Introduction to density-functional theory Introduction to density-functional theory First...

Introduction to density-functional theory


Emmanuel Fromager

Institut de Chimie de Strasbourg - Laboratoire de Chimie Quantique -Université de Strasbourg /CNRS

M2 lecture, Strasbourg, France.

Institut de Chimie, Strasbourg, France


N -electron Schrödinger equation for the ground state

HΨ0 = E0Ψ0

where Ψ0 ≡ Ψ0(x1,x2, . . . ,xN ), xi ≡ (ri, σi) ≡ (xi, yi, zi, σi = ± 12

) for i = 1, 2, . . . , N,

and H = T + Wee + V .

T ≡ −1

2

N∑i=1

∇2ri

= −1

2

N∑i=1

(∂2

∂x2i

+∂2

∂y2i

+∂2

∂z2i

)−→ universal kinetic energy operator

Wee ≡N∑i<j

1

|ri − rj |× −→ universal two-electron repulsion operator

V ≡N∑i=1

v(ri)× where v(r) = −nuclei∑A

ZA

|r−RA|−→ local nuclear potential operator



The Nobel Prize in Chemistry 1998

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Walter Kohn, John Pople

John Pople - Facts

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John A. Pople

Born: 31 October 1925, Burnham-on-Sea, United Kingdom

Died: 15 March 2004, Chicago, IL, USA

Affiliation at the time of the award: Northwestern University, Evanston, IL, USA

Prize motivation: "for his development of computational methods in quantum chemistry"

Field: theoretical chemistry

Prize share: 1/2

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John Pople - Facts http://www.nobelprize.org/nobel_prizes/chemistry/laureates/...

1 of 2 16/05/14 17:48Institut de Chimie, Strasbourg, France


(Fictitious) non-interacting electrons• Solving the Schrödinger equation for non-interacting electrons is easy.

• You “just" have to solve the Schrödinger equation for a single electron.

(T +

N∑i=1

v(ri)×)

Φ0 = E0Φ0 ⇔[−

1

2∇2

r + v(r)×]ϕi(x) = εiϕi(x), i = 1, 2, . . . , N.

Proof: a simple solution to the N -electron non-interacting Schrödinger equation is

Φ0 ≡ ϕ1(x1)× ϕ2(x2)× . . .× ϕN (xN ) =

N∏j=1

ϕj(xj) ← Hartree product !

since(T +

N∑i=1

v(ri)×)

Φ0 =N∑i=1

N∏j 6=i

ϕj(xj)×[−

1

2∇2

ri+ v(ri)×

]ϕi(xi) =

(N∑i=1

εi

)Φ0.



(Real) interacting many-electron problem

• Before addressing the true (interacting) problem we should keep in mind that electrons are fermions.

• Consequently, they should be described by Slater determinants instead of Hartree products.

• Therefore, in the particular case of two electrons, we have

ϕ1(x1)ϕ2(x2) −→ Φ0 ≡1√

2

∣∣∣∣∣∣ ϕ1(x1) ϕ1(x2)

ϕ2(x1) ϕ2(x2)

∣∣∣∣∣∣ =1√

2

[ϕ1(x1)ϕ2(x2)− ϕ1(x2)ϕ2(x1)

].

• When computing the two-electron repulsion energy⟨

Φ0

∣∣∣Wee

∣∣∣Φ0

⟩we then describe both Hartree

and exchange energies.

• Finally, Φ0 cannot be the exact solution to the interacting Schrödinger equation (whatever choice ismade for the spin-orbitals).

• The energy contribution that is missing is referred to as correlation energy.



Mapping the interacting problem onto a non-interacting one

• Is it possible to extract the exact (interacting) ground-state energy from a non-interacting system ?

• If yes, then it would lead to a huge simplification of the problem.

• Nevertheless, the question sounds a bit weird since the two-electron repulsion is completely ignoredin a non-interacting system.

• One way to establish a connection between interacting and non-interacting worlds is to use theelectron density as basic variable (instead of the wavefunction).

• Electron density for a non-interacting system: nΦ0(r) =

∑σ=± 1

2

N∑i=1

|ϕi(r, σ)|2

• Electron density for an interacting system:

nΨ0(r) = N

∑σ=± 1

2

∫dx2 . . .

∫dxN |Ψ0(r, σ,x2, . . . ,xN )|2



Mapping the interacting problem onto a non-interacting one

• There is of course no reason to believe that these two densities are equal.

• However, we may assume that it is possible to adjust the local potential in the non-interacting systemsuch that the two densities become equal.

• This “magical" potential is known as the Kohn–Sham (KS) potential.

• In summary:

interacting problem → non-interacting KS problem

Wee → 0

v(r) → vKS(r)

nΨ0 (r) = nΦKS0

(r)

• Questions to be answered:

(1) If vKS(r) exists, is it unique ? yes!

(2) Does the knowledge of nΨ0(r) gives access (in principle) to E0 ? yes!




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I UNDERSTAND

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Walter Kohn, John Pople

Walter Kohn - Facts

WorkThe structures of molecules and the way they react with one another depends on the movement of electrons and their

distribution in space, which is determined by the laws of quantum mechanics. However, quantum mechanics requires

very complicated calculations for complex systems such as molecules. In 1964 Walter Kohn laid the foundation for a

theory that stated it was not necessary to account for every electron's movement. Instead, one could look at the

average density of electrons in the space. This presented new opportunities for calculations involving chemical

structures and reactions.

Walter Kohn

Born: 9 March 1923, Vienna, Austria

Died: 19 April 2016, Santa Barbara, CA, USA

Affiliation at the time of the award: University of California, Santa Barbara, CA, USA

Prize motivation: "for his development of the density-functional theory"

Field: theoretical chemistry

Prize share: 1/2

Walter Kohn - Facts https://www.nobelprize.org/nobel_prizes/chemistry/laureates/...

1 of 2 23/02/2017, 10:24



Two things to remember before we start ...

• The following expression for the expectation value of the one-electron potential energy in terms of theelectron density will be used intensively in the rest of this lecture:

⟨Ψ

∣∣∣∣∣N∑i=1

v(ri)×

∣∣∣∣∣Ψ⟩

=

∫R3

dr v(r)nΨ(r) = (v|nΨ)

• Note that a constant shift v(r)→ v(r)− µ in the local potential does not affect the ground-statewavefunction (and therefore it does not affect the ground-state density):

(T + Wee +

N∑i=1

(v(ri)− µ

)×)

Ψ0 =(HΨ0

)−Nµ×Ψ0 =

(E0−Nµ

)Ψ0.



First Hohenberg–Kohn theorem

4

¥

.run ,i "

jet ) wavefunctionsno

"C I )

v"

cryno "ci ,r' " Ci ) .

'

i.

i

Potentials that differ Ground -state

by more than a densitiesConstant shift




• Note that v → Ψ0 → E0

→ n0 = nΨ0

• HK1: Hohenberg and Kohn∗ have shown that, in fact, the ground-state electron density fullydetermines (up to a constant) the local potential v. Therefore

n0 → v → Ψ0 → E0

• In other words, the ground-state energy is a functional of the ground-state density: E0 = E[n0].

Proof (part 1):

Let us consider two potentials v and v′ that differ by more than a constant, which means that v(r)− v′(r)

varies with r. In the following, we denote Ψ0 and Ψ′0 the associated ground-state wavefunctions withenergies E0 and E′0, respectively.

∗P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).




If Ψ0 = Ψ′0 then

N∑i=1

(v(ri)− v′(ri)

)×Ψ0 =

N∑i=1

v(ri)×Ψ0 − v′(ri)×Ψ′0

=

(T + Wee +

N∑i=1

v(ri)×)

Ψ0 −(T + Wee +

N∑i=1

v′(ri)×)

Ψ′0

= E0Ψ0 − E′0Ψ′0

=(E0 − E′0

)×Ψ0

so that, in the particular case r1 = r2 = . . . = rN = r, we obtain

v(r)− v′(r) =(E0 − E′0

)/N −→ constant (absurd !)

Therefore Ψ0 and Ψ′0 cannot be equal.




Proof (part 2): Let us now assume that Ψ0 and Ψ′0 have the same electron density n0.

According to the Rayleigh–Ritz variational principle

E0 <

⟨Ψ′0

∣∣∣∣∣T + Wee +

N∑i=1

v(ri)×

∣∣∣∣∣Ψ′0⟩

︸︷︷︸ and E′0 <

⟨Ψ0

∣∣∣∣∣T + Wee +

N∑i=1

v′(ri)×

∣∣∣∣∣Ψ0

⟩︸︷︷︸

E′0 + (v − v′|n0) E0 − (v − v′|n0)

thus leading to

0 < E0 − E′0 − (v − v′|n0) < 0 absurd !

∗P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).



Second Hohenberg–Kohn theorem

HK2: The exact ground-state density n0(r) of the electronic Hamiltonian

H[vne] ≡ T + Wee +N∑i=1

vne(ri)×

minimizes the energy density functional E[n] = F [n] +

∫R3dr vne(r)n(r),

where the Hohenberg–Kohn universal functional F [n] is defined as

F [n] = 〈Ψ[n]|T + Wee|Ψ[n]〉,

and the minimum equals the exact ground-state energy E0:

minnE[n] = E[n0] = E0

Comment: we know from HK1 that n(r) → v[n](r) → Ψ[v[n]] = Ψ[n]︸︷︷︸ground-state wavefunction with density n.



Second Hohenberg–Kohn theorem

Proof:

• for any density n(r), Ψ[n] is well defined according to HK1 and

〈Ψ[n]|H[vne]|Ψ[n]〉 ≥ E0

〈Ψ[n]|T + Wee|Ψ[n]〉︸︷︷︸+

∫R3dr vne(r)nΨ[n](r)︸︷︷︸ ≥ E0

F [n] n(r)

thus leading to E[n] ≥ E0

• When n(r) equals the exact ground-state density n0(r):

n0(r) → vne(r) → Ψ[n0] = Ψ[vne] = Ψ0

E[n0] = 〈Ψ0|T + Wee|Ψ0〉+

∫R3dr vne(r)n0(r) = 〈Ψ0|T + Wee + Vne|Ψ0〉 = E0



Kohn–Sham DFT (KS-DFT)• The HK theorems apply to non-interacting electrons:

interacting problem → non-interacting KS problem

Wee → 0

v[n](r) → vKS[n](r)

Ψ[n] → ΦKS[n]

F [n] → Ts[n] =⟨

ΦKS[n]∣∣∣T ∣∣∣ΦKS[n]

⟩nΨ[n](r) = nΦKS[n](r) = n(r)

• KS decomposition of the universal HK functional:

F [n] = Ts[n] + EHxc[n] with EHxc[n] =1

2

∫R3

∫R3

drdr′n(r)n(r′)|r− r′|

+ Exc[n].

W. Kohn and L. J. Sham, Phys. Rev. A 140, 1133 (1965).



Levy–Lieb constrained search formalism

• For a given density n there is a unique potential vKS[n](r), if it exists ..., such that ΦKS[n] is the

ground state of T +

N∑i=1

vKS[n](ri)× with density n.

• For all normalized wavefunctions Ψ with density n the following inequality is fulfilled:⟨ΦKS[n]

∣∣∣∣∣(T +

N∑i=1

vKS[n](ri)×)∣∣∣∣∣ΦKS[n]

⟩≤⟨

Ψ

∣∣∣∣∣(T +

N∑i=1

vKS[n](ri)×)∣∣∣∣∣Ψ

⟩

−→ Ts[n] ≤ 〈Ψ|T |Ψ〉 −→ Ts[n] = minΨ→n

〈Ψ|T |Ψ〉

• Therefore 〈ΦKS|T |ΦKS〉 = Ts[n0] = minΨ→n0

〈Ψ|T |Ψ〉 but we do not know n0 ...

• Note that, as a consequence of the previous equality, Ts[n0] ≤ 〈Ψ0|T |Ψ0〉 !



Kohn–Sham DFT (KS-DFT)

• For any normalized wavefunction Ψ,

〈Ψ|T |Ψ〉 ≥ Ts[nΨ]

〈Ψ|T + Vne|Ψ〉 ≥ Ts[nΨ] +

∫R3

dr vne(r)nΨ(r)

〈Ψ|T + Vne|Ψ〉+ EHxc[nΨ] ≥ Ts[nΨ] + EHxc[nΨ] +

∫R3

dr vne(r)nΨ(r)︸︷︷︸E[nΨ] ≥ E0

• The exact ground-state energy E0 is recovered when Ψ = ΦKS thus leading to

E0 = minΨ

{〈Ψ|T + Vne|Ψ〉+ EHxc[nΨ]

}

• Note that the minimization can be restricted to single determinantal wavefunctions Φ.



Comparing wavefunction theory with KS-DFT• E0 = min

Ψ

{〈Ψ|T + Vne + Wee|Ψ〉

}= min

Φ

{〈Φ|T + Vne|Φ〉+ EHxc[nΦ]

}↓ ↓

Ψ = ΦHF +∑k

Ckdetk︸︷︷︸Φ = |ϕ2

1ϕ22 . . . ϕ

2N2

|︸︷︷︸multideterminantal wave function single determinant

• Standard approximations to Exc[n]:

ELDAxc [n] =

∫R3

dr f(n(r)

)−→ local density approximation (LDA)

EGGAxc [n] =

∫R3

dr f(n(r), |∇n(r)|

)−→ generalized gradient approximation

Emeta−GGAxc [n] =

∫R3

dr f(n(r), |∇n(r)|,∇2n(r),

N/2∑i=1

|∇ϕi(r)|2)−→ meta-GGA

where f denotes a function.



The uniform electron gas as a model systemBasic idea:

• Let us first consider a single free particle in a box with volume L× L× L and L→ +∞. TheSchrödinger equation is

−1

2∇2

rϕk(r) = εkϕk(r)

and the solutions are ϕk(r) =1√L3

eik.r and εk = k2/2 with k ≡ (kx, ky , kz).

• Note that the wavefunction is complex

• In this case the electron density equals n(r) = |ϕk(r)|2 =1

L3←− constant !

• A large number of electrons N is then introduced into the box. The electron density n(r) = N/L3

is held constant as N → +∞ and L→ +∞

• The KS system is simply obtained when neglecting the electron-electron repulsions

• Analytical expression for the exchange energy: ELDAx [n] = −

3

4

(3

π

)1/3 ∫R3

dr n4/3(r)

• Numerical calculation of the correlation energy (Coupled Cluster, Quantum Monte Carlo)



−1.2

−1.15

−1.1

−1.05

−1

−0.95

−0.9

−0.85

1 2 3 4 5 6 7 8 9 10

Ene

rgy

[Eh]

Interatomic distance [units of a0]

H2 [11Σg+, aug−cc−pVQZ]

FCIHF

LDAPBE

B3LYPCAM−B3LYP



−100

−80

−60

−40

−20

0

20

40

5 6 7 8 9 10

Inte

ract

ion

ener

gy [µ

Eh]


He2 [aug−cc−pVTZ]

LDAMP2

CCSD(T)Expt.



−120

−100

−80

−60

−40

−20

0

20

40

5 6 7 8 9 10

Inte

ract

ion

ener

gy [µ

Eh]



PBExPBEcPBExPW91c

BLYPBP86

BPW91MP2

CCSD(T)Expt.



−120

−100

−80

−60

−40

−20

0

20

40

5 6 7 8 9 10

Inte

ract

ion

ener

gy [µ

Eh]



PBE0BHandHLDA

B3LYPCAM−B3LYPBHandHLYP

MP2CCSD(T)

Expt.



DFT, us (humans), and the machinesExact scaling relations for Ts [n] and Ex[n]

We want to see how (some) universal density functionals are a↵ected by theuniform coordinate scaling.

We start with the simplest one, namely the Hartree functional EH[n].

EXERCISE

Show that the following scaling relation is fulfilled,

EH[n� ] = �EH[n].

It can also be shown that the non-interacting kinetic energy and exact exchangeenergy functionals fulfill the following scaling relations:

Ts [n� ] = �2Ts [n] ,

Ex[n� ] = �Ex[n].EXERCISE

For that purpose, write the variational principle for the KS Hamiltonian

T +PN

i=1 vKS[n](ri)⇥, consider trial wavefunctions with density n [we denote

! n] and conclude that Ts [n] = min !n

h |T | i. Deduce that �KS� [n] = �KS[n� ].

Emmanuel Fromager (UdS) EUR: Theory of extended systems 4 / 6

Uniform coordinate scaling in wavefunctions and densities

Let � > 0 be a scaling factor.

Applying a uniform coordinate scaling consists in multiplying each spacecoordinate by �:

r ⌘ (x, y, z) ! �r ⌘ (�x, �y, �z)

dr = dxdydz ! �3dr

Uniform coordinate scaling applied to the density:

n(r) ! n�(r) = �3n(�r)

Uniform coordinate scaling applied to an N -electron wavefunction [spin isuna↵ected by the scaling]:

(r1, r2, . . . , rN ) ! �(r1, r2, . . . , rN ) = �3N2 (�r1, �r2, . . . , �rN )

Emmanuel Fromager (UdS) EUR: Theory of extended systems 2 / 6Institut de Chimie, Strasbourg, France


ARTICLE

Bypassing the Kohn-Sham equations with machinelearningFelix Brockherde1,2, Leslie Vogt 3, Li Li 4, Mark E. Tuckerman3,5,6, Kieron Burke4,7 & Klaus-Robert Müller1,8,9

Last year, at least 30,000 scientific papers used the Kohn–Sham scheme of density functional

theory to solve electronic structure problems in a wide variety of scientific fields. Machine

learning holds the promise of learning the energy functional via examples, bypassing the need

to solve the Kohn–Sham equations. This should yield substantial savings in computer

time, allowing larger systems and/or longer time-scales to be tackled, but attempts to

machine-learn this functional have been limited by the need to find its derivative. The present

work overcomes this difficulty by directly learning the density-potential and energy-density

maps for test systems and various molecules. We perform the first molecular dynamics

simulation with a machine-learned density functional on malonaldehyde and are able to

capture the intramolecular proton transfer process. Learning density models now allows the

construction of accurate density functionals for realistic molecular systems.

DOI: 10.1038/s41467-017-00839-3 OPEN

1Machine Learning Group, Technische Universität Berlin, Marchstraße 23, 10587 Berlin, Germany. 2Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2,06120 Halle, Germany. 3 Department of Chemistry, New York University, New York, NY 10003, USA. 4Department of Physics and Astronomy, University ofCalifornia, Irvine, CA 92697, USA. 5 Courant Institute of Mathematical Science, New York University, New York, NY 10003, USA. 6NYU-ECNU Center forComputational Chemistry at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China. 7 Department of Chemistry, University of California,Irvine, CA 92697, USA. 8Department of Brain and Cognitive Engineering, Korea University, Anam-dong, Seongbuk-gu, Seoul 136-713, Republic of Korea.9Max-Planck-Institut für Informatik, Stuhlsatzenhausweg, 66123 Saarbrücken, Germany. Correspondence and requests for materials should be addressed toM.E.T. (email:[email protected]) or to K.B. (email:[email protected]) or to K.-R.M. (email:[email protected])

NATURE COMMUNICATIONS |8: �872� |DOI: 10.1038/s41467-017-00839-3 |www.nature.com/naturecommunications 1

28/11/2019 18(01Bypassing the Kohn-Sham equations with machine learning | Nature Communications

Page 1 of 42https://www.nature.com/articles/s41467-017-00839-3

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Article

Open Access

Published: 11 October 2017

Bypassing the Kohn-Sham equations withmachine learning

Felix Brockherde, Leslie Vogt, Li Li, Mark E. Tuckerman , Kieron Burke & Klaus-Robert

Müller

Nature Communications 8, Article number: 872 (2017)

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Abstract

Last year, at least 30,000 scientific papers used the Kohn–Sham scheme of densityfunctional theory to solve electronic structure problems in a wide variety ofscientific fields. Machine learning holds the promise of learning the energyfunctional via examples, bypassing the need to solve the Kohn–Sham equations.This should yield substantial savings in computer time, allowing larger systems



Teaching the HK theorem to a machine

• One can teach the functional Ts[n] to a machine.

• But then it needs to find the value of δTs[n]/δn(r) by itself ...

• ... in order to determine n0 variationally from vne.

• What about learning the map v → n[v] directly ?∗

• If we have vne, the machine will tell us directly what n0 = n[vne] is.

• We can also teach the machine how to compute the energy E[n0] = Ts[n0] + EHxc[n0] + (vne|n0).

∗Brockherde, Felix, Vogt, Leslie, Li ,Li, Tuckerman, Mark E, Burke, Kieron and Muller, Klaus-Robert, Nature Communications 8, 872 (2017).



Teaching the HK theorem to a machine

• Expansion of densities in an orthonormal basis of functions {φl(r)}1≤l≤L:

n[v](r) =

L∑l=1

u(l)[v]× φl(r).

• Kernel Ridge Regression (KRR) method:

u(l)[v] =

M∑j=1

β(l)j × k (v, vj)

k(v, vi) = exp

−∫dr∣∣∣v(r)− vi(r)

∣∣∣22σ2

where {vj}1≤j≤M are the potentials the machine will learn from.

Brockherde, Felix, Vogt, Leslie, Li ,Li, Tuckerman, Mark E, Burke, Kieron and Muller, Klaus-Robert, Nature Communications 8, 872 (2017).



Teaching the HK theorem to a machine• The machine will learn

ni(r) = n[vi](r) =

L∑l=1

u(l)i × φl(r)

• The paramaters to be optimized (learning process) are β ≡{β

(l)j

}1≤j≤M,1≤l≤L

.

• Cost function to be minimized with respect to β:

e (β) =

M∑i=1

∫dr∣∣∣ni(r)− n[vi](r)

∣∣∣2orthonormal basis

=

M∑i=1

L∑l=1

∣∣∣∣∣u(l)i − u

(l)[vi]

∣∣∣∣∣2

=

M∑i=1

L∑l=1

∣∣∣∣∣u(l)i −

M∑j=1

β(l)j × k (vi, vj)

∣∣∣∣∣2



is the average of tautomer atomic positions. For the test set, weuse snapshots from a computationally expensiveBorn–Oppenheimer ab initio MD trajectory at 300 K. Figure 5ashows that the ML-HK map is able to predict DFT energiesduring a proton transfer event (MAE of 0.27 kcal/mol) despitebeing trained on classical geometries that did not include theseintermediate points.

We show, finally, that the ML-HK map can also be used togenerate a stable MD trajectory for malonaldehyde at 300 K(Fig. 5b). In principle, analytic gradients could be obtained for

each timestep, but for this first proof-of-concept trajectory, afinite-difference approach was used to determine atomic forces.The ML-HK-generated trajectory samples the same molecularconfigurations as the ab inito MD simulation (see Fig. 6 andSupplementary Table 1) with a mean absolute energy error of0.77 kcal/mol, but it typically underestimates the energy for out-of-plane molecular fluctuations at the extremes of the classicaltraining set (maximum error of 5.7 kcal/mol, see SupplementaryFig. 4). Even with the underestimated energy values, however, theatomic forces are sufficiently large to return the molecule to the

–0.0165

–0.0105

–0.0045

0.0015

0.0075

0.0135

nPBE − nLDA nPBEb − nPBE nPBEb − nML

nPBEb − nML

–0.0004700–0.0003845–0.0002991–0.0002136–0.0001282–0.00004270.00004270.00012820.00021360.00029910.00038450.0004700

a b c

d e f

−0.015

−0.010

−0.005

0.000

0.005

0.010

0.015nPBE

nLDA − nPBE

nPBE − nPBEb

nPBEb − nML

0.00

0.08

0.16

0.24

0.32

0.40

Fig. 4 The precision of our density predictions using the Fourier basis for the molecular plane of benzene. The plots show a the difference between thevalence density of benzene when using PBE and LDA functionals at the PBE-optimized geometry. b Error introduced by using the Fourier basisrepresentation. c Error introduced by the nML[v] density fitting (a–c on same color scale). d The total PBE valence density. e The density differences along a1D cut of a–c. f The density error introduced with the ML-HK map (same data but different scale, as in c)

0 10 20 30 40 50Simulation time (5 fs)

02468

1012

E (

kcal

/mol

)E

(kc

al/m

ol)

a

0 50 100 150 200

Simulation time (5 fs)

2468

10121416b

Fig. 5 Energy errors of ML-HK along ab initio MD and ML-generated trajectories. a Energy errors of ML-HK along a 0.25 ps ab initio MD trajectory ofmalonaldehyde. PBE values in blue, ML-HK values in red. The ML model correctly predicts energies during proton transfer in frames 7–15 without explicitinclusion of these geometries in the training set. b Energy errors of ML-HK along a 1 ps MD trajectory of malonaldehyde generated by the ML-HK model.ML-HK values in red, PBE values of trajectory snapshots in blue

NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-00839-3 ARTICLE

NATURE COMMUNICATIONS |8: �872� |DOI: 10.1038/s41467-017-00839-3 |www.nature.com/naturecommunications 7

28/11/2019 18(01Bypassing the Kohn-Sham equations with machine learning | Nature Communications

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Article

Open Access

Published: 11 October 2017

Bypassing the Kohn-Sham equations withmachine learning

Felix Brockherde, Leslie Vogt, Li Li, Mark E. Tuckerman , Kieron Burke & Klaus-Robert

Müller

Nature Communications 8, Article number: 872 (2017)

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Abstract

Last year, at least 30,000 scientific papers used the Kohn–Sham scheme of densityfunctional theory to solve electronic structure problems in a wide variety ofscientific fields. Machine learning holds the promise of learning the energyfunctional via examples, bypassing the need to solve the Kohn–Sham equations.This should yield substantial savings in computer time, allowing larger systems


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