Unity of Kohn-Sham density-functional theory and reduced-density-matrix-functional theory

This work presents a rigorous theory to unify the two independent theoretical frameworks of Kohn-Sham (KS) density-functional theory (DFT) and reduced-density-matrix-functional theory (RDMFT). The generalization of the KS orbitals to hypercomplex number systems leads to the hypercomplex KS (HCKS) theory, which extends the search space for the electron density in KS-DFT and reformulates the kinetic energy. The HCKS theory provides a general framework, and different dimensions of the HCKS orbitals lead to different HCKS methods, with KS-DFT and RDMFT being two cases corresponding to the smallest and largest dimensions. Furthermore, a series of tests show that HCKS can capture the multireference nature of strong correlation by dynamically varying fractional occupations, while maintaining the same computational scaling as the KS method. With great potential to overcome the fundamental limitations of the KS method, HCKS creates new possibilities for the development and application of DFT.

Figure 1

Density on real-space grids for the lowest singlet state of C atom. (a) RKS density (purple), (b) cRKS density (green), and (c) HCKS density (blue) are plotted at the isosurface value of 0.2 au. Views along the $z, y$, and $x$ axes are provided respectively. The PBE XC functional is applied for all the calculation.

Figure 2

Triplet-singlet energy gaps of C atom under varying external charged environments. The energies are calculated with two point charges of 0.3 au placed equidistantly on two sides of the C atom, with the distance between the two point charges ranging from 4 to 12 Å. KS and HCKS in use of the same XC functionals (PBE and BLYP) are examined, while CCSDT [79] results are used as reference. For both KS and HCKS, spin-restricted and spin-unrestricted calculations are performed for singlet and triplet states, respectively. All energies are in eV. The occupations of the three $p$ orbitals for the singlet states are also provided in the figure, and the occupations by KS and HCKS (in use of PBE functional) are in round and square brackets, respectively. By comparison, the occupations by KS keep unchanged, while HCKS can provide dynamically varying fractional occupations under different charged environments.

Figure 3

Energy gaps between high- and low-spin states of transition metals. Septet-singlet, quintet-singlet, and triplet-singlet energy gaps for Cr, Fe, and Ni atoms, respectively, are calculated. KS and HCKS in use of the same XC functionals (PBE and BLYP) are examined, while the experimental data [80] are used as reference. For both KS and HCKS, spin-restricted and spin-unrestricted calculations are performed for low- and high-spin states, respectively. All energies are in eV.

Figure 4

Potential energy curves for C-C bond dissociation in ${\mathrm{C}}_2$. The performances of KS and HCKS are evaluated in use of the PBE functional combined with different amounts of correction, that is, $E_{\mathrm{XC}}^{\mathrm{PBE}}+c\mathrm{\Delta }{E}_{\mathrm{XC}}$, with $c$ deciding the amount of correction in the XC functionals. The correction takes the from of $\mathrm{\Delta }{E}_{\mathrm{XC}}=E_{\mathrm{X}}^{\mathrm{Dirac}}-E_{\mathrm{X}}^{\mathrm{TPSS}}$. Here the calculated energies with $c$ ranging from 2.5 to 3.5 are plotted, and the total KS energies of two triplet C atoms are set to zero. All energies are in eV. By comparison, KS and HCKS predict the same energies at the equilibrium distance in use of the same functionals, while the overestimated energies rarely change in KS, which, however, can be effectively reduced by including appropriate amount of correction in HCKS. Note that the XC functionals used in this work are confined to semilocal functionals, similar results can be obtained if the nonlocal correction of $\mathrm{\Delta }{E}_{\mathrm{XC}}=E_{\mathrm{X}}^{\mathrm{Dirac}}-E_{\mathrm{X}}^{\mathrm{HF}}$ is used instead.