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...]
