Editing Path integral formulation
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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)) | :<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> | ||
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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> | ||
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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>) | ||
====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> | ||
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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. | ||
[[Category: Monte Carlo]] | [[Category: Monte Carlo]] | ||
[[category: Quantum mechanics]] | [[category: Quantum mechanics]] |