Energy Density Functional Theory of Many-Electron Systems

ISBN-13: 9789401073813 / Angielski / Miękka / 2011 / 850 str.

Eugene S. Kryachko; Eduardo V. Ludena

Opis

I would like to present to a wide circle of the readers working in quantum chem- istry and solid-state physics, as ,,*ell as in other fields of many-body physics and its interfaces, this book deyoted to density functional theory written by my colleagues Eugene S. Kryachko and Eduardo Y. Ludena. Their ways to this theory are rather different although basically both of them are quantum chemical. Eugene S. Kryachko came to energy density functional theory from the theory of reduced density matrices, and Eduardo '. Ludena dewloped earlier the concept of loges in quantum chemistry. Neyertheless, their earlier interests giw the possibility to consolidate and formulate energy density functional theory in a unified and consistent way, in my opinion. Raymond Daudel Paris ACKNOWLEDGMENTS The authors are indebted to Carl Almbladh, Victor Va. Antonchenko, John Avery, Richard F. W. Bader, Ulf 'on Barth, Jean-Louis Calais, A. John Coleman, Jens P. Dahl, Robert Donnelly, Harold Englisch, Robert 11. Erdahl, Oswaldo Goscinski, John E. Harriman, Gintas Kamuntavichius, Illja G. Kaplan, Jaime Keller, 'alentin Khart- siev, Toshikatsu Koga, Per-Olov Loydin, T. Tung Nguyen-Dang, Ivan Zh. Petkov, Jerome K. Percus, llary Beth Ruskai, John R. Sabin, Zdenek Slanina, 'ladimir Shi- rokov, llario V. Stoitsov, Yoram Tal, and Vaitao Yang, who in one way or another, either through their kind support, help, discussions or valuable comments created the human and intellectual background which made this book possible.

Spis treści

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 Schrödinger’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 Schrödinger 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.