Lecture 8: Introduction to Density Functional Theory

Transcription

1 Lecture 8: Introduction to Density Functional Theory Marie Curie Tutorial Series: Modeling Biomolecules December 6-11, 2004 Mark Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Science 100 Washington Square East New York University, New York, NY 10003

2 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 s top 10 cited papers). 1988: Becke and LYP functionals. DFT useful for some chemistry. 1998: Nobel prize awarded to Walter Kohn in chemistry for development of DFT.

3 Ψ( r, s,..., r, s ) 1 1 N N

4 External Potential: He = Te + Vee + VeN

5 Total Molecular Hamiltonian: H = H + T + V e N NN T V N NN N 1 = 2M = I = 1 2 I N ZZ I J R R I, J> I I J I Born-Oppenheimer Approximation: [ T + V + V ] Ψ 1 = E0 Ψ 1 + NN + 0 = (x,..., x ; R) ( R) (x,..., x ; R) e ee en N N [ TN V E ] χ( R, t) i χ( R, t) t e x = r, s i i i e

6 Hohenberg-Kohn Theorem Two systems with the same number N e of electrons have the same T e + V ee. Hence, they are distinguished only by V en. Knowledge of Ψ 0 > determines V en. Let V be the set of external potentials such solution of [ ] 0 H Ψ = T + V + V Ψ = E Ψ e e ee en yields a non=degenerate ground state Ψ 0 >. Collect all such ground state wavefunctions into a set Ψ. Each element of this set is associated with a Hamiltonian determined by the external potential. There exists a 1:1 mapping C such that C : V Ψ

7 ( T + V + V ) Ψ 0 = E 0 Ψ0 (2) e ee en Ψ =Ψ 0 0

8 ( Te + Vee + VeN) Ψ 0 = E 0 Ψ0

9 Hohenberg-Kohn Theorem (part II) Given an antisymmetric ground state wavefunction from the set Ψ, the ground-state density is given by n() r = N dr dr Ψ(, r s, r, s,..., r, s ) e 2 N N N s s 1 N e e e e 2 Knowledge of n(r) is sufficient to determine Ψ> Let N be the set of ground state densities obtained from N e -electron ground state wavefunctions in Ψ. Then, there exists a 1:1 mapping D : Ψ N D -1 : N Ψ The formula for n(r) shows that D exists, however, showing that D -1 exists Is less trivial.

10 Proof that D -1 exists E = Ψ H Ψ = Ψ T + V + V Ψ 0 0 e 0 0 e ee en 0

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12 [ ] E ( ) ( ) ( ) 0 < E0 d r n r 0 V r ext V r ext (2)

13 (CD) -1 : N V Ψ [ n ] O ˆ Ψ [ n ] = O[ n ] The theorems are generalizable to degenerate ground states!

14 The energy functional The energy expectation value is of particular importance Ψ [ n ] H Ψ [ n ] = E[ n ] 0 0 e From the variational principle, for Ψ> in Ψ: Ψ H Ψ Ψ H Ψ e 0 e 0 Thus, Ψ[ n] H Ψ [ n] = E[ n] E[ n ] e 0 Therefore, E[n 0 ] can be determined by a minimization procedure: En [ ] min En [ ] 0 = n( r ) N

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18 Ψ T + V + V Ψ Ψ T + V + V Ψ n e ee en n 0 e ee en Ψ T + V Ψ + dr n () r V () r Ψ T + V Ψ + dr n 0 () r () r n e ee n 0 ext 0 e ee Ψ T + V Ψ Ψ T + V Ψ n e ee n 0 e ee V ext

19 = min Fn [] + dr n() r Vext () r n( r)

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21 ρ ( rr, ) = N d r d r Ψ ( r, s, r, s,..., r, s ) Ψ( r, s, r, s,..., r, ) N s e Ne * e 2 Ne Ne Ne {} s

22 The Kohn-Sham Formulation Central assertion of KS formulation: Consider a system of N e Non-interacting electrons subject to an external potential V KS. It Is 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 V ext. A non-interacting system is separable and, therefore, described by a set of single-particle orbitals ψ i (r,s), i=1,,n e, such that the wave function is given by a Slater determinant: 1 Ψ (x,...,x ) = det[ ψ (x ) ψ (x )] The density is given by 1 N 1 1 N N N e! N e e e e i= 1 The kinetic energy is given by 2 n( r) = ψ (x) ψ ψ = δ s i i j ij N 1 e T = d ψ (x) ψ (x) r * 2 s i i 2 i= 1 s

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24 n( r ) δ Exc VKS = Vext () r + dr + r r δ n() r

25 1 T ψ ψ N e /2 2 s = i() r i() r 2 i= 1

26 Some simple results from DFT E barrier (DFT) = 3.6 kcal/mol E barrier (MP4) = 4.1 kcal/mol

27 Geometry of the protonated methanol dimer 2.39Å MP G (2d,2p) 2.38 Å

28 Results methanol Dimer dissociation curve of a neutral dimer Expt.: -3.2 kcal/mol

29 Lecture Summary Density functional theory is an exact reformulation of many-body quantum mechanics in terms of the probability density rather than the wave function The ground-state energy can be obtained by minimization of the energy functional E[n]. All we know about the functional is that it exists, however, its form is unknown. Kohn-Sham reformulation in terms of single-particle orbitals helps in the development of approximations and is the form used in current density functional calculations today.