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Some clarifications and minor phrasing changes

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vijay 2020-12-16 11:21:53 +01:00 committed by GitHub
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@ -77,7 +77,7 @@ The determinants in |SetDI| will be characterized as **internal**.
e_\alpha = \frac{ \langle \Psi^{(n)}| {\hat H} | \alpha \rangle^2 }{E^{(n)} - \langle \alpha | {\hat H} | \alpha \rangle }.
`E^{(n)}` is the variational energy of the wave function at the current
iteration. Note that another perturbation theory could be used to estimate
iteration. Note that other perturbation theory baesd methods could be used to estimate
|ealpha|.
#. An estimate of the total missing correlation energy can be computed
@ -96,22 +96,22 @@ The determinants in |SetDI| will be characterized as **internal**.
\{ |D_I \rangle \}^{(n+1)} = \{|D_I\rangle\}^{(n)} \cup \{ |\alpha\rangle \}_\star^{(n)}
#. Go to iteration n+1, or exit on some criterion (number of determinants in
#. Go to iteration n+1, or exit based upon some criteria (number of determinants in
the wave function, low |EPT|, ...).
Of course, such a procedure can be applied on any state and therefore can allow to treat both ground and excited states.
Of course, such a procedure can be applied on any state and therefore can allow the treatment of both ground and excited states.
Stochastic approximations for the selection and the computation of |EPT|
------------------------------------------------------------------------
The simple algorithm would be too slow to make calculations possible. Instead,
the |QP| uses a stochastic algorithm :cite:`Garniron_2017.2` in order to compute
efficiently the |EPT| and to select on-the-fly the best Slater determinants.
The simple algorithm described above would be too slow to make calculations practical. Instead,
|QP| uses a stochastic algorithm :cite:`Garniron_2017.2` in order to compute
|EPT| efficiently and to select the best Slater determinants on-the-fly.
In such a way, the selection step introduces no extra cost with respect to the |EPT| calculation and the |EPT|
itself is unbiased but associated with a statistical error bar rapidly converging.
itself is unbiased but associated with a rapidly converging statistical error bar.
Deterministic approximations for the selection
@ -134,7 +134,7 @@ The default is to use :option:`determinants threshold_generators` = 0.99 for
the generators, and :option:`determinants threshold_selectors` = 0.999 for the
selectors.
This is nothing but the 3-class |CIPSI| approximation to accelerate the selection,
This is nothing but the three-class |CIPSI| approximation to accelerate the selection,
:cite:`Evangelisti_1983` where instead of generating all possible |kalpha|,
we only generate a subset which are likely to be selected.
@ -156,13 +156,13 @@ one will produce a selected |CISD|. If one also changes the rules for the genera
to generate only the double excitations, one will have a selected |CID|.
The generators can also be chosen as determinants belonging to a |CAS|. If the
rules allow only for excitations inside the |CAS|, we obtain a selected
|CAS| |CI|. If the rules allow for excitations in the |FCI| space, we obtain
a selected |CAS-SD|. And if one add the rule to prevent for doing double
excitations with two holes and two particles outside of the active space, one
rules allow for excitations only inside the |CAS|, we obtain a selected
|CAS|-|CI|. If the rules allow for single and double-excitations in the |FCI| space, we obtain
a selected |CAS-SD|. And if one adds the rule to exclude those double
excitations which contain two holes and two particles outside of the active space, one
obtains a selected |DDCI| method.
All such things can be done very easily when programming the |qp|.
All such things can be done very easily when programming within the |qp|.
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