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.
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.
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.
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.
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.
A system of N antisymmetric particles, moving under the influence of a fixed potential and their mutual many-particle interactions, is investigated in the ordinary Hartree-Fock scheme, having the total wave function approximated by a single Slater determinant. It is shown that all the density matrices of various orders, the wave function, and the entire physical situation depends only on a fundamental invariant $\ensuremath{\rho}({x}_{1}, {x}_{2})$, which is identical with the first-order density matrix. The Hartree-Fock equations are expressed in terms of this quantity.The Hartree-Fock equations are also solved by expanding the eigenfunctions in a given complete set, and applications to the MO-LCAO theory of the electronic structure of molecules, and crystals are given. It is shown that, in this scheme, the entire physical situation depends on a charge- and bond-order matrix $R(\ensuremath{\nu}\ensuremath{\mu})$ with respect to the ordinary atomic spin-orbitals involved. The Hartree-Fock equations for this matrix are investigated.Finally, the ionized and excited states are investigated, and it is shown that the Hartree-Fock scheme has a high degree of physical visuality also in case of many-particle interactions. The excitation energy of the system is the difference (${{\ensuremath{\omega}}_{j}}^{\ensuremath{'}}\ensuremath{-}{\ensuremath{\omega}}_{i}$) between two "spin-orbital energies," being eigenvalues to the effective Hamiltonians associated with the two states under consideration.
By introducing a phase field and solving the eigen-functional equation of particles, we obtain the exact expressions of the ground state energy as a functional of the particle density for interacting electron/boson systems, and a two-dimensional electron gas under an external magnetic field, respectively. With the eigen-functionals of the particles, we can construct the ground state wave-function of the systems. Moreover, with the expressions of the ground state energy, we can exactly determine the ground state energy and the ground state particle density of the systems by taking .
We demonstrate that the skeleton of the Fermi surface pertaining to a uniform metallic ground state (corresponding to fermions with spin index σ) is determined by the Hartree–Fock contribution to the dynamic self-energy . That is to say, in order for , it is necessary (but for anisotropic ground states in general not sufficient) that the following equation be satisfied: where stands for the underlying non-interacting energy dispersion and ϵF for the exact interacting Fermi energy. The Fermi surface consists of the set of k points which in addition to satisfying the above equation fulfil where1 The set of k points which satisfy the first of the above two equations but fail to satisfy the second constitute the pseudogap region of the putative Fermi surface of the interacting system. We consider the behaviour of the ground-state momentum distribution function for k in the vicinity of and show that, whereas for the uniform metallic ground states of the conventional single-band Hubbard Hamiltonian, described in terms of an on-site interaction, and (here and denote vectors infinitesimally close to , located respectively inside and outside the underlying Fermi sea), for interactions of non-zero range these inequalities can be violated (without thereby contravening the stability condition ). This aspect is borne out by the pertaining to the normal states of for instance liquid 3He (corresponding to a range of applied pressure) as determined by means of quantum Monte Carlo calculations. We further demonstrate that for Fermi-liquid metallic states of fermions interacting through interaction potentials of non-zero range (e.g. the Coulomb potential), the zero-temperature limit of does not need to be equal to ½ for ; this in strict contrast with the uniform Fermi-liquid metallic states of the single-band Hubbard Hamiltonian (if such states at all exist). This aspect should be taken into account while analysing the deduced from the angle-resolved photoemission data concerning real materials. We discuss, in the light of the findings of the present work, the growing experimental evidence with regard to the ‘frustration’ of the kinetic energy of the charge carriers in the normal states of the copper-oxide-based high-temperature superconducting compounds.