9.9.8 Meyer-Miller Nonadiabatic Dynamics

As an alternative to Tully’s FSSH algorithm (see Section 9.9.7), Q-Chem can also perform trajectory-based electronically nonadiabatic simulations with the vibronic dynamics generated using the classical Meyer-Miller (MM) Hamiltonian. ⁸⁸² Meyer H.-D., Miller W. H.
J. Chem. Phys.
(1979), 70, pp. 3214. Link Additionally, the dynamics can be subject to a symmetrical quasi-classical (SQC) quantization procedure, ²⁵⁹ Cotton S. J., Miller W. H.
J. Phys. Chem. A
(2013), 117, pp. 7190. Link effectively defining the electronic states. ²⁶⁰ Cotton S. J., Miller W. H.
J. Chem. Phys.
(2013), 139, pp. 234112. Link Details of this approach as it pertains to the Q-Chem implementation can be found in Ref.  1252 Talbot J. J., Head-Gordon M., Cotton S. J.
Mol. Phys.
(2023), 121, pp. e2153761. Link . In brief, the Meyer-Miller Hamiltonian maps the electronic degrees of freedom (DOF) in an electronically nonadiabatic process to a coupled set of classical harmonic oscillators, one for each electronic state. Movement of classical vibrational excitation amongst the oscillators then determines the active electronic state, or more precisely, the combination of electronic states which define an effective multi-state potential energy surface (PES) on which the nuclear degrees of freedom are also propagated classically.

In the adiabatic representation (relevant to the Born-Oppenheimer PESs generated by Q-Chem), The Meyer-Miller Hamiltonian is

where $\{x_I,p_I\}$ are the coordinates and momenta of the “electronic oscillators” corresponding to a set of $F$ electronic states, $𝐑,𝐏$ are the coordinates and momenta of the nuclear DOF having reduced masses $𝝁$, $E_I(𝐑)$ is the Born-Oppenheimer PES corresponding to adiabatic state $I$, ${\text{𝐝}}_{IJ}(\text{𝐑})=⟨{\mathrm{\Psi }}_I|{\overrightarrow{\nabla }}_{\text{𝐑}}{\mathrm{\Psi }}_J⟩$ is the standard first-order derivative coupling vector between electronic states $I$ and $J$, and $\{{\gamma }_I\}$ are a set of zero point energy parameters. The evolution of the $F$ classical oscillators in Eq. (9.63) thus describes the electronic configuration in the MM model and, in particular, the classical actions associated with each oscillator.

The symmetric quasi-classical Meyer-Miller (SQC/MM) approach is requested in Q-Chem with the QCMD_METHOD rem variable. Quantization of the classical Hamiltonian dynamics produced by Eq. (9.63) is done symmetrically, i.e., with respect to both the initial and final values of the dynamical electronic variables. This is performed initially by Monte Carlo sampling actions from a “windowing” function. The quantization at the prescribed final times is accomplished by “binning” the time-evolved actions according to the same windowing function. In Q-Chem’s implementation, only the “triangle” style of windowing function is employed, ²⁶¹ Cotton S. J., Miller W. H.
J. Chem. Phys.
(2016), 145, pp. 144108. Link which was found universally superior to the original histogram style windows of Ref.  260 Cotton S. J., Miller W. H.
J. Chem. Phys.
(2013), 139, pp. 234112. Link . Additionally, the option to use the $\gamma$-adjustment protocol of Ref.  262 Cotton S. J., Miller W.H.
J. Chem. Phys.
(2019), 150, pp. 194110. Link is available and requested with the SQC_GAMADJUST keyword. This is generally recommended. The key point of the $\gamma$-adjustment procedure is to set the $\{{\gamma }_I\}$ in Eq. (9.63) per DOF (and per trajectory), so that the initial forces on the nuclei are that of the initial pure quantum state—i.e., the single-PES forces. Ehrenfest simulations are also available where the dynamics of these are equivalent to the SQC calculations, but instead of using symmetric windowing functions for selecting initial conditions and estimating final populations, the Ehrenfest method uses integer initial electronic action variables with $\gamma =0$ and uses the values of these action variables at each desired final time to estimate the electronic state populations.

Sampling distributions for the initial nuclear DOF are requested with the QCMD_INITNUC keyword. Currently, sampling both initial positions and velocity from either a Wigner or Boltzmann distribution is available. Sampling from either of these distributions requires a frequency calculation to be available. Alternatively, the user can input velocities using the $velocity section as described in Section 9.9.2.