scholarly journals Discontinuous behavior of the Pauli potential in density functional theory as a function of the electron number

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
Eli Kraisler ◽  
Axel Schild
Keyword(s):  
Density Functional Theory ◽  
Density Functional ◽  
Electron Number ◽  
Functional Theory ◽  
Pauli Potential

scholarly journals Discontinuous Behavior of the Pauli Potential in Density Functional Theory as a Function of the Electron Number

2019 ◽  
Author(s):  
Eli Kraisler ◽  
Axel Schild
Keyword(s):  
Density Functional Theory ◽  
Density Functional ◽  
Electron Systems ◽  
Electron Number ◽  
Functional Theory ◽  
Fractional Number ◽  
Finite Systems ◽  
Pauli Potential ◽  
Spin Polarized ◽  
Orbital Free

<div>The Pauli potential is an essential quantity in orbital-free density-functional theory (DFT) and in the exact electron factorization (EEF) method for many-electron systems. Knowledge of the Pauli potential allows the description of a system relying on the density alone, without the need to calculate the orbitals.</div><div>In this work we explore the behavior of the exact Pauli potential in finite systems as a function of the number of electrons, employing the ensemble approach in DFT. Assuming the system is in contact with an electron reservoir, we allow the number of electrons to vary continuously and to obtain fractional as well as integer values. We derive an expression for the Pauli potential for a spin-polarized system with a fractional number of electrons and find that when the electron number surpasses an integer, the Pauli potential jumps by a spatially uniform constant, similarly to the Kohn-Sham potential. The magnitude of the jump equals the Kohn-Sham gap. We illustrate our analytical findings by calculating the exact and approximate Pauli potentials for Li and Na atoms with fractional numbers of electrons.</div>

scholarly journals Discontinuous Behavior of the Pauli Potential in Density Functional Theory as a Function of the Electron Number

2019 ◽  
Author(s):  
Eli Kraisler ◽  
Axel Schild
Keyword(s):  
Density Functional Theory ◽  
Density Functional ◽  
Electron Systems ◽  
Electron Number ◽  
Functional Theory ◽  
Fractional Number ◽  
Finite Systems ◽  
Pauli Potential ◽  
Spin Polarized ◽  
Orbital Free

<div>The Pauli potential is an essential quantity in orbital-free density-functional theory (DFT) and in the exact electron factorization (EEF) method for many-electron systems. Knowledge of the Pauli potential allows the description of a system relying on the density alone, without the need to calculate the orbitals.</div><div>In this work we explore the behavior of the exact Pauli potential in finite systems as a function of the number of electrons, employing the ensemble approach in DFT. Assuming the system is in contact with an electron reservoir, we allow the number of electrons to vary continuously and to obtain fractional as well as integer values. We derive an expression for the Pauli potential for a spin-polarized system with a fractional number of electrons and find that when the electron number surpasses an integer, the Pauli potential jumps by a spatially uniform constant, similarly to the Kohn-Sham potential. The magnitude of the jump equals the Kohn-Sham gap. We illustrate our analytical findings by calculating the exact and approximate Pauli potentials for Li and Na atoms with fractional numbers of electrons.</div>

Exact exchange-correlation potentials in spin-density functional theory and their discontinuities at unit electron number

2009 ◽  
Vol 87 (10) ◽  
pp. 1268-1272 ◽  
Author(s):  
John P. Perdew ◽  
Espen Sagvolden
Keyword(s):  
Density Functional Theory ◽  
Spin Density ◽  
Density Functional ◽  
Electron Number ◽  
Functional Theory ◽  
Exchange Correlation ◽  
Exchange Correlation Potential ◽  
Correlation Potential ◽  
Exact Exchange ◽  
Physical Consequences

The exact exchange-correlation potential of Kohn–Sham density functional theory is known to jump discontinuously by a spatial constant as the average electron number, N, crosses an integer in an open system of fluctuating electron number, with important physical consequences for charge transfers and band gaps. We have recently constructed an essentially exact exchange-correlation potential vxc for N electrons (0 ≤ N ≤ 2) in the presence of a –1/r external potential, i.e., for a ground ensemble of H+ ion, H atom, and H– ion densities. That construction illustrates the discontinuity at N = 1, where it equals IH – AH, the positive difference between the ionization energy and the electron affinity of the hydrogen atom. Here we construct the corresponding essentially exact spin-up and spin-down exchange-correlation potentials vxc,↑ and vxc,↓ of the Kohn–Sham spin-density functional theory, more commonly used for electronic structure calculations, for the ground ensemble with most-negative z-component of spin (or equivalently in the presence of a uniform magnetic field of infinitesimal strength). The potentials vxc, vxc,↑, and vxc,↓, which vanish as r → ∞ (except when N approaches an integer from above), are identical for 0 ≤ N ≤ 1 and for N = 2 but not for 1 < N < 2. We find that the majority or spin-down potential has a spatially constant discontinuity at N = 1 equal to IH – AH. The minority or spin-up potential has a discontinuity which is this constant in one order of limits, but is a spatially varying function in a different order of limits. This order-of-limits problem is a consequence of a special circumstance: the vanishing of the spin-up density at N = 1.

scholarly journals Asymptotic behavior of the Hartree-exchange and correlation potentials in ensemble density functional theory

2019 ◽  
Author(s):  
Tim Gould ◽  
Stefano Pittalis ◽  
Julien Toulouse ◽  
Eli Kraisler ◽  
Leeor Kronik
Keyword(s):  
Density Functional Theory ◽  
Quantum Monte Carlo ◽  
Density Functional ◽  
Outer Region ◽  
Electron Number ◽  
Functional Theory ◽  
Exchange Correlation ◽  
Asymptotic Constant ◽  
Coulomb Potentials ◽  
Exchange And Correlation

We report on previously unnoticed features of the exact Hartree-exchange and correlation potentials for atoms and ions treated via ensemble density functional theory, demonstrated on fractional ions of Li, C, and F. We show that these potentials, when treated separately, can reach non-vanishing asymptotic constant values in the outer region of spherical, spin unpolarized atoms. In the next leading order, the potentials resemble Coulomb potentials created by effective charges which have the peculiarity of not behaving as piecewise constants as a function of the electron number. We provide analytical derivations and complement them with numerical results using the inversion of the Kohn-Sham equations for interacting densities obtained by accurate quantum Monte Carlo calculations. The present results expand on the knowledge of crucial exact properties of Kohn-Sham systems, which can guide development of advanced exchange-correlation approximations.<br><br>

scholarly journals What Do We Learn from the Classical Turning Surface of the Kohn-Sham Potential as Electron Number Is Varied Continuously?

2019 ◽  
Author(s):  
Tim Gould ◽  
Benjamin Libereles ◽  
John P. Perdew
Keyword(s):  
Density Functional Theory ◽  
Density Functional ◽  
Principal Quantum Number ◽  
Ionisation Potential ◽  
Electron Number ◽  
Conceptual Density Functional Theory ◽  
Functional Theory ◽  
Atomic Systems ◽  
Additional Information ◽  
Turning Radius

The classical turning radius Rt of an atom can be defined as the radius where the KS potential is equal to the negative ionisation potential of the atom, i.e. where v_s(R_t)=\epsilon_h. It was recently shown [P.N.A.S. 115, E11578 (2018)] to yield chemically relevant bonding distances, in line with known empirical values. In this work we show that extension of the concept to non-integer electron number yields additional information about atomic systems, and can be used to detect the difficulty of adding or subtracting electrons. Notably, it reflects the ease of bonding in open p-shells, and its greater difficulty in open s-shells. The latter manifests in significant discontinuities in the turning radius as the electron number changes the principal quantum number of the outermost electronic shell (e.g. going from Na to Na^{2+}). We then show that a non-integer picture is required to correctly interpret bonding and dissociation in H_2^+. Results are consistent when properties are calculated exactly, or via an appropriate approximation. They can be interpreted in the context of conceptual density functional theory.

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