CASSCF
TeraChem can carry out complete active space self-consistent field calculations, which allow the wavefunction to comprise multiple determinants. The CASSCF method is especially useful for treating multireference problems, such as diradicals, bond-breaking, and electronic excited states. The disadvantages of CASSCF are its lack of attention to dynamic electron correlation (this generally leads to too large S0-S1 excitation energies and to a disfavoring of charge transfer states) and the need to specify an active space. The active space is specified by giving a number of electrons and a number of orbitals, e.g. CAS(2/4) for an active space with 2 electrons in 4 orbitals. All configurations consistent with the active space are included in the calculation.
Configuration state functions and determinants
The CASSCF wavefunction is written as \(\Psi(\Theta) = \sum_I c_I \psi_I(\Theta)\) where \(\Theta\) is a shorthand for the set of molecular orbitals (which are optimized), \(\psi_I\) consists of either Slater determinants or spin-adapted combinations of Slater determinants (CSFs), and \(c_I\) are the configuration interaction coefficients corresponding to the many-electron basis functions.
TeraChem always works with a Slater determinant basis at the fundamental level. However, you may choose to transform the result to a spin-adapted CSF basis. The advantages of working with CSFs are that the basis set is smaller (so the calculation is faster and converges more readily) and there is no ambiguity about spin states. When using determinants, the code can find electronic states of arbitrary spin. When using CSFs, the user specifies the desired spin state (e.g. singlet or triplet) and only states with the desired spin are located.
State-averaging
When searching for excited states, it is often convenient to use state-averaging. The basic idea here is to find orbitals that minimize the weighted average of the energies of a set of electronic states. This ensures that the orbitals are not biased towards any of the given states. The primary advantage is not accuracy (it is widely believed that state-specific CASSCF is more accurate than state-averaged CASSCF, although this certainly depends on the property being computed), but rather stability when one is computing excited states. State-specific CASSCF for excited states is often difficult to converge and excited state solutions tend to "collapse" back to the ground state. When using state-averaging in TeraChem, you must specify how many states are included in the average (these will be selected as the lowest states) and what the weights are for each of the states. Often these weights are chosen to be equal, in order to ensure the correct treatment of conical intersections.
Dynamically-weighted state averaging
Choosing equal weights for all states in a state-averaged CASSCF calculation can be problematic. If one of the states is very high in energy and well-separated from the other electronic states, incuding it in the average might bias the orbitals to describe that stae (which is often not of any interest). One could instead limit the average to one fewer state, but one generally wants the active space and state-averaging to be consistent across the potential energy surface (e.g. when doing dynamics or computing a reaction path). It is often the case that N states are needed at some geometry (e.g. the Franck-Condon region for butadiene where there are two near-degenerate low-lying excited states) but after relaxation on the excited state, only N-1 of these states are relevant (e.g., after torsion of butadiene, one the near-degenerate states is strongly stabilized and intersects with the gorund state). The dynamic weighting procedure makes the weights depend on the energies of the individual states. This gradually removes high-lying states from the state-average and can provide a better global description than the usual insistence on a fixed number of states in an equally-weighted average. It also ensures that states which are near-degenerate (e.g. near a conical intersection) are assigned the same weight, as required.
Files produced by a CASSCF calculation
- Molden files - There are a number of these, with different orbitals
- Singlet.x.molden: The natural orbitals corresponding to the xth singlet state
- state_averaged_natural_orbital.molden: