Three dynamics methods for the same preparation

Set up one common avoided crossing

Use Tully’s 1990 simple avoided crossing with coordinate $R$ and nuclear mass $M=2000$. The real symmetric diabatic matrix has opposite diagonal elements. Its parameters are $A=0.01$, $B=1.6$, $C=0.005$ and $D=1$ in the expressions below.

The electronic state is prepared in the local lower eigenstate at each nuclear position. The incident packet is centered at $R_0=-8$, with position standard deviation $\sigma_R$ and central momentum $p_0$. Both classical methods sample position and momentum from the same scalar minimum-uncertainty nuclear Gaussian Wigner distribution. This particular Wigner distribution is nonnegative and can supply a classical initial ensemble.

Reading: John C. Tully · Tully 1990 original paper · Cigrang et al.

Quantum evolution retains amplitudes at every position

The two nuclear components form $\boldsymbol\chi(R,t)$. Kinetic energy acts on each; the local potential matrix mixes them. A split-operator solver alternates kinetic and potential propagation. The local two-state potential exponential can be evaluated directly.

A symmetric half-potential, full-kinetic, half-potential step has local error starting at the cube of the time step, usually producing quadratic global convergence. Spatial resolution and boundaries introduce separate errors. After splitting, each branch still carries its own phase.

Ehrenfest evolves an electronic mean force

Each trajectory has classical position $R$, momentum $p$ and normalized electronic vector $c$. Electronic evolution uses the potential at the current position. The nuclear force is the expectation of its derivative, including off-diagonal derivatives.

When quantum probability separates onto two surfaces, an individual Ehrenfest trajectory still experiences one average force. More initial trajectories improve sampling of this approximation. Coherent nuclear branching remains a separate approximation error.

Surface hopping gives each trajectory an active surface

Fewest-switches surface hopping, or FSSH, computes nuclear force on an active adiabatic surface and continuously evolves electronic amplitudes. Let $a$ denote the active state, $b$ the target, and $d_{ab}$ their real-basis derivative coupling. A short-time electronic probability flow determines the candidate hop probability.

An accepted hop adjusts nuclear momentum to compensate for the surface-energy change. In one dimension the forward direction is retained and the magnitude changes. An uphill hop requiring more kinetic energy than available is rejected as frustrated. Multidimensional adjustment requires an additional direction convention.

Reading: John C. Tully

Worked example of a closed upper exit

Far from the crossing the two surfaces approach $-A$ and $+A$. Leaving on the upper surface after lower-surface incidence requires energy $2A=0.02$. For $p_0=5$ and $M=2000$, the central kinetic energy is $p_0^2/(2M)=0.00625$.

A narrow incident momentum distribution therefore suppresses the upper far-field exit. A Gaussian has a high-momentum tail and finite-time amplitude may remain near the crossing. The central threshold alone does not prove exactly zero upper probability for the full packet.

Electronic upper-state population, the fraction of trajectories with that active state, and upper-channel outgoing probability are distinct. The last additionally requires arrival in the specified exit. Define the observable before attributing a difference to dynamics.