.. : PyBEST: Pythonic Black-box Electronic Structure Tool : Copyright (C) 2016-- The PyBEST Development Team : : This file is part of PyBEST. : : PyBEST is free software; you can redistribute it and/or : modify it under the terms of the GNU General Public License : as published by the Free Software Foundation; either version 3 : of the License, or (at your option) any later version. : : PyBEST is distributed in the hope that it will be useful, : but WITHOUT ANY WARRANTY; without even the implied warranty of : MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the : GNU General Public License for more details. : : You should have received a copy of the GNU General Public License : along with this program; if not, see : -- .. _quadrupole: Computing the quadrupole moment ############################### Similar to dipoles, PyBEST provides a method-independent wrapper to calculate the total (nuclear + electric) quadrupole moment of some electronic wave functions. The code snippet below shows how to determine the total quadrupole moment for some electronic structure method whose results are stored in the :py:class:`~pybest.io.iodata.IOData` container ``method_output``. The quadrupole moment integrals are stored in the ``quadrupole`` instance as a ``tuple`` (see :ref:`user_molecularham_dipole` for more details). .. code-block:: python quadrupole = compute_quadrupole(gobasis, x=0, y=0, z=0) The above function returns each component of the electric quadrupole moment integrals in the order {olp, :math:`\mu_x`, :math:`\mu_y`, :math:`\mu_z`, :math:`\mu_{xx}`, :math:`\mu_{xy}`, :math:`\mu_{xz}`, :math:`\mu_{yy}`, :math:`\mu_{yz}`, :math:`\mu_{zz}`} . As of now, all quadrupole moment calculations are done strictly taking the origin of the cartesian coordinate system (0,0,0) as the only reference point. .. note:: In order to calculate the quadrupole moment, the 1-RDM has to be available and stored in the :py:class:`~pybest.io.iodata.IOData` container that is passed to the method. In general, the :py:func:`~pybest.wrapper.multipole.compute_quadrupole` function allows you to calculate the quadrupole moment for a 1-RDM expressed in the atomic **or** molecular orbital basis. This functionality can be steered using the ``molecular_orbitals`` keyword argument. In total, 3 keyword arguments can be set :molecular_orbitals: (boolean) if True, the 1-RDM is transformed back to the atomic orbital basis (default False) :scale: (float) the scaling factor for the 1-RDM (default 2.0). All internal 1-RDMs in PyBEST are stored for one set of orbitals. Thus, even in the restricted case, only one spin-component is saved. The total 1-RDM (spin-free) is obtained by multiplying by a factor of 2.0 :name: (str) if provided, the ``name`` is printed in the header of the std output. 1-RDMs expressed in the atomic orbital basis ============================================ PyBEST stores the following 1-RDMs in the atomic orbital basis: - The RHF/UHF 1-RDM The code snippet below shows how to determine the quadrupole moment from an RHF wave function, .. literalinclude:: ../src/pybest/data/examples/multipole/quadrupole_water_scf.py :lines: 48- 1-RDMs expressed in the molecular orbital basis =============================================== All 1-RDMs determined in post-Hartree-Fock methods are stored in the molecular orbital basis. Thus, they need to be transformed back to the atomic orbital basis prior to the calculation of the multipole integrals. This back-transformation is performed automatically if the **keyword argument** ``molecular_orbitals`` is set to ``True``. The current version of PyBEST supports 1-RDMs for - MP2 (relaxed and unrelaxed) - OOpCCD - LCC (LCCD, LCCSD, pCCD-LCCD, pCCD-LCCSD) The code snippet below shows how to determine the quadrupole moment from an OOpCCD wave function assuming the OOpCCD results are stored in ``oopccd_output``, .. literalinclude:: ../src/pybest/data/examples/multipole/quadrupole_water_pccd.py :lines: 50- The main difference is that we have to set the keyword argument ``molecular_orbitals`` to ``True``. Example Python scripts ====================== The quadrupole moment of the water molecule determined from a restricted Hartree-Fock calculation ------------------------------------------------------------------------------------------------- This is a basic example of how to calculate the quadrupole moment in PyBEST. This script performs a restricted Hartree-Fock calculation for the water molecule using the cc-pVDZ basis set. .. literalinclude:: ../src/pybest/data/examples/multipole/quadrupole_water_scf.py :caption: data/examples/multipole/quadrupole_water_scf.py :lines: 3- The quadrupole moment of the water molecule determined from a restricted pCCD calculation ----------------------------------------------------------------------------------------- This is a basic example of how to calculate the quadrupole moment in PyBEST with the RpCCD method. This script performs a restricted Hartree-Fock calculation followed by an orbital-optimized pCCD calculation for the water molecule using the cc-pVDZ basis set. .. literalinclude:: ../src/pybest/data/examples/multipole/quadrupole_water_pccd.py :caption: data/examples/multipole/quadrupole_water_pccd.py :lines: 3- The quadrupole moment of the water molecule from RpCCD-LCCSD calculation ------------------------------------------------------------------------ The example code given below shows how to determine the quadrupole moment from a pCCD-LCCSD wave function. The script details performing a restricted Hatree-Fock calculation followed by an orbital-optimized pCCD calculation and then an LCCSD calculation on top of that. Notice that the :math:`\Lambda`-amplitudes are optimized by setting the keyword argument ``lambda_equations=True`` in the LCC function call. .. literalinclude:: ../src/pybest/data/examples/multipole/quadrupole_water_pccd_lccsd.py :caption: data/examples/multipole/quadrupole_water_pccd_lccsd.py :lines: 3-