ab initio molecular dynamics via the car-parrinello method: basic ideas, theory and algorithms mark...
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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