Path integral molecular dynamics

{{Use American English|date=January 2019}}

{{Use mdy dates|date=January 2019}}

{{Short description|Molecular dynamics simulations augmented with quantum mechanics}}

Path integral molecular dynamics (PIMD) is a method of incorporating quantum mechanics into molecular dynamics simulations using Feynman path integrals. In PIMD, one uses the Born–Oppenheimer approximation to separate the wavefunction into a nuclear part and an electronic part. The nuclei are treated quantum mechanically by mapping each quantum nucleus onto a classical system of several fictitious particles connected by springs (harmonic potentials) governed by an effective Hamiltonian, which is derived from Feynman's path integral. The resulting classical system, although complex, can be solved relatively quickly. There are now a number of commonly used condensed matter computer simulation techniques that make use of the path integral formulation including centroid molecular dynamics (CMD),{{Cite journal | last1 = Cao | first1 = J. | last2 = Voth | first2 = G. A. | doi = 10.1063/1.467175 | title = The formulation of quantum statistical mechanics based on the Feynman path centroid density. I. Equilibrium properties | journal = The Journal of Chemical Physics | volume = 100 | issue = 7 | pages = 5093 | year = 1994 | bibcode = 1994JChPh.100.5093C | url = https://apps.dtic.mil/sti/pdfs/ADA272809.pdf | access-date = April 29, 2018 | archive-date = September 24, 2017 | archive-url = https://web.archive.org/web/20170924192436/http://www.dtic.mil/get-tr-doc/pdf?AD=ADA272809 | url-status = live }}{{Cite journal | last1 = Cao | first1 = J. | last2 = Voth | first2 = G. A. | doi = 10.1063/1.467176 | title = The formulation of quantum statistical mechanics based on the Feynman path centroid density. II. Dynamical properties | journal = The Journal of Chemical Physics | volume = 100 | issue = 7 | pages = 5106 | year = 1994 |bibcode = 1994JChPh.100.5106C }}{{Cite journal | last1 = Jang | first1 = S. | last2 = Voth | first2 = G. A. | doi = 10.1063/1.479515 | title = A derivation of centroid molecular dynamics and other approximate time evolution methods for path integral centroid variables | journal = The Journal of Chemical Physics | volume = 111 | issue = 6 | pages = 2371 | year = 1999 |bibcode = 1999JChPh.111.2371J }}{{Cite journal | last1 = RamíRez | first1 = R. | last2 = LóPez-Ciudad | first2 = T. | doi = 10.1063/1.479666 | title = The Schrödinger formulation of the Feynman path centroid density | journal = The Journal of Chemical Physics | volume = 111 | issue = 8 | pages = 3339 | year = 1999 |arxiv = cond-mat/9906318 |bibcode = 1999JChPh.111.3339R | s2cid = 15452314 }}{{Cite journal | last1 = Polyakov | first1 = E. A. | last2 = Lyubartsev | first2 = A. P. | last3 = Vorontsov-Velyaminov | first3 = P. N. | doi = 10.1063/1.3484490 | title = Centroid molecular dynamics: Comparison with exact results for model systems | journal = The Journal of Chemical Physics | volume = 133 | issue = 19 | pages = 194103 | year = 2010 | pmid = 21090850|bibcode = 2010JChPh.133s4103P }} ring polymer molecular dynamics (RPMD),{{Cite journal | last1 = Craig | first1 = I. R. | last2 = Manolopoulos | first2 = D. E. | doi = 10.1063/1.1777575 | title = Quantum statistics and classical mechanics: Real time correlation functions from ring polymer molecular dynamics | journal = The Journal of Chemical Physics | volume = 121 | issue = 8 | pages = 3368–3373 | year = 2004 | pmid = 15303899|bibcode = 2004JChPh.121.3368C }}{{Cite journal | last1 = Braams | first1 = B. J. | last2 = Manolopoulos | first2 = D. E. | doi = 10.1063/1.2357599 | title = On the short-time limit of ring polymer molecular dynamics | journal = The Journal of Chemical Physics | volume = 125 | issue = 12 | pages = 124105 | year = 2006 | pmid = 17014164|bibcode = 2006JChPh.125l4105B }} and the Feynman–Kleinert quasi-classical Wigner (FK–QCW) method.{{Cite journal|last1 = Smith|first1 = Kyle K. G.|last2 = Poulsen|first2 = Jens Aage|last3 = Nyman|first3 = Gunnar|last4 = Rossky|first4 = Peter J.|date = 2015-06-28|title = A new class of ensemble conserving algorithms for approximate quantum dynamics: Theoretical formulation and model problems|journal = The Journal of Chemical Physics|volume = 142|issue = 24|pages = 244112|doi = 10.1063/1.4922887|pmid = 26133415|issn = 0021-9606|bibcode = 2015JChPh.142x4112S |hdl = 1911/94772|hdl-access = free}}{{Cite journal|last1 = Smith|first1 = Kyle K. G.|last2 = Poulsen|first2 = Jens Aage|last3 = Nyman|first3 = Gunnar|last4 = Cunsolo|first4 = Alessandro|last5 = Rossky|first5 = Peter J.|date = 2015-06-28|title = Application of a new ensemble conserving quantum dynamics simulation algorithm to liquid para-hydrogen and ortho-deuterium|journal = The Journal of Chemical Physics|volume = 142|issue = 24|pages = 244113|doi = 10.1063/1.4922888|pmid = 26133416|issn = 0021-9606|bibcode = 2015JChPh.142x4113S |hdl = 1911/94773| osti=1237171 |hdl-access = free}} The same techniques are also used in path integral Monte Carlo (PIMC).{{Cite journal | last1 = Berne | first1 = B. J. | last2 = Thirumalai | first2 = D. | doi = 10.1146/annurev.pc.37.100186.002153 | title = On the Simulation of Quantum Systems: Path Integral Methods | journal = Annual Review of Physical Chemistry | volume = 37 | pages = 401–424 | year = 1986 |bibcode = 1986ARPC...37..401B }}{{cite book|first=M. J. |last=Gillan |chapter=The path-integral simulation of quantum systems, Section 2.4|title=Computer Modelling of Fluids Polymers and Solids|editor1= C. R. A. Catlow |editor2=S. C. Parker |editor3=M. P. Allen|series= NATO ASI Series C |volume=293|pages=155–188 |year=1990| isbn= 978-0-7923-0549-1}}{{cite journal |first=H. F.|last= Trotter |title=On the Product of Semi-Groups of Operators|journal= Proceedings of the American Mathematical Society |volume=10|pages= 545–551 |year=1959|jstor=2033649 |doi=10.1090/S0002-9939-1959-0108732-6 |issue=4|doi-access=free}}{{Cite journal | last1 = Chandler | first1 = D. | title = Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids | doi = 10.1063/1.441588 | journal = The Journal of Chemical Physics | volume = 74 | issue = 7 | pages = 4078–4095 | year = 1981 |bibcode = 1981JChPh..74.4078C }}{{Cite journal | last1 = Marx | first1 = D. | last2 = Müser | first2 = M. H. | doi = 10.1088/0953-8984/11/11/003 | title = Path integral simulations of rotors: Theory and applications | journal = Journal of Physics: Condensed Matter | volume = 11 | issue = 11 | pages = R117 | year = 1999 |bibcode = 1999JPCM...11R.117M | s2cid = 250913547 }}

There are two ways to calculate the dynamics calculations of PIMD. The first one is the non-Hamiltonian phase space analysis theory{{Cite web |title=Non-Hamilton Theory |url=https://pubs.aip.org/aip/jcp/article-abstract/115/4/1678/451151/Non-Hamiltonian-molecular-dynamics-Generalizing}}, which has been updated to create an "extended system" of isokinetic equations of motion which overcomes the properties of a system that created issues within the community. The second way is by using Nosé–Hoover chain,{{Cite web |date=1992 |title=Nose-Hoover Chains |url=https://www2.stat.duke.edu/~scs/Projects/REMD/NoseHooverChains1992.pdf}} which is a chain of variables instead of a single thermostat of variable.