Path integral formulation - Revision history

95.116.127.139: /* Computer simulation techniques */

2020-09-07T20:04:32Z

Computer simulation techniques

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	<ref>[http://dx.doi.org/10.1063/1.2953308 Thomas E. Markland and David E. Manolopoulos "An efficient ring polymer contraction scheme for imaginary time path integral simulations", Journal of Chemical Physics '''129''' 024105 (2008)]</ref>		<ref>[http://dx.doi.org/10.1063/1.2953308 Thomas E. Markland and David E. Manolopoulos "An efficient ring polymer contraction scheme for imaginary time path integral simulations", Journal of Chemical Physics '''129''' 024105 (2008)]</ref>
	<ref>[http://dx.doi.org/10.1016/j.cplett.2008.09.019 Thomas E. Markland and David E. Manolopoulos "A refined ring polymer contraction scheme for systems with electrostatic interactions" Chemical Physics Letters '''464''' pp. 256-261 (2008)]</ref>)		<ref>[http://dx.doi.org/10.1016/j.cplett.2008.09.019 Thomas E. Markland and David E. Manolopoulos "A refined ring polymer contraction scheme for systems with electrostatic interactions" Chemical Physics Letters '''464''' pp. 256-261 (2008)]</ref>)
			====Car-Parrinello path-integral molecular dynamics====
			Path-integral [[Car-Parrinello technique]]<ref>[http://dx.doi.org/10.1007/BF01312185 Dominik Marx and Michele Parrinello "Ab initio path-integral molecular dynamics", Zeitschrift für Physik B Condensed Matter '''95''' pp. 143-144 (1994)]</ref>
			<ref>[http://dx.doi.org/10.1063/1.471221 Dominik Marx and Michele Parrinello "Ab initio path integral molecular dynamics: Basic ideas", Journal of Chemical Physics '''104''' pp. 4077 (1996)]</ref>
			<ref>[http://dx.doi.org/10.1103/PhysRevE.93.043305 Christopher John, Thomas Spura, Scott Habershon, and Thomas D. Kühne "Quantum ring-polymer contraction method: Including nuclear quantum effects at no additional computational cost in comparison to ab initio molecular dynamics", Physical Review E '''93''' pp. 043305 (2016)]</ref>
	====Normal mode PIMD====		====Normal mode PIMD====
	====Grand canonical Monte Carlo====		====Grand canonical Monte Carlo====
	A path integral version of the [[Widom test-particle method]] for [[grand canonical Monte Carlo]] simulations:		A path integral version of the [[Widom test-particle method]] for [[grand canonical Monte Carlo]] simulations:
	<ref>[http://dx.doi.org/10.1063/1.474874 Qinyu Wang, J. Karl Johnson and Jeremy Q. Broughton "Path integral grand canonical Monte Carlo", Journal of Chemical Physics '''107''' pp. 5108-5117 (1997)]</ref>		<ref>[http://dx.doi.org/10.1063/1.474874 Qinyu Wang, J. Karl Johnson and Jeremy Q. Broughton "Path integral grand canonical Monte Carlo", Journal of Chemical Physics '''107''' pp. 5108-5117 (1997)]</ref>

	==Applications==		==Applications==
	<ref>[http://dx.doi.org/10.1063/1.470898 Jianshu Cao and Gregory A. Voth "Semiclassical approximations to quantum dynamical time correlation functions", Journal of Chemical Physics '''104''' pp. 273-285 (1996)]</ref>		<ref>[http://dx.doi.org/10.1063/1.470898 Jianshu Cao and Gregory A. Voth "Semiclassical approximations to quantum dynamical time correlation functions", Journal of Chemical Physics '''104''' pp. 273-285 (1996)]</ref>

Carl McBride: Cleaned up Cites

2018-05-08T06:57:33Z

Cleaned up Cites

132.239.72.175: /* External links */

2013-05-01T17:24:03Z

External links

← Older revision		Revision as of 19:24, 1 May 2013
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	==External links==		==External links==
	*[http://www.smac.lps.ens.fr/index.php/Programs_Chapter_3:_Density_matrices_and_path_integrals Density matrices and path integrals] computer code on SMAC-wiki.		*[http://www.smac.lps.ens.fr/index.php/Programs_Chapter_3:_Density_matrices_and_path_integrals Density matrices and path integrals] computer code on SMAC-wiki.
			*[http://www.deleramentum.net/codes/repimd/ A simple implementation of PIMD integrator (C++).]
	[[Category: Monte Carlo]]		[[Category: Monte Carlo]]
	[[category: Quantum mechanics]]		[[category: Quantum mechanics]]

Carl McBride: /* Principles */ Added the variable of integration

2013-04-10T11:09:31Z

Principles: Added the variable of integration

← Older revision		Revision as of 13:09, 10 April 2013
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	:<math>Q(\beta, V)= \int {\mathrm d} x_1 \int_{x_1}^{x_1} Dx(\tau)e^{-S[x(\tau)]}</math>		:<math>Q(\beta, V)= \int {\mathrm d} x_1 \int_{x_1}^{x_1} Dx(\tau)e^{-S[x(\tau)]}</math>
	where <math>S[x(\tau)]</math> is the Euclidean action, given by (<ref name="Berne"> </ref> Eq. 2)		where <math>S[x(\tau)]</math> is the Euclidean action, given by (<ref name="Berne"> </ref> Eq. 2)
	:<math>S[x(\tau)] = \int_0^{\beta \hbar} H(x(\tau))</math>		:<math>S[x(\tau)] = \int_0^{\beta \hbar} H(x(\tau)) ~{\mathrm d}\tau</math>
	where <math>x(\tau)</math> is the path in time <math>\tau</math> and <math>H</math> is the [[Hamiltonian]].		where <math>x(\tau)</math> is the path in time <math>\tau</math> and <math>H</math> is the [[Hamiltonian]].
	This leads to (<ref name="Berne"> </ref> Eq. 3)		This leads to (<ref name="Berne"> </ref> Eq. 3)

Carl McBride: /* Additional reading */ Added a link to the Scholarpedia article

2013-04-10T11:01:40Z

Additional reading: Added a link to the Scholarpedia article

← Older revision		Revision as of 13:01, 10 April 2013
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	*[http://dx.doi.org/10.1103/RevModPhys.67.279 D. M. Ceperley "Path integrals in the theory of condensed helium", Reviews of Modern Physics '''67''' 279 - 355 (1995)]		*[http://dx.doi.org/10.1103/RevModPhys.67.279 D. M. Ceperley "Path integrals in the theory of condensed helium", Reviews of Modern Physics '''67''' 279 - 355 (1995)]
	*[http://dx.doi.org/10.1080/014423597230190 Charusita Chakravarty "Path integral simulations of atomic and molecular systems", International Reviews in Physical Chemistry '''16''' pp. 421-444 (1997)]		*[http://dx.doi.org/10.1080/014423597230190 Charusita Chakravarty "Path integral simulations of atomic and molecular systems", International Reviews in Physical Chemistry '''16''' pp. 421-444 (1997)]
			*[http://www.scholarpedia.org/article/Path_integral Jean Zinn-Justin "Path integral" Scholarpedia, 4(2):8674 (2009)]

	==External links==		==External links==

Carl McBride: Tidy of references

2012-05-29T12:15:41Z

Tidy of references

Carl McBride: /* Principles */

2012-05-29T10:43:40Z

Principles

← Older revision		Revision as of 12:43, 29 May 2012
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	where <math>P</math> is the Trotter number. In the Trotter limit, where <math>P \rightarrow \infty</math> these equations become exact. In the case where <math>P=1</math> these equations revert to a classical simulation. It has long been recognised that there is an isomorphism between this discretised quantum mechanical description, and the classical [[statistical mechanics]] of polyatomic fluids, in particular flexible ring molecules<ref>[http://dx.doi.org/10.1063/1.441588 David Chandler and Peter G. Wolynes "Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids", Journal of Chemical Physics '''74''' pp. 4078-4095 (1981)]</ref>, due to the periodic boundary conditions in imaginary time. It can be seen from the first term of the above equation that each particle <math>x_t</math> interacts with is neighbours <math>x_{t-1}</math> and <math>x_{t+1}</math> via a harmonic spring. The second term provides the internal potential energy.		where <math>P</math> is the Trotter number. In the Trotter limit, where <math>P \rightarrow \infty</math> these equations become exact. In the case where <math>P=1</math> these equations revert to a classical simulation. It has long been recognised that there is an isomorphism between this discretised quantum mechanical description, and the classical [[statistical mechanics]] of polyatomic fluids, in particular flexible ring molecules<ref>[http://dx.doi.org/10.1063/1.441588 David Chandler and Peter G. Wolynes "Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids", Journal of Chemical Physics '''74''' pp. 4078-4095 (1981)]</ref>, due to the periodic boundary conditions in imaginary time. It can be seen from the first term of the above equation that each particle <math>x_t</math> interacts with is neighbours <math>x_{t-1}</math> and <math>x_{t+1}</math> via a harmonic spring. The second term provides the internal potential energy.

	The following is a schematic for the interaction between atom <math>i</math> (green) and atom <math>j</math> (orange). Here we show the atoms having five Trotter slices (<math>P=5</math>), forming what can be thought of as a "ring polymer molecule". The harmonic springs between Trotter slices are in yellow, and white/blue bonds represent the [[intermolecular pair potential]].		The following is a schematic for the interaction between atom <math>i</math> (green) and atom <math>j</math> (orange). Here we show the atoms having five Trotter slices (<math>P=5</math>), forming what can be thought of as a "ring polymer molecule". The harmonic springs between Trotter slices are in yellow, and white/blue bonds represent the classical [[intermolecular pair potential]].

	:{\| border="1"		:{\| border="1"

Carl McBride: /* Principles */ Added an image for the classical limit of P=1

2012-05-29T10:31:08Z

Principles: Added an image for the classical limit of P=1

← Older revision		Revision as of 12:31, 29 May 2012
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	(<ref name="Berne">[http://dx.doi.org/10.1146/annurev.pc.37.100186.002153 B. J. Berne and D. Thirumalai "On the Simulation of Quantum Systems: Path Integral Methods", Annual Review of Physical Chemistry '''37''' pp. 401-424 (1986)]</ref> Eq. 1)		(<ref name="Berne">[http://dx.doi.org/10.1146/annurev.pc.37.100186.002153 B. J. Berne and D. Thirumalai "On the Simulation of Quantum Systems: Path Integral Methods", Annual Review of Physical Chemistry '''37''' pp. 401-424 (1986)]</ref> Eq. 1)
	:<math>Q(\beta, V)= \int {\mathrm d} x_1 \int_{x_1}^{x_1} Dx(\tau)e^{-S[x(\tau)]}</math>		:<math>Q(\beta, V)= \int {\mathrm d} x_1 \int_{x_1}^{x_1} Dx(\tau)e^{-S[x(\tau)]}</math>
	where <math>S[x(\tau)]</math> is the ~~Euclidian~~ action, given by (<ref name="Berne"> </ref> Eq. 2)		where <math>S[x(\tau)]</math> is the Euclidean action, given by (<ref name="Berne"> </ref> Eq. 2)
	:<math>S[x(\tau)] = \int_0^{\beta \hbar} H(x(\tau))</math>		:<math>S[x(\tau)] = \int_0^{\beta \hbar} H(x(\tau))</math>
	where <math>x(\tau)</math> is the path in time <math>\tau</math> and <math>H</math> is the [[Hamiltonian]].		where <math>x(\tau)</math> is the path in time <math>\tau</math> and <math>H</math> is the [[Hamiltonian]].
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	:<math>x_{P+1}=x_1</math>		:<math>x_{P+1}=x_1</math>
	and (<ref name="Berne"> </ref> Eq. 4)		and (<ref name="Berne"> </ref> Eq. 4)
	:<math>\Phi_P (x_1...x_P;\beta)= \frac{mP}{2\beta^2 \hbar^2} \sum_{t=1}^P (x_t - x_{t+1})^2 + \frac{1}{P} \sum_{t=1}^P V(x_t)</math>.		:<math>\Phi_P (x_1...x_P;\beta)= \frac{mP}{2\beta^2 \hbar^2} \sum_{t=1}^P (x_t - x_{t+1})^2 + \frac{1}{P} \sum_{t=1}^P V(x_t)</math>

	where <math>P</math> is the Trotter number. In the Trotter limit, where <math>P \rightarrow \infty</math> these equations become exact. In the case where <math>P=1</math> these equations revert to a classical simulation. It has long been recognised that there is an isomorphism between this discretised quantum mechanical description, and the classical [[statistical mechanics]] of polyatomic fluids, in particular flexible ring molecules<ref>[http://dx.doi.org/10.1063/1.441588 David Chandler and Peter G. Wolynes "Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids", Journal of Chemical Physics '''74''' pp. 4078-4095 (1981)]</ref>, due to the periodic boundary conditions in imaginary time. It can be seen from the first term of the above equation that each particle <math>x_t</math> interacts with is neighbours <math>x_{t-1}</math> and <math>x_{t+1}</math> via a harmonic spring. The second term provides the internal potential energy.		where <math>P</math> is the Trotter number. In the Trotter limit, where <math>P \rightarrow \infty</math> these equations become exact. In the case where <math>P=1</math> these equations revert to a classical simulation. It has long been recognised that there is an isomorphism between this discretised quantum mechanical description, and the classical [[statistical mechanics]] of polyatomic fluids, in particular flexible ring molecules<ref>[http://dx.doi.org/10.1063/1.441588 David Chandler and Peter G. Wolynes "Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids", Journal of Chemical Physics '''74''' pp. 4078-4095 (1981)]</ref>, due to the periodic boundary conditions in imaginary time. It can be seen from the first term of the above equation that each particle <math>x_t</math> interacts with is neighbours <math>x_{t-1}</math> and <math>x_{t+1}</math> via a harmonic spring. The second term provides the internal potential energy.
	The following is a schematic for the interaction between atom <math>i</math> (green) and atom <math>j</math> (orange). Here we show the atoms having five Trotter slices, forming what can be thought of as a "ring polymer molecule". The harmonic springs between Trotter slices are in yellow, and white/blue bonds represent the [[intermolecular pair potential]].~~<br>~~
	[[Image:5bead_pathIntegra_ij.png\|~~500px~~]]~~<br>~~		The following is a schematic for the interaction between atom <math>i</math> (green) and atom <math>j</math> (orange). Here we show the atoms having five Trotter slices (<math>P=5</math>), forming what can be thought of as a "ring polymer molecule". The harmonic springs between Trotter slices are in yellow, and white/blue bonds represent the [[intermolecular pair potential]].

			:{\| border="1"
			\| [[Image:1bead_classical_ij.png\|center\|400px]]Classical limit (P=1) \|\| [[Image:5bead_pathIntegra_ij.png\|center\|400px]] Path integral (here with P=5)
			\|}

	In three dimensions one has the ''density operator''		In three dimensions one has the ''density operator''

Carl McBride: /* Principles */ Updated figure

2012-05-29T09:52:26Z

Principles: Updated figure

← Older revision		Revision as of 11:52, 29 May 2012
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	where <math>P</math> is the Trotter number. In the Trotter limit, where <math>P \rightarrow \infty</math> these equations become exact. In the case where <math>P=1</math> these equations revert to a classical simulation. It has long been recognised that there is an isomorphism between this discretised quantum mechanical description, and the classical [[statistical mechanics]] of polyatomic fluids, in particular flexible ring molecules<ref>[http://dx.doi.org/10.1063/1.441588 David Chandler and Peter G. Wolynes "Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids", Journal of Chemical Physics '''74''' pp. 4078-4095 (1981)]</ref>, due to the periodic boundary conditions in imaginary time. It can be seen from the first term of the above equation that each particle <math>x_t</math> interacts with is neighbours <math>x_{t-1}</math> and <math>x_{t+1}</math> via a harmonic spring. The second term provides the internal potential energy.		where <math>P</math> is the Trotter number. In the Trotter limit, where <math>P \rightarrow \infty</math> these equations become exact. In the case where <math>P=1</math> these equations revert to a classical simulation. It has long been recognised that there is an isomorphism between this discretised quantum mechanical description, and the classical [[statistical mechanics]] of polyatomic fluids, in particular flexible ring molecules<ref>[http://dx.doi.org/10.1063/1.441588 David Chandler and Peter G. Wolynes "Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids", Journal of Chemical Physics '''74''' pp. 4078-4095 (1981)]</ref>, due to the periodic boundary conditions in imaginary time. It can be seen from the first term of the above equation that each particle <math>x_t</math> interacts with is neighbours <math>x_{t-1}</math> and <math>x_{t+1}</math> via a harmonic spring. The second term provides the internal potential energy.
	The following is a schematic for the interaction between ~~two~~ (~~purple~~) ~~atoms~~. Here we show the atoms having five Trotter slices, forming what can be thought of as a "ring polymer molecule". The harmonic springs between Trotter slices are in ~~red~~, and white/~~green~~ bonds represent the [[intermolecular pair potential]].<br>		The following is a schematic for the interaction between atom <math>i</math> (green) and atom <math>j</math> (orange). Here we show the atoms having five Trotter slices, forming what can be thought of as a "ring polymer molecule". The harmonic springs between Trotter slices are in yellow, and white/blue bonds represent the [[intermolecular pair potential]].<br>
	[[Image:~~5bead_pathIntegral~~.png\|500px]]<br>		[[Image:5bead_pathIntegra_ij.png\|500px]]<br>
	In three dimensions one has the ''density operator''		In three dimensions one has the ''density operator''

Carl McBride: Undo revision 11385 by RamiroRoberts (talk)

2011-05-12T10:01:42Z

Undo revision 11385 by RamiroRoberts (talk)

← Older revision		Revision as of 12:01, 12 May 2011
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	[[Category: Monte Carlo]]		[[Category: Monte Carlo]]
	[[category: Quantum mechanics]]		[[category: Quantum mechanics]]
	~~[http://customessays.ws/ Custom essays]~~

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	(<ref name="Berne">[http://dx.doi.org/10.1146/annurev.pc.37.100186.002153 B. J. Berne and D. Thirumalai "On the Simulation of Quantum Systems: Path Integral Methods", Annual Review of Physical Chemistry '''37''' pp. 401-424 (1986)]</ref> Eq. 1)		(<ref name="Berne">[http://dx.doi.org/10.1146/annurev.pc.37.100186.002153 B. J. Berne and D. Thirumalai "On the Simulation of Quantum Systems: Path Integral Methods", Annual Review of Physical Chemistry '''37''' pp. 401-424 (1986)]</ref> Eq. 1)
	:<math>Q(\beta, V)= \int {\mathrm d} x_1 \int_{x_1}^{x_1} Dx(\tau)e^{-S[x(\tau)]}</math>		:<math>Q(\beta, V)= \int {\mathrm d} x_1 \int_{x_1}^{x_1} Dx(\tau)e^{-S[x(\tau)]}</math>
	where <math>S[x(\tau)]</math> is the Euclidean action, given by (<ref name="Berne"> </ref> Eq. 2)		where <math>S[x(\tau)]</math> is the Euclidean action, given by (<ref name="Berne"></ref> Eq. 2)
	:<math>S[x(\tau)] = \int_0^{\beta \hbar} H(x(\tau)) ~{\mathrm d}\tau</math>		:<math>S[x(\tau)] = \int_0^{\beta \hbar} H(x(\tau)) ~{\mathrm d}\tau</math>
	where <math>x(\tau)</math> is the path in time <math>\tau</math> and <math>H</math> is the [[Hamiltonian]].		where <math>x(\tau)</math> is the path in time <math>\tau</math> and <math>H</math> is the [[Hamiltonian]].
	This leads to (<ref name="Berne"> </ref> Eq. 3)		This leads to (<ref name="Berne"></ref> Eq. 3)
	:<math>Q_P = \left( \frac{mP}{2 \pi \beta \hbar^2} \right)^{P/2} \int ... \int {\mathrm d}x_1... {\mathrm d}x_P e^{-\beta \Phi_P (x_1...x_P;\beta)}</math>		:<math>Q_P = \left( \frac{mP}{2 \pi \beta \hbar^2} \right)^{P/2} \int ... \int {\mathrm d}x_1... {\mathrm d}x_P e^{-\beta \Phi_P (x_1...x_P;\beta)}</math>
	where the Euclidean time is discretised in units of		where the Euclidean time is discretised in units of
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	:<math>x_t = x(t \beta \hbar/P)</math>		:<math>x_t = x(t \beta \hbar/P)</math>
	:<math>x_{P+1}=x_1</math>		:<math>x_{P+1}=x_1</math>
	and (<ref name="Berne"> </ref> Eq. 4)		and (<ref name="Berne"></ref> Eq. 4)
	:<math>\Phi_P (x_1...x_P;\beta)= \frac{mP}{2\beta^2 \hbar^2} \sum_{t=1}^P (x_t - x_{t+1})^2 + \frac{1}{P} \sum_{t=1}^P V(x_t)</math>		:<math>\Phi_P (x_1...x_P;\beta)= \frac{mP}{2\beta^2 \hbar^2} \sum_{t=1}^P (x_t - x_{t+1})^2 + \frac{1}{P} \sum_{t=1}^P V(x_t)</math>

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	:<math>\hat{H} = \hat{T}^{\mathrm {translational}} + \hat{T}^{\mathrm {rotational}}+ \hat{V}</math>		:<math>\hat{H} = \hat{T}^{\mathrm {translational}} + \hat{T}^{\mathrm {rotational}}+ \hat{V}</math>

	where the rotational part of the kinetic energy operator is given by (<ref name="Marx"> </ref> Eq. 2.2)		where the rotational part of the kinetic energy operator is given by (<ref name="Marx"></ref> Eq. 2.2)

	:<math>T^{\mathrm {rotational}} = \sum_{i=1}^{d^{\mathrm {rotational}}} \frac{\hat{L}_i^2}{2\Theta_{ii}}</math>		:<math>T^{\mathrm {rotational}} = \sum_{i=1}^{d^{\mathrm {rotational}}} \frac{\hat{L}_i^2}{2\Theta_{ii}}</math>

← Older revision		Revision as of 14:15, 29 May 2012
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	:<math>\rho(x',x)= \sqrt{ \frac{m \omega}{2 \pi \hbar \sinh \omega \beta \hbar} } \exp \left( - \frac{m \omega}{2 \hbar (\sinh \omega \beta \hbar)^2 } \left( (x^2 + x'^2 ) \cosh \omega \beta \hbar - 2xx'\right)\right)</math>		:<math>\rho(x',x)= \sqrt{ \frac{m \omega}{2 \pi \hbar \sinh \omega \beta \hbar} } \exp \left( - \frac{m \omega}{2 \hbar (\sinh \omega \beta \hbar)^2 } \left( (x^2 + x'^2 ) \cosh \omega \beta \hbar - 2xx'\right)\right)</math>

	~~'''Related reading'''~~		See also refs
	*[http://dx.doi.org/10.1119/1.18910 Barry R. Holstein "The harmonic oscillator propagator", American Journal of Physics '''66''' pp. 583-589 (1998)]		<ref>[http://dx.doi.org/10.1119/1.18910 Barry R. Holstein "The harmonic oscillator propagator", American Journal of Physics '''66''' pp. 583-589 (1998)]</ref>
	*[http://dx.doi.org/10.1119/1.1715108 L. Moriconi "An elementary derivation of the harmonic oscillator propagator", American Journal of Physics '''72''' pp. 1258-1259 (2004)]		<ref>[http://dx.doi.org/10.1119/1.1715108 L. Moriconi "An elementary derivation of the harmonic oscillator propagator", American Journal of Physics '''72''' pp. 1258-1259 (2004)]</ref>
	==Wick rotation and imaginary time==		==Wick rotation and imaginary time==
	One can identify the [[Temperature#Inverse_temperature \| inverse temperature]], <math>\beta</math> with an imaginary time <math>it/\hbar</math> (see <ref>M. J. Gillan "The path-integral simulation of quantum systems" in "Computer Modelling of Fluids Polymers and Solids" eds. C. R. A. Catlow, S. C. Parker and M. P. Allen, NATO ASI Series C '''293''' pp. 155-188 (1990) ISBN 978-0-7923-0549-1</ref> § 2.4).		Wick rotation <ref>[http://dx.doi.org/10.1103/PhysRev.96.1124 G. C. Wick "Properties of Bethe-Salpeter Wave Functions", Physical Review '''96''' pp. 1124-1134 (1954)]</ref>. One can identify the [[Temperature#Inverse_temperature \| inverse temperature]], <math>\beta</math> with an imaginary time <math>it/\hbar</math> (see <ref>M. J. Gillan "The path-integral simulation of quantum systems" in "Computer Modelling of Fluids Polymers and Solids" eds. C. R. A. Catlow, S. C. Parker and M. P. Allen, NATO ASI Series C '''293''' pp. 155-188 (1990) ISBN 978-0-7923-0549-1</ref> § 2.4).
	*[http://dx.doi.org/10.1103/PhysRev.96.1124 G. C. Wick "Properties of Bethe-Salpeter Wave Functions", Physical Review '''96''' pp. 1124-1134 (1954)]

	==Rotational degrees of freedom==		==Rotational degrees of freedom==
	In the case of systems having (<math>d</math>) rotational [[degree of freedom \| degrees of freedom]] the [[Hamiltonian]] can be written in the form (<ref name="Marx">[http://dx.doi.org/10.1088/0953-8984/11/11/003 Dominik Marx and Martin H Müser "Path integral simulations of rotors: theory and applications", Journal of Physics: Condensed Matter '''11''' pp. R117-R155 (1999)]</ref> Eq. 2.1):		In the case of systems having (<math>d</math>) rotational [[degree of freedom \| degrees of freedom]] the [[Hamiltonian]] can be written in the form (<ref name="Marx">[http://dx.doi.org/10.1088/0953-8984/11/11/003 Dominik Marx and Martin H Müser "Path integral simulations of rotors: theory and applications", Journal of Physics: Condensed Matter '''11''' pp. R117-R155 (1999)]</ref> Eq. 2.1):
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	====Path integral Monte Carlo====		====Path integral Monte Carlo====
	Path integral Monte Carlo (PIMC)		Path integral Monte Carlo (PIMC)
	*[http://dx.doi.org/10.1063/1.437829 J. A. Barker "A quantum-statistical Monte Carlo method; path integrals with boundary conditions", Journal of Chemical Physics '''70''' pp. 2914- (1979)]		<ref>[http://dx.doi.org/10.1063/1.437829 J. A. Barker "A quantum-statistical Monte Carlo method; path integrals with boundary conditions", Journal of Chemical Physics '''70''' pp. 2914- (1979)]</ref>
	====Path integral molecular dynamics====		====Path integral molecular dynamics====
	Path integral molecular dynamics (PIMD)		Path integral molecular dynamics (PIMD)
	*[http://dx.doi.org/10.1063/1.446740 M. Parrinello and A. Rahman "Study of an F center in molten KCl", Journal of Chemical Physics '''80''' pp. 860- (1984)]		<ref>[http://dx.doi.org/10.1063/1.446740 M. Parrinello and A. Rahman "Study of an F center in molten KCl", Journal of Chemical Physics '''80''' pp. 860- (1984)]</ref>
	====Centroid molecular dynamics====		====Centroid molecular dynamics====
	Centroid molecular dynamics (CMD)		Centroid molecular dynamics (CMD)
	*[http://dx.doi.org/10.1063/1.467175 Jianshu Cao and Gregory A. Voth "The formulation of quantum statistical mechanics based on the Feynman path centroid density. I. Equilibrium properties", Journal of Chemical Physics '''100''' pp. 5093-5105 (1994)]		<ref>[http://dx.doi.org/10.1063/1.467175 Jianshu Cao and Gregory A. Voth "The formulation of quantum statistical mechanics based on the Feynman path centroid density. I. Equilibrium properties", Journal of Chemical Physics '''100''' pp. 5093-5105 (1994)]</ref>
	*[http://dx.doi.org/10.1063/1.467176 Jianshu Cao and Gregory A. Voth "The formulation of quantum statistical mechanics based on the Feynman path centroid density. II. Dynamical properties", Journal of Chemical Physics '''100''' pp. 5106- (1994)]		<ref>[http://dx.doi.org/10.1063/1.467176 Jianshu Cao and Gregory A. Voth "The formulation of quantum statistical mechanics based on the Feynman path centroid density. II. Dynamical properties", Journal of Chemical Physics '''100''' pp. 5106- (1994)]</ref>
	*[http://dx.doi.org/10.1063/1.479515 Seogjoo Jang and Gregory A. Voth "A derivation of centroid molecular dynamics and other approximate time evolution methods for path integral centroid variables", Journal of Chemical Physics '''111''' pp. 2371- (1999)]		<ref>[http://dx.doi.org/10.1063/1.479515 Seogjoo Jang and Gregory A. Voth "A derivation of centroid molecular dynamics and other approximate time evolution methods for path integral centroid variables", Journal of Chemical Physics '''111''' pp. 2371- (1999)]</ref>
	*[http://dx.doi.org/10.1063/1.479666 Rafael Ramírez and Telesforo López-Ciudad "The Schrödinger formulation of the Feynman path centroid density", Journal of Chemical Physics '''111''' pp. 3339-3348 (1999)]		<ref>[http://dx.doi.org/10.1063/1.479666 Rafael Ramírez and Telesforo López-Ciudad "The Schrödinger formulation of the Feynman path centroid density", Journal of Chemical Physics '''111''' pp. 3339-3348 (1999)]</ref>
	*[http://dx.doi.org/10.1063/1.3484490 E. A. Polyakov, A. P. Lyubartsev, and P. N. Vorontsov-Velyaminov "Centroid molecular dynamics: Comparison with exact results for model systems", Journal of Chemical Physics '''133''' 194103 (2010)]		<ref>[http://dx.doi.org/10.1063/1.3484490 E. A. Polyakov, A. P. Lyubartsev, and P. N. Vorontsov-Velyaminov "Centroid molecular dynamics: Comparison with exact results for model systems", Journal of Chemical Physics '''133''' 194103 (2010)]</ref>

	====Ring polymer molecular dynamics====		====Ring polymer molecular dynamics====
	Ring polymer molecular dynamics (RPMD)		Ring polymer molecular dynamics (RPMD)
	*[http://dx.doi.org/10.1063/1.1777575 Ian R. Craig and David E. Manolopoulos "Quantum statistics and classical mechanics: Real time correlation functions from ring polymer molecular dynamics", Journal of Chemical Physics '''121''' pp. 3368- (2004)]		<ref>[http://dx.doi.org/10.1063/1.1777575 Ian R. Craig and David E. Manolopoulos "Quantum statistics and classical mechanics: Real time correlation functions from ring polymer molecular dynamics", Journal of Chemical Physics '''121''' pp. 3368- (2004)]</ref>
	*[http://dx.doi.org/10.1063/1.2357599 Bastiaan J. Braams and David E. Manolopoulos "On the short-time limit of ring polymer molecular dynamics", Journal of Chemical Physics '''125''' 124105 (2006)]		<ref>[http://dx.doi.org/10.1063/1.2357599 Bastiaan J. Braams and David E. Manolopoulos "On the short-time limit of ring polymer molecular dynamics", Journal of Chemical Physics '''125''' 124105 (2006)]</ref>
	~~'''~~Contraction scheme~~'''~~		(Contraction scheme
	*[http://dx.doi.org/10.1063/1.2953308 Thomas E. Markland and David E. Manolopoulos "An efficient ring polymer contraction scheme for imaginary time path integral simulations", Journal of Chemical Physics '''129''' 024105 (2008)]		<ref>[http://dx.doi.org/10.1063/1.2953308 Thomas E. Markland and David E. Manolopoulos "An efficient ring polymer contraction scheme for imaginary time path integral simulations", Journal of Chemical Physics '''129''' 024105 (2008)]</ref>
	*[http://dx.doi.org/10.1016/j.cplett.2008.09.019 Thomas E. Markland and David E. Manolopoulos "A refined ring polymer contraction scheme for systems with electrostatic interactions" Chemical Physics Letters '''464''' pp. 256-261 (2008)]		<ref>[http://dx.doi.org/10.1016/j.cplett.2008.09.019 Thomas E. Markland and David E. Manolopoulos "A refined ring polymer contraction scheme for systems with electrostatic interactions" Chemical Physics Letters '''464''' pp. 256-261 (2008)]</ref>)

	====Normal mode PIMD====		====Normal mode PIMD====

	====Grand canonical Monte Carlo====		====Grand canonical Monte Carlo====
	A path integral version of the [[Widom test-particle method]] for [[grand canonical Monte Carlo]] simulations:		A path integral version of the [[Widom test-particle method]] for [[grand canonical Monte Carlo]] simulations:
	*[http://dx.doi.org/10.1063/1.474874 Qinyu Wang, J. Karl Johnson and Jeremy Q. Broughton "Path integral grand canonical Monte Carlo", Journal of Chemical Physics '''107''' pp. 5108-5117 (1997)]		<ref>[http://dx.doi.org/10.1063/1.474874 Qinyu Wang, J. Karl Johnson and Jeremy Q. Broughton "Path integral grand canonical Monte Carlo", Journal of Chemical Physics '''107''' pp. 5108-5117 (1997)]</ref>

	==Applications==		==Applications==
	*[http://dx.doi.org/10.1063/1.470898 Jianshu Cao and Gregory A. Voth "Semiclassical approximations to quantum dynamical time correlation functions", Journal of Chemical Physics '''104''' pp. 273-285 (1996)]		<ref>[http://dx.doi.org/10.1063/1.470898 Jianshu Cao and Gregory A. Voth "Semiclassical approximations to quantum dynamical time correlation functions", Journal of Chemical Physics '''104''' pp. 273-285 (1996)]</ref>
	*[http://dx.doi.org/10.1063/1.1316105 C. Chakravarty and R. M. Lynden-Bell "Landau free energy curves for melting of quantum solids", Journal of Chemical Physics '''113''' pp. 9239-9247 (2000)]		<ref>[http://dx.doi.org/10.1063/1.1316105 C. Chakravarty and R. M. Lynden-Bell "Landau free energy curves for melting of quantum solids", Journal of Chemical Physics '''113''' pp. 9239-9247 (2000)]</ref>
	==References==		==References==
	<references/>		<references/>