Quantal Density Functional Theory of the Density Amplitude (original) (raw)
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Physics Reports, 2014
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A quantum chemical view of density functional theory
The Journal of Physical Chemistry …, 1997
A comparison is made between traditional quantum chemical approaches to the electron correlation problem and the one taken in density functional theory (DFT). Well-known concepts of DFT, such as the exchangecorrelation energy E xc ) ∫F(r) xc (r) dr and the exchange-correlation potential V xc (r) are related to electron correlation as described in terms of density matrices and the conditional amplitude (Fermi and Coulomb holes). The Kohn-Sham one-electron or orbital model of DFT is contrasted with Hartree-Fock, and the definitions of exchange and correlation in DFT are compared with the traditional ones. The exchangecorrelation energy density xc (r) is decomposed into kinetic and electron-electron potential energy components, and a practical way of calculating these from accurate wave functions is discussed, which offers a route to systematic improvement. V xc (r) is likewise decomposed, and special features (bond midpoint peak, various types of step behavior) are identified and related to electronic correlation. X Figure 4. Correlation energy density in He compared to a number of model correlation energy densities: PW, Perdew-Wang; 11 WL, Wilson-Levy; 127 LYP, Lee-Yang-Parr; 8 LW, local Wigner. 126 (a) -F(r) c(r) from r ) 0.0-0.5 bohr. (b) -4π r 2 F(r) c(r) from r ) 0.0-2.0 bohr. Feature Article
Some Fundamental Issues in Ground-State Density Functional Theory: A Guide for the Perplexed
Journal of Chemical Theory and Computation, 2009
Some fundamental issues in ground-state density functional theory are discussed without equations: (1) The standard Hohenberg-Kohn and Kohn-Sham theorems were proven for a Hamiltonian that is not quite exact for real atoms, molecules, and solids. (2) The density functional for the exchange-correlation energy, which must be approximated, arises from the tendency of electrons to avoid one another as they move through the electron density. (3) In the absence of a magnetic field, either spin densities or total electron density can be used, although the former choice is better for approximations. (4) "Spin contamination" of the determinant of Kohn-Sham orbitals for an open-shell system is not wrong but right. (5) Only to the extent that symmetries of the interacting wave function are reflected in the spin densities should those symmetries be respected by the Kohn-Sham noninteracting or determinantal wave function. Functionals below the highest level of approximations should however sometimes break even those symmetries, for good physical reasons. (6) Simple and commonly used semilocal (lower-level) approximations for the exchange-correlation energy as a functional of the density can be accurate for closed systems near equilibrium and yet fail for open systems of fluctuating electron number. The exact Kohn-Sham noninteracting state need not be a single determinant, but common approximations can fail when it is not. (8) Over an open system of fluctuating electron number, connected to another such system by stretched bonds, semilocal approximations make the exchange-correlation energy and hole-density sum rule too negative. (9) The gap in the exact Kohn-Sham band structure of a crystal underestimates the real fundamental gap but may approximate the first exciton energy in the large-gap limit. (10) Density functional theory is not really a mean-field theory, although it looks like one. The exact functional includes strong correlation, and semilocal approximations often overestimate the strength of static correlation through their semilocal exchange contributions. (11) Only under rare conditions can excited states arise directly from a ground-state theory.
Density Functional Analysis: The Theory of Density-Corrected DFT
Journal of Chemical Theory and Computation, 2019
Density-corrected density functional theory (DC-DFT) is enjoying substantial success in improving semilocal DFT calculations in a wide variety of chemical problems. This paper provides the formal theoretical framework and assumptions for the analysis of any functional minimization with an approximate functional. We generalize DC-DFT to allow comparison of any two functionals, not just comparison with the exact functional. We introduce a linear interpolation between any two approximations, and use the results to analyze global hybrid density functionals. We define the basins of density-space in which this analysis should apply, and give quantitative criteria for when DC-DFT should apply. We also discuss the effects of strong correlation on density-driven error, utilizing the restricted HF Hubbard dimer as an illustrative example.
Physical Review A, 2003
The quantal density-functional theory ͑Q-DFT͒ of nondegenerate excited-states maps the pure state of the Schrödinger equation to one of noninteracting fermions such that the equivalent excited state density, energy, and ionization potential are obtained. The state of the model S system is arbitrary in that it may be in a ground or excited state. The potential energy of the model fermions differs as a function of this state. The contribution of correlations due to the Pauli exclusion principle and Coulomb repulsion to the potential and total energy of these fermions is independent of the state of the S system. The differences are solely a consequence of correlation-kinetic effects. Irrespective of the state of the S system, the highest occupied eigenvalue of the model fermions is the negative of the ionization potential. In this paper we demonstrate the state arbitrariness of the model system by application of Q-DFT to the first excited singlet state of the exactly solvable Hookean atom. We construct two model S systems: one in a singlet ground state (1s 2), and the other in a singlet first excited state (1s2s). In each case, the density and energy determined are equivalent to those of the excited state of the atom, with the highest occupied eigenvalues being the negative of the ionization potential. From these results we determine the corresponding Kohn-Sham density-functional theory ͑KS-DFT͒ ''exchangecorrelation'' potential energy for the two S systems. Further, based on the results of the model calculations, suggestions for the KS-DFT of excited states are made.
Quantal Density Functional Theory of Degenerate States
Physical Review Letters, 2003
The treatment of degenerate states within Kohn-Sham density functional theory (KS-DFT) is a problem of longstanding and current interest. We propose a solution to this mapping from the interacting degenerate system to that of the noninteracting fermion model whereby the equivalent density and energy are obtained via the unifying physical framework of quantal density functional theory (Q-DFT). We describe the Q-DFT of both ground and excited degenerate states, and for the cases of both pure state and ensemble v-representable densities. The Q-DFT description further provides a rigorous physical interpretation of the corresponding KS-DFT energy functionals of the density, ensemble density, bidensity and ensemble bidensity , and of their respective functional derivatives. We conclude with examples of the mappings within Q-DFT.
4. QUANTUM CHEMISTRY METHODS: II DENSITY FUNCTIONAL THEORY
The Density Functional Theory (DFT)(Parr, 1989) represents an alternative to the conventional ab initio methods of introducing the effects of electron correlation into the solution to the electronic Schrödinger equation. According to the DFT, the energy of the ground state of a many-electron system can be expressed through the electron density, and in fact, the use of the electron density in place of the wave function to calculate the energy is the foundation of the DFT.
Quantal density functional theory of excited states: Application to an exactly solvable model
International Journal of Quantum Chemistry, 2001
The quantal density functional theory (Q‐DFT) of excited states is the description of the physics of the mapping from any bound nondegenerate excited state of Schrödinger theory to that of the s‐system of noninteracting Fermions with equivalent density ρk(r), energy Ek, and ionization potential Ik. The s‐system may either be in an excited state with the same configuration as in Schrödinger theory or in a ground state with a consequently different configuration. The Q‐DFT description of the s‐system is in terms of a conservative field ℱk(r), whose electron‐interaction ℰee(r) and correlation‐kinetic 𝒵(r) components are separately representative of electron correlations due to the Pauli exclusion principle and Coulomb repulsion, and correlation‐kinetic effects, respectively. The sources of these fields are expectations of Hermitian operators taken with respect to the system wavefunction. The local electron‐interaction potential vee(r) of the s‐system, representative of all the many‐bod...