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Journals >Chinese Journal of Chemical Physics >Volume 33 >Issue 5 >Page 613 > Article

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

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\begin{array}{r} ⟨ \hat{A} (0) \hat{B} (t) ⟩ = \frac{1}{Z} T r ({\hat{A}}^{β} e^{i \hat{H} t / ℏ} \hat{B} e^{- i \hat{H} t / ℏ}) \end{array}

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(1)

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\begin{array}{r} \hat{H} = \frac{1}{2} {\hat{p}}^{T} M^{- 1} \hat{p} + V (\hat{x}) \end{array}

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(2)

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\begin{array}{r} I_{\boldsymbol \dot{μ} \boldsymbol \dot{μ}}^{K u b o} (ω) = \frac{1}{2 π} \int_{- \infty}^{\infty} d t e^{- i ω t} {⟨ \hat{\boldsymbol \dot{μ}} (0) \hat{\boldsymbol \dot{μ}} (t) ⟩}_{K u b o} \end{array}

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(3)

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\begin{array}{r} n (ω) α (ω) = \frac{β π}{3 c V ε_{0}} I_{\boldsymbol \dot{μ} \boldsymbol \dot{μ}}^{K u b o} (ω) \end{array}

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(4)

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\begin{array}{rcl} ⟨ \hat{A} (0) \hat{B} (t) ⟩ & = & \frac{1}{Z} \int d x_{0} \int d p_{0} ρ_{W}^{e q} (x_{0}, p_{0}) \\ \times f_{A^{β}}^{W} (x_{0}, p_{0}) B_{W} (x_{t}, p_{t}) \end{array}

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(5)

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\begin{array}{l} ρ_{W}^{e q} (x, p) = \frac{1}{{(2 π ℏ)}^{N}} \int d Δ x ⟨ x - \frac{Δ x}{2} | \cdot \\ e^{- β \hat{H}} | x + \frac{Δ x}{2} ⟩ e^{i Δ x^{T} p / ℏ} \end{array}

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(6)

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\begin{array}{r} B_{W} (x, p) = \int d Δ x ⟨ x - \frac{Δ x}{2} | \hat{B} | x + \frac{Δ x}{2} ⟩ e^{i Δ x^{T} p / ℏ} \end{array}

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(7)

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\begin{array}{r} f_{A^{β}}^{W} (x, p) = \frac{\int d Δ x ⟨ x - \frac{Δ x}{2} | {\hat{A}}^{β} | x + \frac{Δ x}{2} ⟩ e^{i Δ x^{T} p / ℏ}}{\int d Δ x ⟨ x - \frac{Δ x}{2} | e^{- β \hat{H}} | x + \frac{Δ x}{2} ⟩ e^{i Δ x^{T} p / ℏ}} \end{array}

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(8)

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\begin{array}{l} \frac{1}{Z} ρ_{W}^{e q} (x, p) \approx \frac{1}{Z} ⟨ x | e^{- β \hat{H}} | x ⟩ {(\frac{β}{2 π})}^{N / 2} \\ \times | det (M_{t h e r m}) |^{- 1 / 2} \\ \times \exp [- \frac{β}{2} p^{T} M_{t h e r m}^{- 1} p] \end{array}

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(9)

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\begin{array}{rcl} f_{A^{β}}^{W} (x_{0}, p_{0}) B_{W} (x_{t}, p_{t}) \\ \approx & [p_{0}^{T} M_{t h e r m}^{- 1} (\frac{\partial \boldsymbol μ}{\partial x_{0}})] \cdot \boldsymbol \dot{μ} (x_{t}, p_{t}) \end{array}

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(10)

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$ \left\{

\begin{array}{l} {\dot{x}}_{t} = M^{- 1} p_{t} \\ {\dot{p}}_{t} = - \frac{\partial V_{e f f}^{P I L D} (x_{t}, p_{t})}{\partial x_{t}} \end{array}

\right. $

(11)

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\begin{array}{l} - \frac{\partial V_{e f f}^{P I L D} (x, p)}{\partial x} \\ = \frac{1}{β} M_{t h e r m} M^{- 1} \frac{\partial}{\partial x} \ln ⟨ x | e^{- β \hat{H}} | x ⟩ \end{array}

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(12)

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\begin{array}{l} ⟨ x | e^{- β \hat{H}} | x ⟩ \\ = lim_{P \to \infty} \int d x_{2} \dots {\int d x_{P} (\frac{P}{2 π β ℏ^{2}})}^{N P / 2} {| M |}^{P / 2} \\ \times \exp {- \frac{P}{2 β ℏ^{2}} \sum_{i = 1}^{P} [{(x_{i + 1} - x_{i})}^{T} M (x_{i + 1} - x_{i})] \\ - \frac{β}{P} \sum_{i = 1}^{P} V (x_{i})} \end{array}

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\begin{array}{r} \begin{array}{l} {ξ ξ}_{1} = x_{1} \\ {ξ ξ}_{j} = x_{j} - \frac{(j - 1) x_{j + 1} + x_{1}}{j} (j = \overset{―}{2, P}) \end{array} \end{array}

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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\}$

(15)

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\begin{array}{rcl} ϕ ({\boldsymbol ξ}_{1}, \dots, {\boldsymbol ξ}_{P}) & = & \frac{1}{P} \sum_{i = 1}^{P} V ({\boldsymbol x}_{i}) \end{array}

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(16)

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\begin{array}{rcl} ω_{a d} & = & \frac{1}{β ℏ} \sqrt{\frac{P}{γ_{a d}}} \end{array}

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(17)

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\begin{array}{rcl} {\tilde{M}}_{j} & = & γ_{a d} \frac{j}{j - 1} M (j = \overset{―}{2, P}) \end{array}

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\begin{array}{rcl} \frac{\partial ϕ}{\partial {\boldsymbol ξ}_{1}} & = & \sum_{i = 1}^{P} \frac{\partial ϕ}{\partial x_{i}} = \frac{1}{P} \sum_{i = 1}^{P} \frac{\partial V (x_{i})}{\partial x_{i}} \end{array}

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\begin{array}{rcl} \frac{\partial ϕ}{\partial {\boldsymbol ξ}_{j}} & = & \frac{\partial ϕ}{\partial x_{j}} + \frac{j - 2}{j - 1} \frac{\partial ϕ}{\partial {\boldsymbol ξ}_{j - 1}} (j = \overset{―}{2, P}) \end{array}

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\begin{array}{r} - \frac{1}{β} \frac{\partial}{\partial x} \ln ⟨ x | e^{- β \hat{H}} | x ⟩ \overset{{ξ ξ}_{1} \equiv x_{1} \equiv x}{=} \frac{lim_{P \to \infty} \int d {\boldsymbol ξ}_{2} \dots \int d {\boldsymbol ξ}_{P} \exp {- β [\sum_{j = 2}^{P} \frac{1}{2} ω_{a d}^{2} {\boldsymbol ξ}_{j}^{T} {\tilde{M}}_{j} {\boldsymbol ξ}_{j}^{} + ϕ ({\boldsymbol ξ}_{1}, \dots, {\boldsymbol ξ}_{P})]} \frac{1}{P} \sum_{j = 1}^{P} V^{'} (x_{j})}{lim_{P \to \infty} \int d {\boldsymbol ξ}_{2} \dots \int d {\boldsymbol ξ}_{P} \exp {- β [\sum_{j = 2}^{P} \frac{1}{2} ω_{a d}^{2} {\boldsymbol ξ}_{j}^{T} {\tilde{M}}_{j} {\boldsymbol ξ}_{j}^{} + ϕ ({\boldsymbol ξ}_{1}, \dots, {\boldsymbol ξ}_{P})]}} \end{array}

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(21)

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\begin{array}{rcl} - \frac{1}{β} \frac{\partial}{\partial x} \ln ⟨ x | e^{- β \hat{H}} | x ⟩ \overset{{ξ ξ}_{1} \equiv x_{1} \equiv x}{=} \frac{A}{B} \end{array}

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\begin{array}{l} A = lim_{P \to \infty} \int (\prod_{j = 2}^{P} d ξ_{j} d p_{j}^{}) \\ \times \exp {- β [\sum_{j = 2}^{P} (\frac{1}{2} p_{j}^{T} {\tilde{M}}_{j}^{- 1} p_{j}^{} + \frac{1}{2} ω_{a d}^{2} ξ_{j}^{T} {\tilde{M}}_{j} ξ_{j}^{}) \\ + ϕ (ξ_{1}, \dots, ξ_{P})]} \frac{1}{P} \sum_{j = 1}^{P} V^{'} (x_{j}) \\ B = lim_{P \to \infty} \int (\prod_{j = 2}^{P} d ξ_{j} d p_{j}^{}) \\ \times \exp {- β [\sum_{j = 2}^{P} (\frac{1}{2} p_{j}^{T} {\tilde{M}}_{j}^{- 1} p_{j}^{} \\ + \frac{1}{2} ω_{a d}^{2} ξ_{j}^{T} {\tilde{M}}_{j} ξ_{j}^{}) + ϕ (ξ_{1}, \dots, ξ_{P})]} \end{array}

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()

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$ \left\{

\begin{array}{l} {\boldsymbol \dot{ξ}}_{1} \equiv {\boldsymbol \dot{x}}_{1} \equiv \boldsymbol \dot{x} = M^{- 1} p, \\ {\dot{p}}_{1} \equiv \dot{p} = - M_{t h e r m} M^{- 1} \frac{\partial ϕ}{\partial {\boldsymbol ξ}_{1}}, \\ {\boldsymbol \dot{ξ}}_{j} = {\tilde{M}}_{j}^{- 1} p_{j}, \\ {\boldsymbol \dot{p}}_{j} = - ω_{a d}^{2} {\tilde{M}}_{j} {\boldsymbol ξ}_{j} - \frac{\partial ϕ}{\partial {\boldsymbol ξ}_{j}} (j = \overset{―}{2, P}) \end{array}

\right. $

(23)

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$ \left\{ {

\begin{array}{l} {\dot{ξ}}_{1} \equiv {\dot{x}}_{1} \equiv \dot{x} = M^{- 1} p, \\ {\dot{p}}_{1} \equiv \dot{p} = - M_{t h e r m} M^{- 1} \frac{\partial ϕ}{\partial ξ_{1}}, \\ {\dot{ξ}}_{j} = {\tilde{M}}_{j}^{- 1} p_{j}, \\ {\dot{p}}_{j} = - ω_{a d}^{2} {\tilde{M}}_{j} ξ_{j} - \frac{\partial ϕ}{\partial ξ_{j}} - γ_{L a n g} p_{j} \\ + \sqrt{\frac{2 γ_{L a n g}}{β}} {\tilde{M}}_{j}^{1 / 2} η_{j} (t), (j = \overset{―}{2, P}) \end{array}

} \right. $

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\begin{array}{r} \begin{array}{l} p_{1} \leftarrow p_{1} - M_{t h e r m} M^{- 1} \frac{\partial ϕ}{\partial {\boldsymbol ξ}_{1}} \frac{Δ t}{2}, \\ p_{j} \leftarrow p_{j} - \frac{\partial ϕ}{\partial {\boldsymbol ξ}_{j}} \frac{Δ t}{2} - ω_{a d}^{2} {\tilde{M}}_{j} {\boldsymbol ξ}_{j} \frac{Δ t}{2}, (j = \overset{―}{2, P}) \end{array} \end{array}

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\begin{array}{rcl} {\boldsymbol ξ}_{j} & \leftarrow & {\boldsymbol ξ}_{j} + {\tilde{M}}_{j}^{- 1} p_{j} \frac{Δ t}{2}, (j = \overset{―}{1, P}) \end{array}

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(26)

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\begin{array}{rcl} p_{j} & \leftarrow & c_{1} p_{j} + c_{2} \sqrt{\frac{1}{β}} {({\tilde{M}}_{j})}^{1 / 2} {\boldsymbol η}_{j}, (j = \overset{―}{2, P}) \end{array}

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(27)

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\begin{array}{r} {\boldsymbol ξ}_{j} \leftarrow {\boldsymbol ξ}_{j} + {\tilde{M}}_{j}^{- 1} p_{j} \frac{Δ t}{2}, (j = \overset{―}{1, P}) \end{array}

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\begin{array}{r} \begin{array}{l} p_{1} \leftarrow p_{1} - M_{t h e r m} M^{- 1} \frac{\partial ϕ}{\partial {\boldsymbol ξ}_{1}} \frac{Δ t}{2}, \\ p_{j} \leftarrow p_{j} - \frac{\partial ϕ}{\partial {\boldsymbol ξ}_{j}} \frac{Δ t}{2} - ω_{a d}^{2} {\tilde{M}}_{j} {\boldsymbol ξ}_{j} \frac{Δ t}{2}, (j = \overset{―}{2, P}) \end{array} \end{array}

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\begin{array}{r} \begin{array}{l} c_{1} = \exp [- γ_{L a n g} Δ t], \\ c_{2} = \sqrt{1 - c_{1}^{2}} \end{array} \end{array}

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\begin{array}{r} H_{k l} = \frac{1}{m_{k} m_{l}} \frac{\partial^{2} V}{\partial x_{k} \partial x_{l}} \end{array}

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(31)

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\begin{array}{rcl} H (x + Δ x) & \approx & \frac{1}{2} p^{T} M^{- 1} p + V (x) + {(\frac{\partial V}{\partial x})}^{T} Δ x \\ + \frac{1}{2} Δ x^{T} H Δ x \end{array}

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\begin{array}{r} T^{T} H T = Λ \end{array}

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\begin{array}{r} X = T^{T} M^{1 / 2} x \end{array}

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\begin{array}{r} \begin{array}{l} D_{1} = (1, 0, 0, 4, 0, 0, 1, 0, 0)^{T} \\ D_{2} = (0, 1, 0, 0, 4, 0, 0, 1, 0)^{T} \\ D_{3} = (0, 0, 1, 0, 0, 4, 0, 0, 1)^{T} \end{array} \end{array}

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\begin{array}{r} I = \sum_{i = 1}^{N_{a t o m}} m_{i} ({| r_{i} |}^{2} - r_{i} \cdot {r_{i}}^{T}) \end{array}

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\begin{array}{r} W^{T} I W = Φ \end{array}

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\begin{array}{r} (P_{α})_{i} = r_{i} \cdot W_{α} \end{array}

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\begin{array}{r} \begin{array}{l} D_{4 α, i} = [{(P_{y})}_{i} W_{α, 3} - {(P_{z})}_{i} W_{α, 2}] \sqrt{m_{i}} \\ D_{5 α, i} = [{(P_{z})}_{i} W_{α, 1} - {(P_{x})}_{i} W_{α, 3}] \sqrt{m_{i}} \\ D_{6 α, i} = [{(P_{x})}_{i} W_{α, 2} - {(P_{y})}_{i} W_{α, 1}] \sqrt{m_{i}} \end{array} \end{array}

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\begin{array}{r} L_{v i b}^{T} {(D^{T} H D)}_{v i b} L_{v i b} = Λ_{v i b} \end{array}

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

(41)

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\begin{array}{r} Q (u) = \frac{1}{Q (u_{i})} = \frac{\tanh (u_{i} / 2)}{u_{i} / 2} \end{array}

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\begin{array}{r} ⟨ Q_{k} ⟩ = ⟨ \sum_{j} {| b_{j}^{T} (x) \cdot b_{0 k} (x_{0}) |}^{2} Q_{j} (x) ⟩ \end{array}

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\begin{array}{r} M_{t h e r m} = M^{1 / 2} T_{0} ⟨ Q ⟩ T_{0}^{T} M^{1 / 2} \end{array}

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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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