Density functionals and dimensional renormalization for an exactly solvable model

Abstract

We treat an analytically solvable version of the "Hooke's Law" model for a two-electron atom, in which the electron-electron repulsion is Coulombic but the electron-nucleus attraction is replaced by a harmonic oscillator potential. Exact expressions are obtained for the ground-state wave function and electron density, the Hartree-Fock solution, the correlation energy, the Kohn-Sham orbital, and, by inversion, the exchange and correlation functionals. These functionals pertain to the "intermediate" density regime (r,) 1.4) for an electron gas. As a test of customary approximations employed in density functional theory, we compare our exact density, exchange, and correlation potentials and energies with results from two approximations. These use Becke's exchange functional and either the Lee-Yang-Parr or the Perdew correlation functional. Both approximations yield rather good results for the density and the exchange and correlation energies, but both deviate markedly from the exact exchange and correlation potentials. We also compare properties of the Hooke's Law model with those of two-electron atoms, including the large dimension limit. A renormalization procedure applied to this very simple limit yields correlation energies as good as those obtained from the approximate functionals, for both the model and actual atoms.

Benchmark calculations on the lowest-energy singlet, triplet, and quintet states of the four-electron harmonium atom

“…), the continuous tunability of electron correlation effects makes the two-electron harmonium atom a convenient benchmark that has been already applied to a multitude of electronic structure methods, including those based on DFT [5][6][7][8][9][10][11][12][13][14][15] and other formalisms. [16][17][18] Unfortunately, its general usefulness is severely limited by its failure to provide any new information for those approximate electron correlation methods that are already exact for two-electron systems.…”

Two-Electron Spherical Quantum Dot in a Magnetic Field

“…During last few decades many theoretical and experimental studies on QDs have been performed within parabolic confinement models as well as beyond this approximation. Both the 2D and 3D QD's have been investigated in a framework of strict quantum-mechanical and semiclassical approaches, including also effects of an external magnetic field [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Semiclassical solutions for a two-electron QD in a magnetic field have been investigated in terms of action-angle variables using the classical adiabatic approximation [19].…”

Absorption Model of Hydrogenic Atoms and Ions by Metal Surfaces

“…&~[n2(r1,r 2j\ = 0.645 + 0.037-0.003 + .... The comparison with the calculations of in different models [2] in the frame of different one-particle methods shows good agreement with the first order term. The electron den sity ^(fj) on the metal surface we obtain by the expres sion where the function n2(r1,r 2) is calculated by the vari ational method on the basis of (3).…”

“…The energy func tional is of the form [1] W 2(rl ,r 2)] = r [ n 2(rl ,r 2)] v T J^C I ) + ^( r 2))^2(r1,r 2)d3r 1d3r2 + JlT (r1-r 2)^( r 1,r 2)d3r 1d3r2, (3) where the kinetic energy function 3~\n2 (rx, r2)] is cho sen as a gradient expansion which can be found for any N-electron system (electron gas on the metal sur face, atoms) in the form The convergency of the given gradient expansion we analyze empirically by using Hooke's Law model applied for the Helium like atom. The electronnucleus attraction is replaced by an harmonic oscilla tor potential, but the electron-electron repulsion re mains Coulombic [2]. For the Helium atom the Hamiltonian is H = -' [ -(V 2 + V2) + \ k ( r 2 + r2) + -, 2 2 r X 2 where k is the oscillator spring constant, and r2 are the distances of the electron from the nucleus, and ri2 = k i -r21 ■ For the particular value k = \ the solution can be obtained analytically as ij/(r1,r 2) = N0(\ -I-^)e x p ( -+ r2)) with a normalization constant N0.…”