Density-functional theory

Transcription

1 Density-functional theory Trygve Helgaker Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway The 13th Sostrup Summer School Quantum Chemistry and Molecular Properties July Trygve Helgaker (CTCC, University of Oslo) Density-functional theory 13th Sostrup Summer School (2014) 1 / 78

2 Density-functional theory The solution of the Schrödinger equation is a horrendously difficult task HΨ = EΨ the wave function depends on 3N spatial and N spin coordinates Ψ(x 1, x 2,..., x N ), x i = {r i, σ i } It is possible to build a theory based on the one-electron density ρ(r) = N... Ψ(x, x 2,... x N ) 2 dσdx 2 dx N the density depends on only three spatial coordinates, for all systems N = ρ(r) dr, ρ(r) 0 In density-functional theory (DFT), this approach is taken Literature: E[v] = min ρ (F [ρ] + (v ρ)) DFT combines good accuracy with low cost Parr and Yang, DFT of Atoms and Molecules (Oxford 1989) Koch and Holthausen, A Chemist s Guide to DFT 2nd ed. (Wiley-VCH, 2000) Eschrig, The Fundamentals of DFT 2nd ed. (Eagle 2003) Trygve Helgaker (CTCC, University of Oslo) Introduction 13th Sostrup Summer School (2014) 2 / 78

3 Overview 1 Hohenberg Kohn theory the external potential the Hohenberg Kohn theorem the Hohenberg Kohn functional the Hohenberg Kohn variation principle v-representable densities and ground-state potentials 2 Levy Lieb constrained-search theory N-representable densities the Levy Lieb constrained-search functional 3 Lieb convex-conjugate theory convex functions Legendre Fenchel transform or convex conjugation conjugate and biconjugate functions the Lieb convex-conjugate functional Trygve Helgaker (CTCC, University of Oslo) Introduction 13th Sostrup Summer School (2014) 3 / 78

4 The external potential Consider the electronic Hamiltonian of an N-electron atom or molecule H[v] = T + W + i v(r i ) T and W are system-independent operators T = 1 2 i 2 i, kinetic energy W = i>j r 1 ij, electron repulsion v(r) is the system-dependent multiplicative external potential The Hamiltonian H[v] depends functionally on the external potential v Typically, we are interested in attractive point-charge Coulomb potentials v(r) = Z A A r A More generally, we consider all potentials that can bind N electrons V N = { v H[v] has an N-electron ground state } attractive Coulomb potentials spatially uniform positive background potentials and many others... We shall later also consider potentials that cannot bind N electrons Trygve Helgaker (CTCC, University of Oslo) The external potential 13th Sostrup Summer School (2014) 4 / 78

5 The interaction with the external potential Let us now assume that v V N so that an N-electron ground state exists H[v]Ψ = E[v]Ψ, v V N The ground-state energy may be calculated as an expectation value: E = Ψ H[v] Ψ = Ψ T + W Ψ + Ψ i v(r i ) Ψ We now introduce the electron density ρ(r) = N Ψ(x, x 2,... x N ) 2 dσdx 2 dx N In terms of the density, the expectation value becomes E = Ψ T + W Ψ + (v ρ) where the interaction with the external potential v is given by (v ρ) = v(r)ρ(r) dr interaction or pairing As we shall see, the relationship between v and ρ is at the heart of DFT v ρ Trygve Helgaker (CTCC, University of Oslo) The external potential 13th Sostrup Summer School (2014) 5 / 78

6 Different potentials have different wave functions We denote by Ψ v a normalized ground-state wave function associated with v: H[v]Ψ v = E[v]Ψ v Assume that v 1, v 2 V N have a common ground-state wave function Ψ: } H[v 1 ]Ψ v1 = E[v 1 ]Ψ v1 Ψ H[v 2 ]Ψ v2 = E[v 2 ]Ψ v1 = Ψ v2 = Ψ v2 Subtracting the two Schrödinger equations, we obtain (H[v 1 ] H[v 2 ]) Ψ = i [v 1(r i ) v 2 (r i )] Ψ = (E[v 1 ] E[v 2 ]) Ψ Eliminating Ψ from both sides of the last equation, we find [v 1 (r i ) v 2 (r i )] = E[v 1 ] E[v 2 ] i Identical ground-state wave functions have potentials that are identical up to a constant: Ψ v1 = Ψ v2 = v 1 (r) = v 2 (r) + c Different external potentials have different wave functions v 1 (r) v 2 (r) + c = Ψ v1 Ψ v2 Trygve Helgaker (CTCC, University of Oslo) The external potential 13th Sostrup Summer School (2014) 6 / 78

7 The Hohenberg Kohn theorem Consider two different potentials v 1 and v 2 with ground states Ψ 1 and Ψ 2 : v 1 (r) v 2 (r) + c = Ψ 1 Ψ 2, v i V N are the corresponding densities ρ 1 and ρ 2 also different? Invoking the Rayleigh Ritz variation principle for the two ground states, we obtain E[v 1 ] < Ψ 2 H[v 1 ] Ψ 2 = Ψ 2 H[v 2 ] Ψ 2 + (v 1 v 2 ρ 2 ) = E[v 2 ] + (v 1 v 2 ρ 2 ) E[v 2 ] < Ψ 1 H[v 2 ] Ψ 1 = Ψ 1 H[v 1 ] Ψ 1 (v 1 v 2 ρ 1 ) = E[v 1 ] (v 1 v 2 ρ 1 ) Adding the two inequalities, we arrive at the strict inequality E[v 1 ] + E[v 2 ] < E[v 1 ] + E[v 2 ] + (v 1 v 2 ρ 2 ρ 1 ) This result gives a contradiction unless the two densities are different v 1 (r) v 2 (r) + c = ρ 1 (r) ρ 2 (r) two different potentials thus cannot give rise to the same density The Hohenberg Kohn (HK) theorem Each N-electron density ρ is the ground-state density of at most one external potential v ρ + c, which is uniquely determined up to an additive constant c: ρ 1 (r) = ρ 2 (r) = v 1 (r) = v 2 (r) + c Trygve Helgaker (CTCC, University of Oslo) Hohenberg Kohn theory 13th Sostrup Summer School (2014) 7 / 78

8 The HK mapping between potentials and densities v-representable densities Each density ρ is the ground-state density of at most one external potential v ρ + c. those that arise from some potential v ρ are said to be v-representable those that do not are said to be non-v-representable HK theory thus sets up a mapping between the following two sets: v V N = { v H[v] has an N-electron ground state } ρ A N = { ρ ρ comes from an N-electron ground state } v 1 c Ρ 1 1, Ρ 1 2 v 2 c Ρ 2 1 the density determines the potential up to a constant conversely, the potential determines the density up to a degenerate set Trygve Helgaker (CTCC, University of Oslo) Hohenberg Kohn theory 13th Sostrup Summer School (2014) 8 / 78

9 Hohenberg Kohn functional and variation principle In HK theory, the potential and wave function are determined by the density ρ A N v ρ + c γψ ρ the additive constant c and the phase factor γ with γ γ = 1 are undetermined We now introduce the Hohenberg Kohn functional: F HK [ρ] = Ψ ρ T + W Ψ ρ = E[v ρ] (v ρ ρ), ρ A N it is independent of c and γ and unique also for degenerate systems From the Rayleigh Ritz variation principle, we obtain for arbitrary v V N the inequality F HK [ρ] + (v ρ) = Ψ ρ T + W Ψ ρ + (v ρ) = Ψ ρ H[v] Ψ ρ E[v] which may be written in two equivalent ways: E[v] F HK [ρ] + (v ρ), F HK [ρ] E[v] (v ρ) These inequalities may be sharpened into equalities, yielding two variation principles E[v] = min ρ A N ( FHK [ρ] + (v ρ) ), v V N, Hohenberg Kohn variation principle F HK [ρ] = max v V N ( E[v] (v ρ) ), ρ AN Lieb variation principle Note the duality between E and F HK Trygve Helgaker (CTCC, University of Oslo) Hohenberg Kohn theory 13th Sostrup Summer School (2014) 9 / 78

10 Hohenberg Kohn theory summarized The Hohenberg Kohn functional and the Hohenberg Kohn variation principle F HK [ρ] = Ψ ρ T + W Ψ ρ, E[v] = min ρ A N ( FHK [ρ] + (ρ v) ), ρ A N v V N We may obtain the ground-state energy by a variation over densities there is no need to involve the wave function! However, certain difficulties remain the explicit form of F HK [ρ] is unknown neither of the sets A N and V N are explicitly known Although the exact F HK [ρ] is unknown, good approximations are available we shall consider these approximation later We shall first see how A N and V N may be extended to explicitly known sets the Levy Lieb constrained-search functional the Lieb convex-conjugate functional Trygve Helgaker (CTCC, University of Oslo) Hohenberg Kohn theory 13th Sostrup Summer School (2014) 10 / 78

11 The variation principle for bound and unbound systems In HK theory, we restrict ourselves to electronic eigenstates H[v]Ψ v = E[v]Ψ v, v V N We now broaden our attention and consider the Rayleigh Ritz variation principle E[v] = inf Ψ T + W + Ψ i v(r i ) Ψ, v U VN here U is the set of all potentials such that the energy is finite note: this set is explicitly known: U = L 3/2 + L Since v V N is not assumed, there may not be a minimizing wave function (an eigenstate) we therefore determine an infimum (greatest lower bound) rather than a minimum Example: the oxygen atom has an electronic ground state only for N 9 E v, slope I1 20 E v,1 40 slope I2 50 E v,2 E v,3 E v,4 slope I3 slope I4 for N > 9, no ground state exists and the excess electrons are not bound the infimum is equal to the energy of O, with the excess electrons at rest infinitely far away Trygve Helgaker (CTCC, University of Oslo) Levy Lieb constrained-search theory 13th Sostrup Summer School (2014) 11 / 78

12 The Levy Lieb constrained-search functional We now perform the Rayleigh Ritz variation principle in two nested steps: E[v] = inf Ψ T + W + i v(r i ) Ψ Ψ = inf inf Ψ T + W + ρ i v(r i ) Ψ Ψ ρ [ = inf inf Ψ T + W Ψ + (ρ v)], v U ρ Ψ ρ an outer minimization over ρ and an inner minimization over Ψ ρ Introducing the Levy Lieb constrained-search functional, we obtain F LL [ρ] = inf Ψ T + W Ψ, Ψ ρ ρ I N ( E[v] = inf FLL [ρ] + (ρ v) ), ρ v U where we search over all N-representable densities: I N = {ρ(r) ρ can be obtained from some N-electron wave function Ψ} The set of N-representable densities is explicitly known: I N = {ρ(r) ρ(r) 0, ρ(r) dr = N, ρ 1/2 (r) 2 dr < } among all Ψ ρ, there is always a determinantal wave function Ψ det ρ the unknown set A N is dense in the known set I N Trygve Helgaker (CTCC, University of Oslo) Levy Lieb constrained-search theory 13th Sostrup Summer School (2014) 12 / 78

13 Hohenberg Kohn and Levy Lieb theories compared Hohenberg Kohn theory: F HK [ρ] = Ψ ρ T + W Ψ ρ, E[v] = min ρ A N ( FHK [ρ] + (ρ v) ), ρ A N v V N The Levy Lieb theory: F LL [ρ] = inf Ψ T + W Ψ, ρ Ψ ρ ( E[v] = inf FLL [ρ] + (ρ v) ), ρ I N I N v U We have avoided the unknown domain of Hohenberg Kohn theory search is over all reasonable densities for all reasonable potentials however, this theory is still not fully satisfactory We obtain the energy by a variational minimization of F LL [ρ] + (ρ v) the functional to be minimized should then be as simple as possible In particular, we would like it to at most one minimizer (except by degeneracy) this cannot be guaranteed for the Levy Lieb functional We now turn our attention to a density functional with a unique solution Trygve Helgaker (CTCC, University of Oslo) Levy Lieb constrained-search theory 13th Sostrup Summer School (2014) 13 / 78

14 Outlook: Lieb s convex-conjugate functional The Hohenberg Kohn functional is defined only for v-representable densities unfortunately, this domain is not explicitly known The Levy Lieb functional is a continuation to an explicitly known domain these are the N-representable densities unfortunately, it may give rise to several stationary points We shall now consider the Lieb functional it also has an explicitly known domain it provides unique solutions Lieb s approach requires some understanding of convex analysis we shall briefly review the theory of convex functions Particularly important is the Legendre Fenchel (LF) transform or convex conjugation we obtain Lieb s functional by convex conjugation of the energy the energy and density functional constitute a conjugate pair Trygve Helgaker (CTCC, University of Oslo) Levy Lieb constrained-search theory 13th Sostrup Summer School (2014) 14 / 78

15 Convex functions: the interpolation characterization A function is said to be convex if it satisfies the inequality λf (x 1 ) + (1 λ)f (x 2 ) f (λx 1 + (1 λ)x 2 ), 0 < λ < 1 a linear interpolation always overestimates a convex function f x 1 Λ f x 1 1 Λ f x 2 f Λx 1 1 Λ x 2 f x 2 x 1 Λx 1 1 Λ x 2 x 2 For a strictly convex function, we may replace by > above A function f (x) is concave if f (x) is convex Trygve Helgaker (CTCC, University of Oslo) Convex analysis 13th Sostrup Summer School (2014) 15 / 78

16 Convex functions: continuity and differentiability A convex function is continuous except possibly at the end points of its domain A convex function is not necessarily everywhere differentiable Trygve Helgaker (CTCC, University of Oslo) Convex analysis 13th Sostrup Summer School (2014) 16 / 78

17 Convex functions: supporting lines and stationary points Consider a function f : R R on the real axis a supporting line h to f touches the graph of f and is nowhere above it 1 f 1 1 f A convex function f has a set of supporting lines everywhere in the interior of its domain the slope of a supporting line at (x, f (x)) is called a subgradient of f at x The condition for a (global) minimum at x is the existence of zero subgradient at x all local minima are global minima Trygve Helgaker (CTCC, University of Oslo) Convex analysis 13th Sostrup Summer School (2014) 17 / 78

18 Convex functions constructed from supporting lines I A graph of a convex function f : R R has supporting lines for all x h y (x) = xy g(y) line of slope y and intercept g(y) with the ordinate axis Consider the piecewise linear function f (x) plotted below 4 3 g 1 2 f x h 1 x x 1 g g 3 2 h 3 x 3x h 2 x 2x 2 At each x, the function f (x) is equal to the largest supporting linear function h y (x) f (x) = max y h y (x) = max [xy g(y)] y Note: the convex f is completely described by g, its conjugate function Trygve Helgaker (CTCC, University of Oslo) Convex analysis 13th Sostrup Summer School (2014) 18 / 78

19 Convex functions constructed from supporting lines II A function f : R R is convex if and only if it can be written in the form f (x) = sup [xy g(y)] pointwise supremum of all supporting lines y The plots below illustrate this construction for x 2, x + x 4, x + x 2 and exp(x) the supporting lines (not the functions) have been plotted at intervals of x x x x x exp x Trygve Helgaker (CTCC, University of Oslo) Convex analysis 13th Sostrup Summer School (2014) 19 / 78

20 Convex conjugation For a convex function f : R R, we obtain f (x) = sup [xy g(y)] = f (x) xy g(y) y g(y) xy f (x) = g(y) = sup [xy f (x)] x It can be shown that, for any convex f : R R, there is an associated f = g such that f (x) = sup [xy f (y)] f (y) = sup [xy f (x)] y x f is called the convex conjugate or Legendre Fenchel transform of f illustration: the exponential function and its conjugate y f 1 y 1 2 f Compare with the Hohenberg Kohn and Lieb variation principles: ( ) ( ) E[v] = min F [ρ] + (v ρ) F [ρ]= max E[v] (v ρ) ρ v E and F are convex conjugates: alternative representations of the energy Trygve Helgaker (CTCC, University of Oslo) Convex analysis 13th Sostrup Summer School (2014) 20 / 78

21 Conjugate functionals E[v] F [ρ] We shall later show that the ground-state energy E[v] is concave in v it can therefore be exactly represented by its convex conjugate F [ρ]: E[v] F [ρ] F Ρ sup v E v v Ρ F Ρ F Ρ Ρ min v max E v E v E v inf Ρ F Ρ v Ρ v Ρ v Ρ Both variation principles represent a Legendre Fenchel transformation the essential point is the concavity of E[v] rather than the details of the Schrödinger equation the density functional F [ρ] is more complicated that E[v] Trygve Helgaker (CTCC, University of Oslo) Convex analysis 13th Sostrup Summer School (2014) 21 / 78

22 Conjugate functions L(ẋ) H(p) If f is strictly convex and differentiable, f is called a Legendre transform Legendre transforms are ubiquitous in physics The Lagrangian of classical mechanics is convex in the velocity ẋ L(ẋ) = 1 2 mẋ2 V pot Its Legendre transform (convex conjugate) is the Hamiltonian: H(p) = L (p) = max ẋ the stationary condition identifies the momentum ( pẋ 1 2 mẋ2 + V pot ) p = mẋ ẋ = p m substituting ẋ = p/m into H(p), we obtain the Hamiltonian H(p) = p2 2m + Vpot The reciprocal relation (see next slide) is satisfied: L (ẋ) = mẋ = p & H (p) = p m = ẋ Legendre transforms are also used in thermodynamics Trygve Helgaker (CTCC, University of Oslo) Convex analysis 13th Sostrup Summer School (2014) 22 / 78

23 Fenchel s inequality and the reciprocal relations Conjugate functions are related by the conjugate variation principles: f (x) = sup y [xy f (y)] f (y) = sup [xy f (x)] x From these relations, Fenchel s inequality follows directly: f (x) + f (y) xy valid for all pairs (x, y) Assuming that the maxima can be achieved above, we obtain f (x y ) + f (y x ) = x y y x valid for conjugate pairs (x y, y x ) this may not be possible for all x or all y Differentiation with respect to x y and y x yields the reciprocal relations: f (x y ) = y x (f ) (y x ) = x y we have here assumed differentiability We conclude that the first derivatives of conjugate functions are inverse functions: (f ) 1 = (f ) this relationship holds in a wider sense in the more general case Trygve Helgaker (CTCC, University of Oslo) Convex analysis 13th Sostrup Summer School (2014) 23 / 78

24 Examples of convex conjugate functions { f (x) = 1 + x, f 1, y 0 (y) = +, y > 1 g(x) = { 1 + x 2, g 1 x (y) =, y 0 +, y > 1 f 2 2 f g 2 2 g Trygve Helgaker (CTCC, University of Oslo) Convex analysis 13th Sostrup Summer School (2014) 24 / 78

25 Biconjugation and the convex envelope The conjugate function f is well defined also when f is not convex: f (y) = sup (xy f (x)) x We can therefore always form the biconjugate function: f (x) = sup (xy f (y)) y note: f = f holds only when f is convex We have the following conjugation relationships f (x) f (y) f (x) f is the largest convex lower bound to f, known as its convex envelope: f f (arbitrary f ) f = f (convex f ) Trygve Helgaker (CTCC, University of Oslo) Convex analysis 13th Sostrup Summer School (2014) 25 / 78

26 DFT by convex conjugation: the Lieb functional Consider the variationally optimized ground-state electronic energy E[v] = inf Ψ Ψ H[v] Ψ Concavity of E[v] follows from the variation principle (0 < λ < 1) E[λv 1 + (1 λ)v 2 ] = inf Ψ ( λ Ψ H[v1 ] Ψ + (1 λ) Ψ H[v2 ] Ψ ) λinf Ψ Ψ H[v1 ] Ψ + (1 λ)infψ Ψ H[v2 ] Ψ = λe[v 1 ] + (1 λ)e[v 2 ] the convex functional E[v] therefore has a convex conjugate partner Applying the theory of convex conjugation (generalized to functionals), we obtain ) F [ρ] = sup v (E[v] (v ρ) the Lieb variation principle ) E[v] = inf ρ (F [ρ] + (v ρ) the Hohenberg Kohn variation principle These are alternative attempts at sharpening Fenchel s inequality into an equality F [ρ] E[v] (v ρ) E[v] F [ρ] + (v ρ) these attempts are only successful for ρ A N and v V N, respectively for other ρ and v, the transformations are nevertheless well defined F defined this way is the Lieb functional, which is the convex envelope of F LL F = F LL F LL Trygve Helgaker (CTCC, University of Oslo) Lieb s density functional 13th Sostrup Summer School (2014) 26 / 78

27 Lieb s conjugate density functional The ground-state energy may be represented in two alternative forms: { } F [ρ] = sup v E[v] (v ρ) energy as a functional of density { } E[v] = inf ρ F [ρ] + (v ρ) energy as a functional of potential analogous to the energy represented in terms of velocity L(ẋ) and momentum H(p) The potential v and the density ρ are conjugate variables they belong to dual vector spaces such that (v ρ) = v(r)ρ(r) dr is finite they satisfy the reciprocal relations (assuming well-defined derivatives) δe[v] δv(r) = ρ(r) determines v when calculating F [ρ] from E[v] δf [ρ] = v(r) determines ρ when calculating E[v] from F [ρ] δρ(r) since the functionals are either convex or concave, their solutions are unique Convex conjugation highlights the duality of ρ and v sometimes it is best to work with F [ρ], other times with E[v] DFT parameterizes F [ρ], molecular mechanics parameterizes E[v] Trygve Helgaker (CTCC, University of Oslo) Lieb s density functional 13th Sostrup Summer School (2014) 27 / 78

28 Conjugate functionals E[v] F [ρ] The ground-state energy E[v] is concave in v by the Rayleigh Ritz variation principle it can therefore be exactly represented by its convex conjugate F [ρ]: E[v] F [ρ] F Ρ sup v E v v Ρ F Ρ min Ρ v F Ρ Ρ min v max E v v Ρ E v max v Ρ E v inf Ρ F Ρ v Ρ v Ρ Each variation principle represents a Legendre Fenchel transformation the essential point is the concavity of E[v] rather than the details of the Schrödinger equation we may therefore construct F [v] also for approximate E[v] if these are concave Trygve Helgaker (CTCC, University of Oslo) Lieb s density functional 13th Sostrup Summer School (2014) 28 / 78

29 The Hohenberg Kohn theorem and concavity I The concavity of the energy E [v ] follows from two circumstances: I I the linearity of H[v ] changes the energy linearly from v0 to v1 for fixed Ψ0 the variation principle lowers the energy concavely from Ψ0 to Ψ1 for fixed v1 E E XY1 ÈH@v1 DÈY1 \ = XY0 ÈH@v1 DÈY0 \ XY0 ÈH@v1 DÈY0 \ XY1 ÈH@v1 DÈY1 \ XY0 ÈH@v0 DÈY0 \ XY0 ÈH@v0 DÈY0 \ v0 v1 v v0 v1 v I There are two cases to consider: strict concavity (left) and non-strict concavity (right) I [H[v0 ], H[v1 ]] 6= 0 Ψ1 6= Ψ0 ρ1 6= ρ0 strict concavity [H[v0 ], H[v1 ]] = 0 Ψ1 = Ψ0 ρ1 = ρ0 non-strict concavity note: the density is given by the derivative of the blue curve I We have strict concavity and different densities except if v1 v0 = c is a scalar I I the Hohenberg Kohn theorem: the density determines the potential up to a constant with vector potentials, non-strict concavity occurs more generally Trygve Helgaker (CTCC, University of Oslo) Lieb s density functional 13th Sostrup Summer School (2014) 29 / 78

30 The concave envelope E[v] F [ρ] co E[v] Assume now that E[v] is not concave (not variationally minimized) it still generates a convex F [ρ], conjugate to the concave envelope co E[v] E[v] F Ρ sup v E v v Ρ F Ρ Ρ min v max co E v E v v Ρ E v inf Ρ F Ρ v Ρ v Ρ The concave envelope co E[v] is the least concave upper bound to E[v] excited-state energies are in general not concave approximate electronic ground-state energies are in general not concave Trygve Helgaker (CTCC, University of Oslo) Lieb s density functional 13th Sostrup Summer School (2014) 30 / 78

31 Vector spaces of wave functions, densities and potentials Banach space a Banach space is a complete normed vector space an important special case: Lebesgue space of p-integrable functions L p f p = ( f (r) p dr ) 1/p < + norm in L p Hilbert space a Hilbert space is a Banach space equipped with an inner product f g an important special case: L 2 with norm f = f f To what spaces do wave functions, densities and potentials belong? wave functions belong to the Hilbert space Ψ L 2 (more precisely the Sobolev space H 1 ) densities belong to the Banach space ρ X = L 3 L 1 potentials belong to the dual Banach space v X = L 3/2 + L the duality of X and X guarantees finite interactions (v ρ) < + Trygve Helgaker (CTCC, University of Oslo) Lieb s density functional 13th Sostrup Summer School (2014) 31 / 78

32 The dual spaces of densities and potentials We first identify the Banach space densities, next the space of potentials. The Banach space of densities X The density must be integrable ρ(r) dr = N as a consequence, we require ρ L 1 Next, we require a finite kinetic energy it can be shown that this gives ρ 1/2 L 2 Both conditions are satisfied by requiring ρ X = L 3 L 1 this follows by the Sobolev inequality The Banach space of potentials X We identify the space of potentials by requiring a finite interaction (ρ v). this leads us to the dual space X of X We conclude that v X = L 3/2 + L Finally, we confirm that X contains all Coulomb potentials. Trygve Helgaker (CTCC, University of Oslo) Lieb s density functional 13th Sostrup Summer School (2014) 32 / 78

33 A comparison of all density functionals We have introduced three universal density functionals F HK [ρ] = Ψ ρ T + W Ψ ρ, ρ A N F LL [ρ] = inf Ψ ρ Ψ T + W Ψ, ρ I N { } F [ρ] = sup v X E[v] (v ρ), ρ X = L 3 L 1 These functionals give the same results for ground-state densities: F [ρ] = F LL [ρ] = F HK [ρ], ρ A N Only the Levy Lieb and Lieb functionals are defined for other densities F [ρ] = F LL [ρ] F LL[ρ], ρ I N the Lieb functional is the convex envelop to the Levy Lieb functional the set A N is dense in I N The Lieb functional is the preferred functional it is convex, yielding at most one minimum it is defined on a whole linear space of densities it reflects the fundamental symmetry between densities and potentials Trygve Helgaker (CTCC, University of Oslo) Lieb s density functional 13th Sostrup Summer School (2014) 33 / 78

34 Review and preview We have done the following: proved the Hohenberg Kohn theorem introduced the Hohenberg Kohn functional established the Hohenberg Kohn variation principle introduced the Levy Lieb constrained-search functional introduced the Lieb convex-conjugate functional { } F [ρ] = sup v E[v] (v ρ) energy as a functional of density { } E[v] = inf ρ F [ρ] + (v ρ) energy as a functional of potential We shall now discuss the following topics: ensemble-state DFT the uniform electron gas Thomas Fermi theory the noninteracting reference system and Kohn Sham theory the exchange correlation functional and potential review standard approximations Trygve Helgaker (CTCC, University of Oslo) Overview 13th Sostrup Summer School (2014) 34 / 78

35 Concavity of the ground-state energy and the united atom Consider the variationally optimized ground-state electronic energy E[v] = inf Ψ Ψ H[v] Ψ Concavity of E[v] follows from the variation principle (0 < λ < 1) E[λv 1 + (1 λ)v 2 ] = inf Ψ Ψ H[λv 1 + (1 λ)v 2 ] Ψ = inf Ψ ( λ Ψ H[v 1 ] Ψ + (1 λ) Ψ H[v2 ] Ψ ) λinf Ψ1 Ψ1 H[v 1 ] Ψ 1 + (1 λ)infψ2 Ψ2 H[v 2 ] Ψ 2 = λe[v 1 ] + (1 λ)e[v 2 ] Consider now the following diatomic potential: v mol (r) = Z A Z ( B = λ Z A + Z B r A r B r A where λ = ) + (1 λ) Z A Z A +Z B and 1 λ = 1 Z A Z A +Z B = Z B Z A +Z B. Since 0 < λ < 1, we obtain from the concavity of the energy ( Z ) A + Z B = λv A (r) + (1 λ)v B (r) r B E [v mol ] λe [v A ] + (1 λ)e [v B ] = λe [v A ] + (1 λ)e [v A ] = E [v A ] the energy of the molecule is an upper bound to the energy of the united atom Conclusion: without nuclear nuclear repulsion, all molecules would collapse into atoms Trygve Helgaker (CTCC, University of Oslo) Overview 13th Sostrup Summer School (2014) 35 / 78

36 Energies of the H 2 molecule 2 1 V nn E 0 v V nn E 0 v 2 Trygve Helgaker (CTCC, University of Oslo) Overview 13th Sostrup Summer School (2014) 36 / 78

37 Lieb s theory for N-particle systems The N-electron ground-state energy E N [v] is concave in v: E N [v] = inf Ψ H[v] Ψ minimization over N-electron wave functions Ψ It has a conjugate partner E N [v] F N [ρ] related by Fenchel s inequality: F N [ρ] E N [v] (v ρ) E N [v] F N [ρ] + (v ρ) F N Ρ sup v E N v v Ρ F N Ρ F N Ρ min Ρ v F N Ρ Ρ min v max E N v v Ρ E N v max v Ρ v Ρ Trygve Helgaker (CTCC, University of Oslo) Ensemble DFT 13th Sostrup Summer School (2014) 37 / 78

38 The N-dependence of F N [ρ] Consider the N-electron ground-state energy E N [v + c] where c is a scalar: E N [v + c] = Ψ T + W Ψ + (v + c ρ Ψ ) = Ψ T + W Ψ + (v ρ Ψ ) + Nc = E N [v] + cn gauge transformation Consider next the N-electron Lieb functional F N [ρ] with such scalar variations: F N [ρ] = sup v (E N [v] (v ρ)) = sup (E N [v + c] (v + c ρ)) v,c = sup (E N [v] + cn (v ρ) (c ρ)) v,c = sup (E N [v] (v ρ) + c(n N ρ)) v,c where N ρ is the number of electrons in the density ρ: N ρ = ρ(r) dr When ρ does not contain N electrons, the Lieb functional becomes infinite: N N ρ = F N [ρ] = + apparently, we must construct a separate F N [ρ] for each particle number we need a single density functional valid for all particle numbers N 0 we first need to consider the N-dependence of E N [v] Trygve Helgaker (CTCC, University of Oslo) Ensemble DFT 13th Sostrup Summer School (2014) 38 / 78

39 Ensemble states For a more general DFT, we consider ensemble states we may then lift the restriction on the number of particles we may then also consider nonintegral particle numbers To describe such states, we introduce the ensemble density matrix ˆγ = p in Ψ in Ψ in, in p in 0, in = 1 in this is a convex combination: all coefficients are positive and add up to one it represents a mixed state: a statistical average of two or more pure states Ψ in the coefficients p in play the role of probabilities or weights Here p in is the probability that the system is found in state Ψ in satisfying Ĥ[v] Ψ in = E in [v]ψ in eigenfunction of the Hamiltonian ˆN Ψ in = NΨ in eigenfunction of the number operator Mixed states are useful when there is a classical uncertainty in the description systems in thermal equilibrium systems with an uncertain preparation history In DFT, we cannot distinguish a pure-state density from a mixed-state density an ensemble description is the natural framework for DFT our goal is to construct a universal density F [ρ] valid for all N Trygve Helgaker (CTCC, University of Oslo) Ensemble DFT 13th Sostrup Summer School (2014) 39 / 78

40 The ground-state energy We restrict ourselves to ensembles of (nondegenerate) ground-state density matrices: ˆγ = p K Ψ K Ψ K, p K 0, p K = 1, ĤΨ K = E K Ψ K K K To evaluate expectation values, we need a complete set of states (of all particle numbers): ΦI ΦI = 1, ΦI ΦJ = δij I Expectation values are expressed in terms of traces in this complete basis: tr ˆγĤ[v] = ΦI ˆγĤ[v] Φ I = ΦI p K Ψ K ΨK Ĥ[v] Φ I I I K = p K ΨK Ĥ[v] Φ I ΦI Ψ K = p K ΨK Ĥ[v] Ψ K = p K E K I K K K The N-electron ground-state energy is thus obtained by a constrained minimization of p K : E[v, N] = inf tr ˆγĤ[v] = inf p K E K, p K 0, p K = 1, p K K = N ˆγ N p K K K K The resulting energy E[v, N] is piecewise linear and convex in N see illustration on next slide Trygve Helgaker (CTCC, University of Oslo) Ensemble DFT 13th Sostrup Summer School (2014) 40 / 78

41 The energy as a function of the particle number The energy E[v, N] is piecewise linear and convex in N: E v,0 E v,1 E v,2 At integral N, the ensemble is in a pure state ˆγ[v, N] = Ψ N Ψ N E[v, N] = E N [v] At nonintegral N < N < N +, the ensemble is in a mixed state (0 < p < 1) ˆγ[v, N] = p ΨN ΨN + (1 p) ΨN+ ΨN+ E[v, N] = pe N [v] + (1 p)e N+ [v] We have here assumed that the pure states are nondegenerate Trygve Helgaker (CTCC, University of Oslo) Ensemble DFT 13th Sostrup Summer School (2014) 41 / 78

42 The universal density-functional F [ρ] As for pure states, we can now construct F [ρ, N] conjugate to E[v, N] F [ρ, N] = sup v { E[v, N] (v ρ) } energy as a functional of density E[v, N] = inf ρ { F [ρ, N] + (v ρ) } energy as a functional of potential As for F N [ρ], it can be shown that ρ(r) dr N F [ρ, N] = + We may therefore combine these functionals into a single functional as F [ρ] = inf F [ρ, N] = F [ρ, ρ(r)dr] N this is our universal density-functional, valid for all particle numbers the functional is convex in ρ, without restrictions on the particle number From F [ρ], we may now calculate energies for arbitrary N by the HK variation principle E[v, N] = inf {F [ρ] + (ρ v)} ρ I N the restriction to I N is essential since F [ρ] is now finite for all particle numbers The functional F [ρ] is the convex conjugate to the ensemble energy function G[v] = inf ˆγ tr ˆγĤ[v] Trygve Helgaker (CTCC, University of Oslo) Ensemble DFT 13th Sostrup Summer School (2014) 42 / 78

43 Ionization potentials The ionization potentials are differences in energy bewteen integral N: I N = E[v, N 1] E[v, N] E v, slope I1 20 E v,1 40 slope I2 50 E v,2 E v,3 E v,4 slope I3 slope I4 At nonintegral particle numbers, E[v, N] is differentiable with respect to N: de[v, N] dn = E[v, N +] E[v, N ] = I N+ ionization potential At integral particle numbers, E[v, N] has only one-sided derivatives: E +[v, N] = E[v, N + 1] E[v, N] = I N+1 E [v, N] = E[v, N] E[v, N 1] = I N Trygve Helgaker (CTCC, University of Oslo) Ensemble DFT 13th Sostrup Summer School (2014) 43 / 78

44 Derivative discontinuities for variations in N The ground-state energy for N particles and potential v is given by E[v, N] = inf ρ IN (F [ρ] + (ρ v)) piecewise linear between integral N Writing the N-electron ground-state density of v as ρ = Nρ 1, we obtain E[v, N] = F [Nρ 1 ] + N (ρ 1 v), ρ 1 (r) dr = 1 whose one-sided derivatives with respect to N are given by E ± [v, N] = F ± [Nρ 1] + (ρ 1 v) From the expressions on the previous slide, we obtain F + [Nρ 1] F [Nρ 1] = E + [v, N] E [v, N] = I N+1 ( I N ) We have now established the integer discontinuity of F [ρ] for variations in N: δf [ρ] δf [ρ] δρ(r) N+δ δρ(r) = N = I N I N+1 0, δ > 0 N δ We have learned that the N-conserving derivative of F [ρ] is the negative external potential: δf [ρ] = v(r) but v is only defined up to an additive constant for integral N (HK) δρ(r) Trygve Helgaker (CTCC, University of Oslo) Ensemble DFT 13th Sostrup Summer School (2014) 44 / 78

45 The chemical potential µ Since E[v, N] is convex in N, it has a conjugate function: E[v, N] = sup (G[v, µ] + µn) µ energy as a function of the particle number N G[v, µ] = inf (E[v, N] µn) energy as a function of the chemical potential µ N G[v, µ] is the energy in the potential v µ where µ is the chemical potential G[v, µ] = inf ˆγ tr ˆγĤ[v µ] energy optimized without constraints on N Below, these energies are plotted for the carbon and oxygen atoms: Differentiation (when well defined) yields the reciprocal relations: de[v, N] dn = µ, dg[v, µ] dµ = N Trygve Helgaker (CTCC, University of Oslo) Ensemble DFT 13th Sostrup Summer School (2014) 45 / 78

46 Review and preview We have done the following: proved the Hohenberg Kohn theorem introduced the Hohenberg Kohn functional established the Hohenberg Kohn variation principle introduced the Levy Lieb constrained-search functional introduced the Lieb convex-conjugate functional introduced ensemble-state DFT integer discontinuities The following topics remain: the uniform electron gas Thomas Fermi theory the noninteracting reference system and Kohn Sham theory the exchange correlation functional and potential review standard approximations Trygve Helgaker (CTCC, University of Oslo) Overview 13th Sostrup Summer School (2014) 46 / 78

47 The uniform electron gas (UEG) An important reference system for DFT is the uniform electron gas (UEG) we assume no interactions among the electrons we shall consider the interacting case (uniform electron liquid) later The Schrödinger equation for a free electron is given by ϕ(r) = E ϕ(r) We confine the electron to a cubic box of length l with boundary conditions ϕ(0, y, z) = ϕ(l, y, z) (and likewise in the y and z directions) It has the normalized quantized eigenfunctions and energies { l 3/2 exp(ik r) (inside the box) ϕ k (r) = 0 (outside the box), E k = k2 2, k = 2π l where n x, n y, n z = 0, ±1, ±2,... we consider the limit of large l and close-lying energy levels (nx, ny, nz ) We minimize the energy by doubly occupying orbitals up to the Fermi radius k F : ρ(r) = 2 ϕ k (r) 2 = 2 k k F k k F l 3 We have a uniform distribution of density in space, dependent only on k F Trygve Helgaker (CTCC, University of Oslo) The uniform electron gas 13th Sostrup Summer School (2014) 47 / 78

48 The UEG one-electron density and energy density For a large box, we may replace summation over states by integration: f k = f (k) dn = l3 8π 3 f (k) dk k where we have performed the variable substitution (k x, k y, k z ) = 2π l3 (nx, ny, nz ) dn = l 8π 3 dk The one-electron density and kinetic-energy density are then given by ρ = 2 1 l 3 = 1 4π 3 dk = 1 kf π 2 k 2 dk = k3 F k k 0 3π 2 F E = 2 k k F k2 2l 3 = 1 8π 3 k 2 dk = 1 2π 2 kf 0 k 4 dk = k5 F 10π 2 We may now express the kinetic-energy density in terms of the electron density as E = C F ρ 5/3, C F = 3 ( 3π 2 ) 2/ we shall later use this relation to construct a density functional for the kinetic energy Trygve Helgaker (CTCC, University of Oslo) The uniform electron gas 13th Sostrup Summer School (2014) 48 / 78

49 The UEG two-electron density The UEG gives also insight into the two-electron density The two-electron density is conveniently decomposed in the manner ρ(r 1, r 2 ) = ρ(r 1 )ρ(r 2 ) + ρ(r 1 )h xc(r 1 ; r 2 ) the first term represents an uncorrelated description the second term provides a correction for exchange and correlation Integrating r 2 over all space on both sides of this equation, we obtain ρ(r 1, r 2 ) dr 2 = (N 1)ρ(r 1 ) (N 1) ρ(r 1 ) = Nρ(r 1 ) + ρ(r 1 ) h x(r 1 ; r 2 )dr 2 We conclude that, for each r 1, h x(r 1, r 2 ) integrates to minus one: h xc(r 1 ; r 2 ) dr 2 = 1 we interpret h xc(r 1, r 2) as representing a missing electron (a hole) around r 1 it is called the exchange correlation hole For the noninteracting UEG, there is no correlation and only the exchange remains: ρ(r 1, r 2 ) = ρ(r 1 )ρ(r 2 ) + ρ(r 1 )h x(r 1 ; r 2 ) h x(r 1, r 2) is called the exchange hole or Fermi hole whereas the UEG one-electron density is uniform, the two-electron density is nonuniform Trygve Helgaker (CTCC, University of Oslo) The uniform electron gas 13th Sostrup Summer School (2014) 49 / 78

50 The Fermi hole For the UEG, the Fermi hole has the shape h x(r 1 ; r 2 ) = k3 F γ2 (k F r 12 ) 3 sin x 3x cos x 6π 2, γ(x) = x 3 The Fermi hole (to the left) and its radial distribution (to the right) plotted for k F = 3.09, corresponding to an electron density of one Some properties of the Fermi hole: it is nonpositive and depends only on r 12 it decays in an oscillatory manner as r 4 12 with period π/k F it is 1/2 at r 12 = 0 since only electrons of the same spin contribute Trygve Helgaker (CTCC, University of Oslo) The uniform electron gas 13th Sostrup Summer School (2014) 50 / 78

51 The Thomas Fermi (TF) model Let us now consider an explicit density functional for atoms and molecules atoms and molecules are inhomogeneous systems point-charge potentials rather than a spatially uniform charge distribution Independently, Thomas and Fermi (1927) suggested the following model F TF [ρ] = T TF [ρ] + J[ρ] it precedes the Hartree Fock model by three years The TF universal density functional contains two terms: 1 the Thomas Fermi kinetic energy T TF [ρ] = C F ρ 5/3 (r)dr, C F = 3 ( 10 3π 2 ) 2/ it is obtained by applying locally the UEG expression for kinetic energy this is the principle underlying the local density approximation 2 the Hartree energy J[ρ] = 1 ρ(r1 )ρ(r 2 ) dr 2 1 dr 2 r 12 it is obtained by ignoring the exchange correlation hole in the two-electron density ρ(r 1, r 2) = ρ(r 1)ρ(r 2) + ρ(r 1)h xc(r 1; r 2) ρ(r 1)ρ(r 2) it represents the classical interaction of the density with itself Trygve Helgaker (CTCC, University of Oslo) Thomas Fermi theory 13th Sostrup Summer School (2014) 51 / 78

52 The TF neutral atom Use of the TF density functional in the Hohenberg Kohn variation principle ( ) E TF (Z) = min T TF [ρ] + J[ρ] Z ρ(r)r 1 dr ρ N gives the following stationary condition for the neutral atom: 5 ρ(r 3 C Fρ 2/3 ) (r) + r r Z = µ µ adjusted to give Z electrons r the density becomes infinite at the nucleus, as r 0 the density decays as a power law (not exponentially), as r Plots of the energy E TF [Z] (left) and the radial distribution function 4πr 2 ρ(r) (right) 0,0 N Nc ETF v,nc no derivative discontinuities (left); no shell structure (right) the total energy is given by Z 7/3 too low by 54% for H, 35% for He, and 10% 15% for heavy elements For molecules, there is no binding! (Teller nonbinding theorem) Trygve Helgaker (CTCC, University of Oslo) Thomas Fermi theory 13th Sostrup Summer School (2014) 52 / 78

53 The Thomas Fermi Dirac (TFD) model Thomas and Fermi constructed their model, unaware of exchange The uniform-gas Fermi-hole expression leads to Dirac s exchange formula K D [ρ] = C x ρ 4/3 (r)dr, C x = 3 ( ) 3 1/ π Bloch (1929), Dirac (1930) As a refinement of the TF model, we now include Dirac s functional F TFD [ρ] = T TF [ρ] + J[ρ] K D [ρ] The Thomas Fermi Dirac (TFD) model does not improve on the TF model energy is lowered further still no binding predicted The main error in TF theory is the poor description of kinetic energy the TF functional gives a too low kinetic energy Unlike the TF functional, the Dirac functional is still widely used Trygve Helgaker (CTCC, University of Oslo) Thomas Fermi theory 13th Sostrup Summer School (2014) 53 / 78

54 Scaling relations Consider the electron coordinate scaling: Ψ(r i ) Ψ α(r i ) = α 3N/2 Ψ(αr i ), Ψ α Ψ α = Ψ Ψ It may be shown that Ψ α T Ψ α = α 2 Ψ T Ψ Ψ α W Ψ α = α Ψ W Ψ The corresponding scaling of the electron density gives ρ(r) ρ α(r) = α 3 ρ(αr) Consider the integral: ρ λ α (r) dr = α 3λ ρ λ (αr) dr = α 3(λ 1) ρ λ (r) dr We then obtain scaling relations in agreement with the TFD model: ρα(r) dr = ρ(r) dr = N 5/3 T TF [ρ α] = C F ρ α (r) dr = α 2 T TF [ρ] 4/3 K[ρ α] = C x ρ α (r) dr = αk[ρ] Trygve Helgaker (CTCC, University of Oslo) Thomas Fermi theory 13th Sostrup Summer School (2014) 54 / 78

55 The local-density approximation (LDA) and beyond The TF model is the first example of a local-density approximation (LDA) 1 an inhomogeneous system is viewed as consisting of small boxes 2 in each box, we approximate the density and potential as constant 3 we then apply the results for a homogeneous system in each box separately 4 finally, we make the boxes infinitesimally small E[ρ] E LDA [ρ] = E hom (ρ(r)) dr for example, the TF functional is obtained from the UEG energy density as E UEG (ρ) = C F ρ 5/3 (r) T TF [ρ] C F ρ(r) 5/3 dr The LDA approximation is not always a good one this is particularly true for the kinetic energy As a next step, it is natural to utilize information about the density gradient this idea leads to the gradient-expansion approximation (GEA) to improve the description, a power-type expansion is attempted a large number of such functionals have been developed Trygve Helgaker (CTCC, University of Oslo) Thomas Fermi theory 13th Sostrup Summer School (2014) 55 / 78

56 The von Weizsäcker kinetic-energy functional As an improvement to the TF energy, von Weizsäcker suggested a gradient correction T W [ρ] = 1 ρ 1/2 (r) 2 dr > 0 von Weizsäcker kinetic energy 2 It gives the exact kinetic energy of a one-electron system The condition T W [ρ] < + is used in I N Let us calculate the total TF energy, with an adjustable parameter λ T TFλW [ρ] = T TF [ρ] + λt W [ρ] λ = 1 in von Weizsäcker s original work (rapidly varying densities) λ = 1/9 from gradient-expansion approach (slowly varying densities) λ = 1/5 from fitting to hydrogenic atoms Density is now much improved and binding is predicted finite density at the nucleus and exponential decay far from the nucleus atomic energies (all calculated from the HF density): TF TFD TFD 1 9 W TFD 1 W 5 HF exact Ne Ar Trygve Helgaker (CTCC, University of Oslo) Thomas Fermi theory 13th Sostrup Summer School (2014) 56 / 78

57 Review and preview We have done the following: proved the Hohenberg Kohn theorem introduced the Hohenberg Kohn functional established the Hohenberg Kohn variation principle introduced the Levy Lieb constrained-search functional introduced the Lieb convex-conjugate functional introduced ensemble-state DFT integer discontinuities the uniform electron gas Thomas Fermi theory The following topics remain: the noninteracting reference system and Kohn Sham theory the exchange correlation functional and potential review standard approximations Trygve Helgaker (CTCC, University of Oslo) Overview 13th Sostrup Summer School (2014) 57 / 78

58 The noninteracting kinetic-energy functional T s [ρ] For a fully interacting system, the Levy Lieb universal density functional is F LL [ρ] = inf Ψ T + W Ψ = T [ρ] + W [ρ] Ψ ρ the presence of W makes its evaluation a difficult manybody problem Let us examine a fictitious system of noninteracting electrons FLL 0 [ρ] = inf Ψ T Ψ = Ts[ρ] Ψ ρ The universal density-functional is now the noninteracting kinetic-energy functional T s[ρ] the Levy Lieb functional T s[ρ] can be evaluated exactly it gives much better kinetic energies than does the Thomas Fermi functional T TF [ρ] Expressed in terms of orbitals, the kinetic-energy functional becomes T s[ρ] = min n I φi φ I single-electron kinetic energy ni φ I 2 ρ I where the density is given by: ρ(r) = n I φ I (r) 2, 0 n I 1, φ I φ J = δ IJ, n I = N I I We have here allowed for fractional occupation numbers n I this is equivalent to an ensemble description (and use of the Lieb functional) Trygve Helgaker (CTCC, University of Oslo) Kohn Sham theory 13th Sostrup Summer School (2014) 58 / 78

59 The exchange correlation (XC) energy In the general case, the exact kinetic energy T [ρ] cannot be evaluated as T s[ρ] = min n I φi 1 I 2 2 φ I The noninteracting kinetic energy is a lower bound on the true kinetic energy: T [ρ] T s[ρ] 0 we may nevertheless use T s[ρ] as as a reasonable approximation to T [ρ] We now introduce the Kohn Sham decomposition of the universal density functional: F [ρ] = T [ρ] + W [ρ] = T s[ρ] + J[ρ] + E xc[ρ] We have here introduced the exchange correlation (xc) functional where E xc[ρ] = T [ρ] T s[ρ] + W [ρ] J[ρ] T [ρ] T s[ρ] 0 kinetic-energy correlation correction { two-electron exchange and correlation energy W [ρ] J[ρ] 0 self-interaction correction It can be shown that the exchange correlation energy is negative: E xc[ρ] 0 Trygve Helgaker (CTCC, University of Oslo) Kohn Sham theory 13th Sostrup Summer School (2014) 59 / 78

60 Kohn Sham theory In the Kohn Sham (KS) method (1965), the energy is calculated as ( ) E[v] = min T ρ s[ρ] + (v ρ) + J[ρ] + E xc[ρ] subject to the density constraints ρ(r) = I n I φ I (r) 2, 0 n I 1, φ I φ J = δ IJ, I n I = N This gives the one-electron Kohn Sham equations [ v KS (r) ] φ I (r) = ε I φ I (r), ε 1 ε 2 The noninteracting electrons move in an effective Kohn Sham potential v KS (r) it consists of the external, Coulomb and exchange correlation potentials: v KS (r) = v(r) + v J (r) + v xc(r), v J (r) = δj[ρ] δρ(r) = Orbitals are occupied following the Aufbau principle ρ(r ) r r dr, v xc(r) = δexc[ρ] δρ(r) the highest occupied molecular orbital (HOMO) may be fractionally occupied Trygve Helgaker (CTCC, University of Oslo) Kohn Sham theory 13th Sostrup Summer School (2014) 60 / 78

61 The fictitious noninteracting Kohn Sham system The KS energy may be viewed as obtained in a two-step optimization ( ) E[v] = min T ρ s[ρ] + (v ρ) + J[ρ] + E xc[ρ] T s[ρ] = min φi 1 φi φi ρ I an outer minimization over all densities an inner constrained-search minimization of the noninteracting kinetic energy the Levy Lieb constrained-search approach with only kinetic energy in the search The KS solution provides us with the exact density constructed from orbitals that minimize the noninteracting kinetic energy fictitious noninteracting electrons with the density of the physical system the KS wave function is a determinant constructed from φ I the KS wave function has the physical density and the least kinetic energy To apply Kohn Sham theory, we must provide a recipe for evaluating E xc[ρ] it is customary to divide this contribution into exchange and correlation: E xc[ρ] = E x[ρ] + E c[ρ] by far the largest contribution is usually exchange in Ne, exchange and correlation contribute and 0.39 hartree, respectively We must now devise methods for the evaluation of E x[ρ] and E c[ρ] the success of KS theory relies on our ability to evaluate these terms efficiently and accurately Trygve Helgaker (CTCC, University of Oslo) Kohn Sham theory 13th Sostrup Summer School (2014) 61 / 78

62 Derivative discontinuities in KS theory The total energy is a piecewise linear curve between integral particle numbers this implies a derivative discontinuity in F [ρ] at integral N: δf [ρ] δf [ρ] δρ(r) N+δ δρ(r) N δ there are no derivative discontinuities in TF theory = N = I N I N+1 0, δ > 0 In KS theory, we decompose the universal density functional in the manner F [ρ] = T s[ρ] + J[ρ] + E xc[ρ] a derivative discontinuity is introduced with T s[ρ]: δt s[ρ] δρ(r) N+δ δts[ρ] δρ(r) N δ = KS N the remaining discontinuity jump must be supplied by E xc[ρ]: δe xc[ρ] δρ(r) N+δ δexc[ρ] δρ(r) N δ = ε N+1 ε N = xc N = v + xc (r) v xc (r) The exchange correlation potential jumps by a constant amount at integral N this discontinuity cannot easily be achieved with continuum functionals instead they average over the two limits Trygve Helgaker (CTCC, University of Oslo) Kohn Sham theory 13th Sostrup Summer School (2014) 62 / 78

63 KS and HF theories compared Before discussing approximations to E xc[ρ], let us pause to compare KS and HF theories There are obvious similarities between the KS and HF theories in both methods, a set of orbitals are determined self-consistently both methods use an effective mean-field potential the costs of the methods are similar The HF orbitals give an approximate wave function they minimize the total energy of the determinant they provide an approximate density by design The KS orbitals give the exact density they minimize the noninteracting kinetic energy they provide an exact density in principle Since E xc[ρ] is unknown, the KS description is in practice approximate the best E xc[ρ] forms give results broadly similar to MP2, CCSD and even CCSD(T) results but, no simple systematic procedure for improving E xc[ρ] exists Trygve Helgaker (CTCC, University of Oslo) Kohn Sham theory 13th Sostrup Summer School (2014) 63 / 78

64 The X α method By far the largest contribution to E xc[ρ] is usually exchange in Ne, exchange and correlation contribute and 0.39 hartree, respectively In Slater s X α method, we ignore correlation and approximate exchange: E Xα [ρ] = T s[ρ] + (v ρ) + J[ρ] 3α 2 K D[ρ] with Dirac s (scaled) exchange functional: K D [ρ] = C x ρ 4/3 (r)dr, C x = 3 4 ( ) 3 1/ π The X α method was proposed by Slater in 1951, as an approximate HF theory HF exchange is expensive to evaluate compared with the Coulomb contribution the X α method was developed 14 years before KS theory was proposed in 1965 The parameter α may be treated as an adjustable parameter important special values α = 1 α = 2/3 original value KS-LDA without correlation The X α method may thus be viewed as a special case of KS-LDA The X α method differs from TFD theory only in the kinetic energy Trygve Helgaker (CTCC, University of Oslo) Standard KS models 13th Sostrup Summer School (2014) 64 / 78