Molecular Geometry: 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:
cosγ = (A2 + B2 − C2) / 2AB
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: r • s = r s cos φ Suppose the vectors r and s are S(1)-F(1) and S(1)-F(2), and the angle φ is 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)′′′
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