \mid cd]$ Chemists' notation for the two-electron integral $\int \chi_a(x_1) \chi_b(x_1) \frac{1}{r_{12}} \chi_c(x_2) \chi_d(X_2) \; dx_1 \; dx_2$ .

$(ab \mid cd)$ Chemists' notation for a two-electron integral in terms of spatial orbitals $\int \psi_a(r_1) \psi_b(r_1) \frac{1}{r_{12}} \psi_c(r_2) \psi_d(r_2) \; dr_1 \; dr_2$ .

This page maintained by Brian C. Hoffman