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Molecular Geometry:
II. Bond Angles |
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Molecular Geometry:
II. Bond Angles |
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The mathematical details given in this page are to be treated as
extra-curricular material.
Bond Angles
Method of Calculation
The equation for distance calculations allows us to compute any bond length within a molecule. Given the distances between 3 atoms, one simple method for calculating bond angles is by use of the trigonometric cosine rule:
where A, B, C are the lengths of the
sides of the triangle ABC, and γ
is the angle 
A-C-B.
From the example data, the
distances S(1)-F(1), S(1)-F(2), and F(1)...F(2) may be calculated as
are 1.563 Å, 1.558 Å,
and 2.198 Å, respectively. From this information, and by use of the
above equation, the internal angle
F(1)-S(1)-F(2)
within the sulphur hexafluoride molecule is 89.54°.
Whilst this approach is straightforward, it requires three distances to be
calculated before the angle can be obtained. There are other "short-cuts" to
the calculation of both distances and angles when symmetry is taken into
account. To continue with the example data, the molecule with the sulphur atom
labelled S(1) has the point-group symmetry 2/m. Hence the angle
F(1)-S(1)-F(1)′
(where the prime ′
character indicates a symmetry-related atom) is 180° due to inversion
symmetry; likewise the
F(2)-S(1)-F(2)′ is also 180°.
Whilst these approaches are neat and time-saving for the individual,
they are not the best approach for automation by computer and a more general
approach needs to be considered.
Suppose we want to calculate the bond angle
F(1)-S(1)-F(2)′.
Firstly,
we must be sure that the coordinates (x,y,z)
of the atoms S(1), F(1), and F(2) all belong to the same molecule.
In order to check this, it is necessary to
calculate at least the two distances S(1)-F(1) and S(1)-F(2) from the
coordinates. This information must be kept as it is required later.
Returning to vector mathematics, the angle, φ, between two vectors r and s is given by the vector dot product equation as:
F(1)-S(1)-F(2).
We may then write:
| r = | {xF(1) − xS(1) } a + {yF(1) − yS(1) } b + {zF(1) − zS(1) } c |
| = | Δxr a + Δyr b + Δzr c |
| s = | {xF(2) − xS(1) } a + {yF(2) − yS(1) } b + {zF(2) − zS(1) } c |
| = | Δxs a + Δys b + Δzs c |
| r • s = | ( Δxr a + Δyr b + Δzr c ) • ( Δxs a + Δys b + Δzs c ) |
| = | Δxr Δxs a • a + Δyr Δys b • b + Δzr Δzs c • c + ( Δyr Δzs + Δys Δzr ) b • c + |
| ( Δzr Δxs + Δzs Δxr ) c • a + ( Δxr Δys + Δxs Δyr ) a • b | |
| → cos φ = | {a2 Δxr Δxs + b2 Δyr Δys + c2 Δzr Δzs + bc cos α ( Δyr Δzs + Δys Δzr ) + |
| ca cos β ( Δzr Δxs + Δzs Δxr ) + ab cos γ ( Δxr Δys + Δxs Δyr ) } / r s |
As with the calculation of bond lengths, the real-space metric tensor terms appear in the equation. The actual evaluation of the equation is slow (and prone to error) unless approached systematically! It is best carried out by computer programs, but even then the calculation should be done logically. Firstly, tabulate the real-space metric tensor terms. (These should have already been calculated for bond length calculation.) Secondly, tabulate all the Δ terms (which again should have been done for the bond length calculation). Then calculate the value of the dot product. Finally, divide the result by each of the two bond lengths and take the arc (inverse) cosine function to obtain the bond angle.
In contrast to bond lengths, bond angles can vary widely. For an ideal octahedral molecule such as sulphur hexafluoride, the internal F-S-F bond angles are all approximately 90° (see below). Similarly, an ideal tetrahedral molecule such as carbon tetrafluoride, has all its internal F-C-F bond angles close to 109.47°. However, in the "tetrahedral" molecule bromo-trifluoromethane, the angles differ significantly from 109.47°. This is because less energy is required to bend a bond than to stretch one. The larger bromine atom in bromo-trifluoromethane will bend all of the F-C-F angles such that they become smaller, while the C-F bond length will hardly change in value compared to that observed in carbon tetrafluoride.
Bond-Angle Errors
Just as the equation for calculating bond angles is far more complicated than the one used for bond lengths, the equations for calculating the bond-angle errors are also more complicated than the correspond ones for bond-length errors. The calculation will not be given here, but some typical values are shown in the following table, which lists the values of bond lengths (in Å) and angles (in °) together with esd's for both molecules of SF6 in our case study.
| SF6 (Molecule A) | SF6 (Molecule B) | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Bond | Value | Esd | Angle | Value | Esd | Bond | Value | Esd | Angle | Value | Esd | |||
| S(1)-F(1) | 1.5626 | 0.0027 | F(1)-S(1)-F(1)′ |
180.00 | 0.00 | S(2)-F(3) | 1.5617 | 0.0069 | F(3)-S(2)-F(4) |
178.79 | 0.43 | |||
| S(1)-F(1)′ | 1.5626 | 0.0027 | F(1)-S(1)-F(2) |
89.57 | 0.08 | S(2)-F(4) | 1.5604 | 0.0069 | F(3)-S(2)-F(5) |
89.83 | 0.33 | |||
| S(1)-F(2) | 1.5580 | 0.0021 | F(1)-S(1)-F(2)′ |
89.57 | 0.08 | S(2)-F(5) | 1.5552 | 0.0046 | F(3)-S(2)-F(5)′ |
89.83 | 0.33 | |||
| S(1)-F(2)′ | 1.5580 | 0.0021 | F(1)-S(1)-F(2)′′ |
90.43 | 0.08 | S(2)-F(5)′ | 1.5552 | 0.0046 | F(3)-S(2)-F(6) |
91.11 | 0.25 | |||
| S(1)-F(2)′′ | 1.5580 | 0.0021 | F(1)-S(1)-F(2)′′′ |
90.43 | 0.08 | S(2)-F(6) | 1.5451 | 0.0046 | F(3)-S(2)-F(6)′ |
91.11 | 0.25 | |||
| S(1)-F(2)′′′ | 1.5580 | 0.0021 | F(1)′-S(1)-F(2) |
90.43 | 0.08 | S(2)-F(6)′ | 1.5451 | 0.0046 | F(4)-S(2)-F(5) |
89.31 | 0.25 | |||
| F(1)′-S(1)-F(2)′ |
90.43 | 0.08 | F(4)-S(2)-F(5)′ |
89.31 | 0.25 | |||||||||
| F(1)′-S(1)-F(2)′′ |
89.57 | 0.08 | F(4)-S(2)-F(6) |
89.74 | 0.33 | |||||||||
| F(1)′-S(1)-F(2)′′′ |
89.57 | 0.08 | F(4)-S(2)-F(6)′ |
89.74 | 0.33 | |||||||||
| F(2)-S(1)-F(2)′ |
89.71 | 0.10 | F(5)-S(2)-F(5)′ |
89.75 | 0.31 | |||||||||
| F(2)-S(1)-F(2)′′ |
180.00 | 0.00 | F(5)-S(2)-F(6) |
178.89 | 0.38 | |||||||||
| F(2)-S(1)-F(2)′′′ |
90.29 | 0.10 | F(5)-S(2)-F(6)′ |
89.65 | 0.11 | |||||||||
| F(2)′-S(1)-F(2)′′ |
90.29 | 0.10 | F(5)′-S(2)-F(6) |
89.65 | 0.11 | |||||||||
| F(2)′-S(1)-F(2)′′′ |
180.00 | 0.00 | F(5)′-S(2)-F(6)′ |
178.89 | 0.38 | |||||||||
| F(2)′′-S(1)-F(2)′′′89.71 | 0.10 | F(6)-S(2)-′ |
90.93 | 0.32 | ||||||||||
There are several points to note. Firstly, the number of angles is much larger than the number of bonds. Each N atoms bonded to a central atom results in N(N-1)/2 internal bond angles. Thus for sulphur hexafluoride, where N = 6, the number of angles is 15. In practice, this implies that we need to be much more selective when publishing bond angles than bond lengths. Secondly, it is very important to take into account all of the symmetry-related atoms within the molecule. Thus for S(1), four of the six fluorine atoms are symmetry equivalents of F(1) and F(2). Thirdly, the size of the errors in the bond angles correlates very strongly with the size of the errors in the bond lengths. These in turn correlate with the symmetry of the atoms. Atoms on sites of high point-group symmetry within the unit cell have a smaller error in their absolute position in space; obviously atoms lying on sites with point-symmetry elements -1, -3, -4, or -6 will have no error in their position. Where the fractional coordinate errors are reduced, so are the esd's on bond lengths and bond angles.
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| © Copyright 1997-2006. Birkbeck College, University of London. | Author(s): Jeremy Karl Cockcroft |