mirror of
https://github.com/QuantumPackage/qp2.git
synced 2024-10-31 10:43:38 +01:00
129 lines
5.2 KiB
ReStructuredText
129 lines
5.2 KiB
ReStructuredText
=====
|
|
cipsi
|
|
=====
|
|
|
|
|CIPSI| algorithm.
|
|
|
|
The :c:func:`run_stochastic_cipsi` and :c:func:`run_cipsi` subroutines start with a single
|
|
determinant, or with the wave function in the |EZFIO| database if
|
|
:option:`determinants read_wf` is |true|.
|
|
|
|
The :c:func:`run_cipsi` subroutine iteratively:
|
|
|
|
* Selects the most important determinants from the external space and adds them to the
|
|
internal space
|
|
* If :option:`determinants s2_eig` is |true|, it adds all the necessary
|
|
determinants to allow the eigenstates of |H| to be eigenstates of |S^2|
|
|
* Diagonalizes |H| in the enlarged internal space
|
|
* Computes the |PT2| contribution to the energy stochastically :cite:`Garniron_2017.2`
|
|
or deterministically, depending on :option:`perturbation do_pt2`
|
|
* Extrapolates the variational energy by fitting
|
|
:math:`E=E_\text{FCI} - \alpha\, E_\text{PT2}`
|
|
|
|
The difference between :c:func:`run_stochastic_cipsi` and :c:func:`run_cipsi` is that
|
|
:c:func:`run_stochastic_cipsi` selects the determinants on the fly with the computation
|
|
of the stochastic |PT2| :cite:`Garniron_2017.2`. Hence, it is a semi-stochastic selection. It
|
|
|
|
* Selects the most important determinants from the external space and adds them to the
|
|
internal space, on the fly with the computation of the PT2 with the stochastic algorithm
|
|
presented in :cite:`Garniron_2017.2`.
|
|
* If :option:`determinants s2_eig` is |true|, it adds all the necessary
|
|
determinants to allow the eigenstates of |H| to be eigenstates of |S^2|
|
|
* Extrapolates the variational energy by fitting
|
|
:math:`E=E_\text{FCI} - \alpha\, E_\text{PT2}`
|
|
* Diagonalizes |H| in the enlarged internal space
|
|
|
|
|
|
The number of selected determinants at each iteration will be such that the
|
|
size of the wave function will double at every iteration. If :option:`determinants
|
|
s2_eig` is |true|, then the number of selected determinants will be 1.5x the
|
|
current number, and then all the additional determinants will be added.
|
|
|
|
By default, the program will stop when more than one million determinants have
|
|
been selected, or when the |PT2| energy is below :math:`10^{-4}`.
|
|
|
|
The variational and |PT2| energies of the iterations are stored in the
|
|
|EZFIO| database, in the :ref:`module_iterations` module.
|
|
|
|
|
|
|
|
Computation of the |PT2| energy
|
|
-------------------------------
|
|
|
|
At each iteration, the |PT2| energy is computed considering the Epstein-Nesbet
|
|
zeroth-order Hamiltonian:
|
|
|
|
.. math::
|
|
|
|
E_{\text{PT2}} = \sum_{ \alpha }
|
|
\frac{|\langle \Psi_S | \hat{H} | \alpha \rangle|^2}
|
|
{E - \langle \alpha | \hat{H} | \alpha \rangle}
|
|
|
|
where the |kalpha| determinants are generated by applying all the single and
|
|
double excitation operators to all the determinants of the wave function
|
|
:math:`\Psi_G`.
|
|
|
|
When the hybrid-deterministic/stochastic algorithm is chosen
|
|
(default), :math:`Psi_G = \Psi_S = \Psi`, the full wavefunction expanded in the
|
|
internal space.
|
|
When the deterministic algorithm is chosen (:option:`perturbation do_pt2`
|
|
is set to |false|), :math:`Psi_G` is a truncation of |Psi| using
|
|
:option:`determinants threshold_generators`, and :math:`Psi_S` is a truncation
|
|
of |Psi| using :option:`determinants threshold_selectors`, and re-weighted
|
|
by :math:`1/\langle \Psi_s | \Psi_s \rangle`.
|
|
|
|
At every iteration, while computing the |PT2|, the variance of the wave
|
|
function is also computed:
|
|
|
|
.. math::
|
|
|
|
\sigma^2 & = \langle \Psi | \hat{H}^2 | \Psi \rangle -
|
|
\langle \Psi | \hat{H} | \Psi \rangle^2 \\
|
|
& = \sum_{i \in \text{FCI}}
|
|
\langle \Psi | \hat{H} | i \rangle
|
|
\langle i | \hat{H} | \Psi \rangle -
|
|
\langle \Psi | \hat{H} | \Psi \rangle^2 \\
|
|
& = \sum_{ \alpha }
|
|
\langle |\Psi | \hat{H} | \alpha \rangle|^2.
|
|
|
|
The expression of the variance is the same as the expression of the |PT2|, with
|
|
a denominator of 1. It measures how far the wave function is from the |FCI|
|
|
solution. Note that the absence of denominator in the Heat-Bath selected |CI|
|
|
method is selection method by minimization of the variance, whereas |CIPSI| is
|
|
a selection method by minimization of the energy.
|
|
|
|
|
|
If :option:`perturbation do_pt2` is set to |false|, then the stochastic
|
|
|PT2| is not computed, and an approximate value is obtained from the |CIPSI|
|
|
selection. The calculation is faster, but the extrapolated |FCI| value is
|
|
less accurate. This way of running the code should be used when the only
|
|
goal is to generate a wave function, as for using |CIPSI| wave functions as
|
|
trial wave functions of |QMC| calculations for example.
|
|
|
|
|
|
The :command:`PT2` program reads the wave function of the |EZFIO| database
|
|
and computes the energy and the |PT2| contribution.
|
|
|
|
|
|
State-averaging
|
|
---------------
|
|
|
|
Extrapolated |FCI| energy
|
|
-------------------------
|
|
|
|
An estimate of the |FCI| energy is computed by extrapolating
|
|
|
|
.. math::
|
|
|
|
E=E_\text{FCI} - \alpha\, E_\text{PT2}
|
|
|
|
This extrapolation is done for all the requested states, and excitation
|
|
energies are printed as energy differences between the extrapolated
|
|
energies of the excited states and the extrapolated energy of the ground
|
|
state.
|
|
|
|
The extrapolations are given considering the 2 last points, the 3 last points, ...,
|
|
the 7 last points. The extrapolated value should be chosen such that the extrpolated
|
|
value is stable with the number of points.
|
|
|