Merge pull request #171 from sunqm/multigrid

Multigrid documentation

Author: tberkel

source/modules/pbc.rst

@@ -38,6 +38,7 @@ The list of modules described in this chapter is:
    pbc/scf.rst
    pbc/dft.rst
    pbc/df.rst
+   pbc/multigrid.rst
    pbc/tdscf.rst
    pbc/mp.rst
    pbc/ci.rst

source/modules/pbc/multigrid.rst

@@ -0,0 +1,250 @@
+.. _pbc_dft_multigrid:
+
+pbc.dft.multigrid --- Multigrid Integration
+*******************************************
+
+Multigrid is a numerical integration algorithm optimized for the computation of
+the Coulomb matrix and exchange-correlation (XC) functional. It can take
+advantage of the locality of density and orbitals when computing electron
+density and matrix elements.
+
+The Multigrid algorithm employs different integration grids for different types
+of orbitals. Diffuse orbitals use sparse integration grids, while compact
+orbitals utilize denser grids. The calculations for compact orbitals are only
+performed within smaller spatial regions. The numerical values from different
+grids are then aggregated in reciprocal space using the fast Fourier transform.
+
+Compared to the `get_j` function provided by the FFTDF module and the XC matrix
+evaluation functions offered by the `numint` module, the Multigrid algorithm
+can achieve an order of magnitude improvement in the computation of the Coulomb
+matrix and DFT XC matrix.
+
+Supported Features
+==================
+
+The Multigrid algorithm is designed to accelerate DFT calculations.
+It can also be utilized to speed up derived properties based on DFT, such as
+TDDFT excited state calculations, stability analysis, analytical nuclear
+gradients, and the computation of the orbital Hessian in second-order SCF
+(SOSCF) methods. However, it does not support the computation of the HF exchange
+matrix or the calculation of MO two-electron integrals in post-HF methods.
+
+In terms of supported systems, the Multigrid algorithm can be applied to both
+periodic systems and molecular calculations. For periodic systems, it supports
+single k-point calculations as well as k-point sampling in DFT calculations. For
+molecular systems, additional configurations are required to enable the multigrid
+algorithm, see :ref:`mole_multigrid`.
+
+Generally, the Multigrid algorithm should be used with pseudopotentials and the
+corresponding basis sets to reduce computational costs and accuracy.
+Using the Multigrid algorithm for all-electron calculations typically results in significant overhead.
+
+
+How to enable MultiGrid algorithm
+=================================
+
+In periodic system calculations, enabling the Multigrid algorithm is
+straightforward. The PBC SCF class provides a method `multigrid_numint()`.
+This method creates a new SCF instance which is configured with the Multigrid
+functionality. This new instance can then be used in the same way as a standard
+SCF instance. For example::
+
+    cell = pyscf.M(
+        a = np.eye(3)*3.5668,
+        atom = '''C     0.      0.      0.    
+                  C     0.8917  0.8917  0.8917
+                  C     1.7834  1.7834  0.    
+                  C     2.6751  2.6751  0.8917
+                  C     1.7834  0.      1.7834
+                  C     2.6751  0.8917  2.6751
+                  C     0.      1.7834  1.7834
+                  C     0.8917  2.6751  2.6751''',
+        basis = 'gth-dzv',
+        pseudo = 'gth-pade',
+    )
+    mf = cell.KRKS(xc='pbe', kpts=cell.make_kpts([3,3,3]))
+    mf = mf.multigrid_numint()
+    mf.run()
+
+Once the Multigrid algorithm is enabled, it is automatically applied to other
+methods based on that SCF instance, such as TDDFT and SOSCF. For example::
+
+    mf = cell.KRKS(xc='pbe', kpts=cell.make_kpts([3,3,3]))
+    mf = mf.multigrid_numint()
+
+    mf = mf.soscf().run()
+
+    mf.TDA().run()
+
+    mf.Gradients().run()
+
+The Multigrid algorithm is enabled through the `._numint` attribute of the SCF
+instance You can directly modify the `._numint` attribute of an SCF instance to
+apply the Multigrid algorithm.::
+
+    from pyscf.pbc.dft.numint import MultiGridNumInt
+    mf = cell.KRKS(xc='pbe', kpts=cell.make_kpts([3, 3, 3]))
+    mf._numint = MultiGridNumInt(cell)
+    mf.run()
+
+Configurable Options
+====================
+
+In most scenarios, the Multigrid module can automatically configure parameters such
+as energy cutoff, radius cutoff, sub tasks to balance computational load and
+accuracy. If you need more precise control over the Multigrid algorithm,
+you can modify the attributes of the `mf._numint`. Here are some configurable options.
+
+* `mesh`: It's a 3-element tuple, which controls the upper limit of the mesh
+  points utilized by the algorithm. By default, this value is estimated based on
+  the basis set, the shape, and the size of the lattice. By default, the error in
+  the electron density at each mesh point is controlled to be below `cell.precision`.
+
+* `ntasks`: The maximum number of subgroups. The Multigrid algorithm divides
+  orbital pairs into subgroups and processes the integration for each subgroup
+  separately. Increasing the number of tasks can better utilize the locality of
+  the orbital functions, reducing the number of floating-point operations
+  (FLOPS). However, it can also increase the overhead of other operations, such
+  as FFT. The recommended value for `ntasks` is typically set between 3 and 6.
+  The default value is 4.
+
+* `xc_with_j`: Determines whether the calculation of the XC matrix should also
+  include the calculation of the Coulomb energy and the Coulomb matrix.
+  Combining the calculation of the XC and Coulomb can reduce the computation cost.
+  For GGA functionals, it can reduce the computational load by 30% - 40%.
+  The default setting is `True`, and it is generally recommended to enable this option.
+
+* `ke_ratio`: Controls the ratio of the energy cutoffs among the different
+  subtasks when partitioning the orbital pairs. `ke_ratio` is a number greater
+  than 1. When approaching 1, the task partitioning is fine, and the orbital
+  locality is better utilized. However, this also increases the number of tasks.
+  Regardless of the configuration of this parameter, the maximum number of tasks
+  generated is still controlled by the `ntasks` attribute.
+
+
+Compatibility with NumInt
+=========================
+
+The `NumInt` class offers numerical integration functions for DFT calculations.
+It is assigned to the `_numint` attribute of SCF classes. This module offers
+APIs such as `get_rho` (for computing electron density in real space) and
+`get_vxc` (for computing the value of the XC energy functional and XC matrices),
+for DFT calculations :ref:`dft`. Both molecular and PBC DFT implementations
+follow this API design.
+
+The `pyscf.pbc.dft.multigrid` module offers the `MultiGridNumInt` class, which
+is compatible with the `NumInt` class. Its methods, such as `get_vxc`, `get_fxc`,
+`get_rho`, `cache_xc_kernel`, `nr_rks`, and `nr_uks`, are mostly compatible
+with the corresponding methods in NumInt (with only a few additional keyword
+arguments for controlling multigrid instances). These methods can be
+individually invoked, like those in the `NumInt` class, to compute densities and
+XC matrix elements.
+
+The `pyscf.pbc.dft.multigrid` module also provides the `MultiGridNumInt2` class,
+which further optimizes the implementations of the `MultiGridNumInt` class.
+However, due to differences in the algorithm implementations, the support for
+optimized k-points and non-orthogonal lattices is not as comprehensive as that
+in the `MultiGridNumInt` class. Currently, the `SCF.multigrid_numint()` method
+invokes the `MultiGridNumInt` class. To maximize the multigrid performance, you
+can manually assign the `MultiGridNumInt2` instance to the `mf._numint`
+attribute.
+
+The two classes will be merged into one in the future release.
+
+
+.. _mole_multgird:
+
+How to Apply Multigrid in Molecular DFT
+=======================================
+
+The Multigrid algorithm is currently designed and implemented for data
+structures with periodic boundary conditions. Nevertheless, it can be adapted
+for molecular calculations.
+
+First, we need to initialize molecule within a relatively large periodic lattice.
+A vacuum space needs to be placed between the box and the molecule to
+simulate free boundary conditions::
+
+    cell = pyscf.M(atom='''
+    O    0.000    0.118  0.
+    H    0.758   -0.470  0.
+    H   -0.758   -0.470  0.''',
+    a=np.eye(3)*10,
+    dimension=0,
+    basis='gth-dzvp', pseudo='gth-pade',)
+
+Here, we apply a 10 x 10 x 10 (unit Angstrom) box. The box does not need to be
+excessively large. It only needs to ensure that the electron density of the molecule
+does not leak outside the box. If there are no diffused functions, typically, a
+5 Angstrom margin around the molecule is sufficient. The lattice is a virtual
+box which does not need to be centered on the origin. There is no need to adjust
+the molecule's coordinates to the center of the box.
+
+Alternatively, the lattice can be automatically determined. We can just create a
+`Mole` instance and then call the `Mole.to_cell()` method to convert a `Mole`
+object into a Cell object::
+
+    mol = pyscf.M(atom='''
+    O    0.000    0.118  0.
+    H    0.758   -0.470  0.
+    H   -0.758   -0.470  0.''',
+    basis='gth-dzvp', pseudo='gth-pade')
+    cell = mol.to_cell(dimension=0)
+
+When initializing the Cell, we set `dimension=0`. This setting informs the
+program that the system is a 0-dimensional system (molecule), which allows the
+program to more effectively utilize this property to truncate the long-range
+Coulomb potential and accelerate the computation of certain integrals.
+The system can also be treated as a 3-dimensional crystal, with slightly
+increased computational load.
+
+In this system, we use the GTH pseudo potential and the GTH basis set. This
+basis set does not have very steep GTO functions, making it suitable for the
+multigrid method.
+
+With this `Cell` object, we can initialize the DFT instance as we would for
+typical PBC DFT calculations. The following example demonstrates another
+way which integrates
+the Multigrid XC matrix functionality into the the molecular DFT instances.::
+
+    from pyscf.pbc.dft.multigrid import MultiGridNumInt2
+    mol = pyscf.M(atom='''
+    O    0.000    0.118  0.
+    H    0.758   -0.470  0.
+    H   -0.758   -0.470  0.''',
+    basis='gth-dzvp', pseudo='gth-pade')
+    mf = mol.RKS(xc='pbe')
+
+    cell = mol.to_cell(dimension=0)
+    mf._numint = MultiGridNumInt2(cell)
+    mf._numint.xc_with_j = False
+
+    mf.run()
+
+In this example, we use the `MultiGridNumInt` class for `mf._numint` of the
+molecular DFT instance. This setup invokes the `MultiGridNumInt` algorithm to
+calculate the DFT XC matrix and XC energy, while calls the standard molecular
+`get_jk` functions for Coulomb energy.
+Note the setting `xc_with_j=False`, which disables the computation of Coulomb
+energy by the `MultiGridNumInt` class. This is necessary because the molecular
+DFT program employs a different method to compute nuclear repulsion energy
+compared to the method used in PBC DFT.
+When using the molecular DFT program in conjunction with `MultiGridNumInt`, we
+should not use `MultiGridNumInt` to compute the Coulomb energy.
+
+
+Limitations of Multigrid algorithm
+==================================
+
+* The Multigrid algorithm only supports uniform grids. Currently, it cannot be used with Becke grids.
+
+* The `MultiGridNumInt` class does not support the calculation of analytical nuclear gradients.
+
+* The `MultiGridNumInt2` class does not support k-point
+  calculations, meta-GGA functionals, or the computation of the TDDFT fxc kernel.
+
+
+Examples
+========
+* :source:`examples/pbc/27-multigrid.py`
+* :source:`examples/pbc/27-multigrid2.py`