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Can we derive Tully's surface-hopping algorithm from the semiclassical quantum Liouville equation? Almost, but only with decoherence

### Abstract

In this article, we demonstrate that Tully's fewest-switches surface hopping (FSSH) algorithm approximately obeys the mixed quantum-classical Liouville equation (QCLE), provided that several conditions are satisfied – some major conditions, and some minor. The major conditions are: (1) nuclei must be moving quickly with large momenta; (2) there cannot be explicit recoherences or interference effects between nuclear wave packets; (3) force-based decoherence must be added to the FSSH algorithm, and the trajectories can no longer rigorously be independent (though approximations for independent trajectories are possible). We furthermore expect that FSSH (with decoherence) will be most robust when nonadiabatic transitions in an adiabatic basis are dictated primarily by derivative couplings that are presumably localized to crossing regions, rather than by small but pervasive off-diagonal force matrix elements. In the end, our results emphasize the strengths of and possibilities for the FSSH algorithm when decoherence is included, while also demonstrating the limitations of the FSSH algorithm and its inherent inability to follow the QCLE exactly.

© 2013 AIP Publishing LLC

Received 18 July 2013
Accepted 29 October 2013
Published online 04 December 2013

Acknowledgments:
J.E.S. thanks Ray Kapral for his critical reading of and feedback on this paper. This material is based upon work supported by the (U.S.) Air Force Office of Scientific Research (USAFOSR) PECASE award under AFOSR Grant No. FA9550-13-1-0157. J.E.S. acknowledges an Alfred P. Sloan Research Fellowship and a David and Lucille Packard Fellowship.

Article outline:

I. INTRODUCTION: AN *AD HOC* SURFACE HOPPING VIEW OF NONADIABATIC DYNAMICS
II. THEORY: QCLE AND FSSH IN DETAIL
A. Quantum-classical Liouville equation
B. Tully's FSSH algorithm
1. Classical movement along the adiabatic surfaces
2. Momentum rescaling
3. Hopping rate, consistency, and decoherence
III. THEORY: A NUCLEAR-ELECTRONIC DENSITY MATRIX FOR FSSH
A. On-diagonal full density matrix elements
1. Major condition #1: Unique trajectory assumption
2. Major condition #2: Large velocity assumption
B. Off-diagonal full density matrix elements
C. Consistency of equations: Part 1
1. Major condition #3: Modified electronic propagation that includes decoherence
D. Consistency of equations: Part 2
E. Decoherence by collapse, frozen Gaussian ansatz, and the A-FSSH approximation
IV. DISCUSSION
A. Unique trajectory assumption
B. Large velocity assumption
C. Modified electronic density matrix equation of motion
D. Minor assumptions
E. Immediate future goals
V. CONCLUSION AND FUTURE OUTLOOK