A Tribute to A. John Coleman - The Tame Mathematician.- Reduced Density Matrices: 1929-1989.- Some Aspects on the Development of the Theory of Reduced Density Matrices and the Representability Problem.- Representability Conditions.- On the Diagonal N-Representability Problem.- Fermion N-Representability Conditions Generated by a Decomposition of the 1-Particle Identity Operator onto Mutually Orthogonal Projection Operators.- The Unitarily Invariant Decomposition of Hermitian Operators.- Building Up N-Electron States with Symplectic Symmetry.- Time Dependent Antisymmetrized Geminal Power Theory Using a Coherent State Formulation.- Griffiths Inequalities for Fermion Systems.- Entropy of Reduced Density Matrices.- A Lower Bound to the Ground State Energy of a Boson System With Fermion Source.- Reduced Density Operators, Their Related von Neumann Density Operators, Close Cousins of These, and Their Physical Interpretation.- Theory and Practice of the Spin-Adapted Reduced Hamiltonians (SRH).- Variational Principle with Built-In Pure State N-Representability Conditions. The N-Electron Case.- Wigner Distributions as Representations of the Density Matrix.- Inter-Relationships Between Various Representations of One-Matrices and Related Densities: A Road Map and an Example ..- Current Problems in Density Functional Theory.- The Interface Between Reduced Density Matrices and Density Functional Theory.- The Physics Underlying the Langreth-Mehl Scheme for Non-Uniform Systems.- Understanding Energy Differences in Density Functional Theory.- Density Functional Calculations of Molecular Bond Energies.- Non-Local Effects on Atomic and Molecular Correlation Energies Studied with a Gradient-Corrected Density Functional.- An Evaluation of Local Electron Correlation Corrections and Non-Local Exchange Corrections to the Hartree-Fock-Slater Method from Calculations on Bond Energies and Electronic Spectra of Molecular Systems.- Correlation Energy Functionals of One-Matrices and Hartree-Fock Densities.- Some Remarks on Scaling Relations in Density Functional Theory.- Deduction of Semiempirical MO Methods from Density Functional Theory.- Charge and Spin Densities in Molecular Solids: Local Density Functional Calculations Versus Experiment.- A Functional of the Two-Particle Density Matrix for the Approximate Calculation of the Electronic Correlation Energy.- Extracules, Intracules, Correlation Holes, Potentials, Coefficients and All That.- The Exact Schroedinger Equation for the Electron Density.- Adiabatic Separation, Broken Symmetries and Geometry Optimization.- Asymptotic Results for Density Matrices and Electron Density in Atoms and Nearly Spherical Molecules.- An Algorithm for Calculating Isoelectronic Changes in Energies, Densities, and One-Matrices.- Atoms and Ions in the Limit of Large Nuclear Charge.- Improved Thomas-Fermi Theory for Atoms.- A Bond Energy from Quantum Mechanics.- Measured Electron Densities and Band Structure Calculations.- X-ray Orthonormal Orbital Model for Crystallography.
After a qualitative discussion of the advantages of the density matrix and of the different ways to introduce it (the statistical, quantum mechanical and operational methods of approach), § 2 deals with the general properties of the density matrix, including a discussion of pure cases and mixtures. A brief discussion is given of Green function techniques and of the relation between Green functions and correlation functions. A discussion of recent developments in the evaluation of partition functions concludes the first part of this article dealing with the theory of density matrix techniques. Sections 5 to 9 discuss applications. The first application is the quantum-chemical one to many-body systems in their ground state, that is, systems at absolute zero, and it is shown how the density matrix fits into the Hartree-Fock and Thomas-Fermi schemes. A brief discussion is given of the theory of diamagnetism. This is followed by a discussion of non-equilibrium processes and of Kubo's approach to transport theory. After that the polarization of beams of electrons or of photons is discussed and it is indicated how density matrix techniques can be used to treat scattering processes. Section 9 concludes this part of the paper by a brief account of density matrix theory applications to resonance and relaxation phenomena. Finally, the theory of measurement in quantum mechanics is considered.
For a quantum-mechanical system of
N identical fermions, the
N-representability problem is the problem of recognizing whether, for a given pth-order reduced density matrix ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})$, there exists an antisymmetric
N-particle wave function $\ensuremath{\Psi}(12\ensuremath{\cdots}N)$ such that ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})=({N}{p})\ensuremath{\int}\ensuremath{\Psi}(12\ensuremath{\cdots}N)\ifmmode\times\else\texttimes\fi{}{\ensuremath{\Psi}}^{*}({1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}},p+1\ensuremath{\cdots}N){d}_{{\ensuremath{\tau}}_{p+1}}\ensuremath{\cdots}{d}_{{\ensuremath{\tau}}_{N}}$. It is shown that if the Hamiltonian of a system is time-reversal invariant, and the number of particles,
N, is even, the necessary and sufficient condition that an approximate first-order density matrix corresponding to a nondegenerate energy eigenstate be
N-representable is that its natural spin-orbital occupation numbers be equal in pairs.
Preface. I: Properties of Reduced Density Matrices. 1. RDMs: How Did We Get Here? A.J. Coleman. 2. Some Theorems on Uniqueness and Reconstruction of Higher-Order Density Matrices M. Rosina. 3. Cumulant Expansions of Reduced Densities, Reduced Density Matrices, and Green's Functions P. Ziesche. 4. On Calculating Approximate and Exact Density Matrices R. Erdahl, B. Jin. II: The Contracted Schroedinger Equation. 5. Density Equation Theory in Chemical Physics H. Nakatsuji. 6. Critical Questions Concerning Iterative Solution of the Contracted Schroedinger Equation C. Valdemoro, L.M. Tel, E. Perez-Romero 7. Cumulants and the Contracted Schroedinger Equation D.A. Mazziotti.. III: Density Matrix Functional Theory. 8. Natural Orbital Functional Theory S. Goedecker, C.J. Umrigar. 9. The Pair Density in Approximate Density Functional Theory: The Hidden Agent N.T. Maitra, K. Burke. 10. Functional N-representability in Density Matrix and Density Functional Theory: An Illustration for Hooke's Atom E.V. Ludena, V. Karasiev, A. Artemiev, D. Gomez.. IV: Electron Intracule and Extracule Densities. 11. Intracule and Extracule Densities: Historical Perspectives and Future Prospects E. Valderrama, J.M. Ugalde, R.J. Boyd. 12. Topology of Electron Correlation J. Cioslowski. 13. Electron-Pair Densities of Atoms T. Koga. Index.
The density matrix for a many-electron system has been examined. A formalism has been arrived at which facilitates its evaluation in terms of a basis corresponding to a discrete spectrum of eigenvalues. This allows a representation to be employed associated with a suitably chosen approximation to the Hamiltonian. Thereby, reasonably accurate estimates of the properties of many-electron systems may be anticipated for low orders of approximation.In developing the formalism, attention was focused upon a means for approximating the quantity
exp(zH), where H is the many-particle Hamiltonian and
z is a complex number. It was found possible to represent this quantity exactly as an infinite product of exponential factors, each of which depends upon a portion of the Hamiltonian alone. The relation of this result to statistical mechanical applications is indicated. An approximation procedure is described in terms of which various finite products of exponential factors approach the desired quantity in the limit of indefinitely increasing numbers of factors; this limit is approached in a manner that permits any given approximation to contain all the terms of the previous approximation.The first order of approximation was employed to approximate the many-particle density matrix. The resulting theory was applied to the helium-like systems: He,
Li+,
Be++. The energies of the ground
1S state of helium and the three lowest states of
Li+ and
Be++ were calculated. The agreement with experimental term values is reasonably satisfactory, the discrepancy between calculations and experiments not exceeding one percent. The triplet-singlet splitting for
Li+ was in the order observed; the magnitude calculated was about one-half that observed.The relation between the present theory and the Thomas-Fermi theory is discussed. It is pointed out that the former is the analog of the latter when the density matrix is expressed in other than eigenfunctions of momentum. The quasi-classical character of the approximation is discussed.