Added references and personal reading notee

Manuscript/EPAWTFT.tex

@@ -628,13 +628,13 @@ role of \textit{quasi}-EPs in determining the behaviour of the HF approximation.
 %Through the introduction of non-Hermiticity, we have provided a more general framework in which the complex and diverse characteristics of multiple solutions can be explored and understood.}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{M{\o}ller-Plesset perturbation theory in the complex plane}
+\section{M{\o}ller-Plesset Perturbation Theory in the Complex Plane}
 \label{sec:MP}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
 %=====================================================%
-\subsection{Basics}
+\subsection{Background Theory}
 %=====================================================%
 
 In electronic structure, the HF Hamiltonian \eqref{eq:HFHamiltonian} is often used as the zeroth-order Hamiltonian
@@ -649,7 +649,7 @@ With the MP partitioning, the parametrised perturbation Hamiltonian becomes
     + (1-\lambda) \sum_{i}^{N} v^{\text{HF}}(\vb{x}_i)
     + \lambda\sum_{i<j}^{N}\frac{1}{\abs{\vb{r}_i-\vb{r}_j}}.
 \end{multline}
-Any set of orbitals can be used to define the HF Hamiltonian, although usually either the RHF or UHF orbitals are chosen to 
+Any set of orbitals can be used to define the HF Hamiltonian, although either the RHF or UHF orbitals are usually chosen to 
 define the RMP or UMP series respectively.
 The MP energy at a given order $n$ (\ie, MP$n$) is then defined as
 \begin{equation}
@@ -672,7 +672,7 @@ in the molecular spin-orbital basis\cite{Gill_1994}
       {\abs{\vb{r}_1 - \vb{r}_2}}.
 \end{equation}
 
-While most practical calculations usually consider only the MP2 or MP3 approximations, higher order terms can 
+While most practical calculations generally consider only the MP2 or MP3 approximations, higher order terms can 
 easily be computed to understand the convergence of the MP$n$ series.\cite{Handy_1985}
 \textit{A priori}, there is no guarantee that this series will provide the smooth convergence that is desirable for a
 systematically improvable theory.
@@ -689,19 +689,23 @@ unrestricted reference orbitals.
 %As mentioned above, by searching the singularities of the function $E(\lambda)$, one can get information on the convergence properties of the MP series. 
 %These singularities of the energy function are exactly the EPs connecting the electronic states as mentioned in Sec.~\ref{sec:intro}. 
 %The direct computation of the terms of the series is quite manageable up to fourth order in perturbation, while the fifth and sixth order in perturbation can still be obtained but at a rather high cost. \cite{JensenBook}
-%In order to better understand the behavior of the MP series and how it is connected to the singularity structure, we have to access high-order terms. 
+%In order to better understand the behaviour of the MP series and how it is connected to the singularity structure, we have to access high-order terms. 
 %For small systems, one can access the whole terms of the series using full configuration interaction (FCI). 
 %If the Hamiltonian $H(\lambda)$ is diagonalized in the FCI space, one gets the exact energies (in this finite Hilbert space) and the Taylor expansion with respect to $\lambda$ allows to access the MP perturbation series at any order.
 
 
-Obviously, although practically convenient for electronic structure calculations, the MP partitioning is not the only possibility, and alternative partitioning have been proposed in the literature:
+Although practically convenient for electronic structure calculations, the MP partitioning is not 
+the only possibility and alternative partitionings have been considered including: %proposed in the literature:
 i) the Epstein-Nesbet (EN) partitioning which consists in taking the diagonal elements of $\hH$ as the zeroth-order Hamiltonian. \cite{Nesbet_1955,Epstein_1926} 
 %Hence, the off-diagonal elements of $\hH$ are the perturbation operator,
 ii) the weak correlation partitioning in which the one-electron part is consider as the unperturbed Hamiltonian $\hH^{(0)}$ and the two-electron part is the perturbation operator $\hV$, and 
 iii) the strong coupling partitioning where the two operators are inverted compared to the weak correlation partitioning. \cite{Seidl_2018}
+\hugh{While an in-depth comparison of these different approaches can offer insight into 
+their relative strengths and weaknesses for various situations, we will restrict our current discussion
+to the convergence properties of the MP expansion.}
 
 %=====================================================%
-\subsection{Behavior of the M{\o}ller-Plesset series in molecular systems}
+\subsection{M{\o}ller-Plesset Convergence in Molecular Systems}
 %=====================================================%
 
 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.
@@ -740,7 +744,7 @@ Even if there were still shaded areas in their analysis and that their classific
 
 
 %==========================================%
-\subsection{Behavior of the M{\o}ller-Plesset series in the Hubbard dimer}
+\subsection{M{\o}ller-Plesset Convergence in the Hubbard Dimer}
 %==========================================%
 
 To illustrate the behaviour of the RMP and UMP series, we can again consider the Hubbard dimer.

hugh_notes.txt

@@ -0,0 +1,115 @@
+Bartlett and Silver, JCP (1975):
+--------------------------------
+[Supposedely the first MBPT?]
+Report moderately large molecular calculations using Slater type orbitals.
+
+Pople, Binkley, and Seeger, IJQCS (1976):
+-----------------------------------------
+This paper introduces MP2 as a possible route to incorporating electron correlation. Largely a 
+pioneering paper that lays out the properties of MP2 etc.
+
+Pople, Krishnan, Schlegel, and Binkley, IJQC (1978):
+----------------------------------------------------
+Discusses different correlation techniques for quantum chemistry. This paper is particularly 
+concerned with comparing the MP2 expression with the CC approach which was emerging at the
+time. They show that CCD is equivalent to MP3 (?). 
+
+Krishnan, Frisch, and Pople, JCP (1980):
+----------------------------------------
+Assessed that triple excitations that appear at 4th order are important
+in the quantitative treatment of chemical binding.
+
+Laidig, Fitzgerald, and Bartett, CPL (1984):
+--------------------------------------------
+Investigate convergence of MBPT. They find BH is slowly convergent. HF is also slowly convergent,
+accidentally so since the MBPT(4) is erroneously slow. New excitations are introduced at each even order.
+Introduce Pade approximant to accelerate convergence, giving better accuracy.
+
+Knowles, Somasundram, Handy, and Hirao, CPL (1985):
+---------------------------------------------------
+Apply their FCI code to look at the convergence of MBPT(n).
+
+Rate of convergence and size of terms is heavily system-dependent. Notice different convergence
+behaviour for odd/even terms (oscillatory?). MP4 appears to capture the majority of the correlation
+energy.
+
+Handy, Knowles, and Somasundram, TCA (1985):
+--------------------------------------------
+Apply the FCI framework again to systematically investigate the convergence of the MP series. 
+Attempt to identify whether the MP series is convergent or not, and compare RHF/UHF.
+
+Observe increasingly slow RMP convergence for stretched water with erratic behaviour. For stretched
+geometry with UMP, convergence appears smooth but is very slow. Suggest that this slow convergence 
+probably emerges from spin contamination in the UHF solution.
+
+[IS THERE MORE MBPT LITERATURE TO CONSIDER?]
+
+Laidig, Saxe, and Bartlett, JCP (1986):
+---------------------------------------
+Investigate binding curves for N2 and F2 using multireference CC and MBPT
+
+Divergence in R-MBPT beyond 4 bohr. 
+All finite-order U-MBPT calculations for F2 give an unphysical barrier around 2.8-2.9 bohr.
+Divergence of R-MBPT observed in N2 beyond 3 bohr. Around minimum, the series is oscillatory and
+very slowly convergent. In contrast, the U-MBPT is convergent and non-oscillatory, although 
+low-order expansions give qualitatively wrong energetics (eg. unphysical barriers or second minima). 
+
+Schlegel, JCP, (1986):
+----------------------
+Apply spin-projection to UHF and UMP to obtain improved potential enerrgy curves. Use a
+post-perturbation projection to avoid mixing in higher energy states.
+
+Perturbation corrections do not significantly reduce spin contamination. PUHF has a gradient
+discontinuity at the CFP (but these are PAV). This kink is reduced by adding the perturbation
+correlation. 
+
+Gill and Radom, CP, (1986):
+---------------------------
+Consider ``bottom-up'' approach, where look at successive contributions from HF, MP1, MP2, ...
+Use a recursive approach to higher-order terms.
+
+In \ce{He^2+}, the UHF becomes progressively more spin contaminated for large bond lengths.
+RMP gives a progressively better estimate of the dissociative barrier height. In contrast, UMP
+starts by increasing the barrier, before decrease after 3rd order. They conclude that poor convergence
+can be attributed directly to a poor reference representation of the exact wave function.
+
+While some properties (eg. bond length) might be well-converged, others can be far from convergence.
+Worst barrier height estimate occurs at UMP4, after which there is very slow convergence.
+
+They suspect that UMP problems can be attributed to spin-contamination. Conclude that incorrect
+"qualitative" nature of RMP is not as bad as spin-contamination in UMP.
+
+Gill, Wong, Nobes, and Radom, CPL (1988):
+-----------------------------------------
+Investigate performance of RMP expansions for homolytic bond breaking.
+
+Discuss the fact that the RMP will ultimately be divergent for homolytic bond breaking at 
+large extension, since the orbital energy based denominators will vanish. Propose a (2x2) matrix 
+problem to estimate whether an RMP series will be convergent. They use this metric to determine if
+an RMP series converges rapidly, slowly, or diverges.
+
+Gill, Pople, Radom, and Nobes (1988):
+-------------------------------------
+Investigate the effect of spin-contamination for slow UMP convergence. Spin-projection is 
+difficult to do exactly, and approximate forms can lead to kinks in the potential energy surface.
+
+Above critical point, UHF singles and doubles both mix with HF to give the exact wave function.
+Contribution of singles decreases for complete dissociation.
+
+Rate of UMP convergence slows down after critical point, with less that 3% of total correlation
+captured at UMP4. Increasingly slow convergence not due to singles as the singles contribution to 
+the UCI falls to zero as the rate of convergence becomes slower. It is therefore double 
+contribution that is poorly captured by low-order UMP terms.
+
+Lepetit, Pelissier, and Malrieu, JCP (1988):
+--------------------------------------------
+Investigate the poor convergence of unrestricted many-body perturbation theory.
+
+UHF reference has large and spurious energy shift that dramatically slows the rate of convergence.
+This comes from the localisation of the MOs in large separation and the doubly excited determinants
+that result from spin exchanges in the sigma bond. This effect is seen in N2, and other systems. 
+The EN partitioning avoids this, but the PT terms then become undetermined (zero on numerator and denominator).
+
+Single excitations can interact with the doubly-excited determinants. This matrix elements goes through
+a maximum at intermediate distances. This contribution enters at fourth-order.
+