Samstag, 8. Dezember 2012

QM 004: Two-electron integral manipulation

In the independent-particle model, the electronic energy of an $$N$$-electron system is frequently written as
$$E = \sum_{i}^{N} h_i + 1/2 \sum_{i, j} J_{ij} - K_{ij}.\label{eq:energy}$$
When inserting the expressions for the Coulomb and exchange operators, $$J$$ and $$K$$, and calculating the variation in energy, $$\delta E$$, given by (in physicists notation)
\delta E = \sum_i \langle \delta \phi_i | h_i | \phi_i \rangle + \langle \phi_i | h_i | \delta \phi_i \rangle + 1/2 \sum_{i, j} \left[ \langle \delta \phi_i \phi_j | \phi_i \phi_j \rangle   + \langle \phi_i \delta \phi_j | \phi_i \phi_j  \rangle  + \phi_i \phi_j | \delta \phi_i \phi_j \rangle + \langle \phi_i \phi_j | \phi_i \delta \phi_j \rangle \\
- \langle\delta\phi_i\phi_j | \phi_j\phi_i\rangle - \langle\phi_i\delta\phi_j | \phi_j\phi_i\rangle -  \langle\phi_i\phi_j | \delta\phi_j\phi_i\rangle - \langle\phi_i\phi_j | \phi_j\delta\phi_i\rangle\right], \label{eq:variation}
it can be found that four pairs of terms in the second sum-expression are equal and thus the factor of $$1/2$$ can be cancelled.
For the Coulomb integrals this can be done by considering the definition of the bracket notation
$$\langle \delta\phi_i(1)\phi_j(2) | \phi_i(1) \phi_j(2)\rangle = \int d{\bf x}_1 d{\bf x}_2 \delta\chi_i^*(1)\chi_j^*(2){\bf r}_{12}^{-1} \chi_i(1)\chi_j(2)$$
and noting that when the orbitals are understood to be real, i.e. $$\phi^* = \phi$$, and the ordering of electron labels is considered, the orbitals of an electron $$m \in \{1, 2 \}$$ can be swapped such that
$$\int d{\bf x}_1 d{\bf x}_2 \delta\chi_i(1)\chi_j(2){\bf r}_{12}^{-1} \chi_i(1)\chi_j(2) = \int d{\bf x}_1 d{\bf x}_2 \chi_i(1)\chi_j(2){\bf r}_{12}^{-1} \delta\chi_i(1)\chi_j(2)$$
and thus, in bracket notation,
$$\langle \delta\phi_i\phi_j | \phi_i \phi_j \rangle = \langle\phi_i\phi_j | \delta\phi_i \phi_j \rangle \label{eq:manipulation_coulomb}$$
which we recognize as the the first and third term in the second sum of Eq. \ref{eq:variation}.
For the exchange integrals, a similar expression exists (Szabo, Ostlund, Eq. 2.94)
$$\langle ij | kl \rangle = \langle ji | lk \rangle .$$
To show this, we write the integral explicitly and after first exchanging the dummy variables $$1$$, $$2$$ and then reordering the orbitals in order to restore the conventional $$1, 2$$ order of electrons (keeping "the orbital on the electron"), we obtain
$$\int d{\bf x}_1 d{\bf x}_2 \chi_i^*(1)\chi_j^*(2){\bf r}_{12}^{-1} \chi_k(1)\chi_l(2) = \int d{\bf x}_1 d{\bf x}_2 \chi_j^*(1)\chi_i^*(2){\bf r}_{12}^{-1} \chi_l(1)\chi_k(2). \label{eq:manipulation_exchange}$$
Using this approach
$$\langle \delta\phi_i(1)\phi_j(2) | \phi_j(1)\phi_i(2) \rangle = \langle \phi_j(1)\delta\phi_i(2)|\phi_i(1)\phi_j(2) \rangle = \langle \phi_i(1)\phi_j(2)|\phi_j(1)\delta\phi_i(2) \rangle \label{eq:exchange}$$
where Eq. \ref{eq:manipulation_exchange} was used to obtain the first equality and Eq. \ref{eq:manipulation_coulomb} was used to swap $$\phi_j(1)$$ with $$\phi_i(1)$$ and $$\delta\phi_i(2)$$ with $$\phi_j(2)$$ to obtain the second equality.
The first and last integral are recognized as the first and fourth exchange integral of Eq. \ref{eq:variation}.
Applying these operations on the remaining integrals allows to factor out a factor of $$2$$ which then cancels with the $$1/2$$.