Added paragraph on spin-projected MP convergence.

Manuscript/EPAWTFT.bbl

@@ -6,7 +6,7 @@
 %Control: page (0) single
 %Control: year (1) truncated
 %Control: production of eprint (0) enabled
-\begin{thebibliography}{125}%
+\begin{thebibliography}{131}%
 \makeatletter
 \providecommand \@ifxundefined [1]{%
  \@ifx{#1\undefined}
@@ -890,6 +890,52 @@
   10.1016/0009-2614(87)80545-6} {\bibfield  {journal} {\bibinfo  {journal}
   {Chem. Phys. Lett.}\ }\textbf {\bibinfo {volume} {138}},\ \bibinfo {pages}
   {481} (\bibinfo {year} {1987})}\BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Schlegel}(1986)}]{Schlegel_1986}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {H.~B.}\ \bibnamefont
+  {Schlegel}},\ }\href {\doibase 10.1063/1.450026} {\bibfield  {journal}
+  {\bibinfo  {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {84}},\
+  \bibinfo {pages} {4530} (\bibinfo {year} {1986})}\BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Schlegel}(1988)}]{Schlegel_1988}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {H.~B.}\ \bibnamefont
+  {Schlegel}},\ }\href {\doibase 10.1021/j100322a014} {\bibfield  {journal}
+  {\bibinfo  {journal} {J. Phys. Chem.}\ }\textbf {\bibinfo {volume} {91}},\
+  \bibinfo {pages} {3075} (\bibinfo {year} {1988})}\BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Knowles}\ and\ \citenamefont
+  {Handy}(1988{\natexlab{a}})}]{Knowles_1988a}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {P.~J.}\ \bibnamefont
+  {Knowles}}\ and\ \bibinfo {author} {\bibfnamefont {N.~C.}\ \bibnamefont
+  {Handy}},\ }\href {\doibase 10.1021/j100322a018} {\bibfield  {journal}
+  {\bibinfo  {journal} {J. Phys. Chem.}\ }\textbf {\bibinfo {volume} {92}},\
+  \bibinfo {pages} {3097} (\bibinfo {year} {1988}{\natexlab{a}})}\BibitemShut
+  {NoStop}%
+\bibitem [{\citenamefont {Knowles}\ and\ \citenamefont
+  {Handy}(1988{\natexlab{b}})}]{Knowles_1988b}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {P.~J.}\ \bibnamefont
+  {Knowles}}\ and\ \bibinfo {author} {\bibfnamefont {N.~C.}\ \bibnamefont
+  {Handy}},\ }\href {\doibase 10.1063/1.454397} {\bibfield  {journal} {\bibinfo
+   {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {88}},\ \bibinfo
+  {pages} {6991} (\bibinfo {year} {1988}{\natexlab{b}})}\BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Tsuchimochi}\ and\ \citenamefont {{Van
+  Voorhis}}(2014)}]{Tsuchimochi_2014}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
+  {Tsuchimochi}}\ and\ \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {{Van
+  Voorhis}}},\ }\href {\doibase 10.1063/1.4898804} {\bibfield  {journal}
+  {\bibinfo  {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {141}},\
+  \bibinfo {pages} {164117} (\bibinfo {year} {2014})}\BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Tsuchimochi}\ and\ \citenamefont
+  {Ten-no}(2019)}]{Tsuchimochi_2019}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
+  {Tsuchimochi}}\ and\ \bibinfo {author} {\bibfnamefont {S.~L.}\ \bibnamefont
+  {Ten-no}},\ }\href {\doibase 10.1021/acs.jctc.9b00897} {\bibfield  {journal}
+  {\bibinfo  {journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume}
+  {15}},\ \bibinfo {pages} {6688} (\bibinfo {year} {2019})}\BibitemShut
+  {NoStop}%
 \bibitem [{\citenamefont {Cremer}\ and\ \citenamefont
   {He}(1996)}]{Cremer_1996}%
   \BibitemOpen

Manuscript/EPAWTFT.bib

@@ -6,6 +6,60 @@
 
 %% Saved with string encoding Unicode (UTF-8) 
 
+@article{Tsuchimochi_2019,
+    author ={Takashi Tsuchimochi and Seiichiro L. Ten-no},
+    journal={J. Chem. Theory Comput.},
+    volume ={15},
+    pages  ={6688},
+    title  ={Second-order perturbation theory with spin-symmetry-projected Hartree--Fock},
+    year   ={2019},
+    doi    ={10.1021/acs.jctc.9b00897},
+}
+@article{Tsuchimochi_2014,
+    author ={Takashi Tsuchimochi and Troy {Van Voorhis}},
+    journal={J. Chem. Phys.},
+    volume ={141},
+    pages  ={164117},
+    title  ={Extended {M\oller--Plesset} perturbation theory for dynamical and static correlations},
+    year   ={2014},
+    doi    ={10.1063/1.4898804},
+}
+@article{Knowles_1988a,
+    author ={Peter J. Knowles and Nicholas C. Handy},
+    journal={J. Phys. Chem.},
+    volume ={92},
+    pages  ={3097},
+    title  ={Convergence of projected unrestricted {Hartree--Fock M\oller--Plesset} series},
+    year   ={1988},
+    doi    ={10.1021/j100322a018},
+}
+@article{Knowles_1988b,
+    author ={Peter J. Knowles and Nicholas C. Handy},
+    journal={J. Chem. Phys.},
+    volume ={88},
+    pages  ={6991},
+    title  ={Projected unrestricted {M\oller--Plesset} second-order energies},
+    year   ={1988},
+    doi    ={10.1063/1.454397},
+}
+@article{Schlegel_1986,
+    author ={H. Bernhard Schlegel},
+    journal={J. Chem. Phys.},
+    volume ={84},
+    pages  ={4530},
+    title  ={Potential energy curves using unrestricted {M\oller--Plesset} perturbation theory with spin annihilation},
+    year   ={1986},
+    doi    ={10.1063/1.450026},
+}
+@article{Schlegel_1988,
+    author ={H. Bernhard Schlegel},
+    journal={J. Phys. Chem.},
+    volume ={91},
+    pages  ={3075},
+    title  ={{M\oller--Plesset} perturbation theory with spin projection},
+    year   ={1988},
+    doi    ={10.1021/j100322a014},
+}
 @article{Gill_1988a,
     author ={P. M. W. Gill and M. W. Wong and R. H. Nobes and L. Radom},
     journal={Chem. Phys. Lett.},

Manuscript/EPAWTFT.tex

@@ -147,6 +147,7 @@ In particular, we highlight the seminal work of several research groups on the c
 
 \maketitle
 
+\raggedbottom
 \tableofcontents
 
 %%%%%%%%%%%%%%%%%%%%%%%
@@ -774,6 +775,24 @@ spin-contamination from the reference wave function, limiting their ability to f
 excitations that capture the correlation energy.
 }
 
+% SPIN-PROJECTION SCHEMES
+\hugh{%
+A number of spin-projected extensions have been derived to reduce spin-contamination in the wave function
+and overcome the slow UMP convergence.
+Early versions of these theories, introduced by Schlegel\cite{Schlegel_1986, Schlegel_1988} or 
+Knowles and Handy\cite{Knowles_1988a,Knowles_1988b}, exploited the ``projection-after-variation'' philosophy,
+where the spin-projection is applied directly to the UMP expansion.
+These methods succeeded in accelerating the convergence of the projected MP series and were 
+considered as highly effective methods for capturing the electron correlation at low computational cost.\cite{Knowles_1988b}
+However, the use of projection-after-variation leads to gradient discontinuities in the vicinity of the UHF symmetry-breaking point,
+and can result in spurious minima along a molecular binding curve.\cite{Schlegel_1986,Knowles_1988a}
+More recent formulations of spin-projected perturbations theories have considered the  
+``variation-after-projection'' framework using alternative definitions of the reference 
+Hamiltonian.\cite{Tsuchimochi_2014,Tsuchimochi_2019}
+These methods yield more accurate spin-pure energies without 
+gradient discontinuities or spurious minima.
+}
+
 %When one relies on MP perturbation theory (and more generally on any perturbative partitioning), it would be reasonable to ask for a systematic improvement of the energy with respect to the perturbative order, \ie, one would expect that the more terms of the perturbative series one can compute, the closer the result from the exact energy.
 %In other words, one would like a monotonic convergence of the MP series. Assuming this, the only limiting process to get the exact correlation energy (in a finite basis set) would be our ability to compute the terms of this perturbation series.
 %Unfortunately this is not as easy as one might think because i) the terms of the perturbative series become rapidly computationally cumbersome, and ii) erratic behaviour of the perturbative coefficients are not uncommon. 
@@ -899,6 +918,27 @@ for the two states using the ground-state RHF orbitals is identical.
 %The convergent and divergent series start to differ at fourth order, corresponding to the lowest-order contribution
 %%from the single excitations.\cite{Lepetit_1988}
 
+%%% FIG 3 %%%
+\begin{figure*}
+	\begin{subfigure}{0.32\textwidth}
+	\includegraphics[height=0.75\textwidth]{fig3a}	
+		\subcaption{\label{subfig:UMP_3} $U/t = 3$}
+    \end{subfigure}
+    %
+    \begin{subfigure}{0.32\textwidth}
+	\includegraphics[height=0.75\textwidth]{fig3b}
+		\subcaption{\label{subfig:UMP_cvg}}
+    \end{subfigure}
+    %
+    \begin{subfigure}{0.32\textwidth}
+	\includegraphics[height=0.75\textwidth]{fig3c}	
+		\subcaption{\label{subfig:UMP_7} $U/t = 7$}
+    \end{subfigure}	\caption{
+	Convergence of the UMP series as a function of the perturbation order $n$ for the Hubbard dimer at $U/t = 3$ and $7$.
+	The Riemann surfaces associated with the exact energies of the UMP Hamiltonian \eqref{eq:H_UMP} are also represented for these two values of $U/t$ as functions of $\lambda$.
+	\label{fig:UMP}}
+\end{figure*}
+
 The behaviour of the UMP series is more subtle than the RMP series as the spin-contamination in the wave function
 \hugh{introduces additional coupling between the singly- and doubly-excited configurations.}
 Using the ground-state UHF reference orbitals in the Hubbard dimer yields the \hugh{parametrised} UMP Hamiltonian
@@ -946,7 +986,7 @@ However, the excited-state EP is moved within} the unit cylinder and causes the
 convergence of the excited-state UMP series to deteriorate.
 \hugh{Our interpretation of this effect is that the symmetry-broken orbital optimisation has redistributed the strong 
 coupling between the ground- and doubly-excited states into weaker couplings between all states, and has thus
-sacrificed convergence of the excited-state series so that the chance of ground-state convergence can be maximised.}
+sacrificed convergence of the excited-state series so that the ground-state convergence can be maximised.}
 
 Since the UHF ground state already provides a good approximation to the exact energy, the ground-state sheet of
 the UMP energy is relatively flat and the corresponding EP in the Hubbard dimer always lies outside the unit cylinder.
@@ -974,27 +1014,6 @@ very slowly as the perturbation order is increased.
 %In contrast, the RMP expansion was always convergent for the open-shell excited state (which was a single CSF) while
 %the radius of convergence for the doubly-excited state was identical to the ground-state as this was the only exceptional point.
 
-%%% FIG 3 %%%
-\begin{figure*}
-	\begin{subfigure}{0.32\textwidth}
-	\includegraphics[height=0.75\textwidth]{fig3a}	
-		\subcaption{\label{subfig:UMP_3} $U/t = 3$}
-    \end{subfigure}
-    %
-    \begin{subfigure}{0.32\textwidth}
-	\includegraphics[height=0.75\textwidth]{fig3b}
-		\subcaption{\label{subfig:UMP_cvg}}
-    \end{subfigure}
-    %
-    \begin{subfigure}{0.32\textwidth}
-	\includegraphics[height=0.75\textwidth]{fig3c}	
-		\subcaption{\label{subfig:UMP_7} $U/t = 7$}
-    \end{subfigure}	\caption{
-	Convergence of the UMP series as a function of the perturbation order $n$ for the Hubbard dimer at $U/t = 3$ and $7$.
-	The Riemann surfaces associated with the exact energies of the UMP Hamiltonian \eqref{eq:H_UMP} are also represented for these two values of $U/t$ as functions of $\lambda$. \titou{Please Hugh, modify central figure.}
-	\label{fig:UMP}}
-\end{figure*}
-
 
 %==========================================%
 \subsection{Classifying Types of Convergence Behaviour} % Further insights from a two-state model}

hugh_notes.txt

@@ -1,3 +1,7 @@
++==========================================================+
+| Early MP Convergence Studies (1975-1990)                 |
++==========================================================+
+
 Bartlett and Silver, JCP (1975):
 --------------------------------
 [Supposedely the first MBPT?]
@@ -44,7 +48,7 @@ probably emerges from spin contamination in the UHF solution.
 
 [IS THERE MORE MBPT LITERATURE TO CONSIDER?]
 
-Laidig, Saxe, and Bartlett, JCP (1986):
+Laidig, Saxe, and Bartlett, JCP (1987):
 ---------------------------------------
 Investigate binding curves for N2 and F2 using multireference CC and MBPT
 
@@ -124,3 +128,69 @@ a maximum at intermediate distances. This contribution enters at fourth-order.
 Raghavachari, Pople, Replogle, and Head-Gordon, JPC, (1990):
 -----------------------------------------------------------
 
+
++==========================================================+
+| Spin-Projected MP2                                       |
++==========================================================+
+
+Early works on the convergence of UMP identified that spin-contamination was a driving 
+force behind slow convergence. To alleviate this, some authors considered the use of spin-projected MP2 
+approaches, with varying degrees of success.
+
+Schlegel, JCP (1986):
+---------------------
+First consideration of a spin-projected scheme for MP2. Takes an approximate form of the spin-projection
+operator and applies to project out the spin-contamination in the UHF and UMP energy. This amounts to 
+a PAV scheme, which in turn leads to gradient discontinuities in the binding curves and spurious minima
+for eg LiH. 
+
+Schlegel, JPC (1988):
+---------------------
+This second paper from Schlegel considers the rate of convergence of his spin-projected MP series. 
+He shows that the spin-projection significantly improves the rate of convergence, but that a small 
+slowly convergent term can remain.
+
+Knowles and Handy later argue that Schlegel's approaches are not satisfactory as they do not account
+for the fact that the reference Hamiltonian does not commute with the perturbation operator.
+
+Knowles and Handy, JPC (1988):
+------------------------------
+Consider how to formulate a spin-projected UMP series based on the Lowdin spin-projection operator. 
+Schlegel considered this first, but in a limited fashion where only the contamination from the next highest
+spin state was removed.
+
+This paper considers a spin projection on the previously determined UMP wave function series (determined 
+without spin projection). The challenge is how to incorporate the spin-projection operator without 
+destroying the nice properties of the reference Hamiltonian (eg. reference wave function is an eigenfunction).
+Instead, they use MP theory to build perturbation series for the wave functions, and then apply 
+spin-projection to obtain a series for the energy.
+
+The consider H2O, where they see discontinuities in the perturbed energies at the the CFP. Furthermore, 
+one of their spin-projected MP energies gives rise to a spurious minimum. This is in line with the the 
+results from Schlegel's work. Despite these discontinuities, they see that the spin-projection does
+accelerate the rate of convergence.
+
+Knowles and Handy, JCP (1988):
+-----------------------------
+This paper extends Knowles and Handy's previous approach to show that it is tractable for larger molecules.
+By comparing their results with Schlegel, the authors demonstrate the importance of considering the 
+full projection operator. They conclude by highlighting the remarkable accuracy that can be recovered at
+relatively low cost using this projected MP approach.
+
+Tsuchimochi and Van Voorhis, JCP (2014):
+----------------------------------------
+This paper considers a VAP scheme that is considered to be more cost-effective than the early PAV approaches.
+They define new spin-projected scheme EMP2 that are projected at each expansion order. This PAV method removes
+the discontinuities in the binding curves. However, there is some redundancy in the spin-projected wave functions
+at different orders that probably leads to some level of over counting. They also locate excited-state SUHF 
+states in H2 and demonstrate the the corresponding EMP2 energies also perform well. 
+
+Tsuchimochi and Ten-No, JCTC (2019):
+------------------------------------
+
+This paper brings spin-projected perturbation theory in line with modern CASPT2. They consider a generalised
+Fock operator and construct a first-order wave function ansatz from the spin-projected single and double excitations.
+The resulting SUPT2 provided more accurate binding curves than EMP2, which the authors believe is because
+the SUPT2 approach correctly handles the redundancy of internal rotations in the effective active space of the
+reference spin projection.
+