edit hf theory

Author: fishjojo

source/theory/pbc.rst

@@ -2,3 +2,12 @@
 ****************************
 Periodic boundary conditions
 ****************************
+
+
+
+.. toctree::
+   :maxdepth: 1
+
+   pbc/gto.rst
+   pbc/df.rst
+   pbc/pp.rst

source/theory/pbc/df.rst

@@ -0,0 +1,4 @@
+.. _theory_pbc_df:
+
+Density fitting
+***************

source/theory/pbc/gto.rst

@@ -0,0 +1,4 @@
+.. _theory_pbc_gto:
+
+Crystalline Gaussian-type atomic orbitals
+*****************************************

source/theory/pbc/pp.rst

@@ -0,0 +1,4 @@
+.. _theory_pbc_pp:
+
+Pseudopotentials
+****************

source/theory/scf.rst

@@ -89,7 +89,7 @@ Within the AO representation, the Hartree-Fock equations reduce to the Roothaan-
 
 .. math::
 
-   \mathbf{F}^{\alpha} \mathbf{C}^{\alpha} = \bf{\varepsilon}^{\alpha} \mathbf{S} \mathbf{C}^{\alpha} \;,
+   \mathbf{F}^{\alpha} \mathbf{C}^{\alpha} = \boldsymbol{\varepsilon}^{\alpha} \mathbf{S} \mathbf{C}^{\alpha} \;,
 
 where 
 
@@ -112,6 +112,119 @@ For example, the ground-state electronic energy is expressed as
 
 Periodic boundary conditions
 ----------------------------
+PySCF also alows the user to perform SCF calculations for solids.
+With crystalline Gaussian-type AOs as the underlying single-partial basis (see :numref:`theory_pbc_gto`),
+the molecular SCF code can be easily adapted to the cases where periodic boundary conditions (PBCs) 
+are applied. Instead of solving only one set of Roothaan-Hall or Pople-Nesbet equtions for molecules, 
+it is now necessary to solve them for each k point for solids:
+
+.. math::
+
+   \mathbf{F}(\mathbf{k}) \mathbf{C}(\mathbf{k}) = \boldsymbol{\varepsilon}(\mathbf{k}) \mathbf{S}(\mathbf{k}) \mathbf{C}(\mathbf{k}) \;,
+
+where the Fock matrix is defined (within the restricted formalism) as
+
+.. math::
+
+   \mathbf{F}(\mathbf{k}) = \mathbf{T}(\mathbf{k}) + \mathbf{V}^{\rm PP}(\mathbf{k})
+   +\mathbf{J}(\mathbf{k}) - \frac{1}{2} \mathbf{K}(\mathbf{k}) + \mathbf{V}^{L+J}(\mathbf{k}) \;.
+
+Here, :math:`\mathbf{V}^{\rm PP}` denotes the pseudopotential contribution and 
+:math:`\mathbf{V}^{L+J}` deals with the divergence of local pseudopotential and Hartree potential (see below).
+
+The one-electron overlap, kinetic energy, and local pseudopotential integrals 
+are evaluated through numerical integrations on the real-space grid according to 
+
+.. math::
+
+   S_{\mu\nu}(\mathbf{k}) = \int_\Omega d\mathbf{r} \phi_{\mu\mathbf{k}}^{*}(\mathbf{r}) \phi_{\nu\mathbf{k}}(\mathbf{r}) \;,
+
+.. math::
+
+   T_{\mu\nu}(\mathbf{k}) = -\frac{1}{2} \int_\Omega d\mathbf{r} \phi_{\mu\mathbf{k}}^{*}(\mathbf{r}) 
+   \boldsymbol{\nabla}_{\mathbf{r}}^2 \phi_{\nu\mathbf{k}}(\mathbf{r}) \;,
+
+and 
+
+.. math::
+
+   V_{\mu\nu}^{\rm L-PP}(\mathbf{k}) = \int_\Omega d\mathbf{r} \phi_{\mu\mathbf{k}}^{*}(\mathbf{r}) 
+   v^{\rm L-PP}(\mathbf{r}) \phi_{\nu\mathbf{k}}(\mathbf{r}) \;,
+
+where :math:`\Omega` labels the unit cell volume.
+The non-local part of the pseudopotential is computed in the reciprocal space:
+
+.. math::
+
+   V_{\mu\nu}^{\rm NL-PP}(\mathbf{k}) = \Omega \sum_{\mathbf{G},\mathbf{G}'} \phi_{\mu\mathbf{k}}^{*}(\mathbf{G})
+   v^{\rm NL-PP}(\mathbf{k}+\mathbf{G}, \mathbf{k}+\mathbf{G}') \phi_{\nu\mathbf{k}}(\mathbf{G}') \;,
+
+where
+
+.. math::
+
+   v^{\rm NL-PP}(\mathbf{k}+\mathbf{G}, \mathbf{k}+\mathbf{G}') = \frac{1}{\Omega} \int d\mathbf{r} \int d\mathbf{r}'
+   e^{-i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}} v^{\rm NL-PP}(\mathbf{r},\mathbf{r}') 
+   e^{ i(\mathbf{k}+\mathbf{G}^{'})\cdot\mathbf{r}'} \;.
+
+.. note::
+   The way that the pseudopotential integrals are computed differs in different density fitting schemes and for different 
+   pseudopotentials. Interested readers should refer to :numref:`theory_pbc_df` and :numref:`theory_pbc_pp`.
+
+The Coulomb and exchange matrices are defined similarly as
+
+.. math::
+
+   J_{\mu\nu}(\mathbf{k}) = \int_{\Omega} d\mathbf{r} \phi_{\mu\mathbf{k}}^{*}(\mathbf{r}) v_{\rm H}(\mathbf{r}) \phi_{\nu\mathbf{k}}(\mathbf{r}) \;,
+
+and
+
+.. math::
+
+   K_{\mu\nu}(\mathbf{k}) = \int_{\Omega} d\mathbf{r} \int d\mathbf{r}' \phi_{\mu\mathbf{k}}^{*}(\mathbf{r}) 
+   \frac{\rho(\mathbf{r}, \mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \phi_{\nu\mathbf{k}}(\mathbf{r}') \;.
+
+Here :math:`v_{\rm H}` is the Hartree potential
+
+.. math::
+
+   v_{\rm H}(\mathbf{r}) = \frac{4\pi}{\Omega} \sum_{\mathbf{G}\neq \mathbf{0}} \frac{\rho(\mathbf{G})}{G^2} e^{i\mathbf{G}\cdot\mathbf{r}} \;,
+
+and :math:`\rho(\mathbf{r}, \mathbf{r}')` is the density matrix
+
+.. math::
+
+   \rho(\mathbf{r}, \mathbf{r}') =  \sum_{\mathbf{k}} w_{\mathbf{k}} \sum_{\lambda\sigma} P_{\lambda\sigma}(\mathbf{k}) 
+   \phi_{\lambda\mathbf{k}}(\mathbf{r}) \phi_{\sigma\mathbf{k}}^{*}(\mathbf{r}') \;,
+
+where :math:`w_{\mathbf{k}}` represents the weight of each k point.
+
+Note that the local part of the pseudopotential and the Hartree potential diverge at :math:`G=0`; 
+however, their sum is not, which leads to the :math:`V^{\rm L+J}` term (for charge neutral unit cell):
+
+.. math::
+
+   V_{\mu\nu}^{\rm L+J} (\mathbf{k})  = \frac{S_{\mu\nu}}{\Omega} 
+   \int d\mathbf{r} \left(v^{\rm L-PP}(\mathbf{r}) + \sum_{\alpha} \frac{Z_{\alpha}e^2}{r} \right) \;.
+
+.. note::
+
+   For details about how to compute the Coulomb (:math:`\mathbf{J}`) 
+   and exchange (:math:`\mathbf{K}`) integrals, see :numref:`theory_pbc_df`.
+
+Finally, the total electronic energy differs from the molecular case only by a k-point summation:
+
+.. math::
+
+   E_{\rm HF} = \sum_{\mathbf{k}} w_{\mathbf{k}} E_{\rm HF}(\mathbf{k}) \;,
+
+where
+
+.. math::
+
+   E_{\rm HF}(\mathbf{k}) = \frac{1}{2} \left\{ {\rm Tr}\left[\mathbf{h}(\mathbf{k}) (\mathbf{P}^{\alpha}(\mathbf{k})+\mathbf{P}^{\beta}(\mathbf{k}))\right]
+              + {\rm Tr}\left[\mathbf{F}^{\alpha}(\mathbf{k}) \mathbf{P}^{\alpha}(\mathbf{k})\right] 
+              + {\rm Tr}\left[\mathbf{F}^{\beta}(\mathbf{k}) \mathbf{P}^{\beta}(\mathbf{k})\right] \right\} \;.
 
 
 Initial guess