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. From ISI Citation Report R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985)

Total Cites = 4,812 The “Universal” Hamiltonian M Electrons

N Nuclei ˆ ˆ ˆ ˆ ˆ ˆ HTTVVVe  n  ee  en  nn

Operator Definitions:

Electronic: Nuclear: MN ˆˆ122 1 1 TTe   i n    I 22iI11MI MN1 ZZ VVˆˆ IJ ee nn ˆˆ i jrrˆˆij I J RRIJ

Coupling: MN Z Vˆ  I en  ˆ iI11rRˆiI Molecular energy levels

Electron coordinates Nuclear coordinates

Notation: r r11,..., rMN R R ,..., R

x r1,ssz ,1 ,..., r M , z , M 10    szi, ,  ,   or  , ,   01   

Complete energy level spectrum:

HEˆ (,)(,)x R   x R

TTVVVEˆ ˆ ()(,)()(,)(,)rˆˆ  r R ˆ  R ˆ  x R   x R e n ee en nn Born-Oppenheimer Approximation à la W. H. Flygare, Molecular Structure and Dynamics

Mass disparity: MmH  2000 e Quasi adiabatic separability ansatz for wave function:

(,)(,)()x R x R R

Schrödinger equation separates if 

II()(,)R x R . . . 

Electrons in fixed back- TVVˆ ˆ()(,)(,)()(,)rˆˆ  ˆ r R x R   R  x R ground nuclear geometry R e ee en   

Nuclei on each electronic TVEˆ()()()()RRRR ˆ  ˆ ˆ    hypersurface n nn    Born-Oppenheimer (electronic) surfaces and nuclear energy levels

n ()R

ε2

ε1 (no bound levels)

Vibrations Rotations

ε0

R Classical nuclear motion on an electronic surface

Consider the ground-state electronic surface 0 ()R

Nuclear Hamiltonian: ˆ ˆ ˆ ˆ H TVn 0 ()()RR  nn

“Demote” to a classical Hamiltonian: N 2 PI H (,)()()PRRR 0 Vnn I 1 2MI

Nuclear motion now given by Hamilton’s equations:

HH RPII   PRII Classical nuclei (R,P)

Quantum electrons

0 (,)xR

Hellman-Feynman Theorem Ground-state electronic surface as expectation value:

ˆ (e) ˆˆ(e) 0()()()()RRRR  0H  0 HTVV()()(,)Re  ee rˆˆ  en r R

ˆ (e) 0H   0ˆˆ(e) (e)   0 0()()()()()()RRRRRR  0 HH  0   0 RRRRIIII   

ˆ (e) 0H    0   0 0()RRRRR  0 ()   0()  0 ()   0 () RRRR    IIII  0

 Because 0(RRRR )  0 ( ) 1  0 ( )  0 ( ) 0 RI Kohn-Sham density functional theory

Except for very small systems, we cannot solve for the exact 01(xx ,...,M )

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

Wave function ansatz: 1()()xx 1 1 M

01(xx ,...,M )    

MMM()()xx1 

Single-particle orbitals: i(x )  i  j  ij

Electron density: 2 M /2 n()r M d r d r (,...,) x x () x 2   2M 0 1 M   i s1 sMz i1 s  /2 Kohn-Sham density functional theory Total energy functional:

E[{},]RR Ts [{}]  E H [] n  E xc [] n  E ext [,] n

Energy definitions: 1 1nn (rr ) ( ) T[{ }]   2  E [ n ]  drr d  s i i H  2i 2rr ' n()r E [ n ,Rr ] Z d ext I  I rR I Ground-state energy via constrained minimization  0 (RER ) min [{  }, ]  ij  i  j   ij {}    ij,

Kohn-Sham equations (i are eigenvalues of ij ) 1   2 VVEEE(r ) ( r )   ( r ) ( r )     2KS i i i KSn (r ) H xc ext The Born-Oppenheimer Algorithm

Electrons

Nuclei

 ,n Start with nuclei Add electrons Compute i i ,n  F

Add electrons Propagate nuclei a short time Δt with F

e.g. Verlet: t2 RRRFIIII(tt )  2 (0)  (  )  (0) MI

The Car-Parrinello scheme Avoid explicit minimization with a fictitious adiabatic dynamics for electronic orbitals:

Lagrangian (note μ not a mass! It has units of energy x time2): M 1   2 LME i  i   IRR I  ,    ij  i  j   ij  i1,2 I i j

Equations of motion: EE    i    ij  j M IR I   iIj R

Conditions: 1) “Near” Born-Oppenheimer

MN1   2  i  i M IR I   0 iI112

“Seed” the CP equations of motion with initially minimized orbitals. Energy Conservation in Born-Oppenheimer and Car-Parrinello dynamics

CP 5 a.u CP 10 a.u. BO 10-6, 10 a.u.

BO 10-6, 100 a.u. CP 10 a.u.

BO 10-5, 100 a.u.

BO 10-6, 100 a.u. CP 10 a.u.

BO 10-4, 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

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

System: 8 Silicon atoms Adiabatic Dynamics

Consider a simple 2 degree-of-freedom system:

p p   R R mm R

p F ( R , )  Fbath ( p  ,  bath,  ) pRRRR  F ( R , )  F bath ( p ,  bath, )

 bath, G(  bath,  , p  , T  )  bath,RRR  G (  bath, , p , TR )

Adiabatic conditions:  m mRR T T R Analysis of the dynamics

Full phase-space vector  (,,,,,)p pRR R bath,  bath, evolves according to  iL 

Liouville operator:

p  pR   iL  FR(,)(,)()()   FRRRR  iLTiLTbath,    bath, m  p mRR  R  p

Subdivision of Liouville operator:

p  iLref , iL bath, () T  m 

pR  iLref ,RRR iL bath, () T mRR   iL iLref ,   F  (,)(,) R  FR R pp R

iL iL iLref,R Analysis of dynamics (cont’d)

Evolution of phase space over a time Δt characteristic of nuclear motion: ( te ) iL t  (0)

Trotter factorization: eiL t  eiL t/2 eiLref,R t e iL t /2  O  t3 

Exact Trotter theorem:

n t   t t t  t  FFFF iL t/2RR iLref , 4 n p e lim e4n pRRR e 4 n  p e2n e e 4 n  p n 

Evolution of momentum: t /2 p( t / 2)  p (0)  dt F ( R (0), ( t )) RRR0 Analysis of dynamics (cont’d)

Time-average equated to phase-space average: ( 1/kBRBR T 1/ k T ) d F ( R , ) eVR(,) 21t /2  R  dt FR ( R , ( t )) Z (  ,  ) tR0 VR(,)  de 

Partition function for slow variable:

2 R /  Q( ,  ) dp dR eRRRpm/2  Z ( R ,  ) RR   

Adiabatic method for free-energy profiles: [L. Rosso, et al. JCP 116, 4389 (2002)] 11 ARZRPR()  ln(,)   ln()  R

Annealing property: T0,   2 pR Q(RRR ; )  dp dR exp    min V ( R ,  )   2mR TTR 10 

mmR  5  Model Problem: 2 1 V(,) R D R2  a 2   2   R  0   2

TTR 10  vRR(0) v ( t ) mmR  300 

v(0) v ( t ) Methods: Plane-wave basis sets (periodic box, FFTs)

1iik r k g r 2n 1 2 iikg(r )e c, e g  | g |  E cut V g L 2

11igr 2 n(r ) n ( g ) e | g | 4 Ecut V g 2

orbitals density

EE Car-Parrinello kk ci,,gg k*  ij c i M IR I   ciI,g j R N l = 0 Eliminating ˆ ZI Vext ()r  core electrons I 1 rR I l = 1 M ˆ Eext[] n   i V ext i l = 2 i1 n()r  Zdr  I  I rR Nl I ˆ Vpseud  vlI rR lm lm I10 l  m  l Nl  vl r  R I  vll r  R I  v r  R I  lm lm I10 l  m  l N N l1 l vll r  RI    v l r  R I  v r  R I  lm lm I1 I  1 l  0 m  l

MN E[,{}] n  Vˆ   dr n() r v r  R  E [{},]  R pseud i pseud i l  I NL iI11 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 2

• Common choice – Gaussians

22 i    |rRI | /2 i()(;)(;)r C G  r R I G  r R I N  x y z e ,,,   I 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 igr ii()r  Ceg, V g

Discrete-variable representations (position eigenfunctions): Begin with a set of

N square-integrable orthonormal functions φi(x)

N * ui()()() x a i l x i l x l1

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

ij uxij() ai Expand orbitals as:

i i()()()()r   C lmn u l x u m y u n z l,, m n

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

N 1 2 Force measure: F   FI N I 1 DVR basis sets allow the complete basis set limit to be reached with the ease of plane waves Is Exc = BLYP water overstructured? Plane-wave basis (70-85 Ry cutoff)

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

Mantz, et. al. JPCB 110, 3540 (2006) Pseudopotentials: Troullier-Martins 70 Ry cutoff

292 K Morrone and Car, Gaussians: TZV2P 318 K PRL 101, 017801 (2008) VandeVondele, et. al. Pseudopotentials: JCP 122, 014515 (2005) Troullier-Martins 70 ry cutoff Radial distribution functions for BLYP Water

3, DVR Grid = 75 t =60 ps Neutron X-ray Ensemble: NVT, 300 K, μ = 500 au

3.5

3.0 DZVP DZVP+BSSE-BLYP SCP-BLYP 2.5

2.0

(R) OO g 1.5

1.0

0.5 Grossman, et. al. JCP 120, 300 (2004) 0.0 r(Å) 2 2.5 3 3.5 4 4.5 5 5.5 6 R [Å] H. –S. Lee and MET, JPCA 110, 549 (2006) From Akin-Ojo, et al. JCP 129, 064108 (2008) 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) When basis sets are too small! X-ray: Hura, et. al. Chem. Phys. 113, 9140 (2000) from C. J. Mundy (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.org CP2K: http://cp2k.berlios.de VASP: http://cms.mpi.univie.ac.at/vasp PINY_MD: http://www.nyu.edu/PINY_MD/PINY.html OpenAtom: http://charm.cs.uiuc.edu/OpenAtom NWChem: http://www.emsl.pnl.gov/docs/nwchem/nwchem.html SIESTA: http://www.lrz-muenchen.de/services/software/chemie/siesta