Lecture 9: time-dependent SCFand ab initio molecular dynamics

Runge-Gross theorem, TD-DFT, Coulomb-attenuating methods,TD-HF, symplectic integrators, Krylov propagation, BOMD, CPMD

There are horrible people who, instead of solving a problem, tangle it up and make

it harder to solve for anyone who wants to deal with it. Whoever does not know how

to hit the nail on the head should be asked not to hit it at all.

Friedrich Nietzsche

Dr Ilya Kuprov, University of Southampton, 2012

(for all lecture notes and video records see http://spindynamics.org)

Time-dependent DFTRunge-Gross theorem: there exists a one-to-one mapping between the time-dependent density of an evolving system and the time-dependent potential in whichthe system evolves.

v r r

, ,v r t r t

Hohenberg-Kohn theorem (DFT)

Runge-Gross theorem (TD-DFT)

The potential is defined up to an arbitrary function of time (but not coordinates),which only contributes an overall phase to the wavefunction. Similarly to DFT, theobservables are functionals of the time-dependent density:

ˆ ˆO t t O t O t

By analogy with the classical (Lagrangian) mechanics, the energy minimum principleused in DFT is replaced by the action minimum principle in TD-DFT:

0

ˆ ˆ t

H E t i H t t dt A tt

E. Runge, E.K.U. Gross, http://dx.doi.org/10.1103/PhysRevLett.52.997

Time-dependent DFTThe Kohn-Sham orbitals making up the density become time-dependent:

The potential also becomes time-dependent and now involves the exchange-correlation action rather than energy:

Finally, the Kohn-Sham orbitals now obey a TDSE-like equation:

KS KS KS KS KS KSˆ ˆ , , ,i i i i iH r r r r t iH r t r tt

2 2KS KS= , = ,i i i ii i

r f r r t f r t

KS 3 XC[ ]( )( ) ( ) d( )

ErV r V r rr r r

KS 3 XC[ ]( , )( , ) ( , ) d( , )Ar tV r t V r t r

r r r t

The general task of solving this in the time domain is, however, formidable.

M.A.L. Marques, E.K.U. Gross, http://dx.doi.org/10.1146/annurev.physchem.55.091602.094449

Time-dependent DFT - linearizationIf the time-dependent term in the Hamiltonian is weak, that is:

then TD-DFT equations may be simplified using linear response theory on top of DFT:

ˆ ˆ ˆ ˆ ˆ, , ,H r t H r H r t H r t H r

And so the exchange-correlation energy functionals from the time-independent theorymay be used to approximate the full time- and space-dependent action functional.We can now apply a perturbation (typically an electric field) at specific frequenciesand look for resonances in the linear response. These resonances would correspondto vertical excitation energies.

, , , ,v r v r t r r t v r t r t

0

XC XC XCXC 0 XC 0, , ,

, ,A E vv r r t v r r tr t r t

In the adiabatic approximation to first order in the density perturbation, we have:

S.J.A. van Gisbergen, J.G. Snijders, E.J. Baerends, http://dx.doi.org/10.1016/S0010-4655(99)00187-3

Time-dependent DFT - performancePerformance of TD-DFT for excited states:o Excitation energies to ±0.3 eVo Bond lengths to ±1%o Dipole moments to ±5%o Vibrational frequencies to ±5%o Much better scaling than CI and CCo Not accurate with many solidso Charge transfer excitations are often incorrect

An example of a charge transfer excitation.

Coulomb-attenuated func-tionals (CAM-B3LYP, etc.)must be used for chargetransfer excitations.With conventional function-als (PBE, B3LYP, etc.), theenergies of such excitationsend up being systematicallyunderestimated.

C. Jamorski, M.E. Casida, D.R. Salahub, http://dx.doi.org/10.1063/1.471140

Long-range corrected XC functionalsThe asymptotic behaviour of the exact exchange potential at long distancesmatches the Coulomb law (–r12

–1):

* *1 1 2 2 3HF 3

1 21 2

X1[ ]2

i j j i

ij

r r r rd rd r

rE

r

However, many DFT exchange functionals do not exhibit this asymptotic behaviour:

1/3LDA 4/3X

33 3[ ] ( )4

E r d r

2B88 LDA 4 3

2 4/X 3/3

X , 1

[ ] [ ]E x d r xx

E

This is a serious problem – many long-range properties (particularly non-covalentinteractions and charge transfer excitations) are not properly reproduced by DFT as aresult. Even the generally reliable B3LYP asymptotically behaves as –0.2r12

–1 ...

density part

distance part

B3LYP LDA HF LDXC XC X X

A GGA LDA GGA LDAX C0 X CCXE E a E E a E E a E E

F. Jensen, Introduction to Computational Chemistry, Wiley, 2007.

Long-range corrected XC functionalsAn empirical correction may be introduced by blending the short-range DFTexchange with long-range Hartree-Fock exchange. Ewald split is commonly used:

212 12

12 12 12 0

1 erf erf1 2, erfx

tr rx e dt

r r r

Multiplying the two terms by different functionals creates a smooth transitionbetween approximations. A generalization, known as Coulomb-attenuating method(CAM) appears to be particularly successful:

12 12

12 12 12

1 + erf + erf1 r rr r r

T. Yanai, D.P. Tew, N.C. Hardy, http://dx.doi.org/10.1016/j.cplett.2004.06.011

Time-dependent DFT - performance

Charge transfer excitation energies for the molecule in the previous slide.

D. Jacquemin, V. Wathelet, E.A. Perpete, C. Adamo, http://dx.doi.org/10.1021/ct900298e

Time-dependent Hartree-FockIn TD-HF the time-dependent Schrödinger equation is propagated numericallyforward in time using the Hartree-Fock approximation for the Hamiltonian:

ˆˆ, , , , ,kk

r t iH r t r t A r tt

ext

2

ˆ ˆ, ,

1ˆ ˆ ˆ2

k k

Nk k i j k

N i kkN

r t i f V t r tt

Zf J k K kr

Because the orbitals are orthonormal, they may be propagated separately usingthe Fock operator from the Hartree-Fock theory:

For piecewise-constant Hamiltonians (usually a good approximation), linearequations of Schrödinger type have a “simple” general solution:

ˆ ˆ, , , exp ,r t iH r t r t t iH t r tt

X. Li et al., http://dx.doi.org/10.1039/b415849k

Time-dependent Hartree-FockIn the absence of perturbations, Hartree-Fock orbitals are stationary:

ˆ, , , , ,0ki tk k k k k kr t if r t i r t r t e r

t

With different perturbations, this equation may beused to calculate the dynamic response properties(polarizability, coordinate derivatives, etc.).When used with more complicated reference methods(MCSCF, CC, etc.), this formalism is also known aspolarization propagator (PP) and equation of motion(EOM) methods.PP and EOM methods inherit the scaling of thereference method – this makes TDHF and TDDFT veryattractive in practical calculations.

2 3

ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆexp exp [ , ] [ ,[ , ]] ...2 6t iti f kV t i f kV t iVt f V f f V

k

Under a weak perturbation (deriving this equation is not easy):

T. Korona, http://dx.doi.org/10.1039/C0CP00474J

Unitary propagation problem

Because the exponential of an anti-Hermitian matrix is unitary and inherits all thesymmetries of the Hamiltonian, all conservation laws are guaranteed to hold. Thematrix exponential, however, is often prohibitively expensive:

The time-dependent Schrödinger equation has a “simple” general solution:

0

ˆ ˆ ˆ exp 0t

t iH t t t T i H t dtt

The time-ordered exponential is notoriously hard to compute and most practicalsolutions assume piecewise-constant Hamiltonian with a small enough time step:

ˆexpt t iH t t

0

0ˆ̂ 0

00

! ! ˆ̂! ! !

! ! ˆ̂! ! !

jp

jiHjq

j

p q j piH

p q j p je

p q j piH

p q j p j

0ˆ̂

0 00

ˆ̂2 1iH nn n n

n

e i J T H

0ˆ̂

0

0

ˆ̂( )!

niH

n

iHen

Padé

Taylor

Chebyshev

D. Tannor, Introduction to Quantum Mechanics: a Time-Dependent Perspective, USB, 2007.

Unitary propagation problemThe problem can be partially avoided by re-ordering the multiplication operations:

0 0

i iˆ ˆ ˆ ˆ ˆexp i ...! !

n nn

n n

tH t H H H H

n n

(a slight generalization of this is known as Krylov method) but this only gets thescaling of one stage down from n3 to n2 in matrix dimension. Some simplificationcan be made using symplectic integrators based on CBH formula:

1

1

ˆ ˆ ˆ ˆ ˆexp exp exp expn

NN

n niH t i T U t ic T t id U t O t

symplectic Eulermethod

1

1

1 2

1 2

11

1 1, 2 21, 0

cd

c c

d d

Verletmethod

Symplectic integration is cheaper (though lessaccurate) than exponential propagation, butstill approximately obeys all conservation laws.This is not the case with Euler or RK methods.Still, any integrator other than the Hamiltonianexponential would in practice display energyand momentum drift over long time scales.

D. Tannor, Introduction to Quantum Mechanics: a Time-Dependent Perspective, USB, 2007.

Unitary propagation problem

It is important to understand thatthe trajectory is accurate up to thetruncation error of the method, e.g.:

ˆ ˆˆ ˆ ˆ 32 2t tiT iTiT t iU t iU te e e e O t

posi

tion

timeThe parameter that all approximatemethods quickly lose is phase.

posi

tion

D. Tannor, Introduction to Quantum Mechanics: a Time-Dependent Perspective, USB, 2007.

2

2

ˆmin2

min ,2 i

N NN

N

N Ni N

N

m vL H r

m vL E r

Born-Oppenheimer molecular dynamicsWithin the Born-Oppenheimer approximation, the system Lagrangian is:

density functional theory

ab initio calculation

This produces the following equation of motion for the nuclei:

2

2

2

2

ˆmin

min ,i

kk k N

kk k i N

rm H rtrm E rt

density functional theory

ab initio calculation

This may be solved forward in time using any of the methods mentioned above.BOMD method tends to be expensive because analytical derivatives of the totalmolecular energy with respect to nuclear coordinates are required at each step.

D. Marx, J. Hutter, Modern Methods and Algorithms of Quantum Chemistry, Inst. Comp., 2000.

Car-Parrinello molecular dynamicsThe following Lagrangian was postulated by Car and Parrinello:

2

,2 2

i i iN Ni N ij i j ij

N i ij

m vL E r

electron“kinetic energy”

potentialenergy

nuclear kineticenergy

From Lagrangian mechanics we get the following equations of motion:

N N

i i

d L Ldt v rd L Ldt

N N ij i jijN N

i i ij jji

Em rr rE

orbitalorthogonality

The downside of the CP approach is that the dynamics would deviate from theBorn-Oppenheimer surface. The advantage over BOMD is that expensive energyminimization is no longer required at each step.

R. Car, M. Parrinello, http://dx.doi.org/10.1103/PhysRevLett.55.2471

Car-Parrinello molecular dynamicsThe requirement to maintain adiabaticity (i.e. to stay near the Born-Oppenheimersurface) means that the electron subsystem must be kept “cold” and there shouldbe no exchange of energy between the nuclear and the orbital degrees of freedom:

2 2

, min ,2 2 2 i

i i iN N N Ni N i N

N i N

m v m vE E r E r

To avoid energy exchange, the power spectra of the two subsystems should notoverlap – the lowest orbital frequency must be far from the highest nuclear one:

1/2 1/2min maxe n

2 i j LUMO HOMO

ijE E

The simplest way to achieve this is to reduce the fictitious mass. This, however,requires smaller time steps, because the highest electron frequency also goes up:

1/2max minmax virt occemax

e

t

P.E. Blohl, M. Parrinello, http://dx.doi.org/10.1103/PhysRevB.45.9413

Car-Parrinello molecular dynamicsWhat time step should one choose?

On the one hand: Nyquist-Shannonsampling theorem

On the other hand:CPU time

“correct sampling of a time domain signal requires two points per period of

the fastest oscillation present in the signal”

“the current charging rate at the Oxford Supercomputing Centre is £0.08 per core hour on both clusters and the

shared-memory system”

Commonly used values: μ=300-1500a.u., Δt=0.1-0.2 fs (5-10 a.u.).1. Longer trajectories require smaller

fictitious mass.2. It is sometimes reasonable to inc-

rease nuclear mass.3. A thermostat may be used to drain

the energy from the electrons.4. BOMD allows much greater time

steps due to absence of orbitaldynamics.

P.E. Blohl, M. Parrinello, http://dx.doi.org/10.1103/PhysRevB.45.9413

Car-Parrinello molecular dynamicsThe primary advantage of CPMD over BOMD is cheaper forces:

O N

C C

,ˆ

ˆmin

i Ni i N

N

i ij j N ij i jj ij N

N N k N

E rF H F r

r

F F rr

F r H r

CPMD forces

BOMD force

The following conditions should alwaysbe checked when running CPMD:1. Electron temperature should be

much smaller than the nucleartemperature.

2. Electron temperature should notincrease.

3. The total energy should remainstationary.

BOMD

H.B. Schlegel, Bull. Korean Chem. Soc., 2003, 24, 1-6.

Car-Parrinello molecular dynamics

CPMD has the advantage of accurate gradients.

G. Pastore, E. Smargiassi, F. Buda, http://dx.doi.org/10.1103/PhysRevA.44.6334