We derive a functional form for the energy of interacting many-particle systems from first principles. The effective zero-point energy of a Coulomb-interacting quantum system can be considered in terms of fluctuations around its classical electrostatic interaction and fluctuations of momentum. This perspective provides a controlled and intuitive way to realize the Hohenberg-Kohn theorem. Specifically, an uncertainty equation that relates the interaction energy, kinetic energy and local charge density is derived; it is then combined with the Lieb-Thirring bound on kinetic energy. The combination of these two constraints results in the functional form for total energy of interacting many particle systems that applies for bosons as well as fermions, with appropriate consideration for (anti)symmetry of wavefunctions under coordinate exchange. In the uniform density limit, the functional form is consistent with Quantum Monte Carlo (QMC) data for the bosonic fluid, as well as fermionic systems -- paramagnetic, ferromagnetic and BCC Wigner crystalline.
We derive a functional form for the energy of interacting many-particle systems from first principles. The effective zero-point energy of a Coulomb-interacting quantum system can be considered in terms of fluctuations around its classical electrostatic interaction and fluctuations of momentum. This perspective provides a controlled and intuitive way to realize the Hohenberg-Kohn theorem. Specifically, an uncertainty equation that relates the interaction energy, kinetic energy and local charge density is derived; it is then combined with the Lieb-Thirring bound on kinetic energy. The combination of these two constraints results in the functional form for total energy of interacting many particle systems that applies for bosons as well as fermions, with appropriate consideration for (anti)symmetry of wavefunctions under coordinate exchange. In the uniform density limit, the functional form is consistent with Quantum Monte Carlo (QMC) data for the bosonic fluid, as well as fermionic systems -- paramagnetic, ferromagnetic and BCC Wigner crystalline.
We derive a functional form for the energy of interacting many-particle systems from first principles. Our main contribution is that we define the effective zero-point energy of an interacting system in terms of fluctuations of momentum and the Coulomb electric field. An uncertainty equation can be derived, relating the interaction energy in terms of the kinetic energy and local charge density. The Lieb-Thirring bound is then applied to constrain the kinetic energy which takes particle (anti-)symmetry into account, and the resulting functional form applies to fermionic as well as bosonic systems; in the uniform density case, the functional form agrees with the entire range of available Quantum Monte Carlo (QMC) data of many-particle systems, to within chemical accuracy. Furthermore, the functional form shows promising agreement with QMC phase transition data found in the literature, and in the high density case, where QMC data is not as readily available for a BCC Wigner state, predicts the stability of a BCC Wigner crystal agreeing qualitatively with Unrestricted Hartree-Fock. Seen in this way, many-particle wavefunctions are minimum uncertainty states in coherence with the electromagnetic bath, where the coupling is a self-generated scalar potential.
In pair density-functional theory, the only unknown piece of the energy is the kinetic energy T as a functional of the pair density P(x1,x2). Although T[P] has a simpler structure than the Hohenberg-Kohn functional of conventional density-functional theory, computational requirements are still moderate. In the present work, a particularly convenient model system to represent many-electron pair densities is introduced. This ``boson pair model'' (BPM) approximately treats electron pairs as noninteracting bosons. The resulting explicit model for the kinetic energy T2[P] is shown to be exact for two-electron systems and a lower bound to T[P] for more than two electrons. The one- and two-particle density matrices obtained from the BPM yield upper bounds for the corresponding many-electron quantities. This suggests a partitioning T[P]=T2[P]+Teff[P], where only the remainder ${T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]\ensuremath{\geqslant}0$ needs to be approximated. If the BPM is constrained to yield the exact ground-state pair density, a two-electron Schr\"odinger equation with an effective local two-particle potential results; the latter is identified as a sum of the bare Coulomb interaction and the functional derivative of Teff[P]. This self-consistent scheme to minimize the energy with respect to P is more efficient than previous procedures. Further information on the functional derivative of Teff[P] is derived from a contracted Schr\"odinger equation. Since Teff[P] is explicitly known in the two-electron and noninteracting (Hartree-Fock) limits, the present method provides an alternative to density-matrix functional theories, which can be exact in the same limits and are similar in computational cost.
Based on the Schrodinger equation, exact expressions for the non-relativistic particle energy in the local external field and the external field potential are derived as inhomogeneous density functionals. On this basis, it is shown that, when considering more than two noninteracting electrons, the energy of such a system cannot be an inhomogeneous density functional. This means that the Hohenberg-Kohn lemma which assert that in the ground state to each inhomogeneous density corresponds only one potential of the external field cannot be a justification of the existence of the universal density functional in the general case. At the same time, statements of the density functional theory remain valid when considering any number of noninteracting ground-state bosons due to the Bose condensation effect.
A rigorous derivation of the density functional in the Hohenberg-Kohn theory is presented. With no assumption regarding the magnitude of the electric coupling constant e2 (or correlation), this work provides a firm basis for first-principles calculations. Using the auxiliary field method, in which e2 need not be small, we show that the bosonic loop expansion of the exchange-correlation functional can be reorganized so as to be expressed entirely in terms of the Kohn-Sham single-particle orbitals and energies. The excitations of the many-particle system can be obtained within the same formalism. We also explicitly demonstrate at zero-temperature the single-particle limit, the weak-coupling limit of the energy functional, and its application to homogeneous electron gas.
There has been an enduring interest and controversy about whether or not one can define physically meaningful energy density and stress fields, e(r) and ${\ensuremath{\sigma}}_{\ensuremath{\alpha}\ensuremath{\beta}}(\mathbf{r})$, in quantum systems. A key issue is the kinetic energy since the well-known forms, $\frac{1}{2}{|\mathbf{\ensuremath{\nabla}}\mathrm{\ensuremath{\Psi}}|}^{2}$ and $\ensuremath{-}\frac{1}{2}\mathrm{\ensuremath{\Psi}}{\ensuremath{\nabla}}^{2}\mathrm{\ensuremath{\Psi}}$, lead to different densities, and analogous issues arise for interaction energy terms. This paper considers the ground state of a system of many interacting particles in an external potential, and presents a resolution to the problems in steps. (1) For the kinetic energy all effects of exchange and correlation are shown to be unique functions defined at each point r; all issues of nonuniqueness are relegated to terms that involve only the density n(r) and are equivalent to an effective single-particle problem with ground state wave function s(r)=n(r)/N. (2) Interactions can be considered in two ways: in terms of potentials acting on particles or in terms of the interaction fields, e.g., the Maxwell form in terms of electric fields. In each case, the problem reduces to a mean field part that is a function of the density and a part due to correlation that is uniquely defined; however, it is different for the two cases. (3) The final results follow from the nature of energy and stress. Because the energy determines the ground state itself through the variational principle, the energy density approach leads directly to kinetic energy in the form $\ensuremath{-}\frac{1}{2}s{\ensuremath{\nabla}}^{2}s$ and interactions in terms of potentials acting on the particles. This leads naturally to density functional theory and provides an interpretation in which the energy density e(r) is equilibrated to minimize fluctuations with the same chemical potential at all points r. On the other hand, stress is related to forces, and the only physically acceptable expressions for the stress field involve the combination $\frac{1}{2}[s{\ensuremath{\nabla}}^{2}s\ensuremath{-}{|\mathbf{\ensuremath{\nabla}}s|}^{2}]$, as derived by Schr\"odinger, Pauli, and others, and Coulomb interactions in terms of electric fields, not potentials. Together these results lead to well-defined formulations of energy density and stress fields that are physically motivated and based on a clear set of arguments.
Based on the Schrodinger equation, exact expressions for the non-relativistic particle energy in the local external field and the external field potential are derived as inhomogeneous density functionals. On this basis, it is shown that, when considering more than two noninteracting electrons, the energy of such a system cannot be an inhomogeneous density functional. The result is extended for the system of interacting electrons. This means that the Hohenberg-Kohn lemma which assert that in the ground state to each inhomogeneous density corresponds only one potential of the external field cannot be a justification of the existence of the universal density functional in the general case. At the same time, statements of the density functional theory remain valid when considering any number of noninteracting ground-state bosons due to the Bose condensation effect.