Coming from a DFT background, I'm used to the concept that the DFT eigenvalues do not correspond to excitation energies (i.e., the band gap, ionization potentials, etc.). To correct for these it's common to go to many-body approaches like the GW approximation to the self-energy, which is said to explicitly deal with quasiparticle addition/removal energies.

I'm curious how excited states are calculated within Hartree Fock or CI. Can one do a ground state calculation in HF or CI and then get excitation energies from the difference in these ground-state eigenvalues? Or is this similarly not possible and one needs to explicitly consider excited states in order to get the excitation spectrum (band gap, band eigenvalues for removal/addition of electron,etc.).


1 Answer 1


There are actually quite a few methods post-HF that you can do to calculate excitation energies. The simplest one you will want to look at is Configuration Interaction Singles (CIS). To do this, you express your Hamiltonian in the basis of singly-excited HF determinants and diagonalize it. It doesn't involve correlation and is roughly the same cost as a HF calculation. The matrix elements are given in Szabo and Ostlund, Modern Quantum Chemistry pg 236.

You are already familiar that for DFT, the most common way to get excitation energies is through a time-dependent DFT calculation (TDDFT). You can do a time-dependent Hartree-Fock (TDHF) as well, though you may see this in the literature as the Random Phase Approximation (RPA). They are essentially the same thing. TDHF and TDDFT are pretty much identical save for the fact that TDDFT is based off Kohn-Sham density, and TDHF is based off of Hartree-Fock density. The equations are almost the same, and it isn't that much more expensive to compute.

If you want to include electron correlation in wavefunction-based methods to a greater degree, look for EOM-CCSD like methods. These are based off a HF, followed by a coupled-cluster ground state calculation, followed by an extension to get excited state energies.

My graduate research uses and develops a lot of these methods. In fact, I wrote about these methods not too long ago --- you can read more about it here: http://bit.ly/1anA9dZ

  • $\begingroup$ I should also add that EOM-CC methods allow for direct calculation of ionization and electron addition energies directly. For EOM-CCSD, you'll see these as IP-EOM-CCSD (for ionization potential) and EA-EOM-CCSD (for electron attachment). $\endgroup$
    – jjgoings
    Commented Jan 21, 2014 at 16:58
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    $\begingroup$ +1 for mentioning EOM-CC methods. These methods are so sweet (yes that is my scientific description of them). (FYI, the equations don't have any motion built into them so "Equations of motion" is a bit misleading. Has to do with some traditional thing). $\endgroup$ Commented Jan 21, 2014 at 19:40
  • $\begingroup$ Thanks for your answer. I took a look at p. 236 in my copy of Szabo but will need to read some of the earlier material before fully understanding it. While perusing Szabo I also came across the discussion on p. 127 on Koopman's theorem. I think this partially answers my question as well -- that the orbital energies themselves don't give the EA or the IP unless you assume that the system doesn't relax upon addition/removal of an electron. Thus presumably the orbital energies in Hartree Fock, like DFT, don't have a rigorous meaning as quasiparticle energies. $\endgroup$
    – gammapoint
    Commented Jan 21, 2014 at 20:31

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