3: Density Functional Theory

Transcription

1 The Nuts and Bolts of First-Principles Simulation 3: Density Functional Theory CASTEP Developers Group with support from the ESF ψ k Network

2 Density functional theory Mike Gillan, University College London Ground-state energetics of electrons in condensed matter Energy as functional of density: the two fundamental theorems Equivalence of the interacting electron system to a noninteracting system in an effective external potential Kohn-sham equation Local-density approximation for exchange-correlation energy

3 The problem Hamiltonian H for system of interacting electrons acted on by electrostatic field of nuclei: with T kinetic energy, U mutual interaction energy of electrons, V interaction energy with field of nuclei. To develop theory, V will be interaction with an arbitrary external field: with r i position of electron i. V = + + H T U V N = v( r ) i= 1 Ground-state energy is impossible to calculate exactly, because of electron correlation. DFT includes correlation, but is still tractable because it has the form of a non-interacting electron theory. i

4 Energy as functional of density: the first theorem For given external potential v(r), let many-body wavefunction be ground-state energy E g is: E H V g = Ψ + Ψ Ψ. Then and the electron density n(r) by: where the density operator ^ = Ψ n r Ψ n( r) ( ) ^( ) n r N is defined as: = ^ n( r) ( r r ) i = 1 i Theorem 1: It is impossible that two different potentials give rise to the same ground-state density distribution n(r). Corollary: n(r) uniquely specifies the external potential v(r) and hence the many-body wavefunction. Ψ

5 Convexity of the energy (1) Theorem 1 expresses convexity of the energy E g as function of external potential. v(; r ) v(1; r) Convexity means: For two external potentials and, go along linear path v( λ; r) = (1 λ) v(; r) + λv(1; r) between them; if E ( λ) is ground-state g energy for then: < λ < 1, E ( ) (1 ) E () E (1). λ > λ + λ g g g Proof of perturbation theory: λ > λ + λ E ( ) (1 ) E () E (1) g g g de / d ( ) V ( ) g λ = Ψ λ Ψ λ follows from nd -order d E g Ψ λ Ψ ( ) V ( ) n / dλ = <, E ( ) E ( ) n λ with and wavefns of ground and excited states, and n their energies, and. Ψ ( λ) Ψ ( λ ) E ( λ ) E ( λ ) n V v(1) v() = n λ λ

6 Convexity of the energy () Theorem 1 is equivalent to saying that a change of external potential cannot give a vanishing change of density n( r) λ = v( r ) de / dλ de / d V This follows from convexity. Convexity implies that g at is less than at. But, so that: de / dλ 1 g λ = g λ = Ψ Ψ Ψ Ψ Ψ Ψ (1) V () < () V () so that: Hence: dr v( r) n(1, r) < d r v( r) n(, r). d r v( r ) n( r ) <, n( r) which demonstrates that, and this is Theorem 1.

7 DFT variational principle the second theorem Since ground-state energy E g is uniquely specified by n(r), write it as E g [n(r)]. It s useful to separate out the interaction with the external field, and write: E [ n( r)] dr v( r) n( r) F[ n( r)], g = + Where F[n(r)] is ground-state expectation value of H when density is n(r). Theorem (variational principle): Ground-state energy for a given v(r) is obtained by minimising E g [n(r)] with respect to n(r) for fixed v(r), and the n(r) that yields the minimum is the density in the ground state. Proof: Let v(r) and v (r) be two different external potentials, with ground-state energies E g and E g and ground-state wavefns Ψ and Ψ '. By Rayleigh- Ritz variational principle: Eg < Ψ ' H + V Ψ ' = d r v( r ) n'( r ) + F[ n'( r )], Ψ ' Where n (r) is density associated with. This proves the theorem. The usual assumptions of non-degenerate ground state is needed.

8 The Euler equation Write F[n(r)] as: F[ n] = T[ n] + G[ n], where T[n] is kinetic energy of a system of non-interacting electrons whose density distribution is n(r). Then: E[ n] = d r v( r ) n( r ) + T[ n] + G[ n]. Variational principle: T G E = = d r v( r ) + + n( ), n( r) n( r) r subject to constraint: d r n( r ) =. Handle the constant-number constraint by Lagrange undetermined multiplier, and get: with undetermined multiplier µ T G + v( r) + = µ, n( r) n( r) the chemical potential.

9 Kohn- Sham equation Rewrite the Euler equation for interacting electrons: by defining + + = µ T G v( r) n( r) n( r) v ( r) = v( r) +G / n( r) eff T v ( ) eff r n( r) + =, so that: But this is Euler equation for non-interacting electrons in potential v eff (r), and must be exactly equivalent to Schroedinger equation: with n(r) given by: h µ ψι + v r ψ eff ι = εψ ι ι m = ψ ( ), n( r) ( r). ε < µ Then put n(r) back into G[n(r)] to get total energy: ι ι E [ n( )] d v( ) n( ) T[ n] G[ n]. tot r r r r = + +

10 Self consistency How to do DFT in practice??? We don t know G[n(r)], and probably never will, but suppose we know an adequate approximation to it. Make an initial guess at n(r), calculate = + v ( r) v( r) G / n( r) eff for this initial n(r). and hence Solve the Kohn-Sham equation with this v eff (r) to get the KS orbitals and hence calculate the new n(r): = ψ r n '( r) ( r). ε < µ ι i G / n( ) The output n (r) is not the same as the input n(r). So iterate to reduce residual: 1/ n d n '( ) n( ) r r r. = The whole procedure is called searching for self consistency.

11 Exchange-correlation energy We have already split the total energy into pieces: Now separate out the Hartree energy: E [ n ] = d v ( ) n ( ) + F [ n ] tot r r r = + F[ n] T[ n] G[ n] 1 n( r) n( r') EH[ n( r)] = e d d '. r r r r' Then exchange-correlation energy E xc [n] is defined by: Etot[ n] = d r v( r ) n( r ) + T[ n] + EH[ n] + Exc[ n]. So far, everything is formal and exact. If we knew the exact E xc [n], then we could calculate the exact ground-state energy of any system!

12 Local density approximation There is one extended system for which E xc is known rather precisely: the uniform electron gas (jellium). For this system, we know exchange-correlation energy per electron as a function of density n. ε ( n) xc Local density approximation (LDA): assume the xc energy of an electron at point r is equal to ε ( n( r)) xc for jellium, using the density n(r) at point r. Then total E xc for the whole system is: LDA ( ) = r r ε r E d n ( n( )) xc xc Some kind of justification can be given for LDA (see xxxxxxx). But the main justification is that it works quite well in practice.

13 Kohn- Sham potential in LDA The effective Kohn-Sham effective potential in general is: The Hartree potential is: Exchange-correlation potential in LDA: Where: ( ) ( ) G ( ) EH Exc veff r = v r + = v r + + n( r) n( r) n( r) EH 1 n( r1) n( r) n( r') = e d d e d '. 1 = n( ) n( ) r r r r r r r r r' So in LDA, everything can be straightforwardly calculated! 1 = ε = µ vxc( r) dr1 n( r1) xc( n( r1)) xc( n( r)), n( r ) xc d ( n) = n ( n) xc dn µ ε ( )

14 Useful references Here is a selection of references that contain more detail about DFT: P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964) W. Kohn and L. J. Sham, Phys. Rev. 14, A1133 (1965) N. D. Mermin, Phys. Rev. 137, A1441 (1965) R. O. Jones and O. Gunnarsson, Rev. Mod. Phys., 61, 689 (1989) M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias and J. D. Joannopoulos, Re. Mod. Phys., 64, 145 (199)