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Unformatted text preview: Chapter P2: Equilibrium Geometry and Vibrational
Frequencies
Introduction
Two classes of theoretical models, HartreeFock models and density
functional models, have now been defined. We now use these models to
calculate two closelyrelated molecular properties, equilibrium geometry and
vibrational frequencies. More than anything else, geometry defines a
molecule, whereas vibrational frequencies not only form the basis of infrared
spectroscopy but, as we shall see in Chapter P3, furnish correction to
calculated energies to allow them to be related to measured heats (enthalpies)
and Gibbs energies.
The first objective of this chapter will be to assess the behavior of HartreeFock and B3LYP density functional models with very large “limiting” basis
sets with regard to both equilibrium geometries and vibrational frequencies.
This will allow us to “remove” (or at least minimize) the effects of the LCAO
approximation and concentrate on problems caused by the HartreeFock
approximation and the success of density functional models in eliminating
these problems. The second objective will be to examine the same two classes
of models but with smaller, more practical basis sets. After we establish the
smallest basis set that reliably reproduces limiting behavior, we will be able to
extend assessment to larger molecules and to use the calculations to explore
chemistry. 1 Equilibrium Geometry
The threedimensional structure or geometry of a molecule and with its energy
are arguably its most important characteristics. An equilibrium geometry is
needed in order to calculate all other molecular properties (including the
energy), and providing geometry is normally the first step in any theoretical
investigation. Before we examine methods for determining and verifying
geometry, we review the sources and quality of experimental geometries. This
will provide standards of reliably for experimental data and therefore serve as
a guide to judging how well the different classes of theoretical models actually
perform.
Bond distances and angles of many if not most molecules (at least organic molecules) may
easily be anticipated, at least qualitatively. For example, carboncarbon single bonds
generally fall in the range of 1.45–1.55Å, and within this range vary predictably with the
types of hybrids involved. This leads to the possibility of establishing the 3D geometry of a
molecule based only on its Lewis structure. Given the computational cost involved in
obtaining an equilibrium geometry, it is quite legitimate to ask if this is effort well spent.
Of course, there are situations where a Lewis description may be problematic, for example,
where “correct” Lewis structure is not apparent or where two or more Lewis structures may
contribute. A much more important concern, however, is that Lewis structures are too
“coarse grain” to allow subtle differences among molecules to be revealed. 2 Sources and Quality of Experimental Equilibrium Geometries
There is a wealth of information about equilibrium geometry, accumulated
over nearly a century by a variety of experimental techniques. We shall focus
on only two of these techniques: microwave spectroscopy which is carried out
on a gaseous sample and is arguably the most precise of the available
methods, and Xray diffraction which is carried out on a solid crystalline
sample and is by far the most widely used of the available methods. Other
experimental techniques, in particular, infrared spectroscopy and electron
diffraction in the gas phase and neutron diffraction in the solid, are also able
to provide information about molecular structure. They have certainly proven
to be of value in some instances, but because they have been less commonly
employed they will not be addressed.
Microwave Spectroscopy
Equilibrium geometries for over a thousand small molecules in the gas phase
have been determined by microwave spectroscopy [xxx]. The experiment
involves excitation of rotational energy states by radio frequency radiation and
leads directly to the three principal moments of inertia for each unique set of
atomic masses.
Perhaps the most interesting presentday application of microwave spectroscopy is radio
astromony. Much of what we know about the existence of molecules in interstellar space
comes from assigning the holes in “white radiation” in the radiofrequency range to
molecules in space. The incredibly low concentration of matter in interstellar space (on the
order of x molecules/m3) is offset by an incredibly long path length. A catalog of “holes”
(microwave absorptions) exists and has been used to positively identify several hundred
molecules. Many “holes” have yet to be assigned, meaning that there are more molecules to
be discovered.
A more familiar application of “microwave spectroscopy” is the microwave oven. This
uses a source of radiation specifically tuned to one of the rotational lines of the water
molecule. Absorption causes water molecules (present in most foods) to heat up.
Unfortunately, microwave spectroscopy is now only rarely used for structure
determination. In part, this is because it has been supplanted by theoretical calculations,
which for molecules of the size amenable to experimental investigation leads to geometries
of comparable quality, at far lower expense. Also many if not most of the “easy” molecules
for microwave spectroscopy have already been investigated. We need to emphasize,
however, that microwave data remains the best source of highquality experimental data for
the structures of polyatomic molecules. In the absence of symmetry, the equilibrium geometry for a molecule with N
atoms involves a total of 3N6 bond lengths and angles, and its complete
determination requires as many as N2 individual spectral measurements with
3 different isotopic substitutions. A few elements occur naturally with sizable
populations of more than one isotope, for example, chlorine with two isotopes
in a ratio of 76:24 and bromine with two isotopes in a ratio of 51:49. Here, a
microwave spectrum will yield multiple sets of moments of inertia. However,
most elements, notably hydrogen, carbon, nitrogen and oxygen exist
predominately as a single isotope. Here, the synthesis of specifically labeled
compounds is necessary. At best, this is tedious. At worst, it is synthetically
difficult and very expensive. It is not surprising, therefore, that the use of the
technique has for the most part been limited to very simple (and often highly
symmetric) molecules.
In summary, the primary advantage of microwave spectroscopy is its
precision. Bond lengths are often accurate to within ±0.005Å. Many view
microwave spectroscopy as the “gold standard” of experimental methods for
obtaining molecular geometries. The primary disadvantage of microwave
spectroscopy is the need to examine multiple isotopic substitutions, that is, the
need to prepare multiple molecules with different isotopes. A second serious
disadvantage is the requirement of a permanent dipole moment. Many very
small molecules, for example, all homonuclear diatomic molecules, do not
exhibit a microwave spectrum. Molecules with finite but very small dipole
moments, for example, saturated hydrocarbons, will show only weak
microwave transitions which may be difficult to observe.
Should we get into the difference between re, r0, etc?
XRay Diffraction
Equilibrium geometries for more than 600,000 small to medium size organic,
inorganic and organometallic molecules [xxx] as crystalline solids have been
established by Xray diffraction. In addition, the structures or partial structures
of more than 40,000 proteins and proteinsmall molecule aggregates [xxx] and
several thousand minerals and “materials” [xxx] have been established.
Crystallography has moved into the mainstream for determining the
geometries of “new molecules” (in particular, new organic molecules), and
has become the main source of information about the shapes of proteins and
proteinsmall molecule complexes. Except for proteins and other
macromolecules, bond lengths are typically accurate to ±0.020.04Å.
(Individual bond distances in proteins can be in error by 1Å or more.) Xray
diffraction generally provides a poor account of the positions of hydrogen
atoms. The primary reason is, as previously mentioned in Chapter P1, that
there are relatively few electrons associated with hydrogen atoms. A related
problem is that bonds to heavy atoms are often too short by as much as 0.24 0.3Å, due to the fact that the electron density around hydrogen is shifted
toward the atom to which it is bonded. This is illustrated below for acetylene. Note, however that this is a systematic error, meaning that it may easily be
taken into account.
The fact that very small molecules typically do not easily crystallize (at least
at “normal” temperatures), and molecules that do crystallize may be too large
for full structure determinations with microwave spectroscopy, means that the
threedimensional geometries of only a relatively few molecules are known
both in the gas and in the solid. This makes it difficult to assess the effect of
the crystalline environment on geometry. Some differences for some
molecules are to be expected in order to benefit most from intermolecular
interactions, for example, intermolecular hydrogen bonds, and more generally
from the demands of packing in the crystal. However, the limited data that are
available suggest that changes in bond lengths and angles in going from the
gas to the crystal are likely to be modest, and even inside the error bounds of
the experimental method. Changes in conformation (torsional or dihedral
angles) from the gas to the crystal are likely to be larger and more common.
This will be addressed in Chapter P5.
The primary advantage of Xray diffraction it is easily applied to a wide
variety of molecules and (unlike microwave spectroscopy) requires a single
sample. The primary disadvantage is that this sample must be a crystalline
solid. Some molecules will not crystallize (or will not produce “good”
crystals) and their geometries cannot be determined by Xray diffraction. This
includes many small (low molecular weight) molecules which may be ideal
candidates for microwave spectroscopy. Thus, the two experimental
techniques are in one sense complementary. Xray diffraction lacks the
precision of microwave spectroscopy. Bond lengths in medium sized
molecules are seldom established to better than ±0.02Å (and are commonly
known to no better than twice this amount). Hydrogen positions are poorly
identified and bond lengths to hydrogen are commonly too short by 0.1 Å or
more. Finally, intermolecular interactions associated with the need to pack
into a crystal, may influence bond lengths and angles. 5 Obtaining and Verifying an Equilibrium Structure
An equilibrium structure or equilibrium geometry is a minimum in all
dimensions on a multidimensional potential energy surface. There will likely
be many such locations on such a surface. In chemical terms, these energy
minima may correspond to different isomers or to different conformers of a
particular isomer. For example, cis and gauche1butene and cyclobutane are
three of the many minima on the C4H6 potential energy surface. The first two
are conformers (related by twisting about a single bond) and (collectively) are
isomers of cyclobutane (related by bond reorganization).
CH3
H C
C H H
H
H H C C
C H H C H
C H3 H HH
C
C
HH HH
C
C
HH In order to qualify as an equilibrium structure, two mathematical requirements
must be met. First, the structure must correspond to a stationary point,
meaning that all first derivatives on the 3N6 dimensional (N atoms) potential
surface (the gradient of the energy) must be zero.
∂E(x)/∂xi = 0 for all xi
Second, all terms in the diagonal representation of the matrix of second
derivatives (the Hessian) must be positive. In order to verify this, the full
matrix of second energy derivatives (∂2E(x)/∂xi∂xj) in the original coordinates
first needs to be assembled and then replaced by a new set of coordinates (socalled normal coordinates, ζ) such that the matrix second energy derivatives
in these coordinates is diagonal.
∂2E( ζ) /∂ ζ i∂ ζ j = δij[∂2E( ζ) /∂ ζ i2]
δij is the Kronecker delta function (1 if i=j; 0 otherwise). It will turn out that
the set of second energy derivatives in normal coordinates relate to the
infrared and Raman frequencies (this will be addressed later in this chapter).
For the moment, let us say only that positive (diagonal) Hessian elements
correspond to minima on the energy surface and lead to vibrational
frequencies that are real numbers, while negative elements correspond to
maxima on the surface and lead to vibrational frequencies that are imaginary
numbers. If a vibrational frequency is imaginary, the coordinate motion
6 corresponding to this frequency can be animated pointing the way down to an
energy minimum.
Satisfaction of these two mathematical requirements does guarantee that a
particular structure corresponds to an equilibrium geometry, but does not
guarantee that this structure can actually be experimentally detected, let alone
isolated and characterized. This requires in addition that there are no easily
accessible pathways leading to species that are significantly more stable.
Discussion will be provided in Chapter P4.
In practice, obtaining an equilibrium structure involves an iterative process
starting with a ”guess” at the geometry and terminating only when all first
derivatives fall below a preset tolerance, assuming that neither the energy nor
the atomic coordinates have changed significantly from their values in the
previous iteration. The required number of iterations will typically be of the
same order as the number of independent variables, although this will depend
on how close the guess geometry is to the final geometry.
The reason for this is that each iteration in an optimization involves calculation of an
energy gradient. Assuming a quadratic energy surface, calculation of 3N6 gradients
(one/iteration) is sufficient to project to the energy minimum. While such a procedure is costly in terms of overall computation (an order of
magnitude or more than an energy calculation), it can be fully automated.
Determining an equilibrium geometry is no more difficult in terms of “human
effort” than calculating a property for a fixed geometry.
Because of computational cost, calculation of the full Hessian (the “difficult”
part) and evaluation of vibrational frequencies from this Hessian (the “easy”
part) is generally not undertaken. Rather, the approximate second derivative
matrix formed during course of seeking a zero gradient point (see box above)
is employed by the optimization procedure. If desired, the Hessian and
vibrational frequencies can be calculated following geometry optimization.
Most programs that automatically determine geometry make use of molecular symmetry.
The original motive was to save computer time. This is no longer as relevant not only
because computers are now much faster, but more so because only small molecules
generally possess elements of symmetry. Note, however, that once established (or set)
symmetry elements will be maintained. This means that calculated geometries of molecules
with symmetry elements may not necessarily be energy minima. A good example is 7 provided by ammonia. Starting from a planar (D3h symmetry) structure will not give rise to
the pyramidal (C3v symmetry) equilibrium geometry. Rather, a planar structure will result.
.
Equilibrium Structures Have Real Frequencies: Obtain equilibrium geometries for the
chair and boat forms of cyclohexane using the HF/631G* model and calculate vibrational
frequencies. Are all frequencies real for both molecules? Which if either molecule is not a
minima on the energy surface? Elaborate.
Equilibrium Geometry of Disilylene (H2Si=SiH2): Use the B3LYP/631G* model to
calculate the equilibrium geometry of disilylene, the simplest molecule incorporating a
siliconsilicon double bond. Assume a planar (ethanelike) structure. Calculate vibrational
frequencies. Are all real numbers? If not, perform the following operations: First, animate
any imaginary frequencies to see how disilylene wants to distort to move it to a energy
minimum. Next, distort your structure accordingly. Finally, reoptimize the geometry (of the
distorted molecule) and again calculate vibrational frequencies.
Repeat your calculations and analysis for digermene, H2Ge=GeH2, the simplest molecule
with a germaniumgermanium double bond. Are the two systems similar with regard to
their planarity? CSD results 8 “Limiting” Behavior of HartreeFock and Density Functional Models for
Equilibrium Geometries
We first set out to establish (or at least to estimate) the limiting behavior of
HartreeFock and density functional models with regard to equilibrium
geometries. This will allow us to separate the effects of the LCAO
approximation from effects arising from replacement of the exact manyelectron wavefunction by an approximate HartreeFock or density functional
wavefunction. While it is not possible or at least not practical to actually reach
the limit, it is possible to use a sufficiently large basis set such that the
addition of further functions to the basis should have only a small effect on
calculated equilibrium geometry. The ccpVQZ basis set will be employed.
This is about as large a basis set that be applied for geometry calculations on
molecules with more than a few nonhydrogen (“heavy”) atoms.
We refer to lithium through neon as firstrow elements and sodium through argon as
secondrow elements. Hydrogen and helium may be thought of as comprising the “zeroeth”
row.
For a firstrow element, the ccpVQZ basis set comprises a core made up of 8 stype
Gaussians, and a valence split into four parts made up of 8,1,1,1 s–type Gaussians and
3,1,1,1 sets of ptype Gaussians. Three sets of dtype Gaussians, two sets of ftype
Gaussians and a set of gtype Gaussians are then added. The related smaller ccpVTZ basis
set (a core made up of 7 stype Gaussians, and a valence split into three parts made up of
7,1,1 s–type Gaussians and 3,1,1 sets of ptype Gaussians, supplemented by two sets of dtype Gaussians and a set of ftype Gaussians) is employed to establish the extent to which
geometries change in response to additional basis functions. It is also not practical to explore all the different functionals that have been
proposed. We limit ourselves to a single functional, specifically the B3LYP
functional. This is probably the most commonly employed and thoroughly
documented functional. Finally, it is not practical to explore the behavior of
“limiting” HartreeFock and B3LYP models with the ccpVQZ basis set for
all types of molecules.
As elaborated in Chapter X, analytical forms of some of the integrals arising in density
functional methods are not available and numerical integration procedures are required.
This requires that a “grid” of integration points be defined, introducing another set of
variables (overall number of grid points and the location of each point) into the calculation.
The B3LYP density calculations employed here make use of the so called SG1 grid [xxx].
Experience suggests that equilibrium geometries (and other properties) obtained using this 9 grid are very close to those obtained using much larger (and computationally much more
costly) grids. Errors less than 0.005Ǻ and 0.1o are expected. We restrict ourselves to relatively small molecules comprising first and
secondrow main group elements only. Full details have been provided only
for oneheavyatom hydrides and for hydrocarbons, although summaries in the
form of Excel spreadsheets have been provided for a variety of other types of
molecules. Establishing equilibrium geometries for molecules with transition
metals and for most intermolecular complexes is not practical with a basis set
as large as ccpVQZ, primarily because of the size of the molecules involved.
Geometries obtained from HartreeFock and B3LYP models with smaller
basis sets will be compared with experimental structures later in this chapter.
Signed deviations in AH bond lengths (in Ǻ x 1000) calculated using the
HartreeFock ccpVQZ and B3LYP ccpVQZ models (HF/ccpVQZ and
B3LYP/ccpVQZ, respectively) for oneheavyatom hydrides appear
alongside of experimental distances in Table P21. Unsigned deviations from
calculations using the smaller ccpVTZ basis set in lieu of ccpVQZ are also
provided. This allows us to judge to what extent the ccpVQZ basis set
actually represents a limit.
An Excel spreadsheet containing AH bond distances for both HartreeFock and B3LYP
models with both ccpVTZ and ccpVQZ basis sets is provided on the CDROM
accompanying this text ( AH bond distances). Except for the hydrides of lithium and sodium, “limiting” HartreeFock bond
lengths are consistently shorter than experimental values. This trend and the
two exceptions to it can easily be rationalized by recalling the discussion of
the full configuration method as a means to correlate the motions of electrons
and improve on the HartreeFock model (see Chapter X). Full configuration
interaction involves a weighted mixing of all possible excited states with the
ground state. One way to think about this is in terms of promotion of one or
more electrons from occupied molecular orbitals in the groundstate (HartreeFock) wavefunction to unoccupied molecular orbitals. The lower the
promotion energy, the more likely it is that the resulting excited state will
contribute to the overall mix. Therefore, the most important contributions
should be those in which an electron is removed from one of the few highestoccupied molecular orbitals and placed in one of the few lowestunoccupied
molecular orbitals. 10 Table P21: Signed Errors in “Limiting” HartreeFock and B3LYP AH
Bond Lengths in Hydrogen and in OneHeavyAtom Hydrides AHn (Å x
1000)
HartreeFock molecule to expt to ccpVTZ B3LYP to expt to ccpVTZ expt. H2 8 0 0 0 0.742 LiH
CH4
NH3
H2O
HF 10
10
14
18
20 1
0
1
1
1 7
5
1
2
4 1
1
1
1
1 1.596
1.092
1.012
0.958
0.917 NaH
SiH4
PH3
H2S
HCl 33
6
15
9
9 0
3
3
3
1 10
2
2
7
8 7
3
2
2
1 1.887
1.481
1.420
1.336
1.275 11 The two highestoccupied molecular orbitals of hydrogen fluoride are the
same energy (they are said to be degenerate) and correspond to 2p atomic
orbitals on fluorine. They are not involved in bonding to hydrogen and
electron removal from either or both should have little if any consequence on
the hydrogenfluorine bond distance. Underneath these two orbitals is a σ
bonding orbital, electron removal from which should lead to bond
lengthening. The lowestenergy unoccupied molecular orbital is antibonding
between hydrogen and fluorine. Electron promotion into this orbital, as would
occur in the full CI treatment, should lead to weakening (lengthening) of the
hydrogenfluorine bond. Taken together, this implies the unambiguous
conclusion that the hydrogenfluorine bond distance in the limiting HartreeFock model (prior to full CI) is too short. This is in fact exactly what is
observed. The situation is different (and ambiguous) for lithium hydride. Here both the
highestoccupied and lowestunoccupied molecular orbitals are LiH bonding.
Electron removal from the former and should result in bond weakening
(lengthening), while addition of electrons to the latter should result in bond
strengthening (shortening). Thus, it is not clear whether the effects of a full CI
treatment will be to lengthen or shorten the bond, and it is not clear whether a
limiting HartreeFock calculation will yield too short or too long a bond. In
fact, the calculated bond lengths for both LiH and NaH are longer than
experimental values. “Limiting” B3LYP calculations lead to bond distances that are much closer to
experimental values. The mean absolute error is 0.004Ǻ compared to 0.014Ǻ
for the corresponding HartreeFock calculations. Comparison with
experimental data does not reveal a systematic error in B3LYP bond lengths.
However, with the exception of lithium hydride and sodium hydride,
B3LYP/ccpVQZ bond distances are longer than HF/ccpVQZ lengths.
12 Bond lengths from both HartreeFock and B3LYP calculations with the
smaller ccpVTZ basis set are very similar to those obtained using the larger
ccpVQZ basis set. Differences are 0.001Ǻ or less for firstrow hydrides but
larger for hydrides involving secondrow elements. This suggests that while
the ccpVQZ basis set is sufficiently large to account for limiting behavior of
hydrides with first row elements, it may lack the flexibility needed to deal
with molecules containing heavier elements.
CarbonFluorine and CarbonLithium Bond Lengths: Is the trend in calculated bond
distances for HF and LiH bonds repeated for CF and CLi bonds? The experimental CF
bond distance in methyl fluoride is 1.383Ǻ, but the distance in methyl lithium is not known
(the molecule exists as a tetramer). To decide, examine equilibrium geometries for methyl
fluoride and methyl lithium that have previously been obtained from both HartreeFock and
B3LYP models with the ccpVQZ basis set. These are found in methyl fluoride and methyl
lithium on the Spartan Student CD.
Bond Angles in Ammonia and Water: Examine the bond angles in ammonia and water
that have previously been obtained from both HartreeFock and B3LYP models with the
ccpVQZ basis set. These are found in ammonia and water on the Spartan Student CD.
How do they compare with the experimental HNH bond angle (xxx o) and HOH bond angle
(xxx o)? Are you able to rationalize the differences between calculated HartreeFock and
B3LYP bond angles using the kinds of orbital arguments previously applied for bond
lengths? Elaborate. A particularly interesting comparison involving carboncarbon bond distances
in hydrocarbons is provided in Table P22. “Limiting” HartreeFock lengths
for double and triple bonds are shorter than experimental values by an average
of 0.024Ǻ, with individual deviations ranging from 0.017Ǻ to 0.028Ǻ.
An Excel spreadsheet containing hydrocarbon bond distances for both HartreeFock and
B3LYP models with both ccpVTZ and ccpVQZ basis sets is provided on the CDROM
accompanying this text (hydrocarbon bond distances). Orbital shapes again provide rationale for the trends. For example, the fact
that the highestoccupied molecular orbital of ethylene is strongly CC bonding
while the lowestunoccupied molecular orbital is strongly CC antibonding,
suggests that electron promotion from the former to the latter would lead to
bond lengthening, that is, the HartreeFock bond length) must be too short. 13 Table P22: Signed Errors in “Limiting” HartreeFock and B3LYP Bond
Lengths in Hydrocarbons (Å x 1000)
HartreeFock to B3LYP to to ccpVTZ expt to molecule expt triple bonds
acetylene
propyne
but1yne3ene 24
25
26 1
1
1 8
8
6 1
1
0 1.203
1.206
1.208 double bonds
butatriene C2C3
allene
butatriene C1C2
methylenecyclopropane
propene
ethylene
but1yne3ene
cyclobutene
1,3butadiene 24
17
19
28
21
25
24
23
24 0
1
0
1
0
0
1
1
0 20
8
7
17
10
15
7
8
9 0
0
1
0
0
0
0
1
0 1.283
1.308
1.318
1.332
1.336
1.339
1.341
1.342
1.343 aromatic bonds
benzene 15 1 7 1 1.397 single bonds
but1yne3ene
methylenecyclopropane C2C3
propyne
1,3butadiene
bicyclo[1.1.0]butane C1C3
bicyclo[1.1.0]butane C1C2
propene
cyclopropane
cyclobutene C2C3
propane
ethane
methylenecyclopropane C3C4
cyclobutane
cyclobutene C3C4 5
1
5
3
11
34
2
15
4
1
7
15
5
5 0
1
0
0
1
0
0
1
1
0
0
0
1
0 11
5
4
14
11
3
4
5
2
1
4
5
3
3 0
1
0
0
0
0
0
0
0
0
0
1
0
0 1.431
1.457
1.459
1.467
1.497
1.498
1.501
1.510
1.517
1.526
1.531
1.542
1.548
1.566 14 ccpVTZ expt. Single bond distances are better described from the “limiting” HartreeFock
model than are the lengths of double and triple bonds, although the errors span
a larger range. Most bond lengths are sorter than experimental values (by as
much as 0.034Ǻ for the bridging bond in bicyclo[1.1.0]butane), although
some bonds are actually longer.
Arguments based on the relative bonding/antibonding character of the highestoccupied and lowestunoccupied molecular orbitals are not always successful
in anticipating whether HartreeFock single bond lengths are likely to be too
short or too long. 1,3butadiene provides a particularly simple example. The
HOMO is clearly antibonding between the two center carbons while the
LUMO is clearly bonding. It might be expected, therefore, that transfer of
electrons should lead to bond strengthening (shortening), meaning the limiting
HartreeFock single bond distance should be should be too long. In fact, the “limiting” HartreeFock bond length is 0.003Ǻ longer than the
corresponding B3LYP distance and only slightly (0.003Ǻ) shorter than the
experimental value. The B3LYP/ccpVQZ model also provides a uniform
account of double and triple bond lengths in hydrocarbons. All calculated
lengths are smaller than experimental values and the mean absolute error for
the set is 0.010Ǻ (less than half that for the corresponding HartreeFock
model).
Bond lengths from both HartreeFock and B3LYP models with the ccpVTZ
basis set are nearly identical from those from the corresponding ccpVQZ
models. The differences here are smaller than those previously noted for bond
distances in oneheavyatom hydrides. This suggests that the convergence of
both HartreeFock and B3LYP models depends to some extent on the kind of
bond. 15 Similar results and similar interpretations follow from comparisons of other
classes of compounds. A summary of mean absolute deviations of bond
lengths obtained from HartreeFock and B3LYP models with the ccpVQZ
basis set for diatomic molecules, molecules with bonds between carbon and a
heteroatom, molecules with bonds between two heteroatoms and hypervalent
molecules (in addition to oneheavyatom hydrides and hydrocarbons) is
provided in Table P23.
Excel spreadsheets containing bond distances for both HartreeFock and B3LYP models
with both ccpVTZ and ccpVQZ basis sets for diatomic molecules, molecules with
carbonheteroatom bonds, molecules with heteroatomheteroatom bonds and in hypervalent
molecules are provided on the CDROM accompanying this text (bond distances in
diatomic molecules, carbonheteroatom bond distances, heteroatomheteroatom bond
distances, and bond distances in hypervalent molecules, respectively).
CarbonCarbon Bond Lengths in Cyclopropane and Cyclobutane: The experimental
carboncarbon bond length in cyclopropane is 0.016Ǻ shorter than that in propane whereas
the bond distance in cyclobutane is 0.022Ǻ longer. Provide a rationale for this. 16 Table P23: Mean Absolute Deviations of HF/ccpVQZ and B3LYP/ccpVQZ Bond Lengths Obtained from Experimental Distances
and Bond Lengths Obtained from HF/ccpVTZ and
B3LYP/ccpVTZ Models (Å)
HartreeFock class of compounds B3LYP from expt. from ccpVTZ oneheavyatom hydrides 21 1 6 2 carboncarbon
carbonheteroatom
heteroatomheteroatom
diatomic molecules
hypervalent molecules 15
20
37
34
25 1
2
3
3
5 9
8
16
10
16 0
2
4
4
6 17 from expt. from ccpVTZ Practical HartreeFock and B3LYP Models for Equilibrium Geometry
Calculations
Except for very small molecules, HartreeFock and B3LYP calculations using
the ccpVTZ and ccpVQZ basis sets will not be practical for determining
equilibrium geometries. Large basis set calculations are of value to judge the
limits and ultimately the quality of the underlying models, but smaller basis
sets are needed for routine applications to larger molecules, at least at present.
Here, we ask if equilibrium geometries obtained from HartreeFock and
B3LYP calculations using the smaller 631G* and 6311+G** basis sets are
able to reproduce experimental geometries to an “acceptable” level of
accuracy. Our criterion for bond lengths is ±0.02Å, somewhat less than the
best experimental data (0.005Ǻ for microwave spectroscopy) and comparable
if not somewhat better than that from commonlyavailable data (0.020.04Ǻ
from Xray diffraction).
For a firstrow element, the 6311+G** basis set comprises a core made up of 6 stype
Gaussians, and a valence split into three parts made up of 3,1,1 s–type Gaussians and 3,1,1
sets of ptype Gaussians, s and p Gaussians. A single a set of dtype Gaussians and set of
diffuse Gaussians is added. Hydrogens are represented by a two stype Gaussians and a set
of ptype Gaussians. The smaller 631G* basis set also comprises a core made up of 6 stype Gaussians, but the valence is split into only two parts made up of 3 and 1 s–type
Gaussians and 3 and 1 sets of ptype Gaussians, supplemented by a set of dtype Gaussians.
Hydrogens are represented by two stype Gaussians. Signed deviations from experimental AH bond lengths (Ǻ x 1000) for oneheavyatom hydrides from HartreeFock and B3LYP methods with 631G*
and 6311+G** basis sets are provided in Table P24.
An Excel spreadsheet containing AH bond distances for both HartreeFock and B3LYP
models with 631G*, 6311+G**, ccpVTZ and ccpVQZ basis sets is provided on the CDROM accompanying this text (AH bond distances). In terms of mean absolute error, all four models actually meet the criterion,
and only for LiH and NaH are bond lengths in error by more than 0.02Ǻ.
Bond lengths from HF/6311+G** calculations are better than those from
HF/631G* calculations and those from B3LYP/6311+G** calculations are
better than those from B3LYP/631G* calculations. As expected, HartreeFock models yield bond lengths that are consistently shorter than experimental
values. On the other hand, bond lengths from B3LYP calculations are
typically larger than experimental values. An unexpected (and probably
fortuitous) result is that the B3LYP bond lengths appear to be (slightly) less
sensitive to basis set than HartreeFock lengths. 18 Table P24: Signed Deviations of HartreeFock and B3LYP AH Bond
Lengths in Hydrogen and in OneHeavyAtom Hydrides from
Experimental Bond Lengths (Å x 1000 )
molecule HartreeFock
631G*
6311+G** B3LYP
631G*
6311+G** expt. H2 12 7 1 2 0.742 LiH
CH4
NH3
H2O
HF 42
8
9
11
6 11
8
12
17
20 25
1
7
11
17 4
1
3
4
5 1.596
1.092
1.012
0.958
0.917 NaH
SiH4
PH3
H2S
HCl 30
6
17
9
9 31
4
13
5
6 3
5
4
13
15 3
3
3
12
13 1.887
1.481
1.420
1.336
1.275 mean absolute error 14 12 9 5 – 19 Bond lengths in hydrocarbons offer more subtle criteria with which to judge
the different practical models. While triple bonds are not sensitive to structure,
the carboncarbon double bonds provided in Table P25 below show a range
of 0.06Ǻ (from the central bond in butatriene to the bond in 1,3butadiene)
and single bonds a range of more than 0.13Ǻ (from the bond in but1yne3ene to the C3C4 bond in cyclobutene).
In terms of mean absolute error, all four models are satisfactory with both
B3LYP models being superior to both HartreeFock models. With a few
notable exceptions, trends in carboncarbon bond distances are well
reproduced.
Similar results and similar interpretations follow from comparisons of other
classes of compounds. A summary of mean absolute deviations of bond
lengths obtained from the two HartreeFock models and the two B3LYP
models for diatomic molecules, molecules with bonds between carbon and a
heteroatom, molecules with bonds between two heteroatoms and hypervalent
molecules (in addition to oneheavyatom hydrides and hydrocarbons) is
provided in Table P26.
Excel spreadsheets containing bond distances Excel spreadsheets containing bond
distances for both HartreeFock and B3LYP models with 631G*, 6311+G**, ccpVTZ
and ccpVQZ basis sets for diatomic molecules, molecules with carbonheteroatom bonds,
molecules with heteroatomheteroatom bonds and in hypervalent molecules are provided
on the CDROM accompanying this text (bond distances in diatomic molecules, carbonheteroatom bond distances, heteroatomheteroatom bond distances, and bond distances
in hypervalent molecules, respectively). 20 Table P25: Deviations from Experiment of HartreeFock and B3LYP
Bond Lengths in Hydrocarbons (Å)
HartreeFock molecule 631G* 6311+G** B3LYP 631G* 6311+G** expt. triple bonds
acetylene
propyne
but1yne3ene 18
19
19 20
21
21 2
1
3 4
4
2 1.203
1.206
1.208 double bonds
butatriene C2C3
allene
butatriene C1C2
methylenecyclopropane
propene
ethylene
but1yne3ene
cyclobutene
1,3butadiene 18
12
15
24
18
22
19
20
22 21
13
16
23
16
21
19
18
21 12
1
0
10
3
8
0
1
4 17
5
3
12
5
10
3
3
6 1.283
1.308
1.318
1.332
1.336
1.339
1.341
1.342
1.345 aromatic bonds
benzene 11 11 0 2 1.397 single bonds
but1yne3ene
methylenecyclopropane C2C3
propyne
1,3butadiene
bicyclo[1.1.0]butane C1C3
bicyclo[1.1.0]butane C1C2
propene
cyclopropane
cyclobutene C2C3
propane
ethane
methylenecyclopropane C3C4
cyclobutane
cyclobutene C3C4
mean absolute error 8
5
9
1
30
10
2
12
2
2
4
15
3
3
13 7
6
7
0
25
6
2
10
0
2
4
12
2
3
12 7
13
1
10
4
2
1
1
2
6
0
3
6
6
4 9
10 1.431
1.457
1.459
1.467
1.497
1.498
1.501
1.510
1.517
1.526
1.531
1.542
1.548
1.566
– 21 11
4
3
1
1
2
6
0
0
6
7
5 Table P26: Mean Absolute Deviations from Experiment of HartreeFock and
B3LYP Bond Lengths (Å)
HartreeFock class of compounds
oneheavyatom hydrides
carboncarbon
carbonheteroatom
diatomic molecules
hypervalent molecules 631G* B3LYP 6311+G** 14
13 12
12 22 631G* 6311+G** 9
4 5
5 Can Single Bonds be Shorter than Double Bonds? Suggest a hydrocarbon that
incorporates a single bond as short as possible, and another (or the same) hydrocarbon that
incorporates a double bond as long as possible. Justify your selection. Obtain the
equilibrium geometry for your molecule using the B3LYP/6311+G** model. Are the bond
lengths outside the range of those presented in Table P22? If they are not, refine your
choice of molecules. Have you managed to uncover a single bond that is shorter than a
double bond?
Water Dimer: The water dimer exhibits a structure with a single hydrogen bond and an
OO separation of 2.98Ǻ.
H
O H O H
H Both HartreeFock and B3LYP models with the ccpVQZ basis set show similar overall
geometries, but with different OO distances of 3.03Ǻ and 2.91Ǻ, respectively. Do these
accurately represent the limits of the HartreeFock and B3LYP methods? To tell, obtain
equilibrium geometries for water dimer using the HF/6311+G** and B3LYP/6311+G**
models. 23 Using Calculated Equilibrium Geometries
Combination of HartreeFock and B3LYP Hamiltonians with the 631G* and
6311+G** basis sets leads to four of the simplest available models for the
calculation of equilibrium geometries for small to medium size molecules (up
to 50 heavy atoms). They complement the available experimental techniques,
in the sense of being able to handle molecules larger than possible (or at least
practical) with microwave spectroscopy, and produce results that are more
accurate (or at least as accurate) as those from Xray crystallography. Of
course, many interesting molecules cannot easily be synthesized (or
synthesized at all), and here calculations may be the only choice. At the
extreme, a molecule needs to exist only in the mind of the chemist.
HartreeFock and B3LYP density functional models are unbiased in that they
have not been parameterized to reproduce experimental data, and it is not
unreasonable to expect that they will perform as well for molecules for which
experimental data are unavailable as they will for molecules for which data
exist. Cations, anions, radicals, hydrogenbonded complexes among other
shortlived species may be investigated as easily as “normal” molecules. By
the same token, lack of experimental data complicates assessment of the
models. The structures of ions in particular may not be obtained microwave
spectroscopy, and experimental data will generally be limited to solid phase
structures. Here, different counterions or different ioncounterion
arrangements may lead to different ion geometries.
It could be argued that the some among the present generation of density functional models
are biased in the sense that they have been extensively parameterized to match
experimental data. Discussion is provided in Chapter X. Whether they are used in conjunction with experimental methods or
completely on their own, these four theoretical models offer chemists a
convenient way to probe the geometries of molecules, real or imagined. In
doing so, they provide a means to assess the validity of existing qualitative
models for chemical bonding, for example, the VSEPR model, as well as to
formulate and test new models. Because a molecule need not exist in order for
its structure to be examined, the range of calculations exceeds that of
experimental chemistry. 24 Microwave Spectra of Ions: Why can’t microwave transitions of charged molecules be
observed?
CH Bonds and Hybridization at Carbon: Obtain equilibrium geometries for ethane,
ethylene and acetylene using the HF/631G* model. Do CH bond lengths change with the
hybridization at carbon? If they do, is the magnitude of the changes (in terms of a
percentage) similar to that for CC bonds (see Table P25)?
Diborane: Is it better to depict diborane with or without a boronboron bond?
H
B H B H H H
B B
H H
H To decide, examine a density surface for diborane corresponding to 50% enclosure of the
total number of electrons. Is the surface in the middle of the “bond” convex (suggesting
buildup of electron density) or concave (suggesting depletion of density)? For comparison,
examine the corresponding surface for ethylene. Use the HF/631G* model to first obtain
equilibrium geometries for the two molecules.
Diborane was originally thought to look like ethane. Obtain the geometry of ethanelike
borane (D3d symmetry) and calculate vibrational frequencies. Is this structure an energy
minimum? If not, provide a rationalization. Hint: Examine the highestoccupied orbital(s)
of ethane, and ask what would happen were two electrons to be removed.
Structure of Sulfur Tetrafluoride: VSEPR (Valence State Electron Pair Repulsion)
theory uses two simple rules is able to assign geometry. The first is that the geometry about
an atom is determined by insisting that electron pairs (either lone pairs or bonds) avoid
each other as much as possible. An atom surrounded by two electron pairs assumes a linear
geometry, by three pairs a trigonalplanar geometry, four pairs a tetrahedral geometry, five
pairs a trigonalbipyramidal geometry and six pairs an octahedral geometry. The second
rule is that it is more important to avoid unfavorable lone pairlone pair interactions than it
is to avoid lone pairbond interactions which are in turn more important to avoid than bondbond interactions.
According to the first rule, the the sulfur in sulfur tetrafluoride with five electron pairs (four
bonds and a lone pair) assumes a trigonal bipyramidal geometry. According to the second
rule, the lone pair will prefer to occupy an equatorial rather than axial position. This means
that SF4 adopts a seesaw geometry in which the lone pair is 90° to two of the SF bonds and
120° to the other two bonds, rather than a trigonal pyramidal geometry in which all three
bonds are 90° to the lone pair.
F
F
F F
F
F S•
• S
•• F F
"see saw" trigonal pyramid 25 Use the HF/631G* model, to obtain geometries for both seesaw (C2v symmetry). and
trigonal pyramid (C3v symmetry) forms of SF4. Is the seesaw structure lower in energy
than the trigonalpyramid structure in accord with VSEPR theory? Are both structures
energy minima? Elaborate. If they are, is the energy difference between them small enough
that both would be seen at room temperature?
CaF2. A Failure of VSEPR Theory? According to VSEPR theory, CaF2 should be a linear
molecule. However, in the solid phase the molecule is bent. Is this a failure of the VSEPR
model or is it merely a consequence of crystal packing? Staring with a bent structure, obtain
the equilibrium geometry of CaF2 using the B3LYP/631G* model. Is the molecule bent? If
it is not, then calculate an equilibrium geometry for a molecule that is constrained to having
a FCaF bond angle of 140o and compare its energy to that of "linear" CaF2. What does the
energy difference tell you about the magnitude of the crystal packing energy? If on the
other hand, "free" CaF2 is bent, then calculate the equilibrium geometry of "linear" CaF2 and
compare its energy to that of the bent molecule.
Geometry Change s with Change in the Number of Electrons: A molecule's geometry
depends not only on the constituent atoms, but also on the total number of electrons. Use
the HF/631G* model to obtain equilibrium geometry for 2methyl2 propyl cation (tertbutyl cation), as well as those for the corresponding radical (with one additional electron)
and the anion (with two additional electrons). Describe any changes to the geometry of the
central carbon with increasing number of valence electrons, and speculate on the origin of
these changes.
Radical Cation of Diborane: Diborane incorporates what can only be described as a π
bonding orbital analogous to the familiar π bond in ethylene. Removal of an electron from this orbital should result in elongation of the boronboron
bond, just as removal of an electron from the π orbital in ethylene results in elongation of
the CC bond. However, unlike ethylene, the π orbital in diborane is not the HOMO but
rather an orbital of lower energy. This suggests that the radical cation of diborane (formed
from ionization of diborane) might be quite different than the radical cation of ethylene.
Use the B3LYP/631G* model to obtain the equilibrium geometry of diborane and display
the HOMO. Predict what should happen to the geometry of diborane were an electron to be
26 removed from this orbital. Test your prediction by calculating the equilibrium geometry of
radical cation or diborane. Make certain that you start with a distorted (C1 summetry)
geometry. Compare BB bond lengths for diborane and its radical cation.
Protonated Alkenes and Alkanes: Protonation of a molecule with a localized electron
pair leads to a new σ bond. Its geometry is easy to anticipate by analogy with neutral
molecules containing the same number of electrons. For example, protonated ammonia and
protonated trimethylamine are expected to incorporate a tetrahedral (nitrogen) center, just
as their neutral isoelectronic counterparts, methane and isobutane incorporate tetrahedral
(carbon) centers.
The geometry of a protonated alkene is less clear. For example, is the proton in protonated
ethylene primarily associated with a single carbon does it “bridge” both carbons?
H
H
H C +
C H
H H+
HC
H C H
H Calculate equilibrium geometries for both open and bridged forms of protonated ethylene
(ethyl cation) using the B3LYP/631G* model. Do both structures appear to be minima on
the C2H5+ potential surface or does one of the structures “collapse” to the other? Elaborate.
If there is only one energy minimum, is it open or bridged?
Even alkanes protonate in the gas phase, even though they incorporate no obvious electronrich sites. Use the B3LYP/631G* model to explore possible structures for protonated
methane (CH5+). Calculate vibrational frequencies for whatever you uncover to verify it is
actually and energy minimum. Describe the bonding in terms of a weak complex or a
molecule with a pentavalent carbon.
Repeat you calculations and analysis for protonated ethane (C2H7+).
2Norbornyl Cation: 2Norbornyl cation ranks among the most studied and controversial
molecules in 20th century chemistry. Literally hundreds of papers found their way into the
organic chemical literature, and prompted a lively and sometimes vitriolic debate between
two future Nobel laureates on what became known as the “nonclassical ion problem”. The
observation that led to the debate was that C2 and C6 positions in norbornane substituted in
the 2 position by a good (anionic) leaving group scrambled. This could be accommodated either by invoking a rapid equilibration between two
“classical” cations in which the “positively charged” carbon is tricoordinate, or insisting
that there was only a single “nonclassical” cation incorporating a pentacoordinate carbon. 27 The issue was finally settled by a series of beautiful experiments by George Olah, the most
important being the lowtemperature proton and 13C NMR spectra of the ion. The latter is
shown below. What is the structure of 2norbornyl cation? Use the B3LYP/631G* model to calculate its
geometry, infrared and 13C NMR spectra. Start from a “classical” structure. How do you
know that what you have found is an energy minimum? Does the calculated 13C spectrum
fit what is observed? If so, are the experimental spectral assignments in line with the
calculations? Elaborate.
Dicyclopentadienyl Beryllium: Ferrocene exhibits a beautiful structure in which iron is
sandwiched between two cyclopentadienyl rings. The usual way of writing this is to give
the iron a formal +2 charge and each cyclopentadienyl ring unit negative charge. Why
would such an arrangement be expected to be especially stable?
1 Fe+2
1 It may come as a surprise, then that dicyclopentadienyl beryllium does not look like
ferrocene at all. Rather, it adopts a halfsandwhich structure, with one cyclopentadienyl
ring above the metal (as in ferrocene) but with the other σ bonded to beryllium. Be Be 28 Use the B3LYP/631G* model to obtain geometries and vibrational frequencies for the
sandwich and halfsandwich structures of dicyclopentadienyl beryllium. Is the half
sandwich preferred? Are both energy minima? Elaborate. Provide an explanation for the
change in geometry.
CarbonFluorine Bond Lengths in Fluorosilanes and Fluorogermanes: As discussed
earlier in this chapter, carbonfluorine bond lengths in fluoromethanes, CFnH4n (n=14),
decrease dramatically with increasing number of fluorines, from 1.xxǺ in fluoromethane to
1.xxǺ in tetrafluoromethane. Is the same trend found in the corresponding fluorosilanes,
SiFnH4n (n=14)? Perform B3LYP/631G* calculations to tell. If so, is the percentage bond
length change smaller, larger or of comparable magnitude?
Repeat your calculations and analysis for the fluorogermanes, GeFnH4n (n=14).
Bond Angles in Amines and Ethers: Bond angles about nitrogen in amines and about
oxygen in ethers are typically close to tetrahedral, For example, measured bond angles in
trimethylamine and dimethylether are xxxo and yyyo, repectively. Replacement of methyl
by something bulkier, for example, a tertbutyl group, might be expected to lead to an
increase in nonbonded (steric) repulsion and result in an increase in bond angle. Use the
B3LYP/631G* model to obtain equilibrium geometries for trimethylamine and tri(tertbutyl)amine using the B3LYP/631G* model. Do you observe the expected increase in
CNC bond angle? Are there any other conspicuous structural changes between the two
molecules, in particular, have the CN bonds lengthened? Repeat your calculations for
dimethyl ether and di(tertbutyl) ether. Is there an increase in COC bond angle?
Replacement of methyl by a group capable of drawing electrons away from the lone pair on
nitrogen (two lone pairs on oxygen), for example, a silyl group, might also be expected to
result in an increase in bond angle. Why? Obtain the equilibrium geometry for
trisilylamine. Is the SiNSi bond angle larger than the CNC bond angle in trimethylamine?
Is it as large as the CNC bond angle in tri(tertbutyl)amine? Repeat your calculations and
answer analogous questions for disilyl ether. 29 Lewis Structures and Equilibrium Geometries
At the start of this chapter, we implied that a Lewis structure together with a
table of “standard” bond lengths and angles is not likely to provide a good
enough account of molecular geometry to allow subtle differences in energies
or other properties to be uncovered. Even so, Lewis structures are an
important part of a chemists’ vocabulary, not only because the offer a very
concise way to depict molecular structure but also they provide clues about
“interesting” structures. Benzene provides a good example of the latter. Here,
there are two equivalent ways of placing three single and three double bonds
in a sixmembered ring. The fact that we cannot decide which placement is
“correct”, implies that neither is “correct”. Rather, the proper description of
the geometry of benzene follows from using both. This in turn suggests that
the six carboncarbon bonds in benzene are all identical and midway in length
between normal single and double bonds. This is of course exactly what is
observed. Lewis Structure for D iazomethane: Diazomethane is usually described as a composite
of two Lewis structures, both of which involve separated charges. + N – – N + N N Obtain the geometry of diazomethane using the HF/631G* model. Also obtain the
geometries of methylamine, CH3 NH2, and methyleneimine, H2C=NH, as examples of
molecules incorporating normal CN single and double bonds, respectively, and of trans
diimide, HN=NH, and nitrogen, N≡N, as examples of molecules incorporating normal NN
double and triple bonds, respectively. Which Lewis structure provides the better
description for diazomethane or are both required for adequate representation? Examine
electrostatic charges for diazomethane. Do they suggest the same Lewis structure as the
bond distances?
S tructure of Ozone: Suggest two different Lewis structures for ozone, O3. (One or both
may require nonzero formal charges.) Obtain the equilibrium geometries corresponding to 30 both s tructures using the B3LYP/631G* model. Which structure is lower in energy? Is it in
accord with the experimentally known equilibrium geometry? Calculate the infrared
s pectrum of the higherenergy structure to establish whether or not it is an energy
minimum? Explain you reasoning. If the preferred structure has more than one distinct
oxygen atom, which is most positively charged? Most negatively charged? Is your result
based on electrostatic charges consistent with that based on formal charges?
Which Lewis Structure? Anthracene and Phenanthrene: Whereas the two Lewis
structures for benzene are equivalent and thus need to be weighted equally, the situation is
less clear where the Lewis structures are different. For example, two of the three Lewis
structures that can be written for naphthalene are the same but the third is different. In this case, any conclusions regarding molecular geometry depend on the relative weight
given to each structure. Assigning equal weights to all three structures suggests that four of
the bonds in naphthalene (that are double bonds in two of the three Lewis structures)
should be shorter than the remaining seven bonds, (that are double bonds in only one of the
three Lewis structures). This is in fact what is observed experimentally. 1.42Å 1.43Å
1.42Å
1.38Å Draw the complete set of Lewis structures for anthracene and phenanthrene. anthracene phenanthrene Assuming that each Lewis structure contributes equally, assign which if any of the carboncarbon bonds should be especially short and which if any should be especially long. Next,
obtain equilibrium geometries for the two molecules using the HF/631G* model. Are your
assignments consistent with the results of the calculations? If not, suggest which Lewis
structures need to be weighed more heavily (or which need to be weighed less heavily) in
order to bring the two sets of data into accord. 31 Molecules with Transition Metals
Equilibrium geometries for ~200,000 compounds incorporating transition
metals are known experimentally, almost entirely from Xray crystallography.
While this is similar to the number of structures for maingroup compounds
that are available, relatively little attention has been given to the calculation of
geometries for transitionmetal inorganic and organometallic compounds. In
part, this is no doubt due to the wellknown failure of HartreeFock models to
provide satisfactory geometries for molecules with transition metals.
However, as shown Table P27 for a small selection of organometallic
carbonyl compounds, the B3LYP/631G* model provides a reasonable
successful account. In particular, metalcarbon bond lengths are typically
reproduced to within 0.02Ǻ (the experimental error).
Transition metal inorganic and organometallic chemistry represents an
attractive target for quantum chemistry. 32 Table P27: Comparison of MetalCarbon Bond Distances in Organoiron
Compounds from B3LYP/631G* and Experiment (Å)
organoiron compounds bond B3LYP/631G* ferrocene expt. 2.05 2.06 butadiene iron tricarbonyl butadiene C1
butadiene C2
CO 2.10
2.07
1.78 2.14
2.06
1.76 cyclobutadiene iron tricarbonyl cyclobutadiene
CO 2.04
1.78 2.06
1.79 ethylene iron tetracarbonyl ethylene
COaxial
COequatorial 2.12
1.81
1.79 2.12
1.78
1.81 acetylene iron tetracarbonyl acetylene
COaxial
COequatorial 2.08
1.82
1.79 2.08
1.77
1.76 33 Carbon Monoxide as a Ligand: Carbon monoxide is perhaps the most common ligand in
transitionmetal organometallic compounds. CO molecule acts is to donate an electron pair
into an empty orbital on the metal atom. In return, electrons are donated from an occupied
orbital on the metal atom into lowlying empty orbitals on the CO molecule.
Obtain the equilibrium geometry for carbon monoxide using the B3LYP/631G* model.
Display the HOMO. Is it bonding, antibonding or essentially nonbonding between carbon
and oxygen? What, if anything, would you expect to happen to the CO bond strength as
electrons are donated from the HOMO to the metal atom?
Display the LUMO. (There are actually two equivalent LUMOs, designated LUMO and
LUMO+1, and you can base arguments on either one.) Is the LUMO bonding, antibonding
or essentially nonbonding between carbon and oxygen? What if anything would you
expect to happen to the CO bond strength if electrons were donated from the metal atom
into this orbital? Elaborate. Will this affect the change that results from electrons being
donated from the HOMO of the CO molecule to the metal atom? Elaborate.
Examine the molecular orbitals of Fe(CO)4, arising from loss of CO from Fe(CO)5 to see if
the metal center possesses a high energy filled molecular orbital of proper symmetry to
donate electrons into the LUMO of carbon monoxide. Obtain the equilibrium geometry for
Fe(CO)5 using the B3LYP/631G* model, then delete one of the equatorial CO ligands to
make a Fe(CO)4 fragment. Calculate the energy of the fragment (don’t optimize the
geometry) . Display the HOMO. Does the HOMO in Fe(CO)4 have significant amplitude in
the location where the carbon monoxide ligand will attach? If so, does it have the proper
symmetry to interact with the LUMO in CO? Would you expect electron donation to
occur? Elaborate.
Binding Ethylene to a Metal: Two limiting structures can be drawn to represent ethylene
bonded to a transition metal. The first may be thought of as a weak complex in that it
maintains the carboncarbon double bond, while the second destroys the double bond in
order to form two new metalcarbon σ bonds, leading to a threemembered ring (a
metallacycle). Most likely, real metalalkene complexes will exhibit bonding intermediate
between these two extremes.
M M Optimize the geometry of ethylene using the B3LYP/631G* model and examine both the
HOMO and LUMO. Is the HOMO bonding, antibonding or nonbonding between the two
carbons? What if anything should happen to the carboncarbon bond as electrons are
donated from the HOMO to the metal? Do you expect the carboncarbon bond length to
decrease, increase or remain about the same? Elaborate.
The LUMO is where the next (pair of) electrons will go. Is this orbital bonding,
antibonding or nonbonding between the two carbons? What, if anything, should happen to
the carboncarbon bond as electrons are donated (from the metal) into the LUMO? Is the
expected change in the carboncarbon bond due to this interaction in the same direction or
in the opposite direction as any change due to interaction of the HOMO with the metal?
Elaborate.
34 Optimize the geometry of ethylene iron tetracarbonyl using the B3LYP/631G* model.
Compare the carboncarbon bond to that in ethylene? Based on geometry, how would you
describe the bonding between the ethylene and the metal.
CO
CO
Fe
CO
CO Next, delete the ethylene ligand and calculate the energy calculation on the resulting (iron
tetracarbonyl) fragment (don’t optimize the geometry). Examine both the HOMO and
LUMO of this fragment. Is the LUMO of the iron tetracarbonyl fragment properly situated
to interact with the HOMO of ethylene? Elaborate. Would you expect electron donation
from ethylene to the metal to occur? Does the HOMO of the fragment have the proper
symmetry to interact with the LUMO of ethylene? Elaborate. Would you expect electron
donation from the metal to ethylene to occur?
Is Chromium Tricarbonyl a π Donor or π Acceptor? Chromium tricarbonyl complexes
to one of the faces of benzene leaving the other face exposed for further reaction. Cr
OC CO
CO Is there a significant change in the geometry of benzene as a result of complexing to
chromium tricarbonyl? In particular, is there evidence of bond localization? To decide, use
the B3LYP/631G* model to calculate equilibrium geometries for both benzene and
benzene chromium tricarbonyl. Does the Cr(CO)3 group act to donate electrons leading to
enhanced affinity toward electrophiles or to accept electrons leading to diminished
reactivity? Compare electrostatic potential maps for “free” and complexed benzene. To
establish a point of reference, include electrostatic potential maps (based on equilibrium
geometries from the B3LYP/631G* model for (“free”) aniline (an electronrich arene) and
nitrobenzene (an electronpoor arene). You need to make certain that all four maps are on
the same scale. Do you see the expected trend in electrostatic potential in the maps for
benzene, aniline and nitrobenzene? Elaborate. Where does benzene chromium tricarbonyl
fit in? Classify the Cr(CO)3 group as an electron donor or acceptor. Rank the chromium
tricarbonyl group relative to either the amino group or the nitro group as appropriate.
Benzene Chromium Tricarbonyl vs. Borazine Chromium Tricarbonyl: Borazine,
B3N3H6, is often considered to be a close analogue to benzene in that it contains six π
electrons in a planar sixmembered ring. However, all the electrons (formally) come from
the three nitrogens, suggesting instead that it may not be as “delocalized” as benzene. Like
benzene, borazine forms complexes with chromium tricarbonyl.
B
N B
B OC Cr CO
CO N N
Cr CO
OC
CO 35 Use the B3LYP/631G* model to calculate the geometry of borazine and borazine
chromium tricarbonyl. Does borazine undergo a significant change in geometry as a result
of binding to the metal carbonyl? How does this compare to the change in the geometry of
benzene resulting from its complexation to chromium tricarbonyl (see previous problem)?
Is there evidence of bond localization? If it does, is this bond polarized toward titanium or
toward carbon?
TitaniumCarbon Double Bonds: While double bonds to heavy maingroup carbon
analogues are quite rare, double bonds between carbon and the transition metals, titanium,
zirconium and hafnium are common. Use the B3LYP/631G* model to obtain the
equilibrium geometry of biscyclopentadienyl titanium methylidene, along with that of
tetramethyltitanium as a reference.
Me
Ti
Me Cp Me Ti Me CH2 Cp Is the shrinkage in TiC bond length from single to double comparable to that typically
found for hydrocarbons, either in absolute terms or on a percentage basis? Elaborate.
Display the HOMO of biscyclopentadienyl titanium methylidene. Does it incorporate a π
bond?
bisCyclopentadienyl Titanium Ethylidene: Use the B3LYP/631G* model to obtain the
equilibrium geometry of biscyclopentadienyl titanium ethylidene.
H Cp
Ti C
Me Cp Is the titaniumcarbon bond length shorter, longer or about the same as the bond in biscyclopentadienyl titanium methylidene? If it is significantly different, provide a rationale as
to why. (Hint: examine the TiCMe and TiCH bond angles.)
Transition Metal Carbyne Complexes: While molecules that incorporate a metalcarbon
double bond (“carbenes”) are commonplace, molecules with a metalcarbon triple bond
(“carbynes”) are less frequently encountered. Use the B3LYP/631G* model to obtain the
equilibrium geometry for the chromium carbyne shown below. For comparison, also obtain
the geometry for propyne.
OC
Br
OC CO
Co C CH3 CO Can you identify three metalcarbon binding molecular orbitals? If yes, are these orbitals
qualitatively similar to those in propyne? Elaborate. Is the carboncarbon single bond in the
metal carbyne short, that is, does it reflect sp hybridization at carbon? Compare it to the
analogous CH bond length in propyne. 36 Hybridization and Bond Lengths to Transition Metals: Single bonds to sp2 hybridized
carbon centers are shorter than those to sp3 centers but not as short as those to sp centers.
The differences can be attributed to the fact that valence 2p orbitals extend further from
their atomic centers than the analogous 2s orbital. The same principle should extend to
(transition) metalligand bonding. Here, the hybrids comprise primarily (n) d and (n+1) stype orbitals (The corresponding (n+1) p orbitals are usually assumed to play a lesser role.)
Use the B3LYP/631G* model to obtain equilibrium geometries for the titanium methyl
(X2YTiCH3, methylidene, XYTi=CH2, and methylidyne, XTi≡CH complexes, where
X=Y=H.
X
Y X
Ti CH3 Y T i CH 2 Y Ti CH Y Compare TiH bond lengths for the three compounds. Is there significant variation? If
there is, rationalize this behavior on the basis of the nature (hybridization) of the orbitals on
the metal.
The three compounds you have examined are extremely electron deficient, far short of the
1618 electrons normally needed to satisfy titanium. Any effects that this might have on
their structures can be tempered by substituting one of the hydrogen “ligands” by a Cp
(cyclopentadienyl) ligand. Carry out calculations on the methyl and methylidene systemw
with X=Cp, Y=H. Is the same trend present in these compounds? If it is, is the magnitude
of the change about the same or is it increased or diminished. Rationalize your result.
Repeat both sets of calculations replacing H with a methyl and then with chlorine. Do the
trends that you observed for H maintain? 37 HydrogenBonded and Related Complexes
In the limit of weak metalligand bonding, transitionmetal inorganic and
organometallic compounds may be viewed as intermolecular complexes. That
is to say that the components, the metal fragment and one or more ligands,
maintain their essential identity. There are many other classes of molecules of
this type, most familiar and arguably most important among them being
hydrogenbonded complexes. Here, one molecule incorporates an atom with a
nonbonded electron pair. It is referred to as the hydrogenbond acceptor and
it is the electron donor. The other molecule, referred as the hydrogenbond
donor, incorporates a bond between hydrogen and an electronegative atom.
Because of the difference in electronegativities, the σ molecular orbital
describing this bond is primarily localized on the electronegative atom, while
the corresponding σ* molecular orbital is primarily localized on hydrogen.
Therefore, the hydrogen is electron deficient and should act as an electron
acceptor. Interaction of the filled molecular orbital on the hydrogenbond
acceptor (electron donor) and the empty molecular orbital on the hydrogenbond donor (electron acceptor) is the driving force behind the formation of a
stable complex. Most commonly, hydrogenbond acceptors are nitrogen or
oxygen atoms and hydrogenbond donors involve bonds to nitrogen or
oxygen. Halogens and sulfur may also act as (weak) hydrogenbond acceptors
and SH and PH bond as (weak) hydrogenbond donors.
Consider water dimer. The common terminology refers to the molecule on the
left as a hydrogenbond acceptor and the molecule on the right as a hydrogenbond donor. Alternatively, one might describe the molecule on the left as an
electronpair donor and the molecule on the right as an electronpair acceptor. Note that the water molecule is able to act as both a hydrogenbond acceptor
and hydrogenbond donor. In fact, the same molecule may serve
simultaneously in both roles in hydrogenbonded complexes. Liquid water
offers a good example, with each water molecule contributing up to two
electron pairs (hydrogenbond acceptors) and two OH bonds (hydrogenbond
donors). This leads to a three dimensional network of hydrogen bonds and
gives liquid water its unique properties. A small water cluster is depicted
below.
38 Table P28 compares hydrogen bond lengths for several complexes obtained
from HF/631G* and B3LYP/631G* models with experimental values.
An Excel spreadsheet containing hydrogenbond distances for these and other complexes
from both HartreeFock and B3LYP models with 631G*, 6311+G* and ccpVTZ basis
sets is provided on the CDROM accompanying this text (bond distances in hydrogenbonded complexes). 39 Table P28: Comparison of HartreeFock and B3LYP Connecting Molecules in
Intermolecular Complexes with Experimental Values (Å)
complex geometrical
parameter HF/631G* B3LYP/631G* expt. water dimer OO 2.97 2.98 hydrogen fluoride dimer FF 2.71 2.79 acetic acid dimer OO 2.79 2.66 2.68 adeninethymine NO 3.08 2.94 2.97 NN 3.01 2.88 2.89 ON 2.92 2.81 2.94 NN 3.06 2.96 2.93 NO 3.00 2.92 2.80 guaninecytosine mean absolute error 40 Alternative Structure of the Water Dimer: Water molecule incorporates two electron
pairs that may act as hydrogen bond acceptors and two OH bonds that may act as
hydrogenbond donors. This means that it should be possible to construct a dimer with two
hydrogen bonds.
H
H
O
HO
H Attempt to obtain an equilibrium geometry for the doubly hydrogenbonded structure of
water dimer. Use the B3LYP/631G* model. If you find such a structure, confirm that it is
or is not an energy minimum. If it is an energy minimum, calculate the roomtemperature
Boltzmann distribution of the two different forms of water dimer.
Hydronuim cation: Use the B3LYP/631G* model to calculate the equilibrium geometry
of hydronium cation, H3O+. Guess a structure for the complex between H3O+ and both four
water molecules, and obtain its geometry with the B3LYP/631G* model. Point out any
significant changes that have occurred to the cation a result. In particular, is there evidence
for “sharing” the proton? Is the positive charge primarily localized on the hydronium cation
or has it spread out to the surrounding water molecules?
Chloride Anion: Use the B3LYP/631G* model to calculate the equilibrium geometry for
chloride anion surrounded by four water molecules. Is the negative charge primarily
localized on chlorine or has it spread out to the water molecules?
Hydrogen Positions in Acetic Acid Dimer: Acetic acids forms a symmetrical hydrogenbonded dimer.
HO O
H3C C C
O H CH3 O Calculate equilibrium structures for acetic acid and its dimer using the HF/631G* model.
Point out any significant changes in bond lengths and angles. Have the hydrogens involved
in the hydrogen bonds moved to positions halfway between the oxygens or have they
remained with one oxygen (as in acetic acid)? Do the structural changes (or lack of
structural changes) suggest that hydrogen bonds are comparable to normal (covalent) bonds
or are they weaker? Elaborate.
Repeat your calculations for trifluoroacetic acid and its dimer. Does the fact that it is a
much stronger acid that acetic acid translate into larger structural changes upon
dimeization? Elaborate.
Borane Carbonyl: Borane carbonyl, BH3CO, may be viewed to result from interaction of
the nonbonded electron pair (on carbon) in carbon monoxide and an empty ptype
molecular orbital (on boron) in borane.
H
H
H B C O 41 Borane carbonyl is different from transitionmetal carbonyls in that there is little possibility
for significant back bonding, that is, interaction of a highlying filled molecular orbital on
borane with an empty π* molecular orbital on carbon monoxide. This suggests that the CO
bond in carbon monoxide should not change significantly as a result of complexion.
Use the B3LYP/631G* model to obtain equilibrium geometries for both carbon monoxide
and borane carbonyl. Is there a significant change in CO bond length? If there is, how does
it compare with bond length changes previously for the equatorial and axial CO groups in
iron pentacarbonyl (see earlier problem in the chapter)?
Repeat your calculations and analysis for trifluoroborane carbonyl, BF3CO. Identify any
significant difference between the structure of this and borane carbonyl.
Intramolecular GalliumAmine Complexes: Maingroup elements below boron in the
periodic table generally perfer planar trigonal structures, for example, trimethylaluminum
and trimethyl gallium.
Me
Me Al Me Me
Me Ga Me However, molecules such as these (as well as analogous boron compounds) are known to
form donoracceptor complexes with amines. Here, the nitrogen acts as the electron donor
and the aluminum or gallium as the acceptor.
Me
Me Me
Al
Me
Me Me
Me Me N Ga
Me Me
Me N
Me Use the HF/631G* model to obtain equilibrium geometries for the two complexes shown
above. To what extent does electron transfer actually occur from the donor to the acceptor?
Are either or both of the complexes better represented a zwitterions? Elaborate. A good example of a molecule where the desire to be planar and the benefit of complex
formation come into conflict is the tricyclic galliumamine complex shown below. Is this
an “open” structure allowing the CCC bond angles to be nearly tetrahedral, or a “closed”
structure allowing donoracceptor interaction, or both?
Ga N Use the HartreeFock 631G* model to decide. First, obtain an equilibrium geometry
starting from an open structure, and then (if necessary) start from a closed structure. Do
you find one or two structures? If two structures, which is more stable? If one structure, is
it open or closed? 42 Vibrational Frequencies
In addition to the equilibrium geometry, another important quantity that may
be directly obtained from examination of a potential energy surface is the set
of vibrational frequencies. We have already used frequencies to verify that a
particular point on an energy surface in fact corresponds to an energy
minimum, and we will use them later to connect calculated energy to measure
enthapies and Gibbs energies (Chapter P3) and to verify transitionstate
geometries (Chapter P4). Here, we establish the connection between the
vibrational frequencies of a molecule and its infrared spectrum.
OneDimensional Systems
In one dimension, calculation of the vibrational frequency for a (diatomic)
molecule first requires that the energy be expanded in terms of a Taylor series.
E(x) = E(x0) + (dE(x)/dx) x + (d2E(x)/dx2) x2 + higherorder terms
The harmonic approximation assumes that only the second derivative term
needs to be considered. This term is commonly known as the force constant,
and indicates the relative ease or difficulty of moving away from the
equilibrium structure, that is, the curvature of the surface at the minimum.
E(x0) in the Taylor expansion is a constant, dE(x)/dx (the gradient) is zero at
the equilibrium geometry. (Note that the gradient will not be zero away from
the minimum, making this expression invalid.) Cubic and higherorder terms
(socalled anharmonic terms) are ignored.
The square root of the ratio of the secondderivative term and the reduced
mass (the product of the masses of the two atoms divided by their sum) is
proportional to the frequency.
vibrational frequency √ [(d2E(x)/dx2/)reduced mass] A small second derivative means that distortion away from the equilibrium
position is “easy” and leads to a low frequency, while a large second
derivative means that distortion is “difficult” and leads to a high frequency.
Signed errors (in cm1) for measured vibrational frequencies for diatomic
molecules obtained from “limiting” (ccpVQZ basis set) HartreeFock and 43 B3LYP models are provided in Table P29. Also tabulated are (unsigned)
deviations between calculations with the ccpVTZ and ccpVQZ basis sets.
Except for molecules with lithium and sodium, all HartreeFock vibrational
frequencies are significantly larger than experimental values. This parallels
the trend in calculated bond lengths and follows from our previous discussion
of what might be expected from excitation of electrons from filled to unfilled
molecular orbitals. Frequencies obtained from the B3LYP model are typically
much closer to experimental frequencies, although most are still larger.
Differences in vibrational frequencies from models with the ccvTQZ and
smaller ccpVTZ basis sets are generally much smaller than differences with
experimental frequencies for both HartreeFock and B3LYP models.
ManyDimensional Systems
Generalization to polyatomic molecules is straightforward. The energy is
expanded in the same way as before, the only difference being that a vector
quantity, x, replaces a scalar quantity, x.
E(x) = E(x0) + Σi(∂E(x)/∂xi)xi + ½Σij(∂2E(x)/∂xi∂xj)xixj + higherorder terms
The dimension of x is 3N for a molecule with N atoms. However, the number
of vibrational frequencies is 3N6 (3N5 for a linear molecule), the remaining
6 (5) dimensions corresponding to translation away from and rotation around
the center of mass.
Calculation of the vibrational frequencies involves three steps. In the first
step, the set of 3N x 3N second energy derivatives with respect to the
Cartesian coordinates is obtained. The energy second derivatives then need to
be mass weighted. Diagonal terms (∂2E(x)/∂xi2) are divided by the mass of the
atom associated with xi, and offdiagonal terms (∂2E(x)/∂xi∂xj) are divided by
the product of the square root of the masses of the atoms associated with xi
and xj. These expressions reduce to that already provided for the onedimensional case.
In the second step, the original (Cartesian) coordinates are replaced by a new
set of coordinates (socalled normal coordinates) such that the matrix of massweighted second energy derivatives is diagonal.
[∂2E( ζ) /∂ ζ i∂ ζ j]/(√Mi√Mj) = δij [∂2E( ζ) /∂ ζ i2]/Mi
44 Table P29: Signed Errors (Experiment – Calculated) in “Limiting” (ccpVQZ Basis Set) HartreeFock Vibrational Frequencies for Diatomic
Molecules and Unsigned Deviations Between Frequencies Obtained from
ccpVTZ and ccpVQZ Basis Sets (cm1)
HartreeFock molecule B3LYP to expt to ccpVTZ to expt to ccpVTZ expt. LiF 48 6 8 3 898 LiCl 0 3 3 3 641 CO 284 2 73 2 2143 N2 399 2 117 4 2331 O2 391 18 53 18 1580 F2 372 3 155 7 891 FCl 133 2 9 2 784 NaF 3 9 32 21 536 NaCl 8 1 5 2 363 Cl2 54 1 30 0 560 45 δij is the Kronecker delta function (1 if i=j; 0 otherwise).
In the third step, the six coordinates corresponding to the three translations
and three rotations are removed. This leaves 3N6 internal vibrational
motions.
Anharmonic Effects
Because the actual potential energy in the vicinity of the minimum has been
approximated by a quadratic function, calculated frequencies will almost
always be larger than measured values. This is reasonable because a quadratic
function goes to infinity with increase in distance rather than going
asymptotically to a constant (separated atoms). At least in principle, it is possible to estimate the effect of anharmonic
contributions to measured vibrational frequencies, that is, to extract harmonic
frequencies. This may be accomplished by comparing the actual spacing of
the energy levels associated with the ground and excited states of a particular
vibration (which go to zero with increasing level) with the constant spacing
associated with a quadratic (harmonic oscillator) potential. In practice, such an analysis can only be done for diatomic and for very
simple polyatomic molecules.
An Excel spreadsheet containing vibrational frequencies for hydrogen and oneheavyatom
hydrides calculated from “limiting” (ccpVQZ basis set) HartreeFock and B3LYP models,
both with directly measured frequencies and with “harmonic frequencies” extracted from
46 the experimental spectra is provided on the CDROM accompanying this text (vibrational
frequencies of oneheavyatom hydrides). All frequencies are larger than measured values,
and lead to a mean absolute errors of 225 cm1 (~6%) and 140 cm1 (~4%) for HartreeFock
and B3LYP models, respectively. Most frequencies are also larger than the experimental
harmonic values. Mean absolute errors are significantly reduced; 140 cm1 (~4%) for the
HartreeFock model and 38 cm1 (~1%) for the B3LYP model. It is also possible to account for anharmonic effects by removing the
restriction that cubic and higherorder terms are assumed to be zero. In
practice perturbation theory is used to approximate an expansion of the energy
through fourth order.
E(x)
=
E(x0)
+
Σi[∂E(x)/∂xixi]
+
½Σij[∂2E(x)/∂xi∂xj]xixj
3
4
1/6Σijk[∂ E(x)/∂xi∂xj∂xkxixjxk ] + 1/24Σijkl[∂ E(x)/∂xi∂xj∂xk∂xl]xixjxkxl + Calculating anharmonic corrections requires one to two orders of magnitude
more effort than calculating harmonic frequencies. Because of this, it has been
done only for very small molecules.
Harmonic Frequencies: Measured frequencies corrected for anharmonicity (harmonic
frequencies) are available for a variety of small molecules, including many diatomic
molecules. A sampling (in cm1) include: LiF, 914; CO, 2170; N2, 2360 and F2, 923. For
each, calculate the percentage of the total error of the “limiting” HartreeFock and B3LYP
frequencies given in Table P29 due to the harmonic approximation. 47 Practical HartreeFock and B3LYP Models for Vibrational Frequency
Calculations
Except for very small molecules, HartreeFock and B3LYP models with large
basis sets such as the ccpVTZ and ccpVQZ basis sets are not practical for
calculation of vibrational frequencies. This is the same situation previously
discussed for calculation of equilibrium geometries, but aggravated by the fact
that secondenergy derivatives are much more difficult and their calculation
requires much more computer time than calculation of first derivatives.
Therefore, the primary use of ccpVTZ and ccpVQZ (and even larger) basis
sets is to assess the limits of the underlying (HartreeFock and B3LYP)
models. Smaller basis sets are needed for practical applications. The
performance of two smaller Gaussian basis sets will be examined, 6311+G**
and 631G*. Both are easily applicable for molecules with molecular weights
up to 300 amu, and the second is applicable for molecules with molecular
weights up to 500 amu.
The first comparison involves the full set of (measured) frequencies for just
four molecules: ethane, methylamine, methanol and methyl fluoride (Table
P210). The performance of the four “practical” models described above is
examined as is that of the HartreeFock and B3LYP models with the ccpVTZ
basis set. Errors are quoted in term of a percentage rather than a numerical
value. According to this metric, the three HartreeFock models are
significantly (factor of twothree times) poorer than the corresponding B3LYP
models. This is not surprising in view of our previous discussion. Results from
both HartreeFock and B3LYP models with the 6311+G** basis set offer
significant improvement over the corresponding models with the 631G* basis
set, and are nearly identical to the “limiting” (ccpVTZ basis set) values.
Together these results suggest that the B3LYP/6311+G** model is a suitable
procedure for (absolute) frequency calculation.
A more subtle comparison involves comparisons frequencies associated with
similar motions in closelyrelated molecules. This implies that the vibration of
interest is easily identified and that its frequency is well separated from all
other frequencies. A good example involves frequencies identified with the
stretching of a carboncarbon double bond in molecules with only a single
such bond. Frequencies relative to the CC stretch in ethylene as a standard
from HartreeFock and B3LYP calculations with the 631G*, 6311+G** and 48 ccpvTZ basis sets are compared with experimental values in Table P211.
These span a range (of experimental frequencies) of ~300 cm1, from the CC
stretch in cyclobutene to that in tetrafluoroethylene.
The errors noted here are much smaller than those seen previously for
absolute frequency comparisons. With only a few exceptions, all models
properly order the frequencies for the series of compounds. In terms of mean
absolute error, there is little to distinguish the six models 49 Table P210: Comparison of HartreeFock and B3LYP Vibrational Frequencies for
TwoHeavyAtom Hydrides with Experimental Values (cm1)
symmetry description molecule of vibration of mode CH3CH3 a1g
a1u
a2u
eg
eu CH3NH2 a' a'' CH3OH a' a'' CH3F a1
e HartreeFock B3LYP 631G* 6311+G** ccpVTZ 631G* 6311+G** ccpVTZ expt. CH3 sstretch
CH3 sdeform
CC stretch
torsion
CH3 sstretch
CH3 sdeform
CH3 dstretch
CH3 ddeform
CH3 rock
CH3 dstretch
CH3 ddeform
CH3 rock 3203
1580
1060
330
3197
1548
3247
1650
1338
3272
1644
891 3162
1550
1052
331
3154
1517
3201
1618
1320
3227
1615
881 3158
1548
1048
328
3152
1522
3194
1614
1323
3222
1619
881 3047
1456
1011
314
3048
1434
3098
1532
1237
3123
1538
835 3023
1425
996
310
3023
1410
3069
1504
1219
3094
1506
830 3028
1424
996
308
3029
1409
3072
1504
1220
3098
1506
827 2954
1388
995
289
2986
1379
2969
1468
1190
2985
1469
822 NH2 sstretch
CH3 dstretch
CH3 sstretch
NH2 scis.
CH3 ddeform
CH3 sdeform
CH3 rock
CN stretch
NH2 wag
NH2 astretch
CH3 dstretch
CH3 ddeform
NH2 twist
CH3 rock
torsion 3730
3246
3158
1841
1648
1608
1290
1149
947
3812
3282
1665
1480
1052
342 3737
3201
3122
1801
1618
1582
1265
1138
900
3816
3235
1635
1454
1041
323 3725
3190
3113
1796
1617
1581
1270
1133
912
3803
3221
1635
1458
1041
319 3466
3079
2971
1704
1528
1486
1193
1071
878
3550
3119
1547
1371
988
333 3508
3056
2962
1669
1498
1461
1164
1056
823
3586
3093
1518
1342
974
305 3487
3055
2964
1665
1498
1458
1171
1053
838
3561
3089
1519
1349
976
293 3361
2961
2820
1623
1473
1430
1130
1044
780
3427
2985
1485
1419
1195
268 OH stretch
CH3 dstretch
CH3 sstretch
CH3 ddeform
CH3 sdeform
OH bend
CH3 rock
CO stretch
CH3 dstretch
CH3 ddeform
CH3 rock
torsion 4117
3306
3186
1663
1638
1508
1189
1164
3232
1652
1290
348 4192
3261
3147
1629
1609
1472
1173
1148
3194
1620
1278
318 4174
3243
3137
1632
1610
1484
1174
1156
3179
1621
1279
318 3754
3133
2997
1541
1511
1401
1097
1067
3040
1525
1183
347 3849
3111
2988
1506
1480
1358
1072
1042
3035
1496
1168
299 3834
3104
2985
1512
1480
1373
1081
1044
3027
1495
1170
309 3681
3000
2844
1477
1455
1345
1060
1033
2960
1477
1165
295 CH3 sstretch
CH3 sdeform
CF stretch
CH3 dstretch
CH3 ddeform
CH3 rock 3233
1652
1187
3314
1653
1312 3193
1611
1156
3277
1615
1296 3180
1619
1173
3256
1619
1306 3037
1532
1091
3111
1524
1205 3032
1480
1033
3117
1490
1185 3025
1489
1058
3102
1494
1194 2930
1464
1049
3006
1467
1182 10 5 mean absolute percentage error 12 9 50 3 3  Table P211: Comparison of HartreeFock and B3LYP Relative CarbonCarbon
Double Bond Stretching Frequencies with Experimental Values (cm1)
molecule 631G* HartreeFock
6311+G** ccpVTZ 631G* B3LYP
6311+G** ccpVTZ expt. cyclobutene 52 48 47 60 55 58 53 tetrachloroethylene 32 5 15 91 69 73 52 4 0 1 18 12 11 9 0 0 0 0 0 0 0 cyclopropene 30 34 33 33 43 40 18 propene 25 27 47 19 21 21 33 isobutene 32 35 34 24 27 26 38 tetramethylethylene 55 68 66 31 43 40 60 cis1,2difluoroethylene 115 122 120 71 73 74 92 tetrafluoroethylene 289 315 302 208 226 219 249 mean absolute error 13 21 19 21 14 16 – cyclopentene
ethylene CC stretching frequencies for ethylene are 1856, 1814, 1820, 1720, 1683 and 1693 cm1 for
HartreeFock 631G*, 6311+G** and ccpVTZ and B3LYP 631G*, 6311+G** and ccpVTZ models, respectively. The experimental stretching frequency is 1623 cm1. 51 Acetone: Obtain the equilibrium geometry of acetone using the HF/631G* model and
calculate the infrared spectrum. Locate the line in the spectrum corresponding to the
carbonyl stretch. What is the ratio of the HartreeFock sketching frequency to the
experimental frequency (1731 cm1)? Locate the line in the spectrum corresponding to the
fully symmetric combination of carbonhydrogen stretching motions. Is the ratio of this
frequency to its experimental value (2937 cm1) similar to that for the ratio of calculated
and experimental carbonoxygen stretching frequencies.
Perfluoroacetone: Unlike acetone, the line the infrared spectrum of perfluoroacetone
corresponding to the CO stretch is very weak. Obtain equilibrium geometries and infrared
s pectra for the two molecules using the B3LYP/631G* model. Do the calculations also
show a marked decrease in the intensity of the CO stretching frequency from acetone to
perfluroracetone? Provide an explanation. Hint: compare the change in dipole moment
corresponding to a xx kJ/mol change in energy for motion away from the equilibrium
p osition along the normal mode corresponding to the CO stretch.
S tretching Frequencies for Bonds Involving Electronegative Atoms: It has previously
been pointed out that bond lengths from HartreeFock are almost always shorter than
experimental values. The magnitude of the error generally increases as the elements involved
in the bond move from left to right in the Periodic Table. Thus, the "limiting" CC bond
length in ethane is 0.006Ǻ shorter than the experimental value, while the CF bond length in
methyl fluoride is 0.025Ǻ shorter and the FF bond length is fluorine is 0.087Ǻ shorter. Is
there an analogous trend for stretching frequencies from HartreeFock calculations?
Using data from Tables P29 and P210, establish whether there is a correlation between
errors in CC, CF and FF stretching frequencies and errors in CC, CF and FF bond distances
for ethane, methyl fluoride and fluorine, respectively. Examine both HartreeFock and
B3LYP models with the 6311+G** basis set. No calculations are needed.
CO Stretching Frequencies in Carbonyl Compounds: CO stretching frequencies span a
very narrow range centering around 1750 cm1. Calculate equilibrium geometries and
infrared spectra for transacrolein at the low end of the range (CO stretching frequency =
1724 cm1), methyl formate in the middle of the range (CO stretching frequency = 1754 cm1
) and acetic acid at the top of the range (CO stretching frequency = 1788 cm1) using both
the HF/631G* and B3LYP/631G* models. Locate the line in each of the spectra
corresponding to the carbonyl stretch. Speculate why this particular infrared absorption is a
useful indicator of carbonyl functionality in complex molecules. Do one or both models
reproduce the ordering of carbonyl stretching frequencies? If so, which model better
accounts for the range in frequency variation? Speculate on what causes the variation.
Repeat your calculations using the 6311+G** basis set instead of 631G*. Is there a
significant improvement in results? Elaborate. 52 Vibrations of Dimethysulfoxide: Vibrational motions seldom correspond to isolated
changes in individual bond lengths or bond angles, but rather involve combinations of these
motions. There are of course exceptions, the most notable being the CO stretching motion
in carbonyl compounds (see previous problem). Because the vibrational frequency depends
on the masses of the atoms involved, any change in frequency resulting from a change the
mass (isotope) of one or more atoms can provide insight into the nature of the motion.
Calculate the equilibrium geometry and vibrational spectrum of dimethylsulfoxide,
(CH3)2S=O, using the HF/631G* model. Characterize the motion associated with each
infrared frequency as being primarily bond stretching, angle bending or a combination of
the two. Is bond stretching or angle bending easier? Do the stretching motions each involve
a single bond or do they involve combinations of all three bonds? Next, replace all six
hydrogens in dimethylsulfoxide with six deuteriums and recalculate the vibrational
spectrum. Compare the resulting frequencies with those obtained for the nondeuterated
molecule. Rationalize any differences.
CO Stretching Frequencies in Metal Carbonyls: Carbon monoxide is one of the most
common ligands in transition metal inorganic and organometallic compounds. Chromium
hexacarbonyl, iron pentacarbonyl and nickel tetracarbonyl are representative. Use the B3LYP/631G* model to obtain equilibrium geometries for these there carbonyl
complexes as well as for “free” carbon monoxide. Is there a significant shift in the
frequency of the CO strectch in carbon monoxide as a result of complexation? If there is,
offere a rationalization for the direction of the shift. 53 Calculating Infrared Spectra
Infrared spectroscopy is one of the most commonly used techniques for
identification of molecules. While it does not provide the same level of detail
about molecular structure as NMR spectrometry, it requires much simpler,
smaller and less expensive instrumentation. This means that it is available to
use in environments that are inaccessible to NMR, for example, on the Mars
rovers.
In addition to the set of vibrational frequencies, calculation of an infrared
spectrum requires a matching set of intensities, each of which is proportional
to the change in the electric dipole moment for motion along the
corresponding vibrational coordinate.
formula
This means that any vibrational motion that does lead to a change in dipole
moment will have a zero infrared intensity and no line in the infrared
spectrum will be observed. Extreme cases of this are homonuclear diatomic
molecules which do not give rise to an infrared spectrum. More common and
more important are vibrational motions that lead to very small changes in
dipole moment. Here the line in infrared spectrum may be too weak to be
easily observed.
Benzyne: Benzyne has long been implicated as a possible intermediate in nucleophilic
aromatic substitution, for example, in the reaction of chlorobenzene with strong base.
Cl OH
OH– OH– –H2 O
–Cl– –H2 O
benzyne While the geometry of benzyne has yet to be conclusively established, the results of a 13C
labeling experiment leave little doubt that two (adjacent) positions on the ring are
equivalent.
Cl
* NH2
* NH2 * KNH2 * + NH3
1 : *= 13 C 1 In addition, the infrared spectrum of benzyne has been recorded and a line in the spectrum
at 2085 cm1 assigned to the C≡C stretch.
Obtain equilibrium geometries and calculate infrared spectra for benzyne, 2butyne and
54 benzene using the B3LYP/631G* model. Compare the geometry of benzyne with those of
2butyne and benzene. Does the molecule incorporate a “real” triple bond (as does 2butyne) or is the length closer to that in benzene? Draw an appropriate Lewis structure (or
set of Lewis structures) for benzyne.
Locate the line in the infrared spectrum for 2butyne corresponding to the C≡C stretch.
Calculate the ratio of the experimental C≡C stretching frequency in 2butyne (2240 cm1)
and the calculated value. This will be used to scale the stretching frequency calculated for
benzyne.
Examine the infrared spectrum for benzyne and decide whether or not it is an energy
minimum. Show your reasoning. Locate the line in the spectrum corresponding to the
“C≡C” stretch. Is it weak or intense relative to the other lines in the spectrum? Would you
expect that this line would be easy or hard to observe? Scale the calculated frequency by
the factor you obtained (for 2butyne) in the previous step. Is your (scaled) stretching
frequency in reasonable accord with the reported experimental value of 2085 cm1?
Need to update to current experiments … the old one is incorrect
Infrared Spectrum of Acetic Acid Dimer: Acetic acids forms a symmetrical hydrogenbonded dimer.
O
H3C HO C C
O H CH3 O Use the B3LYP/631G* model to calculate infrared spectra for acetic acid and its dimer.
Point out any significant differences between the frequencies and/or intensities associated
with the (two) OH stretching motions in the dimer and the OH stretching frequency in
acetic acid.
Greenhouse Gases: In order to dissipate the energy that falls on it due to the sun, the earth
“radiates” as a socalled “blackbody” into the universe. The “theoretical curve” is a smooth
distribution peaking around 900 cm1 and decaying to nearly zero around 1500 cm1. This is
in the infrared, meaning that some of the radiation will be intercepted by molecules in the
earth’s gaseous atmosphere. This in turn means that the earth is actually warmer than it
would be were it not to have an atmosphere. This warming is known as the greenhouse
effect, to make the analogy between the earth’s atmosphere and the glass of a greenhouse.
Both allow energy in and both impede its release. The actual distribution of radiated energy
as measured from outside the earth’s atmosphere in the range of 5001500 cm1 is given
below. The overall profile matches that for a blackbody, but the curve is peppered with
holes. 55 Neither nitrogen nor oxygen, which together comprise 99% of the earth’s atmosphere
absorbs in the infrared and causes the “holes”. However, several “minor” atmospheric
components, carbon dioxide most important among them, absorb in the infrared and
contribute directly to greenhouse warming. Its infrared spectrum shows a strong
absorption in the region centering 670 cm1, the location of the most conspicuous hole
in the blackbody radiation profile. Identify three of the top 10 chemicals manufactured worldwide. Use the B3LYP/631G*
model to calculate the infrared spectra for each and comment whether or not you would
expect it to be a significant greenhouse gas. 56 Fingerprinting
One of the most important if not the most important use of infrared spectra is
the identification of compounds through “fingerprinting”. In a sense this takes
advantage of the fact that vibrational motions and vibrational energies are not
readily transferable from one compound to another. The “fingerprint region”
is normally taken to be roughly 600 – 2500 cm1. Measurement of infrared
spectra below this range generally require specialized instrumentation. More
importantly, low frequency vibrations often correspond to torsional motions
and may depend strongly on detailed conformation. Further discussion is
provided in ChapterP5). The region on the spectrum above around 2500 cm1
typically arises from CH stretching motions and is too crowded to be of much
value in distinguishing molecules.
discussion of line shape in order to make calculated infrared spectrum
recognizable 57 ...
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