157 lines
9.3 KiB
ReStructuredText
157 lines
9.3 KiB
ReStructuredText
============================================================
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What is a selected CI caculation ? Some theoretical concepts
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============================================================
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Generalities
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============
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The selected CI algorithm can be seen as a way to compute the energies (and various properties) of a given number of eigenstates
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of a given :term:`target space` (ex : CISD, CAS-CI, DDCI etc ...),
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but by taking the freedom of splitting the wave function of the target space in term
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of :term:`internal determinants` treated variationally and :term:`perturbers` treated perturbatively.
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Why this freedom ? Because in a given :term:`target space` (except some really special cases) most of the information
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is concentrated within a tiny fraction of the :term:`target wave function`, and the remaining part can be reasonabely estimated by perturbtation.
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This splitting of the wave function is not done in one shot, it is done iteratively. The iterative procedure needs a :term:`stopping criterion` to end the calculation and to control the quality of the calculation.
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This :term:`stopping criterion` can be for example the number of determinants in the :term:`intern space`,
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or the value of the :term:`energetic perturbative correction` to estimate the importance of the perturbation, or the convergence of the :term:`estimated target energy`, or anything that can
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be defined in terms of available informations during the calculation.
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The heart of the selected CI algorithm is based on the CIPSI algorithm (ref Malrieu).
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Selected CI in a few words
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==========================
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First you define a :term:`target space`. Once the target space is defined, you define the :term:`stopping criterion`.
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After that, a starting wave function is chosen by the user (HF by default).
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This starting wave function is the first :term:`Internal determinants` wave function.
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After that, one would like to extend this :term:`Internal determinants` wave function by adding
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some :term:`perturbers` determinants.
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How do we select the good :term:`perturbers` ?
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do while (:term:`stopping criterion` is reached)
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1) Generates :term:`perturbers` determinants according to your chosen :term:`target space`.
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:math:`\Rightarrow` generates a set of :term:`perturbers` :math:`\{|D_P\rangle\}`
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2) The :term:`perturbers` importance are estimated by perturbation thanks to the current :term:`internal determinants`.
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3) The most important of the :math:`\{|D_P\rangle\}` are chosen to enter in the :term:`internal determinants`.
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4) You rediagonalize the H matrix with the previous set of :term:`internal determinants` and the chosen :term:`perturbers`.
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:math:`\Rightarrow` create a new wave function and a new set of the :term:`internal determinants`
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5) iterate
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Once the iterative procedure is stopped, the :term:`internal determinants` wave function have a :term:`variational energy`,
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and by adding the :term:`energetic perturbative correction` one have the :term:`estimated target energy`
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which is an approximation of the :term:`target energy`. One should notice that if one takes
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a :term:`stopping criterion` such as the all the determinants of the target space are in the :term:`intern space`,
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the :term:`estimated target energy` is the :term:`target energy`.
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If one is interested of how is built the selected CI wave function into more details, one can read the further section.
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What is a selected CI iteration in practice (and some details)
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==============================================================
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From the previous section we have roughly seen how the selected CI works. Now, getting a bit more into details,
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we will see what is done in practice during a selected CI iteration. To illustrate this, a simple CISD example wil be given.
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The general picture
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^^^^^^^^^^^^^^^^^^^
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The :term:`target space` defines entirely the method that is going to be approximated, and the stopping criterion will be the only approximation.
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This :term:`target space` can always be defined in terms of application of an :term:`H operator`
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(with some :term:`excitation restrictions`) on a given set of determinants that we shall call the :term:`generators` determinants.
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We call :term:`restricted H operator` this precise H operator.
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The target space intirely defines the :term:`restricted H operator`.
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The only flexibility is the perturbation theory to be used to estimate the coeficients of the :term:`perturbers`.
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If the target space is just defines in term of a CI matrix to diagonalize, the standard :term:`Diagonalization EN EG` perturbation will be used.
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If some other constraints are imposed in addition to the CI matrix
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(e.g. some physical conditions of size extensivity such as in the :term:`CISD+SC2` method),
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then the perturbation must be adapted to properly respect the :term:`target space`.
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CISD : the :term:`target space` is here defined intirely by all the single and double excitations acting on the HF determinant.
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So the :term:`generators` subset of determinants here is only the HF determinant and will not change along the iteration.
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If some occupied orbitals are chosen to be frozen (no excitations from those orbitals)
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or some virtuals are chosen to be deleted (no excitations going to these virtuals orbitals),
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this constraint imposes the :term:`excitation restrictions`. So here the :term:`restricted H operator` will be all the single and double excitations except those involving either a frozen core orbital or a deleted virtual orbital.
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Different choices of perturbation theory can be made for the CISD, but the standard :term:`Diagonalization EN EG` can be trustly used.
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Once the :term:`target space` have been defined, what does in practice a selected CI iteration.
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For the sake of simplicity, here we emphasize on the ground state :math:`| \psi_0 \rangle`. At a given iteration, one have a :term:`selected wave function` :math:`|\psi_0\rangle`, and the selected CI algorithm performs :
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do G = 1, N_Generators
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1) Apply the :term:`restricted H operator` on the :math:`|D_G \rangle` :term:`generators` determinant belonging to :math:`| \psi_0 \rangle`
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:math:`\Rightarrow` generates a set of :term:`perturbers` :math:`|D_P\rangle`
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2) Estimate the perturbative importance of each perturbers
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:math:`\Rightarrow` example for the :term:`EN EG` perturbation theory
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.. math::
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c_{D_P}^0= \frac{ \sum_{S=1,N_{\rm selectors}} c_S^0 \langle D_S|H|D_P\rangle}{ \langle \psi_0 |H|\psi_0 \rangle - \langle D_P |H|D_P\rangle } \\
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e_{D_P}^0= \frac{(\sum_{S=1,N_{\rm selectors}} c_S^0 \langle D_S|H|D_P\rangle) ^2}{\langle \psi_0 |H|\psi_0 \rangle - \langle D_P |H|D_P\rangle}
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3) Keep the most important :term:`perturbers` :math:`|D_P \rangle`
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:math:`\Rightarrow` they enter in the :term:`intern space`
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4) Rediagonalize H within this new subset of determinants
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:math:`\Rightarrow` better :term:`selected wave function`
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5) Iterate
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An important point here is that at a given iteration, the estimation of the perturbative coefficients of the :term:`perturbers`
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depends on the quality of the :term:`selected wave function` .
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As the iterations go on, the :term:`selected wave function` becomes closer
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and closer to the :term:`target wave function`, and so the perturbative estimation of the :term:`perturbers` coefficients or energetic contribution becomes more and more precise.
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CISD : At the first iteration, starting from the HF determinant :
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1) By applying H on the :term:`generators` (HF) one generates all singles and doubles
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2) For each :term:`perturbers` you estimate by perturbation its coefficient of energetic contribution.
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i) Here the :term:`selectors` is only the HF determinant.
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ii) If the :term:`Brillouin theorem` is respected, all the singles have zero coefficients since the :term:`selectors` here is only the HF determinant.
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iii) The most important double excitations entered
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iv) The :term:`energetic perturbative correction` is calculated
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v) The :term:`estimated target energy` is just the sum of the HF energy and the :term:`energetic perturbative correction`
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3) H is rediagonlaized in the new set of determinants : HF + the selected doubles
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:math:`\Rightarrow` better :term:`variational energy` and :term:`selected wave function`
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4) The :term:`generators` subset does not change.
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At the second iteration :
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1) By applying H on the :term:`generators` (still HF) one generates all singles and doubles
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2) For each :term:`perturbers` you estimate by perturbation its coefficient of energetic contribution.
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i) Here the :term:`selectors` is now HF + the previously selected doubles
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:math:`\Rightarrow` the :term:`perturbers` now interact with all the previously selected doubles
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:math:`\Rightarrow` better estimation of the coefficients of the :term:`perturbers`
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:math:`\Rightarrow` the singles have non zero coefficients
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ii) The most important :term:`perturbers` enter in the :term:`intern space`
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iv) The :term:`energetic perturbative correction` is re estimated
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v) The :term:`estimated target energy` is now the sum of the variational energy of the :term:`selected wave function` and the :term:`energetic perturbative correction`
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:math:`\Rightarrow` better estimation of the :term:`target energy`
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Iterate untill you reached the desired :term:`stopping criterion`
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