development of new density functionals and new methods for

Development of new density functionals and new methods for

analysis of convergence of ab initio molecular dynamics simulations

• DOE Early Career DE-SC0003871 DOE: DE-FG02-09ER16052

Marivi Fernandez-Serra Michelle Fritz Jose M. Soler

INTRODUCTION SIMULATING WATER

Improving ab initio molecular

dynamics (AIMD) for liquid water

(liquid structure, diffusion coefficient,

eqm. density, etc.)

• Correction of self-interaction errors

- Include exact exchange (PBE0)

+

• Inclusion of vdW interactions (TS-

vdW)

+

• Inclusion of Nuclear Quantum

Effects (NQE)

- Path Integral Molecular

Dynamics (PIMD)

- Temperature increase of 30 K

Results shown are from DiStasio Jr., R. A. et al. The

J. Chem. Phys. 141, 084502 (2014)

Mean absolute deviation (MAD) from Quantum Monte Carlo (QMC)

calculations of liquid water configurations

INTRODUCTION MB-POL

Number of configurations

MB-pol performs better

than all other models

compared.

Medders, G. R. et al. J. Chem. Phys. 143, 104102 (2015).

MB-pol1,2 is the most accurate potential for water available today.

•Incorporates the Partridge and Schwenke water monomer potential3

to describe 1-body energies.

•Fitted to 2- and 3-body energies calculated with CCSD(T).

Many-body expansion of potential energy for N interacting molecules:

•Many-body effects included with polarization model employed by the TTM4-F potential.4

INTRODUCTION MB-POL

2-body energies 3-body energies

1Babin, V., Leforestier, C., and Paesani, F. J. Chem. Theory Comput. 9, 5395 (2013).2Babin, V., Medders, G. R., and Paesani, F. J. Chem. Theory Comput., 10, 1599 (2014).3Partridge, H. and Schwenke, D. W. J. Chem. Phys. 106, 4618 (1997).4Burnham, C. J. et al. J. Chem. Phys. 135, 144502 (2011).

MD results from MB-pol compared with other methods and experiments

INTRODUCTION MB-POL

Medders, G. R. J. Chem. Theory Comput. 10, 2906 (2014).

With nuclear quantum effects (PIMD), MB-pol closely reproduces the experimental structure of water.

• MB-pol also closely reproduces experimental results for theequilibrium density and diffusion constant (D) of water.

• Considering nuclear motion results in an increase of the molecular size due to zero-point energy effects and delocalization, which lower density.

DPPS METHODOLOGY

Motivation: Optimization of a density functional for water

• Hybrid functionals are not yet as accurate as MB-pol and theircomputational cost is still too large.

• We take the MB-pol approach to fit a GGA+vdW-DF functional,using a new method: Data Projection on Parameter Space.

• In practice, there is not one ab initio DFT, but many.

• Trial and error: does a functional fit the data?

• DPPS: what functional fits best the data?

Data Projection onto Parameter Space (DPPS)

II. DATA PROJECTION ONTO PARAMETER SPACE (DPPS)

Optimization of an exchange-correlation density functional for water, Michelle Fritz, Marivi Fernandez-Serra and Jose M. Soler, J. Chem. Phys. 144, 224101 (2016)

Optimization of GGA Exchange-Correlation (XC)

• Change in the XC energy is needed to reproduce reference energies.

• Express density and energy density in terms of 𝒌𝑭 = 𝟑𝝅𝝆 𝟏 𝟑 and 𝒌𝑭 = 𝛁𝝆𝝆 .

DPPS METHODOLOGY

Non –XC energy

> @ > @ > @ > @ nn non XC n XC n XC n refE E E E EU U U U� � � '

Set of electron densities at a given, initial functional

Initial XC ΔEXC to reproduce reference energy

Reference energies

> @ 3( ) ( ( ) , ( ) )

( , ) ( , )

GGAXC XC

GGAF G XC F G F G

E d

k k k k dk dk

U U H U U

U H

' ' �

'

³³ ³

r r r r

Δεxc to reproduce reference energy

Electron density in kF, kG

G

Density histogram for a water monomer

Electron density in real space (e-/Bohr3)

DPPS METHODOLOGY: GGA XC

Electron density in kF ,kG.

Electron density in kF, kG:

xy

z

H2O ( )U r

3 2 1 3( , ) ( ) ( (3 ( )) ) ( ( ) ( ))F G F Gk k d k kU U G S U G U U � � �³ r r r r r

( )( )GkUU�

rr

12 3(3 ( )) ,Fk S U r

• Set interpolation points kFα, kGβ.

• The reference exchange energies arethe projections of the known densitiesonto the set of parameters:

DPPS METHODOLOGY

( , )XC F Gk kDE D EH H

XC energy density

n nref XCE � xH U

Set of parameters

^ `DEH H ^ `n nDEU U

Electron densities in parameter space


𝑘

..

.

..

𝜀(𝑘

,𝑘

.

..

..

𝜀

..

DPPS METHODOLOGY

( , )XC F Gk kDE D EH H

XC energy density

n nref XCE � xH U

Set of parameters

^ `DEH H ^ `n nDEU U

Electron densities in parameter space

( , )XC F Gk kHDPPS

nU

nref XCE �


• Set interpolation points kFα, kGβ.

• The reference exchange energies arethe projections of the known densitiesonto the set of parameters:

Optimization of GGA Exchange

• Change in the exchange energy is needed to reproduce reference energies.

• Exchange energy density is the porduct of the LDA exchange energy densityand a functional of the reduced density gradient 𝒔 ≡ 𝒌𝑮 𝟐𝒌𝑭.

• Express density in terms of s.

DPPS METHODOLOGY

Initial exchange ΔEX to reproduce reference energy

Reference energies

( , ) ( ) ( )LDAX F G X F Xk k k F sH H

Exchange enhancement factor

> @ ( ) ( )LDAX X XE s F s dsU U H' '³

ΔFX to reproduce reference energyElectron density in s

> @ > @ > @ > @ nn non X n X n X n refE E E E EU U U U� � � '

Non –exchange energy


( ) ( , ) ( 2 )F G F G G Fs dk dk k k s k kU U G �³ ³Electron density in s:

Electron density in kF ,kG.

DPPS GGA EXCHANGE OPTIMIZATION FOR WATER

s=0.25

0.51.0

2.0

Reduced density gradient:

Electron density in s.

( , )F Gk kU ( )sU

> @ ( ) ( )LDAX X XE s F s dsU U H' '³

• Set interpolation points sα.

• The reference exchange energies arethe projections of 𝛈 = 𝝆𝜺𝑿𝑳𝑫𝑨 onto theset of parameters:

DPPS METHODOLOGY

( )XF F sD D

)𝑭 𝑿(𝒔

..

.

.

.

s

n nref XE � xF K

Set of parameters

^ `XF D F ^ `n nDK K

Known data (product of 𝝆 and 𝜺𝑿𝑳𝑫𝑨) in parameter space

Optimization of GGA Exchange

𝑭𝑿𝜶

𝜂 = 𝜌𝜀

Theoretical constraints: Bayesian

� �2

2

1

0

( | ) ( | ) ( )

( | ) exp2

1( ) exp ( ) ( , ') ( ') '

2

( ) ( ) ( ) , ( , ') ( ) ( ')

{ PBE, rev

n nref GGA

nn

x x

k k k kx x x x x

k

P theory facts N P facts theory P theory

E EP facts theory

E

P theory F s C s s F s ds ds

F s F s F s C s s F s F s

k

G G

G G G

�

½�° ° �® ¾'° °¯ ¿

½ �® ¾¯ ¿

�

¦

³³¦

PBE, PBEsol, BLYP, PW86, ... }

Probability that a theory is true given that some facts are true

Probability that some facts are true given that a theory is true

Probability that a theory is true

Error bars in the energies

Avg. of exchange functionals

DPPS METHODOLOGY: BAYESIAN CONSTRAINTS

Theoretical constraints: Bayesian DPPS METHODOLOGY: BAYESIAN CONSTRAINTS

Ab initio GGA exchange enhancement factors used as prior information in Bayesian optimizatons.

Optimization of GGA Exchange for Water

• Reference energies 𝑬𝒓𝒆𝒇 𝑿𝒏 are MB-pol energies of water

monomers in 300 geometries.+

• Input densities from initial functional: vdW-DF-cx1

vdW-DF XC general form:

• Apply DPPS to find optimal form of 𝑭𝑿(𝒔).

DPPS METHODOLOGY

> @ > @ > @ > @0 nlXC X C CE E E EU U U U � �

> @ 3 31 ' ( ) ( , ') ( ')2

nlCE d rd rU U I U ³³ r r r r

1K. Berland and P. Hyldgaard, Phys. Rev. B 89, 035412 (2014).

The CUSTOM Fx improves 1-body energies from the initialfunctional vdW-DF-cx.

DPPS EXCHANGE OPTIMIZATION FOR WATER


• As O-H bonds shorten, density moves towards oxygen and density gradient increases.

• Most changes in ρ(kF ,kG) needed are along a constant value of s.


xy

z

Change in H2O density (e-/Bohr3)

Fit of Fx(s) with Bayesian constraintsto 330 2-body 100 3-body energies calculated with MB-pol.


+

DPPS


s=0.25

0.5

1.0

2.0

s=0.25

0.5

1.0

2.0


2-body interaction energy:

H-bonded configuration non-H-bonded configuration


s=0.25

0.51.0

2.0

DPPS GGA EXCHANGE OPTIMIZATION FOR WATER3-body interaction energy:


Fit of Fx(s) with Bayesian constraints

to 330 2-body 100 3-body energies

calculated with MB-pol.

The CUSTOM-MB Fx improves 2-

body and 3-body energies, though

less than CUSTOM-2B and CUSTOM-

3B.

2-body energies 3-body energies


Mean errors of the dataset MB-pol energies used in the optimizations.

• CUSTOM-1B results in significantly larger 2-body and 3-

body errors.

• If a 1-body correction term is added to the functional,

the best functional overall is CUSTOM-MB.


The 1-body energy terms were corrected by the following approach:1

• vdW-DF-cx-Δ1 +1-body corrections

• CUSTOM-2B-Δ1 +2-body corrections

• CUSTOM-MB-Δ1 +3-body corrections

1Patridge, H. and Schwenke, D. W. J. Chem. Phys., 106, 4618 (1997).

1-body-corrected DFT energy

MB-pol 1-body energy

Uncorrected DFT energy

DFT 1-body energy (monomer potential energy surface1 fitted to DFT energies)

III. THERMODYNAMICINTEGRATION (TI) METHOD

Convergence of 𝒈𝑶𝑶(𝒓) withDM tolerance is unclear fromAIMD simulations.

Convergence of 𝒈𝑶𝑶(𝒓) withDM tolerance is unclear fromAIMD simulations.

𝒈 𝑶𝑶(𝒓)

Convergence of 𝑷 withBasis set is unclear fromAIMD simulations.

Thermodynamic Integration (TI)

• More efficient ab initio Molecular Dynamics (AIMD) convergence testing.– Only 1% of calculations in entire AIMD

simulations needs.

• Avoids masking by statistical fluctations.

• Can be used for convergence testing or to quickly test a new functional for a system.

• Consider the linear mix of two potentials:- 𝑽𝟎 (potential used in a full AIMD simulation)- 𝑽𝟐 (new potential for which properties will be estimated)

• Consider a number of snapshots described by potential 𝑽𝝀. The ensemble average of a property 𝜶 is

TI METHODOLOGY

𝛼 = ∫𝑑 𝑥 𝑒 𝛼 (𝑥)

𝛽 = 1𝑘𝑇

𝑍 = 𝑑 𝑥 𝑒

METHODOLOGY

• Consider the linear mix of two potentials:- 𝑽𝟎 (potential used in a full AIMD simulation)- 𝑽𝟐 (new potential for which properties will be estimated)

• Consider a number of snapshots described by potential 𝑽𝝀. The ensemble average of a property 𝜶 is

TI METHODOLOGY

𝛼 = ∫𝑑 𝑥 𝑒 𝛼 (𝑥)

𝛽 = 1𝑘𝑇

𝑍 = 𝑑 𝑥 𝑒

TI METHODOLOGY

• TI applied to convergence testing of parameters in AIMD simulations: – Basis Set– Mesh Cutoff– DM Tolerance

TI CONVERGENCE TESTING

• TI applied to convergence testing of parameters in AIMD simulations: – Basis Set– Mesh Cutoff– DM Tolerance

• Convergence of the following properties was tested: – Average pressure 𝑷– Average order parameter 𝜻 1

– 𝒈𝑶𝑶𝒎𝒊𝒏– 𝒈𝑶𝑶𝒎𝒂𝒙

TI CONVERGENCE TESTING

1Russo, J. and Tanaka, H. Nat. Commun. 5, 3556 (2014).

TI CONVERGENCE TESTINGConvergence with Basis Set

IV. GAS AND LIQUID PHASE WATERRESULTS FROM OPTIMIZEDFUNCTIONALS

Optimization of an exchange-correlation density functional for water, Michelle Fritz, Marivi Fernandez-Serra and Jose M. Soler, J. Chem. Phys. 144, 224101 (2016)

Testing the CUSTOM functional: dimer binding energy curvesRESULTS WATER HEXAMERS

prism

cage

book

bag

cyclic

Relative binding energies of water isomers.

2-body and 3-body energy corrections improve the relative binding

energies of water hexamers.

Testing the CUSTOM functional: dimer binding energy curvesRESULTS ICE LATTICE ENERGIES

Lattice energies of different ice phases, relative to ice Ih, in

meV/molecule at fixed lattice and molecular geometries.

1Dion, M. et al. Phys. Rev. Lett. 92, 246401 (2004).

2Lee, K. et al. Phys. Rev. B 82, 081101(R) (2010).

3Santra, B. et al. M. J. Chem. Phys. 139, 154702 (2013).

CUSTOM-MB is the only functional that

qualitatively reproduces the behavior of

our reference MB-pol energies.

Unit cells of ice phases.1

0

1

2

3

gOO(r) from AIMD with vdW-DF-xc and CUSTOM functionals, comparedwith MB-pol and experimental results.

• CUSTOM-2B: increase in first peak and interstitial region, decrease insecond coordination shell.

• CUSTOM-MB decrease in coordination peaks, in the interstitial region.

• 1-body corrections shift the first coordination peak to the right.

RESULTS AIMD RESULTS

Testing the CUSTOM functional: dimer binding energy curvesRESULTS AIMD RESULTS

• vdW-DF-cx and CUSTOM-MB-Δ1 predict equilibrium densities that are too high.

• Density should improve with NQE and a better basis set.

• The compressibility predicted by CUSTOM-MB-Δ1 is in good agreement with experiments.1

1Wagner, W. and Pruss, A. J. Phys. Chem. Ref. Data 31, 387 (2002).

1

1

CONCLUSIONS

CONCLUSIONS:

•We have designed and implemented a general Bayesian method to optimize GGA exchange-correlation (DPPS).

•We have applied DPPS to the optimization of GGA exchange+vdWfunctional for 2-body and 3-body energies calculated with MB-pol.

•Optimization of 1-body energies is incompatible with that of interaction energies. A 1-body correction term was added to total energies.

•The TI method was introduced and applied to convergence testing for AIMD simulations.

•The functional optimized to 2-body and 3-body energies is shown to perform considerably better than vdW-DF-cx for clusters, ice, and water.

FUTURE WORK:

•Nuclear quantum effects should be taken into account to attempt better agreement with experiments.

•Application of DPPS to larger datasets and other functional forms.

CONCLUSIONS

CONCLUSIONS:

•We have designed and implemented a general Bayesian method to optimize GGA exchange-correlation (DPPS).

•We have applied DPPS to the optimization of GGA exchange+vdWfunctional for 2-body and 3-body energies calculated with MB-pol.

•Optimization of 1-body energies is incompatible with that of interaction energies. A 1-body correction term was added to total energies.

•The TI method was introduced and applied to convergence testing for AIMD simulations.

•The functional optimized to 2-body and 3-body energies is shown to perform considerably better than vdW-DF-cx for clusters, ice, and water.

FUTURE WORK:

•Nuclear quantum effects should be taken into account to attempt better agreement with experiments.

•Application of DPPS to larger datasets and other functional forms.

THANK YOU FOR YOUR ATTENTION!

TI METHODOLOGY

ESTIMATING CHANGES IN PRESSURE WITH FUNCTIONAL

1Vydrov, O. A. and Voorhis, T. V. J. Chem. Phys. 133, 244103 (2010).

1Vydrov, O. A. and Voorhis, T. V. J. Chem. Phys. 133, 244103 (2010).

TI METHODOLOGY

1Wang, J., Román-Pérez, G., Soler, J. M., Artacho, E., and Fernández-Serra, M.-V. J. Chem. Phys.134, 024516 (2011).

11

TI METHODOLOGY

TI METHODOLOGY

development of new density functionals and new methods for

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