Lecture April1 April6

Module-3

Ab Initio Molecular Dynamics

April 01 & April 06

Ab Initio MD: Born-Oppenheimer MD

HBOMD({RI}, {PI}) =NX

I=1

P2I

2MI+ Etot({RI})

=NX

I=1

P2I

2MI+

min{ }

nD

({ri}, {RI})�

�

�

Hel

�

�

�

({ri}, {RI})Eo

+NX

J>I

ZIZJ

RIJ

rRI

D |Hel| |

E=DrRI |Hel| |

E+D |rRI Hel| |

E+D |Hel|rRI |

E

6=D |rRI Hel| |

E

Basis set should be large enough! Convergence of wave function and energy conservation:

Time step (fs)

Convergence (a.u.)

conservation (a.u./ps)

CPU time (s) for 1 ps

trajectory

0.25 10-6 10-6 16590

1 10-6 10-6 4130

2 10-6 6 x 10-6 2250

2 10-4 1 x 10-3 1060

Concerns:Wavefunction optimisation at every MD step is time consuming!

Note: wfn has to be well converged.

Car-Parrinello MD

Basic Idea:

Timescale separation: Electronic part is fastNuclear part is slow

Classical mechanical adiabatic energy-scale separation

making 2 classical subsystems that are adiabatically energy-

scale separated.

1. nuclear coordinates2. orbital coefficients

Lagrangian: LCP =

X

I

1

2

MI˙R2I +

X

i

µi

D˙ i| ˙ i

E

�D 0

��� ˆHel

��� 0

E+ constraints

is the molecular orbital (spatial)

0 is the Slater det.

µ is the fictitious mass for orbitals

i =X

⌫

c⌫i�⌫

Note: LCP ⌘ LCP(RN , RN , n, n)

d

dt

@L@RI

=@L@RI

Equations of motion can be computed as

LCP ⌘ LCP(RN , RN , n, n)

d

dt

�L�D i

���=

�L� h i|

µi¨ i = � �

� h i|D 0

��� ˆHel

��� 0

E+

�

� h i| (constraints)

FI = � @

@RI

D 0

��� ˆHel

��� 0

E+

@

@RI(constraints)

Within the KS theory constraints are due to orthonormality of orbitals:X

i

X

j

⇤ij (h�i|�ji � �ij)

Nuclear temperature(physical temperature): T /

X

I

MIR2I

Fictitious temperature: T orb. /X

i

µi

D�i|�i

E

No energy transfer between physical system and the orbital (quasi-adiabatic separation of dynamics)

Orbitals should move close to the corresponding Born-Oppenheimer solutions: Torb has to be small enough

(close to zero K ⇒“cold electrons”)

Energy transfer from “hot nuclei” to “cold electrons” should be strictly avoided during the dynamics

No overlap in the vibrational density of states(orbital motion has to be much above 4000 cm-1)

Thus motion can be kept adiabatically separable!

Energy is well conserved (no noise due to

SCF procedure!)

Thus, motion about the actual BO surfaceHigh freq. oscillations are not relevant in the timescale of the nuclear dynamics: thus not

only averages, but also time-dependent properties can also be computed

Time step should be small (to sample high freq. motion of orbital degrees of freedom):

usually about 0.06-0.12 fs

(Very small) Fictitious mass should be appropriately chosen (400-700 au)

Energy conservation can be checked to verify the accuracy of dynamics.

CP/5 a.u. /—

CP/10 a.u. /—

BO/10 a.u. /10-6

BO/100 a.u. /10-6

BO/100 a.u. /10-5

BO/100 a.u. /10-4

• Computationally efficient• Better energy conserving

• Not truly an BO MD; dynamics could get affected if parameters are not chosen

properly.• Small tilmestep (~1/10 smaller)

• Adiabatic separation doesn’t work in some cases (zero band gap)

• Better wfn extrapolation algorithms are available today; thus wfns can be

converged fast, and BOMD can be made computationally efficient!