Energy Density Functional Theory of Many-Electron Systems

1. Energy density functional theory: historical and bibliographic sketch.- 1.1. The Thomas-Fermi theory and its sequels.- 1.2. One-electron equations.- 1.3. Bibliographic sketch Monographies and books.- Review articles.- International meetings.- 2. Many-electron wavefunctions, density matrices, reduced density matrices and variational principles.- 2.1. Pure states and emsembles in quantum mechanics.- 2.1.a. The measurement process in quantum mechanics.- 2.1.b. The Liouville formalism.- 2.1.c. Wavefunctions.- 2.l.d. The ATh-order density operator for a pure state.- 2.1.e. The ATh-order density matrix for a pure state.- 2.1.f. Representation of DiNin a continuons coordinate basis.- 2.1.g. The expectation value of an operator.- 2.1.h. The Nth-order density operator for mixed states or emsembles.- 2.1.i. Equivalence between Liouville's and Schrodinger's equation for pure states.- 2.l.j. The case of mixed states or emsembles.- 2.l.k. The Liouvillian as a superoperator.- Problems.- 2.2. Reduced density matrices.- 2.2.a. Definition.- 2.2.b. The case of a single Slater determinant.- 2.2.c. The case of a linear combination of Slater determinants.- 2.2.d. Some properties of D1 and D2.- 2.2.e. Average values of operators.- Problems.- 2.3. Spin structure of wavefunctions and reduced density matrices.- Problems.- 2.4. Variational principle in the Schrodinger picture of quantum chemistry.- 2.4.a. General formulation.- 2.4.b. The expectation value of the Hamiltonian.- 2.4.c. Introduction to point transformations: The virial theorem.- Problems.- 3. The one-electron density.- 3.1. The meaning of the one-electron density.- 3.1.a. The physical interpretation of ?(r) for N identical particles.- 3.1.b. The physical interpretation of ?(r) for N identical particles in the presence of M nuclei.- 3.1.c. The electronic and nuclear density for H2+.- 3.l.d. The evidence for atomic fragments.- 3.1.e. Other properties of the one-electron density.- Asymptotic behavior.- Cusp condition.- Multipole moments.- Bounds.- Problems.- 3.2. The one-electron density and molecular structure.- 3.2.a. Localized orbitals vs. loges.- 3.2.b. Binding and electrostatic forces.- Berlin's regions in the Born-Oppenheimer approximation.- 3.2.c. Forces when the nuclei are treated quantum mechanically.- Berlin's regions in a non-Born-Oppenheimer approximation.- 3.2.d. Generalized forces.- 3.3. Charge distributions and atomic charges.- 3.3.a. The experimental determinations of charges via inductive effects.- 3.3.b. Electron populations analysis.- 3.3.c. Aproximate natural orbitals obtained from molecular orbitals.- Natural atomic orbitals.- 3.3.d. Natural localized molecular orbitals.- 3.4. Quantum mechanics of an atomic fragment.- 3.4.a. Time-independent variational principle for a fragment.- 3.4.b. Time-independent variational principle for a fragment.- 3.5. Molecular structure and its relation to topologic properties of one-electron densities.- 3.5.a. Critical points and gradient paths.- a (3,+3) critical point.- a (3,-3) critical point.- a (3.+1) critical point.- a (3,-1) critical point.- Molecular structure.- 3.5.b. Catastrophe points and their relation to the change in molecular structure.- 4. An Introduction to density functional theory from the perspective of the independent-particle model and its corrections.- 4.0. Preamble.- 4.1. The Hartree-Fock variational approach.- 4.1.a. Introductory remarks.- 4.1.b. The Hartree-Fock method.- Problems.- 4.1.c. General properties of the Hartree-Fock ground state for atoms and ions.- Problems.- 4.1.d. Electron-electron repulsion at the Hartree-Fock level.- 4.1.e. The Hartree potential and direct Coulomb energy.- 4.l.f. The Hartree-Fock exchange energy.- 4.1.g. The Hartree-Fock exchange potential.- Problems.- 4.1.h. Degenerate free-electron gas model at the Hartree-Fock level.- Problem.- 4.2. The exact level.- 4.2.a. Correlation energy.- 4.2.b. Fermi correlation.- Problems.- 4.2.c. Coulomb correlation.- 4.2.d. Concepts in electron correlation theory.- Problems.- 4.2.e. Semiquantitative description of exchange and correlation.- Problems.- 4.3. The kinetic energy term.- Problems.- 4.4. The N-representability problem for D2 and ?.- 5. The Thomas-Fermi energy density functional and its generalization.- 5.1. Formulation of the Thomas-Fermi model for atoms and ions.- Problems.- 5.2. Leading quantum corrections to the Thomas-Fermi atom.- Problems.- 5.3. Post Thomas-Fermi-Dirac-von Weizsaker developments in density functional theory.- 5.3.a. The concept of chemical potential: a density functional point of view.- 5.3.b. The concept of electronegativity from an energy density functional point of view.- 5.3.c. Energy relationships involving electrostatic potential.- 5.3.d. Formulation of equivalent variational principles: in search of the "best" density.- Problems.- 5.4. Molecular structure and molecular interactions from the perspective of the Thomas- Fermi theory and its extensions.- 6. Foundations of density functional theory.- 6.0. Preamble.- 6.1. Correspondence between ground-state one-electron densities and external potentials.- 6.1.a. The first Hohenberg-Kohn theorem.- 6.1.b. From densities to potentials.- 6.1.c. From spectra to potentials: the inverse method in quantum mechanics.- 6.2. v-representability of one-electron densities.- Problems.- 6.3 N-representability of one-electron densities.- Problems.- 6.4. The second Hohenberg-Kohn theorem.- 6.5. Universal functionals for non-v-representable one-electron densities.- 6.5.a. The Levy-Lieb functional.- 6.5.b. The Lieb functional.- 6.5.c. General properties of functional of the one-electron density.- 6.6. Approximate method for the determination of universal functional.- 6.6.a. Freed and Levy's algorithm.- 6.6.b. The constrained variation of Yang and Harriman.- 6.6.c. Westhaus' constrained variational formulation.- Problems.- 6.7. A universal functional of the reduced first-order density operator.- 6.7.a. Pure states and ensembles.- 6.7.b. Variational principle with built-in pure-state N-representability conditions.- 6.7.c. General variational equation for orbitals and occupation numbers.- 6.7.d. Discussion.- 7. A rigorous formulation of the variational principle in density functional theory.- 7.1. Introductory remarks.- 7.1.a. Background: method of local-scaling transformations.- 7.1.b. Point transformations and one-electron densities.- 7.1.c. Topological properties of one-electron densities and local-scaling transformations.- 7.1.d. Local-scaling transformations, electron densities and many-electron wavefunctions.- Problems.- 7.2. Explicit construction of the energy density functional.- 7.2.a. A reformulation of the variational principle.- 7.2.b The energy functional.- Problems.- 7.2.c. Some simple numerical test.- 7.3 Reformulation of the Hohenberg-Kohn theorems.- 7.3.a. N-representability and v-representability of ? (r) revisited.- 7.3.b. Reformulation of the Hohenberg-Kohn first theorem.- Problems.- 7.4. The spin-density functional formalism.- Problems.- 7.5. Density functional theory for excited states.- Problems.- 7.6 The non-adiabatic energy density functional theory.- 7.7. The concept of fractional occupation numbers in density functional theory.- 7.7.a. Preamble.- 7.7.b. The energy density as a functional of occupation numbers.- 7.7.c. Slater's transition state concept.- 7.7.d. Local-scaling transformations and the transition-state concept.- 7.8. N-representability of experimentally determined densities.- (i) Chemical bonds and electron difference densities.- (ii) Local-scaling transformations and the inverse problem.- 7.9. The inverse problem in density functional theory.- 8. The self-consistent field concept in density functional theory.- 8.1. Introductory comments.- 8.2. The Slater-Kohn-Sham ansatz. Self-consistent field version of exchange-only density functional theory.- 8.2.a. The self-consistent field concept at the Llartree-Fock level.- 8.2.b. The concept of exchange potentials in density functional theory.- 8.2.c. Computational schemes.- 8.2.d. The local density approximation.- 8.3. The inverse problem in the Slater-Kohn-Sham ansatz.- 8.3.a. Local density approximation and the nodal structure of orbitals.- 8.3.b. Toward self-interaction free exchange-only density functional.- Problems.- 8.3.c. Interpretation of the one-electron energy eigenvalues.- 8.3.d. Yirial-like relations and related problems.- 8.3.e. Rigorous formulation of the exchange-only self-consistent field concept.- Charge-consistency.- Orbit-consistency.- Orbit-consistency and single-particle equations.- 8.4 The Kohn-Sham ansatz.- 8.4.a. Preamble.- 8.4.b. The Kohn-Sham ansatz: formulation.- 8.4.c. Exchange-correlation energy density functionals based on the electron-gas models.- Problems.- 8.4.d. Nonlocal exchange-correlation energy density functional.- Problems.- 8.4.e. Rigorous formulation of the self-consistent fiel concept with correlation.- 9. Synopsis and future trends.- 9.1 Density functional theory: overview and interfaces.- Many-electron systems in strong magnetic field.- Relativistic energy functional theory.- Temperature-dependent density functional theory.- Time-dependent density functional theory.- The interfaces with quantum chemistry and solid-state physics.- Multicomponent systems.- Theory of nuclear structure.- Statistical mechanics and interface problems.- Molecular properties.- 9.2 Concluding remarks.