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quantum_package/doc/source/wavefunction.rst

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