lecture april1 april6
DESCRIPTION
Molecular Modelling Lecture NotesTRANSCRIPT
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Module-3
Ab Initio Molecular Dynamics
April 01 & April 06
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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
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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
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Concerns:Wavefunction optimisation at every MD step is time consuming!
Note: wfn has to be well converged.
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Car-Parrinello MD
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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
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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)
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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)
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Within the KS theory constraints are due to orthonormality of orbitals:X
i
X
j
⇤ij (h�i|�ji � �ij)
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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
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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!)
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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
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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.
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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
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• 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!