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$: Moreau Yosida Regularised Density-Functional Theoryfolk.uio.no/helgaker/talks/Paris_MYDFT_2015.pdf · Moreau{Yosida Regularised Density-Functional Theory Simen Kvaal y, Ulf Ekstr$
Moreau–Yosida Regularised Density-Functional Theory

Simen Kvaal†, Ulf Ekstrom†, Andy Teale†‡, Trygve Helgaker†

†Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway‡School of Chemistry, University of Nottingham, Nottingham, UK

Laboratoire Jacques–Louis Lions,Universite Pierre et Marie Curie (UPMC) – Sorbonne Universites, Paris, France,

3 April 2015

T. Helgaker (CTCC, University of Oslo) Moreau–Yosida Regularised DFT UPMC, Paris, France, 3 April 2015 1 / 20

Hohenberg–Kohn variation principle

I In DFT, the ground-state energy for a given potential v and electron number N is given by

E(v) = infρ→N

(F (ρ) + (v |ρ)

)Hohenberg–Kohn (HK) variation principle

where ρ is the density and where

F (ρ) = infΨ 7→ρ

〈Ψ |T + W |Ψ〉 Levy constrained-search density functional

(v |ρ) =∫v(r)ρ(r) dr interaction of density with external potential

I The solution of the HK minimisation problem is given by the Euler–Lagrange equation

δF (ρ)

δρ(r)= −v(r)− µ Euler–Lagrange equation with chemical potential µ

I clearly, an underlying assumption is that F is differentiableI however, F is everywhere discontinuous and therefore nondifferentiable

I Outline:

I nondifferentiability of the universal density functional in DFTI Moreau–Yosida regularisation of DFT with differentiable density functionalI consequences for Kohn–Sham theory

I S. Kvaal, U. Ekstrom, A. Teale, and T. Helgaker, JCP 140, 18A518 (2014)


Discontinuity of universal density functional

I For a one-electron system, the universal density functional has a simple explicit form:

F (ρ) = 12

∫ ∣∣∇ρ1/2(r)∣∣2 dr ← one-electron kinetic energy

I A one-electron Gaussian density of unit exponent has a finite kinetic energy:

ρ(r) = π−3/2 exp(−r2

), F (ρ) = 3/4

I Let {ρn} be a sequence that approaches ρ in the norm,

limn→+∞

‖ρ− ρn‖p = 0,

while developing increasingly rapid oscillations of increasingly small amplitude:

I The kinetic energy F (ρn) is driven arbitrarily high in the sequence and F is not continuous:

limn

F (ρn) = +∞ 6= F(

limnρn)

= 3/4

I The universal density functional is everywhere discontinuous and hence nondifferentiable

I P. E. Lammert, Int. J. Quantum Chem. 107, 1943 (2007)


Lieb variation principle

I The nondifferentiability of F makes it awkward to work with

I we shall develop a more regular DFT with a differentiable density functionalI we must first examine the relationship between E and F more carefully

I Lieb showed that the energy and density functional are symmetrically related:

F (ρ) = supv

(E(v)− (v |ρ)

)the Lieb variation principle (1983)

E(v) = infρ

(F (ρ) + (v |ρ)

)the Hohenberg–Kohn variation principle (1964)

I These are alternative attempts at sharpening the same inequality into an equality

E(v) ≤ F (ρ) + (v |ρ) ⇐⇒ F (ρ) ≥ E(v)− (v |ρ)

I In the language of convex analysis, E and F are said to be conjugate functions

E(concave and semi-cont.) ←→ F (convex and semi-cont.)

I such conjugate functions contain the same information, represented in different mannersI each property of one function is exactly reflected in some property of its conjugate function

I Being a conjugate function, F is automatically convex and lower semi-continuous

I however, F is not continuous even though E is continuousI for this, even more is required of E

I Our strategy: to modify E so that F becomes continuous and differentiable


Concavity of the ground-state energyI The ground-state energy is concave in the external potential

E(v) = infΨ〈Ψ|H(v)|Ψ〉

v

E

A

B

C

⟨Ψ0|H(v0)|Ψ0⟩

v0<0 v1<0 v=0

⟨Ψ0|H(v1)|Ψ0⟩

⟨Ψ1|H(v1)|Ψ1⟩

I The concavity of E(v) may be understood in the following two-step manner:

1 from A to B, the energy increases linearly since H(v) is linear in v and Ψ0 is fixed2 from B to C, the energy decreases as the wave function relaxes to the ground state Ψ1


Concavity of the ground-state energyI The concavity of the energy E(v) follows from two circumstances:

I the linearity of H(v) changes the energy linearly from v0 to v1 for fixed Ψ0I the variation principle lowers the energy from Ψ0 to Ψ1 for fixed v1

v

E

XY0 ÈH@v0DÈY0\

v0 v1

XY0 ÈH@v1DÈY0\

XY1 ÈH@v1DÈY1\

v

E

XY0 ÈH@v0DÈY0\

XY1 ÈH@v1DÈY1\ = XY0 ÈH@v1DÈY0\

v0 v1

I There are two cases to consider: strict concavity (left) and nonstrict concavity (right)

[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

I 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 the Hohenberg–Kohn theorem: the density determines the potential up to a constantI with vector potentials, non-strict concavity occurs more generally


Convexity of the universal density functional

I Being conjugate to E , the density functional F is convex and lower semi-continuous

I convexity ensures that all local minima are global in the HK variation principleI lower semi-continuity allows functions to jump down but not up as limits are taken

I However, F is neither continuous nor differentiable

1 2

+∞


Moreau–Yosida regularisation of DFT

I The ground-state energy E(v) is concave but not strictly concave

concave strictly concave

I We obtain a strictly concave energy Eγ by subtracting a term proportional to ‖v‖2:

Eγ(v) = E(v)− 12γ‖v‖2, γ > 0

I such strict concavity is sufficient to guarantee continuity of its conjugate functionI caveat: v must be square integrable (Coulomb potential in a box)

I We now introduce the density functional Fγ in the usual manner, as the conjugate to Eγ :

Fγ(ρ) = supv

(Eγ(v)− (v |ρ)

)Lieb variation principle

Eγ(v) = infρ

(Fγ(ρ) + (v |ρ)

)Hohenberg–Kohn variation principle

I unlike F , the new density functional Fγ is continuous and differentiable

I This procedure is known as Moreau–Yosida (MY) regularisation

I a standard technique in convex optimisation


Moreau envelope Fγ of F

I We have defined Fγ as the conjugate of Eγ by the Lieb variation principle:

Fγ(ρ0)def= sup

v

(Eγ(v)− (v |ρ0)

)I We may express Fγ explicitly in terms of the original density functional F as

Fγ(ρ0) = minρ

(F (ρ) + 1

2γ‖ρ− ρ0‖2

)← Moreau envelope Fγ of F

I for each ρ0, we minimise F (ρ) with a penalty term ‖ρ− ρ0‖2/2γ

∞

F(ρ)

Fγ (ρ0)

1

2 γ∥ρ-ρ0∥

2

ρ0


Moreau envelope Fγ of F

I The regularised density functional Fγ is known as the Moreau envelope of F

I it is an everywhere finite, continuous and differentiable approximation to FI it is a lower bound to F but has the same minimumI it approaches F pointwise from below as γ → 0

Fγ(ρ) = infρ

(F (ρ) + 1

2γ‖ρ− ρ‖2

)I Moreau–Yosida regularisation with large γ (left), medium γ (middle), and small γ (right):

I BLUE curves: strictly concave Eγ (dashed) tends to non-strictly concave E from belowI RED curves: differentiable Fγ (dashed) tends to non-differentiable F from below


Moreau envelope illustration

-3 -2 -1 1 2 3

2

4


Hohenberg–Kohn variation principle in MY and standard DFT

Moreau–Yosida DFT

I Regularised Hohenberg–Kohn variation principle:

Eγ(v) = infρ

(Fγ(ρ) + (v |ρ)

)I The Euler–Lagrange equation is now well defined:

−v(r)− µ =δFγ(ργv )

δρ(r)⇐⇒ Eγ(v) = Fγ(ργv ) + (v |ργv )

Standard DFT

I Hohenberg–Kohn variation principle:

E(v) = infρ

(F (ρ) + (v |ρ)

)I Optimality conditions in terms of subdifferential of F :

−v(r)− µ ∈ ∂F (ρv ) ⇐⇒ E(v) = F (ρv ) + (v |ρv )

I The regularised and true energies are related in a trivial manner:

E(v) = Eγ(v) + 12γ‖v‖2


Moreau–Yosida quasi-densities

I Densities in MY DFT are not physical densities but quasi-densitiesI they may be negative in some region and need not integrate to the correct number of electronsI they are not obtained from a wave function in the usual way (like amplitudes in RPA)

I However, with each ργv , there is an associated physical density ρv :

ρv = arg minρ

(F (ρ) + 1

2γ‖ρ− ργv ‖2

)← proximal density

I ρv is the energetically most favoured nearby ground-state density to ργvI ρv minimises ρ 7→ ‖ρ− ργv ‖

2 over all physical densities ρ with penalty function 2γF (ρ)

I In fact, the proximal density ρv is a ground-state density with associated potential

v = 1γ

(ρv − ργv ) ← proximal potential

I In short, each Moreau–Yosida quasi-density ργv may be uniquely decomposed in the manner

ργv = ρv − γv ← “depot” = density–potential

where ρv is the ground-state density associated with the external potential v


Kohn–Sham theoryI Usually, we work within the Kohn–Sham formalism:

F (ρ) = Ts(ρ) + J(ρ) + Ex(ρ) + Ec(ρ)

I From Fλ(ρ) calculated at different interaction strengths 0 ≤ λ ≤ 1,

Fλ(ρ) = supv

(Eλ(v)− (v |ρ)

)we have access to all Kohn–Sham components:

Ts(ρ) = F0(ρ) ← noninteracting kinetic energy (zero order)

J(ρ) + Ex(ρ) = F ′0 (ρ) ← Coulomb and exchange energies (first order)

Ec(ρ) =∫ 1

0F ′λ(ρ) dλ− F ′0 (ρ) ← correlation energy (infinite order)

I Each point on the adiabatic-connection (AC) curve 0 ≤ λ ≤ 1 contributes to Ec(ρ) (H2):I the curve reflects wave-function adjustments to an increasing interaction λ for a fixed densityI correlation contribution increases with λ but flattens out because of the fixed-density constraint

0.0 0.2 0.4 0.6 0.8 1.0

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

1.4 bohr

5 bohr

10 bohr


MY regularised Kohn–Sham theory

I KS theory assumes a noninteracting system with the same density as the physical system

I this assumption is unfounded: the noninteracting representability problem

I In Moreau–Yosida DFT, the two systems share the same quasi-density:

1 the interacting Euler–Lagrange equation determines the quasi-density ργv :

δFγ(ργv )

ργv (r)= −v(r)− µ interacting system with differentiable Fγ

2 the noninteracting Euler–Lagrange equation determines the Kohn–Sham potential vγs :

δ(Ts)γ(ργv )

ργv (r)= −vγ

s (r)− µs noninteracting system with differentiable (Ts)γ

I A regularised Kohn–Sham potential vγs exists for each external potential v :

I same quasi-density but different physical densities:

ρv = ργv + γv interacting density and potential

ργs = ργv + γvγs noninteracting density and potential

I the physical densities differ by an amount proportional to γ


Different metrics in MY DFT

I It is possible to work with a more general MY regularisation, with a weighted penalty:

Eγ(v) = E(v)− 12γ〈v |M|v〉 ← energy with damped potential

Fγ(ρ) = minρ

(F (ρ) + 1

2γ 〈ρ− ρ|M−1|ρ− ρ〉

)← energy of favoured nearby density

I two choices of metric: M = I (overlap) and M = T (kinetic)

I The kinetic metric gives the regularisation of Heaton–Burgess, Bulat and Yang (2007)

〈v |T|v〉 =∫|∇v(r)|2 dr ← small for smooth v

〈∆ρ|T−1|∆ρ〉 =∫∫

∆ρ(r1)r−112 ∆ρ(r2) dr1dr2 ← small for oscillatory ∆ρ = ρ− ρ

I In Lieb variation principle, potential becomes smoother as density oscillations are removed:

Fγ(ρ) = supv

(Eγ(v)− (v |ρ)

)I Regularised optimised effective potential (OEP) at zero interaction strength (Wu–Yang)


Conclusions

I Universal density functional of DFT is an exceedingly complicated function of the density

I it is discontinuous and nondifferentiableI ill-defined Euler–Lagrange equations and Kohn–Sham representability problem

I Moreau–Yosida regularisation gives a regular but exact DFT

I we can build arbitrarily accurate differentiable universal density functionalsI well-defined Euler–Lagrange equations and no Kohn–Sham representability problem

I Publication:

I S. Kvaal, U. Ekstrom, A. Teale, and T. Helgaker, JCP 140, 18A518 (2014)

I Acknowledgements:

I Ulf Ekstrom, Simen Kvaal, and Andy TealeI ERC advanced grant ABACUSI Norwegian Research Council for Centre of Excellence CTCC


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