Multi-reference perturbation theory

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In contrast to the Epstein-Nesbet scheme traditionally used in individually selecting MRCI calculations the Møller-Plesset partitioning [4] of the Hamiltonian usually performs much better with respect to accuracy. However, the generalization of the Møller-Plesset theory to the multi reference case needs more attention. There are three implications from the generalization to the multi reference case [5,6,7,8]: 1. The mean field approximation of the Fock operator within the SCF procedure has to be generalized with respect to the multi reference state. Accordingly the generalized Fock operator depends on the reference state and has the following matrix elements:

$$\rangle \sum_{{rs}}^{} \rangle \left[\vphantom{(pq\vert rs) - {1\over 2} (pr\vert qs)}\right. {1\over 2} \left.\vphantom{(pq\vert rs) - {1\over 2} (pr\vert qs)}\right]$$ (6)

with h_pq the representation of the core hamiltonian, D_rs^{| 0$\rangle$} the one-electron density matrix of the reference state | 0$\rangle$, (pq| rs) the two-electron integrals. 2. The Raighley-Schrödinger perturbation theory needs the reference state to be an eigenfunction of the $\hat{{H}}_{0}^{}$ operator. The introduction of the projection operators

$$\hat{{P}}_{0}^{} \rangle \langle \hat{{P}}_{{\bar 0}}^{} \hat{{P}}_{0}^{}$$ (7)

and the definition of $\hat{{H}}_{0}^{}$ as

$$\hat{{H}}_{0}^{{{\vert\> }}}_{{}} \rangle \hat{{P}}_{0}^{} \hat{{P}}_{{\bar 0}}^{} \hat{{F}}^{{\vert\> }}_{} \hat{{P}}_{{\bar 0}}^{}$$ (8)

with E^{| 0$\rangle$} the reference energy solves this problem from the other side as it defines the $\hat{{H}}_{0}^{}$ operator in a way that | 0$\rangle$ is an eigenfunction to $\hat{{H}}_{0}^{}$. 3. Due to the equations (7) and (9) the excited slater determinants in the expansion space are no longer eigenfunctions of $\hat{{H}}_{0}^{}$. As a consequence the $\hat{{H}}_{0}^{}$-matrix is no longer diagonal and the very easily performed division by the orbital energy differences within the evaluation of the single reference MP2 is transformed into a solution of a linear equation system:

$$\sum_{j}^{} \langle \hat{{H}}_{0}^{} \rangle \rangle \langle \hat{{H}} \Psi^{{\hat {H}_0}}_{} \rangle$$ (9)

or in short notation

$$\bf\sf H_{0}^{} \rangle \bf\sf 1 \vec{x} \vec{b}$$ (10)

Due to the size of $\bf\sf H_{0}^{}$ (11) can be solved by means of an iterative procedure only. The MP2 energy is then simply given by

$$\vec{x} \vec{b}$$ (11)