Question

7. A BF$_3$ molecule lies in the xy-plane (so all the z-positions are 0); the xy-positions of the atoms are (in Å) $\vec{r}_B = \begin{pmatrix} 0.000\\0.000 \end{pmatrix}$, $\vec{r}_{F1} = \begin{pmatrix} 1.313\\0.000 \end{pmatrix}$, $\vec{r}_{F2} = \begin{pmatrix} -0.657\\1.137 \end{pmatrix}$, $\vec{r}_{F3} = \begin{pmatrix} -0.657\\-1.137 \end{pmatrix}$ a) Sketch the structure of the molecule in a 3D coordinate system. b) Calculate the F1$-$B$-$F2 bond angle, $\alpha$. c) Show that rotating the molecule about the z-axis by $2\alpha$ is the same as rotating it by $-\alpha$. That is, show that $D(\alpha_z)^2 = D(\alpha_z)^{-1}$. The required rotation matrix is $D(\alpha_z) = \begin{pmatrix} \cos \alpha & -\sin \alpha\\ \sin \alpha & \cos \alpha \end{pmatrix}$

          7.
A BF$_3$ molecule lies in the xy-plane (so all the z-positions are 0); the xy-positions of the
atoms are (in Å)
$\vec{r}_B = \begin{pmatrix} 0.000\\0.000 \end{pmatrix}$, $\vec{r}_{F1} = \begin{pmatrix} 1.313\\0.000 \end{pmatrix}$, $\vec{r}_{F2} = \begin{pmatrix} -0.657\\1.137 \end{pmatrix}$, $\vec{r}_{F3} = \begin{pmatrix} -0.657\\-1.137 \end{pmatrix}$
a) Sketch the structure of the molecule in a 3D coordinate system.
b) Calculate the F1$-$B$-$F2 bond angle, $\alpha$.
c) Show that rotating the molecule about the z-axis by $2\alpha$ is the same as rotating
it by $-\alpha$. That is, show that
$D(\alpha_z)^2 = D(\alpha_z)^{-1}$.
The required rotation matrix is
$D(\alpha_z) = \begin{pmatrix} \cos \alpha & -\sin \alpha\\ \sin \alpha & \cos \alpha \end{pmatrix}$

7.
A BF3 molecule lies in the xy-plane (so all the z-positions are 0); the xy-positions of the
atoms are (in Å)
r⃗B =
< p m a t r i x >, r⃗F1 =
< p m a t r i x >, r⃗F2 =
< p m a t r i x >, r⃗F3 =
< p m a t r i x >
a) Sketch the structure of the molecule in a 3D coordinate system.
b) Calculate the F1-B-F2 bond angle, α.
c) Show that rotating the molecule about the z-axis by 2α is the same as rotating
it by -α. That is, show that
D()^2 = D()^-1.
The required rotation matrix is
D() =
< p m a t r i x >

Added by Kent F.

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