Computational Spectroscopy
II. ab initio Methods
Chemistry 713
Updated: January 29, 2008
(b) Optimized Structures and
Rotational Constants
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Calculation of the ab initio energy at a single molecular
geometry my involve significant computational effort.
To find a relative minimum, the geometry is changed along the
line of steepest decent (fall line).
The program calculates the energies for slight changes in the
geometry along each atomic coordinate to determine the fall
line.
Second derivatives of the energy w.r.t. the coordinates are used
to determine how far to go along the fall line. (Some methods
allow analytic second derivatives: HF, B3LYP, MP2, CASSCF;
for others the derivatives must be computed numerically).
Process is repeated until a minimum is found. (all 1st
derivatives equal zeros and second derivatives are positive.)
Gaussian allows various convergence criteria to be set.
G03 example files
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Gaussian provides over 700 example input files and the corresponding
output files.
Mac (UNIX)
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Input files:
Output files:
Test file index:
/Applications/g03/tests/com/testxxx.com
/Applications/g03/tests/rs6k/testxxx.log
/Applications/g03/tests/test.idx
Windows
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Input files:
Output files:
Test file index:
C:\G03W\tests\gif\testxxx.gif
… missing
C:\G03W\tests\tests.idx
Example: …/g03/tests/com/test083.com
Water dimer geometry optimization
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Input file: “.com” or “.gif” =>
#p rhf/6-31g* opt=tight test
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Gaussian Test Job 83:
Water dimer optimization
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Input file viewed with
GaussView
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Distance matrix (angstroms):
1
2
3
4
5
1 O .000000
2 O 2.983080 .000000
3 H .948390 3.393962 .000000
4 H .948390 3.393962 1.513015 .000000
5 H 3.362029 .946860 3.869192 3.869192 .000000
6 H 2.031411 .951730 2.491320 2.491320 1.505701
(Initial geometry from top of log file)
0,1
O
O,1,ROO
X,1,1.,2,X3O
H,1,RO1H,3,HOX3,2,90.,0
H,1,RO1H,3,HOX3,2,-90.,0
X,2,1.,1,52.5,3,180.,0
H,2,RO2H1,6,H7OX,1,180.,0
H,2,RO2H2,6,H8OX,1,0.,0
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ROO=2.98308
RO1H=0.94839
X3O=120.2827
HOX3=52.90868
RO2H1=0.94686
RO2H2=0.95173
H7OX=52.98178
H8OX=51.9632
Example: …/g03/tests/rs6k/test083.log
Water dimer geometry optimization
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Output file: “.log” or “.out” =>
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#p rhf/6-31g* opt=tight test
…
Gaussian Test Job 83: Water dimer optimization
…
Berny optimization.
Internal Forces: Max
.000871642 RMS
.000359230
Search for a local minimum.
Step number 1 out of a maximum of 23
…
Item
Value
Threshold
Maximum Force
.000851
.000015
RMS Force
.000275
.000010
Maximum Displacement
.013238
.000060
RMS Displacement
.005575
.000040
Predicted change in Energy=-5.784113D-06
…
Converged?
NO
NO
NO
NO
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Distance matrix (angstroms):
1
2
1 O .000000
2 O 2.971263 .000000
3 H .948352 3.360777
4 H .948352 3.360777
5 H 3.422316 .946706
6 H 2.025317 .951801
(from bottom of log file)
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3
4
5
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.000000
1.512866 .000000
3.905657 3.905657 .000000
2.481287 2.481287 1.510657
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GradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradG
Berny optimization.
Internal Forces: Max
.000000441 RMS
.000000186
Search for a local minimum.
Step number 18 out of a maximum of 23
….
Item
Value
Threshold
Converged?
Maximum Force
.000000
.000015
YES
RMS Force
.000000
.000010
YES
Maximum Displacement
.000008
.000060
YES
RMS Displacement
000004
.000040
YES
Predicted change in Energy=-2.034343D-12
Optimization completed.
-- Stationary point found
Hints: How to cope with Gaussian output

THE BEST WAY TO
CONVINCE A FOOL
THAT HE IS WRONG
IS TO LET HIM
HAVE HIS OWN
WAY.
JOSH BILLINGS
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“.log” or “.out” are plain text files.
may be over 100 pages long!
If your input file was “MyMolecule1.com” (or “… .gif”), then
the ouput file is named “MyMolecule1.log” (or “… .out”).
ALWAYS us a different file name for every job, so that you
don’t over-write any of your old outputs.
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The next job might be “MyMolecule2.com”, etc.
NEVER delete any of your input or output files!
Use a spreadsheet (or lab notebook) to keep a log of your
Gaussian (& Spartan) jobs.
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One line of a spreadsheet can summarize the essential info:
Date, File name, Molecule, Purpose, Command lne
(e.g., #p rhf/6-31g* opt=tight test), Any problem encountered.
If you don’t keep meticulous records of your calculations, you
will risk drowning in your data!
Hints: How to read Gaussian output
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… from file test083,log
1\1\GINC-JERRY\FOpt\RHF\6-31G(d)\H4O2\FRISCH\07-Apr-2004\0\\#P RHF/6-3
1G* OPT=TIGHT TEST\\Gaussian Test Job 83: Water dimer optimization\\0,
1\O,-0.0029670399,0.,-1.4285923793\O,-0.0012160547,0.,1.5426696978\H,0
.504461364,0.7564331492,-1.6925928276\H,0.504461364,-0.7564331492,-1.6
925928276\H,-0.8866727134,0.,1.8776611243\H,-0.0887852583,0.,0.5949059
822\\Version=IBM64-G03RevC.02\State=1-A'\HF=-152.0304565\RMSD=7.932e-0
9\RMSF=1.778e-07\Dipole=0.0596938,0.,-1.0577951\PG=CS [SG(H2O2),X(H2)]
\\@
THERE'S SMALL CHOICE IN A BOWL OF ROTTEN APPLES.
SHAKESPEARE
Job cpu time: 0 days 0 hours 0 minutes 36.7 seconds.
File lengths (MBytes): RWF= 18 Int=
0 D2E=
0 Chk= 10 Scr=
Normal termination of Gaussian 03 at Wed Apr 7 15:20:15 2004.
Open your “.log” or “.out” files
with a simple text editor such as
NotePad (Windows) or TextEdit
(Mac).
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1
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Microsoft Word is NOT
recommended because, for this
purpose, it is big, clumsy, and
slow.
It you find a cryptic bit of
philosophy at the bottom of the
file, the Gaussian thinks it
completed the job successfully.
You will also find the CPU time
and a densely packed summary
of the results.
Hints: How to read Gaussian output
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Use your text editor to search for certain words or phrases that will take
you directly to the part of the output that you need.
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E.g., “#p”, “Job CPU time”, “Optimization completed”
Some phases like “Converged?” occur after each step of the optimization.
With 100 pages of output, this is MUCH faster than scrolling through the
file.
Item
Value
Threshold Converged?
Maximum Force
.000000
.000015
YES
RMS
Force
.000000
.000010
YES
Maximum Displacement
.000008
.000060
YES
RMS
Displacement
.000004
.000040
YES
Predicted change in Energy=-2.034343D-12
Optimization completed.
-- Stationary point found.
----------------------------
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You can also open any Gaussian input or output or . chk file with
Gaussview, but less information is accessible that way.
R.N. Zare, Angular Momentum, pp 253-258.
Rotational constants
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The rotational Hamiltonian is H  AJa  BJ b  CJa
with A = 1/2Ia, etc., where Ia, Ib, and Ic are the principal monents of
inertia. (Beware of units!)
Ia   mi ri2  zi2  where i labels the atoms in the molecule.
2
i
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
2
2

Gaussian calculates the rotational constants, A, B, and C:
Standard orientation:
---------------------------------------------------------------------------Center Atomic Atomic Coordinates (Angstroms)
Number Number Type
X
Y
Z
---------------------------------------------------------------------------1
8
0
.003321 -1.427751 .000000
2
8
0
.003321 1.555329 .000000
3
1
0
-.490596 -1.716172 .756507
4
1
0
-.490596 -1.716172 -.756507
5
1
0
.915824 1.808076 .000000
6
1
0
.012237
.603641 .000000
----------------------------------------------------------------------------Rotational constants (GHZ): 215.8670694
6.1508112
A
B
6.1482646
C
(c) Infrared and Raman Spectra
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Vibrations of Diatomic Molecules
Vibration in Polyatomic Molecules: Normal
Modes
Infrared Spectra
Raman Spectra
“Frequency” Calculations in Gaussian
Vibrations of diatomic molecules
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Chemical bonds will break at large
interatomic distances. (red curve)
2
1
A harmonic potential V  2 k(x  x e )
approximates the true potential near the
equilibrium geometry. (orange)
The real potential is ANHARMONIC.

The Harmonic Oscillator
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Harmonic oscillator wavefunctions are shown
in red, n = 0, 1,2, ….
Allowed optical (infrared) transitions are
n = ±1. (thick arrows)
Real molecules are anharmonic, therefore
overtone transitions (n = ±2, 3, …) are also
allowed, but are 1 or 2 orders of magnitude
weaker for each higher overtone. (thinner
arrows)
At room temperature, most of the molecules are
in n = 0, so the n = 1  0 “fundamental”
transition dominates the observed IR spectrum.
Note that anharmonicity shifts the observed IR
fundamental transition to a frequency lower
than the harmonic frequency by about 1 - 5%.
Vibrations in polyatomic molecules
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A nonlinear molecule with N atoms has 3N-6 vibrations:
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Methyl fluoride
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Each vibration is assumed to be a harmonic oscillator and
treated as an independent “normal” mode.
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Similar movies can be
obtained from Spartan 04 or
06 without additional software.
For each mode, all atoms move in hase with the same
frequency.
The commonly used functional group vibrations (e.g., C-F
stretch) are only rough descriptions.
The n = 1  0 “fundamental” transitions (one for each
normal mode) dominate the IR spectrum.

Movies created by GaussView,
captured by Snapz Pro X, and
edited by QuickTime.
3N degrees of freedom
minus 3 translations and 3 rotations
Overtone and combination bands are often observed as small
peaks in the experimental spectrum, but are NOT included in
the simulated spectra produced by Spartan and Gaussian.
As with diatomics, anharmocity shifts the obesrved IR
fundamentals are shift to lower frequency.
Calculation of molecular vibrations

In the harmonic approximation, the vibrational
frequencies depend on:
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all of the second derivatives of the potential, including the
diagonal and mixed second derivatives
The molecular geometry, including all bond lengths, angles,
and dihedral angles
The atmoic masses
The calculation is only valid at a stationary point
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A place where all of the first derivatives are zero
Local minima, maxima, saddlepoints
MUST optimize the structure first, and then run the
frequency calculation at ezacly the same level of theory and
same basis set.
Calculation of molecular vibrations:
More on the harmonic approximation

Expand the potential in a multidimensional Taylor series”:
V (q1,q2 ,...)  Veq 
3N 6

i1
V
qi
3N 6
 2V
qi  qieq   q q
i
j
i, j1
eq
q  q q
1
2
i
ieq
j
 q jeq  anharmonic terms
eq
The higher derivatives are neglected in most
calculations causing most calculated
frequencies to be too high.
These 2nd derivatives (force constants) are used to
determine the normal mode frequencies in the harmonic
approximation.
The the 1st derivatives (forces) are all zero at the equilibrium geometry.
The energy of the molecule at it equilibrium geometry.
Porcedure for calculating vibrational frequencies
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In Gaussian, optimized the molecular geometry first (OPT)
Then run a frequency calculation (FREQ) at the same level of theory.
You can run both at the same time, but it is not recommended until you
have had some experience.
Analytical second derivatives are available for the following methods:


HF, B3LYP, MP2, CASSCF
2nd derivatives are calculated numericaly for other methods, and are
MUCH more time consuming.
Porcedure for
calculating vibrational
frequencies
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In Gaussview:
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Build the molecule.
From the Edit menu, click “Symmetrize”
Job type is “OPT” ,
then “FREQ”.
From the Results menu, select
“Vibrations …”.
Vibrations in
Gaussian
output
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Check that the Point Group is
what you expect.
The information for each
vibrational mode is given.
Gaussian assumes that each
atom is the most abundant
isotope of each element, but
other istoppes can be set with
the ReadIsotopes option to te
FREQ command.
Thermochemical data, e.g., U,
CV, S are included. The
default temperature is 298 K,
but this can be changed
# opt freq hf/3-21g geom=connectivity
. . .
Full point group
D2H
NOp
8
. . .
Harmonic frequencies (cm**-1), IR intensities (KM/Mole), Raman scattering
activities (A**4/AMU), depolarization ratios for plane and unpolarized
incident light, reduced masses (AMU), force constants (mDyne/A),
and normal coordinates:
1
2
3
B2U
B3U
B2G
Frequencies -941.6104
1114.4412
1156.0372
Red. masses -1.0439
1.1607
1.5215
Frc consts -.5453
.8494
1.1980
IR Inten
-1.3937
133.9954
.0000
Raman Activ -.0000
.0000
4.9745
Depolar (P) -.3940
.0000
.7500
Depolar (U) -.5653
.0000
.8571
Atom AN
X
Y
Z
X
Y
Z
X
Y
Z
1
6
.00
-.04
.00
.08
.00
.00
.15
.00
.00
2
6
.00
-.04
.00
.08
.00
.00
-.15
.00
.00
3
1
.00
.24
-.44
-.50
.00
.00
-.49
.00
.00
4
1
.00
.24
.44
-.50
.00
.00
-.49
.00
.00
5
1
.00
.24
-.44
-.50
.00
.00
.49
.00
.00
6
1
.00
.24
.44
-.50
.00
.00
.49
.00
.00
4
5
6
AU
B3G
AG
Frequencies -- 1164.9569
1386.3705
1521.8389
Red. masses -1.0078
1.5294
1.2818
Frc consts -.8059
1.7319
1.7491
IR Inten
-.0000
.0000
.0000
Raman Activ -.0000
.9300
45.6702
Depolar (P) -.0000
.7500
.4294
Depolar (U) -.0000
.8571
.6008
Atom AN
X
Y
Z
X
Y
Z
X
Y
Z
1
6
.00
.00
.00
.00
.15
.00
.00
.00
.11
2
6
.00
.00
.00
.00
-.15
.00
.00
.00
-.11
3
1
.50
.00
.00
.00
-.15
.47
.00
-.19
.46
4
1
-.50
.00
.00
.00
-.15
-.47
.00
.19
.46
5
1
.50
.00
.00
.00
.15
-.47
.00
.19
-.46
6
1
-.50
.00
.00
.00
.15
.47
.00
-.19
-.46
. . .
E (Thermal)
CV
S
KCal/Mol
Cal/Mol-Kelvin
Cal/Mol-Kelvin
Total
36.426
7.387
52.029
Electronic
.000
.000
.000
Translational
.889
2.981
35.927
Rotational
.889
2.981
15.788
Vibrational
34.649
1.425
.313
. . .
Infrared Intensities
Infrared intensities are proportional to
2
2

I 
 upper   lower ,   x, y,z


The transition moment vectors I  Ix ,Iy ,Iz 
are represented graphically by GaussView (amber).

The infrared fundamental bands are allowed or forbidden

according to the irreducible representation
 table for the point group.
(e.g., B2u) in the character

Y
Z
X
QuickTime™ and a
decompressor
are needed to see this picture.
1: 941.6 cm-1 B2u
Peter Bernath, Spectra of Atoms and Molecules,
Oxford University Press, New York, 1995, p
381.
Infrared
allowed
fundamentals
Scaling calculated infrared spectra




Calculated harmonic frequencies may be higher or lower than the true harmonic
frequencies, according to the level of theory and basis set.
The true harmonic frequencies are higher than the experimental frequencies because of
the neglect of anharmonicity.
To compare with experiment, the calculated frequencies are SCALED empirically
(fudged) according to the level of theory and basis set.
It is advisable to check the literature on what scaling people have done for similar
systems.
QuickTime™ and a
decompressor
are needed to see this picture.
Foresman and Frisch, p 64.
Conventions of vibrational spectroscopy

Conventionally
numbered
Ethylene modes
Ag 1
2
3
B2g 4
B3g 5
6
Au 7
B1u 8
9
B2u 10
11
B2u 12
3329.6 cm-1
1841.8 cm-1
1521.8 cm-1
1156.0 cm-1
3373.1 cm-1
1386.3 cm-1
1164.9 cm-1
3307.8 cm-1
1604.0 cm-1
3404.5 cm-1
941.6 cm-1
1114.4 cm-1



The frequencies are from our
calculation above.
Less
symmetry
By convention, the highest frequency mode of the highest symmetry (most
symmetric irreducible representation) is 1, the next highest of that symmetry
is 2 etc., through all th emodes of that symmetry.
Then the modes of the next greatest symmetry get the next available mode
numbers from highest frequenct to lowest, until all modes have a number.
Gaussian does NOT follow this convention, but the literature does, and you
should.
The z-axis is always the highest symmetry axis.
Figures from Peter Bernath, Spectra of Atoms
and Molecules, Oxford University Press, New
York, 1995, p 284, 285.
Raman Spectroscopy

Most routine Raman spectra are Q-branches in the
Stokes vibrational Raman band.



The scattered light is shifted to lower frequencies.
The dfference between the scattered and incident
frequencies corresponds to a vibrational excitation of
the molecules
Many nonlinear Raman rechniques are also
available.
Selection rules for Raman spectroscopy

Raman spectra derive their intensity from the dependence of the molecular polarizability
on the various vibrational coordinates.



The polarizability at the equilibrium geometry gives rise to Rayleigh scattering (light scattered
at the same frequency as the incident light), which is MUCH more intense that the Raman
scattered light.
The polarizability components multiply bilinear combinations of the the coordinates,
which are listed in point group character tables.
For molecules with a center of symmetry, vibrational bands that are allowed in Raman
are forbidden in the infrared, and vice versa.
QuickTime™ and a
decompressor
are needed to see this picture.
Raman allowed
Ag: Raman allowed;
Infrared forbidden
Infrared
allowed
Calculation of Raman spectra



Calculated in Gaussian by default with FREQ
command. (No Raman intensities in Spartan).
Raman spectra can be polarized or depolarized, or both
according to whether the scattered light has the same or
different polarization as the incident laser beam.

A useful aid in assignments
The same scalling as for IR spectra may be applied.
QuickTime™ and a
decompressor
are needed to see this picture.
Advantages and disadvantages
of Raman spectroscopy*

Advantages



Complements IR spectroscopy: Provides access to vibrations that are weak or forbidden in IR.

Systematic studies of molecular vibrations always involve both IR and Raman spectra.
The effective resolution of routine spectra is higher.

Gas phase rotational structure: The Q-branch (J=0) is strong and produces as a sharp,
narrow peak for each vibration, whereas in the infrared, the P, Q, and R branches ( J=1,0,+1) all have substantial intensity resulting in broad bands that may overlap the bands of
other vibrations. O and S branches (J=-2, +2) are allowed in Raman, but are weak and very
speard out, and so usually disappear into the baseline.

Condensed phase: Rotations become librations, but the qualitative effect is the same.
Disadvantages



The Raman effect is weak, so the signal levels are low.
It is much harder to observe overtone and combination bands.
Raman spectra are less routinely available because a Raman spectrometer requires a laser light
source and so is more expensive. (For both IR and Raman, current spectrometers are usually
Fourier transform spectrometers.)
* Common Old-fashioned Ordinary Raman Spectroscopy (COORS). Other specialized forms include Coherent Antistokes Raman
Spectroscopy (CARS), Coherent Stokes Raman Spectroscopy (CSRS), Box-CARS, and Resonance Raman Spectroscopy.