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