α-CASSCF

α-CASSCF is an efficient, empirical correction to state-averaged CASSCF (SA-CASSCF) that closely approximates the potential energy surfaces (PESs) produced by multistate complete-active-space second-order perturbation theory (MS-CASPT2) — but at essentially the cost of the underlying SA-CASSCF calculation. It is intended primarily for nonadiabatic dynamics and photochemistry on systems too large for MS-CASPT2.

Motivation

SA-CASSCF is a workhorse for excited states and nonadiabatic dynamics because it treats several states on an equal footing and correctly describes the topology of the PES near conical intersections. However, it largely neglects dynamic electron correlation. As a result its excitation energies and state splittings are systematically too large, and its PESs can differ qualitatively from a more accurate reference such as MS-CASPT2. Methods that recover dynamic correlation explicitly (MS-CASPT2, MRCI) are far more expensive and quickly become intractable for large chromophores in their environment.

α-CASSCF observes that SA-CASSCF tends to overestimate the splitting of the individual states about their state-averaged energy by an approximately constant multiplicative factor. Scaling that splitting back down recovers much of the missing correlation effect empirically.

The scaling scheme

Write the SA-CASSCF energy of the $\Theta$-th state as the state-averaged energy plus its splitting from that average,

\[ E_\Theta^{\mathrm{CAS}} = E_{\mathrm{SA}}^{\mathrm{CAS}} + \Delta E_\Theta . \]

α-CASSCF leaves the state-averaged energy untouched and scales only the splitting by a single empirical parameter $\alpha$:

\[ E_\Theta^{\alpha\mathrm{CAS}} = E_{\mathrm{SA}}^{\mathrm{CAS}} + \alpha\,\Delta E_\Theta = \alpha\,E_\Theta^{\mathrm{CAS}} + (1-\alpha)\,E_{\mathrm{SA}}^{\mathrm{CAS}} . \]

This is exactly the operation TeraChem applies to each state energy. A few consequences follow directly:

Excitation energies are scaled by $\alpha$ at a given geometry: $E_\Theta^{\alpha\mathrm{CAS}} - E_\Phi^{\alpha\mathrm{CAS}} = \alpha\left(E_\Theta^{\mathrm{CAS}} - E_\Phi^{\mathrm{CAS}}\right)$.
Gradients combine the state-specific and state-averaged SA-CASSCF gradients, $\dfrac{\partial E_\Theta^{\alpha\mathrm{CAS}}}{\partial x} = \alpha\,\dfrac{\partial E_\Theta^{\mathrm{CAS}}}{\partial x}
(1-\alpha)\,\dfrac{\partial E_{\mathrm{SA}}^{\mathrm{CAS}}}{\partial x}$, both of which the SA-CASSCF response machinery already provides.
Nonadiabatic coupling vectors are unchanged. Because $\alpha$ rescales only the energies and not the wavefunction parameters, the derivative coupling $\langle\Theta|\partial/\partial x|\Phi\rangle$ is identical to its SA-CASSCF value.
Setting $\alpha = 1$ recovers ordinary SA-CASSCF exactly.

Because α-CASSCF is just a scaling of SA-CASSCF, it has the same computational cost and scaling as the underlying SA-CASSCF calculation — on the order of 100 times faster than MS-CASPT2 for combined energy-and-gradient evaluations — while producing PESs with low nonparallelity error relative to MS-CASPT2.

What does and does not change

The location of the conical-intersection seam is preserved by the scaling. However, the relative energy of points along a seam, barrier heights, and the detailed topology in the immediate vicinity of an intersection (e.g. sloped vs. peaked) may change. The relative energy between two different molecular geometries is not simply scaled by $\alpha$, because the state-averaged energy is itself geometry-dependent; α-CASSCF can therefore be qualitatively different from SA-CASSCF, not merely a rigid shift.

Choosing α

$\alpha$ is an empirical parameter. It is system-specific but not geometry-specific, and it also depends on the choice of active space and on the number and weighting of the states being averaged — so those must be held fixed across the PES. Optimal values are typically in the range $\alpha \approx 0.7$–$0.8$ (strikingly close to the canonical Hartree–Fock vibrational frequency-scaling factor of 0.89, i.e. $\alpha \approx 0.79$).

The most robust way to determine $\alpha$ is to minimize the nonparallelity error (or RMS deviation) between α-CASSCF and a higher-level reference (MS-CASPT2, experiment, etc.) over a set of critical points on the PES. When that is too expensive, $\alpha$ may be fit to a single observable — for example, the Franck–Condon excitation energy or an absorption maximum — or determined on a smaller representative chromophore and transferred to the full system. Reported values include $\alpha = 0.73$ for the anionic GFP chromophore (HBDI) and $\alpha = 0.80$ for p-coumaric acid (the PYP chromophore).

Following the SA-$N$-CAS($m$,$n$) convention, an α-CASSCF wavefunction is written ($\alpha=a$)-$N$-CAS($m$,$n$) — for example, ($\alpha=0.73$)-2-CAS(4,3).

Input keywords

α-CASSCF is layered on top of an ordinary SA-CASSCF setup. Enable state averaging and the desired states first, then add the keywords below.

Keyword	Type	Default	Description
`alphacas`	bool	no	Enable the α-CASSCF correction. Each state energy is replaced by $\alpha E_\Theta^{\mathrm{CAS}} + (1-\alpha)E_{\mathrm{SA}}^{\mathrm{CAS}}$.
`alpha`	float	1.0	The empirical scaling parameter $\alpha$. The default (1.0) reproduces ordinary SA-CASSCF; physically meaningful values are typically ≈ 0.7–0.8.
`alphacas_scale_coupling`	bool	no	Whether to use the α-scaled energy gap when forming the interstate (nonadiabatic) coupling vector. When `no` (default), the coupling is computed with the unscaled SA-CASSCF energy gap; when `yes`, the α-scaled gap is used, consistent with the scaled energy splittings.

Warning

alpha has no effect unless alphacas yes is also set. Conversely, setting alphacas yes with the default alpha 1.0 performs an ordinary SA-CASSCF calculation.

Example

A two-singlet-state α-CASSCF gradient on a CAS(2,2) active space with $\alpha = 0.64$ (as used in a QM/MM input):

casscf                 yes
closed                 7
active                 2
cassinglets            2
castarget              1
castargetmult          1
alphacas               yes
alpha                  0.64

Caveats

α-CASSCF is an empirical scaling scheme, not a replacement for an explicit correlation treatment. It still inherits the deficiencies of SA-CASSCF that arise from the wavefunction itself — for instance, state inversion or energy discontinuities far from the reference points used to fit $\alpha$. It also does not always succeed (for example, it performs poorly for uracil). It is best validated against a higher-level reference before being used for production dynamics. Where even a single MS-CASPT2 reference is impossible, fitting $\alpha$ on a representative subsystem is recommended.

Reference

J. W. Snyder Jr., R. M. Parrish, and T. J. Martínez, "α-CASSCF: An Efficient, Empirical Correction for SA-CASSCF To Closely Approximate MS-CASPT2 Potential Energy Surfaces," J. Phys. Chem. Lett. 8, 2432–2437 (2017). doi:10.1021/acs.jpclett.7b00940