Index Conventions

There are two main conventions in the quantum-chemistry community to index two-electron integrals which also determines how to write the matrix elements.

For a given first quantised two-electron operator $g^c(x_1, x_2)$, we can define integrals over the orbitals $\phi$ using the so called chemist’s notation $g_{PQRS}$ or the physicist’s notation $U_{PRQS}$ by: $$\begin{equation} g_{PQRS} = U_{PRQS} = \int \int \phi^*_{P}(x_1) \phi^*_{R}(x_2) g^c(x_1, x_2) \phi_{Q}(x_1) \phi_{S}(x_2) \, \mathrm{d} x_1 \mathrm{d} x_2 \quad. \end{equation}$$

With these integrals we can write the second-quantised two electron operator in both notations as: $$\begin{equation} \hat{g} = \sum_{PQRS} a^\dagger_P a^\dagger_R a_S a_Q g_{PQRS} = \sum_{PQRS} a^\dagger_P a^\dagger_R a_S a_Q U_{PRQS} \end{equation}$$

Typical textbooks that assume the chemist’s notation are the purple book[1] or Szabo-Ostlund[2].

The FCIDUMP file that transfers the electronic integrals from Molpro or Molcas to NECI assumes the chemist’s notation.

Internally NECI uses the physicist’s notation, i.e. the function get_umat_el that returns the stored two-electronic integrals uses the indexing of $U_{PRQS}$. (Which is also why it is called umat in the first place.)

If we write a double excitation with second quantised operators, we have the following commutation relation: $$\begin{equation} a^\dagger_P a^\dagger_R a_S a_Q = -a^\dagger_P a^\dagger_R a_Q a_S = -a^\dagger_R a^\dagger_P a_S a_Q \end{equation}$$ to remove any ambiguity about the sign we assume $P < R$ and $S > Q$ and call this a canonical excitation.

In NECI any excitation of rank $n$ is represented by a $2 \times n$ matrix, where the first row contains all the source indices (particles which are to be annihilated) and the second row all target indices (particles which are to be created).

A canonical excitation $a^\dagger_P a^\dagger_R a_S a_Q, (P < R \land Q < S )$ in NECI is given by a matrix, where each row is ascendingly sorted. $$\begin{equation} \begin{bmatrix} \texttt{src}_1 & \texttt{src}_2 \\ \texttt{tgt}_1 & \texttt{tgt}_2 \end{bmatrix} = \begin{bmatrix} Q & S \\ P & R \end{bmatrix} \quad, \quad \texttt{src}_1 < \texttt{src}_2, \quad \texttt{tgt}_1 < \texttt{tgt}_2 \end{equation}$$

This means that the operator, i.e. the matrix elements in NECI, is given by: $$\begin{equation} \hat{g} = \sum_{PQRS} a^\dagger_P a^\dagger_R a_S a_Q g_{PQRS} = \sum_{PQRS} a^\dagger_P a^\dagger_R a_S a_Q U_{PRQS} = \sum \texttt{tgt}_1^\dagger \texttt{tgt}_2^\dagger \texttt{src}_2 \texttt{src}_1 U_{ \texttt{tgt}_1 \texttt{tgt}_2 \texttt{src}_2 \texttt{src}_1 } \end{equation}$$