van der Waals interactions from non-empirical

Density Functional Theory: developments and applications

Stefano de GironcoliSISSA

RBI TWINNING – Zagreb, 5 Feb 2016

van der Waals

Van der Waals interaction isrelatively weak but widespread in nature.An important source of stability for molecular solids and physisorption of molecules on surfaces.

Density Functional Theory

depends only on the GS density(3D function) and satisfies a variational principle [Hohenberg and Kohn, 1964]

HK:

It is useful to introduce a ficticious system of non-interacting electrons

HK:

This defines Exc

KS:

The energy becomes

Self-consistent equations [Kohn-Sham, 1965]

It is as simple as a Mean-field approach but it is exact !

Self-consistent equations [Kohn-Sham, 1965]

It is as simple as a Mean-field approach but it is exact !

Walter Kohn got the Nobel prize (in Chemistry) in 1998 for the develoment of DFT

Self-consistent equations [Kohn-Sham, 1965]

It is as simple as a Mean-field approach but it is exact !

is not known exactly → approximations

Density Functional Theory

KS self consistent eqs.

Density Functional Theory

Formally correct, but very expensive, Adiabatic Coupling Fluctuation Dissipation approach

Practical approximations involve many simplifications

Impact of Density Functional Theory

K. Burke perspective on DFT J. Chem. Phys. 136, 150901 (2012);

Impact of Density Functional Theory

LDA/GGA and vdW

DFT within LDA and GGA functionals has been extremely successful in predicting structural, elastic, vibrational properties of materials bound by metallic, ionic, covalent bonds.

These functionals focus on the properties of the electron gas around a single point in space.

As such they do not describe vdW interaction.

The same is true for Hybrids, DFT+U and SIC etc...

vdW : non local correlation

vdW : non local correlation

vdW : non local correlation

Failure of semilocal functionals

Graphite

c

a

Truly non-local functionals

6 dimIntegral !

Several Non-Local Functionals

vdWDF

vdWDF2

SLA+PW+RPBE- 18.5%

SLA+PW+RPW86- 60.9 %

vdWDF-09 SLA+PW+RPBE- 10.4 %

vv10 SLA+PW+RPW86+PBC 10.7 %

error

vdWDF M. Dion et al., PRL 92, 246401 (2004)vdWDF2 Lee et al., Phys. Rev. B 82, 081101 (2010)VdWDF-09 V. R. Cooper, Phys. Rev. B 81, 161104(R) (2010)vv10 O.A. Vydrov, T.J. Van Voorhis, Chem. Phys. 133, 244103 (2010)

Efficient integrationRoman-Perez Soler [PRL 103, 096102 (2009)] interpolation:If it's possible to express the complex density dependence on r, r' via a single function of q(r) ( and q(r') ) then ...

The vdW energy can be expressed as a sum of simple 3d integrals

We implemented this scheme in the Quantum Espresso distribution and extended it to the calculation of forces, stress tensor and dynamical properties.

Alanine evolution with Pressure

R Sabatini, E Kucukbenli, B Kolb, T Thonhauser, SdG, J Phys Condens Matter 24,424209 (2012)

Alanine evolution with Pressure

R Sabatini, E Kucukbenli, B Kolb, T Thonhauser, SdG, J Phys Condens Matter 24,424209 (2012)

Correct ordering of Glycine polymorphes

R Sabatini, E Kucukbenli, B Kolb, T Thonhauser, SdG, J Phys Condens Matter 24,424209 (2012)

Several Non-Local Functionals

vdWDF

vdWDF2

SLA+PW+RPBE- 18.5%

SLA+PW+RPW86- 60.9 %

vdWDF-09 SLA+PW+RPBE- 10.4 %

vv10 SLA+PW+RPW86+PBC 10.7 %

error

vdWDF – functional can exploit the Roman-Perez Soler interpolationvv10 – functional does not fulfill the needed conditions

VV10 functional

Separable !

R. Sabatini T. Gorni, SdG, Phys.Rev.B 87, 041108 (2013).

VV10 vs rVV10

The error in the kernel is small except when the density itself is very small !

rVV10 validationS22 – hydrogen bonded S22 – Mixed complexes S22 – dispersion dominated

rVV10 applications

a c

vdW-DF 2.48 7.19

vdW-DF2 2.47 7.06

rVV10 2.46 6.72

exp 2.46 6.71

Argon dimer

Graphite cel parameters (A)

c

a

Noble gas dimer are classical examples of dispersion dominated systems where the quality of different functionals can be explored.

Phonons in Graphite

LDA vdW-DF vdW-DF2 rVV10

MAE (cm-1) 39.86 24.57 28.29 18.29

MARE (%) 3.21 1.85 2.04 1.36

Comparison of DFPT results at high symmetry points

Stiff intralayer modes

R Sabatini, C H Pham E Kucukbenli SdG unpublished

Phonons in Graphite

LDA vdW-DF vdW-DF2 rVV10

MAE (cm-1) 5.50 13.50 10.00 7.50

MARE (%) 10.51 28.17 22.50 13.63

Soft interlayer modes

Adiabatic Connection Fluctuation Dissipation

RPA: RPA in QE.: V.H. Nguyen , SdG, PRB 79, 165406 (2009).Scf RPA: N.L. Nguyen, N. Colonna, SdG, PRB 90, 045138 (2014).

OK for vdW interaction, overestimates correlation energy in absolute terms

RPAx: RPAx in QE.: N. Colonna,M. Hellgren, SdG, PRB 90, 125150 (2014).

Much more accurate correlation energy, displays instabilities at low densityScfRPAx still to be implemented.

Dyson Equation for the response function

Advanced xc functionals from ACFD formalism

RPA: OK for vdW interaction but overestimates correlation energy

RPAx: much better energy, ...

Homogeneous Electron Gas

small rs ~ weak interactionLarge rs ~ srtrong interaction

Advanced xc functionals from ACFD formalism

RPA: OK for vdW interaction but overestimates correlation energy

RPAx: much better energy, displays instabilities at low density

Homogeneous Electron Gas

small rs ~ weak interactionLarge rs ~ srtrong interaction

Some kind of non perturbativre treatment is needed

Conclusions

Truly non-local vdW-aware functionals are now available.

Qualitatively improved description of the structural properties of molecular crystals can be obtained with vdW-avare functionals.

Several functional flavors are now available. Among these VV10 is very promising but cannot be evaluated efficiently in its original formulation.

We have developed a simple revision of VV10 (rVV10) that can be efficiently evaluated.

Results for S22 benchmark set, noble gas dimers, graphite are presented. Vibrational properties also available.

-Adiabatic Coupling Fluctuation Dissipation treatment of XCRPA, RPAx and beyond