update cc theory

doc/src/week47/LatexSlides/cctheory.tex

@@ -252,11 +252,10 @@
   Morten Hjorth-Jensen}
 \institute[ORNL, University of Oslo and MSU]{
   Department of Physics and Center of Mathematics for Applications\\
-  University of Oslo, N-0316 Oslo, Norway and\\
-  National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA }
+  University of Oslo, N-0316 Oslo, Norway}
 
   
-\date[UiO]{Fall 2024}
+\date[UiO]{Week 48, November 25-29}
 \subject{Coupled-cluster theory}
 
 \begin{document}
@@ -271,173 +270,36 @@
 
 \frame[containsverbatim]
 {
-  \frametitle{Configuration interaction (CI) theory}
+  \frametitle{Plans for week 48: coupled-cluster theory and summary of course}
 \begin{small}
 {\scriptsize
-An alternative way to derive the last equation is to start from 
-\[
-(\hat{H} -E)|\Psi_0\rangle = (\hat{H} -E)\sum_{P'H'}C_{H'}^{P'}|\Phi_{H'}^{P'} \rangle=0, 
-\]
-and if this equation is successively projected against all $\Phi_H^P$ in the expansion of $\Psi$, then the last equation on the previous slide
-results.   As stated previously, one solves this equation normally by diagonalization. If we are able to solve this equation exactly (that is
-numerically exactly) in a large Hilbert space (it will be truncated in terms of the number of single-particle states included in the definition
-of Slater determinants), it can then serve as a benchmark for other many-body methods which approximate the correlation operator
-$\hat{C}$.  
-
-For reasons to come (link with Coupled-Cluster theory and Many-Body perturbation theory), 
-we will rewrite Eq.~(\ref{eq:fullci}) as a set of coupled non-linear equations in terms of the unknown coefficients $C_H^P$. 
+  \begin{itemize}
+  \item Short repetition from last week
+  \item How to write your own coupled-cluster theory code, pairing model example
+  \item Coupled-cluster theory for singles and doubles excitations using a diagrammatic derivation
+  \item Summary and discussion of final oral exam
+  \item Suggested literature: Shavitt and Bartlett chapters 9 and 10
+ \end{itemize}   
  }
  \end{small}
  }
 
 
-\frame[containsverbatim]
-{
-  \frametitle{Configuration interaction (CI) theory}
-\begin{small}
-{\scriptsize
-To see this, we look at $ \langle \Phi_H^P | = \langle \Phi_0 |$ in  Eq.~(\ref{eq:fullci}), that is we multiply with $\langle \Phi_0 |$
-from the left in 
-\[
-(\hat{H} -E)\sum_{P'H'}C_{H'}^{P'}|\Phi_{H'}^{P'} \rangle=0, 
-\]
-and we assume that we have a two-body operator at most.  Using Slater's rule gives then and equation for the 
-correlation energy in terms of $C_i^a$ and $C_{ij}^{ab}$.  We get then
-\[
-\langle \Phi_0 | \hat{H} -E| \Phi_0\rangle + \sum_{ai}\langle \Phi_0 | \hat{H} -E|\Phi_{i}^{a} \rangle C_{i}^{a}+
-\sum_{abij}\langle \Phi_0 | \hat{H} -E|\Phi_{ij}^{ab} \rangle C_{ij}^{ab}=0,
-\]
-or 
-\[
-E-E_0 =\Delta E=\sum_{ai}\langle \Phi_0 | \hat{H}|\Phi_{i}^{a} \rangle C_{i}^{a}+
-\sum_{abij}\langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab} \rangle C_{ij}^{ab},
-\]
-where the $E_0$ is the reference energy and $\Delta E$ becomes the correlation energy.
-We have already computed the expectation values $\langle \Phi_0 | \hat{H}|\Phi_{i}^{a} $ and $\langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab}\rangle$.
- }
- \end{small}
- }
-
-
-\frame[containsverbatim]
-{
-  \frametitle{Configuration interaction (CI) theory}
-\begin{small}
-{\scriptsize
-We can rewrite
-\[
-E-E_0 =\Delta E=\sum_{ai}\langle \Phi_0 | \hat{H}|\Phi_{i}^{a} \rangle C_{i}^{a}+
-\sum_{abij}\langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab} \rangle C_{ij}^{ab},
-\]
-as
-\[
-\Delta E=\sum_{ai}\langle i| \hat{f}|a \rangle C_{i}^{a}+
-\sum_{abij}\langle ij | \hat{v}| ab \rangle C_{ij}^{ab}.
-\]
-This equation determines the correlation energy but not the coefficients $C$. 
-We need more equations. Our next step is to set up
-\[
-\langle \Phi_i^a | \hat{H} -E| \Phi_0\rangle + \sum_{bj}\langle \Phi_i^a | \hat{H} -E|\Phi_{j}^{b} \rangle C_{j}^{b}+
-\sum_{bcjk}\langle \Phi_i^a | \hat{H} -E|\Phi_{jk}^{bc} \rangle C_{jk}^{bc}+
-\sum_{bcdjkl}\langle \Phi_i^a | \hat{H} -E|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=0,
-\]
-as this equation will allow us to find an expression for the coefficents $C_i^a$ since we can rewrite this equation as 
-\[
-\langle i | \hat{f}| a\rangle +\langle \Phi_i^a | \hat{H}-E|\Phi_{i}^{a} \rangle C_{i}^{a}+ \sum_{bj\ne ai}\langle \Phi_i^a | \hat{H}|\Phi_{j}^{b} \rangle C_{j}^{b}+
-\sum_{bcjk}\langle \Phi_i^a | \hat{H}|\Phi_{jk}^{bc} \rangle C_{jk}^{bc}+
-\sum_{bcdjkl}\langle \Phi_i^a | \hat{H}|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=0.
-\]
- }
- \end{small}
- }
-
 
-\frame[containsverbatim]
-{
-  \frametitle{Configuration interaction (CI) theory}
-\begin{small}
-{\scriptsize
-We rewrite this equation as
-\[
-C_{i}^{a}=-(\langle \Phi_i^a | \hat{H}-E|\Phi_{i}^{a})^{-1}\left(\langle i | \hat{f}| a\rangle+ \sum_{bj\ne ai}\langle \Phi_i^a | \hat{H}|\Phi_{j}^{b} \rangle C_{j}^{b}+.\right
-\]
-\[
-.\left
-\sum_{bcjk}\langle \Phi_i^a | \hat{H}|\Phi_{jk}^{bc} \rangle C_{jk}^{bc}+
-\sum_{bcdjkl}\langle \Phi_i^a | \hat{H}|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}\right).
-\]
-Since these equations are solved iteratively ( that is we can start with a guess for the coefficients $C_i^a$), it is common to start the  iteration 
-by setting 
-\[
- C_{i}^{a}=-\frac{\langle i | \hat{f}| a\rangle}{\langle \Phi_i^a | \hat{H}-E|\Phi_{i}^{a}\rangle},
-\]
-and the denominator can be written as
-\[
-  C_{i}^{a}=\frac{\langle i | \hat{f}| a\rangle}{\langle i | \hat{f}| i\rangle-\langle a | \hat{f}| a\rangle+\langle ai | \hat{v}| ai\rangle-E}.
-\]
-The observant reader will however see that we need an equation for $C_{jk}^{bc}$ and $C_{jkl}^{bcd}$ as well.
-To find equations for these coefficients we need then to continue our multiplications from the left with the various
-$\Phi_{H}^P$ terms. 
- }
- \end{small}
- }
 
 
+\section{Coupled Cluster theory}
 
-\frame[containsverbatim]
-{
-  \frametitle{Configuration interaction (CI) theory}
-\begin{small}
-{\scriptsize
-For $C_{jk}^{bc}$ we need then
-\[
-\langle \Phi_{ij}^{ab} | \hat{H} -E| \Phi_0\rangle + \sum_{kc}\langle \Phi_{ij}^{ab} | \hat{H} -E|\Phi_{k}^{c} \rangle C_{k}^{c}+
-\sum_{cdkl}\langle \Phi_{ij}^{ab} | \hat{H} -E|\Phi_{kl}^{cd} \rangle C_{kl}^{cd}+
-\]
-\[
-\sum_{cdeklm}\langle \Phi_{ij}^{ab} | \hat{H} -E|\Phi_{klm}^{cde} \rangle C_{klm}^{cde}+\sum_{cdefklmn}\langle \Phi_{ij}^{ab} | \hat{H} -E|\Phi_{klmn}^{cdef} \rangle C_{klmn}^{cdef}=0,
-\]
-and we can isolate the coefficients $C_{kl}^{cd}$ in a similar way as we did for the coefficients $C_{i}^{a}$. 
-At the end we can rewrite our solution of the Schr\"odinger equation in terms of $n$ coupled equations for the coefficients $C_H^P$.
-This is a very cumbersome way of solving the equation. However, by using this iterative scheme we can illustrate how we can compute the
-various terms in the wave operator or correlation operator $\hat{C}$. We will later identify the calculation of the various terms $C_H^P$
-as parts of different many-body approximations to full CI. In particular, we will relate this non-linear scheme with Coupled Cluster theory and
-many-body perturbation theory.
- }
- \end{small}
- }
-
-\frame[containsverbatim]
-{
-  \frametitle{Configuration interaction (CI) theory}
-\begin{small}
-{\scriptsize
-If we use a Hartree-Fock basis, how can one   simplify the equation
-\[
-\Delta E=\sum_{ai}\langle i| \hat{f}|a \rangle C_{i}^{a}+
-\sum_{abij}\langle ij | \hat{v}| ab \rangle C_{ij}^{ab}?
-\]
-And what about
-\[
-\langle \Phi_i^a | \hat{H} -E| \Phi_0\rangle + \sum_{bj}\langle \Phi_i^a | \hat{H} -E|\Phi_{j}^{b} \rangle C_{j}^{b}+
-\sum_{bcjk}\langle \Phi_i^a | \hat{H} -E|\Phi_{jk}^{bc} \rangle C_{jk}^{bc}+
-\sum_{bcdjkl}\langle \Phi_i^a | \hat{H} -E|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=0,
-\]
-and
-\[
-\langle \Phi_{ij}^{ab} | \hat{H} -E| \Phi_0\rangle + \sum_{kc}\langle \Phi_{ij}^{ab} | \hat{H} -E|\Phi_{k}^{c} \rangle C_{k}^{c}+
-\sum_{cdkl}\langle \Phi_{ij}^{ab} | \hat{H} -E|\Phi_{kl}^{cd} \rangle C_{kl}^{cd}+
-\]
-\[
-\sum_{cdeklm}\langle \Phi_{ij}^{ab} | \hat{H} -E|\Phi_{klm}^{cde} \rangle C_{klm}^{cde}+\sum_{cdefklmn}\langle \Phi_{ij}^{ab} | \hat{H} -E|\Phi_{klmn}^{cdef} \rangle C_{klmn}^{cdef}=0?
-\]
- }
- \end{small}
- }
 
+\include{src/ccsd_H2summary}
+\include{src/ccsd_H2diagrams}
+\include{src/ccsd_barh_expansion}
+\include{src/ccsd_factoring}
+\include{src/ccsd_algorithm}
 
 
 
+\end{document}
 
 
 \frame
@@ -476,20 +338,6 @@
 
 
 
-\section{Coupled Cluster theory}
-
-
-\include{src/ccsd_H2summary}
-\include{src/ccsd_H2diagrams}
-\include{src/ccsd_barh_expansion}
-\include{src/ccsd_factoring}
-\include{src/ccsd_algorithm}
-
-
-
-\end{document}
-
-
 
 \frame
 {

doc/src/week47/LatexSlides/src/ccsd_algorithm.tex

@@ -1,2 +1,2 @@
 \input{src/ccsd_algorithm01}
-\input{src/ccsd_algorithm02}
+%\input{src/ccsd_algorithm02}