3.4.3: Molecular Orbitals from p Orbitals
- Page ID
- 459515
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Some Assumptions
Other diatomic molecules can be constructed in a similar way to that of H2. Consider dinitrogen, N2.
Nitrogen has five valence electrons, and these electrons are found in the 2s and 2p levels. There are three possible atomic orbitals in the 2p level where some of these electrons could be found: px, py and pz. We need to look at the interaction between the s and px, py and pz orbitals on one nitrogen atom with the s and px, py and pz orbitals on the other nitrogen. That process could be extremely complicated, but is simplified because orbital interactions are governed by symmetry. Orbitals interact most easily with other orbitals that have the same element of symmetry. For now, we can simplify this idea and say that orbitals on one atom only interact with the same type of orbitals on the other atom.
- s orbitals interact with s orbitals. We can already see how that will work out in dinitrogen, because that is what happened in dihydrogen.
- px orbitals interact with px orbitals.
- py orbitals interact with py orbitals.
- pz orbitals interact with pz orbitals.
Another complication here is that the s and p orbitals do not start out at the same energy level. When the orbitals mix, one combination goes up in energy and one goes down. Does the s antibonding combination go higher in energy than the combinations from p orbitals? Do the p bonding combinations go lower in energy than the combinations from s orbitals? We will simplify and assume that the s and p levels remain completely separate from each other. This is not always true, but the situation varies depending on what atoms we are dealing with.
- The combination of one s orbital with another is just like in hydrogen. Two original orbitals will combine and rearrange to produce two new orbitals.
- There is a bonding combination in which the orbitals are in phase. The new orbital produced has a longer wavelength than the original orbital. It is lower in energy.
- There is an antibonding combination in which the orbitals are out of phase. The new orbital produced has a shorter wavelength than the original orbital. It is higher in energy.
Sigma bonding with p-orbitals
In considering the interaction of two p orbitals, we have to keep in mind that p orbitals are directional. A p orbital lies along a particular axis: x, y or z. The three p orbitals on nitrogen are all mutually perpendicular (or orthogonal) to each other. That situation is in contrast to s orbitals, which are spherical and thus look the same from any direction.
We first need to define one axis as lying along the N-N bond. It does not really matter which one. However, by convention we say the N-N bond lies along the z axis. The pz orbitals have a different spatial relationship to each other compared to the py and px. The pz orbitals lie along the bond axis, whereas the py and px are orthogonal (perpendicular) to it.
As the nitrogen atoms are brought together, one lobe on one pz orbital overlaps strongly with one lobe on the other pz orbital. The other lobes point away from each other and do not interact in any obvious way.
As with the s orbital, the pz orbitals can be in-phase or out-of-phase. The in-phase combination results in constructive interference. (Here, "in-phase" means the lobes that overlap are in-phase; for that to happen the two p orbitals are actually completely out-of-phase with each other mathematically, so that one orbital is the mirror image of the other.) This combination is at a longer wavelength than the original orbital. It is a lower energy combination.
The out-of-phase combination (meaning in this case that the overlapping lobes are out-of-phase) results in destructive interference. This combination is at shorter wavelength than the original orbital. It is a higher energy combination.
As a result, we have two different combinations stemming from two different p orbitals coming together in two different ways. We get a low-energy, in-phase, bonding combination and a high-energy, out-of-phase, antibonding combination.
Pi-bonding with p-orbitals
As the nitrogen atoms are brought together, one lobe on one pz orbital overlaps strongly with one lobe on the other pz orbital. The other lobes point away from each other and do not interact in any obvious way.
As with the s orbital, the pz orbitals can be in-phase or out-of-phase. The in-phase combination results in constructive interference. (Here, "in-phase" means the lobes that overlap are in-phase; for that to happen the two p orbitals are actually completely out-of-phase with each other mathematically, so that one orbital is the mirror image of the other.) This combination is at a longer wavelength than the original orbital. It is a lower energy combination.
As the nitrogen atoms are brought together, the parallel (but not collinear), p orbitals can also interact with each other. The px orbital on one atom will interact with the px orbital on the other atom. The py orbital on one atom will interact with the py orbital on the other atom. As the two atoms approach along the internuclear axis (z axis), the orbitals would approach each other side by side, with their lobe interacting above and below the bond axis between the two atoms. Although the px and py orbitals can be close enough to each other to overlap, they do not overlap as strongly as pz orbitals lying along the bond axis.
The interaction of a pair of px (or py) orbitals produces a bonding and an antibonding combination. The in-phase combination (bonding) is shown below, followed by the out-of-phase combination. The resulting orbitals contain nodes along the bond axis, and the electron density is found above and below the bond axis. This bond is called a p (pi) bond.
The illustration above is for one set of p orbitals that are orthogonal to the bond axis. The second picture shows the result of the constructive (or destructive) interference. A similar picture could be shown for the other set of p orbitals.
An important consequence of the spatial distribution or "shape" of a p orbital is that it is not symmetric with respect to the bond axis. An s orbital is not affected when the atom at one end of the bond is rotated with respect to the other. A p orbital is affected by such a rotation. If one atom turns with respect to the other, the p orbital would have to stretch to maintain the connection. The orbitals would not be able to overlap, so the connection between the atoms would be lost.
Problems
Draw an MO cartoon of a sigma bonding orbital formed by the overlap of two p orbitals between two oxygen atoms. Label the positions of the oxygen nuclei with the symbol "O". Label the O-O bond axis.
- Answer
The combinations of ______________ atomic orbitals leads to σ orbitals.
Draw pictures.
- Answer
-
The combinations of s + s; s + p; p + p; s + d; p + d atomic orbitals can lead to σ orbitals.
The combinations of ______________ atomic orbitals leads to π orbitals.
Draw pictures.
- Answer
-
The combinations of side by side p + p or p + d atomic orbitals leads to π orbitals.
Which molecular orbital is typically the highest in energy?
a. p
b. σ
c. π*
d. π
e. σ*
- Answer
-
e. σ*
Why would a core 1s orbital not interact with a valence 2s orbital?
Hint: Why is a Li2O bond stronger than a K2O bond?
- Answer
-
Li+ and O2- are more similar in size than K+ and O2-, so the bond between Li+ and O2- is stronger.
The energy difference between any core orbitals and valence orbitals is too large, so they cannot interact. In order for orbitals to interact, the orbitals need to have the same symmetry, be in the same plane, and be similar in energy.
Add a few words to explain the ideas conveyed in these drawings.
- Answer
-
When two parallel p orbitals combine out-of-phase, destructive interference occurs.
There is a node between the atoms.
The energy of the electrons increases.
When two parallel p orbitals combine in-phase, constructive interference occurs.
There is no node between the atoms; the electrons are found above and below the axis connecting the atoms.
The energy of the electrons decreases.
Attribution
Chris P Schaller, Ph.D., (College of Saint Benedict / Saint John's University)
Curated or created by Kathryn Haas