Homonuclear Diatomic Molecules

Other homonuclear diatomic molecules (e.g., $ Li_2, \ O_2, \ He_2, \ F_2, \ N_2,$ ...) can be described according to the LCAO approach introduced with the study of the $ H_2^+$ molecule. A general feature of the LCAO method is that a combination of two atomic orbitals on different centers gives two molecular orbitals (MO). One of these molecular orbitals is called bonding and the other one is called antibonding. The bonding state is more stable than the system of infinitely separated atomic orbitals. On the other hand, the antibonding state is less stable than the isolated atomic orbitals. The description of the $ H_2^+$ molecule discussed in previous sections can be summarized by the following diagram:


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\linethickness{1pt}\thinlines\par\put(0,20){\circ...
...\text{1S}$}}
\put(110,1.5){\makebox(0,0)[t]{$\sigma_g \text{1S}$}}
\end{picture}

This diagram introduces the nomenclature of states of homonuclear diatomic molecules, which is determined by the following aspects:

1. Nature of the atomic orbitals in the linear combination (e.g., 1S orbitals in the study of the $ H_2^+$ molecules).

2. Eigenvalue of $ \hat{L}_z$, with $ z$ the internuclear axis (e.g., such eigenvalue is zero for the $ H_2^+$ molecule and, therefore, the orbital is called $ \sigma$).

3. Eigenvalue of the inversion operator through the center of the molecule (e.g., g when the eigenvalue is 1, and u when the eigenvalue is -1).

4. Stability with respect to the isolated atoms (e.g., an asterisk indicates that the state is unstable relative to the isolated atoms).

Other homonuclear diatomic molecules involve linear combinations of $ p$ orbitals. Such linear combinations give rise to $ \sigma$ type orbitals when there is no component of the angular momentum in the bond axis (e.g., we choose the bond axis to be the z axis). An example of such linear combination is represented by the following diagram:


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...akebox(0,0)[t]{$\oplus$}}
\put(97,1.5){\makebox(0,0)[t]{$\oplus$}}
\end{picture}

In order to classify molecular states according to eigenvalues of $ \hat{L}_z$, we make linear combinations of eigenfunctions of $ \hat{L}_z$ with common eigenvalues. There are four possible states:

m= 1:          $ 2P_{+1}(A) \pm 2P_{+1}(B), \qquad \overbrace{\pi_{u} 2P_{+1} }^{\oplus}, \ \overbrace{\pi_g^* 2P_{+1} }^{\ominus},$

m=-1:          $ 2P_{-1}(A) \pm 2P_{-1}(B), \qquad \pi_{u} 2P_{-1} , \ \pi_g^* 2P_{-1}.$

All of these linear combinations are $ \pi$ states, because $ \lambda =\vert m\vert = 1$ for all of them. In order to justify their symmetry properties with respect to inversion we analyze the following particular case,

$\displaystyle \pi_{u} 2 P_{+1} = 2P_{+1}(A) +2P_{+1}(B) = \frac{1}{8\sqrt{\pi}} (\frac{z}{a})^{5/2}(e^{i\phi_A}e^{-\frac{z r_A}{2a}}r_A$   sin$\displaystyle \theta_A +e^{i\phi_B}e^{-\frac{z r_B}{2a}}r_B$   sin$\displaystyle \theta_B),$

which is represented by the following diagram:


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...kebox(0,0)[t]{$\hspace{1.cm} z \hspace{.20cm} \text{nodal line}$}}
\end{picture}

This diagram shows that under inversion through the origin, coordinates are transformed as follows,

$ r_A \rightarrow r_B, \qquad \theta_A \rightarrow \theta_B, $

$ r_B \rightarrow r_A, \qquad \theta_B \rightarrow \theta_A, $

$ \phi_A = \phi_B = \phi,$

$ \phi \rightarrow \phi + \pi,$

$ e^{i(\phi+\pi)} =e^{i\phi}e^{i\pi} = -e^{i\phi}$, because $ e^{i\pi} = \underbrace{\text{Cos}\pi}_{-1} + i \underbrace{\text{Sin} \pi}_{0}.$

The states constructed with orbitals $ P_{-1}$ differ, relative to those constructed with orbitals $ p_{+1}$, only in the sign of phase $ \phi$ introduced by the following expression,

$\displaystyle \pi_g^* 2 P_{+1} = \frac{1}{8\sqrt{\pi}} (\frac{z}{a})^{5/2}e^{i\phi}( e^{-\frac{z r_A}{2a}}r_A$   sin$\displaystyle \theta_A + e^{-\frac{z r_B}{2a}}r_B$   sin$\displaystyle \theta_B).$

This function has a nodal xy plane and is described by the following diagram:


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\linethickness{1pt}\thinlines\par\put (0,0){\vect...
...1){40}{$.$}
\put(43,30){\makebox(0,0)[t]{$\text{nodal xy plane}$}}
\end{picture}

Since atomic orbitals $ 2p_x$, and $ 2p_y$ are linear combinations of atomic orbitals $ 2p_{+1}$ and $ 2p_{-1}$ molecular orbitals $ \pi_{u}2p_{+1}$ and $ \pi_{u}2p_{-1}$ can be combined to construct molecular orbitals $ \pi_{u}2p_x$, and $ \pi_{u}2p_y$ as follows,

$ \pi_{u}2p_x = 2p_x(A) + 2p_x(B),$

$ \pi_{u}2p_y = 2p_y(A) + 2p_y(B).$

Note, however, that molecular orbitals $ \pi_{u}2p_x$, and $ \pi_{u}2p_y$ are not eigenfunctions of $ \hat{L}_z$.

The order of increasing energy for homonuclear diatomic orbitals is described by the following diagram:


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\linethickness{1pt}\thinlines\put(20,-20){\line(1...
...(0,0)[t]{$\sigma_g2p$}}
\put(80,13){\makebox(0,0)[t]{$\pi_{u}2p$}}
\end{picture}

The electronic structure of homonuclear diatomic molecules can be approximated to zeroth order by filling up the unperturbed states according to the Pauli exclusion principle. However, we should always keep in mind that we are using the $ H_2^+$ molecular orbitals (i.e., the unperturbed states) and, therefore, we are neglecting the repulsive interaction between electrons.

This is the same kind of approximation implemented in the construction of zeroth order wave functions of atoms according to hydrogenlike atomic orbitals, where the repulsion energy between electrons was disregarded and the electronic configuration was constructed by filling up hydrogenlike atomic orbitals according to the Pauli exclusion principle.

Exercise 55:

(A) Predict the multiplicity of the ground state of $ O_2$.

(B) Show that the ground electronic state of $ C_2$ is a singlet.