# 26: Molecular Orbitals and Diatomics

- Page ID
- 65258

Lecture 26

Last Lecture focused on the LCAO approximation for the \(H_2^+\) molecule. We solved for the bonding and antibonding LCAO-MOs, which was not hard. Solving for the energies of these molecular orbitals was a bit harder and depended on the bond length and three integrals (defined below for the \(\{| 1s \rangle \}\) basis set):

- The Overlap Integral: \[S(R)= \left \langle 1s_A | 1s_B \right \rangle\]
- The Coulomb Integral: \[J(R) = - \left \langle 1s_A | \dfrac {e^2}{4 \pi \epsilon _0 r_B } | 1s_A \right \rangle\]
- The Exchange Integral: \[K(R)= - \left \langle 1s_A | \dfrac {e^2}{4 \pi \epsilon _0 r_A } | 1s_B \right \rangle \]

The expression for the electronic energy (of a single electron in a single molecular orbital in this system) of the \(H_2^+\) molecule

\[E_{\pm} = E_H + \dfrac {e^2}{4\pi \epsilon _0 R} + \dfrac {J \pm K}{1 \pm S} \label {9.4.10}\]

The results formulas were for the bonding orbital

\[E_{+}= E_H + \dfrac {e^2}{4\pi \epsilon _0 R} + \dfrac{J+K}{1+S}\ \ \ \ \ (bonding)\nonumber \]

and for the antibonding orbital

\[E_{-}= E_H + \dfrac {e^2}{4\pi \epsilon _0 R} + \dfrac{J-K}{1-S}\ \ \ \ \ (antibonding) \nonumber\]

The energy of the molecular orbital (Equation \ref{9.4.10}) is a function of \(R\) since the three integrals outlines above are functions of \(R\) as is the Coulombic repulsion term. Equation \(\ref{9.4.10}\) tells us that the energy of the \(H_2^+\) molecule is

- the energy of a hydrogen atom plus \[E_H \label{eq1}\]
- the Coulombic repulsive energy of two protons plus \[\dfrac {e^2}{4\pi \epsilon _0 R} \label{eq2}\]
- some additional electrostatic interactions of the electron with the protons. \[\dfrac {J \pm K}{1 \pm S} \label{eq3}\]

These bonding interactions are given by Equation \ref{eq3}. To get a chemical bond (i.e., a stable H_{2}^{+} molecule):

- \(\Delta E_{\pm} = E_{\pm} - E_H\) must be less than zero (otherwise it is lower energy to be a \(H\) atom and a \(H^+\) free proton, which no energy)
- The energy of the molecule must have a
**minimum**, i.e. the bonding energy (Equation \ref{eq3})**must**be sufficiently negative to overcome the positive repulsive energy of the two protons (Equation \ref{eq2} for some value of \(R\). For large \(R\), these both terms (Equations \ref{eq2} and \ref{eq3}) are zero, and for small \(R\), the Coulombic repulsion (Equation \ref{eq2}) of the protons rises to infinity.

The \(\psi_+\) molecular orbital has a lower energy than either of the hydrogen 1*s* atomic orbitals. Conversely, the \(\psi_-\) molecular orbital has a *higher* energy than either of the hydrogen 1*s* atomic orbitals.

Equation \(\ref{9.4.10}\) can be rewritten as

\[ \Delta E_{\pm} = E_{\pm} - E_H = \dfrac {e^2}{4\pi \epsilon _0 R} + \dfrac {J \pm K}{1 \pm S} \label {10.31}\]

which shows energy of H_{2}^{+} relative to the energy of a separated hydrogen atom and a proton. For the electron in the \(\psi_-\) orbital, the energy of the molecule, \(E_{el}(R)\), **always **is greater than the energy of the separated atom and proton.

Nature wants to minimize total energy

For the electron in the \(\psi_-\) orbital, the energy of the molecule, \(E_{el}(R)\), **always **is greater than the energy of the separated atom and proton. That is, nature will prefer a bond to be formed to lower the energy of the system.

## Secular Equations in Molecular Orbitals (MOs within the linear Variational Method)

This picture of bonding in H_{2}^{+} in the previous section is very simple, but gives reasonable results when compared to an exact calculation. The equilibrium bond distance is 134 pm compared to 106 pm (exact), and a dissociation energy is 1.8 eV compared to 2.8 eV (exact). One way to improve this approach is to use a bigger basis set, e.g.,

\[| \psi_J \rangle = c_{J,1} 1s_A + c_{J,2} 1s_B + c_{J,3} 2s_A + c_{J,4} 2s_B + c_{J,5} 2p_{z,A} + c_{J,6} 2p_{z,B} \label{bigger}\]

Above is a 6-element (atomic orbital) basis set using both n=1 and n=2 atomic orbitals. The expressions above for the energy of \(H_2^+\) can be formatted in terms of a basis set of any number of functions (only six are in the LCAO expansion in Equation \ref{bigger}).

Following the LCAO approximation, the j^{th} molecular orbital can be expressed as a linear combination of many atomic orbitals {\(\{\phi_i\}\)}:

\[\psi_J = \sum_i a_{J,i} \phi_i \label{9.5.1}\]

The energy of the non-normalized molecular orbital can be calculated from the expectation value integral of the Hamiltonian,

\[E_{J} = \dfrac{\left \langle \psi _{J} | \hat {H} _{elec} | \psi _{J} \right \rangle}{\left \langle \psi _{J} | \psi _{J} \right \rangle} \label {9.5.2}\]

In this case, the denominator (\(\left \langle \psi _{J} | \psi _{J} \right \rangle\)) is left in; if the MOs are normalized we do not need to explicitly address.

After substituting the LCAO expansion for \(\phi_J\) (Equation \ref{bigger}) into the energy expression (Equation \ref{9.5.2}):

\[\begin{align} E_{J} &= \dfrac{\left \langle \displaystyle \sum_i a_{J,i}^* \phi_i | \hat {H} _{elec} | \displaystyle \sum_j a_{J,i} \phi_j \right \rangle}{\left \langle \displaystyle \sum_i a_{J,i}^* \phi_j | \displaystyle \sum_j a_{J,j} \phi_j\right \rangle} \label {9.5.3} \\[4pt] &= \dfrac{\displaystyle \sum_{i,j} a_{J,i}^* a_{J,j} \left \langle \phi_i | \hat {H} _{elec} | \phi_j \right \rangle}{\displaystyle \sum_{i,j} a_{J,i}^* a_{J,j} \left \langle \phi_i | \phi_j\right \rangle} \label {9.5.4} \\[4pt] &= \dfrac{ \displaystyle \sum_{i,j} a_{J,i}^* a_{J,j} H_{ij}}{ \displaystyle \sum_{i,j} a_{J,i}^* a_{J,j} S_{ij} } \label {9.5.5} \end{align}\]

Following the varitional theorem, to determine the coefficients of the LCAO expansion \(a_i\), we need to minimize \(E_J\)

\[ \dfrac{\partial E_J}{\partial a_k} = 0 \label{9.5.6}\]

for all \(k\). This requires solving \(N\) linear equations to hold true (where N is the number of atomic orbitals in the basis)

\[ \sum_{i=1}^{N} a_i (H_{ki} - ES_{ki}) = 0 \label{9.5.7}\]

These equations are the **secular equations** and were discussed previously in the context of linear variational method approximation.

For the two function basis set, like that discussed above, these secular equtations can be written out as

\[\begin{array}{rcl} a_1(H_{11} - ES_{11}) + a_2(H_{12} - ES_{12}) & = & 0 \\ a_1(H_{12} - ES_{12}) + a_2(H_{22} - ES_{22}) & = & 0 \end{array} \label{9.5.8}\]

where \(a_1\) and \(a_2\) are the coefficients in the linear combination of the atomic orbitals used to construct the molecular orbital. Writing this set of homogeneous linear equations in matrix form gives

\[\begin{pmatrix} H_{11} - ES_{11} & H_{12} - ES_{12} \\ H_{12} - ES_{12} & H_{22} - ES_{22} \end{pmatrix} \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \label{9.5.9}\]

Solving this requires finding the N roots of a polynomial of order \(N\) (i.e., N different satisfactory values of E). For each \(E_J\) there is a different set of coefficients, \( { a_{ij} } \) where i runs over all functions in the basis functions, and \(J\) runs over molecular orbitals, each having energy \(E_J\). Solve the set of linear equations using that specific \(E_J\) to determine \(a_{ij}\) values.

Steps in a Solving the Secular Equations

- Select a set of N basis functions
- Determine all N( N –1)/2 values of both \(H_{ij}\) and \(S_{ij}\)
- Form the secular determinant; determine \(N\) roots \(E_j\) of secular equation
- For each \(E_J\) solve the set of linear equations to determine the basis set coefficients \(a_{ij}\) for the j-th molecular orbital

## Molecules follow the Pauli Exclusion Principle and Hund's Rules

Just as for atoms, each electron in a molecule can be described by a product of spin-orbitals. Since electrons are fermions, the electronic wavefunction must be **antisymmetric **with respect to the permutation of any two electrons. A Slater determinant containing the molecular spin orbitals produces the antisymmetric wavefunction. For example for two electrons,

\[\Psi (r_1, r_2) = \dfrac{1}{\sqrt{2}} \begin {vmatrix} \psi _A (r_1) \alpha (1) & \psi _B (r_1) \beta (1) \\ \psi _A (r_2) \alpha (2) & \psi _B (r_2) \beta (2) \end {vmatrix} \label{9.6.2}\]

Solving the Schrödinger equation in the orbital approximation will produce a set of spatial molecular orbitals, each with a specific energy, \(\epsilon\). Following the Aufbau Principle, two electrons with different spins ( \(\alpha\) and \(\beta\), consistent with the Pauli Exclusion Principle discussed for muliti-electron atoms) are assigned to each spatial molecular orbital in order of increasing energy. For the ground state of the 2n electron molecule, the n lowest energy spatial orbitals will be occupied, and the electron configuration will be given as \(\psi ^2_1 \psi ^2_2 \psi ^2_3 \dots \psi ^2_n\). The electron configuration also can be specified by an orbital energy level diagram as shown in Figure \(\PageIndex{1}\). Higher energy configurations exist as well, and these configurations produce excited states of molecules. Some examples:

Molecular Orbitals are often characterized by their Symmetry

**Reflection in 1D**

Before we discussed the symmetry of wavefunctions upon **refection **(i.e., \(x \rightarrow -x\) and called them "odd" or "even" (or neither, but let's ignore that symmetry).

\[\hat{R} | \psi(x) \rangle = |\psi(-x) \rangle = (+1) | \psi(x) \rangle \,\,\, even\]

and

\[\hat{R} | \psi(x) \rangle = |\psi(-x) \rangle = (-1) | \psi(x) \rangle \,\,\, odd\]

**Circular Symmetry (\(\sigma\) symmetry)**

Circular symmetry is a type of continuous symmetry for a planar object that can be **rotated **by any arbitrary angle and map onto itself.

\[\hat{\Theta} | \psi(r, \,\theta, \,\phi) \rangle = \psi(r, \,\theta+\theta_1, \,\phi) \rangle = | \psi(r, \,\theta, \,\phi) \rangle \,\,\, \text{circular symmetric}\]

If a molecular orbital is symmetrical with respect to circular symmetry, it is called a sigma (\(\sigma\)) orbital.

**Reflection or 180° Rotation Symmetry (\(\pi\) symmetry)**

Circular symmetry is a type of continuous symmetry for a planar object that can be **rotated **by any arbitrary angle and map onto itself.

\[\hat{\Pi} | \psi(r, \,\theta, \,phi) \rangle = \psi(r, \,\theta+\pi, \,\phi) \rangle = (-1) | \psi(r, \,\theta, \,\phi) \rangle \,\,\, \text{circular symmetric}\]

A MO will have π symmetry if the orbital is **asymmetric **with respect to a \(\pi\) rotation (180°) about the internuclear axis. This means that rotation of the MO about the internuclear axis will result in a phase change.

If a molecular orbital is symmetrical with respect to circular symmetry, it is called a pi (\(\pi\)) orbital.

**Center of Inversion**

We can introduce another symmetry: the center of inversion (i.e., \(x, \,y, \,z \rightarrow -x, \,-y, \,-z\). Under this operator, we may have "gerade" and "ungerade" symmetry (or neither, but let's ignore that symmetry).

\[\hat{I} | \psi(x, \,y, \,z) \rangle = \psi(-x, \,-y, \,-z) \rangle = (+1) | \psi(x, \,y, \,z) \rangle \,\,\, gerade\]

and\[\hat{I} | \psi(x, \,y, \,z) \rangle = \psi(-x, \,-y, \,-z) \rangle = (-1) | \psi(x, \,y, \,z) \rangle \,\,\, ungerade\]

If a molecular orbital is symmetrical with respect to inversion, it has a subscript g (**gerade**, for even). If it is asymmetrical with respect to inversion, it is given a subscript u (**ungerade**, for uneven).

## Bond Order

Bond order is the net number of bonding electrons in a molecule. is defined as one-half the *net* number of bonding electrons:

\[ \text{bond order}=\dfrac{\text{number of bonding electrons} - \text{number of antibonding electrons}}{2} \label{9.8.1} \]

To calculate the bond order of H_{2}, we see from Figure 9.8.1 that the σ_{1}_{s} (bonding) molecular orbital contains two electrons, while the \( \sigma _{1s}^{\star } \) (antibonding) molecular orbital is empty. The bond order of H_{2} is therefore

\( \dfrac{2-0}{2}=1 \label{9.8.2} \)

This result corresponds to the single covalent bond predicted by Lewis dot symbols. Thus molecular orbital theory and the Lewis electron-pair approach agree that a single bond containing two electrons has a bond order of 1. Double and triple bonds contain four or six electrons, respectively, and correspond to bond orders of 2 and 3. We can use energy-level diagrams to describe the bonding in other pairs of atoms and ions where *n* = 1, such as the H_{2}^{+} ion, the He_{2}^{+} ion, and the He_{2} molecule. Again, we fill the lowest-energy molecular orbitals first while being sure not to violate the Pauli principle or Hund’s rule.

Species |
Electron Configuration |
Bond Order |
Bond Length (pm) |
Bond Energy (kJ/mol) |
---|---|---|---|---|

H_{2}^{+} |
\((σ_{1s})^1\) | 1/2 | 106 | 269 |

H_{2} |
\((σ_{1s})^2\) | 1 | 74 | 436 |

He_{2}^{+} |
\( \left (\sigma _{1s} \right )^{2}\left (\sigma _{1s}^{\star } \right )^{1} \) | 1/2 | 108 | 251 |

He_{2} |
\( \left (\sigma _{1s} \right )^{2}\left (\sigma _{1s}^{\star } \right )^{2} \) | 0 | 5,500 | \(4.6 \times 10^{−5}\) |

With a total of four valence electrons, both the σ_{1}_{s} bonding and \( \sigma _{1s}^{\star } \) antibonding orbitals must contain two electrons. This gives a \( \left (\sigma _{1s} \right )^{2}\left (\sigma _{1s}^{\star } \right )^{1} \) electron configuration, with a predicted bond order of (2 − 2) ÷ 2 = 0, which indicates that the He_{2} molecule has no net bond and is not a stable species (covalent bonding-wise at least).