Permutational Symmetries of One- and Two-Electron Integrals

C. David Sherrill
Department of Chemistry
University of California, Berkeley
September 1996

Most algorithms in ab initio electronic structure theory compute quantities in terms of one- and two-electron integrals. Using the notation of Szabo and Ostlund, Modern Quantum Chemistry, the one-electron integrals over spin orbitals in physicist's notation are defined as

$$\langle i \vert h \vert j \rangle = \int d{\mathbf x}_1 \chi... ...}({\mathbf x}_1) \hat{h}({\mathbf r}_1) \chi_j({\mathbf x}_1)$$ (1)

where the one-electron Hamiltonian operator ${\hat h}({\mathbf r}_1)$ is defined as

$${\hat h}({\mathbf r}_1) = - \frac{1}{2} \nabla_1^2 - \sum_{A} \frac{Z_A}{r_{1A}}$$ (2)

It is immediately obvious that

$$\langle i \vert h \vert j \rangle = \langle j \vert h \vert i \rangle^*$$ (3)

or

$$[i \vert h \vert j] = [j \vert h \vert i]^*$$ (4)

in chemists' notation, which is equivalent to physicists' notation for one-electron integrals. For real orbitals, then,

$$\langle i \vert h \vert j \rangle = \langle j \vert h \vert i \rangle$$ (5)

$$[i \vert h \vert j] = [j \vert h \vert i]$$ (6)

These symmetries are unchanged if the spin is integrated out to yield spatial orbitals.

Permutational symmetries in the two-electron integrals are somewhat more interesting. The two-electron integral in physicists' notation is

$$\langle ij \vert kl \rangle = \int d{\mathbf x}_1 d{\mathbf ... ... \frac{1}{r_{12}} \chi_k({\mathbf x}_1) \chi_l({\mathbf x}_2)$$ (7)

while in chemists' notation it is written

$$[ij \vert kl] = \int d{\mathbf x}_1 d{\mathbf x}_2 \chi_i^{... ...ac{1}{r_{12}} \chi_k^{*}({\mathbf x}_2) \chi_l({\mathbf x}_2)$$ (8)

Clearly the integral is unchanged if the dummy indices of integration are permuted. This leads to the symmetry

$$\langle ij \vert kl \rangle = \langle ji \vert lk \rangle$$ (9)

Furthermore, the complex conjugate of the integral is

$$\langle ij \vert kl \rangle = \langle kl \vert ij \rangle^*$$ (10)

Combining these two symmetries leads to one further equality, namely

$$\langle ij \vert kl \rangle = \langle lk \vert ji \rangle^*$$ (11)

Therefore, in the general case we have

$$\langle ij \vert kl \rangle = \langle ji \vert lk \rangle = \langle kl \vert ij \rangle^* = \langle lk \vert ji \rangle^*$$ (12)

or

$$[ij \vert kl] = [kl \vert ij] = [ji \vert lk]^* = [lk \vert ji]^*$$ (13)

For the case of real orbitals, we can clearly remove the complex conjugations in the equations above, leading to a four-fold permutational symmetry in the two-electron integrals. However, an additional symmetry arises if the orbitals are real: in that case, the same integral is obtained if $i$ and $k$ (or $j$ and $l$)are swapped in $\langle ij \vert kl \rangle$. It is trivial to verify that this leads to an overall eightfold permutational symmetry,

$$\langle ij \vert kl \rangle = \langle ji \vert lk \rangle = \langle kl \vert ij \rangle = \langle lk \vert ji \rangle =$$ (14)

$$\langle kj \vert il \rangle = \langle li \vert jk \rangle = \langle il \vert kj \rangle = \langle jk \vert li \rangle$$

or

$$[ij \vert kl] = [kl \vert ij] = [ji \vert lk] = [lk \vert ji] =$$ (15)

$$[ji \vert kl] = [lk \vert ij] = [ij \vert lk] = [kl \vert ji]$$

Finally, it is worthwhile to consider the permutational symmetries in the antisymmetrized two-electron integral, $\langle ij \vert\vert kl \rangle$,defined as

$$\langle ij \vert\vert kl \rangle = \langle ij \vert kl \rangle - \langle ij \vert lk \rangle$$ (16)

$$= [ik \vert jl] - [il \vert jk]$$ (17)

In the general case, the permutational symmetries are

$$\langle ij \vert\vert kl \rangle = \langle ji \vert\vert lk \rangle = \langle kl \vert\vert ij \rangle^* = \langle lk \vert\vert ji \rangle^* =$$ (18)

$$-\langle ij \vert\vert lk \rangle = -\langle ji \vert\vert kl \rangle = -\langle lk \vert\vert ij \rangle^* = -\langle kl \vert\vert ji \rangle^*$$

One consequence of these relationships is that

$$\langle ii \vert\vert jk \rangle = \langle ij \vert\vert kk \rangle = 0$$ (19)