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Department of Chemistry, University of Montreal Ma6hias Ernzerhof Introduction to density functional theory and to approximate funtionals

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Page 1: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

Department of Chemistry,  University of Montreal

Ma6hias Ernzerhof

Introduction to density functional theory and to approximate funtionals

Page 2: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

Introduc=on

Time‐dependant Schrödinger equa=on:

                                                     

For molecules, solids, etc.               

                                                 N=number of electrons

                                                                  n=number of nuclei Density‐Func=onal Theory of Atoms and Molecules, Parr, Yang, Oxford University Press (1989).Density func=onal theory, Dreizler, Gross, Springer Verlag, Berlin (1990).Perdew, Kurth, in Density func=onals: Theory and applica=ons, Lecture notes in Physics 500,Springer Verlag, Berlin (1998). Quantum Theory of the Electron Liquid, Giuliani, Vignale, Cambridge (2005).

Page 3: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

We use atomic units

For sta=onary states

Page 4: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

Wave func=on methods

Varia=onal methods

Perturba=on theoryCoupled‐Cluster theory

Many‐body (field theore=cal)methods

Green’s func=on methods

Page 5: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For
Page 6: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

Electron density

Page 7: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

The ground‐state varia=onal method in 

Hohenberg, Kohn, PR 136B, 864 (1964)M. Levy, Proc. Natl. Acad. Sci, USA 79, 6062 (1979)M. Levy, PRA 26, 1200 (1982)

Thomas&Fermi (1927)

Page 8: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

.

Slater determinant

Page 9: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

Hartree‐Fock energy

Kohn‐Sham expression for the ground‐state energy

Kohn, Sham, PR 140, A1133 (1965)

Page 10: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

Exchange-correlation energy

Single-particle kinetic energy

Interaction of the electron densitywith itself

Page 11: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

Kohn‐Sham equa=ons

Page 12: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

AtomHomogeneous electron gasρ

Local density approxima=on to the exchange‐correla=on energy

Page 13: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

Local density approxima=on to the exchange energy 

Page 14: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

Gradient expansion approxima=on (GEA)

Page 15: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

Generalized gradient approxima=on (GGA)

with

Page 16: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For
Page 17: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

Perdew‐Burke‐Ernzerhof (PBE)

PBE generalized gradient approximation:

Perdew, Burke, Ernzerhof,Phys. Rev. Lett. 77, 3865 (1996)

Page 18: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

Results

Page 19: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

Dependence of the energy on N

Perdew, Parr, Levy, Balduz, PRL 49, 1691 (1982)

Page 20: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

Dependence of the energy on N

Janak, PRB 18, 7165 (1978)

N

Page 21: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

Exchange‐correla=on holesurrounding an electron 

Ernzerhof, Perdew, Burke, Topics in Current Chemistry Vol. 180, Springer (1996)Burke, Perdew, Ernzerhof, J. Chem. Phys. 109, 3760 (1998)

Page 22: Introduction to density functional theory and to …...Introduction to density functional theory and to approximate funtionals Introducon Time‐dependant Schrödinger equaon: For

Acknowledgments

Funding&support:     NSERC, CFI, GAUSSIAN

John Perdew


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