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
\begin{equation}E = \sum_{i}^{N} h_i + 1/2 \sum_{i, j} J_{ij} - K_{ij}.\label{eq:energy}\end{equation}
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)
\begin{equation}\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}\end{equation}
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,
\begin{equation} \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}\end{equation}
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
\begin{equation} \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}\end{equation}
Using this approach
\begin{equation} \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}\end{equation}
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 \(1/2\).