# 9.7: Molecular Orbitals Can Be Ordered According to Their Energies

- Page ID
- 204577

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The LCAO-MO method that we used for H_{2}^{+} can be applied qualitatively to homonuclear diatomic molecules to provide additional insight into chemical bonding. A more quantitative approach also is helpful, especially for more complicated situations, like heteronuclear diatomic molecules and polyatomic molecules. When two atoms are close enough for their valence orbitals to overlap significantly, the filled inner electron shells are largely unperturbed; hence they are often ignored in constructing molecular orbitals. This means that we can focus our attention on the molecular orbitals derived from valence atomic orbitals.

## Molecular Orbitals Formed from *ns* Orbitals

The molecular orbitals diagrams formatted for the dihydrogen species are similar to the diagrams to any **homonuclear diatomic molecule** with two identical alkali metal atoms (Li_{2} and Cs_{2}, for example) is shown in part (a) in Figure \(\PageIndex{1}\), where M represents the metal atom. Only two energy levels are important for describing the valence electron molecular orbitals of these species: a σ_{ns} bonding molecular orbital and a σ^{*}_{ns} antibonding molecular orbital. Because each alkali metal (M) has an *ns*^{1} valence electron configuration, the M_{2} molecule has two valence electrons that fill the σ_{ns} bonding orbital. As a result, a bond order of 1 is predicted for all homonuclear diatomic species formed from the alkali metals (Li_{2}, Na_{2}, K_{2}, Rb_{2}, and Cs_{2}). The general features of these M_{2} diagrams are identical to the diagram for the H_{2} molecule. Experimentally, all are found to be stable in the gas phase, and some are even stable in solution.

Similarly, the molecular orbital diagrams for homonuclear diatomic compounds of the alkaline earth metals (such as Be_{2}), in which each metal atom has an *ns*^{2} valence electron configuration, resemble the diagram for the He_{2} molecule. As shown in Figure \(\PageIndex{1b}\), this is indeed the case. All the homonuclear alkaline earth diatomic molecules have four valence electrons, which fill both the σ_{ns} bonding orbital and the *σ _{ns}** antibonding orbital and give a bond order of 0. Thus Be

_{2}, Mg

_{2}, Ca

_{2}, Sr

_{2}, and Ba

_{2}are all expected to be unstable, in agreement with experimental data.In the solid state, however, all the alkali metals and the alkaline earth metals exist as extended lattices held together by metallic bonding. At low temperatures, \(Be_2\) is stable.

Example \(\PageIndex{1}\): Sodium Dimer Ion

Use a qualitative molecular orbital energy-level diagram to predict the valence electron configuration, bond order, and likely existence of the Na_{2}^{−} ion.

**Given: **chemical species

**Asked for: **molecular orbital energy-level diagram, valence electron configuration, bond order, and stability

**Strategy**

- Combine the two sodium valence atomic orbitals to produce bonding and antibonding molecular orbitals. Draw the molecular orbital energy-level diagram for this system.
- Determine the total number of valence electrons in the Na
_{2}^{−}ion. Fill the molecular orbitals in the energy-level diagram beginning with the orbital with the lowest energy. Be sure to obey the Pauli principle and Hund’s rule while doing so. - Calculate the bond order and predict whether the species is stable.

**Solution**

**A** Because sodium has a [Ne]3*s*^{1} electron configuration, the molecular orbital energy-level diagram is qualitatively identical to the diagram for the interaction of two 1*s* atomic orbitals.

**B** The Na_{2}^{−} ion has a total of three valence electrons (one from each Na atom and one for the negative charge), resulting in a filled σ_{3}_{s} molecular orbital, a half-filled σ_{3s}_{* }and a \( \left ( \sigma _{3s} \right )^{2}\left ( \sigma _{3s}^{\star } \right )^{1} \) electron configuration.

**C** The bond order is (2-1)÷2=1/2 With a fractional bond order, we predict that the Na_{2}^{−} ion exists but is highly reactive.

Exercise \(\PageIndex{1}\): Calcium Dimer Cation

Use a qualitative molecular orbital energy-level diagram to predict the valence electron configuration, bond order, and likely existence of the Ca_{2}^{+} ion.

**Answer**-
Ca

_{2}^{+}has a \( \left ( \sigma _{4s} \right )^{2}\left ( \sigma _{4s}^{\star } \right )^{1} \) electron configurations and a bond order of 1/2 and should exist.

## Molecular Orbitals Formed from n*p* Orbitals

Atomic orbitals other than *ns* orbitals can also interact to form molecular orbitals. Because individual *p*, *d*, and *f* orbitals are not spherically symmetrical, however, we need to define a coordinate system so we know which lobes are interacting in three-dimensional space. Recall that for each *np* subshell, for example, there are *np*_{x}, *np*_{y}, and *np*_{z} orbitals. All have the same energy and are therefore degenerate, but they have different spatial orientations.

\[ \sigma _{np_{z}}=np_{z}\left ( A \right )-np_{z}\left ( B \right ) \label{9.7.1}\]

Just as with *ns* orbitals, we can form molecular orbitals from *np* orbitals by taking their mathematical sum and difference. When two positive lobes with the appropriate spatial orientation overlap, as illustrated for two *np*_{z} atomic orbitals in part (a) in Figure \(\PageIndex{2}\), it is the mathematical *difference* of their wave functions that results in *constructive* interference, which in turn increases the electron probability density between the two atoms. The difference therefore corresponds to a molecular orbital called a \( \sigma _{np_{z}} \) *bonding molecular orbital* because, just as with the σ orbitals discussed previously, it is symmetrical about the internuclear axis (in this case, the *z*-axis):

\[ \sigma _{np_{z}}=np_{z}\left ( A \right )-np_{z}\left ( B \right ) \label{9.7.2}\]

The other possible combination of the two *np*_{z} orbitals is the mathematical sum:

\[ \sigma _{np_{z}}=np_{z}\left ( A \right )+np_{z}\left ( B \right ) \label{9.7.3}\]

In this combination, shown in part (b) in Figure \(\PageIndex{2}\), the positive lobe of one *np*_{z} atomic orbital overlaps the negative lobe of the other, leading to *destructive* interference of the two waves and creating a node between the two atoms. Hence this is an antibonding molecular orbital. Because it, too, is symmetrical about the internuclear axis, this molecular orbital is called a \( \sigma _{np_{z}}=np_{z}\left ( A \right )-np_{z}\left ( B \right ) \) *antibonding molecular orbital*. Whenever orbitals combine, *the bonding combination is always lower in energy* (more stable) than the atomic orbitals from which it was derived, and *the antibonding combination is higher in energy* (less stable).

The remaining *p* orbitals on each of the two atoms, *np*_{x} and *np*_{y}, do not point directly toward each other. Instead, they are perpendicular to the internuclear axis. If we arbitrarily label the axes as shown in Figure \(\PageIndex{3}\), we see that we have two pairs of *np* orbitals: the two *np*_{x} orbitals lying in the plane of the page, and two *np*_{y} orbitals perpendicular to the plane. Although these two pairs are equivalent in energy, the *np*_{x} orbital on one atom can interact with only the *np*_{x} orbital on the other, and the *np*_{y} orbital on one atom can interact with only the *np*_{y} on the other. These interactions are side-to-side rather than the head-to-head interactions characteristic of σ orbitals. Each pair of overlapping atomic orbitals again forms two molecular orbitals: one corresponds to the arithmetic sum of the two atomic orbitals and one to the difference. The sum of these side-to-side interactions increases the electron probability in the region above and below a line connecting the nuclei, so it is a bonding molecular orbital that is called a pi (π) orbital (a bonding molecular orbital formed from the side-to-side interactions of two or more parallel np atomic orbitals). The difference results in the overlap of orbital lobes with opposite signs, which produces a nodal plane perpendicular to the internuclear axis; hence it is an antibonding molecular orbital, called a pi star (π*) orbital An antibonding molecular orbital formed from the difference of the side-to-side interactions of two or more parallel *np* atomic orbitals, creating a nodal plane perpendicular to the internuclear axis..

\[ \pi _{np_{x}}=np_{x}\left ( A \right )+np_{x}\left ( B \right ) \label{9.7.4}\]

\[ \pi ^{\star }_{np_{x}}=np_{x}\left ( A \right )-np_{x}\left ( B \right ) \label{9.7.5}\]

The two *np*_{y} orbitals can also combine using side-to-side interactions to produce a bonding \( \pi _{np_{y}} \) molecular orbital and an antibonding \( \pi _{np_{y}}^{\star } \) molecular orbital. Because the *np*_{x} and *np*_{y} atomic orbitals interact in the same way (side-to-side) and have the same energy, the \( \pi _{np_{x}} \) and \( \pi _{np_{y}} \)molecular orbitals are a degenerate pair, as are the \( \pi _{np_{x}}^{\star } \) and \( \pi _{np_{y}}^{\star } \) molecular orbitals.

## Energies for Homonuclear Diatomic Molecules

We now describe examples of systems involving period 2 homonuclear diatomic molecules, such as N_{2}, O_{2}, and F_{2}. When we draw a molecular orbital diagram for a molecule, there are four key points to remember:

- The number of molecular orbitals produced is the same as the number of atomic orbitals used to create them.
- As the overlap between two atomic orbitals increases, the difference in energy between the resulting bonding and antibonding molecular orbitals increases.
- When two atomic orbitals combine to form a pair of molecular orbitals, the bonding molecular orbital is stabilized about as much as the antibonding molecular orbital is destabilized.
- The interaction between atomic orbitals is greatest when they have the same energy.

Figure \(\PageIndex{4}\) is an energy-level diagram that can be applied to two *identical *interacting atoms that have three *np* atomic orbitals each. There are six degenerate *p* atomic orbitals (three from each atom) that combine to form six molecular orbitals, three bonding and three antibonding. The bonding molecular orbitals are lower in energy than the atomic orbitals because of the increased stability associated with the formation of a bond. Conversely, the antibonding molecular orbitals are higher in energy, as shown. The energy difference between the σ and σ* molecular orbitals is significantly greater than the difference between the two π and π* sets. The reason for this is that the atomic orbital overlap and thus the strength of the interaction are greater for a σ bond than a π bond, which means that the σ molecular orbital is more stable (lower in energy) than the π molecular orbitals.

The number of molecular orbitals is always equal to the total number of atomic orbitals we started with.

We illustrate how to use these points by constructing a molecular orbital energy-level diagram for F_{2}. We use the diagram in Figure \(\PageIndex{5a}\); the *n* = 1 orbitals (σ_{1}_{s} and σ_{1}_{s}***) are located well below those of the *n* = 2 level and are not shown. As illustrated in the diagram, the σ_{2}_{s} and σ_{2}_{s}*** molecular orbitals are much lower in energy than the molecular orbitals derived from the 2*p* atomic orbitals because of the large difference in energy between the 2*s* and 2*p* atomic orbitals of fluorine. The lowest-energy molecular orbital derived from the three 2*p* orbitals on each F is \( \sigma _{2p_{z}} \) and the next most stable are the two degenerate orbitals, \( \pi _{2p_{x}} \) and \( \pi _{2p_{y}} \). For each bonding orbital in the diagram, there is an antibonding orbital, and the antibonding orbital is destabilized by about as much as the corresponding bonding orbital is stabilized. As a result, the \( \sigma ^{\star }_{2p_{z}} \) orbital is higher in energy than either of the degenerate \( \pi _{2p_{x}}^{\star } \) and \( \pi _{2p_{y}}^{\star } \) orbitals. We can now fill the orbitals, beginning with the one that is lowest in energy.

Each fluorine has 7 valence electrons, so there are a total of 14 valence electrons in the F_{2} molecule. Starting at the lowest energy level, the electrons are placed in the orbitals according to the Pauli principle and Hund’s rules. Two electrons each fill the σ_{2}_{s} and σ_{2}_{s}*** orbitals, 2 fill the \( \sigma _{2p_{z}} \) orbital, 4 fill the two degenerate π orbitals, and 4 fill the two degenerate π^{*} orbitals, for a total of 14 electrons.

For period 2 diatomic molecules to the left of N_{2} in the periodic table, a slightly different molecular orbital energy-level diagram is needed because the \( \sigma _{2p_{z}} \) molecular orbital is slightly *higher* in energy than the degenerate \( \pi ^{\star }_{np_{x}} \) and \( \pi ^{\star }_{np_{y}} \) orbitals. The difference in energy between the 2*s* and 2*p* atomic orbitals increases from Li_{2} to F_{2} due to increasing nuclear charge and poor screening of the 2*s* electrons by electrons in the 2*p* subshell. The bonding interaction between the 2*s* orbital on one atom and the 2*p _{z}* orbital on the other is most important when the two orbitals have similar energies. This interaction decreases the energy of the σ

_{2}

_{s}orbital and increases the energy of the \( \sigma _{2p_{z}} \) orbital. Thus for Li

_{2}, Be

_{2}, B

_{2}, C

_{2}, and N

_{2}, the \( \sigma _{2p_{z}} \) orbital is higher in energy than the \( \sigma _{3p_{z}} \) orbitals, as shown in Figure \(\PageIndex{6}\). Experimentally, the energy gap between the

*ns*and

*np*atomic orbitals

*increases*as the nuclear charge increases (Figure \(\PageIndex{6}\)). Thus for example, the \( \sigma _{2p_{z}} \) molecular orbital is at a lower energy than the \( \pi _{2p_{x,y}} \) pair.

Example \(\PageIndex{2}\): Diatomic Sulfur

Use a qualitative molecular orbital energy-level diagram to predict the electron configuration, the bond order, and the number of unpaired electrons in S_{2}, a bright blue gas at high temperatures.

**Given: **chemical species

**Asked for: **molecular orbital energy-level diagram, bond order, and number of unpaired electrons

**Strategy:**

- Write the valence electron configuration of sulfur and determine the type of molecular orbitals formed in S
_{2}. Predict the relative energies of the molecular orbitals based on how close in energy the valence atomic orbitals are to one another. - Draw the molecular orbital energy-level diagram for this system and determine the total number of valence electrons in S
_{2}. - Fill the molecular orbitals in order of increasing energy, being sure to obey the Pauli principle and Hund’s rule.
- Calculate the bond order and describe the bonding.

**Solution:**

**A** Sulfur has a [Ne]3*s*^{2}3*p*^{4} valence electron configuration. To create a molecular orbital energy-level diagram similar to those in Figures \(\PageIndex{6}\) and \(\PageIndex{7}\), we need to know how close in energy the 3*s* and 3*p* atomic orbitals are because their energy separation will determine whether the \( \pi _{3p_{x,y}} \) or the \( \sigma _{3p_{z}} \)> molecular orbital is higher in energy. Because the *ns*–*np* energy gap *increases* as the nuclear charge increases, the \( \sigma _{3p_{z}} \) molecular orbital will be lower in energy than the \( \pi _{3p_{x,y}} \) pair.

**B** The molecular orbital energy-level diagram is as follows:

Each sulfur atom contributes 6 valence electrons, for a total of 12 valence electrons.

**C** Ten valence electrons are used to fill the orbitals through \( \pi _{3p_{x}} \) and \( \pi _{3p_{y}} \), leaving 2 electrons to occupy the degenerate \( \pi ^{\star }_{3p_{x}} \) and \( \pi ^{\star }_{3p_{y}} \) pair. From Hund’s rule, the remaining 2 electrons must occupy these orbitals separately with their spins aligned. With the numbers of electrons written as superscripts, the electron configuration of S_{2} is \( \left ( \sigma _{3s} \right )^{2}\left ( \sigma ^{\star }_{3s} \right )^{2}\left ( \sigma _{3p_{z}} \right )^{2}\left ( \pi _{3p_{x,y}} \right )^{4}\left ( \pi _{3p ^{\star }_{x,y}} \right )^{2} \) with 2 unpaired electrons. The bond order is (8 − 4) ÷ 2 = 2, so we predict an S=S double bond.

Exercise \(\PageIndex{2}\)

Use a qualitative molecular orbital energy-level diagram to predict the electron configuration, the bond order, and the number of unpaired electrons in the peroxide ion (O_{2}^{2−}).

**Answer**-
\( \left ( \sigma _{2s} \right )^{2}\left ( \sigma ^{\star }_{2s} \right )^{2}\left ( \sigma _{2p_{z}} \right )^{2}\left ( \pi _{2p_{x,y}} \right )^{4}\left ( \pi _{2p ^{\star }_{x,y}} \right )^{4} \) bond order of 1; no unpaired electrons

## Molecular Orbitals Formed from *ns* with n*p* Orbitals

Although many combinations of atomic orbitals form molecular orbitals, we will discuss only one other interaction: an *ns* atomic orbital on one atom with an *np*_{z} atomic orbital on another. As shown in Figure \(\PageIndex{7}\), the sum of the two atomic wave functions (*ns* + *np*_{z}) produces a σ bonding molecular orbital. Their difference (*ns* − *np*_{z}) produces a σ* antibonding molecular orbital, which has a nodal plane of zero probability density perpendicular to the internuclear axis.