174 lines
7.3 KiB
ReStructuredText
174 lines
7.3 KiB
ReStructuredText
Selected Configuration Interaction
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==================================
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.. default-role:: cite
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These methods rely on the same principle as the usual |CI| approaches, except
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that determinants aren't chosen *a priori* based on an occupation or
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excitation criterion, but selected *on the fly* among the entire set of
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determinants based on their estimated contribution to the |FCI| wave function.
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It has been noticed long ago that, even inside a predefined subspace of
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determinants, only a small number significantly contributes to the wave
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function. `Bytautas_2009,Anderson_2018` Therefore, an *on the fly*
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selection of determinants is a rather natural idea that has been proposed
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in the late 60's by Bender and Davidson `Bender_1969` as well as Whitten
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and Hackmeyer. `Whitten_1969`
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The approach we are using in the |qp| is based on |CIPSI| developed by Huron,
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Rancurel and Malrieu, `Huron_1973` that iteratively selects *external*
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determinants (determinants which are not present in the variational space)
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using a perturbative criterion.
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There is however a computational downside. In *a priori* selected
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methods, the rule by which determinants are selected is known *a
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priori*, and therefore, one can map a particular determinant to some row or
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column index. `Knowles_1984` As a consequence, it can be systematically
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determined to which matrix element of :math:`\hat H` a two-electron integral
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contributes. This allows for the implementation of so-called
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*integral-driven* methods, that work essentially by iterating over
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integrals and are very fast.
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On the contrary, in selected methods an explicit list of determinants has to be
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kept, and there is no immediate way to know whether a determinant has been
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selected, or what its index is in the list. Consequently, a
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*determinant-driven* approach will be used, in which the loops run over
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determinants rather than integrals. This can be a lot more computationally
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expensive since the number of determinants is typically much larger than the
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number of integrals.
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What makes *determinant-driven* approaches possible here is:
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- the fact that selected |CI| methods will keep the number of determinants small
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enough, orders of magnitude smaller than in *a priori* selected methods for
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wave functions with equal energies,
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- an efficient way to compare determinants in order to extract the
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corresponding excitation operators `Scemama_2013`,
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- an intense filtering of the internal space to avoid as much as possible
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determinant comparisons of disconnected determinants,
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- a fast retrieval of the corresponding two-electron integrals in memory.
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Simple Algorithm
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----------------
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.. default-role:: math
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.. |SetDI| replace:: `\{|D_I\rangle\}^{(n)}`
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.. |Psi_n| replace:: `|\Psi^{(n)}\rangle`
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.. |kalpha_star| replace:: `\{ |\alpha \rangle \}_\star ^{(n)}`
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.. |ealpha| replace:: `e_\alpha`
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.. |EPT| replace:: `E_\text{PT2}`
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The variational wave function |Psi_n| is defined over a set of determinants
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|SetDI| in which we diagonalize |H|.
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.. math::
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|\Psi^{(n)}\rangle = \sum_{I} c_I^{(n)} |D_I\rangle
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The determinants in |SetDI| will be characterized as **internal**.
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#. For all **external** determinants |kalpha| `\notin` |SetDI|, compute the
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Epstein-Nesbet second-order perturbative contribution to the energy
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.. math::
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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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iteration. Note that other perturbation theory baesd methods could be used to estimate
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|ealpha|.
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#. An estimate of the total missing correlation energy can be computed
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by summing all the |ealpha| contributions
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.. math::
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E_\text{PT2} & = \sum_{\alpha} e_\alpha \\
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E_\text{FCI} & \approx E + E_\text{PT2}
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#. |kalpha_star|, the subset of determinants |kalpha| with the largest
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contributions |ealpha|, is added to the variational space
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.. math::
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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 based upon some criteria (number of determinants in
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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 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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------------------------------------------------------------------------
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The simple algorithm described above would be too slow to make calculations practical. Instead,
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|QP| uses a stochastic algorithm :cite:`Garniron_2017.2` in order to compute
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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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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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----------------------------------------------
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The following description was used in a previous version of the |CIPSI| algorithm
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which was less efficient. Nonetheless, it introduces the notions of **generator** and **selector** determinants
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which are much more general than the |CIPSI| algorithm that targets the |FCI| and can be used to realize virtually
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**any kind of CI in a selected way**.
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We define **generator** determinants, as determinants of the internal space
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from which the |kalpha| are generated.
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We then define **selector** determinants, a truncated wave function
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used in the computation of |ealpha|.
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For calculations in the |FCI| space, the determinants are sorted by decreasing
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`|c_I|^2`, and thresholds are used on the squared norm of the wave function.
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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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selectors.
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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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we only generate a subset which are likely to be selected.
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The computation of |EPT| using a truncated wave function is biased,
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so if an accurate estimate of the |FCI| energy is desired, it is preferable
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to recompute |EPT| with the hybrid deterministic/stochastic algorithm
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:cite:`Garniron_2017b` which is unbiased (this is the default).
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Modifying the selection space
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-----------------------------
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By changing the definition of generators, and the rules for the generation of
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the |kalpha|, it is easy to define selected variants of traditional |CI| methods.
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For example, if one defines the |HF| determinant as the only generator,
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one will produce a selected |CISD|. If one also changes the rules for the generation
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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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rules allow for excitations only inside the |CAS|, we obtain a selected
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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 adds the rule to exclude those double
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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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All such things can be done very easily when programming within the |qp|.
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-----------------------------------
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.. bibliography:: selected.bib
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:style: unsrt
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:labelprefix: A
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