# The correlation energy from frozen natural orbtials (FNOs) using CCSD and MPBT(2)?

In other works and from the paper: Scaling Up Electronic Structure Calculations on Quantum Computers: The Frozen Natural Orbital Based Method of Increments.

I have seen that the corrected CCSD correlation energy ($$E_{c}^{CCSD}(corrected)$$) is calculated from a CCSD computation of the truncated virtual space ($$E_{c}^{FNO(CCSD)}$$) correlation energy, which is added to a MPBT2 correction term ($$\Delta E_{c}^{MPBT(2)}$$):

$$E_{c}^{CCSD}(correct)$$ = $$E_{c}^{FNO(CCSD)} + \Delta E_{c}^{MPBT(2)}$$

where, the correction term ($$\Delta E_{c}^{MPBT(2)}$$) is the MBPT(2) correlation energy in the full molecular orbital space $$E_{c}^{MO(MPBT(2))}$$ minus the MBPT(2) correlation energy in the truncated frozen natural orbtial (FNO) space $$E_{c}^{FNO(MPBT(2)}$$:

$$\Delta E_{c}^{MPBT(2)}$$ = $$E_{c}^{MO(MPBT(2))} - E_{c}^{FNO(MPBT(2)}$$

This strikes me as combining two different levels of theory. Why is this acceptable?

Here, I have interpreted the equations as: The correlation energy of the occupied block is given by $$\Delta E_{c}^{MPBT(2)}$$ at the MPBT(2) level, which is added to the correlation energy of the truncated FNO block $$E_{c}^{FNO(CCSD)}$$ at the CCSD level.

To assist in my learning, I may have misinterpreted by what is meant as frozen natural orbitals (FNOs). The paper states that FNOs are transformed and ranked virtual molecular orbitals, which enable one to truncate the virtual space. Therefore, are the orbitals in the truncated frozen natural orbital (FNO) space from which $$E_{c}^{FNO(MPBT(2)}$$ is calculate actually the remaining active virtual orbitals after truncation?

The idea in the FNO method is to do a MP2 calculation for the one-particle density matrix and diagonalize it to get the natural orbitals (NOs) and natural orbital occupation numbers (NOONs).

So, starting from (usually) a Hartree-Fock reference wave function, you figure out which orbitals are the most strongly correlated, and which ones yield only small contributions.

The next step is that you take the most strongly correlated NOs, and run CCSD in that orbital space; truncating the virtual space saves you a lot since CCSD has a $$v^4$$ term.

Why does the scheme work so well? This is a good question, which arises also with composite methods like G-$$n$$ theories and more modern versions like W-$$n$$ theories. It just does :D

The main point is that the error in the approximation is controllable by the NOON cutoff value. If you use a small cutoff (note that MP2 NOONs can also be negative - in which cases the scheme is probably unreliable!), you should be very close to the full CCSD energy.