Ab initio molecular dynamics via the Car-Parrinello method: Basic ideas, theory and

algorithms

Mark E. Tuckerman

Dept. of Chemistry

and Courant Institute of Mathematical Sciences

New York University, 100 Washington Sq. East

New York, NY 10003

1808: “We are perhaps not far removed from the time when we shall be able to submit the bulk of chemical phenomena to calculation.”

Joseph Louis Gay-Lussac (1778-1850)

“The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact solution of these laws leads to equations much to complicated to be soluble.”

Paul Dirac on Quantum Mechanics (1929).

“Every attempt to refer chemical questions to mathematical doctrines must be considered, now and always, profoundly irrational, as being contrary to the nature of the phenomena.”

August Comte, 1830

Motivation• Car-Parrinello is a method for performing molecular dynamics with

forces obtained from electronic structure calculations performed “on the fly” as the simulation proceeds. This is known as ab initio molecular dynamics (AIMD).

• As a result, AIMD calculations are considerably more expensive than force-field calculations, which only involve evaluation of simple functions of position.

• Force fields, although useful, are, with notable exceptions, unable to treat chemical bond breaking and forming events.

• Force fields often lack transferability to thermodynamic situations in which they are not designed to work.

• Polarization and manybody interactions included implicitly.

Total Cites = 4,812

R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985)

From ISI Citation Report

The “Universal” Hamiltonian

N Nuclei

M Electrons

ˆ ˆ ˆ ˆ ˆ ˆe n ee en nnH T T V V V

2 2

1 1

1 1

1 1 1ˆ ˆ 2 2

1ˆ ˆ ˆ ˆˆ ˆ

ˆ ˆˆ

M N

e i n Ii I I

M NI J

ee nni j I Ji j I J

M NI

eni I i I

T TM

Z ZV V

ZV

r r R R

r R

Operator Definitions:

Electronic: Nuclear:

Coupling:

Molecular energy levels

Complete energy level spectrum:

ˆ ( , ) ( , )H E x R x R

1 1

1 ,1 ,

,..., ,...,

, ,..., ,M N

z M z Ms s

r r r R R R

x r r

Notation:

Electron coordinates Nuclear coordinates

ˆ ˆ ˆ ˆˆ ˆ( ) ( , ) ( ) ( , ) ( , )e n ee en nnT T V V V E r r R R x R x R

,

1 0, , or , ,

0 1z is

Born-Oppenheimer Approximation

HMass disparity: 2000 eM m

Quasi adiabatic separability ansatz for wave function:

( , ) ( , ) ( ) x R x R R

Schrödinger equation separates if

( ) ( , )I I R x R . . .

ˆ ˆ ˆˆ ˆ( ) ( , ) ( , ) ( ) ( , )e ee enT V V r r R x R R x RElectrons in fixed back-ground nuclear geometry R

ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )n nnT V E R R R RNuclei on each electronichypersurface

à la W. H. Flygare, Molecular Structure and Dynamics

ε0

ε2

ε1 (no bound levels)

R

( )n R

Born-Oppenheimer (electronic) surfaces and nuclear energy levels

Vibrations

Rotations

Classical nuclear motion on an electronic surface

Consider the ground-state electronic surface 0 ( ) R

Nuclear Hamiltonian:

0ˆ ˆ ˆˆ ( ) ( )n nnT V R RH

“Demote” to a classical Hamiltonian:2

01

( , ) ( ) ( )2

NI

nnI I

VM

PP R R RH

Nuclear motion now given by Hamilton’s equations:

I II I

R P

P R H H

Classical nuclei (R,P)

Quantum electrons

0 ( , ) x R

Hellman-Feynman Theorem

Ground-state electronic surface as expectation value:

(e)0 0 0

ˆ( ) ( ) ( ) ( )H R R R R (e)ˆ ˆ ˆ ˆ( ) ( ) ( , )e ee enH T V V R r r R

(e)(e) (e)0 0 0

0 0 0 0

ˆˆ ˆ( ) ( ) ( ) ( ) ( ) ( )

I I I I

HH H

R R R R R R

R R R R

(e)0 0 0

0 0 0 0 0

ˆ( ) ( ) ( ) ( ) ( )

I I I I

H

R R R R R

R R R R

0

Because 0 0 0 0( ) ( ) 1 ( ) ( ) 0 I

R R R R

R

Kohn-Sham density functional theory

Except for very small systems, we cannot solve for the exact 0 1( ,..., )M x x

Density functional theory represents a compromise between accuracy and computational cost.

Wave function ansatz:1 1 1

0 1

1

( ) ( )

( ,..., )

( ) ( )

M

M

M M M

x x

x x

x x

1

/ 22 2

2 0 11 / 2

( ) ( ,..., ) ( )M z

M

M M is s i s

n M d d

r r r x x x

Single-particle orbitals: ( ) i i j ij x

Electron density:

Kohn-Sham density functional theory

Total energy functional:

[{ }, ] [{ }] [ ] [ ] [ , ]s H xc extE T E n E n E n R R

21 1 ( ) ( )[{ }] [ ]

2 2 '

( ) [ , ]

s i i Hi

ext II I

n nT E n d d

nE n Z d

r rr r

r r

rR r

r R

0{ }

,

( ) min [{ }, ] ij i j iji j

R E R

21( ) ( ) ( ) ( )

2 ( )KS i i i KS H xc extV V E E En

r r r r

r

Energy definitions:

Ground-state energy via constrained minimization

Kohn-Sham equations ( are eigenvalues of )i ij

Nuclei

Electrons

Start with nuclei Compute ,i n ,i n F

Propagate nuclei ashort time Δt with F

Add electrons

Add electrons

The Born-Oppenheimer Algorithm

2

( ) 2 (0) ( ) (0)I I I II

tt t

M

R R R F

e.g. Verlet:

The Car-Parrinello scheme

Avoid explicit minimization with a fictitious adiabatic dynamics for electronic orbitals:

2

1 ,

1,

2

M

i i I I ij i j iji I i j

L M E

R R

Lagrangian (note μ not a mass! It has units of energy x time2):

i ij j I Iji I

E EM

R

R

Equations of motion:

2

1 1

1

2

M N

i i I Ii I

M R 0

Conditions: 1) “Near” Born-Oppenheimer

“Seed” the CP equations of motion with initially minimized orbitals.

Energy Conservation in Born-Oppenheimer and Car-Parrinello dynamics

CP 5 a.uCP 10 a.u.

BO 10-6, 10 a.u.

CP 10 a.u.

BO 10-6, 100 a.u.

BO 10-5, 100 a.u.

BO 10-4, 100 a.u.

CP 10 a.u.

BO 10-6, 100 a.u.

System: 8 Silicon atoms Marx and Hutter, Modern Methods and Algorithms of Quantum Chemistry (NIC Series) 1, J. Grotendorst, ed. (Forschungszentrum, Jülich, 2000)

Energy conservation timing comparison

System: 8 Silicon atoms

Marx and Hutter, Modern Methods and Algorithms of Quantum Chemistry (NIC Series) 1, J. Grotendorst, ed. (Forschungszentrum, Jülich, 2000)

Adiabatic Dynamics

Consider a simple 2 degree-of-freedom system:

bath bath, bath bath,

bath, bath, bath, bath,

( , ) ( , ) ( , ) ( , )

( , , ) ( , ,

R

R

R R R R

R R R

p pR

m m

p F R F p p F R F p

G p T G p T

)R

Adiabatic conditions:

R Rm m T T R

bath, bath,( , ) ( , ) ( ) ( )RR R R

R R

p piL F R F R iL T iL T

m p m R p

Analysis of the dynamics

Liouville operator:

Subdivision of Liouville operator:

ref , bath,

ref , bath,

ref ,

( )

( )

( , ) ( , )

RR R R

R

RR

piL iL T

m

piL iL T

m R

iL iL F R F Rp p

Full phase-space vectorbath, bath,( , , , , , )R Rp p R

iL

evolves according to

ref ,RiL iL iL

Analysis of dynamics (cont’d)

( ) (0)iL tt e

ref,/ 2 / 2 3RiL t iL tiL tiL te e e e O t

ref ,/ 2 44 4 42limR R

R R R

ntt t tt FF F FiLiL t n pn p n p n pn

ne e e e e e

/ 2

0( / 2) (0) ( (0), ( ))

t

R R Rp t p dt F R t

Evolution of phase space over a time Δt characteristic of nuclear motion:

Trotter factorization:

Exact Trotter theorem:

Evolution of momentum:

Analysis of dynamics (cont’d)

( , )/ 2

( ,)0

( , )2 1 ( , ( )) ( , )

V Rt R

R V R

d F R edt F R t Z

t Rd e

2 // 2( , ) ( , )R

R R Rp mR RQ dp dR e Z R

1 1( ) ln ( , ) ln ( )

R

A R Z R P R

2

( ; ) exp min ( , )2

RR R R

R

pQ dp dR V R

m

Time-average equated to phase-space average:

Partition function for slow variable:

Adiabatic method for free-energy profiles: [L. Rosso, et al. JCP 116, 4389 (2002)]

Annealing property: 0, T

( 1/ 1/ )B R B Rk T k T

22 2 20

1( , )

2V R D R a R

Model Problem:

(0) ( )R Rv v t

(0) ( )v v t

10

5R

R

T T

m m

10

300

R

R

T T

m m

Methods: Plane-wave basis sets (periodic box, FFTs)

2, cut

1 2 1( ) | |

2i i

i ie c e ELV

k r k g rk g

g

nr g g

2cut

1 1( ) ( ) | | 4

2in n e E

V g r

g

r g g

, ,*,

i ij i I Iji I

E Ec c M

c

k k

g gkg

RR

Car-Parrinello

orbitals density

1

1 1 0

N N l l

I l I Il lI I l m l

v v v lm lm

r R r R r R

1

1

ˆ ( )

ˆ[ ]

( )

NI

extI I

M

ext i ext ii

II I

ZV

E n V

nZ d

rr R

rr

r R

1 1

ˆ[ ,{ }] ( ) [{ }, ]M N

pseud i pseud i I NLli I

E n V d n v E

r r r R R

pseud1 0

ˆN l

l II l m l

V v lm lm

r R

1 0

N l

l I I Il lI l m l

v v v lm lm

r R r R r R

l = 0

l = 1

l = 2

Eliminating core electrons

Why a real-space basis?

• Plane-waves are elegant but scale as N 2M

• Slow convergence of plane waves to the basis set limit.

• Ease of localizing orbitals.

• Ease of representing position-dependent operators.

• Exact representation of

• Common choice – Gaussians

2

2 2| | / 2

, , ,

( ) ( ; ) ( ; ) Iii I I

I

C G G N x y z e

r Rr r R r R

Selecting a real-space basis (why not Gaussians?)

• Retain simplicity of plane waves.

• Systematic convergence to the basis-set limit.

• Spatially localized for possible linear-scaling.

• Position independence and orthonormality.

• No BSSE

• For flexibility of use, seek noncompact support.

• Choice: Discrete variable representations (DVRs).

J. C. Light, et al. J. Chem. Phys. 82, 1400 (1985); Edwards, Tuckerman, Friesner, Sorensen, J. Comp. Phys. 110, 82 (1994).R. A. Friesner, Chem. Phys. Lett. 116, 39 (1985);Bacic and Light, Ann. Rev. Phys. Chem. 40, 469 (1989); J. T. Muckerman, Chem. Phys. Lett. 173,200 (1990); Colbert and Miller, J. Chem. Phys. 96, 1982 (1992); Light and Carrington, Adv. Chem.Phys. 114, 263 (2000); Littlejohn and Cargo, J. Chem. Phys. 117, 27, 37, 59 (2002); Varga, et al. Phys. Rev. Lett. 93, 176403 (2004).

Definition of a DVR

Plane-waves (at the Γ (k=0)-point) -- momentum eigenfunctions:

,

1( ) i

i iC eV

g rg

g

r

Discrete-variable representations (position eigenfunctions): Begin with a set ofN square-integrable orthonormal functions φi(x)

*

1

( ) ( ) ( )N

i i l i ll

u x a x x

On an appropriately chosen quadrature grid {x1,…,xN}

( ) iji j

i

u xa

Expand orbitals as:

, ,

( ) ( ) ( ) ( )ii lmn l m n

l m n

C u x u y u z r

Y. Liu, D. Yarne and MET, PRB 68, 125110 (2003); H. –S. Lee and MET, JPCA 110, 5549 (2006)

DVR convergence for a 32 water box vs. plane-waves with TM PPs

Force measure: 2

1

1 N

II

FN

F

DVR basis sets allow the complete basis set limit to be reached with the ease of plane waves

Is Exc = BLYP water overstructured?

Mantz, et. al. JPCB 110, 3540 (2006)Pseudopotentials: Troullier-Martins70 Ry cutoff

Grossman, et. al. JCP 120, 300 (2004)Pseudopotentials: Hamann (1989)85 Ry cutoff

Plane-wave basis (70-85 Ry cutoff)

Morrone and Car, PRL 101, 017801 (2008)Pseudopotentials: Troullier-Martins70 ry cutoff

Gaussians: TZV2PVandeVondele, et. al.JCP 122, 014515 (2005)

292 K318 K

Radial distribution functions for BLYP Water

DVR

Neutron

X-ray

H. –S. Lee and MET, JPCA 110, 549 (2006)H. –S. Lee and MET JCP 125, 154507 (2006).H. –S. Lee and MET JCP 126, 164501 (2007).Neutron: Soper, et. al. JCP 106, 247 (1997)X-ray: Hura, et. al. Chem. Phys. 113, 9140 (2000)

Grid = 753, t =60 ps

Ensemble: NVT, 300 K, μ = 500 au

r(Å)0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

2 2.5 3 3.5 4 4.5 5 5.5 6

DZVPDZVP+BSSE-BLYPSCP-BLYP

gO

O(R

)

R [Å]

When basis sets are too small!from C. J. Mundy (2008)

Grossman, et. al. JCP 120, 300 (2004)

From Akin-Ojo, et al. JCP 129, 064108 (2008)

Selected References

1. R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985)2. D. K. Remler and P. Madden, Mol. Phys. 70, 921 (1990)3. G. Galli and M. Parrinello in Computer Simulations in Chemical Physics (NATO ASI Series C) 397, 261 (1993)4. M. Parrinello, Solid State Commun. 102, 107 (1997)5. D. Marx and J. Hutter, Modern Methods and Algorithms of Quantum Chemistry (NIC Series) 1, J. Grotendorst, ed. (Forschungszentrum, Jülich, 2000)6. M. E. Tuckerman, J. Phys. Condens. Matter, 14, R1297 (2002)7. F. Krajewski and M. Parrinello, Phys. Rev. B 73, 041105 (2006)8. T. D. Kunhe, M. Krack, F. R. Mohamed and M. Parrinello, Phys. Rev. Lett. 98, 066401 (2007)9. H. –S. Lee and M. E. Tuckerman, J. Phys. Chem. A 110, 5549 (2006); J. Chem. Phys. 125, 154507 (2006); J. Chem. Phys. 126, 164501 (2007).10. E. Bohm, et. al. IBM J. Res. Devel. 52, 159 (2008)

Ab initio molecular dynamics codes:

CPMD: http://www.cpmd.orgCP2K: http://cp2k.berlios.deVASP: http://cms.mpi.univie.ac.at/vaspPINY_MD: http://www.nyu.edu/PINY_MD/PINY.htmlOpenAtom: http://charm.cs.uiuc.edu/OpenAtomNWChem: http://www.emsl.pnl.gov/docs/nwchem/nwchem.htmlSIESTA: http://www.lrz-muenchen.de/services/software/chemie/siesta