Commit 3fb673b6 by Thomas Ple

Modified method description

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 ... ... @@ -23,21 +23,26 @@ Purpose of Module ================= Module **PIM_qtb** generates trajectories based on several classical and semi-classical stochastic methods: - Langevin classical dynamics Module **PIM_qtb** generates trajectories according to one of the following stochastic methods: - Classical Langevin dynamics - Quantum Thermal Bath [Dam]_ - Adaptive Quantum Thermal Bath [Man]_ These trajectories can be used to sample initial conditions for intramolecular vibrational-energy redistribution (IVR) dynamics. These trajectories can be used to sample initial conditions for Linearized Semi-Classical Initial Value Representation (LSC-IVR) calculations. Description of the module ========================= The module implements various methods based on Langevin dynamics to sample initial conditions for IVR or to directly exploit the generated trajectories. The methods implemented are: classical Langevin dynamics, The module implements different methods based on Langevin dynamics. The trajectories generated can be exploited directly or used to sample initial conditions for LSC-IVR calculations. The methods implemented are: classical Langevin dynamics, Quantum Thermal Bath (QTB) and two variants of adaptive QTB (adQTB-r and adQTB-f). ... ... @@ -47,15 +52,15 @@ Classical Langevin dynamics Classical Langevin dynamics is described by a stochastic differential equation : .. math:: m \dot v = -\nabla U -m \gamma v + F(t) .. math:: \dot p = -\nabla U - \gamma p + F(t) :label: eqLGV where :math:-\nabla U is the vector forces exerted on the sets of atoms, :math:m the mass of atoms, :math:\gamma the damping term determining the strength of a viscous force and :math:F(t) a stochastic noise. The stochastic noise is gaussian and delta-correlated. To ensure the classical fluctuation-dissipation theorem, one might write its autocorrelation function spectra as : where :math:p is the momentum vector of the set of atoms, interacting via the potential :math:U , :math:\gamma is the damping coefficient and :math:F(t) is the stochastic force. The random force :math:F(t) is gaussian and delta-correlated. To enforce the classical fluctuation-dissipation theorem, its autocorrelation spectrum is given by : .. math:: C_{FF}(\omega)= \int \limits_{-\infty}^{+\infty} dt \langle F(t) F(t+\tau) \rangle e^{-i \omega t} = 2m \gamma k_B T ... ... @@ -65,12 +70,10 @@ temperature. Quantum Thermal Bath (QTB) -------------------------- Quantum Thermal Bath is an approximate semi-classical method [Dam]_ consisting in modifying the stochastic noise in eq. :eq:eqLGV in order to mimic the energy distribution of a set of quantum harmonic oscillators. In QTB dynamics, stochastic noise is no longer delta-correlated but is colored with the following form : The Quantum Thermal Bath uses a generalized Langevin equation in order to approximate nuclear quantum effects[Dam]_ . In QTB dynamics, the stochastic force is no longer delta-correlated but is colored according to the following formula : .. math:: C_{FF}(\omega)=2m \gamma \theta(\omega,T) :label: eqQTB ... ... @@ -79,23 +82,25 @@ with .. math:: \theta(\omega,T) = \hbar \omega \left[\frac{1}{2}+\frac{1}{e^{\hbar \beta \omega}-1}\right] where :math:\beta = \frac{1}{k_BT} and :math:2 \pi \hbar is Planck constant. :math:\theta(\omega,T) function represents the energy distribution of a quantum harmonic oscillators of angular frequency :math:\omega at a temperature :math:T (Bose function) and notably, its zero-point energy. This method is known to lead to qualitatively good results [Bri]_ but as most of the semi-classical methods, it suffers from zero-point energy leakage [Hern]_. where :math:\beta = \frac{1}{k_BT} and :math:2 \pi \hbar is the Planck constant. The function :math:\theta(\omega,T) describes the energy of a quantum harmonic oscillators of angular frequency :math:\omega at a temperature :math:T . The colored random force allows to approximate zero-point energy contributions to the equilibrium properties of the system. The QTB method is known to lead to qualitatively good results [Bri]_ but as many semi-classical methods, it suffers from zero-point energy leakage (ZPEL) [Hern]_. Adaptive Quantum Thermal Bath (adQTB-r/f) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Adaptive Quantum Thermal Bath is a variant of QTB to ensure for each degree of freedom and each frequency the energy distribution provided by the quantum fluctuation-dissipation theorem all along the trajectories. The Adaptive Quantum Thermal Bath is an extension of the QTB method, designed to eliminate the zero-point energy leakage by enforcing the energy distribution prescribed by the quantum fluctuation-dissipation theorem for each degree of freedom and each frequency, all along the trajectories [Man]_ . In practice, one minimizes during QTB dynamics a fluctuation-dissipation relation :math:\Delta_{FDT} [Man]_ defined as: In practice, this is done by minimizing the fluctuation-dissipation spectrum :math:\Delta_{FDT} defined as: .. math:: \Delta_{FDT} (\omega) = {\rm{Re}} \left[C_{vF}(\omega)\right] - m \gamma_{r} (\omega) C_{vv} (\omega) :label: eqDFDT ... ... @@ -105,28 +110,31 @@ spectrum, :math:C_{vv} the velocity autocorrelation spectrum and :math:\gamma_{r} a set of damping coefficients dependent (or not) on the frequency. This minimization is carried out by dissymetrizing the system-bath coefficients between the injected and the extracted energy distribution. One can do it either by directly modifying the spectrum of the random noise :math:F(t) with frequency dependent damping term This minimization is carried out on the fly during the QTB simulation by dissymetrizing the system-bath coupling coefficients corresponding to the damping force (dissipation) and to the random force (energy injection). This can be done either by directly modifying the random force spectrum :math:F(t) with frequency dependent damping term :math:\gamma_r(\omega) (adQTB-r variant) or by modifying the memory dissipative kernel :math:\gamma_{f} (\omega) within the framework of a generalized Langevin equation (adQTB-f variant). kernel of the dissipative force :math:\gamma_{f} (\omega) within the framework of a non-Markovian generalized Langevin equation (adQTB-f variant). The coefficients :math:\gamma_r or :math:\gamma_f are slowly adjusted with a first-order differential equation and a damping term adjusted with a first-order differential equation and an adaptation coefficient :math:A_\gamma : .. math:: \frac{d }{dt}\gamma_{r/f} (\omega) \propto A_\gamma \gamma \Delta_{FDT,r/f}(\omega,t) :label: eqgammadapt during a “thermalization time” until they reach convergence. Then, observables are computed by keeping on active the adaptive process. Further and more precise implementation details can be found in ref.[Man]_. during a preliminary “adaptation time” until they reach convergence. Observables are then computed while the adaptive process is kept active. Further informations and precise implementation details can be found in ref.[Man]_. Two implementations are currently available in PaPIM: #. Random force adaptive QTB (adQTB-r) In this variant, the dissipation #. Random force adaptive QTB (adQTB-r) In this variant, the dissipation kernel is left unchanged, i.e. :math:\gamma_{f}(\omega) = \gamma while the random force is modified according to a frequency-dependent set of damping coefficients :math:\gamma_r(\omega) to satisfy ... ... @@ -139,15 +147,17 @@ This method is applicable only if the initial damping coefficient :math:\gamma is large enough to compensate effects of a possible zero-point energy leakage. #. Dissipative kernel adaptive QTB (adQTB-f) In this approach, the random force is not modified (i.e. :math:\gamma_{r} (\omega) = \gamma which remains the same as in QTB formalism(eq. :eq:eqQTB)) but the dissipation term is not only represented as a mere damping viscous term (:math:-m \gamma v) but as a dissipative memory kernel. It leads to a generalized Langevin equation: #. Dissipative kernel adaptive QTB (adQTB-f) In this approach, the random force is not modified, i.e. :math:\gamma_{r} (\omega) = \gamma and remains the same as in the standard QTB method (eq. :eq:eqQTB)) but the dissipation term is not described by a viscous damping term anymore (:math:-m \gamma v) but corresponds to a non-Markovian dissipative force. This leads to the following generalized Langevin equation: .. math:: m \dot v = -\nabla U -m \int_0^\infty \ \gamma_f(\tau) v(t-\tau) \ d\tau + F(t) .. math:: \dot p = -\nabla U - \int_0^\infty \ \gamma_f(\tau) p(t-\tau) \ d\tau + F(t) :label: eqgenlgv In order to avoid solving with brute force this integro-differential ... ... @@ -160,7 +170,7 @@ equally spaced (:math:\Delta \omega) lorentzian functions of width :label: eqlorentzgenlgv The parameter :math:\gamma_{f,j} are then modified to satisfy :math:\Delta_{FDT} = 0 (eq. :eq:eqDFDT). In this method, one should take care of checking results convergence by decreasing the :math:\alpha parameter. :math:\Delta_{FDT} = 0 (eq. :eq:eqDFDT). Input file ========== ... ...
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