Mixed quantum-classical dynamics

In the Born-Oppenheimer approximation, the ensemble of electrons of a molecule or supramolecular system can have several discrete states. The potential energy of each of these electronic states depends on the position of the nuclei, forming multidimensional surfaces.

Under usual conditions (room temperature, for instance), the molecular system is in the ground electronic state (the electronic state of lowest energy). In this stationary situation, nuclei and electrons are in equilibrium, and the molecule naturally vibrates near harmonically due to the zero-point energy.

Particle collisions and photons with wavelengths in the range from visible to X-ray can promote the electrons to electronically excited states. Such events create a non-equilibrium between nuclei and electrons, which leads to an ultrafast response (picosecond scale) of the molecular system. During the ultrafast evolution, the nuclei may reach geometric configurations where the electronic states mix, allowing the system to transfer to another state spontaneously. These state transfers are nonadiabatic phenomena.

Nonadiabatic dynamics is the field of computational chemistry that simulates such ultrafast nonadiabatic response.

In principle, the problem can be exactly addressed by solving the time-dependent Schrödinger equation (TDSE) for all particles (nuclei and electrons). Methods like the multiconfigurational self-consistent Hartree (MCTDH) have been developed to do such task. Nevertheless, they are limited to small systems with two dozen degrees of freedom due to the enormous difficulties of developing multidimensional potential energy surfaces and the costs of the numerical integration of the quantum equations.

NA-MQC dynamics methods have been developed to reduce the burden of these simulations by profiting from the fact that the nuclear dynamics is near classical. Treating the nuclei classically allows simulating the molecular system in full dimensionality. The impact of the underlying assumptions depends on each particular NA-MQC method.

Most of NA-MQC dynamics methods have been developed to simulate internal conversion (IC), the nonadiabatic transfer between states of the same spin multiplicity. The methods have been extended, however, to deal with other types of processes like intersystem crossing (ISC; transfer between states of different multiplicities){{cite journal |last1=Granucci |first1=Giovanni |last2=Persico |first2=Maurizio |last3=Spighi |first3=Gloria |title=Surface hopping trajectory simulations with spin-orbit and dynamical couplings |journal=The Journal of Chemical Physics |date=14 December 2012 |volume=137 |issue=22 |pages=22A501 |doi=10.1063/1.4707737|pmid=23249038 |bibcode=2012JChPh.137vA501G }} and field-induced transfers.{{cite journal |last1=Mitrić |first1=Roland |last2=Petersen |first2=Jens |last3=Wohlgemuth |first3=Matthias |last4=Werner |first4=Ute |last5=Bonačić-Koutecký |first5=Vlasta |last6=Wöste |first6=Ludger |last7=Jortner |first7=Joshua |title=Time-Resolved Femtosecond Photoelectron Spectroscopy by Field-Induced Surface Hopping |journal=The Journal of Physical Chemistry A |date=28 April 2011 |volume=115 |issue=16 |pages=3755–3765 |doi=10.1021/jp106355n|pmid=20939619 |bibcode=2011JPCA..115.3755M }}

NA-MQC dynamics has been often used in theoretical investigations of photochemistry and femtochemistry, especially when time-resolved processes are relevant.{{cite journal |last1=Akimov |first1=Alexey V. |last2=Prezhdo |first2=Oleg V. |title=Large-Scale Computations in Chemistry: A Bird's Eye View of a Vibrant Field |journal=Chemical Reviews |date=8 April 2015 |volume=115 |issue=12 |pages=5797–5890 |doi=10.1021/cr500524c|pmid=25851499 }}{{cite journal |last1=Brunk |first1=Elizabeth |last2=Rothlisberger |first2=Ursula |title=Mixed Quantum Mechanical/Molecular Mechanical Molecular Dynamics Simulations of Biological Systems in Ground and Electronically Excited States |journal=Chemical Reviews |date=16 April 2015 |volume=115 |issue=12 |pages=6217–6263 |doi=10.1021/cr500628b|url=http://infoscience.epfl.ch/record/207721 |pmid=25880693 }}

NA-MQC dynamics is a general class of methods developed since the 1970s. It encompasses:

1. Trajectory surface hopping (TSH; FSSH for fewest switches surface hopping);{{cite journal |last1=Tully |first1=John C. |title=Molecular dynamics with electronic transitions |journal=The Journal of Chemical Physics |date=15 July 1990 |volume=93 |issue=2 |pages=1061–1071 |doi=10.1063/1.459170|bibcode=1990JChPh..93.1061T |s2cid=15191625 }}
2. Mean-field Ehrenfest dynamics (MFE);{{cite journal |last1=Tully |first1=John C. |title=Mixed quantum–classical dynamics |journal=Faraday Discussions |date=1998 |volume=110 |pages=407–419 |doi=10.1039/A801824C|bibcode=1998FaDi..110..407T }}
3. Coherent Switching with Decay of Mixing (CSDM; MFE with Non-Markovian decoherence and stochastic pointer state switch);{{cite journal |last1=Truhlar |first1=Chaoyuan Z. |title=Coherent Switching with Decay of Mixing: An Improved Treatment of Electronic Coherence for Non-Born-Oppenheimer Trajectories |journal=Journal of Chemical Physics |date=2004 |volume=121 |issue=16 |pages=7658–7670 |doi=10.1063/1.1793991 |pmid=15485225 |bibcode=2004JChPh.121.7658Z |doi-access=free }}
4. Multiple spawning (AIMS for ab initio multiple spawning; FMS for full multiple spawning);{{cite journal |last1=Curchod |first1=Basile F. E. |last2=Martínez |first2=Todd J. |title=Ab Initio Nonadiabatic Quantum Molecular Dynamics |journal=Chemical Reviews |date=21 February 2018 |volume=118 |issue=7 |pages=3305–3336 |doi=10.1021/acs.chemrev.7b00423|pmid=29465231 |url=http://dro.dur.ac.uk/24099/1/24099.pdf }}
5. Coupled-Trajectory Mixed Quantum-Classical Algorithm (CT-MQC);{{cite journal |last1=Agostini |first1=Federica |last2=Min |first2=Seung Kyu |last3=Abedi |first3=Ali |last4=Gross |first4=E. K. U. |title=Quantum-Classical Nonadiabatic Dynamics: Coupled- vs Independent-Trajectory Methods |journal=Journal of Chemical Theory and Computation |date=19 April 2016 |volume=12 |issue=5 |pages=2127–2143 |doi=10.1021/acs.jctc.5b01180|pmid=27030209 |arxiv=1512.04638 |s2cid=31630792 }}
6. Mixed quantum−classical Liouville equation (QCLE);{{cite journal |last1=Kapral |first1=Raymond |last2=Ciccotti |first2=Giovanni |title=Mixed quantum-classical dynamics |journal=The Journal of Chemical Physics |date=8 May 1999 |volume=110 |issue=18 |pages=8919–8929 |doi=10.1063/1.478811|bibcode=1999JChPh.110.8919K }}
7. Mapping approach;{{cite journal |last1=Thoss |first1=Michael |last2=Stock |first2=Gerhard |title=Mapping approach to the semiclassical description of nonadiabatic quantum dynamics |journal=Physical Review A |date=January 1999 |volume=59 |issue=1 |pages=64–79 |doi=10.1103/PhysRevA.59.64|bibcode=1999PhRvA..59...64T }}
8. Nonadiabatic Bohmian dynamics (NABDY);{{Cite journal|last1=Curchod|first1=Basile F. E.|last2=Tavernelli|first2=Ivano|date=2013-05-14|title=On trajectory-based nonadiabatic dynamics: Bohmian dynamics versus trajectory surface hopping|journal=The Journal of Chemical Physics|language=en|volume=138|issue=18|pages=184112|doi=10.1063/1.4803835|pmid=23676034 |bibcode=2013JChPh.138r4112C |issn=0021-9606}}
9. Multiple cloning; (AIMC for ab initio multiple cloning){{cite journal |last1=Makhov |first1=Dmitry V. |last2=Glover |first2=William J. |last3=Martinez |first3=Todd J. |last4=Shalashilin |first4=Dmitrii V. |title=multiple cloning algorithm for quantum nonadiabatic molecular dynamics |journal=The Journal of Chemical Physics |date=7 August 2014 |volume=141 |issue=5 |pages=054110 |doi=10.1063/1.4891530|pmid=25106573 |doi-access=free }}
10. Global Flux Surface Hopping (GFSH);{{cite journal |last1=Wang |first1=L. |last2=Trivedi |first2=D. J. |last3=Prezhdo |first3=O. V. |title=Global Flux Surface Hopping Approach for Mixed Quantum-Classical Dynamics |journal=Journal of Chemical Theory and Computation |date=12 June 2014 |volume=10 |issue=9 |pages=3598–3605 |doi=10.1021/ct5003835|pmid=26588504 |url=http://pubs.acs.org/doi/abs/10.1021/ct5003835 |url-access=subscription }}
11. Decoherence Induced Surface Hopping (DISH){{cite journal |last1=Jaeger |first1=H. M. |last2=Fischer |first2=S. |last3=Prezhdo |first3=O. V. |title=Decoherence-induced surface hopping |journal= The Journal of Chemical Physics |date=15 Oct 2012 |volume=137 |issue=22 |pages= 22A545 |doi=10.1063/1.4757100|pmid=23249082 |bibcode=2012JChPh.137vA545J |url=http://aip.scitation.org/doi/10.1063/1.4757100 |url-access=subscription }}

The classical trajectories can be integrated with conventional methods, as the Verlet algorithm. Such integration requires the forces acting on the nuclei. They are proportional to the gradient of the potential energy of the electronic states and can be efficiently computed with diverse electronic structure methods for excited states, like the multireference configuration interaction (MRCI) or the linear-response time-dependent density functional theory (TDDFT).

In NA-MQC methods like FSSH or MFE, the trajectories are independent of each other. In such a case, they can be separately integrated and only grouped afterward for the statistical analysis of the results. In methods like CT-MQC or diverse TSH variants,{{cite journal |last1=Wang |first1=Linjun |last2=Akimov |first2=Alexey |last3=Prezhdo |first3=Oleg V. |title=Recent Progress in Surface Hopping: 2011–2015 |journal=The Journal of Physical Chemistry Letters |date=23 May 2016 |volume=7 |issue=11 |pages=2100–2112 |doi=10.1021/acs.jpclett.6b00710|pmid=27171314 }} the trajectories are coupled and must be integrated simultaneously.

In NA-MQC dynamics, the electrons are usually treated by a local approximation of the TDSE, i.e., they depend only on the electronic forces and couplings at the instantaneous position of the nuclei.

File:Na-mqc-types.jpg

There are three basic algorithms to recover nonadiabatic information in NA-MQC methods:

1. Spawning - new trajectories are created at regions of large nonadiabatic coupling.
2. Hopping - trajectories are propagated on a single potential energy surface (PES), but they are allowed to change surface near regions of large nonadiabatic couplings.
3. Averaging - trajectories are propagated on a weighted average of potential energy surfaces. The weights are determined by the amount of nonadiabatic mixing.

NA-MQC dynamics are approximated methods to solve the time-dependent Schrödinger equation for a molecular system. Methods like TSH, in particular in the fewest switches surface hopping (FSSH) formulation, do not have an exact limit.{{cite journal |last1=Ou |first1=Qi |last2=Subotnik |first2=Joseph E. |title=Electronic Relaxation in Benzaldehyde Evaluated via TD-DFT and Localized Diabatization: Intersystem Crossings, Conical Intersections, and Phosphorescence |journal=The Journal of Physical Chemistry C |date=19 September 2013 |volume=117 |issue=39 |pages=19839–19849 |doi=10.1021/jp405574q}} Other methods like MS or CT-MQC can in principle deliver the exact non-relativistic solution.

In the case of multiple spawning, it is hierarchically connected to MCTDH,{{cite journal |last1=Worth |first1=Graham A. |last2=Hunt |first2=Patricia |last3=Robb |first3=Michael A. |title=Nonadiabatic Dynamics: A Comparison of Surface Hopping Direct Dynamics with Quantum Wavepacket Calculations |journal=The Journal of Physical Chemistry A |date=February 2003 |volume=107 |issue=5 |pages=621–631 |doi=10.1021/jp027117p|bibcode=2003JPCA..107..621W }} while CT-MQC is connected to the exact factorization method.

The most common approach in NA-MQC dynamics is to compute the electronic properties on-the-fly, i.e., at each timestep of the trajectory integration. Such an approach has the advantage of not requiring pre-computed multidimensional potential energy surfaces. Nevertheless, the [https://barbatti.org/2017/03/13/whats-the-biggest-system-we-can-do-dynamics/ costs associated with the on-the-fly approach] are significantly high, leading to a systematic level downgrade of the simulations. This downgrade has been shown to lead to qualitatively wrong results.{{cite journal |last1=Plasser |first1=Felix |last2=Crespo-Otero |first2=Rachel |last3=Pederzoli |first3=Marek |last4=Pittner |first4=Jiri |last5=Lischka |first5=Hans |last6=Barbatti |first6=Mario |author-link6=Mario Barbatti|title=Surface Hopping Dynamics with Correlated Single-Reference Methods: 9H-Adenine as a Case Study |journal=Journal of Chemical Theory and Computation |date=13 March 2014 |volume=10 |issue=4 |pages=1395–1405 |doi=10.1021/ct4011079|pmid=26580359 |hdl=11858/00-001M-0000-0024-A689-7 |hdl-access=free }}

The local approximation implied by the classical trajectories in NA-MQC dynamics also leads to failing in the description of non-local quantum effects, as tunneling and quantum interference. Some methods like MFE and FSSH are also affected by decoherence errors.{{cite journal |last1=Subotnik |first1=Joseph E. |last2=Jain |first2=Amber |last3=Landry |first3=Brian |last4=Petit |first4=Andrew |last5=Ouyang |first5=Wenjun |last6=Bellonzi |first6=Nicole |title=Understanding the Surface Hopping View of Electronic Transitions and Decoherence |journal=Annual Review of Physical Chemistry |date=27 May 2016 |volume=67 |issue=1 |pages=387–417 |doi=10.1146/annurev-physchem-040215-112245|pmid=27215818 |bibcode=2016ARPC...67..387S |doi-access=free }} New algorithms have been developed to include tunneling{{cite journal |last1=Zheng |first1=Jingjing |last2=Xu |first2=Xuefei |last3=Meana-Pañeda |first3=Rubén |last4=Truhlar |first4=Donald G. |title=Army ants tunneling for classical simulations |journal=Chem. Sci. |date=2014 |volume=5 |issue=5 |pages=2091–2099 |doi=10.1039/C3SC53290A|s2cid=17600447 }} and decoherence effects.{{cite journal |last1=Granucci |first1=Giovanni |last2=Persico |first2=Maurizio |last3=Zoccante |first3=Alberto |title=Including quantum decoherence in surface hopping |journal=The Journal of Chemical Physics |date=7 October 2010 |volume=133 |issue=13 |pages=134111 |doi=10.1063/1.3489004|pmid=20942527 |bibcode=2010JChPh.133m4111G }}{{cite journal |last1=Jain |first1=Amber |last2=Alguire |first2=Ethan |last3=Subotnik |first3=Joseph E. |title=An Efficient, Augmented Surface Hopping Algorithm That Includes Decoherence for Use in Large-Scale Simulations |journal=Journal of Chemical Theory and Computation |date=7 October 2016 |volume=12 |issue=11 |pages=5256–5268 |doi=10.1021/acs.jctc.6b00673|pmid=27715036 }} Global quantum effects can also be considered by applying quantum forces between trajectories.

Survey of NA-MQC dynamics implementations in public software.

^a Development version.

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Category:Computational chemistry