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update week43
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mhjensen committed Oct 25, 2023
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5 changes: 0 additions & 5 deletions doc/pub/week43/html/._week43-bs000.html
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None,
'density-functional-theory'),
('Hohenberg-Kohn theorem', 2, None, 'hohenberg-kohn-theorem'),
('$\\mathbf{k} = $ collection of quantum numbers',
3,
None,
'mathbf-k-collection-of-quantum-numbers'),
('More reading', 3, None, 'more-reading')]}
end of tocinfo -->

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<!-- navigation toc: --> <li><a href="._week43-bs001.html#hartree-fock-ground-state-energy-for-the-electron-gas-in-three-dimensions" style="font-size: 80%;"><b>Hartree-Fock ground state energy for the electron gas in three dimensions</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs001.html#density-functional-theory" style="font-size: 80%;"><b>Density functional theory</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs001.html#hohenberg-kohn-theorem" style="font-size: 80%;"><b>Hohenberg-Kohn theorem</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs001.html#mathbf-k-collection-of-quantum-numbers" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;$\mathbf{k} = $ collection of quantum numbers</a></li>
<!-- navigation toc: --> <li><a href="._week43-bs001.html#more-reading" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;More reading</a></li>

</ul>
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54 changes: 11 additions & 43 deletions doc/pub/week43/html/._week43-bs001.html
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Original file line number Diff line number Diff line change
Expand Up @@ -54,10 +54,6 @@
None,
'density-functional-theory'),
('Hohenberg-Kohn theorem', 2, None, 'hohenberg-kohn-theorem'),
('$\\mathbf{k} = $ collection of quantum numbers',
3,
None,
'mathbf-k-collection-of-quantum-numbers'),
('More reading', 3, None, 'more-reading')]}
end of tocinfo -->

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<!-- navigation toc: --> <li><a href="#hartree-fock-ground-state-energy-for-the-electron-gas-in-three-dimensions" style="font-size: 80%;"><b>Hartree-Fock ground state energy for the electron gas in three dimensions</b></a></li>
<!-- navigation toc: --> <li><a href="#density-functional-theory" style="font-size: 80%;"><b>Density functional theory</b></a></li>
<!-- navigation toc: --> <li><a href="#hohenberg-kohn-theorem" style="font-size: 80%;"><b>Hohenberg-Kohn theorem</b></a></li>
<!-- navigation toc: --> <li><a href="#mathbf-k-collection-of-quantum-numbers" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;$\mathbf{k} = $ collection of quantum numbers</a></li>
<!-- navigation toc: --> <li><a href="#more-reading" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;More reading</a></li>

</ul>
Expand Down Expand Up @@ -723,10 +718,10 @@ <h2 id="density-functional-theory" class="anchor">Density functional theory </h2
Carlo methods.
</p>

<p>The electronic energy \( E \) is said to be a \emph{functional} of the
<p>The electronic energy \( E \) is said to be a functional of the
electronic density, \( E[\rho] \), in the sense that for a given function
\( \rho(r) \), there is a single corresponding energy. The
\emph{Hohenberg-Kohn theorem} confirms that such a functional exists,
Hohenberg-Kohn theorem confirms that such a functional exists,
but does not tell us the form of the functional. As shown by Kohn and
Sham, the exact ground-state energy \( E \) of an \( N \)-electron system can
be written as
Expand All @@ -742,7 +737,7 @@ <h2 id="density-functional-theory" class="anchor">Density functional theory </h2
\end{equation*}
$$

<p>with \( \Psi_i \) the \emph{Kohn-Sham} (KS) \emph{orbitals}.</p>
<p>with \( \Psi_i \) the Kohn-Sham (KS) orbitals.</p>

<p> The ground-state charge density is given by</p>
$$
Expand All @@ -753,11 +748,11 @@ <h2 id="density-functional-theory" class="anchor">Density functional theory </h2
$$

<p>where the sum is over the occupied Kohn-Sham orbitals. The last term,
\( E_{EXC}[\rho] \), is the \emph{exchange-correlation energy} which in
\( E_{EXC}[\rho] \), is the exchange-correlation energy which in
theory takes into account all non-classical electron-electron
interaction. However, we do not know how to obtain this term exactly,
and are forced to approximate it. The KS orbitals are found by solving
the \emph{Kohn-Sham equations}, which can be found by applying a
the Kohn-Sham equations, which can be found by applying a
variational principle to the electronic energy \( E[\rho] \). This approach
is similar to the one used for obtaining the HF equation.
</p>
Expand All @@ -774,7 +769,7 @@ <h2 id="density-functional-theory" class="anchor">Density functional theory </h2
$$

<p>where \( \epsilon_i \) are the KS orbital energies, and where the
\emph{exchange-correlation potential} is given by
exchange-correlation potential is given by
</p>

$$
Expand All @@ -792,7 +787,7 @@ <h2 id="density-functional-theory" class="anchor">Density functional theory </h2
</p>

<p>The main source of error in DFT usually arises from the approximate
nature of \( E_{EXC} \). In the \emph{local density approximation} (LDA) it
nature of \( E_{EXC} \). In the \local density approximation (LDA) it
is approximated as
</p>

Expand Down Expand Up @@ -826,7 +821,7 @@ <h2 id="hohenberg-kohn-theorem" class="anchor">Hohenberg-Kohn theorem </h2>
\end{eqnarray}
$$

<p>\( \hat{\Psi} \), \( \hat{\Psi}^{\dagger } = \) annihilation, creation \emph{field operators}</p>
<p>\( \hat{\Psi} \), \( \hat{\Psi}^{\dagger} = \) annihilation, creation field operators</p>
$$
\begin{equation*}
\hat{\Psi}(\mathbf{r})\equiv \sum_{\mathbf{k}}\psi_{\mathbf{k}}(\mathbf{r})a_{\mathbf{k}} \nonumber
Expand All @@ -835,31 +830,6 @@ <h2 id="hohenberg-kohn-theorem" class="anchor">Hohenberg-Kohn theorem </h2>
\hat{\Psi}^{\dagger }(\mathbf{r})\equiv \sum_{\mathbf{k}}\psi_{\mathbf{k}}^{*}(\mathbf{r})a_{\mathbf{k}}^{\dagger } \nonumber
\end{equation*}
$$
<h3 id="mathbf-k-collection-of-quantum-numbers" class="anchor">\( \mathbf{k} = \) collection of quantum numbers </h3>

$$
\begin{align}
\hat{T} &= \text{kinetic energy operator} \nonumber \\
\hat{V} &= \text{external single-particle potential operator} \nonumber \\
\hat{W} &= \text{two-particle interaction operator} \nonumber
\end{align}
$$

<p>\( \mathcal{V} = \) set of external single-particle {potentials} \( v \) so that </p>
$$
\begin{equation*}
\hat{H}\ket{\phi} = \left(\hat{T}+\hat{V}+\hat{W}\right)=E\ket{\phi},\qquad \hat{V}\in \mathcal{V},\nonumber
\end{equation*}
$$

<p>gives a {non-degenerate} N-particle ground state \( \ket{\Psi } \)</p>
$$
\begin{equation*}
\Longrightarrow \qquad C:\mathcal{V}(C)\longrightarrow \Psi \qquad \text{{surjective},} \nonumber
\end{equation*}
$$

<p>where \( \Psi = \) set of ground states (GS) \( \ket{\Psi } \)</p>

<p>The density </p>
$$
Expand All @@ -878,12 +848,10 @@ <h3 id="mathbf-k-collection-of-quantum-numbers" class="anchor">\( \mathbf{k} =
<p>where \( \mathcal{N} = \) set of GS densities.</p>
<h3 id="more-reading" class="anchor">More reading </h3>
<ol>
<li> R. van Leeuwen: \emph{Density functional approach to the many-body problem: key concepts and exact functionals}, Adv. Quant. Chem. \textbf{43}, 25 (2003). (Mathematical foundations of DFT)</li>
<li> R. M. Dreizler and E. K. U. Gross: \emph{Density functional theory: An approach to the quantum many-body problem}. (Introductory book)</li>
<li> W. Koch and M. C. Holthausen: \emph{A chemist's guide to density functional theory}. (Introductory book, less formal than Dreizler/Gross)</li>
<li> R. van Leeuwen: Density functional approach to the many-body problem: key concepts and exact functionals, Adv. Quant. Chem. \textbf{43}, 25 (2003). (Mathematical foundations of DFT)</li>
<li> R. M. Dreizler and E. K. U. Gross: Density functional theory: An approach to the quantum many-body problem. (Introductory book)</li>
<li> W. Koch and M. C. Holthausen: A chemist's guide to density functional theory. (Introductory book, less formal than Dreizler/Gross)</li>
<li> E. H. Lieb: Density functionals for Coulomb systems, Int. J. Quant. Chem. \textbf{24}, 243-277 (1983). (Mathematical analysis of DFT)</li>
<li> J. P. Perdew and S. Kurth: In \emph{A Primer in Density Functional Theory: Density Functionals for Non-relativistic Coulomb Systems in the New Century}, ed. C. Fiolhais \emph{et al}. (Introductory course, partly difficult, but interesting points of view)</li>
<li> E. Engel: In \emph{A Primer in Density Functional Theory: Orbital-Dependent Functionals for the Exchange-Correlation Energy}, ed. C. Fiolhais \emph{et al}. (Introductory lectures, only about orbital-dependent functionals)</li>
</ol>
<p>
<!-- navigation buttons at the bottom of the page -->
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5 changes: 0 additions & 5 deletions doc/pub/week43/html/week43-bs.html
View file
Original file line number Diff line number Diff line change
Expand Up @@ -54,10 +54,6 @@
None,
'density-functional-theory'),
('Hohenberg-Kohn theorem', 2, None, 'hohenberg-kohn-theorem'),
('$\\mathbf{k} = $ collection of quantum numbers',
3,
None,
'mathbf-k-collection-of-quantum-numbers'),
('More reading', 3, None, 'more-reading')]}
end of tocinfo -->

Expand Down Expand Up @@ -98,7 +94,6 @@
<!-- navigation toc: --> <li><a href="._week43-bs001.html#hartree-fock-ground-state-energy-for-the-electron-gas-in-three-dimensions" style="font-size: 80%;"><b>Hartree-Fock ground state energy for the electron gas in three dimensions</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs001.html#density-functional-theory" style="font-size: 80%;"><b>Density functional theory</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs001.html#hohenberg-kohn-theorem" style="font-size: 80%;"><b>Hohenberg-Kohn theorem</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs001.html#mathbf-k-collection-of-quantum-numbers" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;$\mathbf{k} = $ collection of quantum numbers</a></li>
<!-- navigation toc: --> <li><a href="._week43-bs001.html#more-reading" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;More reading</a></li>

</ul>
Expand Down
55 changes: 11 additions & 44 deletions doc/pub/week43/html/week43-reveal.html
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Original file line number Diff line number Diff line change
Expand Up @@ -946,10 +946,10 @@ <h2 id="density-functional-theory">Density functional theory </h2>
Carlo methods.
</p>

<p>The electronic energy \( E \) is said to be a \emph{functional} of the
<p>The electronic energy \( E \) is said to be a functional of the
electronic density, \( E[\rho] \), in the sense that for a given function
\( \rho(r) \), there is a single corresponding energy. The
\emph{Hohenberg-Kohn theorem} confirms that such a functional exists,
Hohenberg-Kohn theorem confirms that such a functional exists,
but does not tell us the form of the functional. As shown by Kohn and
Sham, the exact ground-state energy \( E \) of an \( N \)-electron system can
be written as
Expand All @@ -967,7 +967,7 @@ <h2 id="density-functional-theory">Density functional theory </h2>
$$
<p>&nbsp;<br>

<p>with \( \Psi_i \) the \emph{Kohn-Sham} (KS) \emph{orbitals}.</p>
<p>with \( \Psi_i \) the Kohn-Sham (KS) orbitals.</p>

<p> The ground-state charge density is given by</p>
<p>&nbsp;<br>
Expand All @@ -980,11 +980,11 @@ <h2 id="density-functional-theory">Density functional theory </h2>
<p>&nbsp;<br>

<p>where the sum is over the occupied Kohn-Sham orbitals. The last term,
\( E_{EXC}[\rho] \), is the \emph{exchange-correlation energy} which in
\( E_{EXC}[\rho] \), is the exchange-correlation energy which in
theory takes into account all non-classical electron-electron
interaction. However, we do not know how to obtain this term exactly,
and are forced to approximate it. The KS orbitals are found by solving
the \emph{Kohn-Sham equations}, which can be found by applying a
the Kohn-Sham equations, which can be found by applying a
variational principle to the electronic energy \( E[\rho] \). This approach
is similar to the one used for obtaining the HF equation.
</p>
Expand All @@ -1003,7 +1003,7 @@ <h2 id="density-functional-theory">Density functional theory </h2>
<p>&nbsp;<br>

<p>where \( \epsilon_i \) are the KS orbital energies, and where the
\emph{exchange-correlation potential} is given by
exchange-correlation potential is given by
</p>

<p>&nbsp;<br>
Expand All @@ -1023,7 +1023,7 @@ <h2 id="density-functional-theory">Density functional theory </h2>
</p>

<p>The main source of error in DFT usually arises from the approximate
nature of \( E_{EXC} \). In the \emph{local density approximation} (LDA) it
nature of \( E_{EXC} \). In the \local density approximation (LDA) it
is approximated as
</p>

Expand Down Expand Up @@ -1063,7 +1063,7 @@ <h2 id="hohenberg-kohn-theorem">Hohenberg-Kohn theorem </h2>
$$
<p>&nbsp;<br>

<p>\( \hat{\Psi} \), \( \hat{\Psi}^{\dagger } = \) annihilation, creation \emph{field operators}</p>
<p>\( \hat{\Psi} \), \( \hat{\Psi}^{\dagger} = \) annihilation, creation field operators</p>
<p>&nbsp;<br>
$$
\begin{equation*}
Expand All @@ -1074,37 +1074,6 @@ <h2 id="hohenberg-kohn-theorem">Hohenberg-Kohn theorem </h2>
\end{equation*}
$$
<p>&nbsp;<br>
<h3 id="mathbf-k-collection-of-quantum-numbers">\( \mathbf{k} = \) collection of quantum numbers </h3>

<p>&nbsp;<br>
$$
\begin{align}
\hat{T} &= \text{kinetic energy operator} \nonumber \\
\hat{V} &= \text{external single-particle potential operator} \nonumber \\
\hat{W} &= \text{two-particle interaction operator} \nonumber
\end{align}
$$
<p>&nbsp;<br>

<p>\( \mathcal{V} = \) set of external single-particle {potentials} \( v \) so that </p>
<p>&nbsp;<br>
$$
\begin{equation*}
\hat{H}\ket{\phi} = \left(\hat{T}+\hat{V}+\hat{W}\right)=E\ket{\phi},\qquad \hat{V}\in \mathcal{V},\nonumber
\end{equation*}
$$
<p>&nbsp;<br>

<p>gives a {non-degenerate} N-particle ground state \( \ket{\Psi } \)</p>
<p>&nbsp;<br>
$$
\begin{equation*}
\Longrightarrow \qquad C:\mathcal{V}(C)\longrightarrow \Psi \qquad \text{{surjective},} \nonumber
\end{equation*}
$$
<p>&nbsp;<br>

<p>where \( \Psi = \) set of ground states (GS) \( \ket{\Psi } \)</p>

<p>The density </p>
<p>&nbsp;<br>
Expand All @@ -1127,12 +1096,10 @@ <h3 id="mathbf-k-collection-of-quantum-numbers">\( \mathbf{k} = \) collection o
<p>where \( \mathcal{N} = \) set of GS densities.</p>
<h3 id="more-reading">More reading </h3>
<ol>
<p><li> R. van Leeuwen: \emph{Density functional approach to the many-body problem: key concepts and exact functionals}, Adv. Quant. Chem. \textbf{43}, 25 (2003). (Mathematical foundations of DFT)</li>
<p><li> R. M. Dreizler and E. K. U. Gross: \emph{Density functional theory: An approach to the quantum many-body problem}. (Introductory book)</li>
<p><li> W. Koch and M. C. Holthausen: \emph{A chemist's guide to density functional theory}. (Introductory book, less formal than Dreizler/Gross)</li>
<p><li> R. van Leeuwen: Density functional approach to the many-body problem: key concepts and exact functionals, Adv. Quant. Chem. \textbf{43}, 25 (2003). (Mathematical foundations of DFT)</li>
<p><li> R. M. Dreizler and E. K. U. Gross: Density functional theory: An approach to the quantum many-body problem. (Introductory book)</li>
<p><li> W. Koch and M. C. Holthausen: A chemist's guide to density functional theory. (Introductory book, less formal than Dreizler/Gross)</li>
<p><li> E. H. Lieb: Density functionals for Coulomb systems, Int. J. Quant. Chem. \textbf{24}, 243-277 (1983). (Mathematical analysis of DFT)</li>
<p><li> J. P. Perdew and S. Kurth: In \emph{A Primer in Density Functional Theory: Density Functionals for Non-relativistic Coulomb Systems in the New Century}, ed. C. Fiolhais \emph{et al}. (Introductory course, partly difficult, but interesting points of view)</li>
<p><li> E. Engel: In \emph{A Primer in Density Functional Theory: Orbital-Dependent Functionals for the Exchange-Correlation Energy}, ed. C. Fiolhais \emph{et al}. (Introductory lectures, only about orbital-dependent functionals)</li>
</ol>
</section>

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