Transcription of Lecture 8: Introduction to Density Functional Theory
1 Lecture 8: Introduction to Density Functional TheoryMarie Curie Tutorial Series: Modeling BiomoleculesDecember 6-11, 2004 Mark TuckermanDept. of Chemistry and Courant Institute of Mathematical Science100 Washington Square EastNew York University, New York, NY 10003 Background 1920s: Introduction of the Thomas-Fermi model. 1964: Hohenberg-Kohn paper proving existence of exact DF. 1965: Kohn-Sham scheme introduced. 1970s and early 80s: LDA. DFT becomes useful. 1985: Incorporation of DFT into molecular dynamics (Car-Parrinello)(Now one of PRL stop 10 cited papers). 1988: Beckeand LYP functionals. DFT useful for some chemistry. 1998: Nobel prize awarded to Walter Kohn in chemistry fordevelopment of (,,..,,)NNss rreeeeeNHTVV=++External Potential:Total Molecular Hamiltonian:eNNNHHTV=++21,12||NNIIINIJNN IJIIJTMZZV=>= = RRBorn-Oppenheimer Approximation: []1010(x.)
2 ,x;)()(x,..,x;)[](,)(,)eeeeeeNNNNNNTVVET VE titt ++ = ++= RRRRRx,iiis=rHohenberg-Kohn Theorem Two systems with the same number Neof electrons have the sameTe+ Vee. Hence, they are distinguished only by Ven. Knowledge of | 0> determines Ven. Let V be the set of external potentials such solution of yields a non=degenerate ground state | 0>. Collect all such ground state wavefunctionsinto a set . Each element of this set is associated with a Hamiltonian determined by the external potential. There exists a 1:1 mapping Csuch thatC: V []0eeeeeNHTVVE =++ = 00 = ()000 (2)eeeeNTVVE ++ = ()000eeeeNTVVE ++ = Hohenberg-Kohn Theorem (part II)Given an antisymmetricground state wavefunctionfrom the set , the ground-state Density is given by122122()(,,,,..,,)eeeNeeNNNssnNddsss= rrrrrr""Knowledge of n(r)is sufficient to determine | >Let Nbe the set of ground state densities obtained from Ne-electron groundstate wavefunctionsin.
3 Then, there exists a 1:1 mappingD-1: N D : NThe formula for n(r)shows that D exists, however, showing that D-1existsIs less that D-1exists00000eeeeeNEHTVV = = ++ []000()()() (2)extextEEdnVV < rrrr(CD)-1: N V00000 [][][]nOnOn =The theorems are generalizableto degenerate ground states!The energy functionalThe energy expectation value is of particular importance00000[][][]enHnEn =From the variationalprinciple, for | > in :00eeHH Thus,0[][][][]enHnEnEn = Therefore, E[n0]can be determined by a minimization procedure:0()[]min[]nEnEn =rN0000000000000()() ()neeeeNneeeeNneeenexteeeneeeneeeTVVTVVT VdnVTVdnTVTV ++ ++ + + + + + + rrrr0()extVrr() min[] ()()extnFndnV =+ rrrr*2122122{}(,,,,..,,)(,,,,eeeeNNNsNdd sssss= rrrrrrr"(,)..,,)eeNNs rrrThe Kohn-Sham FormulationCentral assertion of KS formulation:Consider a system of NeNon-interacting electrons subject to an external potential VKS.
4 ItIs possible to choose this potential such that the ground state Density Of the non-interacting system is the same as that of an interacting System subject to a particular external potential non-interacting system is separable and, therefore, described by a setof single-particle orbitals i(r,s), i=1,..,Ne, such that the wave function isgiven by a Slater determinant:1111(x,..,x)det[(x)(x)]!eeeN NNeN ="The Density is given by21()(x) eNiijijisn === rThe kinetic energy is given by*211(x)(x)2eNsiiisTd == rKS()() ()xcextEnVVdn =++ rrrrrr/2211()()2eNsiiiT == rrSome simple results from DFTE barrier(DFT) = kcal/molEbarrier(MP4) = kcal/molGeometry of the protonatedmethanol MP2 6-311G (2d,2p) Results methanolDimer dissociation curve of a neutral dimerExpt.: kcal/molLecture Summary Density Functional Theory is an exact reformulation of many-bodyquantum mechanics in terms of the probability Density rather thanthe wave function The ground-state energy can be obtained by minimization of theenergy Functional E[n].
5 All we know about the Functional is thatit exists, however, its form is unknown. Kohn-Sham reformulation in terms of single-particle orbitalshelpsin the development of approximations and is the form used in current Density Functional calculations today.
