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Chinese Journal of Chemical Physics
Vol. 33, Issue 5, 613 (2020)

Path Integral Liouville Dynamics Simulations of Vibrational Spectra of Formaldehyde and Hydrogen Peroxide^†

Zhi-jun Zhang, Zi-fei Chen, and Jian Liu^*

DOI: 10.1063/1674-0068/cjcp2006099 Cite this Article

Zhi-jun Zhang, Zi-fei Chen, Jian Liu. Path Integral Liouville Dynamics Simulations of Vibrational Spectra of Formaldehyde and Hydrogen Peroxide^†[J]. Chinese Journal of Chemical Physics, 2020, 33(5): 613 Copy Citation Text

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Abstract

Formaldehyde and hydrogen peroxide are two important realistic molecules in atmospheric chemistry. We implement path integral Liouville dynamics (PILD) to calculate the dipole-derivative autocorrelation function for obtaining the infrared spectrum. In comparison to exact vibrational frequencies, PILD faithfully captures most nuclear quantum effects in vibrational dynamics as temperature changes and as the isotopic substitution occurs.

Keywords

Path integral Liouville dynamics Quantum correlation function Vibrational spectrum

$ \begin{eqnarray} \langle {\hat A(0)\hat B(t)} \rangle = \frac{1}{Z}{{\rm Tr}}\left( {{{\hat A}^\beta }{ {\rm e}^{i\hat Ht/\hbar }}\hat B{ {\rm e}^{ - i\hat Ht/\hbar }}} \right) \end{eqnarray} $

(1)

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$ \begin{eqnarray} \hat H = \frac{1}{2}{{\bf{\hat p}}^T}{{\bf{M}}^{ - 1}}{\bf{\hat p}} + V\left( {{\bf{\hat x}}} \right) \end{eqnarray} $

(2)

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$ \begin{eqnarray} I_{\mathit{\boldsymbol{{{\dot{{{\rm \mu }}}}}}} \mathit{\boldsymbol{\dot{{{\rm \mu }}}}}}^{{{\rm Kubo}}}(\omega ) = \frac{1}{{2\pi }}\int_{ - \infty }^\infty { {\rm d}t{{\rm }}{ {\rm e}^{ - i\omega t}}{{\langle {{\hat {\mathit{\boldsymbol{\dot{{{\rm \mu }}}}}}}(0){\hat {\mathit{\boldsymbol{\dot{{{\rm \mu }}}}}}}(t)} \rangle }_{{{\rm Kubo}}}}} \end{eqnarray} $

(3)

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$ \begin{eqnarray} n(\omega )\alpha (\omega ) = \frac{{\beta \pi }}{{3cV{\varepsilon _0}}}I_{{{\mathit{\boldsymbol{\dot{{{\rm \mu }}}}}}}{{\mathit{\boldsymbol{\dot{{{\rm \mu }}}}}}}}^{{{\rm Kubo}}}(\omega ) \end{eqnarray} $

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$ \begin{eqnarray} \langle {\hat A(0)\hat B(t)} \rangle & = & \frac{1}{Z}\int {{ {\rm d}}{{\bf{x}}_0}\int { {\rm d}{{\bf{p}}_0}\rho _W^{{{\rm eq}}}} } \left( {{{\bf{x}}_0}, {{\bf{p}}_0}} \right) \\ &&\times f_{{A^\beta }}^W\left( {{{\bf{x}}_0}, {{\bf{p}}_0}} \right){B_W}\left( {{{\bf{x}}_t}, {{\bf{p}}_t}} \right) \end{eqnarray} $

(5)

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$ \begin{array}{l} \rho _W^{{\rm{eq}}}\left( {{\bf{x}},{\bf{p}}} \right) = \frac{1}{{{{\left( {2\pi \hbar } \right)}^N}}}\int {\rm{d}} \Delta {\bf{x}}\left\langle {{\bf{x}} - \frac{{\Delta {\bf{x}}}}{2}| \cdot } \right.\\ \left. {{{\rm{e}}^{{\rm{ - }}\beta {\rm{\hat H}}}}{\rm{|}}{\bf{x}}{\rm{ + }}\frac{{\Delta {\bf{x}}}}{{\rm{2}}}} \right\rangle {{\rm{e}}^{{{i}}\Delta {{\bf{x}}^{{T}}}{\bf{p}}{\rm{/}}\hbar }} \end{array} $

(6)

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$ \begin{eqnarray} {B_W}\left( {{\bf{x}}, {\bf{p}}} \right) = \int { {\rm d}\Delta {\bf{x}}\langle {{\bf{x}} - \frac{{\Delta {\bf{x}}}}{2}\left| {\hat B} \right|{\bf{x}} + \frac{{\Delta {\bf{x}}}}{2}} \rangle } { {\rm e}^{i\Delta {{\bf{x}}^T}{\bf{p}}/\hbar }} \end{eqnarray} $

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$ \begin{eqnarray} f_{{A^\beta }}^W\left( {{\bf{x}}, {\bf{p}}} \right) = \frac{ {{\int { {\rm d}\Delta {\bf{x}}\langle {{\bf{x}} - \frac{{\Delta {\bf{x}}}}{2}\left| {{{\hat A}^\beta }} \right|{\bf{x}} +\frac{{\Delta {\bf{x}}}}{2}} \rangle } { {\rm e}^{i\Delta {{\bf{x}}^T}{\bf{p}}/\hbar }}}}}{ {{\int{ {\rm d}\Delta {\bf{x}}\langle {{\bf{x}} -\frac{{\Delta {\bf{x}}}}{2}\left| {{ {\rm e}^{ - \beta \hat H}}} \right|{\bf{x}} +\frac{{\Delta {\bf{x}}}}{2}} \rangle } { {\rm e}^{i\Delta {{\bf{x}}^T}{\bf{p}}/\hbar }}}}}\\ \end{eqnarray} $

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$ \begin{array}{l} \frac{1}{Z}\rho _W^{{\rm{eq}}}\left( {{\bf{x}},{\bf{p}}} \right) \approx \frac{1}{Z}\langle {\bf{x}}|{{\rm{e}}^{ - \beta \hat H}}|{\bf{x}}\rangle {\left( {\frac{\beta }{{2\pi }}} \right)^{N/2}}\\ \times |\det \left( {{{\bf{M}}_{{\rm{therm}}}}} \right){|^{ - 1/2}}\\ \times \exp \left[ { - \frac{\beta }{2}{{\bf{p}}^T}{\bf{M}}_{{\rm{therm}}}^{ - 1}{\bf{p}}} \right] \end{array} $

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$ \begin{eqnarray} &&f_{{A^\beta }}^W\left( {{{\bf{x}}_0}, {{\bf{p}}_0}} \right){B_W}\left( {{{\bf{x}}_t}, {{\bf{p}}_t}} \right) \\ &\approx& \left[ {{\bf{p}}_0^T{\bf{M}}_{{{\rm therm}}}^{ - 1}\left( {\frac{{\partial {\mathit{\boldsymbol{{{\rm \mu }}}}}}}{{\partial {{\bf{x}}_0}}}} \right)} \right] \cdot {\mathit{\boldsymbol{\dot{{{\rm \mu }}}}}}\left( {{{\bf{x}}_t}, {{\bf{p}}_t}} \right) \end{eqnarray} $

(10)

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$ \left\{ \begin{array}{l} {{{\bf{\dot x}}}_t} = {{\bf{M}}^{ - 1}}{{\bf{p}}_t}\\ {{{\bf{\dot p}}}_t} = - { \frac{\partial V_{{{\rm eff}}}^{{{\rm PILD}}}\left( {{{\bf{x}}_t}, {{\bf{p}}_t}} \right)}{{\partial {{\bf{x}}_t}}}} \end{array} \right. $

(11)

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$\begin{array}{l} - \frac{{\partial V_{{\rm{eff}}}^{{\rm{PILD}}}\left( {{\bf{x}},{\bf{p}}} \right)}}{{\partial {\bf{x}}}}\\ = \frac{1}{\beta }{{\bf{M}}_{{\rm{therm}}}}{{\bf{M}}^{ - 1}}\frac{\partial }{{\partial {\bf{x}}}}\ln \langle {\bf{x}}\left| {{{\rm{e}}^{ - \beta \hat H}}} \right.|{\bf{x}}\rangle \end{array}$

(12)

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$ \begin{array}{l} \langle {\bf{x}}\left| {{{\rm{e}}^{ - \beta \hat H}}} \right|{\bf{x}}\rangle \\ = \mathop {\lim }\limits_{P \to \infty } \int {{\rm{d}}{{\bf{x}}_2} \cdots {{\int {{\rm{d}}{{\bf{x}}_P}\left( {\frac{P}{{2\pi \beta {\hbar ^2}}}} \right)} }^{NP/2}}} {\left| {\bf{M}} \right|^{P/2}}\\ \times \exp \left\{ { - \frac{P}{{2\beta {\hbar ^2}}}\sum\limits_{i = 1}^P {\left[ {{{\left( {{{\bf{x}}_{i + 1}} - {{\bf{x}}_i}} \right)}^T}{\bf{M}}({{\bf{x}}_{i + 1}} - {{\bf{x}}_i})} \right]} } \right.\\ \left. { - \frac{\beta }{P}\sum\limits_{i = 1}^P {V\left( {{{\bf{x}}_i}} \right)} } \right\} \end{array} $

(13)

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$ \begin{eqnarray} \begin{array}{l} {{\bf{ \pmb{\mathsf{ ξ}} }}_1} = {{\bf{x}}_1}\\ {{\bf{ \pmb{\mathsf{ ξ}} }}_j} = {{\bf{x}}_j} - \frac{ {(j - 1){{\bf{x}}_{j + 1}} + {{\bf{x}}_1}}}{j}{{\rm }}\left( {j = \overline {2, P} } \right) \end{array} \end{eqnarray} $

(14)

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$ \langle {\bf{x}}\left| {{{\rm{e}}^{ - \beta \hat H}}} \right.|{\bf{x}}\rangle \\ = \mathop {\lim }\limits_{P \to \infty } {\int { {\rm d}{{\mathit{\boldsymbol{\xi}}}_2} \cdots \int { {\rm d}{{\mathit{\boldsymbol{\xi}}}_P}\left( {\frac{P}{{2\pi \beta {\hbar ^2}}}} \right)} } ^{NP/2}}{\left| {\bf{M}} \right|^{P/2}} \\ \times \exp \left\{ { - \beta \left[ {\frac{1}{2}\omega _{{{\rm ad}}}^2\sum\limits_{i = 2}^P {{\mathit{\boldsymbol{\xi}}}_i^T{{{\bf{\tilde M}}}_i}{{\mathit{\boldsymbol{\xi}}}_i}} + \phi \left( {{{\mathit{\boldsymbol{\xi}}}_1}, \cdots , {{\mathit{\boldsymbol{\xi}}}_P}} \right)} \right]} \right\}$

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$ \begin{eqnarray} \phi \left( {{{\mathit{\boldsymbol{\xi}}}_1}, \cdots , {{\mathit{\boldsymbol{\xi}}}_P}} \right) & = & \frac{1}{P}\sum\limits_{i = 1}^P {V\left( {{{\mathit{\boldsymbol{x}}}_i}} \right)} \end{eqnarray} $

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$ \begin{eqnarray} {\omega _{{{\rm ad}}}}& = &\frac{1}{{\beta \hbar }}\sqrt {\frac{P}{{{\gamma _{{{\rm ad}}}}}}} \end{eqnarray} $

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$ \begin{eqnarray} {{\bf{\tilde M}}_j} & = & {\gamma _{{{\rm ad}}}}\frac{j}{{j - 1}}{\bf{M}}{{\rm }}\left( {j = \overline {2, P} } \right) \end{eqnarray} $

(18)

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$ \begin{eqnarray} \frac{{\partial \phi }}{{\partial {{\mathit{\boldsymbol{\xi}}}_1}}}& = &\sum\limits_{i = 1}^P {\frac{{\partial \phi }}{{\partial {{\bf{x}}_i}}}} = \frac{1}{P}\sum\limits_{i = 1}^P {\frac{{\partial V\left( {{{\bf{x}}_i}} \right)}}{{\partial {{\bf{x}}_i}}}} \end{eqnarray} $

(19)

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$ \begin{eqnarray} \frac{{\partial \phi }}{{\partial {{\mathit{\boldsymbol{\xi}}}_j}}}& = & \frac{{\partial \phi }}{{\partial {{\bf{x}}_j}}} + \frac{{j - 2}}{{j - 1}}\frac{{\partial \phi }}{{\partial {{\mathit{\boldsymbol{\xi}}}_{j - 1}}}}{{\rm }}\left( {j = \overline {2, P} } \right) \end{eqnarray} $

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$ \begin{eqnarray} - {\frac{1}{\beta }}\frac{\partial }{{\partial {\bf{x}}}}\ln \langle {\bf{x}}|{{\rm{e}}^{ - \beta \hat H}}|{\bf{x}}\rangle \mathop = \limits^{{{\bf{ \pmb{\mathsf{ ξ}} }}_1} \equiv {{\bf{x}}_1} \equiv {\bf{x}}} \frac{{\mathop {\lim }\limits_{P \to \infty } \int { {\rm d}{{\mathit{\boldsymbol{\xi}}}_2}} \cdots \int { {\rm d}{{\mathit{\boldsymbol{\xi}}}_P}} \exp \left\{ { -\beta \left[ {\sum\limits_{j = 2}^P {\frac{1}{2}\omega _{ {\rm ad}}^2{\mathit{\boldsymbol{\xi}}}_j^T{{{\bf{\tilde M}}}_{{\rm j}}}{\mathit{\boldsymbol{\xi}}}_j^{}} + \phi \left( {{{\mathit{\boldsymbol{\xi}}}_1}, \cdots , {{\mathit{\boldsymbol{\xi}}}_P}} \right)} \right]} \right\}\frac{1}{P}\sum\limits_{j = 1}^P {V'\left( {{{\bf{x}}_j}} \right)} }}{{\mathop {\lim }\limits_{P \to \infty } \int { {\rm d}{{\mathit{\boldsymbol{\xi}}}_2}} \cdots \int { {\rm d}{{\mathit{\boldsymbol{\xi}}}_P}} \exp \left\{ { - \beta \left[ {\sum\limits_{j = 2}^P {\frac{1}{2}\omega _{ {\rm ad}}^2{\mathit{\boldsymbol{\xi}}}_j^T{{{\bf{\tilde M}}}_{{\rm j}}}{\mathit{\boldsymbol{\xi}}}_j^{}} + \phi \left( {{{\mathit{\boldsymbol{\xi}}}_1}, \cdots , {{\mathit{\boldsymbol{\xi}}}_P}} \right)} \right]} \right\}}} \, \, \, \, \, \end{eqnarray} $

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$ \begin{eqnarray} && - \frac{1}{\beta }\frac{\partial }{{\partial {\bf{x}}}}\ln \langle {\bf{x}}|{{\rm{e}}^{ - \beta \hat H}}|{\bf{x}}\rangle \;\mathop = \limits^{{{\bf{ \pmb{\mathsf{ ξ}} }}_1} \equiv {{\bf{x}}_1} \equiv {\bf{x}}} \frac{A}{B} \end{eqnarray} $

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$ \begin{array}{l} A = \mathop {\lim }\limits_{P \to \infty } \int {\left( {\prod\limits_{j = 2}^P {{\rm{d}}{{\bf{\xi }}_j}{\rm{d}}{\bf{p}}_j^{}} } \right)} \\ \times \exp \{ - \beta [\sum\limits_{j = 2}^P {\left( {\frac{1}{2}{\bf{p}}_j^T{\bf{\tilde M}}_j^{ - 1}{\bf{p}}_j^{} + \frac{1}{2}\omega _{{\rm{ad}}}^2{\bf{\xi }}_j^T{{{\bf{\tilde M}}}_{\rm{j}}}{\bf{\xi }}_j^{}} \right)} \\ + \phi \left( {{{\bf{\xi }}_1}, \cdots ,{{\bf{\xi }}_P}} \right)]\} \frac{1}{P}\sum\limits_{j = 1}^P {} V'\left( {{{\bf{x}}_j}} \right)\\ B = \mathop {\lim }\limits_{P \to \infty } \int {\left( {\prod\limits_{j = 2}^P {{\rm{d}}{{\bf{\xi }}_j}{\rm{d}}{\bf{p}}_j^{}} } \right)} \\ \times \exp {\rm{\{ }} - \beta {\rm{[}}\sum\limits_{j = 2}^P {} (\frac{1}{2}{\bf{p}}_j^T{\bf{\tilde M}}_j^{ - 1}{\bf{p}}_j^{}\\ + \frac{1}{2}\omega _{{\rm{ad}}}^2{\bf{\xi }}_j^T{{{\bf{\tilde M}}}_{\rm{j}}}{\bf{\xi }}_j^{}) + \phi \left( {{{\bf{\xi }}_1}, \cdots ,{{\bf{\xi }}_P}} \right)]\;\} \end{array} $

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$ \left\{ \begin{array}{l} {{{\mathit{\boldsymbol{\dot \xi}}}}_1} \equiv {{{\mathit{\boldsymbol{\dot x}}}}_1} \equiv {\mathit{\boldsymbol{\dot x}}} = {{\bf{M}}^{ - 1}}{\bf{p}}, \\ {{{\bf{\dot p}}}_1} \equiv {\bf{\dot p}} = - {{\bf{M}}_{{{\rm therm}}}}{{\bf{M}}^{ - 1}} {\frac{\partial \phi }{{\partial {{\mathit{\boldsymbol{\xi}}}_1}}}}, \\ {{{\mathit{\boldsymbol{\dot \xi}}}}_j} = {\bf{\tilde M}}_j^{ - 1}{{\bf{p}}_j}, \\ {{{\mathit{\boldsymbol{\dot p}}}}_j} = - \omega _{{{\rm ad}}}^2{{{\bf{\tilde M}}}_j}{{\mathit{\boldsymbol{\xi}}}_j} - { \frac{{\partial \phi }}{{\partial {{\mathit{\boldsymbol{\xi}}}_j}}}}{{\rm }}\left( {j = \overline {2, P} } \right) \end{array} \right. $

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$ \left\{ {\begin{array}{*{20}{l}} {{{{\bf{\dot \xi }}}_1} \equiv {{{\bf{\dot x}}}_1} \equiv {\bf{\dot x}} = {{\bf{M}}^{ - 1}}{\bf{p}},}\\ {{{{\bf{\dot p}}}_1} \equiv {\bf{\dot p}} = - {{\bf{M}}_{{\rm{therm}}}}{{\bf{M}}^{ - 1}}\frac{{\partial \phi }}{{\partial {{\bf{\xi }}_1}}},}\\ {{{{\bf{\dot \xi }}}_j} = {\bf{\tilde M}}_j^{ - 1}{{\bf{p}}_j},}\\ {{{{\bf{\dot p}}}_j} = - \omega _{{\rm{ad}}}^2{{{\bf{\tilde M}}}_j}{{\bf{\xi }}_j} - \frac{{\partial \phi }}{{\partial {{\bf{\xi }}_j}}} - {\gamma _{{\rm{Lang}}}}{{\bf{p}}_j}}\\ {\quad {\rm{ + }}\sqrt {\frac{{2{\gamma _{{\rm{Lang}}}}}}{\beta }} {\bf{\tilde M}}_j^{1/2}{{\bf{\eta }}_j}(t),\quad \left( {j = \overline {2,P} } \right)} \end{array}} \right. $

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$ \begin{eqnarray} \begin{array}{l} {{\bf{p}}_1} \leftarrow {{\bf{p}}_1} - {{\bf{M}}_{{{\rm therm}}}}{{\bf{M}}^{ - 1}} {\frac{{\partial \phi }}{{\partial {{\mathit{\boldsymbol{\xi}}}_1}}}}\frac{{\Delta t}}{2}, \\ {{\bf{p}}_j} \leftarrow {{\bf{p}}_j} - {\frac{{\partial \phi }}{{\partial {{\mathit{\boldsymbol{\xi}}}_j}}}}\frac{{\Delta t}}{2} - \omega _{{{\rm ad}}}^2{{{\bf{\tilde M}}}_j}{{\mathit{\boldsymbol{\xi}}}_j}\frac{{\Delta t}}{2}{{\rm }}, \quad \left( {j = \overline {2, P} } \right) \end{array} \end{eqnarray} $

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$ \begin{eqnarray} {{\mathit{\boldsymbol{\xi}}}_j} &\leftarrow& {{\mathit{\boldsymbol{\xi}}}_j} + {\bf{\tilde M}}_j^{ - 1}{{\bf{p}}_j}\frac{{\Delta t}}{2}{{\rm }}, \quad \left( {j = \overline {1, P} } \right) \end{eqnarray} $

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$ \begin{eqnarray} {{\bf{p}}_j} &\leftarrow& {c_1}{{\bf{p}}_j} + {c_2}\sqrt {\frac{1}{\beta }} {\left( {{{{\bf{\tilde M}}}_j}} \right)^{1/2}}{}{{\mathit{\boldsymbol{\eta}}}_j}, \quad {{\rm }}\left( {j = \overline {2, P} } \right) \end{eqnarray} $

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$ \begin{eqnarray} {{\mathit{\boldsymbol{\xi}}}_j} \leftarrow {{\mathit{\boldsymbol{\xi}}}_j} + {\bf{\tilde M}}_j^{ - 1}{{\bf{p}}_j}\frac{{\Delta t}}{2}, \quad \left( {j = \overline {1, P} } \right) \end{eqnarray} $

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$ \begin{eqnarray} \begin{array}{l} {{\bf{p}}_1} \leftarrow {{\bf{p}}_1} - {{\bf{M}}_{{{\rm therm}}}}{{\bf{M}}^{ - 1}} {\frac{{\partial \phi }}{{\partial {{\mathit{\boldsymbol{\xi}}}_1}}}}\frac{{\Delta t}}{2}, \\ {{\bf{p}}_j} \leftarrow {{\bf{p}}_j} - { \frac{{\partial \phi }}{{\partial {{\mathit{\boldsymbol{\xi}}}_j}}}}\frac{{\Delta t}}{2} - \omega _{{{\rm ad}}}^2{{{\bf{\tilde M}}}_j}{{\mathit{\boldsymbol{\xi}}}_j}\frac{{\Delta t}}{2}{{\rm }}, \quad \left( {j = \overline {2, P} } \right) \end{array} \end{eqnarray} $

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$ \begin{eqnarray} \begin{array}{l} {c_1} = \exp \left[ { - {\gamma _{{{\rm Lang}}}}\Delta t} \right], \\ {c_2} = \sqrt {1 - c_1{^2}} \end{array} \end{eqnarray} $

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$ \begin{eqnarray} {\mathcal{H}_{kl}} = \frac{1}{{{m_k}{m_l}}}\frac{{{\partial ^2}V}}{{\partial {x_k}\partial {x_l}}} \end{eqnarray} $

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$ \begin{eqnarray} {H}({\bf{x}} + \Delta {\bf{x}}) &\approx& \frac{1}{2}{{\bf{p}}^T}{{\bf{M}}^{ - 1}}{\bf{p}} + V({\bf{x}}) + {\left( {\frac{{\partial V}}{{\partial {\bf{x}}{{\rm }}}}} \right)^T}\Delta {\bf{x}} \\ &&+ \frac{1}{2}\Delta {{\bf{x}}^T}\mathcal{H}\Delta {\bf{x}} \end{eqnarray} $

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$ \begin{eqnarray} {{\bf{T}}^T}\mathcal{H}{\bf{T}} = {\bf{\Lambda }} \end{eqnarray} $

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$ \begin{eqnarray} {\bf{X}} = {{\bf{T}}^T}{{\bf{M}}^{1/2}}{\bf{x}} \end{eqnarray} $

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$ \begin{eqnarray} \begin{array}{l} {{\bf{D}}_1} = {(1, 0, 0, 4, 0, 0, 1, 0, 0)^T}\\ {{\bf{D}}_2} = {(0, 1, 0, 0, 4, 0, 0, 1, 0)^T}\\ {{\bf{D}}_3} = {(0, 0, 1, 0, 0, 4, 0, 0, 1)^T} \end{array} \end{eqnarray} $

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$ \begin{eqnarray} {\bf{I}} = {{\rm }}\sum\limits_{i = 1}^{{N_{ {\rm atom}}}} {{m_i}\left( {{{\left| {{{\bf{r}}_i}} \right|}^2} - {{\bf{r}}_i} \cdot {{\bf{r}}_i}^T} \right)} \end{eqnarray} $

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$ \begin{eqnarray} {{\bf{W}}^T}{\bf{IW}} = {\bf{\Phi }} \end{eqnarray} $

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$ \begin{eqnarray} ({P_{\alpha }})_i = {{\bf{r}}_i} \cdot {{\bf{W}}_\alpha } \end{eqnarray} $

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$ \begin{eqnarray} \begin{array}{l} {D_{4\alpha , i}} = \left[ {{{\left( {{P_y}} \right)}_i}{W_{\alpha , 3}} - {{\left( {{P_z}} \right)}_i}{W_{\alpha , 2}}} \right]\sqrt {{m_i}} \\ {D_{5\alpha , i}} = \left[ {{{\left( {{P_z}} \right)}_i}{W_{\alpha , 1}} - {{\left( {{P_x}} \right)}_i}{W_{\alpha , 3}}} \right]\sqrt {{m_i}} \\ {D_{6\alpha , i}} = \left[ {{{\left( {{P_x}} \right)}_i}{W_{\alpha , 2}} - {{\left( {{P_y}} \right)}_i}{W_{\alpha, 1}}} \right]\sqrt {{m_i}} \end{array} \end{eqnarray} $

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$ \begin{eqnarray} {\bf{L}}_{ {\rm vib}}^T{\left( {{{\bf{D}}^T}{\bf{HD}}} \right)_{ {\rm vib}}}{{\bf{L}}_{ {\rm vib}}} = {{\bf{\Lambda }}_{ {\rm vib}}} \end{eqnarray} $

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$ Q(u) = \frac{1}{{Q({u_i})}} = \frac{{\tanh ({u_i}/2)}}{{{u_i}/2}} $

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$ \begin{eqnarray} Q(u) = \frac{1}{Q({u_i})} = \frac{{\tanh ({u_i}/2)}}{{{u_i}/2}} \end{eqnarray} $

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$ \begin{eqnarray} \langle {{Q_k}} \rangle = \langle {\sum\limits_j {{{\left| {{\bf{b}}_j^T({\bf{x}}) \cdot {{\bf{b}}_{0k}}({{\bf{x}}_0})} \right|}^2}{Q_j}({\bf{x}})} } \rangle \end{eqnarray} $

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$ \begin{eqnarray} {{\bf{M}}_{{{\rm therm}}}} = {{\bf{M}}^{1/2}}{{\bf{T}}_0}\langle {\bf{Q}} \rangle {\bf{T}}_0^T{{\bf{M}}^{1/2}} \end{eqnarray} $

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Zhi-jun Zhang, Zi-fei Chen, Jian Liu. Path Integral Liouville Dynamics Simulations of Vibrational Spectra of Formaldehyde and Hydrogen Peroxide^†[J]. Chinese Journal of Chemical Physics, 2020, 33(5): 613

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