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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?

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    $\begingroup$ Not sure about the right answer, maybe it is related to mixing the term "occupied" for Hartree Fock $\endgroup$ – user1420303 May 27 '20 at 22:49
  • $\begingroup$ @user1420303 I thinking along the lines that occupation numbers are associated with single, double, excitations into the virtual orbitals. Though I would need clarification on this. $\endgroup$ – Wychh May 28 '20 at 8:03
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    $\begingroup$ I mean that when natural orbitals are constructed, their occupation number is not integer. Many of them are very close to zero or one. But they are not exactly zero or one (specially in multireference systems). Then you can classify nat orbs as occupied when they are essentially 1 or unoccupied when they are close to 0 (and eventually you can find orbitals with intermediate occupation numbers). $\endgroup$ – user1420303 May 28 '20 at 13:24
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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.

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