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CASCI and FOMO-CASCI

CASCI: CASSCF without orbital optimization

CASCI (complete active space configuration interaction) is the same many-electron ansatz as CASSCF — the wavefunction is a full CI expansion within a chosen active space — but without the orbital-optimization step. In CASSCF the molecular orbitals and the CI coefficients are optimized together (see Theory and implementation); in CASCI the orbitals are held fixed at a set provided up front, and only the CI problem is solved: the Hamiltonian is diagonalized in the space of all configurations consistent with the active space.

Because there is no orbital relaxation, CASCI is considerably cheaper than CASSCF and avoids the sometimes-difficult convergence of the orbital optimization. The trade-off is that the result depends entirely on the quality of the input orbitals — CASCI cannot relax a poor set of orbitals to compensate. The orbitals typically come from a preceding self-consistent calculation (Hartree–Fock or, very commonly, FOMO-SCF — see below). CASCI is enabled with casci yes.

How CASSCF parameters carry over to CASCI

CASCI and CASSCF share the same active-space machinery, so most of the keywords documented on the CASSCF overview apply unchanged. The table below summarizes which carry over.

Keyword group Applies to CASCI? Notes
Active space — closed, active, activeorbs Yes Define the active space exactly as for CASSCF. closed/active have the same meaning.
States — cassinglets, casdoublets, castriplets, … Yes Number of CI roots to compute per multiplicity.
Target state — castarget, castargetmult Yes Select the state whose gradient/properties are computed.
CI solver — ci_solver Yes The CI eigenproblem is solved identically; the same solvers are available.
Initial guess — casguess, casaopguess Yes Used to supply/align the (fixed) orbitals that CASCI runs in.
Output / analysis — casgradprint, cascharges, cas_ntos, casrelaxeddipoles, cassoc Yes Same state-resolved analysis as CASSCF.
Orbital optimization — casscfmaxiter, casscfconvthre, casscfenergyconvthre, casscforbnriter, casscfnriter, casscftrust*, casscflambda* No There is no orbital optimization in CASCI, so these have no effect.
State-averaging — casweights, dynamicweights, dwbandwidth No State-averaging exists to bias the orbital optimization toward a balanced set of states. With fixed orbitals there is nothing to average over; all requested roots are obtained directly from one CI diagonalization.
Response equations — cpsacasscf*, zvecguess Only for gradients Needed when an analytic gradient requires the orbital response; not used for single-point CASCI energies.

In short: define the active space and the states exactly as you would for CASSCF, drop the orbital-optimization and state-averaging controls, and add casci yes.

FOMO-CASCI

A well-known difficulty with plain CASCI is that its results can become unstable when orbitals are (near-)degenerate around the Fermi level — for example along a bond-dissociation coordinate or near a conical intersection. When two orbitals are nearly degenerate at the active-space boundary, an infinitesimal change in geometry can swap which orbital is "occupied" versus "active," causing the CASCI energy to jump discontinuously. This makes plain CASCI poorly suited to potential energy surfaces, geometry optimizations, and dynamics.

FOMO-CASCI (floating-occupation molecular orbital CASCI) addresses this by generating the orbitals from a FOMO-SCF calculation — a self-consistent field (FON-HF, or the analogous DFT) in which the orbitals near the Fermi level are given fractional occupations smeared by a finite electronic temperature — and then performing CASCI in those orbitals. The fractionally occupied orbitals are the CASCI active space: the closed/active/virtual partition of the CASCI coincides exactly with the integer-/fractional-/zero-occupied orbitals of the FOMO-SCF. Performing CASCI in these orbitals helps in two distinct ways:

  1. It smooths over degeneracies near the Fermi level. Because the occupations vary continuously with orbital energy, the orbitals (and hence the CASCI energies built from them) change smoothly as the geometry changes, rather than switching abruptly. This restores the continuity that plain CASCI lacks.
  2. It makes the virtual orbitals more appropriate for CASCI. In a standard Hartree–Fock calculation the virtual orbitals are unoccupied and tend to be diffuse and poorly suited to describing the correlated states targeted by CASCI. Giving the active-space orbitals partial occupation pulls the would-be virtuals in to a more compact, valence-like form, so the fixed CASCI orbital set is much better balanced between the occupied and virtual character that the multireference wavefunction needs.

A useful practical consequence of restricting the fractional occupations to the (small) active space is that the results become fairly insensitive to the choice of electronic temperature, so FOMO-CASCI needs little of the calibration that a fully smeared finite-temperature method would require.

FOMO-CASCI is therefore not merely a patch for CASCI but a well-defined, inexpensive alternative to SA-CASSCF: it sidesteps the nonlinear orbital optimization (with its convergence difficulties, local minima, and root flipping) while still producing a multireference wavefunction that correctly describes the topology of potential energy surfaces around conical intersections — including intersections with the ground state, which linear-response CIS and TDDFT cannot describe. Its analytic gradients and nonadiabatic coupling vectors (implemented on GPUs in TeraChem) cost about the same as CIS or TDDFT. Because the FOMO orbitals vary continuously with geometry, FOMO-CASCI also avoids the energy discontinuities that active-space orbital rotations can introduce in CASSCF along a reaction path or trajectory, making it well suited to geometry optimization and nonadiabatic molecular dynamics. Like SA-CASSCF, it neglects dynamic correlation and so tends to overestimate excitation energies.

FOMO-CASCI is requested by combining casci yes with the fractional-occupation keywords (fon yes, etc.). The active space used by CASCI is the same set of fractionally occupied orbitals produced by the FOMO-SCF step — closed counts the fully (doubly) occupied orbitals and active counts the fractionally occupied orbitals.

Where the FOMO details live

The fractional-occupation procedure itself — the choice of smearing distribution (fon_method), the electronic temperature (fon_temperature), annealing, and so on — is a property of the SCF, not of CASCI. Those keywords and the underlying theory are documented with FOMO-SCF in the SCF documentation. This page only describes how FOMO orbitals are used as the basis for a CASCI calculation.

Example

A two-state singlet FOMO-CASCI on a CAS(2,2) active space:

casci                  yes
fon                    yes
fon_temperature        0.25
closed                 7
active                 2
cassinglets            2
castarget              0
castargetmult          1

References

The GPU-accelerated FOMO-CASCI method in TeraChem — including its analytic gradients — and its use for nonadiabatic dynamics are described in:

  1. E. G. Hohenstein, M. E. F. Bouduban, C. Song, N. Luehr, I. S. Ufimtsev, and T. J. Martínez, "Analytic first derivatives of floating occupation molecular orbital-complete active space configuration interaction on graphical processing units," J. Chem. Phys. 143, 014111 (2015). doi:10.1063/1.4923259
  2. D. Hollas, L. Šištík, E. G. Hohenstein, T. J. Martínez, and P. Slavíček, "Nonadiabatic Ab Initio Molecular Dynamics with the Floating Occupation Molecular Orbital-Complete Active Space Configuration Interaction Method," J. Chem. Theory Comput. 14, 339–350 (2018). doi:10.1021/acs.jctc.7b00958