Independent electrons in an effective potential
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1 ABC of DFT
2 Adiabatic approximation Independent electrons in an effective potential Hartree Fock Density Functional Theory MBPT - GW
3 Density Functional Theory in a nutshell Every observable quantity of a stationary quantum mechanical system is determined by the system ground-state density alone The ground-state density of the interacting system of interest can be calculated as ground-state density of an auxiliary non-interacting system in an effective potential There are prescriptions to determine the auxiliary effective potential with useful accuracy
4 Many Properties are within the scope of DFT... -Energetics, relative stability of different structures, phase diagrams; -Structural properties, geometry; -Response to external perturbations: elastic, dielectric, piezoelectric constants; -Vibrational properties; -Chemical potentials, ionization energies, electron affinities, Fermi energies; -Defect energetics and transport properties. and many are NOT -Electronic excitations (unless ); -Fermi surfaces and band structures; -Defect induces electronic structure changes; -Electronic transport properties.
5 There is nothing so practical as a good theory - Kurt Lewin founder of social psychology
6 elk
7
8 Adiabatic approximation Independent electrons in an effective potential Hartree Fock Density Functional Theory MBPT - GW
9 Adiabatic approximation Independent electrons in an effective potential Hartree Fock Density Functional Theory - Hohenberg-Kohn theorem - Kohn-Sham equations - Exc functionals MBPT - GW
10 Hohenberg - Kohn Theorem Pierre C Walter
11 Hohenberg and Kohn Theorem The GS density can be uses as basic variable to describe the status of a quantum many-body system. All properties of the system are therefore functionals of the GS density.
12
13 p.864
14 Hohenberg and Kohn Theorem
15 Levy's constrained search formulation of DFT M Levy PNAS 76, 6062 (1979); M Levy, PRA 26, 1200 (1982) The original HK theorem was obtained for densities that are (non degenerate) GS densities of some potential V... These density are named V-representable But what are they? How can we define the domain of definition for the HK theorem? It can be shown that very reasonable densities are not V-representable The extension is defined for all N-representable densities
16 Hohenberg and Kohn Theorem An analogy with a well known case... Consider the structural stability problem: Volume V and pressure P are equally legitimate variables to describe the status of the system, each with its own thermodynamic potential, linked by a Legendre transform
17 Hohenberg and Kohn Theorem In QM potential V(r) determines the GS density n(r)
18 Hohenberg and Kohn Theorem In QM potential V(r) determines the GS density n(r) Performing a Legendre transform we obtain The GS density n(r) determines the potential V(r)
19 DFT variational principle HK: depends only on the GS density (3D function) and satisfies the variational principle [Hohenberg and Kohn, 1964]
20 Kohn - Sham equations Walter Lu J
21 n(r) is a smooth and simple function of r, so...
22 F[n] is a very non trivial functional of n(r)!
23 p.1133
24 HK: It is useful to introduce a ficticious system of non-interacting electrons KS: This defines Exc The energy becomes
25 Thomas-Fermi approximation to Ts EH Lieb, Rev Mod Phys 53, 605 (1981) Thomas-Fermi and related theories of atoms and molecules
26 Simple approximations for Ts are only qualitative
27
28 - interacting system Kohn-Sham equations - non-interacting system
29
30 Self-consistent equations [Kohn-Sham, 1965] The density is exact if is, however & are auxiliary functions w/o clear physical meaning
31
32 Self-consistent equations [Kohn-Sham, 1965] It is as simple as a Mean-field approach but it is exact! is not known exactly approximations
33 Simple approximations to Exc are possible
34 Local Density Approximation The simplest approximation is LDA that exploits nearsightedness of the electronic matter W. Kohn, PRL 76,3168 (1996) Analogous to the Thomas-Fermi approximation for the Kinetic Energy term but applied to the much smaller Exchange-Correlation term In many cases it works very nicely
35 Local Density Approximation The simplest approximation is LDA that exploits nearsightedness of the electronic matter W. Kohn, PRL 76,3168 (1996) From accurate DMC data Ceperley-Alder PRL 45, 566 (1980)
36 Total Energy expression in DFT
37 Exchange-Correlation functionals Kieron Burke, Perspective on density functional theory JCP 136 (2012)
38 Jacob's ladder of Density Functional Theory John P. Perdew
39 Jacob's ladder of Density Functional Theory
40
41 THE END
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