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Molecular Modelling Lecture Notes
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CHM695March 11
Connection to frequency:
∂2E∂l2
6= k
ni =1
2p
skiµ
Imaginary freq: ∂2E∂l2
i< 0
if molecule is not having a stable structure; motion along will decrease energy.
li
3N-5 vibrational modes if planar
3N-6 vibrational modes if non-planarvibrational
mode
Frequency Calculation Using Gaussian: H2 example
%Chk=h2.chk #P HF/3-21G Freq Opt=Tight
Frequency Calculation of H2
0 1 H H 1 hh
hh 0.8
optimize the structure
keyword to do freq.
calculation
visualize vibrations using molden: http://www.cmbi.ru.nl/molden/vibration.html
visualize normal modes using gaussview: http://www.gaussian.com/g_tech/gv5ref/results.htm
FDD DD F
DD F
For TS, frequency is complex along the reaction coordinate. Along the all other modes, freq. is real
TS
Vibrational freq.
• Quadratic approximation:
f (x) = f (x0) +
✓d f
dx
◆
x0
(x � x0) +12
✓d
2f
dx
2
◆
x0
(x � x0)2
+16
✓d
3f
dx
3
◆
x0
(x � x0)3
f(x) x
f (x) = f (x0) +
✓d f
dx
◆
x0
(x � x0) +12
✓d
2f
dx
2
◆
x0
(x � x0)2
+16
✓d
3f
dx
3
◆
x0
(x � x0)3
a number zero
higher terms are ignored
E ⇡ 12
3N
Âi
3N
Âj
∂2E
∂qi∂qj
!
0
qiqj
qi =p
mi
⇣qi � q0
i
⌘
=12
3N�6
Âi
∂2E∂l2
i
!
0
l2i
ni =1
2p
pki
k is mass weighted
ki
Application of Vibrational Analysis
qvib
=exp(�bhn/2)
1 � exp(�bhn)
from vib. partition function => vibrational contributions to thermodynamic properties
frequency from vib. analysis
There are models to incorporate anharmonicity
Comparison to IR/Raman spectrum
Planar Ammonia at HF/STO-3G level: Compute the frequencies, and characterise the normal
modes and their frequencies.
Explain why one of the normal modes have imaginary frequency.
What does motion along the normal mode indicate?
Hint: create an z-matrix for planar ammonia. Use z-matrix input and optimize the structure of planar ammonia. By specifying the value of an internal coordinate within the z-mat will constrain the structure to that value. You may fix angles (120deg.) and t o r s i o n s ( 1 8 0 d e g . ) d u r i n g optimisation.
N
N
HHdNHdNH
120
dNH120120
torsion H-N-H-H=180 deg.
Structure Optimization
q
E
q0
E(q) =12
k(q � q0)2
E(qn) =12
k(qn � q0)2
✓dEdq
◆
q=qn
= k (qn � q0)
qn
q1
q2
gradient (as arrows in the left figure) has the direction of greatest rate of increase of E
q0 = qn �1k
✓dEdq
◆
q=qn
q
E
q0qn
But, k is not known!
q0 = qn �1k
✓dEdq
◆
q=qn
qn+1 = qn � c✓
dEdq
◆
q=qn
scaling parameter
q
E
q0qn
Steepest descent method(“line search”)
q
E
q0qn
qn+1 = qn �✓
d2Edq2
◆�1
q=qn
✓dEdq
◆
q=qn
q
E
quadratic
quadratic assumption
qn+1 = qn � c✓
d2Edq2
◆�1
q=qn
✓dEdq
◆
q=qn
q
E
qn+1 = qn � cH�1n gn
In multi-dimensions:
Hessian gradient
BFGS Method (Quasi-Newton methods): Here H is not computed explicitly!Make an initial guess of H
Keeps on improving this by appropriate update based on gradients.
Based on the change in energy, one can on-the-fly compute appropriate c
BFGS is also usually done together with line-search method to improve the efficiency
Newton-Raphson
Hessian computation: usually numerical but is computationally expensive!
E
q
Local minima and global minima on PES can occur like above.
Standard optimizations algorithms find the local minimum, which may not be the true global minimum. Thus different
starting structures may be experimented.
Optimizing the TS
min.max.
qn+1 = qn � c✓
d2Edq2
◆�1
q=qn
✓dEdq
◆
q=qn+
normal mode with imaginary freq.q
%chk=job.chk #hf/3-21G opt=(ts,CalcFc,Z-matrix) freq
TS OPT
-1 1 c h 1 hc h 1 hc 2 120.0 h 1 hc 3 120.0 2 180.0 cl 1 clc 2 90.0 4 90.0 cl 1 clc 2 90.0 3 90.0
hc 1.089000 clc 2.000000
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