FROZEN-DENSITY EMBEDDING:TESTING THE ACCURACY OF PRESENT-DAY KINETIC-ENERGY DENSITY FUNCTIONALS

SAMUEL FUXa, KARIN KIEWISCH

a, CHRISTOPH R. JACOBa,

JOHANNES NEUGEBAUERb, MARKUS REIHER

a

a Laboratorium fur Physikalische Chemie, ETH Zurich, Wolfgang-Pauli-Strasse 10,8093 Zurich, Switzerland

b Gorlaeus Laboratories, Universiteit Leiden, 2300 RA Leiden, Netherlands

{samuel.fux, markus.reiher}@phys.chem.ethz.ch

Ammoniaborane - subsystems connectedby coordination bonds

-2 -1 0 1 2

-1

0

1

2

3

-2 -1 0 1 2

-1

0

1

2

3

-2 -1 0 1 2

-1

0

1

2

3

-2 -1 0 1 2

-1

0

1

2

3

x

y

ρKS−DFT(r) ρemb(r)

ρKS−DFT(r) −ρemb(r)ρKS−DFT(r) −ρfrag(r)

BCP1

BCP3

H3H1, H2

BCP2

H4, H5

B ( 0.00 / 0.00 )

H6

Subsystem 1

Subsystem 2

N

• Comparison of electron densities from KS-DFT calculations to elec-tron densities from embedding calculations.

• The difference density is a good measure for the accuracy of the ap-proximation of T nadd

s [ρ1, ρ2] by the PW91k density functional.

rx,BCP ry,BCP ρ(r) L(r)BCP 3 sup 0.53 0.00 0.71 −1.77

emb 0.60 0.00 0.87 1.09diff −0.07 0.00 −0.16 −2.86

• Subsystems are connected by a coordination bond which partiallyexhibits covalent character.

• B−N bonding region (connection between the subsystems) is rea-sonably well described, although the negative Laplacian has thewrong sign at BCP3.

Titaniumtetrachlorideionic bonds as a challenge for PW91k

-2 -1 0 1 2 3 4

-2

-1

0

1

2

3

4

-2 -1 0 1 2 3 4

-2

-1

0

1

2

3

4

-2 -1 0 1 2 3 4

-2

-1

0

1

2

3

4

-2 -1 0 1 2 3 4

-2

-1

0

1

2

3

4ρ

KS−DFT(r) ρ

emb(r)

ρKS−DFT

(r) −ρ (r)frag

ρKS−DFT

(r) −ρemb

(r)

BCP2

BCP1

y

x

Cl1

Cl2

Ti ( 0.00 / 0.00)

Cl3, Cl4

Subsystem 1

Subsystem 2

• Unphysical charge transfer from Cl− to TiCl+3 , which can be over-come by applying a long-distance correctiona.

• Good agreement of the electron density and its negative Laplacian.

a C. R. Jacob, M. Beyhan, L. Visscher, J. Chem. Phys 2007, 126, 234116

ChromiumhexacarbonylFDE and π-backdonation

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4ρKS−DFT

(r) ρemb

(r)

ρ (r)frag

ρKS−DFT

(r) − ρemb

(r)ρKS−DFT

(r) −

BCP1

BCP2

BCP3

BCP4

BCP5BCP6

BCP7BCP8

y

x

C

O

Subsystem 1

Subsystem 2

• Results are not reliable, because the expected orbital order (from aKS-DFT reference calculation) could not be reproduced.

• π-backdonation could not be described reasonably.

Acknowledgments

This work has been supported by the swiss federal institute of technologyZurich (Grant TH-26 07-3).

FDE theory II

Minimization of the bifunctional for the total energy:

• Minimization condition with N1 denoting the number of electronsin subsystem 1:

δ

δρ1

[

Etot[ρ1 + ρ2] − µ

(∫

ρ1(r)d3r − N1

)]

= 0

• Corresponding Euler–Lagrange equation:

µ = vnuc1 (r) + vnuc

2 (r) +

∫

ρ1(r′)

|r − r′|

d3r′ +

∫

ρ2(r′)

|r − r′|

d3r′

+δExc[ρ1 + ρ2]

δρ1+

δTs[ρ1]

δρ1+

δT nadds [ρ1, ρ2]

δρ1.

• The electron density, obtained from the minimization of the bifunc-tional, is expressed in terms of canonical Kohn–Sham orbitals.

ρ1(r) = 2

N1/2∑

i=1

|φ(1)i (r)|2.

• These orbitals can be evaluated by solving the Kohn–Sham equa-tions with constraint electron density (KSCED):

[

−1

2∇2 + vKSCED

eff [ρ1, ρ2](r)

]

φ(1)i (r) = εiφ

(1)i (r); i = 1, . . . , N1/2

• The effective potential can now be divided into a sum of a KS effec-tive potential, and an effective embedding potential:

vKSCEDeff [ρ1, ρ2](r) = vKS

eff [ρ1](r) + vembeff [ρ1, ρ2](r)

with the effective embedding potential (representing the frozensubsystem)

vKSeff [ρ1](r) = vnuc

1 (r) +

∫

ρ1(r′)

|r − r′|

d3r′ +δExc[ρ1]

δρ1

vembeff [ρ1, ρ2](r) = vnuc

2 (r) +

∫

ρ2(r′)

|r − r′|

d3r′ +δEnadd

xc [ρ1, ρ2]

δρ1+

δT nadds [ρ1, ρ2]

δρ1

• The kinetic-energy component of vembeff [ρ1, ρ2](r) is defined as:

vT[ρ1, ρ2] =δT nadd

s [ρ1, ρ2]

δρ1=

δTs[ρtot]

δρtot−

δTs[ρ1]

δρ1≈

δTs[ρtot]

δρtot−

δTs[ρ1]

δρ1

FDE theory I

Basic ideas:

• Partitioning of the total electron density into subsystem densities:

ρtot(r) = ρ1(r) + ρ2(r)

• Expressing the total energy Etot[ρtot] as a bifunctional of ρ1 and ρ2:

Etot[ρ1, ρ2] = Enuc.rep. +

∫

(ρ1(r) + ρ2(r))(vnuc1 (r) + vnuc

2 (r))d3r

+1

2

∫∫

(ρ1(r) + ρ2(r))(ρ1(r′) + ρ2(r

′))

|r − r′|

d3r d3r′

+Exc[ρ1 + ρ2] + Ts[ρ1 + ρ2]

• Exc[ρ1 +ρ2] has to be approximated by a density functional as in theKS-DFT framework.

• Partitioning of the kinetic energy:

Ts[ρtot] = Ts[ρ1 + ρ2] = Ts[ρ1] + Ts[ρ2] + T nadds [ρ1, ρ2]

• The non-additive part of the kinetic energy T nadds [ρ1, ρ2] has to be

approximated by a kinetic-energy density functional (e.g. PW91k):

T nadds [ρ1, ρ2] ≈ Ts[ρtot] − Ts[ρ1] − Ts[ρ2]

Introduction

• Frozen-density embedding (FDE) is a subsystem formulation withinKohn–Sham density functional theory (KS-DFT), in which the totalelectron density is expressed as a superposition of subsystem elec-tron densities.

• The non-additive part of the kinetic energy T nadds [ρ1, ρ2] is approx-

imated by a kinetic-energy density functional (only additional ap-proximation when comparing to KS-DFT).

• In principle exact if Exc[ρ] and Ts[ρ] are known.

• Main focus of the work lies on the investigation of the accuracy ofpresent-day kinetic-energy density functionals, regarding local cri-teria like the electron density and the kinetic-energy potential.

• A systematic investigation of the electron densities fromsubsystem-DFT in comparison to Kohn-Sham-DFT was carriedout.

• A topological analysis of the electron densities was performed atthe bond critical points of the test systems.

• Attention was paid to the intermediate case of subsystems con-nected via coordination bonds.

• Furthermore kinetic-energy potentials obtained from different den-sity functionals were compared to KS-DFT reference potentials, us-ing the noble gas atoms helium and neon as test systems.

Kinetic-energy potentials:approximations from density functionals

Generalized gradient expansion:

TGGAs =

3

10(3π2)2/3

∫

ρ(r)5/3Φ(s)d3r with s =|∇ρ(r)|

2ρ(r)4/3(3π2)1/3

functional Φ(s)

Thomas–Fermi 1

von Weizsacker5

3s2

TF9W / TSGA 1 +5

27s2

PW91ka 1 + A1s sinh−1(As) + (A2 − A3 exp(−A4s2))s2

1 + A1s sinh−1(As) + B1s4

aA = 76.320, A1 = 0.093907, A2 = 0.26608, A3 = 0.0809615, A4 = 100.00, B1 =0.57767 · 10−4

Approximate kinetic-energy potentials:

• If the electron density of a system is known, δTs[ρ]δρ(r) can directly be calcu-

lated according to the formula shown above.

KS-DFT reference for the kinetic-energy potential:

• References for the kinetic-energy potential can direcly be calculated fromthe Euler–Lagrange equation.

δTs[ρ]

δρ(r)= µs − vext(r) −

∫

ρ(r)

|r − r′|

d3r′ −δExc[ρ]

δρ(r)

→ The approximate kinetic-energy potentials can now be compared to the KS-

DFT reference potentials.

Noble gas atoms: vT[ρ] vs. vT[ρ]

Helium

r r

rr

0 1 2 3 4

r @a.u.D

0

2

4

6

8

10

12

14

v @a.u.D

0 1 2 3 4

r @a.u.D

0

2

4

6

8

10

12

14

v @a.u.D

0 1 2 3 4

r @a.u.D

0

2

4

6

8

10

12

14

v @a.u.D

0 1 2 3 4

r @a.u.D

0

2

4

6

8

10

12

14

v @a.u.D

r[a.u.] r[a.u.]

r[a.u.]r[a.u.]

Thomas−Fermi

KS−DFT KS−DFT

von Weizsäcker

KS−DFT

PW91k

KS−DFT

TF9W

4π 2 v (r)T

[a.u.]

4π 2 v (r)T

[a.u.]

4π 2 v (r)T

[a.u.]

4π 2 v (r)T

[a.u.]

Neon

r r

rr

0 1 2 3 4

r @a.u.D

-10

0

10

20

30

v @a.u.D

0 1 2 3 4

r @a.u.D

-10

0

10

20

30

v @a.u.D

0 1 2 3 4

r @a.u.D

-10

0

10

20

30

v @a.u.D

0 1 2 3 4

r @a.u.D

-10

0

10

20

30

v @a.u.D

r[a.u.] r[a.u.]

r[a.u.]r[a.u.]

Thomas−Fermi

KS−DFT KS−DFT

von Weizsäcker

KS−DFT

PW91k

KS−DFT

TF9W

4π 2 v (r)T

[a.u.]

4π 2 v (r)T

[a.u.]

4π 2 v (r)T

[a.u.]

4π 2 v (r)T

[a.u.]

• Kinetic-energy potentials, obtained from approximate kinetic-energy den-sity functionals are compared to KS-DFT reference potentials.

• Approximate kinetic-energy potentials exhibit only small differences nearthe nucleus and far away from the nucleus, when compared to the KS-DFTreference potentials.

• Large deviations are observed at intermediate distances.

ConclusionFDE

• Local criteria like the electron density and the kinetic-energy potential al-low a spatial resolution of the error which arises from the approximationof Tnadd

s [ρ1, ρ2] by a kinetic-energy density functional.

• Employing PW91k in an embedding calculation yields reasonably good re-sults for TiCl4.

• FDE fails in the description of π-backdonation for Cr(CO)6 and the coordi-nation bond in BH3NH3.

Kinetic-energy potentials

• Approximate kinetic-energy potentials only yield large deviations at inter-mediate distances.

• Functionals, that contain the full Thomas-Fermi term, yield too large val-ues in regions where the change in the electron density is small.

• Regarding the results for neon, the number of maxima in the potential istoo large, compared to the reference potential.

References• Analysis of electron density distributions from subsystem DFT:

S. Fux, K. Kiewisch, C. R. Jacob, J. Neugebauer, M. Reiher, Chem. Phys. Lett. 461,

353-359 (2008).