Toward a Resolution of the Static Correlation Problem in Density Functional Theory from Semidefinite Programming

Abstract

Kohn-Sham density functional theory (DFT) has long struggled with the accurate description of strongly correlated and open shell systems and improvements have been minor even in the newest hybrid functionals. In this Letter we treat the static correlation in DFT when frontier orbitals are degenerate by the means of using a semidefinite programming (SDP) approach to minimize the system energy as a function of the N -representable, non-idempotent 1-electron reduced density matrix. While showing greatly improved singlet-triplet gaps for linear density approximation and generalized gradient approximation (GGA) functionals, the SDP procedure reveals flaws in modern meta and hybrid GGA functionals, which show no major improvements when provided with an accurate electron density.

Foundation of one-particle reduced density matrix functional theory for excited states

“…Most importantly, RDMFT can describe strong correlation effects in a more directly manner since they manifest themselves in the form of fractional occupation numbers of the 1RDM. This distinguishes RDMFT compared to DFT as a more suitable approach to strongly correlated many-fermion systems and explains why RDMFT has become an active research field in recent years [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. While the accuracy of ground state calculations compares favourably to those of DFT [21], no proper foundation for calculating energies of excited states exists yet.…”

Density Functional Theory Transformed into a One-electron Reduced Density Matrix Functional Theory for the Capture of Static Correlation

“…Written in this novel format, we observe that the correction converts the DFT functional to the 1-RDMFT functional and hence, must accomplish two separate but related adjustments: (1) addition of the full kinetic energy correction to the explicit part of the functional (part of the functional that depends explicitly upon the 1-RDM) and (2) removal of the kinetic energy correction from the universal part of the functional. In recent work we accomplished (1) by expressing the self-consistentfield Kohn-Sham equations as a semidefinite program (SDP) [42]. A semidefinite program is a special type of optimization in which a linear objective functional of the matrix M is minimized with respect to both linear equalities and the constraint that M is positive semidefinite (nonnegative eigenvalues), M 0 [31][32][33]43].…”

“…The caveat "when molecular orbitals in the Kohn-Sham Hamiltonian are degenerate" introduces a significant restriction that limits applicability of our recent SDP-DFT [42] to static (multi-reference) correlation arising from orbitals that are nearly but not exactly energetically degenerate. Here we remove this restriction by implementing (2) above: removing the kinetic energy from the universal part of the functional.…”

Ensemble Reduced Density Matrix Functional Theory for Excited States and Hierarchical Generalization of Pauli’s Exclusion Principle

“…The particle density namely does not directly reflect the correlation strength, in striking contrast to the full one-particle reduced density matrix (1RDM) with fractional occupation numbers in case of strong correlations. This motivates one-particle reduced density matrix functional theory (RDMFT) [30] as a more suitable approach to strongly correlated quantum systems and explains why RDMFT has become an active field of research in recent years [31][32][33][34][35][36][37][38][39][40][41][42][43][44]. While the accuracy of ground state calculations compares favourably to those of DFT [45], no proper foundation for targeting excited states within RDMFT exists yet.…”