Lecture March8

7/24/2019 Lecture March8

1/28

CHM695March 9


2/28

s-p hybridization on Be:

H HBe

H H

Be

Homework 1:Work out the bonding in CH4 based on similar

analysis using HF/STO-3G basis.

a) identify non-bonding MOsb) identify canonical MOsc) obtain, canonincal to localized MOsd) Based on that, show that C is sp3hybridised.

Homework 2: Do the same for acetylene


3/28

If you do a similar calculation in ethene,

localisation yields

Banana bonds

C C C C

CH2=CH2

instead of 1 pi and 1 sigma bonds

Similar ly, there are molecules, where3-centre 2-e bonds

are formed!

e.g. B2H6


4/28

Natural Bonding Orbitals

(NBO)

A self consistent procedure to obtain localised

orbitals (called natural orbitals) from thewavefunction (either HF or KS-DFT)

http://www.cup.uni-muenchen.de/ch/compchem/pop/nbo2.html

Workout example using Gaussian:

Read more:Reed, Curtiss and Weinhold, Chem. Rev. (1988) 88, 899


5/28

(r1, r01) = nZ (1, 2,

, n) (10, 2,

, n)d2d3

dn

one particle density matrix

natural orbitals (Loewdin), are eigenfunctions

i = ii

()ij =Z

d1d0

1

(1)j(1

0

)

matrix {i}is AO (basis)


6/28

Basis on

atom A

Basis on atom

B

Find eigenvectors of

each block

(A)

i

eigenvectors of eachblocks are not

orthogonal to each other.

Apply orthogonalization

S1/2i = i


7/28

sigma bonds: bonding

sigma bonds: antibonding


8/28

NBO analysis with

Gaussian: First Step

%Chk=ch4.chk#PHF/3-21GPop=(NBORead,SaveNLMOs)

Methane:NMOAnalysis;writeNLMOstoCheckpointfile

01CH1hcH1hc2hchH1hc2hch3dih0H1hc2hch3-dih0

hc1.08618hch109.47122dih120.

$NBOAONLMO$END


9/28

NBO analysis with

Gaussian: Second Step

%Chk=ch4.chk#PHF/3-21GGuess=(Read,Only)

Methane:NMOAnalysis;writeNLMOstoCheckpointfile

01

Use molden to read the output file of the above run


10/28

Module 2Potential Energy Surface


11/28

H

r(HH)=3!

r(HH)

-13.6 eV

U(r)equilibrium distance

(1.06!)

1!g1

r(HH)=1.06!

H

e

1!g1 is the electronic configurationof the electronic ground state of H2+

-16.39 eV

H + H+

1!u1

there are bound states


12/28

FDD DD F

transition state

there are bound statesperpendicular to the

minimum energy

pathways


13/28

Complex PES for large molecules!

Need not obtain the full PES: interestingparts are minima and maxima

I = 1, ,N

Multiple minima/maxima could present in PES

maximum

minimum

2E

RIRJ< 0

2E

RIRJ> 0

E

RI= 0

E

RI= 0


14/28

Example of H2

stationary point on the PES

E(x1,y1,z1, x2,y2,z2) =E(q)

E

x1=

E

y1= =

E

z2= 0


15/28

Usually, in programs like Gaussian, this ischecked by

(hartree/bohr)max

E

qi

< 1 10

4i=1,,N

vuut

1

3N

"3N

i

E

qi 2#< 1 10

5

parameters; can beadjusted by input

keywords

RMS of

grad.

2 2 2


16/28

Hessian

Di!cult to judge if non-diagonal elements are non-zero!

h =

2E

l21

0 0

0 2E

l22

0

... ...

. . . ...

0 2E

l

2

6

H =

2Ex2

1

2Ex1y1

2Ex1z2

2Ey1x1

2Ey2

1

2Ey1z2

... ... . . . ...

2Ez2x1

2E

z22

eigenvalue

eigenvector

(normal coordinates)

h = LHL


17/28

2E

l21

=

2E

l22

= =

2E

l25

= 0

2E

l26

> 0

(asymptote)

(minimum)

At equilibrium


18/28

Eigenvector matrix elements:

(L)ij = lj

qj!

2E

l2

i! = qj

li

2E

qjqk!qk

lias,

Thus, a normal coordinateis

i =

3N

j

liqj

!qj


19/28

ithnormal mode is displayed by drawing arrows (of

arbitary length) on each atom (j ) in the directions

lixj

!,

liyj

!,

lizj

!

l1 l2 l3

l4 l5

l6


20/28

Connection to frequency:

2E

l26

= k

i =1

2

ski

Imaginary freq: 2E

l2i

< 0

if molecule is not having a stable structure; motion along will

decrease energy.i

3N-5 vibrational

modes if planar

3N-6 vibrationalmodes if non-planarvibrational

mode


21/28

Frequency Calculation Using Gaussian: H2example

%Chk=h2.chk#PHF/3-21GFreqOpt=Tight

FrequencyCalculationofH2

01

HH1hh

hh0.8

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
http://www.gaussian.com/g_tech/gv5ref/results.htmhttp://www.cmbi.ru.nl/molden/vibration.html


22/28

FDD DD F

DD F

For TS, frequency is complex along the reaction coordinate.

Along the all other modes, freq. is real

TS


23/28

Planar Ammonia at HF/STO-3G level:Compute the frequencies, and characterise the normal

modes and the ir frequencies.

Explain why one of the normal modes have imaginaryfrequency.

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 internalcoordinate 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

HH

dNHdNH

120

dNH

120120

torsion H-N-H-H=180 deg.


24/28

Structure Optimization

q

E

q0

E(q) =

1

2k(q

q0)2

E(qn) = 1

2k(qn q0)

2

dE

dq

q=qn

=k(qn q0)

qn

q1

q2

gradient (as arrows in the lef t figure)

has the direction of greatest rate of increase of E

q0 = qn 1

k

dEdq

q=qn


25/28

q

E

q0qn

But, kis not known!

q0 = qn 1

k

dE

dq

q=qn

qn+1 = qn c

dE

dq

q=qn

scalingparameter

q

E

q0qn

Steepest descent method(line search)


26/28

q

E

q0qn

qn+1 = q

nd

2E

dq21

q=qndEdqq=qn

q

E

quadratic

quadratic assumption

qn+1 = qn cd2E

dq2

1

q=qndE

dqq=qn

q

E


27/28

qn+1 = qn cH1n gn

In multi-dimensions:

Hessian gradient

BFGS Method (Quasi-Newton methods): Here H isnot computed explicitly!Make an initial guess of H

Keeps on improving this by appropriate updatebased 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!


28/28

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.

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Lecture March8