Editing Path integral formulation
Jump to navigation
Jump to search
The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.
Latest revision | Your text | ||
Line 1: | Line 1: | ||
The ''' | The '''Path integral formulation''' is an elegant method by which [[quantum mechanics | quantum mechanical]] contributions can be incorporated | ||
within a classical | within a classical simulation using Feynman path integrals (Refs. 1-7). 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]]. | ||
==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. 4 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. 4 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 ( | This leads to (Ref. 4 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 | ||
Line 14: | Line 13: | ||
:<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 ( | and (Ref. 4 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 | 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. 3), 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> | ||
Line 44: | Line 35: | ||
:<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 ( | The density matrix for a harmonic oscillator is given by (Ref. 1 Eq. 10-44) | ||
:<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== | ||
==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 ( | 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. 8 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 ( | where the rotational part of the kinetic energy operator is given by (Ref. 8 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. 9 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)] | |||
*[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)] | |||
====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== | ||
#R. P. Feynman and A. R. Hibbs "Path-integrals and Quantum Mechanics", McGraw-Hill, New York (1965) ISBN 0-07-020650-3 | |||
#R. P. Feynman "Statistical Mechanics", Benjamin, Reading, Massachusetts, (1972) ISBN 0-201-36076-4 Chapter 3. | |||
#[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)] | |||
#[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)] | |||
#[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)] | |||
# 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 | |||
#[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)] | |||
#[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)] | |||
==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]] |