mirror of https://github.com/QMCPACK/qmcpack.git
Merge pull request #944 from anbenali/Manual_LCAO_features
Manual for PBC gaussians for gamma Kpoint
This commit is contained in:
Paul R. C. Kent
GitHub
commit
1b028a62fa
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@ -24,6 +24,7 @@ different file formats and QMCPACK (e.g. ppconvert and convert4qmc),
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\section{Converters}
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\input{convert4qmc}
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\subsection{pw2qmcpack.x}
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\label{sec:pw2qmcpack}
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pw2qmcpack.x is an executable that converts PWSCF wavefunctions to QMCPACK readable
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HDF5 format. This utility is built alongside the Quantum Espresso post-processing utilities.
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This utility is written in Fortran90, and is distributed as a patch of the Quantum Espresso
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@ -1,16 +1,18 @@
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@article{Ceperley1978,
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title = {Ground state of the fermion one-component plasma: A Monte Carlo study in two and three dimensions},
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author = {Ceperley, D.},
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journal = {Phys. Rev. B},
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volume = {18},
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issue = {7},
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pages = {3126--3138},
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numpages = {0},
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year = {1978},
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month = {Oct},
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% Encoding: UTF-8
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@Article{Ceperley1978,
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author = {Ceperley, D.},
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title = {Ground state of the fermion one-component plasma: A Monte Carlo study in two and three dimensions},
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journal = {Phys. Rev. B},
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year = {1978},
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volume = {18},
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pages = {3126--3138},
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month = oct,
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doi = {10.1103/PhysRevB.18.3126},
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issue = {7},
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numpages = {0},
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publisher = {American Physical Society},
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doi = {10.1103/PhysRevB.18.3126},
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url = {https://link.aps.org/doi/10.1103/PhysRevB.18.3126}
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url = {https://link.aps.org/doi/10.1103/PhysRevB.18.3126},
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}
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@ -582,3 +584,5 @@ eprint = {
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pages={},
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year={2018},
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}
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@Comment{jabref-meta: databaseType:bibtex;}
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@ -0,0 +1,297 @@
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\chapter{Periodic LCAO for solids}
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\label{chap:LCAO}
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\section{Introduction}
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QMCPACK implements linear combination of atomic orbitals (LCAO) and Gaussian
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basis sets in periodic boundary conditions. This method uses orders of
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magnitude less memory than the real space spline wavefunction. While
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the spline scheme enables very fast evaluation of the wavefunction, it may
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require too much on-node memory for a large complex cell. The periodic
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Gaussian evaluation provides a fallback that will definitely fit in
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available memory, but at significantly increased computational
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expense. Well designed Gaussian basis sets should be used to accurately
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represent the wavefunction, typically
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including both diffuse and high angular momentum functions.
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The initial implementation is limited to the $\Gamma$-point using
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trial wavefunctions generated by PySCF\cite{Sun2018}, but other codes such as
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Crystal can be interfaced on request.
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%\subsection{Single Particle Orbitals}
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%
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%In QMC the many-body trial wavefunction is expressed as the product of an antisymmetric part and a correlating Jastrow factor:
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% \begin{equation}
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%\Psi_T(\vec{R}) = \mathcal{A}(\vec{R}) \exp\left[\sum_i J_i(\vec{R})\right]
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%\end{equation}
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%
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%Where $\Psi_T(\vec{R})$ is the trial wave function, $\vec{R}$ is a space spin coordinates, $J(\vec{R})$ the jastrow function and $\mathcal{A}(\vec{R})$ the antisymmetric wavefunction. $\mathcal{A}(\vec{R})$ is traditionally obtained from methods such as DFT, Hartree Fock, MCSCF or CI expansion. Many trial-wavefunctions forms have been explored, but the most popular and effective general form remains the Slater Jastrow form
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% \begin{equation}
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%\Psi_T(\vec{R}) = \exp\left[\sum_i J_i(\vec{R})\right]\sum_k^M C_kD_k^{\uparrow}(\varphi)D_k^{\downarrow}(\varphi)
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%\end{equation}
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%Where $D_k^{\downarrow}(\varphi)$ is a slater determinant expressed in terms of single particle orbitals (SPO) $\varphi_i=\sum^{N_b}_l C_l ^i \Phi_l$ . The choice of SPO representation is crucial for QMC as the cost of computing $\Phi_l$ scales linearly with the number of basis functions evaluation. The scaling grows with the system size and the total evaluation of the N SPOs scales as $ \mathcal{O}(N)^3$ per Monte Carlo step. In the QMCPACK parallelization scheme, SPOs are stored in read only memory replicated on each node or GPU, limiting the size of the systems to the available memory per node.
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%
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%In real space QMC methods it is standard to use a real space b-spline scheme or a closely related method, due to the considerable speedup over plane-waves while retaining simple convergence properties. Use of atomic orbitals and Gaussians that include more physics or chemistry results in much more efficient basis sets, but gives up easy convergence properties.
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%
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%\subsubsection{B-splines}
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% 3D tricubic B-splines provide a basis in which only
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%64 elements are nonzero at any given point in space.
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%The one-dimensional cubic B-spline is given by,
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%\begin{equation}
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%f(x) = \sum_{i'=i-1}^{i+2} b^{i'\!,3}(x)\,\, p_{i'},
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%\label{eq:SplineFunc}
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%\end{equation}
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%where $b^{i}(x)$ are $p_i$ the piecewise cubic polynomial basis functions
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%and $i = \text{floor}(\Delta^{-1} x)$ is the index of
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%the first grid point $\le x$. Constructing a tensor product in each Cartesian
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%direction, we can represent a 3D orbital as
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%\begin{equation}
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% \phi_n(x,y,z) =
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% \!\!\!\!\sum_{i'=i-1}^{i+2} \!\! b_x^{i'\!,3}(x)
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% \!\!\!\!\sum_{j'=j-1}^{j+2} \!\! b_y^{j'\!,3}(y)
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% \!\!\!\!\sum_{k'=k-1}^{k+2} \!\! b_z^{k'\!,3}(z) \,\, p_{i', j', k',n}.
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%\label{eq:TricubicValue}
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%\end{equation}
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%This allows the rapid evaluation of each orbital in constant time.
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%Furthermore, this basis is systematically improvable with a single spacing
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%parameter, so that accuracy is not compromised and convergence checks are simple.
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%
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%Trial wavefunctions for materials are commonly produced using plane wave codes such as Quantum Espresso. The conversion to real space b-splines is straightforward. Compared to directly evaluating Fourier series, b-splines are approximately one order of magnitude faster, with the speedup increasing with system size.
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%
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%\subsubsection{Linear Combination of Atomic Orbitals (LCAO)}
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LCAO schemes use physical considerations to construct a highly
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efficient basis set compared to plane waves. Typically only a few tens
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of basis functions per atom are required compared to thousands of
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plane-waves. Many forms of LCAO schemes exist and are being
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implemented in QMCPACK. The details of the already implemented methods
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will be described in the following section of the manual.
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\noindent \textbf{Gaussian Trial Orbitals (GTOs):}
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The Gaussian basis functions follow a radial-angular decomposition
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\begin{equation}
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\phi (\mathbf{r} )=R_{l}(r)Y_{lm}(\theta ,\phi )
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\end{equation}
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where $ Y_{{lm}}(\theta ,\phi )$ is a spherical harmonic, $l$ and $m$
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are the angular momentum and its $z$ component, and $r,\theta ,\phi $
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are spherical coordinates. In practice they are atom centered and the
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$l$ expansion typically includes 1-3 additional channels compared to
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the formally occupied states of the atom. e.g. 4-6 for a nickel atom with
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occupied $s$, $p$, and $d$ electron shells.
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The evaluation of GTOs within PBC differs slightly from evaluating
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GTOs in Open Boundary Conditions (OBC). The orbitals are evaluated at
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a distance $r$ in the primitive cell (similar to OBC) and then the
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contributions of the periodic images are added by evaluating the
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orbital at a distance $r+T$ where T is a translation of the cell
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lattice vector. This requires loops over the periodic images until the
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contributions are orbitals $\Phi$. In the current implementation, the
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number of periodic images is an input parameter named
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\textit{PBCimages}, which takes three integers corresponding to the
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number of periodic images along the supercell axes (X, Y and Z axes
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for a cubic cell). By default these parameters are set to
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\textit{PBCimages= 5 5 5} but they \textbf{require manual convergence
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checks}. Convergence checks can be performed by checking the total
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energy convergence with respect to \textit{PBCimages}, similar to checks
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performed for plane wave cutoff energy and b-spline grids. Use of
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diffuse Gaussians may require these parameters to be increased, while
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sharply localized Gaussians may permit a decrease. The cost of
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evaluating the wavefunction increases sharply as \textit{PBCimages} is
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increased. This input parameter will be replaced by a tolerance
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factor and numerical screening in future.
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\section{Generating and using periodic gaussian trial wavefunctions
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using PySCF}
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Similar to any QMC calculation, using periodic GTOs requires the
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generation of a periodic trial wavefunction. QMCPACK is currently
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interfaced to PySCF which is a
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multipurpose electronic structure written mainly in Python with key
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numerical functionality implemented via optimized C and C++
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libraries\cite{Sun2018}. Such a wavefunction can be generated
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following the example
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for a 2x1x1 supercell, below. Note that the current
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implementation and example covers only the use of the Gamma ($\Gamma$)
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point. More general and multiple k-points (real and complex) will be
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supported in future releases.
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\begin{lstlisting}[caption=Example PySCF input for single k-point calculation for a 2x1x1 Carbon supercell.]
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#!/usr/bin/env python
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import numpy
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from pyscf.pbc import gto, scf, dft
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from mpi4pyscf.pbc import df
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from pyscf.pbc.tools.pbc import super_cell
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nmp = [2, 1, 1]
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cell = gto.Cell()
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cell.a = '''
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3.37316115 3.37316115 0.00000000
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0.00000000 3.37316115 3.37316115
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3.37316115 0.00000000 3.37316115'''
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cell.atom = '''
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C 0.00000000 0.00000000 0.00000000
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C 1.686580575 1.686580575 1.686580575
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'''
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cell.basis='bfd-vtz'
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cell.ecp = 'bfd'
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cell.unit='B'
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cell.drop_exponent=0.1
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cell.verbose = 5
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cell.build()
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supcell = super_cell(cell, nmp)
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mydf = df.FFTDF(supcell)
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mydf.auxbasis = 'weigend'
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mf = dft.RKS(supcell)
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mf.xc = 'lda'
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mf.exxdiv = 'ewald'
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mf.with_df = mydf
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e_scf=mf.kernel()
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print 'e_scf',e_scf
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kpts=[]
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title="C_Diamond"
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from PyscfToQmcpack import savetoqmcpack
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savetoqmcpack(supcell,mf,title=title,kpts=kpts)
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\end{lstlisting}
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Note that the last 4 lines of the file
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\begin{lstlisting}
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kpts=[]
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title="C_Diamond"
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from PyscfToQmcpack import savetoqmcpack
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savetoqmcpack(supcell,mf,title=title,kpts=kpts)
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\end{lstlisting}
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contains an empty list of k-points (since this is a gamma point
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calculation) and a title. The title variable will be the name of the
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HDF5 file where all the data needed by QMCPACK will be stored. The
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function \textit{savetoqmcpack} will be called at the end of the
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calculation and will generate the HDF5 similarly to the non-periodic
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PySCF calculation in section~\ref{sec:convert4qmc} (convert4qmc). The
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function is distributed with QMCPACK and located in the
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qmcpack/src/QMCTools directory under the name
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\textit{PyscfToQmcpack.py}. In order for the script to work, you need
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to specify the path to the file in your PYTHONPATH such as
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\begin{lstlisting}
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export PYTHONPATH=QMCPACK_PATH/src/QMCTools:$PYTHONPATH
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\end{lstlisting}
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In order to generate QMCPACK input files, you will need to run \textit{convert4qmc} exactly as specified in section ~\ref{sec:convert4qmc};
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\begin{lstlisting}
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convert4qmc -pyscf C_Diamond.h5
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\end{lstlisting}
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This tool can be used with any option described in convert4qmc. Since the HDF5 contains all the information needed for a QMC run, there is no need to specify any other specific tag for periodicity.
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Running such a command will generate 3 input files;\\
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\begin{lstlisting}[caption=CDiamond.structure.xml. This file contains the geometry of the system.]
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<?xml version="1.0"?>
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<qmcsystem>
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<simulationcell>
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<parameter name="lattice">
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6.74632230000000e+00 6.74632230000000e+00 0.00000000000000e+00
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0.00000000000000e+00 3.37316115000000e+00 3.37316115000000e+00
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3.37316115000000e+00 0.00000000000000e+00 3.37316115000000e+00
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</parameter>
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<parameter name="bconds">p p p</parameter>
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<parameter name="LR_dim_cutoff">15</parameter>
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</simulationcell>
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<particleset name="ion0" size="4">
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<group name="C">
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<parameter name="charge">4</parameter>
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<parameter name="valence">4</parameter>
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<parameter name="atomicnumber">6</parameter>
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</group>
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<attrib name="position" datatype="posArray">
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0.0000000000e+00 0.0000000000e+00 0.0000000000e+00
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1.6865805750e+00 1.6865805750e+00 1.6865805750e+00
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3.3731611500e+00 3.3731611500e+00 0.0000000000e+00
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5.0597417250e+00 5.0597417250e+00 1.6865805750e+00
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</attrib>
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<attrib name="ionid" datatype="stringArray">
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C C C C
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</attrib>
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</particleset>
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<particleset name="e" random="yes" randomsrc="ion0">
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<group name="u" size="8">
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<parameter name="charge">-1</parameter>
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</group>
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<group name="d" size="8">
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<parameter name="charge">-1</parameter>
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</group>
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</particleset>
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</qmcsystem>
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\end{lstlisting}
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As one can see, the cell has been extended to contain 4 atoms in a 2x1x1 Carbon Cell.
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\begin{lstlisting}[caption=CDiamond.wfj-Twist0.xml. This file contains the trial wavefunction.]
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<?xml version="1.0"?>
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<qmcsystem>
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<wavefunction name="psi0" target="e">
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<determinantset type="MolecularOrbital" name="LCAOBSet" source="ion0" transform="yes" twist="0 0 0" href="C_Diamond.h5" PBCimages="5 5 5">
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<slaterdeterminant>
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<determinant id="updet" size="8">
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<occupation mode="ground"/>
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<coefficient size="116" spindataset="0"/>
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</determinant>
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<determinant id="downdet" size="8">
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<occupation mode="ground"/>
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<coefficient size="116" spindataset="0"/>
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</determinant>
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</slaterdeterminant>
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</determinantset>
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<jastrow name="J2" type="Two-Body" function="Bspline" print="yes">
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<correlation size="10" speciesA="u" speciesB="u">
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<coefficients id="uu" type="Array"> 0 0 0 0 0 0 0 0 0 0</coefficients>
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</correlation>
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<correlation size="10" speciesA="u" speciesB="d">
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<coefficients id="ud" type="Array"> 0 0 0 0 0 0 0 0 0 0</coefficients>
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</correlation>
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</jastrow>
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<jastrow name="J1" type="One-Body" function="Bspline" source="ion0" print="yes">
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<correlation size="10" cusp="0" elementType="C">
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<coefficients id="eC" type="Array"> 0 0 0 0 0 0 0 0 0 0</coefficients>
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</correlation>
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</jastrow>
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</wavefunction>
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</qmcsystem>
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\end{lstlisting}
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This files contains information related to the trial wavefunction. It is identical to the input file from an Open Boundary Conditions calculation to the exception of the following tags:\\
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\begin{table}[h]
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\begin{center}
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\begin{tabularx}{\textwidth}{l l l l l }
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\hline
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\multicolumn{5}{l}{*.wfj.xml specific tags} \\
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\hline
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%\multicolumn{2}{l}{Outputfiles} & \multicolumn{3}{l}{}\\
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& \bfseries tag & \bfseries tag type & \bfseries default & \bfseries description \\
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& \texttt{twist } & 3 doubles & Gamma ( 0 0 0)& coordinate of the twist to compute\\
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& \texttt{href } & string & default& name of the HDF5 file generated by\\
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& & & & PySCF and used for convert4qmc\\
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& \texttt{PBCimages } & 3 Integer & 5 5 5 & Number of periodic images to evaluate the orbitals\\
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\hline
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\end{tabularx}
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\end{center}
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\end{table}
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Other files containing QMC methods (such as optimization, VMC and DMC blocks) will be generated and will behave in a similar fashion regardless of the type of SPO in the trial wavefunction.
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@ -138,7 +138,7 @@ The full QMCPACK pseudopotential is also included in \texttt{oxygen\_atom/refere
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\label{sec:lqb_dft}
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With the pseudopotential in hand, the next step toward a QMC calculation is to obtain the Fermionic part of the wavefunction, in this case a single Slater determinant constructed from DFT-LDA orbitals for a neutral oxygen atom. If you had trouble with the pseudopotential conversion step, pre-converted pseudopotential files are located in the \texttt{oxygen\_atom/reference} directory.
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Quantum ESPRESSO input for the DFT-LDA ground state of the neutral oxygen atom can be found in \texttt{O.q0.dft.in} and also listing \ref{lst:O_q0_dft} below. Setting \texttt{wf\_collect=.true.} instructs Quantum Espresso to write the orbitals to disk at the end of the run. Option \texttt{wf\_collect=.true.} may be a potential problem in large simulations, it is recommended to avoid it and use the converter pw2qmcpack in parallel, see details in Sec.~\ref{app:pw2qmcpack}. Note that the plane-wave energy cutoff has been set to a reasonable value of 300 Ry here (\texttt{ecutwfc=300}). This value depends on the pseudopotentials used, and in general should be selected by running DFT$\rightarrow$(orbital conversion)$\rightarrow$VMC with increasing energy cutoffs until the lowest VMC total energy and variance is reached.
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Quantum ESPRESSO input for the DFT-LDA ground state of the neutral oxygen atom can be found in \texttt{O.q0.dft.in} and also listing \ref{lst:O_q0_dft} below. Setting \texttt{wf\_collect=.true.} instructs Quantum Espresso to write the orbitals to disk at the end of the run. Option \texttt{wf\_collect=.true.} may be a potential problem in large simulations, it is recommended to avoid it and use the converter pw2qmcpack in parallel, see details in Sec.~\ref{sec:pw2qmcpack}. Note that the plane-wave energy cutoff has been set to a reasonable value of 300 Ry here (\texttt{ecutwfc=300}). This value depends on the pseudopotentials used, and in general should be selected by running DFT$\rightarrow$(orbital conversion)$\rightarrow$VMC with increasing energy cutoffs until the lowest VMC total energy and variance is reached.
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\begin{lstlisting}[caption={Quantum ESPRESSO input file for the neutral oxygen atom (\texttt{O.q0.dft.in})\label{lst:O_q0_dft}}]
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&CONTROL
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|
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@ -1,5 +1,15 @@
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\section{Obtaining pseudopotentials}
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\subsection{Pseudopotentiallibrary.org}
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An open website collecting community developed and tested
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pseudopotentials for QMC and other many-body calculations is being
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developed at \url{https://pseudopotentiallibrary.org}. This site
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includes potentials in QMCPACK format and an increasing range of
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electronic structure and quantum chemistry codes. We recommend using
|
||||
potentials from this site if available and suitable for your science
|
||||
application.
|
||||
|
||||
\subsection{Opium}
|
||||
\label{subsec:opium}
|
||||
|
||||
|
|
@ -40,7 +50,7 @@ The QMC code CASINO also makes available its pseudopotentials available at the w
|
|||
\begin{lstlisting}[caption=Convert CASINO-formatted pseudopotential to Quantum ESPRESSO UPF format]
|
||||
ppconvert --casino_pot pp.data --casino_us awfn.data_s1_2S --casino_up awfn.data_p1_2P --casino_ud awfn.data_d1_2D --upf Li.TN-DF.upf
|
||||
\end{lstlisting}
|
||||
QMCPACK can directly read in the CASINO-formated pseudopotential (\texttt{pp.data}), but four parameters found in the pseudopotential summary file must be specified in the pseudo element (\texttt{l-local}, \texttt{lmax}, \texttt{nrule}, \texttt{cutoff})[see Section~\ref{subsec:pseudopotentials} for details]:
|
||||
QMCPACK can directly read in the CASINO-formated pseudopotential (\texttt{pp.data}), but four parameters found in the pseudopotential summary file must be specified in the pseudo element (\texttt{l-local}, \texttt{lmax}, \texttt{nrule}, \texttt{cutoff})[see Section~\ref{sec:nlpp} for details]:
|
||||
\begin{lstlisting}[caption=XML syntax to use CASINO-formatted pseudopotentials in QMCPACK]
|
||||
<pairpot type="pseudo" name="PseudoPot" source="ion0" wavefunction="psi0" format="xml">
|
||||
<pseudo elementType="Li" href="Li.pp.data" format="casino" l-local="s" lmax="2" nrule="2" cutoff="2.19"/>
|
||||
|
|
|
|||
|
|
@ -191,12 +191,14 @@
|
|||
|
||||
\input{output_overview}
|
||||
\input{analysis}
|
||||
|
||||
\input{gaussian_orbitals_solids}
|
||||
\input{selected_ci}
|
||||
|
||||
\input{afqmc}
|
||||
\input{integrals_for_afqmc}
|
||||
|
||||
|
||||
|
||||
\input{examples}
|
||||
|
||||
% labs: import each as a separate chapter for now
|
||||
|
|
|
|||
|
|
@ -5,8 +5,19 @@ In this section we describe the use of localized basis sets to expand the \textt
|
|||
\begin{equation}
|
||||
\phi_i(\vec{r}) = \sum_k C_{i,k} \ \eta_k(\vec{r}),
|
||||
\end{equation}
|
||||
where $\{\eta_k(\vec{r})\}$ is a set of M atom-centered basis functions and $C_{i,k}$ is a coefficient matrix. This \texttt{sposet} should be used in calculations of finite systems employing an atom-centered
|
||||
basis set and is typically generated by the \textit{convert4qmc} converter. (While it is possible to use this \texttt{sposet} on calculations of periodic systems, this feature is not currently implemented in QMCPACK.) Examples include calculations of molecules using gaussian basis sets or slater-type basis functions. Even though this section is called "Gaussian basis set" (by far the most common atom-centered basis set), QMCPACK works with any atom-centered basis set built based on either spherical harmonic angular functions or cartesian angular expansions. The radial functions in the basis set can be expanded in either gaussian functions, slater-type functions or numerical radial functions.
|
||||
where $\{\eta_k(\vec{r})\}$ is a set of M atom-centered basis
|
||||
functions and $C_{i,k}$ is a coefficient matrix. This \texttt{sposet}
|
||||
should be used in calculations of finite systems employing an
|
||||
atom-centered basis set and is typically generated by the
|
||||
\textit{convert4qmc} converter. Examples include calculations of
|
||||
molecules using gaussian basis sets or slater-type basis
|
||||
functions. Initial support for periodic systems is described in Sec.
|
||||
\ref{chap:LCAO}. Even though this section is called "Gaussian basis
|
||||
sets" (by far the most common atom-centered basis set), QMCPACK works
|
||||
with any atom-centered basis set built based on either spherical
|
||||
harmonic angular functions or cartesian angular expansions. The radial
|
||||
functions in the basis set can be expanded in either gaussian
|
||||
functions, slater-type functions or numerical radial functions.
|
||||
|
||||
In this section we describe the input sections for the atom-centered basis set and the \texttt{sposet} for a single slater determinant trial wavefunction. The input sections for multideterminant trial wavefunctions are described in section \ref{sec:multideterminants}. The basic structure for the input block of a single slater determinant is given in Listing \ref{listing:lcaosposet}.
|
||||
A list of options for \texttt{determinantset} associated with this \texttt{sposet} is given in Table \ref{table:determinantset}.
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
\subsection{Spline basis sets}
|
||||
\label{sec:spo_spline}
|
||||
In this section we describe the use of spline basis sets to expand the \texttt{sposet}.
|
||||
Spline basis sets are designed to work seamless with plane wave DFT code, e.g.\ Quantum ESPRESSO as a trial wavefunction generator.
|
||||
Spline basis sets are designed to work seamlessly with plane wave DFT code, e.g.\ Quantum ESPRESSO as a trial wavefunction generator.
|
||||
|
||||
In QMC algorithms, all the SPOs $\{\phi(\vec{r})\}$ need to be updated every time a single electron moves.
|
||||
Evaluating SPOs takes very large portion of computation time.
|
||||
|
|
|
|||
Loading…
Reference in New Issue