This program is intended to automate many of the tedious manipulations that one encounters when working in second quantization. To use it, be sure that your $PYTHONPATH variable includes this directory. Then, in the Python2 interpreter or in a Python2 script, use a command such as:

import sqa_plus

To see how to use core SecondQuantizationAlgebra capabilities, such as the definition of indexes, operators and terms, and algebraic functions, such as Normal Ordering, please visit the original TUTORIAL file.

Dyall Hamiltonian (sqa_plus.dyallH), Effective Hamiltonians up to second-order (sqa_plus.Heff), T amplitudes operators (sqa_plus.Tamplitude) and V perturbation operators (sqa_plus.Vperturbation) can be obtained automatically.

Spin-integrated effective Hamiltonians and operators can now be obtained by functions implemented in sqaHeff.py. Dyall Hamiltonian (sqa_plus.dyallH), Effective Hamiltonians up to second-order (sqa_plus.Heff), T amplitudes operators (sqa_plus.Tamplitude) and V perturbation operators (sqa_plus.Vperturbation) are included. The spin-integrated basis can be globally selected by using sqa_plus.options.spin_integrated = True. By default, sqa_plus assumes the spin-orbital basis (sqa_plus.options.spin_orbital = True).

sqa_plus.options.spin_integrated = True
terms_Heff0_si = sqa_plus.Heff(0)        # Spin-integrated zeroth-order Heff

In a similar design to the choice of the spin basis, the CVS approximation can be selected by using sqa_plus.options.cvs_approach = True. It will set globally all the implemented operators (sqaHeff.py, sqa_plus.dyallH, sqa_plus.Heff, sqa_plus.Tamplitude, and sqa_plus.Vperturbation) to assume the CVS approximation.

In the SQA+ language, the CVS approximation decouple the options.core_type type indices in two new types of indices: options.cvs_core_type and options.cvs_valence_type. Then, core indices defined by the user must be of these types.

Besides the usage of sqa_plus.index class to define dummy indices, we provide an automated way to generate unique indices. The sqa_plus.indexLists object can be used to create dummy indices, free of conflict with SQA+ implemented operators. New indices can be obtained using the new_index() method.

# Examples of sqa_plus.indexLists usage
cvs_core_ind_1 = sqa_plus.indexLists.cvs_core.new_index()
active_ind_1 = sqa_plus.indexLists.active.new_index()
virtual_alpha_ind_1 = sqa_plus.indexLists.virtual_alpha.new_index()

• Spin-orbital basis:

  • indexLists.core
  • indexLists.active
  • indexLists.virtual

• Spin-orbital basis using CVS:

  • indexLists.cvs_core
  • indexLists.cvs_valence
  • indexLists.active
  • indexLists.virtual

• Spin-integrated basis:

  • indexLists.core_alpha

  • indexLists.active_alpha

  • indexLists.virtual_alpha

  • indexLists.core_beta

  • indexLists.active_beta

  • indexLists.virtual_beta

• Spin-integrated basis using CVS:

  • indexLists.cvs_core_alpha

  • indexLists.cvs_valence_alpha

  • indexLists.active_alpha

  • indexLists.virtual_alpha

  • indexLists.cvs_core_beta

  • indexLists.cvs_valence_beta

  • indexLists.active_beta

  • indexLists.virtual_beta

The function sqa_plus.convertSpinIntegratedToAdapted allows the automated conversion of spin-integrated terms to quantitites in the spin-adapted formulation. Include the spin-adaptation of 1e- and 2e- integrals, and reduced density matrices (up to 4-RDMs).

# Define spin-integrated Zeroth-order Heff
sqa_plus.options.spin_integrated = True
terms_Heff0_si = sqa_plus.Heff(0)

# Evaluate terms
terms_Heff0_si = sqa_plus.matrixBlock(terms_Heff0_si)

# Convert to spin-adapted formulation
terms_Heff0_sa = sqa_plus.convertSpinIntegratedToAdapted(terms_Heff0_si)

The evaluated terms of the spin-integrated effective Hamiltonian in terms_Heff0_si are:

 (   1.00000) E_fc(Const.) 
 (   1.00000) h(x,y) cre(x) des(y) 
 (   1.00000) h(x,y) cre(x) des(y) 
 (   1.00000) v(i,x,i,y) cre(x) des(y) 
 (   1.00000) v(i,x,i,y) cre(x) des(y) 
 (   1.00000) v(i,x,i,y) cre(x) des(y) 
 (   1.00000) v(i,x,i,y) cre(x) des(y) 
 (  -0.25000) v(x,y,z,w) cre(x) cre(y) des(z) des(w) 
 (  -1.00000) v(x,y,z,w) cre(x) cre(y) des(z) des(w) 
 (  -0.25000) v(x,y,z,w) cre(x) cre(y) des(z) des(w) 

After the spin-adaptation procedure, terms_Heff0_sa should contain:

 (   1.00000) E_fc(Const.) 
 (   1.00000) h(x,y) rdm(x,y) 
 (   2.00000) v(i,i,x,y) rdm(x,y) 
 (  -1.00000) v(i,x,y,i) rdm(x,y) 
 (   0.50000) v(x,y,z,w) rdm(x,z,y,w) 

Note that sqa_plus.convertSpinIntegratedToAdapted will globally set sqa_plus.options.spin_adapted = True) and changes 2e- integrals and RDMs to the chemists' notation (sqa_plus.options.chemists_notation = True).

Automates the evaluation of terms applying normal-order to all active-space creation ahd annihilation operators with respect to physical vacuum, normal-order core creation and annihilation operators relative to the Fermi vaccum, and evaluate expectation values with respect to the active-space states.

# Define spin-orbital Zeroth-order Heff
terms_Heff0 = sqa_plus.Heff(0)

# Evaluate terms
terms_Heff0 = matrixBlock(terms_Heff0)

Intermediates code requires NumPy scientific computation package installed.

Translates terms and SQA+ objects to Numpy's Einsum notation, ready to be used in Python codes. The sqa_plus.genEinsum function has options setted by positional arguments.

sqa_plus.genEinsum(terms, lhs_string = None, indices_string = None, suffix = None,
				   trans_indices_string = None, intermediate_list = None, help = False, **tensor_rename):

• lhs_string: Variable name to store the results of np.einsum operations;
• indices_string: String containing external indices of the summation;
• suffix: Add suffix to variables;

Aditionally, options for genEinsum can be selected using the object sqa_plus.options

• sqa_plus.options.genEinsum.lhs_string: Variable name to store the results of np.einsum operations;

• sqa_plus.options.genEinsum.indices_string: String containing external indices of the summation;

• sqa_plus.options.genEinsum.spin_integrated_tensors: Express spin-integrated quantities by slices of spin-orbital objects; (default: False)

• sqa_plus.options.genEinsum.keep_user_defined_dummy_names: (default: False)

• sqa_plus.options.genEinsum.trans_rdm: Use transition reduced density matrices (tRDMs); (default: False)

• sqa_plus.options.genEinsum.trans_indices_string: String containing tRDMs indices;

• sqa_plus.options.genEinsum.suffix: Add suffix to variables;

• sqa_plus.options.genEinsum.cvs_indices_list: List of core indices to be included in Core-Valence Separation (CVS) space;

• sqa_plus.options.genEinsum.valence_indices_list: List of core indices not included in Core-Valence Separation (CVS) space;

• sqa_plus.options.genEinsum.cvs_tensors: Generate expressions using CVS-ready variables; (default: True)

• sqa_plus.options.genEinsum.remove_trans_rdm_constant: Remove tRDMs constants (default: False)

• sqa_plus.options.genEinsum.remove_core_integrals: Remove double-counted contributions to core terms when using the effective Hamiltonian (default: True)

• sqa_plus.options.genEinsum.intermediate_list: List of intermediates tensors;

• sqa_plus.options.genEinsum.opt_einsum_terms: Select optEinsum or built-in Numpy einsum function; (default: True)

• sqa_plus.options.genEinsum.optimize: Configure optimize flag in einsum;(default: True)

SecondQuantizationAlgebra was originally developed by Eric Neuscamman [email protected]

In addition, any modification or use of this software should cite the following paper:

  E. Neuscamman, T. Yanai, and G. K.-L. Chan.
  J. Chem. Phys. 130, 124102 (2009)

• Alexander Yu. Sokolov [email protected]
• Koushik Chatterjee [email protected]

• Ilia Mazin [email protected]

• Carlos E. V. de Moura [email protected]

• Carlos E. V. de Moura [email protected]
• Ilia Mazin [email protected]