# Analytic Hessian at the level of full CI?

This is a follow-up question to my previous question: Computing analytic derivatives of molecular Hamiltonians obtained from solving Hartree-Fock equations.

After looking at various quantum chemistry packages, such as PySCF, Psi4, and GAMESS, it is now clear that one can compute analytic molecular Hessians. However, it seems as though Hessians can only be computed analytically within the Hartree-Fock approximation, with more precise treatments using a finite-difference approach.

I was wondering: is it possible to use any of the above (or other) quantum chemistry libraries to compute Hessians analytically at the level of Full CI?

GAMESS also should be able to calculate analytic hessian for the full CI case. From the GAMESS manual:

\$FORCE group

This group controls the computation of the hessian
matrix (the energy second derivative tensor, also known
as the force constant matrix), and an optional harmonic
vibrational analysis.  This can be a very time consuming
calculation.  ...

METHOD = chooses the computational method:
= ANALYTIC is a fully analytic calculation.  This
is implemented for SCFTYP=RHF, UHF, ROHF,
GVB (for NPAIR=0 or 1, only), and
MCSCF (for CISTEP=ALDET or ORMAS, only).
R-DFT and U-DFT are also analytic.
This is the default for these cases.
= SEMINUM  does numerical differentiation of
analytically computed first derivatives. ...


(I have removed some extra text, emphasis mine)

So GAMESS can do analytic hessians with MCSCF (which means CASSCF, which means full CI can be done). The CISTEP keyword just indicates which scheme is used to generate the excited configurations. (ALDET is Ames Lab determinant full CI, and ORMAS is Occupation Restricted Multiple Active Space determinant CI.)

Analytical Hessians have been derived for pretty much everything; they just aren't available in most programs.

CFOUR has analytical vibrational frequencies i.e. nuclear Hessians for HF and the following post-HF methods: MP2, MP3, MP4, CC2, CCD, QCISD, CCSD, QCISD(T), CCSD(T), CCSDT-n (n=1-4), CC3, CCSDT.

Kállay's MRCC program has analytical Hessians for arbitrary single- or multireference configuration interaction or coupled-cluster approach (note that this includes full CI by definition!) in J. Chem. Phys. 120, 6841 (2004).

Analytical Hessians are available for at least the simpler methods in many other codes, as well. For instance, analytic MP2 hessians can probably be found in any older quantum chemistry program since MP2 was very popular in the 1990s before DFT came along.

• A side note: any program that supports CASSCF Hessian should support FCI Hessian, as long as the program properly treats the edge case that there is no inactive orbital and no virtual orbital. This gives one more programs to choose from. Jul 7 at 7:12