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Path integral formulation - Revision history
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95.116.127.139: /* Computer simulation techniques */
2020-09-07T20:04:32Z
<p><span dir="auto"><span class="autocomment">Computer simulation techniques</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:04, 7 September 2020</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l88">Line 88:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><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></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><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></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><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>)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><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>)</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">====Car-Parrinello path-integral molecular dynamics====</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">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></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><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></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><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></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Normal mode PIMD====</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Normal mode PIMD====</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Grand canonical Monte Carlo====</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Grand canonical Monte Carlo====</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>A path integral version of the [[Widom test-particle method]] for [[grand canonical Monte Carlo]] simulations:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>A path integral version of the [[Widom test-particle method]] for [[grand canonical Monte Carlo]] simulations:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><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></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><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></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Applications==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Applications==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><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></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><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></div></td></tr>
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95.116.127.139
http://www.sklogwiki.org/SklogWiki/index.php?title=Path_integral_formulation&diff=20065&oldid=prev
Carl McBride: Cleaned up Cites
2018-05-08T06:57:33Z
<p>Cleaned up Cites</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 07:57, 8 May 2018</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l5">Line 5:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>(<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)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>(<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)</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>Q(\beta, V)= \int {\mathrm d} x_1 \int_{x_1}^{x_1} Dx(\tau)e^{-S[x(\tau)]}</math> </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>Q(\beta, V)= \int {\mathrm d} x_1 \int_{x_1}^{x_1} Dx(\tau)e^{-S[x(\tau)]}</math> </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where <math>S[x(\tau)]</math> is the Euclidean action, given by (<ref name="Berne"> </ref> Eq. 2)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where <math>S[x(\tau)]</math> is the Euclidean action, given by (<ref name="Berne"></ref> Eq. 2)</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>S[x(\tau)] = \int_0^{\beta \hbar} H(x(\tau)) ~{\mathrm d}\tau</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>S[x(\tau)] = \int_0^{\beta \hbar} H(x(\tau)) ~{\mathrm d}\tau</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where <math>x(\tau)</math> is the path in time <math>\tau</math> and <math>H</math> is the [[Hamiltonian]].</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where <math>x(\tau)</math> is the path in time <math>\tau</math> and <math>H</math> is the [[Hamiltonian]].</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>This leads to (<ref name="Berne"> </ref> Eq. 3)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>This leads to (<ref name="Berne"></ref> Eq. 3)</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<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></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<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></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where the Euclidean time is discretised in units of </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where the Euclidean time is discretised in units of </div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l14">Line 14:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>x_t = x(t \beta \hbar/P)</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>x_t = x(t \beta \hbar/P)</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>x_{P+1}=x_1</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>x_{P+1}=x_1</math></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>and (<ref name="Berne"> </ref> Eq. 4)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>and (<ref name="Berne"></ref> Eq. 4)</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<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></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<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></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l59">Line 59:</td>
<td colspan="2" class="diff-lineno">Line 59:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>\hat{H} = \hat{T}^{\mathrm {translational}} + \hat{T}^{\mathrm {rotational}}+ \hat{V}</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>\hat{H} = \hat{T}^{\mathrm {translational}} + \hat{T}^{\mathrm {rotational}}+ \hat{V}</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where the rotational part of the kinetic energy operator is given by (<ref name="Marx"> </ref> Eq. 2.2)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where the rotational part of the kinetic energy operator is given by (<ref name="Marx"></ref> Eq. 2.2)</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>T^{\mathrm {rotational}} = \sum_{i=1}^{d^{\mathrm {rotational}}} \frac{\hat{L}_i^2}{2\Theta_{ii}}</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>T^{\mathrm {rotational}} = \sum_{i=1}^{d^{\mathrm {rotational}}} \frac{\hat{L}_i^2}{2\Theta_{ii}}</math></div></td></tr>
</table>
Carl McBride
http://www.sklogwiki.org/SklogWiki/index.php?title=Path_integral_formulation&diff=13661&oldid=prev
132.239.72.175: /* External links */
2013-05-01T17:24:03Z
<p><span dir="auto"><span class="autocomment">External links</span></span></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==External links==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==External links==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*[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.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*[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.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[http://www.deleramentum.net/codes/repimd/ A simple implementation of PIMD integrator (C++).]</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category: Monte Carlo]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category: Monte Carlo]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[category: Quantum mechanics]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[category: Quantum mechanics]]</div></td></tr>
</table>
132.239.72.175
http://www.sklogwiki.org/SklogWiki/index.php?title=Path_integral_formulation&diff=13553&oldid=prev
Carl McBride: /* Principles */ Added the variable of integration
2013-04-10T11:09:31Z
<p><span dir="auto"><span class="autocomment">Principles: </span> Added the variable of integration</span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 12:09, 10 April 2013</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>Q(\beta, V)= \int {\mathrm d} x_1 \int_{x_1}^{x_1} Dx(\tau)e^{-S[x(\tau)]}</math> </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>Q(\beta, V)= \int {\mathrm d} x_1 \int_{x_1}^{x_1} Dx(\tau)e^{-S[x(\tau)]}</math> </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where <math>S[x(\tau)]</math> is the Euclidean action, given by (<ref name="Berne"> </ref> Eq. 2)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where <math>S[x(\tau)]</math> is the Euclidean action, given by (<ref name="Berne"> </ref> Eq. 2)</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:<math>S[x(\tau)] = \int_0^{\beta \hbar} H(x(\tau))</math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:<math>S[x(\tau)] = \int_0^{\beta \hbar} H(x(\tau)) <ins style="font-weight: bold; text-decoration: none;">~{\mathrm d}\tau</ins></math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where <math>x(\tau)</math> is the path in time <math>\tau</math> and <math>H</math> is the [[Hamiltonian]].</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where <math>x(\tau)</math> is the path in time <math>\tau</math> and <math>H</math> is the [[Hamiltonian]].</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This leads to (<ref name="Berne"> </ref> Eq. 3)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This leads to (<ref name="Berne"> </ref> Eq. 3)</div></td></tr>
</table>
Carl McBride
http://www.sklogwiki.org/SklogWiki/index.php?title=Path_integral_formulation&diff=13552&oldid=prev
Carl McBride: /* Additional reading */ Added a link to the Scholarpedia article
2013-04-10T11:01:40Z
<p><span dir="auto"><span class="autocomment">Additional reading: </span> Added a link to the Scholarpedia article</span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 12:01, 10 April 2013</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*[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)]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*[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)]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*[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)]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*[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)]</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[http://www.scholarpedia.org/article/Path_integral Jean Zinn-Justin "Path integral" Scholarpedia, 4(2):8674 (2009)]</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==External links==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==External links==</div></td></tr>
</table>
Carl McBride
http://www.sklogwiki.org/SklogWiki/index.php?title=Path_integral_formulation&diff=12706&oldid=prev
Carl McBride: Tidy of references
2012-05-29T12:15:41Z
<p>Tidy of references</p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<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> </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<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> </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">'''Related reading'''</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">See also refs</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[http://dx.doi.org/10.1119/1.18910 Barry R. Holstein "The harmonic oscillator propagator", American Journal of Physics '''66''' pp. 583-589 (1998)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref></ins>[http://dx.doi.org/10.1119/1.18910 Barry R. Holstein "The harmonic oscillator propagator", American Journal of Physics '''66''' pp. 583-589 (1998)]<ins style="font-weight: bold; text-decoration: none;"></ref></ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref></ins>[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)]<ins style="font-weight: bold; text-decoration: none;"></ref></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Wick rotation and imaginary time==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Wick rotation and imaginary time==</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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> &sect; 2.4).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">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>. </ins>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> &sect; 2.4).</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*[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)]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Rotational degrees of freedom==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Rotational degrees of freedom==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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):</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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):</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l72">Line 72:</td>
<td colspan="2" class="diff-lineno">Line 70:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Path integral Monte Carlo====</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Path integral Monte Carlo====</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Path integral Monte Carlo (PIMC)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Path integral Monte Carlo (PIMC)</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref></ins>[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)]<ins style="font-weight: bold; text-decoration: none;"></ref></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Path integral molecular dynamics====</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Path integral molecular dynamics====</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Path integral molecular dynamics (PIMD)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Path integral molecular dynamics (PIMD)</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref></ins>[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)]<ins style="font-weight: bold; text-decoration: none;"></ref></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Centroid molecular dynamics====</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Centroid molecular dynamics====</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Centroid molecular dynamics (CMD)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Centroid molecular dynamics (CMD)</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref></ins>[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)]<ins style="font-weight: bold; text-decoration: none;"></ref></ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref></ins>[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)]<ins style="font-weight: bold; text-decoration: none;"></ref></ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref></ins>[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)]<ins style="font-weight: bold; text-decoration: none;"></ref></ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref></ins>[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)]<ins style="font-weight: bold; text-decoration: none;"></ref></ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref></ins>[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)]<ins style="font-weight: bold; text-decoration: none;"></ref></ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Ring polymer molecular dynamics====</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Ring polymer molecular dynamics====</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Ring polymer molecular dynamics (RPMD)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Ring polymer molecular dynamics (RPMD)</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref></ins>[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)]<ins style="font-weight: bold; text-decoration: none;"></ref></ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref></ins>[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)]<ins style="font-weight: bold; text-decoration: none;"></ref></ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">'''</del>Contraction scheme<del style="font-weight: bold; text-decoration: none;">'''</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">(</ins>Contraction scheme</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref></ins>[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)]<ins style="font-weight: bold; text-decoration: none;"></ref></ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref></ins>[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)]<ins style="font-weight: bold; text-decoration: none;"></ref>)</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Normal mode PIMD====</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Normal mode PIMD====</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Grand canonical Monte Carlo====</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Grand canonical Monte Carlo====</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>A path integral version of the [[Widom test-particle method]] for [[grand canonical Monte Carlo]] simulations:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>A path integral version of the [[Widom test-particle method]] for [[grand canonical Monte Carlo]] simulations:</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref></ins>[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)]<ins style="font-weight: bold; text-decoration: none;"></ref></ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Applications==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Applications==</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref></ins>[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)]<ins style="font-weight: bold; text-decoration: none;"></ref></ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref></ins>[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)]<ins style="font-weight: bold; text-decoration: none;"></ref></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td></tr>
</table>
Carl McBride
http://www.sklogwiki.org/SklogWiki/index.php?title=Path_integral_formulation&diff=12705&oldid=prev
Carl McBride: /* Principles */
2012-05-29T10:43:40Z
<p><span dir="auto"><span class="autocomment">Principles</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 11:43, 29 May 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19">Line 19:</td>
<td colspan="2" class="diff-lineno">Line 19:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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]].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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 <ins style="font-weight: bold; text-decoration: none;">classical </ins>[[intermolecular pair potential]].</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:{| border="1"</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:{| border="1"</div></td></tr>
</table>
Carl McBride
http://www.sklogwiki.org/SklogWiki/index.php?title=Path_integral_formulation&diff=12704&oldid=prev
Carl McBride: /* Principles */ Added an image for the classical limit of P=1
2012-05-29T10:31:08Z
<p><span dir="auto"><span class="autocomment">Principles: </span> Added an image for the classical limit of P=1</span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 11:31, 29 May 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l5">Line 5:</td>
<td colspan="2" class="diff-lineno">Line 5:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>(<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)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>(<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)</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>Q(\beta, V)= \int {\mathrm d} x_1 \int_{x_1}^{x_1} Dx(\tau)e^{-S[x(\tau)]}</math> </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>Q(\beta, V)= \int {\mathrm d} x_1 \int_{x_1}^{x_1} Dx(\tau)e^{-S[x(\tau)]}</math> </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where <math>S[x(\tau)]</math> is the <del style="font-weight: bold; text-decoration: none;">Euclidian </del>action, given by (<ref name="Berne"> </ref> Eq. 2)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where <math>S[x(\tau)]</math> is the <ins style="font-weight: bold; text-decoration: none;">Euclidean </ins>action, given by (<ref name="Berne"> </ref> Eq. 2)</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>S[x(\tau)] = \int_0^{\beta \hbar} H(x(\tau))</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>S[x(\tau)] = \int_0^{\beta \hbar} H(x(\tau))</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where <math>x(\tau)</math> is the path in time <math>\tau</math> and <math>H</math> is the [[Hamiltonian]].</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where <math>x(\tau)</math> is the path in time <math>\tau</math> and <math>H</math> is the [[Hamiltonian]].</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l15">Line 15:</td>
<td colspan="2" class="diff-lineno">Line 15:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>x_{P+1}=x_1</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>x_{P+1}=x_1</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>and (<ref name="Berne"> </ref> Eq. 4)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>and (<ref name="Berne"> </ref> Eq. 4)</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:<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><del style="font-weight: bold; text-decoration: none;">.</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:<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></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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]].<del style="font-weight: bold; text-decoration: none;"><br></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[Image:5bead_pathIntegra_ij.png|<del style="font-weight: bold; text-decoration: none;">500px</del>]]<del style="font-weight: bold; text-decoration: none;"><br></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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 <ins style="font-weight: bold; text-decoration: none;">(<math>P=5</math>)</ins>, 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]].</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:{| border="1"</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| [[Image:1bead_classical_ij.png|center|400px]]Classical limit (P=1) || </ins>[[Image:5bead_pathIntegra_ij.png|<ins style="font-weight: bold; text-decoration: none;">center|400px</ins>]] <ins style="font-weight: bold; text-decoration: none;">Path integral (here with P=5) </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In three dimensions one has the ''density operator''</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In three dimensions one has the ''density operator''</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
</table>
Carl McBride
http://www.sklogwiki.org/SklogWiki/index.php?title=Path_integral_formulation&diff=12701&oldid=prev
Carl McBride: /* Principles */ Updated figure
2012-05-29T09:52:26Z
<p><span dir="auto"><span class="autocomment">Principles: </span> Updated figure</span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 10:52, 29 May 2012</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The following is a schematic for the interaction between <del style="font-weight: bold; text-decoration: none;">two </del>(<del style="font-weight: bold; text-decoration: none;">purple</del>) <del style="font-weight: bold; text-decoration: none;">atoms</del>. 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 <del style="font-weight: bold; text-decoration: none;">red</del>, and white/<del style="font-weight: bold; text-decoration: none;">green </del>bonds represent the [[intermolecular pair potential]].<br></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The following is a schematic for the interaction between <ins style="font-weight: bold; text-decoration: none;">atom <math>i</math> (green) and atom <math>j</math> </ins>(<ins style="font-weight: bold; text-decoration: none;">orange</ins>). 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 <ins style="font-weight: bold; text-decoration: none;">yellow</ins>, and white/<ins style="font-weight: bold; text-decoration: none;">blue </ins>bonds represent the [[intermolecular pair potential]].<br></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[Image:<del style="font-weight: bold; text-decoration: none;">5bead_pathIntegral</del>.png|500px]]<br></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[Image:<ins style="font-weight: bold; text-decoration: none;">5bead_pathIntegra_ij</ins>.png|500px]]<br></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In three dimensions one has the ''density operator''</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In three dimensions one has the ''density operator''</div></td></tr>
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</table>
Carl McBride
http://www.sklogwiki.org/SklogWiki/index.php?title=Path_integral_formulation&diff=11387&oldid=prev
Carl McBride: Undo revision 11385 by RamiroRoberts (talk)
2011-05-12T10:01:42Z
<p>Undo revision 11385 by <a href="/SklogWiki/index.php/Special:Contributions/RamiroRoberts" title="Special:Contributions/RamiroRoberts">RamiroRoberts</a> (<a href="/SklogWiki/index.php?title=User_talk:RamiroRoberts&action=edit&redlink=1" class="new" title="User talk:RamiroRoberts (page does not exist)">talk</a>)</p>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[http://customessays.ws/ Custom essays]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
</table>
Carl McBride