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It is well known that the Hartree–Fock (HF) approximation predicts a wrong order for the last two molecular orbitals of N$_{2}$. The calculations indicate that the 1$\pi_{u}$ orbital has higher energy than the 3$\sigma_{g}$, while the experiment suggests the opposite. As pointed out in the book "Modern Quantum Chemistry" by Szabo and Ostlund, this error can be corrected by including correlation effects, for example using the configuration interaction (CI) method.

My question is: how to perform this calculation with CI? For example, I could calculate the energy of the neutral molecule and its anion with CI and subtract one from the other, this would give me the ionization energy. However, how can I deduce the correct energies of the molecular orbitals from this difference? I really have no idea how to perform this calculation in practice using a quantum chemistry software like "Gaussian16".

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The incorrect ordering of molecular orbitals given by Hartree-Fock (Koopmans' defect) can be resolved with methods based on electron propagator/one-electron Green's function theories. Such methods provide both Dyson orbitals and correlated electron binding energies without directly computing CI wavefunctions. The correlated binding energies or ionization potentials (IPs) should, in principle, provide information about the correct molecular electronic configuration. An advantage of electron propagator theory is that it includes correlation and relaxation effects missing in Hartree-Fock while still retaining familiar one-electron concepts.

See:

B.T. Pickup and O. Goscinski. Mol. Phys. 26. 4 (1973) 1013–1035

Y. Öhrn and G. Born. Advances in Quantum Chemistry. Vol. 13. (1981) 1–88

J.V. Ortiz. WIREs Comput. Mol. Sci. 3. 2 (2013) 123–142

Chapter 7 of Szabo and Ostlund is also useful.

The inadequacy of Koopmans' theorem in the case of the nitrogen molecule is well known:

L.S. Cederbaum, G. Hohlneicher, W. von Niessen. Chem. Phys. Lett. 18. 4 (1973) 503-508

H.H. Corzo. Chemical Reactivity. Ch. 1 (2023) 1-26

The correct assignment of the HOMO of nitrogen can be achieved through second-order self-energy corrections to the Hartree-Fock eigenvalues. The correlated IPs indicate that $3\sigma_{g}$ and $\pi_{u}$ orbitals should be swapped.

Electron propagator calculations may be performed in Gaussian. Various diagonal self-energy approximations are available. Here are example inputs:

Second-order (D2)

%chk=n2.chk
#p ept(ep2) cc-pvtz window=full 

nitrogen

0 1
N 0.0 0.0 0.0
N 1.098 0.0 0.0

Renormalized partial third-order (P3+)

%chk=n2.chk
#p ept(p3,readOrbitals) cc-pvtz window=full 

nitrogen

0 1
N 0.0 0.0 0.0
N 1.098 0.0 0.0

1 10

You can also replace "p3" with "ovgf" to try the outer valence Green's function (OVGF) method. Keep in mind that the quality of the basis set will affect the results. Pole strengths (Dyson orbital norms) less than 0.85 indicate that the diagonal approximations are not reliable and that more accurate methods are required.

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