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CHM695March 9
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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
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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
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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
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(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)
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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
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sigma bonds: bonding
sigma bonds: antibonding
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NBO analysis with
Gaussian: First Step
%Chk=ch4.chk#PHF/3-21GPop=(NBORead,SaveNLMOs)
Methane:NMOAnalysis;writeNLMOstoCheckpointfile
01CH1hcH1hc2hchH1hc2hch3dih0H1hc2hch3-dih0
hc1.08618hch109.47122dih120.
$NBOAONLMO$END
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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
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Module 2Potential Energy Surface
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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
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FDD DD F
transition state
there are bound statesperpendicular to the
minimum energy
pathways
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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
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Example of H2
stationary point on the PES
E(x1,y1,z1, x2,y2,z2) =E(q)
E
x1=
E
y1= =
E
z2= 0
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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
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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
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2E
l21
=
2E
l22
= =
2E
l25
= 0
2E
l26
> 0
(asymptote)
(minimum)
At equilibrium
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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
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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
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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
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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.html7/24/2019 Lecture March8
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FDD DD F
DD F
For TS, frequency is complex along the reaction coordinate.
Along the all other modes, freq. is real
TS
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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.
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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
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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)
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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
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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!
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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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