Download Chemistry 3810 Lecture 9 SPECTROSCOPY The following

Chemistry 3810
Lecture 9
SPECTROSCOPY
The following sequence of lectures are designed to aid the laboratory course. Its aims are to
provide a brief background to the various spectroscopic techniques used in the inorganic
chemistry laboratory. A brief theory will be presented; a note on experimental technique,
followed by examples and judgement calls on the interpretation of the results.
PART I: VIBRATIONAL SPECTROSCOPY
References:
SAL 3.7 - 3.8, pp. 134 - 143
Butler and Harrod, Chapter 6.
R.S. Drago, "Physical methods in inorganic chemistry", Reinhold,
1965
This term covers the two common techniques, infra-red and Raman spectroscopy, which
deal with absorption by the vibrations of molecules and materials. There are a few other
highly-specialized methods which observe bond vibrations which we will not deal with.
1.
What vibrational spectroscopy can tell us
Both IR and Raman spectroscopy deal with the interaction of vibrational energy levels in
molecules with photons of light. It is found that those photons which excite molecular
vibrations are in the infra-red region of electromagnetic radiation. That means photons
with considerably less energy than visible light.
These interactions are useful to chemists because they can be used to diagnose chemical
structure. The application of vibrational spectroscopy depends on the type of compound
being investigated. First and foremost, vibrational spectroscopy requires the presence of
chemical bonds. Simple ionic solids do not have IR spectra - an important fact which
allows us to use ionic crystals as transparent windows for measuring spectra. One key
thing which you must learn in Chemistry 3810 is to learn to exercise judgement as to how
to apply vibrational spectroscopy to a particular compound.
1.1
Gas-phase spectra of small molecules
Typically, only small molecules will exist as gases at reasonably low temperatures,
i.e. below the temperature at which bonds break. At low pressures (≤20 mmHg),
we can observe essentially independent molecules in the gas phase with little or no
effect due to the interactions between molecules. Under these special conditions,
it is observed that IR light not only excites molecular vibrations, but also
molecular rotation. Thus for gas-phase molecules, not only are the vibrational
states of molecules quantized, the way the whole molecule spins around its centre
of gravity is also quantized.
Typically what one observes for gas-phase molecules are a small number of
vibrational bands, but each of these are usually split into many sub-bands due to
the rotational behaviour, which occurs at considerably lower energy (≤ 20 cm-1)
than the vibrations (typically 100 - 4000 cm-1).
A mathematical treatment of small molecules allows one to develop models which
"predict" IR spectra. By adjusting the models till they fit with experiment, it is
possible for small gas-phase molecules to determine the shapes of molecules
and/or bond lengths. The bond length data comes from the rotational fine
structure.
Page 1
Chemistry 3810
1.2
Lecture 9
Solid and liquid samples of high symmetry or small size
The symmetry arguments which we will develop for gas-phase molecules also
apply to solid and liquid samples. However, due to intermolecular interactions, the
rotational motion in such samples is normally completely quenched. This means
that IR bands in condensed phases are often sharper than they are for gases! When
the molecules being observed are relatively small, or have high symmetry, it is
often possible to distinguish among possible structures or isomers by examining
the patterns of some of the IR or Raman bands, or by counting the number of bands
present. However,since the rotational lines are absent, no information is available
on bond lengths from condensed-phase spectra.
1.3
Large solid and liquid samples with distinctive units
When molecules get very large, or have low symmetry, the number of bands can
be so large, that it is impossible to assign their structure from their vibrational
spectra. However, specific types of bond units may absorb radiation within a
narrow range of possible energies. In the language of organic chemistry, these are
often called functional groups. The identity of a certain type of chemical species
may be possible if the presence or absence of such functional groups can be
determined. Functional groups are just as important in inorganic chemistry, and a
list of typical inorganic group frequencies is given in Table 9-3.
1.4
Large solid and liquid samples without distinctive units
The unique combination of vibrational absorptions of a molecule or substance
leads to a complex pattern of vibrational bands that is found to be characteristic for
only that specific compound. For technical reasons, the majority of bands for
compounds composed of the 2nd period elements tend to fall in the region 500
cm-1 to 1500 cm-1, and this region is often called the fingerprint region, because
it can often be too complex to analyze precisely, but is a unique identifier of the
compound. Comparing the fingerprint region of a sample against a known
standard is considered proof of identity for a chemical compound.
The fingerprinting method represents the least sophisticated application of
vibrational spectroscopy in chemistry, requiring no theory at all! It is however
very important, and can always be used if the above three methods can not. I will
say nothing further about using the fingerprinting method, except to mention that
chemists report vibrational data in the following manner:
IR (KBr pellet): 1604(m), 1413(w), 1296(w), 1259(w), 1211(s), 1198(vs), 1155(w),
1041(w), 924(m), 901(s), 847(w), 765(m), 686(w), 547(w), 289(m) cm-1.
Where (w), (m), (s) and (vs) stand for weak, medium, strong and very strong
intensity. This assignment is a matter of judgement.
2.
Background theory to vibrational spectroscopy
Consider two atoms. The bond between them can be considered as a simple spring, with
force constant k.
Page 2
Table 9-1
Table 9-2
Some typical force constants for chemical bonds
Regions in which various solvents transmit at least 25% of the incident radiation
(so-called Open Regions)
Chemistry 3810
Lecture 9
ν =
k
1
2
m1
m2
µ=
1
k
2π c
µ
cm-1
m1 m 2
m1 + m 2
The term µ is called the reduced mass. It of course has the units of kg. Note that c is the
velocity of light, and must be in units of cm @ s-1 for the equation to be valid. The
quantity ν is called the wavenumber, and by convention implies units of cm-1. All
modern IR and Raman instruments are calibrated in cm-1, which means that they are
frequency linear. However, older instruments and text-books record IR data in microns,
and this is a wavelength linear method.
For some examples of the presentation of vibrational spectroscopic data, see Fig. 9-1 9-4.
Consider the example spectra in the notes which are presented in both formats (Fig. 9-2
and 9-3). Do you see the difference? Actually, since most of the specific information is in
the fingerprint region, which is much expanded in the micron format, one could make a
very valid argument that the older form of presentation is more useful for chemical
characterization by IR! The main advantage of the wavenumber method is its more direct
correlation with the energy of the vibration, which for a given set of nuclei (i.e. constant
m1 and m2) is a direct measure of the force constant k. Some examples of force constants
are given in Table 9-1
The key point of this equation is that the frequency of a vibrational band for a bond
depends in an interrelated manner on:
(1) mass of attached nuclei
(2) strength of the bond
Increase the masses, and ν decreases; increase the bond strength, and ν increases.
So far we have touched on the choice in the presentation of the x-axis, i.e. in frequency or
wavelength. The other choice deals with the y-axis, for which we may choose either
%-transmittance, or absorbance. In days gone by, many spectrophotometers were
calibrated in transmittance (e.g. the Spectronic 20’s used in the Freshman labs) because
these instruments has very simplistic electronic readouts. Modern spectrometers,
however, read out directly in absorbance, since this is linear with concentration.
Remember the definition of this term in the Beer-Lambert law:
%T =
I
Io
log10


I
Io


= A = ebc
However, most IR data is still presented in the old fashioned manner. Fig. 9-1 shows the
same gas-phase spectrum of SO2 in both the %T and absorbance modes. With the advent
of FT-IR there has been a concerted movement towards the presentation of IR data as
Page 3
Figure 9-1
%Transmittance versus Absorbance presentation of IR data
Figure 9-2
Gas-Phase spectrum of SO2 from 3000 - 450 cm–1
Fig. 9-3
Same as Fig. 9-2 but using a wavelength scale rather than wavenumbers
Fig. 9-4
Expansion of ν 1 (P+Q branches) and ν 3 (P-Q-R branches) bands of SO2
Chemistry 3810
Lecture 9
absorbance. Raman data has almost always been presented in this mode. It has the
distinct advantage of emphasizing the weaker bands at the expense of the stronger bands.
3.
Harmonic and anharmonic oscillator
The above equation is called the equation for the classical harmonic oscillator. Such an
oscillator follows a plot or energy vs. internuclear separation given by the left hand plot
below. To apply in a valid manner to molecules, we need two major corrections:
quantization and anharmonicity.
3.1
The quantum-mechanical harmonic oscillator
Quantum-mechanics applies to most chemically bonded vibrations. We can
consider quantization as an add-on to the classical oscillator. Remember that E =
hν, the basic statement of quantization by Planck. Then we get the following
equations:
Eν = (ν + 4)
When ν = 0, then Eν = 4
h
k
2π c
µ
h
k
2π c
µ
ν = 0, 1, 2, ....
. This is called the zero-point energy, and
and is lowest vibrational energy state of
the bond. In other words, all bonds vibrate!
Vibrational spectroscopy is governed by a selection rule:
∆ν = ±1
Hence all absorptions for harmonic oscillators lead to the same absorption peak,
given by the original classical equation! Selection rules express the probability of
transitions between quantized states.
3.2
The quantum-mechanical anharmonic oscillator
At high-enough energy, bonds will of course break. This means that they are not
true harmonic oscillators. In fact they are anharmonic oscillators. However, as the
figure shows, there is considerable similarity left between the two models at low
energy. At higher energy, however, the spacing between the levels decreases as
the potential well gets wider.
The practical consequence of anharmonicity is that instead of all the absorptions
overlapping to give a single sharp peak, multiple peaks at slightly off-set energies
occur, which in practice leads to line-broadening. This is most significant for light
elements: as much as +100 cm-1 for H- bonds; as little as +1 cm-1 for
heavy-element bonds.
Another consequence is the breakdown of the strict selection rule. Now some
transitions will occur with ∆ν = ±2, ±3, etc. These events lead to so-called
overtones, which are always found at 2@, 3@, etc. of the primary band. Overtones
are always of much weaker intensity than the fundamental vibration peaks.
Page 4
Chemistry 3810
Classical Harmonic Oscillator
Lecture 9
Quantized Harmonic Oscillator
E
N
E
R
G
Y
ν= 6
ν= 5
ν= 4
ν= 3
ν= 2
ν= 1
ν= 0
Quantized Anharmonic Oscillator
ν= 6
ν= 5
ν= 4
ν= 3
ν= 2
ν= 1
ν= 0
Internuclear separation, r
Fig. 9-5
3.3
Degrees of freedom and normal modes
The total number of degrees of freedom for the motion of any molecule is 3N,
where N is the number of atoms in the molecule. This is simply the result of
allowing x,y,z motion for each atom in the ensemble.
In general, there will be 3N - 6 vibrational degrees of freedom for any molecule
(except linear molecules, for which there are 3N - 5 vibrations.) The other 6 are
taken by 3 translational and 3 rotational (2 for linear molecules) degrees of
freedom, i.e. the number of coordinates required to describe the movement of the
whole molecule and the tumbling of the whole molecule. It is these rotational
degrees of freedom that are responsible for the rotational fines structure seen in gs
phase spectra, e.g. in SO2. Translational motion is NOT observed by IR
spectroscopy.
For molecules of three or more nuclei, more than one vibration is possible. For
example, for a bent triatomic, there are vibrations which stretch both bonds at
once, ones which stretch one while compressing the other, and also vibrations
which bend the molecule more or less. These are called the symmetric stretch,
asymmetric stretch and scissoring vibrations.
It is possible to develop for small molecules a set of idealized vibrations which
fully describe the vibrational degrees of freedom. These unique vibrations are
called normal modes of vibration, that is "orthogonal" vibrations which as a set
express all the possible motion of the molecule. Normal modes are defined such
that neither the centre of gravity nor the orientation of the molecule are altered
during the vibrations (since the latter are expressed by the translational and
rotational degrees of freedom.)
Page 5
Chemistry 3810
4.
Lecture 9
Examples of using normal-mode analysis
Normal mode analysis can be used to determine chemical structure if the model devised
by this method is compared to the experimental data. Under ideal conditions the
complete structure can be obtained in this way. In general we will use pre-calculated
normal modes as given in Table 9-4. However, I will show you for a simple example how
these are generated.
4.1
Application of normal-mode analysis to SO2 vapour
Fig. 9-6 shows the three possible vibrations for the triatomic molecule SO2:
Normal modes of vibration
of SO2
:O
..
..
O
.. :
S
S
O
..
S
O
S
O
O
O
S
O
S
O
O
O
O
ν1
Symmetric stretch
S
O
ν3
ν2
Asymmetric stretch
Bend (scissoring)
Figure 9-6 Normal mode analysis for SO2
Now consider the gas-phase IR spectrum of SO2 presented in Fig. 9-2 and 9-4.
High resolution work has established the following assignments:
Band, cm-1
519
506
1151
1361
1871
2305
2499
Assignment
ν2
ν1 - ν2
ν1
ν3
ν2 + ν3
2 ν1
ν1 + ν3
Comments
Rotational lines resolvedm
Weak, obscured by ν2
Rotational lines resolvedm
Intense*
Very weak, not observed
Weak "overtone" band of ν1
Weak combination band
This is a gas-phase spectrum, which illustrates what I stated in section (1.1). Each
vibrational band shows rotational fine structure. The form of these bands is
referred to as the P - Q - R pattern.
* The Q branch is seen whenever the vibration induces a dipole moment change
Page 6
O
Chemistry 3810
Lecture 9
which is 9 to the principal axis of rotation, Cn.
m The Q branch is not seen whenever the vibration induces a dipole moment
change which is z to the principal axis of rotation, Cn.
The presence of low intensity overtone and combination bands is due to
anharmonicity, as mentioned in section (3.2). Their frequencies can be obtained
by adding or multiplying those of the fundamentals, so that for example, 2ν1 is
found at 2305 cm-1, which is very close to 2 @ 1151 (ν1) = 2302 cm-1.
Usually, stretches have higher energy than bends, and this is clearly the case for
SO2, where ν2 has half the energy of ν1 and ν3. Detailed studies have been done on
SO2 to confirm the above assignments.
However, once a molecule’s spectrum has been assigned in this manner, it is
possible to prove its structure, and even to calculate the bond lengths from the
gas-phase vibrational spectrum! In the laboratory this is illustrated in Experiment
3, where you make GeH4, and analyse its gas-phase spectrum, from which the
Ge-H bond length can be determined.
4.2
Limitations of normal-mode analysis
Large molecules simply have too many bands to make assignments possible.
Remember that we have NOT gone into the work involved in assigning the bands
in SO2. This process becomes too complex to solve for large molecules. As an
example, consider the number of normal modes one gets for a medium-sized
molecule such as ethane:
Fig. 9-8
This 8-atom molecule already has 12 normal modes. You can imagine how this
grows with the size of the molecule. Also, the more bands there are, the more they
overlap, and you soon get the familiar "fingerprint" complexity of the IR spectra
you saw in 2500 and 2600.
So in general, small molecules can at times be completely characterized by their IR
spectra, usually in the gas phase. Complex molecules can be analysed (1) by the
group-frequency approach or (2) by the fingerprint method.
Page 7
Table 9-4B
Table 9-4C (and Fig. 9-8)
Chemistry 3810
Lecture 10
SPECTROSCOPY - II: VIBRATIONAL, CONCLUDED
5.
IR spectroscopy - instrumentation
The IR frequency range is considered to be 4000-30 cm-1. The BOMEM instrument can
detect signals in the range 6000-200 cm-1. The range above 4000 is called the
"near-visible" range, and is not very useful. Below 200 cm-1, the optical materials of the
instrument absorb all the light (CsI). I don’t believe there is any instrument known which
can observe the complete IR range in one go. The lower region, called the "far IR"
requires special instrumentation.
The optical techniques used are (1) scanning (or dispersive), where a grating or a prism is
slowly turned through the spectrum, and the intensity of transmitted light is measured. A
newer technique is (2) the Fourier transform, in which all the wavelengths of light are
detected simultaneously. However, half the light is rendered out-of-phase with the other
half, and the degree of this phase shift is altered by moving a mirror inside the instrument.
The resulting pattern is called an interferogram. This can by analyzed by the
mathematical technique of the Fourier transform, in which the shape of the interferogram
is ratioed to the speed at which the mirror moves. The computer then presents the data in
the traditional way. The advantage is speed. In our runs we typically do 10 scans, in
which the true signal is additive, but background noise tends to cancel out. We get a much
better quality spectrum.
6.
IR spectroscopy - sampling techniques
GAS
Gases are measured at low pressure in long-path length cells which can hold a vacuum.
The path length most common is 10 cm, although for special purposes paths of several
meters can be devised!
LIQUIDS
Liquids are usually measured as smears (thin films) held between two plates of alkali
halide, sometimes with a spacer to get a definite path length.
SOLIDS
I think all the compounds in this lab are solids. Solids can be measured by three common
methods:
6.1
Mulls
Grind the sample to particles ≤ λ, i.e. to several microns in size. This means
vigorous grinding in a hard mortar like agate. Then scrape the fine solid off with a
spatula. To minimize dispersion of the light, mix with oils like Nujol or
Fluorolube to the consistency of a thick paste. Spread a thin film of this paste
between the plates.
Nujol is a hydrocarbon oil, with strong absorption around 2900 and 1400 cm-1.
Fluorolube is a completely fluorinated oil, giving a clear window above 1600 cm-1,
and a mess below.
6.2
KBr pellets
Grind both the KBr powder (dry! desiccator storage) and the sample (1-2% only
Page 1
Chemistry 3810
Lecture 10
required) to the same size as for mulls. Load into the pellet press, and apply 20
tons pressure for 5 minutes. Remove the thin pressed pellet (or plate) into the
appropriate sample holder. There are now no oil peaks to obscure the sample!
6.3
Solutions
Choose a solvent with a clear window in the region you wish to measure. Avoid
solvents with a strong affinity for water, because these will damage the alkali
halide cells! Run as a reference the cell containing this solvent. Then make a
solution of your compound in that solvent, and run this as a transmittance
spectrum. This will have the effect of subtracting the solvent peaks out of the
spectrum. However, the regions where the solvent absorbs most strongly will have
funny noise peaks. This is because so little IR light gets through in this region that
the instrument gets a little fooled.
Table 9-2 in the notes gives useful information on the "open regions" for common
IR solvents.
Window materials:
7.
NaCl
6000-600 cm-1
KBr
6000-450 cm-1
CsI
6000-200 cm-1
Polyethylene < 600 cm-1
Group frequency method
Where the detailed analysis is not possible, we can sometimes use the group frequencies
approximation. This use is common in organic chemistry, where certain regions of the
spectrum are associated with certain organic functional groups. In Table 9-3 in your
notes, I have given you an inorganic group frequencies table. Use this to assign your
laboratory spectra.
The basic assumption of the group frequencies is that a certain normal mode of vibration
of the molecule will be dominated by motion in the bond of interest. Also, it will
dominate in all or most derivatives with that same functional group. E.g. the C=O stretch
in acetone dominates one normal mode. This frequency can be used to identify other C=O
groups in other molecules, where this is again a dominant contributor.
In general the group frequencies approach is valid so long as the normal mode is 80-90%
dominated by the group being identified. Be particularly aware of special bonding
scenarios which might alter this assumption (e.g. a strong contribution by another
resonance isomer in the Lewis structure.)
Going back to the original force-constant equation, recognize that all 3rd period and over
atoms are very heavy (22 mass units or more!) As a rule, the element-ligand stretch will
be found as a weak set of peaks at very low wavenumber. (The exception will be for
metal hydrides, i.e. with a direct M-H bond.) Also remember that many of the
constituents of your molecules will be "organic" fragments, which may have very similar
vibrations to their parent organic compounds.
8.
Raman spectroscopy: principles and techniques
Consider Fig. 10-1. This is a Raman spectrum of CCl4, named after the discoverer of the
phenomenon. An intense, monochromatic light is shined on a sample, and a detector is
mounted at 90‘, so that the direct beam of light is not detected. Fig. 10-3
Page 2
Figure 10-1
Figure 10-2
Chemistry 3810
Lecture 10
Illustrations: Figure 10-1 and 10-2 (from B&H)
SCHEMATIC OF THE RAMAN EXPERIMENT
Direct beam
Laser
Sample
Scattered light
Monochromator
Fig. 10-3
Some of the light is scattered by the sample, in fact some is actually scattered at the
molecular level, which actually involves the absorption and re-emission of the light by the
compounds in the sample.
MOST of the light is re-emitted at the same wavelength as the incident light. This is
called Raleigh scattering. (Intensity is 10-3 of the incident light.)
But a very small amount of the re-emitted light is returned with slightly different
wavelength. Let’s talk in terms of frequency. The light whose frequency is less than the
incident has lost part of its energy to vibrational excitement of the molecule. This is
called the Stokes scattering.
Also some of the light is re-emitted with higher frequency. This light has gained energy
from the vibrations of the molecule. This is called the Anti-Stokes scattering. The
intensity of these lines are usually lower, since more molecules are in low vibrational
states. Thus they have more to accept, and less to give away! The combination of the two
sets of lines is called the Raman scattering of the molecule, and has intensity of 10-6 of the
incident beam.
In Fig. 10-2 the IR and Raman of the same compound are compared.
9.
Applications of Raman and IR together to solve structure
An IR absorption will only occur if the molecular vibration involved in some way alters
the net dipole moment of the molecule. Some of the classic examples are the simple
diatomics: N2 and O2 retain the same, zero dipole moment when the bond is stretched.
Therefore they do no absorb IR light! However, CsO does absorb in the IR, since any
change in the bond length will change the dipole moment (i.e. polarity) of the molecule.
Raman scattering occurs only when the net polarizability of the molecule is altered during
the vibration. The two techniques are therefore complementary.
Lets take a real example: BF3 (which is a mixture of 10BF3 {19%} and 11BF3 {81%}).
There are two reasonable structures for this compound, i.e. trigonal pyramidal C3v, and
trigonal planar, D3h. Which of these is predicted by VSEPR?
Page 3
Chemistry 3810
Lecture 10
F
B
F
F
B
F
F
F
Now we do a symmetry analysis of these two possibilities:
Step 1: Look at the normal modes of vibration in Table 9-4
Step 2: Tabulate the results
Mode
C3v
activity
D3h
ν1
ν2
ν3
ν4
a1
a1
e
e
IR-Raman
IR-Raman
IR-Raman
IR-Raman
a’1
a"2
e’
e’
Raman
IR
IR-Raman
IR- Raman
Step 3: Now look at the character tables. Normal modes which have the coordinates x, y,
and z next to them in the tables are IR active. Those with square terms, xy, xz, x2, y2 etc
are Raman active. (Because polarizability is a tensor property, whereas dipole moment is
a vector quantity.)
Step 4: look at the spectra to see what is observed.
IR 10B
11B
Raman 10B
11B
Assignment
1505
1454
1505
888
1454
888
791
482
691
480
482
480
ν3
ν1
ν2
ν4
Step 5: We see that the spectrum has 1 IR active band, 1 Raman active band, and two
which are IR & Raman active. This clearly fits only the D3h geometry. We conclude the
molecule probably is trigonal planar.
Such detailed insight into structure is usually restricted to relatively small molecules, and
specifically to structures with well-defined symmetry constraints.
BRAIN TEASERS
1) What is the difference in positions of 10B and 11B bands due to?
2) Why does ν1 not show this difference?
Page 4
Chemistry 3810
Lecture 11
BRIEF INTRODUCTION TO INORGANIC NMR SPECTROSCOPY
References:
1)
The course text, pp. 47-49, 273, 292-296
2)
Butler and Harrod, Ch 7, pp. 198-213
3)
"Multinuclear NMR", Joan Mason (NY: Plenum, 1987). [QD96.N8.M85 on reserve.]
4)
For spectral assignment of 1H and 13C, see the book by Silverstein (in the Organic labs).
1.
Nuclear spin
Nuclei may have spins of zero or greater than zero in units of 4. In the handout is a table
of the common isotopes for NMR spectroscopy, which presents various nuclear properties.
Another source of data on isotopes is the CRC Handbook, which has a table of isotopes
(stable and artificial) which is extremely handy for NMR, ESR and Mass spectroscopy.
See Table 11-1 (two pages)
Those nuclei with zero spin angular momentum have no NMR signals. The best
candidates for NMR are those nuclei with spin of 4. Nuclei with spins of 1 or more are
often called the quadrupolar nuclei. The signals of quadrupolar nuclei tend to be
extremely broad except for the most symmetrical of compounds, e.g. 33S in SF6. If
insensitive, can often be hard to detect at all.
Spin-4 nuclei generally give sharp, easily detected NMR signals. The most common for
NMR use are:
1H
19F
31P
13C
15N
77Se
29Si
119Sn
183W
195Pt
250 MHz @ 5.89 Tesla
235.2
101.2
62.9
25.3
47.7
49.7
93.2
10.3
53.7
99.985% abundant
100%
100%
1.11%
0.3%
7.50%
4.7%
8.6%
14.4%
33.8%
Commonly observed quadrupolar nuclei (those with small quadrupole moments) include:
2H
10B
11B
14N
17O
2.
38.4
26.9
80.4
18.1
33.9
1.6 @ 10-2%
18.8%
81.2%
99.6%
3.7 @ 10-2%
Relative sensitivity
The %abundance is not the only factor in the sensitivity of an NMR nucleus. To
understand this, we must delve a little into the essence of the NMR experiment.
Unfortunately, NMR theory is a terrible muddle of quantum and classical models. The
heart of the technique, however, can be understood in the usual quantum-mechanical
Page 1
Chemistry 3810
Lecture 11
method for any spectroscopy.
The nuclear energy levels become distinct only when the atom is inside the magnetic field.
ml +4
E
N
E
R
G
Y
∆E
ml -4
MAGNETIC FIELD
The size of this energy gap is quite small, hence the spectroscopy takes place in the FM
radio frequency range. (Broadcast FM ranges from about 88 to 108 MHz) The size of the
energy gap is given by the equation:
∆E
=
hν0
=
γ
2π
B0
The size of the energy gap is directly related to two parameters: B0 which is the strength
of the applied field. The second is γ , the magnetogyric ratio. Both of these factors affect
sensitivity for the following reason. The difference in the population of the two energy
levels is very small, and is controlled by the Boltzman distribution. The larger the gap, the
fewer nuclei populate the higher energy level, and thus the more nuclei are able to absorb
a quantum of radiation, which leads to an NMR signal.
Compared to other spectroscopic techniques, NMR is inherently insensitive. Stronger
magnetic fields increase sensitivity. Hence the push for larger and larger magnets. For
example, the new instrument has a field strength of 5.89 T vs the 1.4 Tesla of the Varian
EM 360 iron magnet instrument. The largest commercially available NMR magnets are
currently about 14 Tesla.
The magnetogyric ratio is an inherent property of the nucleus. This value coupled with the
natural abundance determines the inherent sensitivity of the nucleus.
See Table 11-1, Reltive sensitivity at constant field
3.
Chemical shift
Think of a large frequency spread (at fixed magnetic field).
Page 2
Table 11-1 (on this page and the next)
Chemistry 3810
Lecture 12
INORGANIC NMR SPECTROSCOPY, PART II
4.
Spin-spin coupling constants
As you all know, close nuclei in molecules can couple with each other to give the familiar
patterns of NMR signals.
Fig. 12-1
Note that the Pascal’s triangle for coupling only holds if the coupled nuclei are spin 4!
For example, the text gives the spectrum of HD measured in the 1H region.
Fig. 12-2
Here coupling to a single 2H nucleus (I = 1, 2I + 1 = 3) gives a 1:1:1 triplet. Similar
results are obtained with boron compounds.
So far your are familiar with First-Order Coupling. This is a limiting situation, whenever
the chemical shift of the coupled nuclei are sufficiently different, then coupling occurs
according to the Pascal triangle. However, when the chemical shift separation is small,
second-order coupling is often observed. This leads to funny patterns, which often have
to be computer-analyzed to be fully understood or to measure the actual coupling
constants.
Sometimes these patterns can be deceiving: they look almost like Pascal triangle coupling,
but there is no logical reason for it to be that way. Read up on the references before trying
to interpret the NMR of these compounds.
Consider the simplest example of second order coupled NMR signals. This is the case of
two isolated spin 4 nuclei coupled together. Called AX and AB.
(Nomenclature: if the letters are close in the alphabet, they are close in chemical shift. If
far apart in shift, so in letter.)
See Fig. 12-3 for an example
A
J=0
∆AB ≠ 0
J≠0
∆AB ≠ 0
J≠0
∆AB ≠ 0
J ~ ∆AB
J≠0
∆AB ≠ 0
J > ∆AB
J≠0
∆AB = 0
J << ∆AB
B
J
J
J
Go through the various scenarios of J vs. δ, an appearance of spectra.
Page 1
Chemistry 3810
Lecture 12
Heteronuclear coupling, e.g. between 1H and 31P will always be first-order, because the
base frequencies of the nuclei are so vastly different.
5.
Satellites
If the nuclei under observation is coupled to a spin 4 nucleus which is not 100% abundant,
then two separate NMR signals will appear, one the uncoupled one, the other the
spin-coupled multiplet. But because they are centered on the same frequency, we talk
about these signals as having "satellites".
Fig. 12-3 & 12-4
This occurs also for quadrupolar nuclei, e.g. the 1H NMR of GeH4. Spin 9/2, therefore 10
lines in proton NMR.
Fig. 12-5
6.
Decoupling
Decoupling is accomplished by selectively irradiating at a select frequency. This
saturates the NMR signal of the irradiated nucleus, and other nuclei coupled to them lose
their coupling. It is a useful technique for unraveling complex NMR patterns, although
nowadays it is largely superseded by so-called two-dimensional NMR experiments.
Still in common use today is the concept of broadband decoupling, which, for example,
irradiates the whole 1H region when observing 13C or 31P. This has the effect of greatly
simplifying the spectra of these nuclei, but often also causes a signal enhancement, which
is advantageous for insensitive nuclei like 13C. This type of decoupling is indicated by the
symbolism:
{1H}-31P
7.
or
{1H}-13C
NMR and timescale
SAL Section 3.2, page 109 - 113
All the spectroscopic methods have a definite interaction time. As a rule, the timeframe
of the interaction varies inversely with the energy of the photons. Because NMR is such a
low-energy process (hence its attractiveness in medical imaging) it also has a long
interaction time.
Whereas most other spectroscopies give us a snapshot of a particular molecule in a
particular stage of motion (like using a fast shutter-speed to "freeze" an athlete in
mid-motion), NMR often occurs at the timescale similar to the timescale of molecular
motion. This means that it is possible for NMR signals to be averaged by molecular
motion (just like using a long exposure-time when a moving object is photographed: the
athlete looks blurred). Examples of these effects are given in the text. Another example is
given by the 19F NMR spectra of SF4.
Fig. 12-6
This effect plays a role in the laboratory in Expt. 7 (Cp2TiS5 - in which inversion of the
cyclohexane-like boat conformation of the TiS5 ring renders the two Cp rings equivalent.)
Page 2
Figure 12-1 and Spectral Analysis Worksheet
Figure 12-2
A schematic 1H-NMR spectrum of HD
Fig. 12-3
CH2 peaks in 1H NMR of [PtOS5Me(Cl)PPh3] at 100 MHz
Note the presence of two sub-spectra, one due to the 66% of the molecules without a spin-active Pt
nucleus, the other due to the 33% containing 195Pt with I = ½.
Fig. 12-4
The 12-line 31P spectrum of [PtOS5Me(PPh3)2]BF4
Note the presence of two sub-spectra, one due to the 66% of the molecules without a spin-active Pt
nucleus, the other due to the 33% containing 195Pt with I = ½. Also note the large range of chemical
shifts and the huge size of some of the spin-spin coupling constants.
Fig. 12-5
The 1H-NMR spectrum of GeH4 showing the satellites due to 73Ge with I = 9/2
Fig. 12-6a
The relationship between lifetime of a state and its energy. A state with an infinite
lifetime has a precisely defined energy; a state that has a short lifetime has an energy that may be
anywhere with a range.
Fig.12-6b
When the lifetime τ of a
conformation of a molecule is infinite, the
spectrum shows narrow absorptions. (b) As
the lifetimes shorten, the absorptions
broaden and merge. (c) When the lifetime is
very short, a single sharp absorption is
obtained at the mean of the two frequencies.
Fig. 12-6c
Temperature dependence of the 19F
spectrum of SF4
Tris-(2,6-diisopropylphenyl)-guanidine
N
H
N
N
H
A,B,C
180°C
A,B
120°C
C
B
C
C
A
70°C
A
C
C
A
A
B
30°C
B,C
AC A
B
-40°C
5.0
Figure 12-7
4.0
3.0
2.0
ppm
Variable Temperature 1H-NMR spectrum of the Guanidine shown in the diagram
Figure 12-8 and Spectral Analysis Worksheet
A BRIEF INTRODUCTION TO ELECTRON PARAMAGNETIC RESONANCE SPECTROSCOPY
1. ELECTRON SPIN.
2. INSTRUMENTATION
3. PRESENTATION OF ESR SPECTRA
4. HYPERFINE SPLITTING
5. EPR SPECTRA OF TRANSITION METAL COMPLEXES
6. SOME APPLICATIONS
1
2
3
5
8
9
EPR detects unpaired electrons of materials contained in a homogeneous magnetic field using microwave
radiation to establish a resonance condition that can be detected by suitable microwave detectors. It is often
called electron spin resonance (ESR) since many systems with unpaired electrons are free radicals. As an
alternative to these disputed names, some spectroscopists have suggested the replacement term electron
magnetic resonance. EPR is a technique used in chemistry and physics, in materials science (e.g. to
characterize polymers, magnetic and conducting materials, crystalline defects, etc.) and increasingly in
biology and medicine. EPR can also be used for magnetic resonance imaging where it has the special
advantage of only detecting unpaired electrons contained, usually, in a specific spin-labeled drug.
The origin of the signal is therefore the electron. In fact, since many students of chemistry are much more
familiar with NMR, one way of describing EPR is as if it were NMR with the electron as the observe
nucleus. To extend this analogy a little further:
NMR property
EPR equivalent
• observe nucleus (e.g. 1H, 77Se)
• one or more unpaired electrons
• magnetic dipole vs. magnetic quadrupole
• ms = ± ½ (i.e. resembles spin-½ nuclei)
• chemical shift (i.e. field/frequency ratio)
• g-value (i.e. field/frequency ratio)
• homo-nuclear coupling
• normally not seen - all electrons equivalent
• hetero-nuclear coupling (always first-order)
• hyperfine splitting (always first-order)
References:
1.
•
•
Shriver, Atkins, Langford, Inorganic Chemistry, 2nd edition(Freeman, 1994) p. 606 - 608
Buttler and Harrod, Inorganic Chemistry, (Benjamin/Cummings, 1989) p. 221-225.
J.A. Weil, J.R. Bolton and J.E. Wertz, Electron Paramagnetic Resonance, (Wiley, 1994).
Electron spin.
Electron spin energy levels become distinct only when the atom is inside the magnetic field.
The size of this energy gap is larger than for nuclear spin, and the spectroscopy takes place in the
microwave frequency range. The size of the energy gap is given by the equation:
∆E = hυ0 = gµ BB0
R. T. Boeré
Brief Introduction to EPR Spectroscopy
Here g is called the Landé splitting energy, usually called the "g value." This is 2.0023 for the free
electron, and for compounds is often quoted in Gauss relative to this. (1 Tesla = 10,000 Gauss.)
ml = +1/2
E
N
E
R
G
Y
∆E
ml = -1/2
MAGNETIC FIELD
µ B is the Bohr magneton, µ B = e;/2me (value in back of text).
B0 is the applied magnetic field (B and H are almost, but not quite interchangeable measures of
magnetic field strength. Note also that we use here cgs units! There is a fundamental difference in the
definitions of magnetism laws using cgs and SI units!)
The size of ∆E corresponds to photon energies in the microwave region of the electromagnetic
spectrum. Thus EPR spectrometers combine microwave sources and detectors with powerful
magnets, much as NMR spectrometers combine radio frequency technology with magnets.
2.
Instrumentation
Most EPR spectrometers are CW instruments, similar to the original instrumentation used for NMR
before the development of pulsed Fourier Transform spectroscopy. Most also use electromagnets
rather than permanent magnets or cryomagnets. There are good reasons for this. First of all, EPR
does not suffer from the crushing sensitivity problems of NMR, since the experiment is inherently
more sensitive. (Why?) There is therefore not as strong an incentive to use the FT approach. Also,
the spectral width of many EPR experiments can be much greater than in NMR; broad frequency
pulses are always a challenge to pulsed-mode spectroscopy. Nonetheless there are key advantages to
pulsed-mode spectroscopy, and this is one of the hot areas of research by EPR experts.
The advantage of higher fields, i.e. those requiring cryomagnets, is also not as obvious in EPR,
because the nature of microwave resonant cavities is such that their size must reflect the wavelength
of the radiation. Thus as the wavelength gets shorter with higher field strength, the samples must get
smaller. Cryomagnet systems for W-band EPR are now commercially available, and are used for
systems where the samples are inherently small, especially in biomedical applications.
Page 2
R. T. Boeré
Brief Introduction to EPR Spectroscopy
But for now, CW
instruments are the
norm for most users
in chemistry and
materials
science.
This Department’s
new
EPR
spectrometer
is
therefore a very
advanced, computer
controlled,
CW
instrument.
The
design
of
such
instruments
represents
highlyrefined development
of
the
basic
instrumental design
that has been around
for a long time.
Consider
the
following schematic
diagram of a CW
EPR spectrometer:
The main components one sees in the EPR lab are (1) the magnet power supply; (2) the coolant recirculator; (3) the iron-core electromagnet; (4) the microwave bridge; (5) the electronic system
console; (6) a Pentium II computer; (7) printers and other peripherals.
The microwave bridge is connected via wave-guides (rectangular, gold-coated tubes) to the resonant
cavity. This device is suspended between the pole-faces of the magnet, and is the place that the
sample is inserted. Samples in EPR are not spun, because the requirements for field homogeneity are
not as strict as in NMR. More fundamentally, the design of the resonant cavity may preclude sample
spinning. There are a wide variety of such cavities in use, and each has its special advantages and
disadvantages.
The cavity on the University of Calgary’s Bruker EMX 10/12 spectrometer operates in the TE102
mode, and is of rectangular cross section. The orientation of the sample is extremely important, and
it would not be desirable to have the magnetized nuclei moving rapidly from the magnetic to the
electric field region of the spectrometer. The use of EPR flat cells is predicated on keeping the in the
node of the electric field, since it is the electric field that leads to energy loss through dielectric
heating of the solvent. Flat cells are used for solvents of high dielectric constant.
Page 3
R. T. Boeré
Brief Introduction to EPR Spectroscopy
Page 4
R. T. Boeré
3.
Brief Introduction to EPR Spectroscopy
Presentation of ESR spectra
Traditional spectrometers used a detection circuitry that naturally provided a dispersion signal. This
means that the spectra appear as the “first-derivative” of the absorption peak. Since ESR spectra
often have quite broad lines, there is a distinct advantage to this type of data presentation, and for this
reason the convention has been retained. Consider
the spectrum at the right, taken from a real sample of
a C-N-S ring system. The left-hand curve is the
absorption peak; the right-hand is the original
dispersion pattern! Obviously the dispersion signal is
much easier to interpret, e.g. by counting the number
of lines.
The centre of an ESR spectrum is quoted in Gauss units as the "g-value". This corresponds to the
chemical shift of the signals in an NMR spectrum. In virtually all cases, one species only gives one
ESR signal, hence there is normally only one g-value! The g-value measures how far the magnetic
environment of the unpaired electrons differ from that of a free, gas-phase electron. Reference
compounds such as "Fremy's salt" are used to set the g-value. If you compare the left-hand spectrum
with a typical 1H-nmr spectrum you will see another important aspect of EPR: the bands are broad
w.r.t. the difference in g-value, and although the possible range of g-values is huge, most condensedphase main-group free-radicals in fact have very little variation in their g-value. For such compounds,
it is the rule rather than the exception to have overlapping spectra if there is more than one compound
present in a solution.
4.
Hyperfine splitting
What closely relates ESR to NMR is that the electron spin energy levels are split by interaction of the
nuclear spins of attached nuclei. This results in a multi-line pattern, which for each set of N
equivalent nuclei gives:
No. of lines = 2NI + 1
where N = the number of equivalent nuclei the electron
couples to, and I = nuclear spin
We first consider a very simple example: a deuterium
atom, which of course has a single electron. The spin of
deuterium is I = 1. Looking back at the splitting induced
by the magnetic field (first figure), the effect of the
hyperfine splitting is that of additional small “nuclear
magnets”. The amount of the splitting depends on the
value of I. For this simple example, the resulting splitting
turns out to be:
Critical to the appearance of the spectrum is the EPR
selection rule (the same as in NMR), which is that ∆Ms
(the electron spin quantum number) = ±1, while ∆MI (the
Page 5
R. T. Boeré
Brief Introduction to EPR Spectroscopy
nuclear spin quantum number) = 0. These transitions therefore predict a 1:1:1 triplet spectrum. This
is of course the same result as that predicted by the 2NI + 1 rule: 2 × 1 × 1 + 1 = 3
The intensity ratio’s can also be predicted without developing detailed energy-level diagrams because
they follow simple multinomial expansions, as shown in the following table:
Page 6
R. T. Boeré
Brief Introduction to EPR Spectroscopy
Page 7
R. T. Boeré
Brief Introduction to EPR Spectroscopy
Consider as an example the •CH2OH radical for
which the EPR spectrum is:
In ESR spectroscopy alone, this splitting due to
nuclear spin multiplicity is called "hyperfine
splitting". Despite the odd name, it means
nothing else than splitting in an NMR signal!
Note that such splitting is by definitions
“heteronuclear”, i.e. between the electron and a nucleus, species with very different resonance
frequencies.
When one or more of the spin-active nuclei is found to be less than 100% abundant, the spectra
obtained are superposition’s of more than one spectrum. Such spectra are usually referred to as
having “satellites”. An explanation of the origin of such spectra is contained in the box entitled “EPR
spectra with satellites”.
5.
EPR spectra of transition metal complexes
Consider the EPR spectrum of bis(salicylaldiminato)copper(II):
63
Cu: I = 3/2, (2NI + 1) = 4, 0 4 groups of lines
HA:
I = 1/2, two equivalent ones, (2NI + 1) = 3 lines times
N:
I = 1, two equivalent ones, (2NI + 1) = 5 lines
Since 3 × 5 = 15, each group should be 15 lines!
By far the largest splitting
constant is that due to the 63Cu
nucleus (100% enriched in this
HB
HA
spectrum). This implies that the
9
O
N
unpaired electron due to the d
2+
Cu
electron configuration of Cu is
N
O
mostly concentrated on the
HB
HA
metal. The presence of the
other signals implies that some
of the electron density is spread onto the ligands, but only as far as the immediate coordination
sphere.
As a second example, consider the
radical obtained by air oxidation of a
Cr porphyrin, whose spectrum is
shown below. Can you interpret this
spectrum? (53Cr is I = 3/2 and is 9.5%
natural abundance).
Page 8
R. T. Boeré
Brief Introduction to EPR Spectroscopy
6.
Some Applications
The following examples illustrate several aspects of EPR drawn from my own research field of C-N-S
heterocycles, which have a high tendency to form stable free radicals.
A)
The following EPR spectrum belongs to a 4-aryl-1,2,3,5-dithiadiazole. We observe five lines in the I
= 1 multinomial expansion, i.e. in a 1:2:3:2:1 intensity ratio (and not the Pascal ratio 1:4:6:4:1 seen
for pentets in 1H NMR spectra). The energy level shows the origin of this more complex pattern.
N S
.
N S
Why is no coupling to the S atoms observed? Why is no coupling to the aryl H-atoms observed?
B)
Inequivalent nitrogen atoms are observed in a 4-CF3-1,2,7-benzodithiazole radical.
N
.S
CF3
S
What nuclei does the electron couple to here?
C)
The effect of sample concentration: 3-CF3-5-Ph-1,2,4,6-thiatriazine:
N
S
.
N
N
CF3
High concentration
Lower concentration
No further changes are seen on greater dilution.
concentration spectrum!
Lowest concentration
Explain the coupling in the limiting lowPage 9
R. T. Boeré
Brief Introduction to EPR Spectroscopy
(this diagram should be located at the end of section 3)
The following diagram shows the range of field strengths at the X-band (9.5 GHz) and Q-band (36
GHz) for different kinds of molecules and elements:
Page 10
Fig. 13-1
Comparison of spectral presentation as absorption (a and c) and derivative (b and d)
curves
(c)
Fig. 13-2
(a) EPR spectrum of the methyl radical CH3. In a methane matrix at about 4 K; (b)
EPR spectrum of the NH2. Radical; (c) EPR spectrum of .CH2OH, a radical derived
from methanol. All spectra are shown as derivative curves
Chemistry 3810
Lecture 14
MASS SPECTROSCOPY
I assume you all have some familiarity with mass spectroscopy.
1.
Instrumentation
Fig. 14-1
Its useful to consider the instrumentation, because this explains the technique. The sample
is introduced through a tiny port into a high-vacuum chamber. Here it can be heated if
necessary to volatilize it. In a typical electron impact experiment, the sample is
bombarded with high energy electrons, which strip off even more electrons to make
positive ions:
M + e-
H
M.+ + 2e-
The mass of the positive ion is then measured by analyzing its kinetic energy, e.g. by
relative curvature of the path through a bent magnetic field. There are many other
analysis techniques to get this information.
Note that the read-out of a mass spectrometer is properly:
mass/charge
This is often expressed as m/e-. This means that a doubly charged ion will appear half as
heavy in the mass spectrometer!
We call the M.+ ion the parent ion. Usually, these ions are unstable and we also see
signals due to many fragment ions in the spectrum. For well-established chemistries, as in
standard organic chemistry, the nature of the fragments produced can be used to
distinguish, e.g. among different isomers of compounds with the same molecular mass.
Nevertheless, one should use cautious scepticism in this branch of spectra interpretation.
2.
Accuracy
There are two main classes of mass spectra: low resolution and exact mass or high
resolution.
In low resolution work, we are interested to the type of fragments produced and their
masses within one amu. We restrict ourselves to this type of mass spectroscopy here.
What we are more interested in is the isotopic patterns in the peaks due to the parent and
daughter ions.
3.
Spectra and interpretation
The best way to learn mass spectroscopy is to interpret examples. The key thing is to
remember that mass spectroscopy is an isotope sensitive technique. Information on
isotopes is most easily gotten by looking in the Table of Isotopes in the CRC handbook.
This is a detailed table tucked in between the tables of inorganic and organic compounds
in this handbook.
Examples:
(a) Low resolution spectrum of CF3CN2Se2 Fig. 14-2 & 14-3
Page 1
Figure 14-2
Mass spectrum of CF3CN2Se2 under low-resolution (1 amu) conditions
Figure 14-3
Calculation of isotopic peak composition of parent ion of CF3CN2Se2 in mass spectrum
Figure 14-4
Mass spectrum of CF3CN2Se2 under high-resolution conditions (0.0001 amu)
Figure 14-5
Extract of peak report for high-resolution mass spectrum of CF3CN2Se2
(Agreement between measured and calculated given in parts per million!)
Chemistry 3810
Lecture 14
(b) High resolution spectrum of CF3CN2Se2 Fig. 14-4 & 14-5
II
ELECTRONIC ABSORPTION SPECTROSCOPY
As the name implies, this technique involves the absorption of photons to alter the
electron structure of a molecule or atom.
1.
Electronic transitions
You will probably be most familiar with this type of spectroscopy in the study of the H
atom in Chemistry 1000. An atom or molecule can absorb a photon of energy if the
photon exactly matches the energy difference between a filled and an empty, higher lying,
orbital. Upon the absorption, the electron is promoted to the higher lying energy level.
In molecules, we call the most energetic filled orbital the HOMO, and the least energetic
empty orbital the LUMO. Thus the electronic absorption band of lowest energy (longest
wavelength) corresponds to the HOMO-LUMO transition.
Schematic illustration of electron promotion on photon absorption
LUMO
From the lenghts of the arrows it is clear that the lowest
HOMO
2.
energy transition is that between the HOMO and the LUMO.
Lambert-Beer law
Most measurements of electronic absorption spectroscopy involve a measurement of the
wavelength of the absorption maximum, λmax, as well as the intensity of the signal. The
latter is expressed in terms of the molar absorptivity, using the Lambert-Beer law:
%T =
I
Io
log10


I
Io


= A = ebc
Most UV-visible spectrophotometers read-out directly in absorbance units, since that is
linear with concentration. We typically use precision fused quartz cells with path-lengths
of exactly 1.00 cm.
Note that UV-visible bands in the condensed phase are usually very broad, Gaussian-type
curves. We measure A at the λmax point only.
For some typical examples, see Fig. 14-6 and 14-7.
Page 2
UV-vis Absorption Spectra - Some Examples
Since the spectra measure electron promotion from filled to empty MO's, an MO diagram is used to explain the phenomenon.
Here are two examples of closely related inorganic ring compounds, one of which has higher symmetry (S3N3–) and therefore
shows only a single band due to the 2e"→2a2" promotion, and the other has lower symmetry (Me2PN3S2), so that two bands are
observed close together in energy (in this case corresponding to 3b1→4b1 and 2a2→4b1 transitions. The HOMO-LUMO gap in
the second compound is significantly smaller than in the first, so that the 2a2→4b1 transition is found at lower energy (18400
cm–1 compared to 28000 cm–1 in S3N3–.
S
N
N
S
S
N
A note on units: ε is the Absorbance per unit concentration, while A is the absorbance of the actual sample (use Beer's law).
The x-axis is usually measured in nanometers (nm), but is often converted to wavenumbers (cm–1) for purposes of presentation
or analysis, since the latter are energy units and thus directly comparable to the gaps in the MO diagram. Sometimes the cm–1
are themselves converted to eV.