We illustrate the importance of restrictions in improving ground state energy lower bounds of a model of correlated electrons on a lattice. A reduced density matrix (RDM) formalism is employed. The restrictions are derived from closely related and exactly solved models. Such conditions raise the estimates without resorting to increasing the size of the physical space, thus improving computational efficiency. Our main motivation for this work is the problematic picture of Hohenberg–Kohn–Sham density functional theory for strongly correlated fermions. We find that using small cluster representations, errors can be reduced by more than 50% depending on the nature of the model and parameter regime studied. We obtain results for one- and two-dimensional lattices at half filling in the thermodynamic limit, although the method could be easily adapted to finite molecular structures as well.We illustrate the importance of restrictions in improving ground state energy lower bounds of a model of correlated electrons on a lattice. A reduced density matrix (RDM) formalism is employed. The restrictions are derived from closely related and exactly solved models. Such conditions raise the estimates without resorting to increasing the size of the physical space, thus improving computational efficiency. Our main motivation for this work is the problematic picture of Hohenberg–Kohn–Sham density functional theory for strongly correlated fermions. We find that using small cluster representations, errors can be reduced by more than 50% depending on the nature of the model and parameter regime studied. We obtain results for one- and two-dimensional lattices at half filling in the thermodynamic limit, although the method could be easily adapted to finite molecular structures as well.

The Journal of Chemical PhysicsMay 1, 19966 citations

Some Recent Advances in Density Matrix Theory

R. McWeeny

Reviews of Modern PhysicsApr 1, 19601,460 citations

Constraints of reduced density-matrix functional theory for the two-dimensional homogeneous electron gas

A. Putaja, E. Räsänen

Reduced density-matrix functional theory (RDMFT) has become an appealing alternative to density-functional theory to describe electronic properties of strongly correlated systems. Here we derive exact conditions for the suitability of RDMFT to describe the two-dimensional homogeneous electron gas, which is the base system for semiconductor quantum dots and quantum Hall devices, for example. Following the method of Cioslowski and Pernal [J. Chem. Phys. 111, 3396 (1999)] we focus on the properties of power functionals of the form $f(n,{n}^{\ensuremath{'}})={(n{n}^{\ensuremath{'}})}^{\ensuremath{\alpha}}$ for the scaling function in the exchange-correlation energy. We show that in order to have stable and analytic solutions, and for $f$ to satisfy the homogeneous scaling constraint, the power is restricted to $1/4\ensuremath{\leqslant}\ensuremath{\alpha}\ensuremath{\leqslant}3/4$. Applying a reasonable ansatz for the momentum distribution and the lower bound for the exchange-correlation energy tightens the physical regime further to $0.64\ensuremath{\lesssim}\ensuremath{\alpha}\ensuremath{\leqslant}0.75$.Reduced density-matrix functional theory (RDMFT) has become an appealing alternative to density-functional theory to describe electronic properties of strongly correlated systems. Here we derive exact conditions for the suitability of RDMFT to describe the two-dimensional homogeneous electron gas, which is the base system for semiconductor quantum dots and quantum Hall devices, for example. Following the method of Cioslowski and Pernal [J. Chem. Phys. 111, 3396 (1999)] we focus on the properties of power functionals of the form $f(n,{n}^{\ensuremath{'}})={(n{n}^{\ensuremath{'}})}^{\ensuremath{\alpha}}$ for the scaling function in the exchange-correlation energy. We show that in order to have stable and analytic solutions, and for $f$ to satisfy the homogeneous scaling constraint, the power is restricted to $1/4\ensuremath{\leqslant}\ensuremath{\alpha}\ensuremath{\leqslant}3/4$. Applying a reasonable ansatz for the momentum distribution and the lower bound for the exchange-correlation energy tightens the physical regime further to $0.64\ensuremath{\lesssim}\ensuremath{\alpha}\ensuremath{\leqslant}0.75$.

arXiv:1105.2473Physical Review BJul 15, 20117 citations

The variational calculation of reduced density matrices

Mitja Rosina, Claude Garrod

Journal of Computational PhysicsJul 1, 197536 citations

Variational reduced density matrix theory : successes and failures

Paul W. Ayers, Dimitri Van Neck, Patrick Bultinck +2 more

Because the molecular Hamiltonian contains only one-body and two-body operators, the two-electron reduced density matrix contains all the information needed to evaluate the energy, as well as most of the other properties of interest to chemists and molecular physicists. A straightforward minimization of the energy is confounded by the N-representability problem, which can only be addressed approximately. The resulting theory has both advantages and disadvantages compared to more-traditional wavefunction-based approaches. The biggest advantage is that it performs well even when the molecule of interest has strong multireference character and the “Hartree-Fock plus correction” wavefunction paradigms fail. Also, as a lower-bound method, it provides a complementary tool to variational wavefunction approaches. The biggest disadvantages are the computational cost (which may yet be surmounted) and problems with dissociation and degeneracy that seem to afflict all approaches based on a “reduced” description of the system.Because the molecular Hamiltonian contains only one-body and two-body operators, the two-electron reduced density matrix contains all the information needed to evaluate the energy, as well as most of the other properties of interest to chemists and molecular physicists. A straightforward minimization of the energy is confounded by the N-representability problem, which can only be addressed approximately. The resulting theory has both advantages and disadvantages compared to more-traditional wavefunction-based approaches. The biggest advantage is that it performs well even when the molecule of interest has strong multireference character and the “Hartree-Fock plus correction” wavefunction paradigms fail. Also, as a lower-bound method, it provides a complementary tool to variational wavefunction approaches. The biggest disadvantages are the computational cost (which may yet be surmounted) and problems with dissociation and degeneracy that seem to afflict all approaches based on a “reduced” description of the system.

Jan 1, 20090 citations

f

to satisfy the homogeneous scaling constraint, the power is restricted to $1/4\ensuremath{\leqslant}\ensuremath{\alpha}\ensuremath{\leqslant}3/4$. Applying a reasonable ansatz for the momentum distribution and the lower bound for the exchange-correlation energy tightens the physical regime further to $0.64\ensuremath{\lesssim}\ensuremath{\alpha}\ensuremath{\leqslant}0.75$.