Parametric2RDM - Maple Help

QuantumChemistry

 Parametric2RDM
 compute the ground-state energy of a molecule as a parametric functional of the two-electron reduced density matrix (2-RDM)

 Calling Sequence Parametric2RDM(molecule, options)

Parameters

 molecule - list of lists; each list has 4 elements, the string of an atom's symbol and atom's x, y, and z coordinates options - (optional) equation(s) of the form option = value where option is one of functional, nuclear_gradient, return_rdm, return_t2t1, populations, symmetry, unit, frozen, conv_tol, max_memory, max_cycle, conv_tol_hf, diis_hf, diis_space_hf, diis_start_cycle_hf, direct_scf_hf, direct_scf_tol_hf, level_shift_hf, max_cycle_hf, nuclear_gradient_hf, populations_hf

Description

 • The parametric 2-RDM method computes the ground-state energy of a molecule as a parametric functional of the two-electron reduced density matrix (2-RDM).  The parametrization approximately enforces N-representability conditions, which are necessary constraints for the 2-RDM to represent an N-electron density matrix.
 • Energies and properties from the parametric 2-RDM method typically have an accuracy between those from coupled cluster with single and double excitations (CCSD) and those from coupled cluster with single, double, and perturbative triple excitations [CCSD(T)].
 • The optional functional keyword controls the parametric functional employed in the calculation.  It can be set to the following strings: "CEPA", "K", and "M" (default).
 • The optional return_rdm keyword controls whether or not the spin-free 1- and/or 2-RDMs are returned.  If set to "rdm1" (default), the 1-RDM is returned, if set to "rdm1_and_rdm2", the 1- and 2-RDMs are returned, and if set to "none", RDMs are not returned.
 • The optional frozen keyword can be provided to prevent some orbitals from being correlated.  The keyword can be assigned to a set {} containing the indices of the molecular orbitals to be treated as frozen.  If the frozen keyword is not assigned, then all of the molecular orbitals are considered active, and a parametric 2-RDM calculation with all orbitals is performed.

Outputs

The table of following contents:

 ${t}\left[{\mathrm{e_tot}}\right]$ - float -- total electronic energy of the system ${t}\left[{\mathrm{e_corr}}\right]$ - float -- the difference between the variational 2-RDM method's energy and the Hartree-Fock energy ${t}\left[{\mathrm{mo_coeff}}\right]$ - Matrix -- coefficients expressing natural molecular orbitals (columns) in terms of atomic orbitals (rows) ${t}\left[{\mathrm{mo_occ}}\right]$ - Vector -- molecular (natural) orbital occupations ${t}\left[{\mathrm{group}}\right]$ - string -- name of the molecule's point group symmetry ${t}\left[{\mathrm{aolabels}}\right]$ - Vector -- string label for each atomic orbital consisting of the atomic symbol and the orbital name ${t}\left[{\mathrm{active_orbitals}}\right]$ - list -- list of integers and/or integer ranges indicating the molecular orbitals that are active for correlation ${t}\left[{\mathrm{rdm1}}\right]$ - Matrix -- one-particle reduced density matrix (1-RDM) in molecular-orbital (MO) representation ${t}\left[{\mathrm{rdm2}}\right]$ - Matrix -- two-particle reduced density matrix (2-RDM) in molecular-orbital (MO) representation ${t}\left[{\mathrm{dipole}}\right]$ - Vector -- dipole moment according to its x, y and z components ${t}\left[{\mathrm{populations}}\right]$ - Matrix -- atomic-orbital populations ${t}\left[{\mathrm{charges}}\right]$ - Vector -- atomic charges from the populations ${t}\left[{\mathrm{nuclear_gradient}}\right]$ - Matrix -- analytical nuclear gradient ${t}\left[{\mathrm{t2t1}}\right]$ - Vector -- one- and two-electron transition amplitudes ${t}\left[{\mathrm{t2t1_indices}}\right]$ - Matrix -- the first 4 indices of each row give the indices of either t2 (or t1 if the last two integers are 0) The 5th integer indicates the spin block.  For t2: 2 = αα, 1 = αβ, 0 = ββ; for t1: 1 = α, 0 = β.

Options

 • basis = string -- name of the basis set.  See Basis for a list of available basis sets.  Default is "sto-3g".
 • spin = nonnegint -- twice the total spin S (= 2S). Default is 0.
 • charge = nonnegint -- net charge of the molecule. Default is 0.
 • symmetry = string/boolean -- is the Schoenflies symbol of the abelian point-group symmetry which can be one of the following:  D2h, C2h, C2v, D2, Cs, Ci, C2, C1. true (default) finds the appropriate symmetry while false does not use symmetry.
 • unit = string -- "Angstrom" or "Bohr". Default is "Angstrom".
 • functional = string -- "CEPA", "K", or "M". Default is "M".
 • frozen = set -- set of orbitals to be frozen.
 • return_rdm = string -- options to return the 1-RDM and/or 2-RDM: "none", "rdm1", "rdm1_and_rdm2". Default is "rdm1".
 • return_t2t1 = boolean -- option to return the one- and two-electron transition amplitudes.  Default is false.
 • populations = string -- atomic-orbital population analysis: "Mulliken" and "Mulliken/meta-Lowdin". Default is "Mulliken".
 • nuclear_gradient = boolean -- option to return the analytical nuclear gradient if available. Default is false.
 • conv_tol = float -- converge threshold. Default is 5*${10}^{-5}.$
 • max_memory = posint/boolean -- allowed memory in MB. Default is 4000.
 • verbose = posint -- positive integer between 1 and 5 that controls printing. Default is 1.
 • Attributes for Hartree Fock:
 • conv_tol_hf = float -- converge threshold. Default is ${10}^{-10}.$
 • diis_hf = boolean -- whether to do diis. Default is true.
 • diis_space_hf = posinut -- diis's space size. By default, 8 Fock matrices and errors vector are stored.
 • diis_start_cycle_hf = posint -- the step to start diis. Default is 1.
 • direct_scf_hf = boolean -- direct SCF in which integrals are recomputed is used by default.
 • direct_scf_tol_hf = float -- direct SCF cutoff threshold. Default is ${10}^{-13}.$
 • level_shift_hf = float/int -- level shift (in a.u.) for virtual space. Default is $0.$
 • max_cycle_hf = posint -- max number of iterations. Default is 50.
 • nuclear_gradient_hf = boolean -- option to return the analytical nuclear gradient. Default is false.
 • populations_hf = string -- atomic-orbital population analysis: "Mulliken" and "Mulliken/meta-Lowdin". Default is "Mulliken".

References

 1 D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008). "Parametrization of the two-electron reduced density matrix for its direct calculation without the many-electron wave function"
 2 D. A. Mazziotti, Phys. Rev. A 81, 062515 (2010). "Parametrization of the two-electron reduced density matrix for its direct calculation without the many-electron wave function: Generalizations and applications"
 3 J. J. Foley IV and D. A. Mazziotti, J.Phys. Chem. A 117, 6712 (2013). "Cage versus prism: electronic energies of the water hexamer"
 4 A. J. Valentine and D. A. Mazziotti, J. Phys. Chem. A 117, 9746 (2013). "Theoretical prediction of the structures and energies of olympicene and its isomers"

Examples

 > $\mathrm{with}\left(\mathrm{QuantumChemistry}\right):$

A parametric 2-RDM calculation of the  molecule

 >
 ${\mathrm{molecule}}{≔}\left[\left[{"H"}{,}{0}{,}{0}{,}{0}\right]{,}\left[{"F"}{,}{0}{,}{0}{,}{0.95000000}\right]\right]$ (1)
 >
 ${\mathrm{table}}{}\left({\mathrm{%id}}{=}{18446744382196970078}\right)$ (2)
 >