We intend to compute some material properties of a metallic hydride with crystalline structure.

Introduction

The meanfield Lagrangian (free-entropy) has the following structure: \(\mathcal{L}_0[\{h_0\}, \{\gamma\}]\ =\ k_{\rm B} \log{\Xi}_0\ +\ k_B \{ \beta \}^T \Big( \sum_{i\in{I_\text{M}}} \langle \bar{h}_{i} \rangle_0 \Big)\ -\ k_B \{ \gamma \}^T \{ x \} ,\) where $\Xi_0$ is the meanfield grand-canonical partition function: \(\mathnormal{\Xi}_0\ =\ \left(\prod_{i\in{I_\text{M}}} \Big(\dfrac{\bar{\sigma}_i\sqrt{k_{\text{B}} \Theta m_{\text{M}}}}{\hbar}\Big)^3 \right) \left( \prod_{i\in{I_\text{H}}} \Big(\dfrac{\bar{\sigma}_i\sqrt{k_{\text{B}} \Theta m_{\text{H}}}}{\hbar}\Big)^3 \Big( 1 + {\rm e}^{\bar{\gamma}_{i}+\gamma_i}\Big) \right),\) $k_B{ \beta } \equiv (k_B \beta_i){i=1}^{N}$ is a Lagrange multiplier devoted to enforce finite temperature conditions, under isothermal conditions: \(k_B \beta_i\ =\ k_B \beta_i \frac{1}{k_{\rm B} \Theta}\ =\ \frac{1}{\Theta} \quad \forall\ i,\) where $k{\text{B}}$ is the Boltzmann constant and $\Theta$ the system\textquotesingle s temperature. $\langle \bar{h}{i} \rangle_0$ is the meanfield Hamiltonian: \(\langle \bar{h}_{i} \rangle_0\ =\ \langle V_i \rangle_0\ +\ \frac{1}{2m_i} |\bar{p}_i|^2\ +\ \frac{1}{2} k_{\rm B} \Theta,\) and $k_B{ \gamma }\equiv (k_B {\gamma}_i){i=1}^{N}$ is a Lagrange multiplier devoted to enforce the chemical potential with $\gamma_i = (\gamma_{ik})_{k=1}^M$ and: \(\sum_{i \in I_\text{M}} \gamma_i = 0.\) After some simplifications and algebraic manipulations, $\mathcal{L}_0$ reduces to \(\mathcal{L}_0[\{h_0\}, \{\gamma\}] = k_{\rm B} \log{\Xi}_0 + \frac{1}{T} \langle V \rangle_0 + \frac{1}{2} N k_{\rm B} - k_B \sum_{i \in I_\text{H}} \gamma_i \frac{ e^{\bar{\gamma}_{i}+\gamma_i}}{1 + {\rm e}^{\bar{\gamma}_{i}+\gamma_i}},\) where $\log{\Xi}_0$ is \(\log{\Xi}_0\ =\ \sum_{i \in I_\text{M}} 3 \log{\dfrac{\bar{\sigma}_i\sqrt{k_{\text{B}}\, \Theta\, m_{\text{M}}}}{\hbar}}\ +\ \sum_{i \in I_\text{H}} 3 \log{\dfrac{\bar{\sigma}_i\sqrt{k_{\text{B}}\, \Theta\, m_{\text{H}}}}{\hbar}}\ +\ \log{\Big( 1 + {\rm e}^{\bar{\gamma}_{i}+\gamma_i}\Big)}.\)

Thermostat

This function aims to find the value of $\textbf{U}$ on a NVT system such that is the solution to the following minimization problem: \(\min_{\textbf{U}, \bar{\sigma}} \psi (\Theta, \{h_0\}, \{\gamma\})\) For this particular case, the \textit{reference} configuration correspond to the undeformed system at 0 K. Therefore, $\psi$ is: \(\psi\ =\ \frac{1}{\Omega} \left(\mathcal{F}_0(\bar{\textbf{q}},\bar{\sigma}) - V \right),\) where $V$ is the total energy of the system at 0 K. The corresponding Euler-Lagrange optimality conditions evaluate to: \(\begin{split} \frac{\partial \psi}{\partial \textbf{U}}\ &=\ \frac{1}{\Omega} \frac{\partial \mathcal{F}_0(\bar{\textbf{q}},\bar{\sigma})}{\partial \textbf{U}}\, -\, \psi \textbf{U}^{-T}\, =\, 0 \\ \frac{\partial \psi}{\partial \bar{\sigma}_i}\ &=\ \frac{1}{\Omega} \frac{\partial \mathcal{F}_0(\bar{\textbf{q}},\bar{\sigma})}{\partial \bar{\sigma}_i}\, =\, 0 \end{split}\)