Update paper.md

paper/paper.md

@@ -51,7 +51,7 @@ The Jastrow factor and the sum of Slater determinants are then multiplied to yie
 
 QMC simulations use samples of the electronic density to approximate the total energy of the system. In `QMCTorch`, Markov-Chain Monte-Carlo (MCMC) techniques, namely Metropolis-Hasting and Hamiltonian Monte-Carlo, are used to obtained those sample. Each sample, $R_i$, contains the positions of all the electrons contained in the system. MCMC techniques require the calculation of the density for a given positions of the electrons: $\rho(R_i) = |\Psi(R_i)|^2$ that can simply obtained by squaring the result of a forward pass of the network described above.
 
-The value of local energy of the system is then computed at each sampling point and these values are summed up to compute the total energy of the system: $E = \sum_i \frac{H\Psi(R_i)}{\Psi(R_i)}$, where $H$ is the Hamiltonian of the molecular system: $H = -\frac{1}{2}\sum_i \Delta_i + V_{ee} + V_{en}$, with $\Delta_i$ the Laplacian w.r.t the i-th electron, $V_{ee}$ the coulomb potential between the electrons and $V_{en}$ the electron-nuclei potential. in `QMCTorch`, the calculation of the Laplacian of the Slater determinants can be performed using automatic differentiation but analytical expressions have also been implemented as they are computationally more robust and less expensive [@jacobi_trace]. The gradients of the total energy w.r.t the variational parameters of the wave function, i.e. $\frac{\partial E}{\partial \theta_i}$ are simply obtained via automatic differentiation. Thanks to this automatic differentiation, users can define new kernels for the backflow transformation and Jastrow factor without having to derive analytical expressions of the energy gradients. 
+The value of local energy of the system is then computed at each sampling point and these values are summed up to compute the total energy of the system: $E = \sum_i \frac{H\Psi(R_i)}{\Psi(R_i)}$, where $H$ is the Hamiltonian of the molecular system: $H = -\frac{1}{2}\sum_i \Delta_i + V_{ee} + V_{en}$, with $\Delta_i$ the Laplacian w.r.t the i-th electron, $V_{ee}$ the coulomb potential between the electrons and $V_{en}$ the electron-nuclei potential. In `QMCTorch`, the calculation of the Laplacian of the Slater determinants can be performed using automatic differentiation but analytical expressions have also been implemented as they are computationally more robust and less expensive [@jacobi_trace]. The gradients of the total energy w.r.t the variational parameters of the wave function, i.e. $\frac{\partial E}{\partial \theta_i}$ are simply obtained via automatic differentiation. Thanks to this automatic differentiation, users can define new kernels for the backflow transformation and Jastrow factor without having to derive analytical expressions of the energy gradients. 
 
 Any optimizer included in `PyTorch` (or compatible with it) can then used to optimize the wave function. This gives users access to a wide range of optimization techniques that they can freely explore for their own use cases. Users can also decide to freeze certain variational parameters or defined different learning rates for different layers. Note that the positions of atoms are also variational parameters, and therefore one can perform geometry optimization using `QMCTorch`. At the end of the optimization, all the information relative to the simulations are dumped in a dedicated HDF5 file to enhance reproducibility of the simulations.