Mel Levy's research while affiliated with Tulane University and other places

Ground-state Kohn-Sham density functional theory provides, in principle, the exact ground-state energy and electronic spin densities of real interacting electrons in a static external potential. In practice, the exact density functional for the exchange-correlation (xc) energy must be approximated in a computationally efficient way. About 20 mathematical properties of the exact xc functional are known. In this work, we review and discuss these known constraints on the xc energy and hole. By analyzing a sequence of increasingly sophisticated density functional approximations (DFAs), we argue that ( a) the satisfaction of more exact constraints and appropriate norms makes a functional more predictive over the immense space of many-electron systems and ( b) fitting to bonded systems yields an interpolative DFA that may not extrapolate well to systems unlike those in the fitting set. We discuss both how the class of well-described systems has grown along with constraint satisfaction and the possibilities for future functional development. Expected final online publication date for the Annual Review of Physical Chemistry, Volume 74 is April 2023. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

The Hartree-Fock (HF) approximation has been an important tool for quantumchemical calculations since its earliest appearance in the late 1920s,and remains the starting point of most single-reference methods in use today.Intuition suggests that the HF kinetic energy should not exceed theexact kinetic energy, but no proof of this conjecture exists,despite a near century ofdevelopment.Beginning from a generalized virial theorem derived from scalingconsiderations, we derive a general expression forthe kinetic energy difference that applies to all systems. For any atom or ion this trivially reducesto the well-known result that the total energy is the negative of the kineticenergy and since correlation energies are never positive, proves theconjecture in this case. Similar considerations apply to moleculesat their equilibrium bond lengths.We use highly precise calculations onHooke's atom (two electrons in a parabolic well)to test the conjecture in a non-trivial case, and to parameterizethe difference between density-functional and HF quantities,but find no violations of the conjecture.

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 300 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 776 entries, the paper represents a broad snapshot of DFT, anno 2022.

Ground-state Kohn-Sham density functional theory provides, in principle, the exact ground-state energy and electronic spin-densities of real interacting electrons in a static external potential. In practice, the exact density functional for the exchange-correlation (xc) energy must be approximated in a computationally efficient way. About twenty mathematical properties of the exact xc functional are known. In this work, we review and discuss these known constraints on the xc energy and hole. By analyzing a sequence of increasingly sophisticated density functional approximations (DFAs), we argue that: (1) the satisfaction of more exact constraints and appropriate norms makes a functional more predictive over the immense space of many-electron systems; (2) fitting to bonded systems yields an interpolative DFA that may not extrapolate well to systems unlike those in the fitting set. We discuss how the class of well-described systems has grown along with constraint satisfaction, and the possibilities for future functional development.

The Hartree-Fock (HF) approximation has been an important tool for quantum chemical calculations since its earliest appearance in the late 1920s, and remains the starting point of most single-reference methods in use today. Intuition suggests that the HF kinetic energy should not exceed the exact kinetic energy, but no proof of this conjecture exists, despite a near century of development. Beginning from a generalized virial theorem derived from scaling considerations, we derive a general expression for the kinetic energy difference that applies to all systems. For any atom or ion this trivially reduces to the well-known result that the total energy is the negative of the kinetic energy and since correlation energies are never positive, proves the conjecture in this case. Similar considerations apply to molecules at their equilibrium bond lengths. We use highly precise calculations on Hooke's atom (two electrons in a parabolic well) to test the conjecture in a non-trivial case, and to parameterize the difference between density-functional and HF quantities, but find no violations of the conjecture.

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 300 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 776 entries, the paper represents a broad snapshot of DFT, anno 2022.

It has recently been shown how to replace spin-wave functions with purely spatial wave functions in the formulation of spin-density functional theory (SDFT), without any change to the resultant density functional expressions [F. Zahariev and M. Levy, Phys. Rev. A 100, 062507 (2019)]. The purely spatial wave functions that are obtained are much more convenient to use and thus allow for a relatively easy decomposition and manipulation of terms in the constrained-search formulation of SDFT. All the essential ingredients of SDFT, including the kinetic, exchange, and correlation contributions, are explicitly defined in terms of the universal functional expressed in this spin-free manner. Constrained-search derivations of the Oliver-Perdew relations for the kinetic and exchange terms are presented. The up-down spin component of the correlation term is found to contain the up-down spin component of the Hartree term. A spin-dependent generalization of the adiabatic connection is put forth and a connection with spin-dependent coordinate scaling is established for the correlation energy.

Exact density functionals for the exchange and correlation energies are approximated in practical calculations for the ground-state electronic structure of a many-electron system. An important exact constraint for the construction of approximations is to recover the correct non-relativistic large-Z expansions for the corresponding energies of neutral atoms with atomic number Z and electron number N = Z, which are correct to the leading order (−0.221Z5/3 and −0.021Z ln Z, respectively) even in the lowest-rung or local density approximation. We find that hydrogenic densities lead to Ex(N, Z) ≈ −0.354N2/3Z (as known before only for Z ≫ N ≫ 1) and Ec ≈ −0.02N ln N. These asymptotic estimates are most correct for atomic ions with large N and Z ≫ N, but we find that they are qualitatively and semi-quantitatively correct even for small N and N ≈ Z. The large-N asymptotic behavior of the energy is pre-figured in small-N atoms and atomic ions, supporting the argument that widely predictive approximate density functionals should be designed to recover the correct asymptotics. It is shown that the exact Kohn–Sham correlation energy, when calculated from the pure ground-state wavefunction, should have no contribution proportional to Z in the Z → ∞ limit for any fixed N.

Exact density functionals for the exchange and correlation energies are approximated in practical calculations for the ground-state electronic structure of a many-electron system. An important exact constraint for the construction of approximations is to recover the correct non-relativistic large-$Z$ expansions for the corresponding energies of neutral atoms with atomic number $Z$ and electron number $N=Z$, which are correct to leading order ($-0.221 Z^{5/3}$ and $-0.021 Z \ln Z$ respectively) even in the lowest-rung or local density approximation. We find that hydrogenic densities lead to $E_x(N,Z) \approx -0.354 N^{2/3} Z$ (as known before only for $Z \gg N \gg 1$) and $E_c \approx -0.02 N \ln N$. These asymptotic estimates are most correct for atomic ions with large $N$ and $Z \gg N$, but we find that they are qualitatively and semi-quantitatively correct even for small $N$ and for $N \approx Z$. The large-$N$ asymptotic behavior of the energy is pre-figured in small-$N$ atoms and atomic ions, supporting the argument that widely-predictive approximate density functionals should be designed to recover the correct asymptotics.