170 lines
7.6 KiB
ReStructuredText
170 lines
7.6 KiB
ReStructuredText
=====================================
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Selection, perturbation ... keywords
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=====================================
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.. |CISD| replace:: :abbr:`CISD (Configuration Interaction with Single and Double excitations)`
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.. |HF| replace:: :abbr:`HF (Hartree Fock)`
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.. |CAS-CI| replace:: :abbr:`CAS-CI (Complete Active Space Configuration Interaction)`
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.. |DDCI| replace:: :abbr:`DDCI (Difference Dedicated Configuration Interaction)`
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.. glossary::
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:sorted:
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Energetic perturbative correction
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Corresponds to the correction to the energy at the second order of a given perturbtation theory
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to a given state m.
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By convention it noted :math:`E_{PT2}^m`
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Variational energy
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Corresponds to the variational energy of the :term:`selected wave function` for a given state .
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By convention it noted :math:`E_{Var}^m` for the mth eigenvector.
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.. math::
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E_{Var}^m = \langle \psi_m |H|\psi_m \rangle
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Estimated target energy
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Corresponds to the estimation of the target energy for a given :term:`selected wave function` and a given state.
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By convention it noted :math:`E_{Target}^m`.
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Its mathematical expression is :
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.. math::
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E_{Target}^m = E_{Var}^m + E_{PT2}^m
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Selected wave function
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Corresponds to the wave function that have been previously selected for a given state m at a current iteration.
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This wave function is defined by the set of the :term:`internal determinants` and by their coefficients
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on the state m.
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By convention it is noted :math:`|\psi_m\rangle`
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.. math::
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| \psi_m \rangle = \sum_{I=1,N_{selected}} c_I^m | D_I \rangle
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EN EG
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Stands for Eipstein Nesbet with EigenValues zeroth order energy perturbation theory.
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It is a state specific 2nd order perturbation theory. Here m is the index of the eigenstate.
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The :math:`H_0` of this PT is defined as the diagonal part of the Hamiltonian such as
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the :math:`E_m` is equal to the average value of the Hamiltonian on the :term:`selected wave function`
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and the :math:`E_P` is equal to the average value of the Hamiltonian on the :term:`perturbers`
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This perturbation have bad formal properties but some nice numerical features of convergence.
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From the definition, one get the first order coefficient and its related second order energetic contribution of a a perturber :
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.. math::
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c_{D_P}^m= \sum_{S=1,N_{\rm selectors}} \frac{c_S^m \langle D_S|H|D_P\rangle}{ \langle \psi_m |H|\psi_m \rangle - \langle D_P|H| D_P \rangle } \\
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e_{D_P}^m= \frac{(\sum_{S=1,N_{\rm selectors}} c_S^m \langle D_S|H|D_P\rangle)^2}{\langle \psi_m |H|\psi_m \rangle - \langle D_P|H| D_P \rangle }
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Stopping criterion
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Condition decided by the user to stop the calculation.
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This criterion might be on the :term:`Energetic perturbative correction`, on the number of :term:`internal determinants` N_selected_max
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or on the stability of the :term:`estimated target energy`
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The user can also send a Ctrl+C to stop the calculation, and it will kill itself properly, saving the datas that need to be saved.
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Target wave function
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Wave function of the :term:`target space`
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Target space
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Target of the CI calculation. Defining a method (CISD, CAS-CI and so on) is equivalent to define the :term:`target space`.
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The target space defines the rules to define the :term:`Generators` ,
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the rules of the :term:`excitation restrictions`,
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and the perturbation theory to be used.
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There are two type of methods/:term:`target space` proposed in the code :
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#) the CAS-CI type methods where you do not restrict any kind of excitation degree within a given list of orbitals.
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#) the singles and doubles excitations on the top of a given reference wave function (:term:`CISD`, :term:`CISD+SC2`, :term:`CAS+SD`, :term:`CAS+DDCI`, :term:`CAS+MRPT2`)
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Their is a great difference between those two types of method in the way it is implemented.
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In the CAS-CI method, when you have chosen an :term:`active space` (so a list of orbitals and electrons to make a FCI within this active space),
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all the :term:`Internal determinants` that have been selected and that form the :term:`selected wave function`
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can potentially be part of the :term:`generators`, by mean that the :term:`restricted H operator`
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could be potentially applyed on all the :term:`internal determinants` to generate some other :term:`perturbers`.
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In the singles and doubles excitation on the top of a given reference wave function, the subset of :term:`generators`
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and so the rules to recognize them, is fixed at the begining of the method. Those :term:`generators` are precisely
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all the determinants forming the :term:`reference wave function`.
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There are the different :term:`target space` that are available :
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#) :term:`CISD`
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#) :term:`CISD+SC2`
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#) :term:`CASCI`
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#) :term:`CASCI+S`
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#) :term:`CASCI+SD`
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#) :term:`CASCI+DDCI`
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#) :term:`CASCI+DDCI+(2h-2p)PT2`
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#) :term:`CAS-CI+MRPT2`
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Target energy
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Energy of the target space.
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H operator
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Hamiltonian operator defined in terms of creation and anihilation operators in the spin orbital space.
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Excitation restrictions
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Restriction in the :term:`H operator` that the user imposes to define the target sapce.
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For example :
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1) If one freeze some core orbitals or delete some virtuals, it is an :term:`excitation restrictions`
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2) If one prohibits the pure inactive double excitations in a CAS+SD one get a DDCI
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3) any kind of restriction in the full application of the :term:`H operator`
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Restricted H operator
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:term:`H operator` taking into account the :term:`Excitation restrictions`
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CISD+SC2
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Method developped by JP. Malrieu that can be seen as a cheap approximation of the CCSD.
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It makes a CISD size consistant and separable for closed shell systems.
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It is based on a CISD calculation
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where the diagonal part of the H matrix is dressed by the repeatable correlation energy previsously obtained.
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So it is a CISD dressed by the disconnected triples and quadruples.
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Generators
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Set of generator determinants.
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By convention a generator is written as :math:`|D_G\rangle` .
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A generator determinant is a determinant on which
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the :term:`restricted H operator` is being applied for the selection and/or the perturbation.
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Internal determinants
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Selected determinants in terms of integers keys.
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By convention an Internal determinant is written as :math:`|D_I\rangle` .
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By convention, the :term:`Generators` are at the begining of the array.
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Intern space
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Set of all the :term:`internal determinants`.
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Perturbers
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Determinants within the :term:`target space` but taht are not already included in the :term:`intern space`.
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They are created from the :term:`Generators` that belongs :term:`Intern space` for a given :term:`selected wave function`.
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By convention a perturber is written as :math:`|D_P\rangle`.
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Selectors
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Determinants that are used to compute the perturbative properties of the :term:`perturbers`.
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By convention a selector is written as :math:`|D_S\rangle` .
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The selectors are a subset of determinant of the total wave function (that is the :term:`Internal determinants`).
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This subset contains at least the :term:`Generators` determinants.
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The perturbative properties (energy, coefficient or else) of the :term:`perturbers` are calculated on all the :term:`selectors` :math:`|D_S\rangle`
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.. math::
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c_{D_S}= \sum_{S=1,N_{\rm selectors}} \frac{c_S\langle D_S|H|D_P\rangle}{\Delta E_{P,S}} \\
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e_{D_S}= \frac{(\sum_{S=1,N_{\rm selectors}} c_S \langle D_S|H|D_P\rangle) ^2}{\Delta E_{P,S}}
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