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The '''path integral formulation''', here from the [[statistical mechanics | statistical mechanical]] point of view, is an elegant method by which [[quantum mechanics | quantum mechanical]] contributions can be incorporated | The '''path integral formulation''', here from the [[statistical mechanics | statistical mechanical]] point of view, is an elegant method by which [[quantum mechanics | quantum mechanical]] contributions can be incorporated | ||
within a classical [[Computer simulation techniques |simulation]] using Feynman path integrals (see the [[Path integral formulation#Additional reading|additional reading ]] section). Such simulations are particularly applicable to light atoms and molecules such as [[hydrogen]], [[helium]], [[neon]] and [[argon]], as well as quantum rotators such as [[methane]] and | within a classical [[Computer simulation techniques |simulation]] using Feynman path integrals (see the [[Path integral formulation#Additional reading|additional reading ]] section). Such simulations are particularly applicable to light atoms and molecules such as [[hydrogen]], [[helium]], [[neon]] and [[argon]], as well as quantum rotators such as [[methane]] and [[water]]. From a more idealised point of view path integrals are often used to study [[quantum hard spheres]]. | ||
==Principles== | ==Principles== | ||
In the path integral formulation the canonical [[partition function]] (in one dimension) is written as | In the path integral formulation the canonical [[partition function]] (in one dimension) is written as | ||
(<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 | where <math>S[x(\tau)]</math> is the Euclidian action, given by (<ref name="Berne"> </ref> Eq. 2) | ||
:<math>S[x(\tau)] = \int_0^{\beta \hbar} H(x(\tau)) | :<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]]. | ||
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>. | ||
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. Thus in three dimensions one has the ''density operator'' | ||
:<math>\hat{\rho} (\beta) = \exp\left[ -\beta \hat{H} \right]</math> | :<math>\hat{\rho} (\beta) = \exp\left[ -\beta \hat{H} \right]</math> | ||
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:<math>\langle K_P \rangle = \frac{3NP}{2\beta}- \langle U_{\mathrm {spring}} \rangle </math> | :<math>\langle K_P \rangle = \frac{3NP}{2\beta}- \langle U_{\mathrm {spring}} \rangle </math> | ||
==Harmonic oscillator== | ==Harmonic oscillator== | ||
The density matrix for a harmonic oscillator is given by (<ref>R. P. Feynman and A. R. Hibbs "Path-integrals and Quantum Mechanics", McGraw-Hill, New York (1965) ISBN 0-07-020650-3</ref> Eq. 10-44) | The density matrix for a harmonic oscillator is given by (<ref>R. P. Feynman and A. R. Hibbs "Path-integrals and Quantum Mechanics", McGraw-Hill, New York (1965) ISBN 0-07-020650-3</ref> Eq. 10-44) | ||
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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''' | |||
*[http://dx.doi.org/10.1119/1.18910 Barry R. Holstein "The harmonic oscillator propagator", American Journal of Physics '''66''' pp. 583-589 (1998)] | |||
*[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)] | |||
==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). | |||
*[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): | ||
:<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> | ||
where <math>\hat{L}_i</math> are the components of the angular momentum operator, and <math>\Theta_{ii}</math> are the moments of inertia. | where <math>\hat{L}_i</math> are the components of the angular momentum operator, and <math>\Theta_{ii}</math> are the moments of inertia. For a rigid three dimensional asymmetric top the kernel is given by (<ref>[http://dx.doi.org/10.1103/PhysRevLett.77.2638 M. H. Müser and B. J. Berne "Path-Integral Monte Carlo Scheme for Rigid Tops: Application to the Quantum Rotator Phase Transition in Solid Methane", Physical Review Letters '''77''' pp. 2638-2641 (1996)]</ref> Eq. 5): | ||
:'' | :<math>\rho(\omega,\omega'; \beta/P) = \sum_{JM\tilde{K}} \left( \frac{2J+1}{8\pi^2}\right) d_{MM}^J (\tilde{\theta'} ) | ||
\cos \left[ M(\tilde{\phi}' + \tilde{\chi}') \right] \left| A_{\tilde{K}M}^{JM} \right|^2 \exp \left( - \frac{\beta}{P} E_{\tilde{K}}^{JM} \right)</math> | |||
where <math>\omega</math> are the [[Euler angles]], <math>d_{MM}^J </math> is the [[Wigner D-matrix]] and <math>E_{\tilde{K}}^{JM}</math> are the eigenenergies. | |||
==Techniques== | |||
====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)] | |||
====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)] | |||
====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)] | |||
*[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)] | |||
*[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)] | |||
*[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)] | |||
====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)] | |||
*[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)] | |||
'''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)] | |||
*[http://dx.doi.org/10.1016/j.cplett.2008.09.019 Thomas E. Markland, David E. Manolopoulos "A refined ring polymer contraction scheme for systems with electrostatic interactions" Chemical Physics Letters '''464''' pp. 256-261 (2008)] | |||
====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)] | |||
==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)] | |||
*[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)] | |||
==References== | ==References== | ||
<references/> | <references/> | ||
==Additional reading== | ==Additional reading== | ||
*R. P. Feynman "Statistical Mechanics", Benjamin, Reading, Massachusetts, (1972) ISBN 0-201-36076-4 Chapter 3. | *R. P. Feynman "Statistical Mechanics", Benjamin, Reading, Massachusetts, (1972) ISBN 0-201-36076-4 Chapter 3. | ||
*[http://dx.doi.org/10.1016/0370-1573(75)90030-7 F. W. Wiegel "Path integral methods in statistical mechanics", Physics Reports '''16''' pp. 57-114 (1975)] | *[http://dx.doi.org/10.1016/0370-1573(75)90030-7 F. W. Wiegel "Path integral methods in statistical mechanics", Physics Reports '''16''' pp. 57-114 (1975)] | ||
*[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)] | ||
==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. | ||
[[Category: Monte Carlo]] | [[Category: Monte Carlo]] | ||
[[category: Quantum mechanics]] | [[category: Quantum mechanics]] |