Associated occupation numbers of frozen natural orbitals (FNOs) from MP2 method

Question

The paper Frozen Virtual Natural Orbitals for Coupled-Cluster Linear-Response Theory gives the following (simplified) description of frozen natural orbitals (FNOs).

In the MP2 method, the unrelaxed one-electron density matrix can be written in terms of spin orbitals

$\gamma_{pq}=\langle\Psi^{1}|\{a_{p}^{\dagger}a_{q}\}|\Psi^{1}\rangle$

where $|\Psi^{1}\rangle$ is the first-order correction to the Hartree-Fock wavefunction. In the MP2 based NO method, the virtual-virtual block is constructed:

$\gamma_{ab}=\frac{1}{2}\sum t_{ij}^{ac}t_{ij}^{bc}$

and then diagonalised:

$\gamma V=nV$

The eigenvectors V are the virtual NOs, and the eigenvalues n are the associated occupation numbers.

If the eigenvalues are found from a virtual-virtual (unoccupied-unoccupied) block, how can there be an associated occupation number?

Answer 1

Virtual or unoccupied refers to the Hartree-Fock solution - the virtual orbitals are thus unoccupied in Hartree-Fock, i.e. have occupation number 0, whereas the occupied orbitals have occupation number 2 (spin-traced occupation in restricted HF) or 1 (for a given spin in unrestricted HF). The first-order MP2 correction to the wave-function will have different occupation numbers - the occupied orbitals will become slightly less occupied, whereas the virtual orbitals become slightly occupied by the same total amount as a result of electron correlation. This doesn't change the nomenclature: occupied and virtual orbital still refers to the HF reference wave-function.