EPAWTFT/hugh_notes.txt

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| Early MP Convergence Studies (1975-1990)                 |
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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 (1987):
---------------------------------------
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 energy 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.

Nobes, Pople, Radom, Handy and Knowles, CPL (1987):
---------------------------------------------------
Look at the MP convergence in the cyanide anion (CN-) at this is a molecule with a symmetry-broken
solution at equilibrium. They observe very slow convergence in these UMP series too, with more than
MP4 being needed for accurate energies. This confirms that spin-contamination, rather than extended 
bonds, provides the driving force for slow UMP convergence. 


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.

Raghavachari, Pople, Replogle, and Head-Gordon, JPC, (1990):
-----------------------------------------------------------


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| Spin-Projected MP2                                       |
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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 (1988a):
------------------------------
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 (1988b):
-----------------------------
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.

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| Classifying Convergence Behaviour                        |
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Cremer and He, JPC (1996):
--------------------------

Consider the MP6 energy as this is the next even order after MP2 and MP4 so introduces new excitations 
(in this case pentuples and hextuples). 

They decompose their MPn correlation into pair-pair, pair pair pair, etc terms to try and understand the 
convergence behaviour:
   SDQ = SS + SD + DD + DQ + QQ (singles, doubles, quadruples)
     T = ST + DT + TT + TQ (terms including triple excitations)

They intend to show: 
Class A) Monotonic convergence expected for systems in which the electron pairs are well-separated and weakly couple.
         including eg BH, NH2, CH3, CH2 etc
         Generally include well-separated electron pairs such that three-electron correlation effects are weak.

Class B) Initial oscillatory convergence with strong pair and three-electron correlation effects.
         eg. Ne, F, F^-, FH
         In these systems, there are closely spaced electron pairs that cluster in a small region of space. 
         One might imagine that this requires greater orbital relaxation, perhaps ``breathing'' relaxation, 
         to allow the electron pairs to become separated? Or maybe that it generally introduces stronger
         dynamic correlation effects? Orbital relaxation plus pair correlation comes through T1 T2 terms.

They observe both E(SDQ) and E(T) terms negative in Class A systems, but E_MP5(SDTQ) terms generally positive 
in Class B. Oscillations in the T correlation terms drive the oscillatory convergence behaviour. This behaviour
does not appear to be caused by multiconfigurational effects, but may be amplified by them.

Class B has more important orbital relaxation effects and three-electron correlation than Class A.

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| Moller-Plesset Critical Point                            |
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Stillinger, JCP (2000):
-----------------------
Convergence appears to fall into two categories:
    1) Convergent (eg. BH, CH2)
    2) Divergent with even-odd sign alternation (eg Ne, HF, H2O)
This second type is characterisitic of a singularity on the negative real axis.

Aim of this paper is to show that this singularity emerges from a multielectron 
autoionization process.

Introduce the idea of the positive SCF energy component for negative lambda.

SCF is essentially a negative charge cloud that is spatially distributed by the extent of the orbitals.
For sufficiently positive lambda, this field converts to diffuse attraction surrounding the nucleus and
electron pairs become increasingly repulsive. On the negative axis, this field becomes repulsive but the 
electron-electron interactions become positive. This allows the electrons to form a bound state away from 
the nucleus, leading to autoionization. 

This autoionization threshold is analogous to the Z^-1 expansion for the two-electron atom. (Baker 1990)
They illustrate this process using the two-electron atom again, for which the find the ionization threshold
at \lc = -1.33. This is outside the radius of convergence, so the MP series is predicted to be absolutely
convergent. This singularity moves further from the origin for larger Z, but for H- the threshold is -0.08!!!

Overall, this paper concludes that the MP convergence will be affected by a fundamental critical point on 
the negative real axis. The form of this singularity is, at this stage, unclear.

Goodson, JCP (2000a):
---------------------
Introduces some approximants... [to be read later].

If dominant branch points are complex-conjugate pairs in the negative half plane, then they correspond to
regios of alternating signs with a pattern broken periodically by consecutive terms of the same sign,
If in the positive half plane, then there are regions of only one sign alternating with regions of only
the opposite sign.

Goodson, JCP (2000b):
---------------------
Class A: branch point connects the eigenstate with the next higher eigenstate of the same symmetry.
Class B: branch point lies on the negative real axis.

Goodson and Sergeev, AQC (2004):
--------------------------------
This review considers what is currently known about singular points in the complex-lambda plane and how
this affects the convergence of the perturbation series. Aim to connect the singularity structure and the
different convergence ``classes''.

E(l) is a complicated function with a `rich structure' of singular points.

Behaviour of Stillinger singularity is different to a branch point. E(z) will acquire an imaginary part as
it passes through the singularity as the eigenstate becomes a state in the scattering continuum. Expect
the derivative to be continuous through the critical point.

Following Goodson previous work, they draw connections between Cremer and He's classes the singularity 
structure:
    Class B - corresponds to dominant singularity on the negative real axis (eg Stillinger critical point)
    Class A - corresponds to dominant singularity on the positive real aixs.
If the imaginary part of Class A singularities is sufficiently small, then oscillations can have such
a long period that they may appear to converge monotonically to very high orders. This is what we observe
in the UMP series of the Hubbard dimer.

Period can be given by n0 = pi / arctan(|Im(z01) / Re(z01)|)

Physical connection to Cremer and He arises because if valence orbitals are clustered in a relatively small
region of space, then the autoionization will be favoured at small |z| and the this is likely to be the
dominant singularity.

The relationship between these singularities and the basis set also matches as Class A is relatively insensitive
to basis, but class B gets worse with larger basis sets. It is also possible to get a branch point in the negative half plane, and this leads to the worst type of convergence (eg N2, C2, CN+).

> Resummation:

Can use either Pade or quadratic approximants. Pade can't describe branch points, so quadratic are more
suitable. Quadratics fit more complicated branch points using clusters of square-root branch points.

> Examples

They use these approximants to identify the dominant singularities. As expected, they find the dominant
singularity in the Class A systems lie on the positive half plane with relatively large imaginary component.

BH is classic Class A, and F- is a classic Class B.

For Class B, the quadratic approximants gain an imaginary part beyond the critical point. The rational 
approximant maps the branch point using alternating zeros and poles along the real axis. The quadratic
approximants cluster a number of branch points around the critical point, suggesting a fundamental difference
to a branch point in the positive plane.

From this analysis, all complex conjugate branch points are defined as `class \alpha', and the critical 
point is defined as `class \beta'.

Better to actually classify with respect to the dominant singularity in the negative/positive half planes
to give eg alpha/alpha ....

Sergeev, Goodson, Wheeler, and Allen, JCP (2005):
-------------------------------------------------
Olsen showed that the F- series is divergent with diffuse functions, but convergent with compact functions.
This paper considers Stillinger's conjecture for the noble gases by analysing the singularity structure.

Increasing Z increases barrier for the electrons to escape, but the well in the nuclear region narrows. 
Eventually the electrons can escape by tunneling through the barrier. It is also possible to get a critical
point in the positive real z-axis corresponding to one-electron ionization. THIS would correspond to the 
two-electron critical point.

In a finite basis set, the singular (branch points) must occur in complex conjugate pairs. They show that 
increasing the basis set size leads to a cluster of very tight avoided crossings for negative z. These
avoided crossings are modelling the continuum and the critical point. They add
a ghost atom to allow the electrons to dissociate, and show that these lead to greater clustering of
negative avoided crossings. This ghost atom can then be replaced by a real atom (eg Ne -> HF), and then 
the valence electrons will jump to the hydrogen, leading to a critical point (as shown by a plot of the 
dipole moment). These two systems therefore have similar convergence behaviour. Without this ghost atom, 
one gets complete dissociation rather than an electron cluster formation.

Indeed similar clustering is seen in the positive real z values, eg in Ar. The argument is that the valence
electrons are farther from the nucleus than in Ne, so the mean-field potential is less able to counter
the increased interelectron repulsion than in Ne.

Analysis resolves a disagreement between Stillinger and Olsen. Olsen found Class B resulted in square-root 
branch points, but this is only because the 2x2 matrix is insufficient. 

A key result from this paper is that critical points can also occur on the positive real axis, and these
correspond to one-electron ionisations. Origin is an avoided crossing with high-energy Rydberg state.

Sergeev and Goodson, JCP (2006):
--------------------------------
Further explore the singularity structure of a set of systems to classify using the alpha/beta scheme.

Systems with a low-lying excited state that mixes strongly with the ground state, such that a 
single-reference HF determinant gives a poor descriptions of the wave function, will have a class \alpha
singularity in the positive half plane slightly beyond the physical point z=1.

Goodson and Sergeev, PLA (2006):
--------------------------------
This paper considers how to understand the singularity structure using only up to MP4. Argument is that E_FCI(z)
must always have complex-conjugate branch points, so cannot accurately model the true critical point E(z). 
Instead, it models these critical points with a cluster of square-root branch point pairs with small imaginary
components. (See Sergeev et al. 2005)

This paper introduces further approximants to model these singularities using only MP4 information. It can
then consider larger systems. They also use some conformal mapping and other tricks to improve the representation
of the singularities and improve convergence.

Herman and Hagedorn, IJQC (2008):
---------------------------------
Consider convergence or divergence of MP is considered for two-electrons with variable nuclear charge.
In particular, they look to extend Goodson analysis to see how the singularity changes for increasingly more
exact Hamiltonians. 

They use a `delta-function model' for He-like atoms, where the delta functions replace the Coulomb potentials.
This is advantageous as the problem becomes one-dimensional. They introduce a second model for the e-e cluster.
The Stillinger critical point is then a point where the two energies cross.

[ALL GETS A BIT INVOLVED... SKIPPING TO CONCLUSIONS...]

[[TO BE HONEST, NOT SURE WHAT ALL THIS SHOWS...]]


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| Miscellaneous (or category currently unclear)            |
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Fink, JCP (2016):
-----------------