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Molecular geometry and the VSEPR theory

The problem of determining an accurate molecular geometry is rather complex. Given an arrangement of $N$ atoms with coordinates ${\bf r}_1,...,{\bf r}_N$, the total energy, $E$, of the assembly is a function of these $N$ coordinate vectors and is determined by the electronic structure:

\begin{displaymath}
E = E({\bf r}_1,...,{\bf r}_N)
\end{displaymath}

The optimal geometry of the molecule will be determined by those particular values of the coordinates for which the total energy is a minimum. Thus, in principle, one should compute the derivative of the energy function with respect to each coordinate and set all these derivatives equal to 0 (which is how one does a minimization problem):

\begin{displaymath}
{\partial E \over \partial {\bf r}_i} = 0
\end{displaymath}

which would lead to a set of $3N$ equations in $3N$ unknowns. The solution of such a problem is extremely difficult and is an active area of research.



Fortunately, there exists a simple theory for determining molecular geometries at a qualitative level, which is useful for understanding the basic shapes of molecules. This theory is known as the VSEPR theory, which is an acronym for the valence shell electron pair repulsion theory. It is useful for estimating the shapes of molecules for which there is a central atom bonded to several other atoms surrounding it.



The basic principle of the VSEPR theory is that molecular geometry can be predicted based on the notion that electron pairs in molecules tend to repel each other and achieve a maximum separation from each other. This applies both to bonding electrons as well as lone pairs.



In order to use the VSEPR principle, one needs to compute a number known as the steric number. This is given by

\begin{displaymath}
SN = ({\rm number\ of\ atoms\ bonded\ to\ central\ atom}) +
({\rm number\ of\ lone\ pairs\ on\ central\ atom})
\end{displaymath}

The second number can be determined directly from the Lewis structure.



Once the steric number is known, the basic shape of the molecule is given by the following assignment of shapes:

    $\displaystyle SN=2\;\;\;\;\;\;\;\;\;\;{\rm linear}$  
    $\displaystyle SN=3\;\;\;\;\;\;\;\;\;\;{\rm trigonal\ planar}$  
    $\displaystyle SN=4\;\;\;\;\;\;\;\;\;\;{\rm tetrahedral}$  
    $\displaystyle SN=5\;\;\;\;\;\;\;\;\;\;{\rm trigonal\ bipyramidal}$  
    $\displaystyle SN=6\;\;\;\;\;\;\;\;\;\;{\rm octahedral}$  



The basic shapes are shown in the left column of the figure below:

Figure 7: VSEPR shapes
\includegraphics[scale=1.0]{VSEPR_shapes.eps}

The reason for these assignments will become clear through a study of several examples:

1.
CO$_2$

The Lewis diagram for CO$_2$ is

\begin{displaymath}
\stackrel{..}{\stackrel{{\rm O}}{..}}\;::\;{\rm C}\;::\;
\stackrel{..}{\stackrel{{\rm O}}{..}}
\end{displaymath}

Thus, the steric number of the central carbon is

\begin{displaymath}
SN = 2 + 0 = 2
\end{displaymath}

and, according to the assignment table, the shape is linear:

\begin{displaymath}
{\rm O}={\rm C}={\rm O}
\end{displaymath}

The linear shape can be understood on the basis of the electron repulsion principle: The central carbon is surrounded by two sets of shared electron pairs, which want to achieve a maximum separation between them. The maximum separation they can achieve is at a 180$^{\rm o}$ angle between the pairs, giving rise to the linear configuration.



2.
ClO$_3^+$

The Lewis diagram for this cation is was shown earlier. The steric number of the central chlorine is

\begin{displaymath}
SN = 3 + 0 = 3
\end{displaymath}

and the geometry, according to the table, is trigonal planar:


Figure 8:
\begin{figure}\begin{center}
\leavevmode
\epsfbox{lec11_fig9.ps}
{\small}
\end{center}\end{figure}

Again, this geometry can be seen to achieve a maximum separation between the three pairs of shared electrons around the chlorine.

3.
NO$_2^-$

The Lewis structure for this anion is a resonant form:

\begin{displaymath}
\left\{\left[\stackrel{..}{\stackrel{:{\rm O}}{..}}\;:\;\sta...
...:\;
\stackrel{..}{\stackrel{{\rm O:}}{..}}\right]^{-1}\right\}
\end{displaymath}

The steric number of the nitrogen is

\begin{displaymath}
SN = 2 + 1 = 3
\end{displaymath}

Hence the basic shape should be trigonal planar. However, one of the atoms will be missing from this structure and replaced by the lone pair on the nitrogen. Hence its shape will actually be bent, with a bond angle less than 120$^{\rm o}$ owing to the fact that the lone pair is spatially more delocalized than the bonding pairs. This spatial delocalization of the lone pair causes the bonding pair to be repelled from the lone pair at greater distances than could be achieved by a 120 degree angle, giving rise to the smaller bend angle in the structure:


Figure 9:
\begin{figure}\begin{center}
\leavevmode
\epsfbox{lec11_fig10.ps}
{\small}
\end{center}\end{figure}

Because of the resonant structure, the bond lengths are the same.




next up previous
Next: About this document ... Up: lecture_11 Previous: Valence shell expansion
Mark E. Tuckerman 2011-11-05