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