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DFT Notes (#44)
* backup notes * back up notes * good enough for now * try again * fixed?
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‎docs/README.md‎

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~ limitations under the License.
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-->
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General instructions for building documentation found throughout the NWChemEx project are available at:
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General instructions for building documentation found throughout the NWChemEx
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project are available at:
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https://github.com/NWChemEx/NWChemEx/blob/master/docs/README.md

‎docs/source/background/dft_notes.rst‎

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.. Copyright 2025 NWChemEx-Project
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..
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.. Licensed under the Apache License, Version 2.0 (the "License");
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.. you may not use this file except in compliance with the License.
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.. You may obtain a copy of the License at
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..
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.. http://www.apache.org/licenses/LICENSE-2.0
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..
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.. Unless required by applicable law or agreed to in writing, software
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.. distributed under the License is distributed on an "AS IS" BASIS,
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.. WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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.. See the License for the specific language governing permissions and
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.. limitations under the License.
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#########
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DFT Notes
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#########
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.. |e_xc| replace:: :math:`E^{XC}\left[\rho\left(\vec{r}\right)\right]`
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.. |v_xc| replace:: :math:`V^{XC}\left[\rho\left(\vec{r}\right)\right]`
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.. |edensity| replace:: :math:`f\left[\rho\left(\vec{r}\right), \cdots\right]`
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.. |eparticle| replace:: :math:`\epsilon\left[\rho\left(\vec{r}\right), \cdots\right]`
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.. |rho| replace:: :math:`\rho\left(\vec{r}\right)`
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.. |rho_i| replace:: :math:`\rho_i`
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.. |drho| replace:: :math:`\left|\bigtriangledown\rho\left(\vec{r}\right)\right|^2`
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.. |dr| replace:: :math:`d\vec{r}`
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These are notes.
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Algorithmically, DFT will differ from Hartee-Fock primarily in the fact that we
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need to compute two additional quantities:
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1. the exchange-correlation (XC) energy, |e_xc|, and
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2. the XC potential, |v_xc|,
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both of which are functionals of the electron density of the system, |rho|.
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By definition, |v_xc| is:
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.. math::
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\newcommand{\density}{\rho\left(\vec{r}\right)}
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\newcommand{\exc}[1]{E^{XC}\left[#1\right]}
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\newcommand{\vxc}[1]{V^{XC}\left[#1\right]}
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\vxc{\density} \equiv
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\frac{\partial \exc{\density}}{\partial \density}
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or rearranging for |e_xc|:
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.. math::
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\exc{\density} = \int \vxc{\density}\density d\vec{r}.
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Conceptually, |e_xc| is the XC contribution to the electronic energy and
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|v_xc| is the potential needed to make the non-interacting system have the same
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density as the real system. Unfortunately, we do not know the analytic form of
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|e_xc| (or equivalently |v_xc|) and likely never will.
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To make progress, we need to introduce an ansantze for either |e_xc| or |v_xc|.
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To do this we define a quantity |edensity| called the XC energy density
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which is the XC energy per infinitesimal volume |dr|. In terms of
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|edensity|, |e_xc| is written as:
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.. math::
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\newcommand{\edensity}[1]{f\left[#1\right]}
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\newcommand{\eparticle}[1]{\epsilon\left[#1\right]}
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\exc{\density} = \int \edensity{\density,\cdots} d\vec{r}.
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Related to |edensity| is a quantity |eparticle| which is the XC energy per
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unit particle. The exact relationship between the two is:
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.. math::
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\edensity{\density,\cdots} = \eparticle{\density,\cdots}\density.
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It should be noted that, unlike |v_xc| and |e_xc|, |edensity| and |eparticle|
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will in general depend on additional functionals beyond the density.
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.. warning::
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Colloquially speaking, "a DFT functional" (e.g., when someone says they
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used "the BLYP DFT functional") can refer to the analytic form of |edensity|
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or |eparticle|. Making matters worse, :math:`\epsilon` is commonly used to
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denote both quantities. When comparing equations it is critical to
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distinguish between these two quantities.
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XC functionals are typically classified by the parameters that |edensity| or
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|eparticle| depend on. More specifically dependence on:
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- only |rho| defines the local density approximation (LDA),
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- |rho| and |drho| (the square of the gradient of |rho|) defines the
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generalized gradient approximation (GGA), and
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- |rho|, |drho|, the Laplacian of |rho|, and the kinetic energy density defines
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a meta GGA.
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For an LDA we have:
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.. math::
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\vxc{\density} =& \frac{\partial \exc{\density}}{\partial \density}\\
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=& \frac{\partial \edensity{\density}}{\partial \density}
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*******************
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Introduction of AOs
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*******************
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In Kohn-Sham DFT we solve the Kohn-Sham equations in an orbital basis that is
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obtained as a linear combination of atomic orbitals (AOs). Assume that there
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are :math:`N_b` AOs and let :math:`\phi_\mu\left(\vec{r}\right)` be the
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:math:`\mu`-th AO. The equation for |e_xc| remains unchanged other than |rho|
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is now:
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.. math::
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\newcommand{\bf}[1]{\phi_{#1}\left(\vec{r}\right)}
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\density = \sum_{\mu}^{N_b}\sum_{\nu}^{N_b} \bf{\mu}P_{\mu\nu}\bf{\nu}
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where :math:`P_{\mu\nu}` is the :math:`\mu\nu`-th element of the atomic
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density matrix. In the AO basis set the :math:`\mu\nu`-th element of |v_xc| is:
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.. math::
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V^{XC}_{\mu\nu} = \int \bf{\mu} \vxc{\density} \bf{\nu} d\vec{r}.
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In the LDA this becomes:
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.. math::
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V^{XC}_{\mu\nu} = \int \bf{\mu}
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\frac{\partial \edensity{\density}}{\partial \density} \bf{\nu} d\vec{r}.
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For most DFT functionals, analytic solutions for the above integrals are not
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known and |e_xc| and |v_xc| must be evaluated by quadrature.
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**************************
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Introduction of Quadrature
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**************************
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In solving an integral by quadrature, we make the following approximation:
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.. math::
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\int g(\vec{r}) d\vec{r} \approx \sum_{i=1}^{N_g} w_i g(\vec{r_i}).
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That is we define a quadrature rule :math:`\mathcal{Q}` which is a set of
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:math:`N_g` pairs of the form :math:`\lbrace w_i, \vec{r}_i\rbrace`. Here,
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:math:`w_i` and :math:`\vec{r_i}` are respectively the quadrature weight and
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real-space location of the :math:`i`-th grid point.
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At this point it is helpful to define:
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.. math::
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\newcommand{\densityg}[1]{\rho_{#1}}
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\newcommand{\bfg}[1]{\phi_{#1}}
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\densityg{i}\equiv&\rho\left(\vec{r_i}\right)\\
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\bfg{\mu i}\equiv&\phi_{\mu}\left(\vec{r_i}\right)
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which respectively are the values of the density and the :math:`\mu`-th AO
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evaluated at the :math:`i`-th grid point. Similarly, we define:
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.. math::
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\newcommand{\edensityg}[1]{f_{#1}}
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\newcommand{\dedensitygdrho}[1]{f_{#1}^{\left(\rho\right)}}
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\edensityg{i}\equiv&\edensity{\densityg{i}}\\
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\dedensitygdrho{i}\equiv&
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\left.
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\frac{\partial \edensity{\density{}}}
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{\partial \density{}}
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\right|_{\density{}=\densityg{i}}
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which are the energy density, and the "derivative of the energy density with
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respect to the density" evaluated at |rho_i|.
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Using these quantities, |rho_i| is then given by:
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.. math::
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\densityg{i} =& \sum_{\mu}^{N_b}
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\sum_{\nu}^{N_b} \bfg{\mu i}P_{\mu \nu}\bfg{\nu i}\\
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=& \sum_{\mu}^{N_b} \bfg{\mu i}X_{\mu i}
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where in the second line we defined the common intermediate (the collocation
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matrix):
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.. math::
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X_{\mu i} = \sum_{\nu}^{N_b} P_{\mu\nu}\bfg{\nu i}
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Using :math:`\mathcal{Q}`, |e_xc| becomes:
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.. math::
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\exc{\density{}} = \sum_i^{N_g} w_i\edensityg{i}
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and |v_xc| becomes:
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.. math::
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V_{\mu\nu}^{XC} =&
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\sum_i^{N_g} w_i \bfg{\mu i} \dedensitygdrho{i} \bfg{\nu i}\\
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=& \sum_i^{N_g} w_i \bfg{\mu i} Z_{\nu i}
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where we defined the intermediate:
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.. math::
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Z_{\mu i} =\dedensitygdrho{i} \bfg{\mu i}.
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***********************
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As a Sparse Map Problem
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***********************
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While the last sections have described DFT as a tensor problem it's usually not
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solved as one. DFT is not usually treated as a tensor problem because:
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- Large tensors. Grids minimally use about 1000 grid points per atom (higher-
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quality grids tend to be order 10,000) and most AO basis sets have order 10
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basis functions per atom. Tensors like :math:`\phi_{\mu i}` then have
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minimally "10,000 times number of atoms squared" elements, meaning the tensor
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for 100 atoms already requires gigabytes of memory.
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- Sparsity. Most DFT quantities are local. So if basis functions for a tensor
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element are spatially far a part, the element is usually close to zero.
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To describe the sparsity we introduce sparse maps. Given two basis sets,
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:math:`A` and :math:`B`, the sparse map :math:`L` maps each basis
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function in :math:`A` to a subset of the basis functions in :math:`B`. Assume
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we have some tensor with elements :math:`T_{ab}` where :math:`a` indexes basis
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functions in :math:`A` and :math:`b` indexes basis functions in :math:`B`.
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For a given value of :math:`a`, the non-zero elements of :math:`T_{ab}` are
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those such that :math:`b` is in :math:`L(a)`.
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In DFT, we use atom-centered grids and AOs. It is therefore common to define
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sparse maps :math:`L(A\rightarrow i)` and :math:`L(A\rightarrow \mu)` which
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respectively map atom indices to grid points and atom indices to AOs. Using
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these maps the equation for the density becomes:
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.. math::
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\densityg{i_A} = \sum_{\mu_A} \bfg{\mu_A i_A}X_{\mu_A i_A}
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where an index like :math:`i_A` is shorthand for restricting the value of
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:math:`i` to those afforded by the sparse map :math:`L(A\rightarrow i)`.
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Applying the same logic to the other DFT quantities:
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.. math::
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X_{\mu_A i_A} =& \sum_{\nu_A} P_{\mu_A\nu_A}\bfg{\nu_A i_A}\\
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Z_{\mu_A i_A} =& \dedensitygdrho{i_A} \bfg{\mu_A i_A}\\
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V_{\mu_A\nu_A}^{XC} =& \sum_{i_A} w_{i_A}\bfg{\mu_A i_A} Z_{\nu_A i_A}.
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Finally, the equation for |e_xc| becomes:
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.. math::
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\exc{\density{}} = \sum_{A}\sum_{i_A} w_{i_A}\edensityg{i_A}
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Of note, for a given grid we expect the number of grid points associated with
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an atom to be roughly constant. Similarly, for a given AO basis set we expect
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the number of AOs associated with an atom to also be roughly constant. This
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means that cost to form all quantities will scale linearly with the number of
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atoms.

‎docs/source/background/index.rst‎

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.. Copyright 2025 NWChemEx-Project
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..
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.. Licensed under the Apache License, Version 2.0 (the "License");
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.. you may not use this file except in compliance with the License.
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.. You may obtain a copy of the License at
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..
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.. http://www.apache.org/licenses/LICENSE-2.0
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..
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.. Unless required by applicable law or agreed to in writing, software
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.. distributed under the License is distributed on an "AS IS" BASIS,
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.. WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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.. See the License for the specific language governing permissions and
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.. limitations under the License.
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##########
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Background
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##########
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.. toctree::
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:maxdepth: 2
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:caption: Contents
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dft_notes

‎docs/source/index.rst‎

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.. See the License for the specific language governing permissions and
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.. limitations under the License.
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15-
#########
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###
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SCF
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#########
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###
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.. toctree::
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:maxdepth: 2
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:caption: Contents
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background/index
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.. toctree::
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:maxdepth: 2

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