CHEM 834: Computational Chemistry

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CHEM 834 Computational Chemistry
Properties
April 6, 2009
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Topics
last time

• semi-empirical molecular orbital methods

• molecular mechanics/force-fields

• multi-level methods

today

• properties

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Properties
two main categories of methods
2. Property Calculation Methods

• evaluate specific properties of a system

• use information from energy calculation methods

There are three main categories of properties

• single molecule properties

• experimental observables ? have well-defined
  quantum mechanical operators

• could (in principle) be obtained from an
  experiment on an individual molecule

• include

• NMR coupling constants and chemical shifts

• EPR hyperfine constants

• rotational, vibrational, and electronic spectra

• electron affinity and ionization potential

• dipole moments and electrostatic potentials

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Properties
two main categories of methods
2. Property Calculation Methods

• evaluate specific properties of a system

• use information from energy calculation methods

There are three main categories of properties

• thermodynamic properties

• experimental observables ? defined for large
  numbers of molecules

• include

• equilibrium constants

• rate constants

• heats of formation

• kinetic isotope effects

• acidity and basicity

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Properties
two main categories of methods
2. Property Calculation Methods

• evaluate specific properties of a system

• use information from energy calculation methods

There are three main categories of properties

• non-observable quantities

• quantities that are not experimental observables
  ? no quantum mechanical operators or rigorous
  definitions

• include numerous useful theoretical concepts

• partial atomic charges

• bond orders

• reaction concertedness

• aromaticity

• molecular orbitals

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Electronic Structure
understanding how the electron density is
distributed in a molecule can provide a lot of
information

• ab initio, DFT and semi-empirical methods give us
  one-electron orbitals

• since the orbitals are orthoganol and each
  contain one electron, we can write the density as

• if we write the molecular orbitals as linear
  combinations of basis functions

• we can express the density as

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Electronic Structure
understanding how the electron density is
distribution in a molecule can provide a lot of
information
integrating the density over all space gives us
the total number of electrons
we define
1. density matrix
2. overlap matrix
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Electronic Structure
the density matrix and overlap integrals tell us
how the electron density is shared between basis
functions ? and ?
how much do basis functions ? and ? overlap with
each other?
how much do basis functions ? and ? contribute to
the molecular orbitals that contain the electrons?

• if the basis functions dont overlap, they wont
  share any electrons

• if the basis functions dont contribute to the
  molecular orbitals, they wont contain any
  electron density

high overlap
low overlap
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Electronic Structure
recall that Gaussian basis functions are centered
on specific nuclei
So, we can have 2 situations
1. basis functions ? and ? are both on atom A

• all of the electrons represented by N?? can be
  assigned to atom A

• this is useful in calculating atomic charges

2. basis function ? is on atom A and basis
function ? is on atom B

• the electrons represented by N?? are shared
  between atoms A and B

• this is useful in calculating atomic charges and
  bond orders

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Partial Atomic Charges
if we assign electrons to each atom and know the
atomic charge, we can evaluate atomic charges
Mulliken charges

• if basis functions ? and ? are both on atom A,
  assign all electron density shared between the
  electrons to atom A

• if basis function ? is on atom A and basis
  function ? is on atom B, split the shared
  electron density in half

• subtract from the nuclear charge to get the
  partial atomic charge

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Partial Atomic Charges
Mulliken charges are easy to calculate and
provide useful qualitative insight into the
electron distributions
Mulliken charges

• are usually qualitatively correct

• are calculated by default with virtually all
  quantum chemical programs

• are useful in reproducing trends in charges

e.g. usually the correct trend will be observed
as the system goes from reactants to transition
state to products

• can have problems with diffuse basis functions

electron density around atom B represented by a
diffuse function are assigned to atom A
B
A
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Atomic Spin Densities
Mulliken populations can also be used to
determine the net spin on an atom
spin density matrices

• the net spin is given by the number of ?
  electrons minus the number of ? electrons

• only relevant in calculations on open-shell
  systems

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Mulliken Population Analysis in Gaussian
Gaussian calculates Mulliken charges and net spin
densities by default
output of a Hartree-Fock calculation of methanol
hydrogens bonded to carbon have different values
at finite temperature, rotation around the C-O
bond would make these symmetric
oxygen is negative
can include the charges on the hydrogen atoms in
the heavy atoms
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Mulliken Population Analysis in Gaussview
Gaussview will print Mulliken charges on the atoms
Results ? Charges
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Mulliken Population Analysis in Gaussview
Gaussview will print Mulliken charges on the atoms
Results ? Charges
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Mulliken Population Analysis in Gaussview
Gaussview will print Mulliken charges on the atoms
Results ? Charges
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Mulliken Population Analysis in Gaussian
Gaussian calculates Mulliken charges and net spin
densities by default
output of a UHF calculation of a methyl radical
alpha electron localized on carbon
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Multipole Moments
molecules exhibit dipole, quadrupole, octapole,
etc. moments
Dipole moment

• for a set of point charges, the dipole moment is

• this is a vector that points from negative to
  positive charge

• most chemistry texts use the opposite sign
  convention, but this is whats used in programs
  like Gaussian

example
P
H
F
positive end
negative end
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Multipole Moments
molecules exhibit dipole, quadrupole, octapole,
etc. moments
Dipole moment

• for point charges (nuclei) and electron density,
  the dipole moment is

• this can be computed exactly with quantum
  chemical methods that provide molecular orbitals

• higher order multipole moments can also be
  calculated

• these quantities provide insight into the charge
  distribution in a molecule

e.g. the dipole in water points away from oxygen,
so we know the oxygen atom is negative
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Multipole Moments
Gaussian calculates multipole moments by default
at the end of the output file
x, y, and z components of the dipole moment vector
magnitude of the dipole moment in units of Debye
higher order multipole moments, represented as
tensors
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Bond Orders
density matrix provides information regarding how
electrons are shared between atoms
Wiberg bond orders

• no real theoretical basis (bond orders are not
  real quantities)

• tend to provide intuitively correct values

single bonds have values of 1, double bonds 2,
triple bond 3, etc.

• straightforward to calculate

• can be sensitive to basis sets like Mulliken
  charges

• trends in Wiberg bond orders are usually more
  relevant than the actual values

• other definitions of bond orders exist, too

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Bond Orders
Wiberg bond orders for the reactants, transition
state, and product of a Diels-Alder cycloaddition
2.0248
1.0052
1.5644
1.0055
0.3602
1.1294
1.9419
1.0127
1.3500
1.5253
concerted reaction
1.8819
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Bond Orders in Gaussian
Gaussian uses the NBO program to calculate bond
orders
you will have to modify the input manually
add popnboread to the route line
add nbo end section after geometry input
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Bond Orders in Gaussian
Gaussian uses the NBO program to calculate bond
orders
Wiberg bond orders from a calculation on acetone
CO bond order 1.8368 2 C-C bond orders
1.0041 1
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MESP Charges
it might be more reasonable to base the charges
on a physical observable
Molecular Electrostatic Potential (MESP)

• the electrostatic potential that a positive test
  charge would feel at a position around a molecule

• in regions where the MESP is negative, a positive
  charge experiences an attractive potential
  vice-versa when the MESP is negative

• the MESP is an experimentally measurable quantity

negative region
positive region
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MESP Charges
it might be more reasonable to base the charges
on a physical observable
Molecular Electrostatic Potential (MESP)

• using the wavefunction, the MESP at some point r
  is

• using a series of partial atomic charges, the
  MESP at r is

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MESP Charges
it might be more reasonable to base the charges
on a physical observable
so, to get partial atomic charges we can

• select values of QI that reproduce the MESP on a
  grid of points around the molecule

• MESP is evaluated at each point on grid

• different methods of choosing grids and fitting
  values exist

• standard methods include CHELP, CHELPG, and RESP

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MESP Charges
it might be more reasonable to base the charges
on a physical observable
so, to get partial atomic charges we can

• select values of QI that reproduce the MESP on a
  grid of points around the molecule

partial charges obtained from the MESP are

• relatively insensitive to the basis set

• based on physical observables

• charges of atoms deep inside a molecule can be
  assigned unreasonable values in fitting procedure

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MESP Charges in Gaussian/Gaussview
Gaussian can calculate MESP charges according to
several different fitting procedures

• popchelpg on the route line

• can also evaluate MESP charges by setting pop to

MK or ESP or MerzKollman
chelp

• setting pop(method,dipole) forces the total
  charges to reproduce the molecular dipole moment,
  in addition to the MESP

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MESP Charges in Gaussian/Gaussview
Gaussian can calculate MESP charges according to
several different fitting procedures

• MESP partial atomic charges in Gaussian output

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MESP Charges in Gaussian/Gaussview
Gaussview will print ESP charges on the atoms
Results ? Charges
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Molecular Orbitals
quantum chemical methods focus on solving for the
molecular orbitals
coefficients define orbitals for a given basis set
In Hartree-Fock

• the orbitals represent one-electron wavefunctions

• can be treated as approximate wavefunctions for
  each electron

In DFT

• the Kohn-Sham orbitals are completely artificial
  constructs used to get the density

• but they often resemble our intuitive picture of
  molecular orbitals

we can learn a lot about molecules by studying
the molecular orbitals
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Molecular Orbitals
we can learn a lot about molecules by studying
the molecular orbitals
Quantum chemical programs provide
1. Molecular orbital shapes and locations

• can determine which regions of a molecule may be
  reactive

• can determine how orbitals change during reactions

• can identify transitions involved in electronic
  excitations

2. Molecular orbital energies

• can evaluate properties like HOMO-LUMO gaps

• can estimate ionization potential and electron
  affinity

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In Gaussian
you have to set popfull on the route line to get
the orbital coefficients
In the output (HF/3-21G(d,p) of ethene)
orbital symmetries
orbital energies
molecular orbital coefficients

• each column gives the coefficients for 1 orbital

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In Gaussian
you have to set popfull on the route line to get
the orbital coefficients
In the output (HF/3-21G(d,p) of ethene)
O occupied V virtual
orbital energy in au
basis functions on atom 1
coefficient of basis function 1 (1s on Carbon
atom 1) in molecular orbital 1 c1,1
basis functions on atom 2
coefficient of basis function 17 (px orbital on
Hydrogen atom 3) in molecular orbital 1 c17,1
basis functions on atom 3
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Plotting Molecular Orbitals
you can plot the molecular orbitals in Gaussview
(and many other programs)
1. run a calculation with popfull on the route
line
2. read in the .chk file from the Scratch
directory
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Plotting Molecular Orbitals
you can plot the molecular orbitals in Gaussview
(and many other programs)
3. generate a cube file

• a cube file is a grid with the property of
  interest calculated at each grid point

Results ? Surfaces
select New Cube from the Cube Actions Menu
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Plotting Molecular Orbitals
you can plot the molecular orbitals in Gaussview
(and many other programs)
3. generate a cube file

• a cube file is a grid with the property of
  interest calculated at each grid point

set the property to Molecular Orbital
pick the orbital of interest
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Plotting Molecular Orbitals
you can plot the molecular orbitals in Gaussview
(and many other programs)
4. generate the surface
select isosurface value
select New Surface under Surface Actions
Isosurface

• surface on which all the values of some property
  are equal

• smaller isosurface values larger orbitals

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Plotting Molecular Orbitals
you can plot the molecular orbitals in Gaussview
(and many other programs)
5. look at the orbital
different colours indicate different phases of
the wavefunction
HOMO of ethene, isosurface value 0.02 au
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Plotting Molecular Orbitals
you can plot the molecular orbitals in Gaussview
(and many other programs)
5. look at the orbital
HOMO of ethene, isosurface value 0.1 au
HOMO of ethene, isosurface value 0.02 au
typical isosurface values are between 0.01 and
0.1 au
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Plotting Molecular Orbitals
sometimes molecular orbitals are consistent with
our ideas of bonding and anti-bonding
interactions, sometimes they are delocalized
LUMO of ethene C-C ? bond
HOMO-1 of ethene C-H ? bonds
HOMO of ethene C-C ? bond
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Applications of Molecular Orbital Analysis
molecular orbital analysis can be useful in
understanding how reactions proceed, and
optimizing reaction conditions
cyclopropane diester
nitrone
1,2 oxazine
we wanted to consider various mechanisms
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Applications of Molecular Orbital Analysis
molecular orbital analysis can be useful in
understanding how reactions proceed, and
optimizing reaction conditions
Concerted half-chair pathway
Nitrone HOMO Cyclopropane LUMO
Nitrone LUMO Cyclopropane HOMO
Orbitals are of correct symmetry for concerted
bond formation.
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Applications of Molecular Orbital Analysis
molecular orbital analysis can be useful in
understanding how reactions proceed, and
optimizing reaction conditions
Concerted chair pathway
HOMO nitrone LUMO cyclopropane
LUMO nitrone HOMO cyclopropane
Orbitals are not of the correct symmetry for
concerted bond formation.
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Applications of Molecular Orbital Analysis
molecular orbital analysis can be useful in
identifying the orbitals involved in electronic
excitations
CIS/6-31G(d,p) calculation of formaldehyde
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Applications of Molecular Orbital Analysis
molecular orbital analysis can be useful in
identifying the orbitals involved in electronic
excitations
excitation 1 involves transition from orbital 8
into orbitals 9 and 13
MO 9
MO 8
characterized as n ? ? transition
MO 13
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Applications of Molecular Orbital Analysis
molecular orbital eigenvalues can be used to
predict ionization potentials and electron
affinities
orbital energies
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Applications of Molecular Orbital Analysis
molecular orbital eigenvalues can be used to
predict ionization potentials and electron
affinities
Koopmans theorem

• the Hartree-Fock HOMO energy is equal to the
  negative of the ionization potential

• orbital energies are calculated relative to
  infinitely separated nuclei and electrons

• so orbital energy is the stabilization arising
  from bringing in an electron from infinitely away

energy

• ionization potential is the energy needed to move
  an electron infinitely away -IP

• ignores relaxation effects of the rest of
  electronic system when electron is removed

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Applications of Molecular Orbital Analysis
molecular orbital eigenvalues can be used to
predict ionization potentials and electron
affinities
Koopmans theorem

• the Hartree-Fock HOMO energy is equal to the
  negative of the ionization potential

• generally accurate for HOMO because lack of
  relaxation effects and lack of electronic
  relaxation cancel out

energy

• can also be applied to core states (e.g. core
  binding energies), but with less accuracy

• LUMO energy is often used to estimate electron
  affinity, but errors do not cancel, so results
  are typically inaccurate

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IR/Raman
IR/Raman probes the vibrational excitations of
molecules
consider a 1-D case

• transitions between energy levels decrease with n

• the system is bound for low values of n and the
  bond vibrates

potential energy

• the system is unbound for high values of n and
  the bond dissociates

n3
n2
n1
n0
nuclear separation (RAB)
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IR/Raman
IR/Raman probes the vibrational excitations of
molecules
consider a 1-D case

• in IR/Raman, we use light to induce vibrational
  excitations

• energies of transitions give us the vibrational
  wavelengths/frequencies

• intensities are determined by how strongly the
  dipole moment changes with the vibration (IR) or
  the polarizability changes with vibration (Raman)

potential energy
frequency of light
n3
n2
n1
?E h?
n0
nuclear separation (RAB)
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IR/Raman
IR/Raman probes the vibrational excitations of
molecules
intensities
energies of vibrational excitations
we can model IR spectra with frequency
calculations
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IR/Raman
we can model vibrations using a Taylor series
expansion of the potential energy
potential energy
equilibrium bond distance, Req
nuclear separation (RAB)
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IR/Raman
we can truncate the Taylor series after the
second term
and we have
potential energy
giving
equilibrium bond distance, Req
nuclear separation (RAB)
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IR/Raman
we model vibrations with the harmonic oscillator
the quantum mechanical solution of the harmonic
oscillator tells us
1.
potential energy
only transitions of ?n 1 are allowed
2.
n3
n2
so, all transitions are of equal energy and
determined by the vibrational frequency, ?
n1
n0
nuclear separation (RAB)
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IR/Raman
to simulate vibrational excitations, we need the
vibrational frequencies
force constant
reduced mass of atoms involved in bond
most molecules are polyatomic, so we need to deal
with normal modes instead of bonds
O
bend
H
H

• collective vibrational motions of atoms

symmetric stretch
O

• simplest vibrational motions in a molecule

H
H

• probed in IR/Raman experiments

• 3N-6 modes/molecule (3N-5 if linear)

asymmetric stretch
O
H
H
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IR/Raman
recall our discussion of frequency/normal mode
calculations in the first lecture
P
H
K

• H Hessian matrix of second derivatives of the
  energy with respect to nuclear displacements

• P matrix containing nuclear displacements along
  the 3N-6 vibrational modes, plus 3 translation
  and 3 rotation motions

• K force constants for each normal mode

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IR/Raman
recall our discussion of frequency/normal mode
calculations in the first lecture

• diagonal elements are force constants, k

• k1 k6 correspond to translations/rotations, k7
  k3N are vibrations

• mass-weighting gives the vibrational frequencies

you get much of the information you need to model
vibrational excitations every time you do a
normal mode calculation
You can do frequency calculations with many
methods including

• force-fields

• MP2

• semi-empirical methods

• DFT

• Hartree-Fock

• CCSD

(some methods may take a really long time)
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IR/Raman
you get much of the information you need to model
vibrational excitations every time you do a
normal mode calculation
you can estimate the positions of peaks and
associate the peaks with specific vibrational
modes
intensities?
energies of vibrational excitations
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IR/Raman
the harmonic oscillator approximation does have
some limitations
1. it is only valid at stationary points

• frequency calculations within the harmonic
  oscillator approximation are only meaningful at
  minima and transition state

potential energy

• since, geometries change with the method and
  basis set, frequency calculations have to be
  performed at the same level of theory as a
  geometry optimization

2. will not capture Fermi resonance

• two modes with similar frequencies are shifted to
  higher and lower energies

nuclear separation (RAB)
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IR/Raman
the harmonic oscillator approximation does have
some limitations
3. can never be exact

• even with exact wavefunction, the frequencies
  will be too high

• any computational method that exhibits
  frequencies that are too low has other errors
  that cancel this out

potential energy

• harmonic oscillator energy is higher than the
  real potential energy

• as a result, k will be too high

nuclear separation (RAB)
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IR/Raman
scaling the frequencies can give better agreement
with experiment
example
Hartree-Fock calculations consistently overbind

• force-constants and vibrational frequencies are
  too high

• usually the frequencies are 15 too high

• so, multiply all the frequencies by 0.85

there is no theoretical justification for this,
it just works because of systematic errors with
Hartree-Fock

• in general, every method/basis set combination
  needs a different scaling factor

• published lists of these exist

Scott Radom, J. Phys. Chem. (1996) 100, 16502.
Wong, Chem. Phys. Lett. (1996) 256, 391.

• if your specific basis set isnt listed, you can
  probably use a scaling factor for a similar basis
  set with the same method

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IR/Raman
scaled frequencies are reasonably accurate

• frequencies for 1066 vibrations in 122 molecules

method
scaling factor
RMS error (cm-1)
AM1
0.9532
126
PM3
0.9761
159
HF/3-21G
0.9085
87
HF/6-311G(d,p)
0.9051
54
MP2/6-311G(d,p)
0.9434
63
QCISD/6-311G(d,p)
0.9537
37
BLYP/6-31G(d)
0.9945
45
BP86/6-31G(d)
0.9914
41
B3LYP/6-31G(d)
0.9614
34
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IR/Raman
what about the intensities?
IR intensities are related to the change in the
dipole moment

• easy to calculate as part of a frequency
  calculation

Raman intensities are related to the change in
the polarizability

• these end up involving third derivatives
  (expensive)

• simple solutions exist with Hartree-Fock

Raman and IR peaks occur at the same frequencies
(but some may be inactive)
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IR/Raman
intensities will be calculated as sharp peaks,
not broad bands
intensities
energies of vibrational excitations

• IR Raman calculations are usually most useful in
  a qualitative sense

• trends are generally more important than actual
  values (this is particularly true for intensities)

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IR/Raman in Gaussian
Gaussian performs frequency calculations with the
freq keyword on the route line
Useful options
1. Scale

• tells Gaussian to use a scaling factor

B3LYP/6-31G(d) freq scale0.9614
2. Freq(Raman)

• tells Gaussian to calculate Raman in addition to
  IR

• IR is always performed by default

B3LYP/6-31G(d) freq(Raman)

• Raman is performed by default with Hartree-Fock,
  for other methods you are in for a long,
  expensive calculation

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IR/Raman in Gaussian
Gaussian performs frequency calculations with the
freq keyword on the route line
Output
information regarding IR/Raman frequencies and
intensities
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IR/Raman in Gaussview
you select frequency under job type
select whether to calculate Raman intensities
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IR/Raman in Gaussview
you can look at the frequencies under
Results/Vibrations
same information as in the output
animate modes
visualize spectrum
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IR/Raman in Gaussview
you can plot the spectra
line widths are completely artificial