Commit 89e27803 by Thomas Ple

### Fixed bugs in description of the method

parent 3fb673b6
Pipeline #5498 failed with stages
in 1 minute and 48 seconds
 ... ... @@ -58,8 +58,7 @@ equation : 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, is a Gaussian white noise: 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 ... ... @@ -71,8 +70,8 @@ Quantum Thermal Bath (QTB) -------------------------- 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 to approximate nuclear quantum effects [Dam]_ . In QTB dynamics, the stochastic force is no longer a white noise but is colored according to the following formula : .. math:: C_{FF}(\omega)=2m \gamma \theta(\omega,T) ... ... @@ -85,8 +84,8 @@ with 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. :math:\omega at a temperature :math:T . The colored random force allows approximating 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]_. ... ... @@ -104,6 +103,7 @@ In practice, this is done by minimizing the fluctuation-dissipation spectrum .. math:: \Delta_{FDT} (\omega) = {\rm{Re}} \left[C_{vF}(\omega)\right] - m \gamma_{r} (\omega) C_{vv} (\omega) :label: eqDFDT where :math:C_{vF} is the velocity random force cross-correlation spectrum, :math:C_{vv} the velocity autocorrelation spectrum and ... ... @@ -128,14 +128,14 @@ adjusted with a first-order differential equation and an adaptation coefficient 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]_. 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 kernel is left unchanged, i.e. :math:\gamma_{f}(\omega) = \gamma #. 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 :math:\Delta_{FDT} = 0 (eq. :eq:eqDFDT): ... ... @@ -143,14 +143,14 @@ In this variant, the dissipation .. math:: C_{FF}(\omega)=2m \gamma_r(\omega) \theta(\omega,T) :label: eqadQTBr 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. 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. 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 ... ... @@ -160,15 +160,16 @@ In this approach, the .. 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 equation, the dissipative memory kernel is expressed as a sum of equally spaced (:math:\Delta \omega) lorentzian functions of width :math:\alpha : In order to avoid solving with brute force this integro-differential equation, the dissipative memory kernel is expressed as a sum of equally spaced (:math:\Delta \omega) lorentzian functions of width :math:\alpha : .. math:: \gamma_f(\omega) = \frac{\Delta \omega}{\pi}\sum_{j=0}^{n_\omega} \frac{ \gamma_{f,j} }{\alpha + i(\omega-\omega_j)} +\frac{ \gamma_{f,j}}{\alpha + i(\omega+\omega_j)} :label: eqlorentzgenlgv The parameter :math:\gamma_{f,j} are then modified to satisfy :math:\Delta_{FDT} = 0 (eq. :eq:eqDFDT). ... ...
Markdown is supported
0% or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!