Generalized Kohn-Sham system in one-matrix functional theory

A system of electrons in a local or nonlocal external potential can be described with one-matrix functional theory (1MFT), which is similar to density-functional theory (DFT) but takes the one-particle reduced density matrix (one-matrix) instead of the density as its basic variable. Within 1MFT, Gilbert derived [Phys. Rev. B 12, 2111 (1975)] effective single-particle equations analogous to the Kohn-Sham (KS) equations in DFT. The self-consistent solution of these 1MFT-KS equations reproduces not only the density of the original electron system but also its one-matrix. While in DFT it is usually possible to reproduce the density using KS orbitals with integer (0 or 1) occupancy, in 1MFT reproducing the one-matrix requires in general fractional occupancies. The variational principle implies that the KS eigenvalues of all fractionally occupied orbitals must collapse at self-consistency to a single level. We show that as a consequence of the degeneracy, the iteration of the KS equations is intrinsically divergent. Fortunately, the level-shifting method, commonly introduced in Hartree-Fock calculations, is always able to force convergence. We introduce an alternative derivation of the 1MFT-KS equations that allows control of the eigenvalue collapse by constraining the occupancies. As an explicit example, we apply the 1MFT-KS scheme to calculate the ground state one-matrix of an exactly solvable two-site Hubbard model.

Figure 1

(Color online) The density variable $m=\left(n_1-n_2\right)/2$ is shown with respect to the dimensionless external potential $\nu =V/B$ $\left(B=\sqrt{T^2+U^2}\right)$. The curves for $U/T=\left(1/16,1/4,1,4\right)$ are shown as solid (blue), dotted (green), dash-dotted (light blue), and dashed (red) curves, respectively.

Figure 2

(Color online) The iteration map [Eq. (52)] is shown for $t=1$, $U=1$, $V=0$, and $A=A_{\text{gs}}-0.02$. The solid (black) curve is the left-hand side of Eq. (52). The dashed (red) curve is the right-hand side. The dotted (blue) curve demonstrates the first two iterations. The iteration map does not converge to the ground state fixed point $\theta =\pi /2$.

Figure 3

(Color online) The iteration map for $t=1$, $U=5$, $V=-2.5$, and $A=A_{\text{gs}}-0.1$. The solid (black) curve is the left-hand side of Eq. (52). The dash-dotted (blue), dotted (green), and dashed (red) curves are the right-hand side with level shift $\zeta =\left(0,3,6\right)$. The threshold level shift for convergence is ${\zeta }_c\approx 4.07$, which may be calculated with Eq. (34).

Figure 4

(Color online) The “KS energy” for $t=1$, $U=3.5$, and $V=0$ is shown as a function of the deviation $\left(\delta {\gamma }_x,\delta {\gamma }_z\right)$ from the ground state. The two surfaces represent the two branches of the KS energy. The space curves show the energy as a function of $\theta$ for fixed occupation numbers. Optimization of the orbitals corresponds to moving along one of these curves to the stationary point. The ground state is the point of conic intersection at the origin.

Figure 5

The iteration map for the external potential $V$ is shown for a weakly interacting system with $U=t$. The left- and right-hand sides of Eq. (72) are shown as solid and dashed curves, respectively.

Figure 6

The iteration map for the external potential $V$ is shown for a strongly interacting system with $U=4t$. The left- and right-hand sides of Eq. (72) are shown as solid and dashed curves, respectively.