from tddft to molecular dynamics

From time-dependent density functional theoryto molecular dynamics

Xavier Andrade†∗

in collaboration with

J. L. Alonso‡, F. Falceto‡, D. Prada‡, P. Echenique‡ and A. Rubio†

†ETSF and Universidad del Paıs Vasco‡Universidad de Zaragoza

Benasque, September 2008

∗[email protected]. Andrade (UPV/EHU) From TDDFT to MD Benasque 2008 1 / 18

Outline

1 Introduction: Molecular dynamics2 New method for molecular dynamics.3 Results and comparisons.4 Conclusions.

X. Andrade (UPV/EHU) From TDDFT to MD Benasque 2008 2 / 18

Outline

Introduction

Molecular dynamics (MD): Simulate the movement of the ions of asystem.Describe many properties.Classical MD:

Ions interact by classical forces.Parametrized force fields.Treat large systems.

Ab-initio MD:More precise.Access to electronic properties, including excited states.Limited system size and simulation times.

Introduction

Born-Oppenheimer molecular dynamics

Ions move in the Born-Oppenheimer surface (following classicalforces).Solve Kohn-Sham equations for each ionic configuration.Cubic scaling with system size.Time steps limited by ionic motion.

Car-Parrinello molecular dynamics†

A more efficient way: propagate wave functions.

Lagrangian

L =N∑

∫12µcp φ

2j dr +

R2I − E[φ,R]

µcp controls adiabaticity.Fictitious “newton like” electron dynamics.Wave function orthogonality has to be imposed.Cubic scaling with system size.Widely used method.

†Car and Parrinello, Phys. Rev. Lett. 55 2471 (1985)X. Andrade (UPV/EHU) From TDDFT to MD Benasque 2008 5 / 18

Lagrangian

L =N∑

∫12µcp φ

2j dr +

R2I − E[φ,R]

Lagrangian

L =N∑

∫12µcp φ

2j dr +

R2I − E[φ,R]

Lagrangian

L =N∑

∫12µcp φ

2j dr +

R2I − E[φ,R]

Lagrangian

L =N∑

∫12µcp φ

2j dr +

R2I − E[φ,R]

Lagrangian

L =N∑

∫12µcp φ

2j dr +

R2I − E[φ,R]

Lagrangian

L =N∑

∫12µcp φ

2j dr +

R2I − E[φ,R]

Ehrenfest dynamics

Ehrenfest equations of motion

i φj =[−1

2∇2 + veff(r, t; R)

MIRI = −N∑

〈φj |∂veff

∂RI|φj〉

Electrons propagated by real-time TDDFT.Ions follow Hellmann-Feynman forces.Non-adiabatic dynamics, adiabatic for large gap systems.Some nice properties for molecular dynamics:

Propagative scheme.Conservation of the energy.Conservation of wave function orthogonality.

Fast electrons: not practical for molecular dynamics.

Ehrenfest dynamics

i φj =[−1

2∇2 + veff(r, t; R)

MIRI = −N∑

〈φj |∂veff

∂RI|φj〉

Ehrenfest dynamics

i φj =[−1

2∇2 + veff(r, t; R)

MIRI = −N∑

〈φj |∂veff

∂RI|φj〉

Ehrenfest dynamics

i φj =[−1

2∇2 + veff(r, t; R)

MIRI = −N∑

〈φj |∂veff

∂RI|φj〉

Ehrenfest dynamics

i φj =[−1

2∇2 + veff(r, t; R)

MIRI = −N∑

〈φj |∂veff

∂RI|φj〉

Ehrenfest dynamics

i φj =[−1

2∇2 + veff(r, t; R)

MIRI = −N∑

〈φj |∂veff

∂RI|φj〉

Ehrenfest dynamics

i φj =[−1

2∇2 + veff(r, t; R)

MIRI = −N∑

〈φj |∂veff

∂RI|φj〉

Ehrenfest dynamics

i φj =[−1

2∇2 + veff(r, t; R)

MIRI = −N∑

〈φj |∂veff

∂RI|φj〉

Modified Ehrenfest dynamics‡

Given a positive parameter µ.

Lagrangian

L = iµ

N∑j=1

∫ (φ∗j φj − φ∗jφj

R2I − E[φ,R]

If µ→ 0 then L → Lbo

Equations of motion

i µ φj =[−1

2∇2 + veff(r, t; R)

MIRI = −N∑

〈φj |∂veff

∂RI|φj〉

‡Alonso, Andrade et al, Phys. Rev. Lett. 101 096403 (2008)X. Andrade (UPV/EHU) From TDDFT to MD Benasque 2008 7 / 18

Lagrangian

L = iµ

N∑j=1

R2I − E[φ,R]

Equations of motion

i µ φj =[−1

2∇2 + veff(r, t; R)

MIRI = −N∑

〈φj |∂veff

∂RI|φj〉

Lagrangian

L = iµ

N∑j=1

R2I − E[φ,R]

Equations of motion

i µ φj =[−1

2∇2 + veff(r, t; R)

MIRI = −N∑

〈φj |∂veff

∂RI|φj〉

Lagrangian

L = iµ

N∑j=1

R2I − E[φ,R]

Equations of motion

i µ φj =[−1

2∇2 + veff(r, t; R)

MIRI = −N∑

〈φj |∂veff

∂RI|φj〉

Physical interpretation

Electronic time

d (t/µ)

Two time scales: real for ions and fictitious for electrons.

Maximum time step

∆tmax(µ) = µ∆tmax(µ = 1)

Scaling of electronic excitation energies

ωi(µ) =1µωi(µ = 1)

Electronic time

d (t/µ)

Maximum time step

ωi(µ) =1µωi(µ = 1)

Electronic time

d (t/µ)

Maximum time step

ωi(µ) =1µωi(µ = 1)

Electronic time

d (t/µ)

Maximum time step

ωi(µ) =1µωi(µ = 1)

Properties

µ = 1Ehrenfest - TDDFTClose to Born-Oppenheimerfor large gap systems.

µ→ 0Excitation energies→ ∞Born-Oppenheimer

µ > 1µ times faster than TDDFT.Excitation energies get closer to vibrational modes.How close to the adiabatic regime?

µmax ∼ lowest excitation energyhighest vibrational frequency

Properties

Numerical properties

Simple to implement.No orthogonalization:

Quadratic scaling with system size.Naturally parallelizable in states:Each processor handles a group of orbitals.

Preserves time reversibility.Complex wave functions required.

Our implementation

Octopus code§.Ehrenfest dynamics:

Propagators from real-time TDDFT¶.Approximated enforced time reversal symmetry.Taylor approximation for the exponential.

Car-Parrinello:Velocity Verlet/RATTLE propagation.

Velocity verlet for the ions.Parallelized in domains and states.

§http://www.tddft.org/programs/octopus¶Castro et al, J. Chem. Phys.121 3425 (2004)

Our implementation

Vibration of the N2 molecule

1.9 2 2.1 2.2 2.3Interatomic distance (b)

-541.4

-541.2

-540.8

-540.6

-541.4

-541.2

-540.8

-540.6

gsBOµ=1

-541.4

-541.2

-540.8

-540.6

gsBOµ=1

-541.4

-541.2

-540.8

-540.6

gsBOµ=1

-541.4

-541.2

-540.8

-540.6

gsBOµ=1

-541.4

-541.2

-540.8

-540.6

gsBOµ=1

-541.4

-541.2

-540.8

-540.6

gsBOµ=1

-541.4

-541.2

-540.8

-540.6

gsBOµ=1µ=10

-541.4

-541.2

-540.8

-540.6

gsBOµ=1µ=10µ=20

-541.4

-541.2

-540.8

-540.6

gsBOµ=1µ=10µ=20µ=30

-541.4

-541.2

-540.8

-540.6

-541.4

-541.2

-540.8

-540.6

-541.4

-541.2

-540.8

-540.6

µmax = 27

-541.4

-541.2

-540.8

-540.6

Vibrational frequencies (cm−1)Experimental 2331µ = 1 2352µ = 10 2332µ = 20 2274µ = 30 2194

Comparison with Car-Parrinello

µ: same role as fictitious electronic mass in CP.Numerical values are not directly comparable:

Different units.Car-Parrinello: ∆tmax ∝

√µcp

Ehrenfest: ∆tmax ∝ µDeviation from BO:

Equivalence of µ and µcp.

Numerical cost per unit of simulated time:System size.Parallelizability.

√µcp

Infrared spectrum for benzene

Exp. [NIST Chemistry WebBook]

EhrenfestCar-Parrinello

0 1000 2000 3000

Frequency [cm-1

µ=1 µCP

µ=5 µCP

µ=10 µCP

µ=15 µCP

Computational cost comparison

Artificial system: array of Benzene moleculesµ = 15 and µcp = 750

100 200 300 400 500 600Number of atoms

Fast EhrenfestCar-Parrinello

Serial performance

0 500 1000 15000

Extrapolation

Xeon E5345 processor.

250 500 750 1000Number of atoms

Fast EhrenfestCar-Parrinello

Parallel performance

SGI Altix 32 Itanium 2 processors.

0 20 40 60 80 100 120Number of processors

IdealFast EhrenfestCar-Parrinello

Parallel scalability

480 Benzene molecules, SGI Altix Itanium 2 processors.

Conclusions

Simple and scalable method for ab-initio molecular dynamics.Adjustable parameter µ that controls adiabaticity.Retain good properties of Ehrenfest MD.Suitable for large scale simulations.Fictitious electron dynamics.

Conclusions

Prospects

Small gap and metallic systems.Understand thermodynamical limit.Import technical improvements from CP.Non-adiabatic dynamicsExcited states dynamics.Massively parallel implementation.

Prospects

Acknowledgements

European Theoretical Spectroscopy Facility.Nanoquanta Network of Excellence.Gobierno Vasco.Ministerio Educacion y Ciencia, Espana.Barcelona Supercomputing Center.

from tddft to molecular dynamics

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