Physics 250-06 “Advanced Electronic Structure”

Density Functional Molecular Dynamics

Contents:

1. Methods of Molecular Dynamics

2. Car-Parrinello Method.

3. Examples of Simulations.

4. Challenges and Future DevelopmentsFor a review, see, Ab initio molecular dynamics: Theory and Implementation by Dominik Marx and Jurg HutterModern Methods and Algorithms of Quantum Chemistry,J. Grotendorst (Ed.), John von Neumann Institute for Computing, Julich, NIC Series, Vol. 1, pp. 301-449 (2000) (available on the WEB)

Foundations: Motions of Nuclei

Non-relativistic hamiltonian including nuclei and electrons

Wave functions become functionals of electronic and nucleardegrees of freedom

Born-Oppenheimer approximation separates slowand fast degrees of freedom (MR>>me)

2 2 22 2 '

'

1 1 1 1

2 | | 2 ' | | |R R R

R iR i i j R R iRR e i j i

e Z Z e Z eH

M m r r R R r R

1 1 1 1( ... ... ) ( ... ... )M N M NH r r R R E r r R R

1 1 1 1( ... ... ) ( ... ) ( ... )M N M Nr r R R r r R R

Molecular Dynamics

Classical equations of motions for the nuclei need to be solved

({ }{ }) ({ }{ }) ({ }{ })R RM R r R H r R r R dr FR

where forces on nuclei are given by the derivative of the average total electronic energy with respect to the nucleirposition

1

1

{ ... } ({ }{ }) ({ }{ }) ({ }{ })

{ ... }i

N

R Ni

E R R r R H r R r R dr

F E R RR

Forces

Thus, integrating classical equation of motionswe will be able to predict trajectories of atomsand answer the questions of equilibrium atomicconfigurations, melting temperatures, phase diagrams,diffusions processes, and so on.

The major difficulty is to find forces which evolve as nuclearcoordinates change their positions:

1( ) { ( )... ( )}R R NM R t F R t R t

Classical Molecular Dynamics

1

(1) (2) (3)1

' ' ''

{ ... } ({ }{ }) ({ }{ }) ({ }{ })

( ... } ( ) ( ') ( ' '') ...

N

NR RR RR R

E R R r R H r R r R dr

V R R V R V RR V RR R

In classical molecular dynamics, the average electronicenergy as a functional of nuclear coordinates is parametrizedin terms of various (one-,two-,three-,etc- body) interatomic potentials

which, for example, can be found using various fits to observable properties.

Typical two-body interactions (Lennard-Jonesvan der Waals), look like that

( ')V R R

'R R

Equilibrium

( ') ~| ' | | ' |M N

a bV R R

R R R R

M N

Simulation of interaction energy of argon dimer

is represented by van der Waals interactions

Dimensionality bottleneck

1 1 1 500 500 500( ... }x y z x y zV R R R R R R

Decomposition of global potential energy onto pair-likepotentials is crucial if one wants to avoid dimensionalitybottleneck. 500 argon atoms need a fit of a function of 1500coordinates

while pair-potential representation needs a fit of a functionin one dimension only:

(2)1 500

'

( ... } ( ')RR

V R R V R R

Ab Initio Molecular Dynamics

1 1{ ... } ({ }{ }) ({ }{ }) ({ }{ }) [ ( ),{ ... }]N DFT NE R R r R H r R r R dr E r R R

In ab initio molecular dynamics, the average electronicenergy as a functional of nuclear coordinates is recomputedusing quantum mechanical methods such as Density Functional Theory

There are two intrinsically different methods to integrateelectronic degrees of freedom while performing simulations:

• Born-Oppenheimer Molecular Dynamics

• Ehrenfest Molecular Dynamics

Born-Oppenheimer Molecular Dynamics

In Born-Oppenheimer molecular dynamics, electronicstructure problem is solved self-consistently each timefor a given set of configurations of nuclei

00 0

0 0 0

min { } | | { }

({ }{ }) ({ }{ })

RM R R H RR

H r R E r R

In other words electrons are allowed to fully relax and reachits minimum for a given position of nuclei.

Consider calculation of force using Hellmann-Feynman Theorem

0 0 0 0

0 00 0 0 0

0 00 0 0 0 0 0

0 0 0 0 0 0 0

| | | |

| | | | | |

| | | |

| | | | |

RF H HR RH

H HR R RH

E ER R RH H

ER R R

Recall the expression for the hamiltonian2 2 2

2 2 '

'

1 1 1 1

2 | | 2 | ' | | |R R R

R iR i i j R R iRR e i j i

e Z Z e Z eH

M m r r R R r R

its derivative with respect to R is trivial

2 2'

' | ' | | |R R R

R RR i i

H Z Z e Z e

R R R r R

and the force is given by

0 0

2 2'

0 1 0 1 1'

2 2'

'

| |

( ... ) ( ... ) ...| ' | | |

( )| ' | | |

R

R R RR M R M M

R i i

R R RR R

R

HF

R

Z Z e Z er r r r dr dr

R R r R

Z Z e Z er dr

R R r R

where we used a defintion of electronic density:

0 2 0 2 2( ) ( , ... ) ( , ... ) ...M M Mr r r r r r r dr dr

2 2

( ) ( ) ( ) ( )| | | |

R RR R R R

Z e Z er dr r dr V R Z E R

r R R r

Remarkably that the electronic contribution to the force on a given nuclei is represented by a classical electrostatic force made by electronic chargedensity cloud on the charge ZR centered at R:

Thus, density functional theory is ideally suited for performingab initio molecular dynamics simulations!

Born-Oppenheimer molecular dynamics simulations using DFTinvolve the following steps:

1. Fix positions of nuclei, {R1…RN}, solve DFT equationsself-consistently

22

2

( ') '[ ( )] ( ) ( )

| | | ' |

( ) | ( ) |

Rxc i i i

R

i ii

Z e r drV r r r

r R r r

r f r

2. Find electrostatic force on each atom:2 2

'

'

( )| ' | | |

R R RR R R

R

Z Z e Z eF r dr

R R r R

3. Integrate equation of motion for the nuclei: perform a timestep and find new positions of nuclei

1( ) { ( )... ( )}R R NM R t F R t R t

MD

Sim

ula

tion

Born-Oppenheimer molecular dynamics became popular since many electronic structure codes are available

Obvious drawback: necessity to fully relax electronicsubsystem while moving the atoms. This makes it computationally very slow.

Full self-consistency at each MD step may not be necessaryespecially when system is far from its equilibrium, onesimply needs a rough idea on the force field for a given atomic configuration

Construction of potential energy surface is avoided sinceforces are found “on the fly”

Ehrenfest Molecular Dynamics

In Ehrenfest molecular dynamics, electronic structure problem is solved using time-dependent Schrodinger equation andboth equations for nuclei and electrons are solved simultaneously

( ,{ }) | | ( ,{ })

( ,{ }{ })( ,{ }{ })

RM R t R H t RR

t r Ri H t r R

t

Construction of potential energy surface is avoided by findingforces “on the fly”

Let expand the electronic wave functions into full set of Slaterdeterminants ( ,{ }{ }) ( ) ({ }{ })

({ }{ }) ({ }{ })

k kk

k k k

t r R c t r R

H r R E r R

Thus, Ehrenefest molecular dynamics tracks the time dependence of the coefficients ck(t) as the system evolves with time. The corresponding equations of motions read as 2

,

2 2

| ( ) | ( ) ( )( ) ( )

( ) ( ) { } ( ) ( )

( )| | | |

R k k k l k l klk k l

k k k l klR l

k R Rk k k

M R c t E c t c t E E d RR

ic t c t E R i c t Rd R

E Z e Z er

R r R r R

( ( ))k l k ld R tR

Equations are coupled via

Thus the approach includes all non-adiabatic transitions between various states Ek during nuclear motions

If only one state with the lowest energy is kept, we obtain

0 0

00

{ } | | { }

({ }{ }){ ( }) ({ }{ })

RM R R H RR

r Ri H R t r R

t

which needs to be integrated simultaneously sincethe hamiltonian depends on time via nuclear coordinateswhich are time dependent!

The last point makes the Ehrenfest molecular dynamics fundamentally different from Born-Oppenheimermolecular dynamics.

Ehrenfest molecular dynamics simulations using time dependent version of DFT involve simultaneous integration:

22

2

( ', ) ' ( , )[ ( , )] ( , )

| ( ) | | ' |

( , ) | ( , ) |

R ixc i

R

ii

Z e r t dr r tV r t r t i

r R t r r t

r t r t

where electrostatic force on each atom can be evaluated:

2 2'

1'

{ ( )... ( )} ( , )| ( ) '( ) | | ( ) |

R R RR N R R

R

Z Z e Z eF R t R t r t dr

R t R t r R t

1( ) { ( )... ( )}R R NM R t F R t R t

Comparing Ehrenfest molecular dynamics simulations using time dependent DFT and Born-Oppenheimer moleculardynamics using static DFT we see that in TD-DFT Kohn-Sham states are represented by frequency integrals

( , ) ( , ) i ti ir t r e d

while in static DFT frequency-dependent functions are peakedat a given Kohn-Sham eigenvalue:

( , ) ( , ) ( )

( , ) ( , ) i

i i i i

i ti i i

r r

r t r e

The main task achieved in Ehrenfest dynamics is simply to keep the wavefunction automatically minimized as the nuclei are propagated. This, however, might be achieved in principle by another sort of dynamics, namely Car-Parrinello molecular dynamics. In summary, the “Best of all Worlds Method" should

(i) integrate the equations of motion on the (long) time scale set by the nuclear motion but nevertheless

(ii) take intrinsically advantage of the smooth time evolution of the dynamically evolving electronic subsystem as much as possible.

The second point allows to circumvent explicit diagonalization or minimization to solve the electronic structure problem for the next molecular dynamics step as it is done in Born-Oppenheimer Molecular dynamics. Car-Parrinello molecular dynamics is an efficient method to satisfy requirement (ii) in a numerically stable fashion and makes an acceptable compromise concerning the length of the time step (i).

Car-Parrinello Molecular Dynamics

In CPMD a two-component quantum / classical problem is mapped onto a two-component purely classical problem with two separate energy scales at the expense of loosing the explicit time dependence of the quantum subsystem dynamics.

Now, in classical mechanics the force on the nuclei is obtained from the derivative of a Lagrangian with respect to the nuclear positions. This suggests that a functional derivative with respect to the orbitals, which are interpreted as classical fields, might yield the force on the orbitals, given a suitable Lagrangian. In addition, possible constraints within the set of orbitals have to be imposed, such as e.g.orthonormality (or generalized orthonormality conditions that include an overlap matrix).

Car-Parrinello Lagrangian

1 1| { } ( | )

2 2CP R i i i DFT i ij i i ijR i ij

L M R E

Car and Parrinello (1985) have postulated the following Lagrangian

where i are fictitious masses and i are classical fields. Classical action needs to be minimized which results in equation of motions

( )

( ) ( )

( ) ( )

( , ) ( , )

CP

CP CP

CP CP

i i

S L t dt

d dL t dL t

dt dR dRd dL t dL t

dt d r t d r t

Kinetic Energy Potential Energy Constraints

2[{ },{ }]( , )

| ( ) |

[{ },{ }]( , ) ( , )

( , )

( , ) ( , )

DFT i RR

DFT ii i ij j

ji

DFT i ij jj

E R Z eM R r t dr

R r R t

E Rr t r t

r t

H r t r t

Car-Parrinello Equations of Motions

Car and Parrinello (1985) have derived equations of motions

which are obviously transformed back to Born-Oppenheimer molecular dynamics if fictitious masses for the electrons i

At the eqilibrium, there are no forces on electrons, thereforeGround state of density functional theory is reached with theeigenvalues being the Kohn Sham eigenstates.

Why does the Car-Parrinello Method works?

Conserved Energy in CPMD is not a physical energy but supplemented with a small fictitious kinetic term

1{ }

2

1 1| [{ }{ }]

2 2

1|

2

phys R DFT iR

cons R i i i DFT i phys fictR i

fict i i ii

E M R E

E M R E R E T

T

Various Energies extracted from CPMD for a model system.

Tfict

EDFT

The fictitious kinetic energy of the electrons is found to perform bound oscillations around a constant, i.e. the electrons do not heat up“ systematically in the presence of the nuclei; note that Tfict is a measure for deviations from the exact Born-Oppenheimer surface. Closer inspection shows actually two time scales of oscillations: the one visible in the Figure stems from the drag exerted by the moving nuclei on the electrons and is the mirror image of the EDFT fluctuations. As a result the physical energy (the sum of the nuclear kinetic energy and the electronic total energy which serves as the potential energy for the nuclei) is essentially constant on the relevant energy and time scales.

Given the adiabatic separation and the stability of the propagation, the centralquestion remains if the forces acting on the nuclei are actually the “correct" onesin Car-Parrinello molecular dynamics. As a reference serve the forces obtainedfrom full self-consistent minimizations of the electronic energy at each time step, i.e. Born-Oppenheimer molecular dynamics with extremely well converged wavefunctions.

How to control adiabaticity?

Since the electronic degrees of freedom are described by much heavier masses than the electronic masses, time step to perform CPMD simulations needs not to be too small as compared to Ehrenfest molecular dynamics.

For a system with a gap in the spectrum, the lowest possible frequency of “fictitious” electronic oscillations

2min

1/ 2

min

~

~

gap

gap

E

E

To guarantee adiabatic separation this frequency should bemuch larger than the typical phonon energy and/or the gapin the spectrum which would make sure that the electronsfollow the nuclei adiabatically. Hence fictitious mass .

At the same time, small fictitious mass would implysmaller and smaller time step because maximum fictitious electronic frequency is proportional to the plane-wave cutoff energy

2

1/ 2

max

1/ 2

max

~

~

~

max cut

cut

cut

E

E

tE

As a result a compromise fictitious mass needs to be foundin CPMD simulations.

For metals gap is zero and zero frequency “fictitious” electronicmodes occur in the spectrum overlapping with the phonon spectrum. Thus, a well-controlled Born-Oppenheimer approach can only be recommended

| |H

n nn

c

2

| |

| | 1

n n m mnm

nnm

c H c

c

Consider variational principle which can be used to find an upper bound for the lowest eigenstates of the hamiltonian

using basis set expansion

CP Method as dynamical solution of DFT equations

CP Method invented a new way to solve Kohn-Shamequations alternative to diagonalization.

CPMD offers a way to determine the coefficients withoutreduction to the eigenvalue problem.

In traditional molecular dynamics the system heated attemperature T is gradually cooled and find its minimumenergy configuration.

(after Payne et.al, RMP 1992)

In CMPD scheme the total energy is a functional of the coefficients which expands the wave function in some basisset.

Each coefficient is regarded as a coordinate of aclassical particle. To minimize the KS functional each particleis given some kinetic energy and the system is gradually cooledUntil the set of “coordinates” c reaches its values that minimizethe functional.

Thus the problem of solving KS equations is reduced to solving “fictitious” classical equations of motions

n nn

c

(after Payne et.al, RMP 1992)

Flowchart of the CPMD algorithm (Payne et.al, RMP 1992)

How KS states converge: (CP, PRL 1985)

What about Hellmann-Feynman forces?

Derivation assumes that the wave functions are exact solutionsof the Schrodinger equation. On the language of the DFT

0 0

0 00 0 0 0

0 0 0 0 0 0 0

2

| |

| | | | | |

| | | | |

( )| |

RF HRH

H HR R RH H

ER R R

Zer

r R

[ ,{ }][ ,{ }] [ ,{ }] DFT

R DFT DFT

d E RF E R E R

dR R R

The last contribution is equal to zero only if self-consistencyis reached

Non-Self-Consistency Force

In general, the force is made of Hellmann-Feynman contributionand contribution due to non-self-consistency (also due to incompleteness of the basis set which we discuss later):

R HF NSCF F F

2

( )| |

( )[ ]

HF

NSC SCF NSC

ZeF r

r R

F r V V

So, in Born-Oppenheimer MD, well convergent self-consistentcalculations are needed to eliminate the non-self-consistency correction.

The crucial point is, however, that in Car-Parrinello as well as in Ehrenfest molecular dynamics it is not the minimized expectation value of the electronic Hamiltonian that yields the consistent forces. What is merely needed is to evaluate the expression with the Hamiltonian and the associated wavefunction available at a certain time step In other words, it is not required (concerning the present discussion of the contributions to the force!) that the expectation value of the electronic Hamiltonian is actually completely minimized for the nuclear configuration at that time step. Whence, full self-consistency is not required for this purpose in the case of Car-Parrinello (and Ehrenfest) molecular dynamics. As a consequence, the non-self-consistency correction to the force is irrelevant in Car-Parrinello(and Ehrenfest) simulations.Heuristically one could also argue that within Car-Parrinello dynamics the non-vanishing non-self-consistency force is kept under control or counterbalanced bythe non-vanishing “mass times acceleration term"

which is small but not identical to zero and oscillatory. This is sufficient to keep the propagation stable, whereas , i.e. an extremely tight minimization,

is required by its very definition in order to make the Born-Oppenheimer approachstable. Thus, also from this perspective it becomes clear that the fictitious kinetic energy of the electrons is a measure for the departure from the exact Born-Oppenheimersurface during Car-Parrinello dynamics.

i i

0i i

00 0min { } | | { }R H R

Error Cancellation in Hellmann-Feynman forces

Publications on Molecular Dynamics

Examples of Simulations: Carbon (Galli, et.al)

Examples of Simulations: Water (Sprik et.al.)

Current Challenges

• Simulations with basis sets different from plane waves. Difficulties in determining forces.

• Molecular dynamics of strongly correlated systems:better than DFT functionals needed. Again, d-, and f-electronsare better represented with local orbital basis sets, determinations of forces would complicate the simulations.