Surface hopping is a mixed quantum-classical technique that incorporates quantum mechanical effects into molecular dynamics simulations.
[1][2][3][4] Traditional molecular dynamics assume the Born-Oppenheimer approximation, where the lighter electrons adjust instantaneously to the motion of the nuclei.
Molecular dynamics simulations numerically solve the classical equations of motion.
These simulations, though, assume that the forces on the electrons are derived solely by the ground adiabatic surface.
Solving the time-dependent Schrödinger equation numerically incorporates all these effects, but is computationally unfeasible when the system has many degrees of freedom.
To tackle this issue, one approach is the mean field or Ehrenfest method, where the molecular dynamics is run on the average potential energy surface given by a linear combination of the adiabatic states.
When the difference between the adiabatic states is large, then the dynamics must be primarily driven by only one surface, and not an average potential.
The probability of these hops are dependent on the coupling between the states, and is generally significant only in the regions where the difference between adiabatic energies is small.
corresponds to the electronic degree of freedom, light atoms such as hydrogen, or high frequency vibrations such as O-H stretch.
The forces in the molecular dynamics simulations are derived only from one adiabatic surface, and are given by: where
This effect is incorporated in the surface hopping algorithm by considering the wavefunction of the quantum degrees of freedom at time t as an expansion in the adiabatic basis: where
are given by The adiabatic surface can switch at any given time t based on how the quantum probabilities
Based on this, the probability of hopping from state j to n is proposed to be This criterion is known as the "fewest switching" algorithm, as it minimizes the number of hops required to maintain the population in various adiabatic states.
Whenever a hop takes place, the velocity is adjusted to maintain conservation of energy.
This shows that the nuclear forces acting during the hop are in the direction of the nonadiabatic coupling vector
[2] Another suggestion is not to change the adiabatic state, but reverse the direction of the component of the velocity along the nonadiabatic coupling vector.
[6] Ignoring forbidden hops without any form of velocity reversal does not recover the correct scaling for Marcus theory in the nonadiabatic limit, but a velocity reversal can usually correct the errors [7] Surface hopping can develop nonphysical coherences between the quantum coefficients over large time which can degrade the quality of the calculations, at times leading the incorrect scaling for Marcus theory.
[8] To eliminate these errors, the quantum coefficients for the inactive state can be damped or set to zero after a predefined time has elapsed after the trajectory crosses the region where hopping has high probabilities.
is given by the phase space of all the classical particles, the quantum amplitudes, and the adiabatic state.
The classical positions and velocities are chosen based on the ensemble required.
Compute forces using Hellmann-Feynman theorem, and integrate the equations of motion by time step
Integrate the Schrödinger equation to evolve quantum amplitudes from time
Generate a random number, and determine whether a switch should take place.
Go back to step 2, till trajectories have been evolved for the desired time.
The method has been applied successfully to understand dynamics of systems that include tunneling, conical intersections and electronic excitation.
[9][10][11][12] In practice, surface hopping is computationally feasible only for a limited number of quantum degrees of freedom.
In addition, the trajectories must have enough energy to be able to reach the regions where probability of hopping is large.
Most of the formal critique of the surface hopping method comes from the unnatural separation of classical and quantum degrees of freedom.
Recent work has shown, however, that the surface hopping algorithm can be partially justified by comparison with the Quantum Classical Liouville Equation.
[13] It has further been demonstrated that spectroscopic observables can be calculated in close agreement with the formally exact hierarchical equations of motion.