Molecular terms

Molecular terms refer to MO-LCAO theory of molecular orbitals.

Homonuclear molecules

A molecule is said to be homonuclear if it's made of two identical atoms. For this kind of molecules there are special symmetries in the hamiltonian:

• symmetry with respect to the center : $\left[H,P\right]=0$$\left[H,P\right]=0$
• symmetry with respect to the bound axis, which we choose to be the z axis: $\left[H,L_{z}\right]=0$$\left[H,L_{z}\right]=0$
• symmetry with respect to every plane orthogonal to the bound axis : $\left[H,P_{\pi }\right]=\left[H,P_{xy}\right]=0$$\left[H,P_{\pi}\right]=\left[H,P_{xy}\right]=0$. However $\left[L_{z},P_{\pi }\right]\neq 0$$\left[L_z,P_{\pi}\right]\neq 0$ and we can't construct a C.S.C.O of eigenstates common to ${H,L_{z},P,P_{\pi }}$${H,L_z,P,P_{\pi}}$. We have to choose ${H,L_{z},P}$${H,L_z,P}$ or ${H,L_{z},P_{\pi }}$${H,L_z,P_{\pi}}$, but since ${L_{z},P_{\pi }}=L_{z}P_{\pi }+P_{\pi }L_{z}=0\quad \left(L_{z}{\mbox{anticommutes with}}P_{\pi }\right)$${L_z,P_{\pi}} = L_zP_{\pi} + P_{\pi}L_z = 0 \quad \left(L_z \mbox{anticommutes with} P_{\pi}\right)$ the eigenstates of $L_{z}$$L_z$ with $m=0$$m=0$ are also eigenstates of $P_{\pi }$$P_{\pi}$ and we can classify these states with an additional index

A molecular orbital is a linear combination of atomic orbitals of atom $\left(1\right)$$\left(1\right)$ with that of atom $\left(2\right)$$\left(2\right)$ and is described by a greek small letter identifying m-value plus a subscript for the parity, $g$$g$ for even orbitals or $u$$u$ for odd orbitals.

$${\begin{aligned}m=0\rightarrow \sigma \\m=\pm 1\rightarrow \pi \\m=\pm 2\rightarrow \delta \\\dots \\{\mbox{even orbitals}}\quad \left[\sigma ,\pi ,\dots \right]_{g}\\{\mbox{odd orbitals}}\quad \left[\sigma ,\pi ,\dots \right]_{u}\end{aligned}}$$

$${\begin{aligned}m=0\rightarrow \sigma \\m=\pm 1\rightarrow \pi \\m=\pm 2\rightarrow \delta \\\dots \\{\mbox{even orbitals}}\quad \left[\sigma ,\pi ,\dots \right]_{g}\\{\mbox{odd orbitals}}\quad \left[\sigma ,\pi ,\dots \right]_{u}\end{aligned}}$$

Furthermore we have to keep in mind that every pair of atomic orbitals gives rise, when combined, to a bonding orbital and an antibonding orbital (denoted with a superscript $*$$*$). Once we have found molecular orbitals we fill them with the electrons. The molecular terms are then given by the assignment of the total angular momentum projection $M_{z}$$M_z$,a superscript on the left for spin multiplicity and the subscript for the overall parity, plus the $\pm$$\pm$ superscript for $\Sigma$$\Sigma$ terms.

$$^{2s+1}\left[\Sigma ,\Pi ,\Delta ,\Phi ,\dots \right]_{g/u}$$

$$^{2s+1}\left[\Sigma,\Pi,\Delta,\Phi,\dots\right]_{g/u}$$

Let us discuss an example, the $C_{2}$$C_{2}$ molecule.

• Write the expression for each atom electronic configuration: $C_{\mbox{ground state}}=[He]2s^{2}2p^{2}$$C_{\mbox{ground state}} = [He]2s^22p^2$
• Since $[H,P]=0$$[H,P]=0$ and parity is governed by $l$$l$-values we can combine atomic orbitals with $\mathbf {l=l'}$$\mathbf{l=l'}$. Furthermore since $[H,L_{z}]=0$$[H,L_z] =0$ we must have $\mathbf {m=m'}$$\mathbf{m=m'}$. Thus we are allowed to combine $2s_{C_{1}}$$2s_{C_1}$ with $2s_{C_{2}}$$2s_{C_2}$ and $2p_{C_{1}}$$2p_{C_1}$ with $2p_{C_{2}}$$2p_{C_2}$.
  $${\begin{aligned}2s_{1}+2s_{2}\rightarrow 1\sigma _{g}\quad 1\sigma _{u}^{*}\\2p_{z1}+2p_{z2}\rightarrow 2\sigma _{g}\quad 2\sigma _{u}^{*}\end{aligned}}$$

  $${\begin{aligned}2s_{1}+2s_{2}\rightarrow 1\sigma _{g}\quad 1\sigma _{u}^{*}\\2p_{z1}+2p_{z2}\rightarrow 2\sigma _{g}\quad 2\sigma _{u}^{*}\end{aligned}}$$
  $${\begin{aligned}\left({\mbox{linear combination of}}p_{x1}{\mbox{and}}p_{y1}\right)+\left({\mbox{linear combination of}}p_{x2}{\mbox{and}}p_{y2}\right)\rightarrow 1\pi _{u}\quad 1\pi _{g}^{*}\end{aligned}}$$

  $${\begin{aligned}\left({\mbox{linear combination of}}p_{x1}{\mbox{and}}p_{y1}\right)+\left({\mbox{linear combination of}}p_{x2}{\mbox{and}}p_{y2}\right)\rightarrow 1\pi _{u}\quad 1\pi _{g}^{*}\end{aligned}}$$
  $$\pi _{u}\underbrace {\rightarrow } _{\mbox{composed of}}\pi _{u}^{+}\quad m=1\quad +\quad \pi _{u}^{-}\quad m=-1$$

  $$\pi_u \underbrace{\rightarrow}_{\mbox{composed of}} \pi_u^+ \quad m=1 \quad + \quad \pi_u^- \quad m= -1$$
• Put the electrons in the so-found molecular orbitals starting from the lowest energy orbital. There isn't a general rule which regulates the energy scale of molecular orbitals. However for homonuclear molecules experiments show that:

• 

  energy diagram of $B_{2}$$B_2$. The order of filling is the same for $C_{2},N_{2}$$C_2,N_2$

• 

  energy diagram of $O_{2}$$O_2$. The order of filling is the same for $F_{2}$$F_2$

In our example the electronic configuration of $C_{2}$$C_2$ is then

$$[He]1\sigma _{g}^{2}1\sigma _{u}^{2}1\pi _{u}^{4}$$

$$[He]1\sigma_g^2 1\sigma_u^2 1\pi_u^4$$

• Closed shells do not contribute to the molecular terms, so consider just not completely full shells. Find the possible values of $M=\sum {m_{i}}$$M = \sum{m_i}$ and $s$$s$ looking at the allowed configurations.For each configuration find the corresponding $g/u$$g/u$ and,eventually,$\pm$$\pm$ indexes with the help of the following rules:
  $${\begin{aligned}g\cdot u=u\cdot g=u\\g\cdot g=u\cdot u=g\\+\cdot -=-\cdot +=-\\+\cdot +=-\cdot -=+\end{aligned}}$$

  $${\begin{aligned}g\cdot u=u\cdot g=u\\g\cdot g=u\cdot u=g\\+\cdot -=-\cdot +=-\\+\cdot +=-\cdot -=+\end{aligned}}$$
  . The molecular term for $C_{2}$$C_2$ ground state is then $^{1}\Sigma _{g}^{+}$$^1\Sigma_g^+$.

Let us discuss the problem of finding molecular terms for the excited configuration of $C_{2}^{+}$$C_2^+$ : $1\sigma _{g}^{2}1\sigma _{u}^{2}1\pi _{u}^{2}1\pi _{g}^{1}$$1\sigma_g^2 1\sigma_u^2 1\pi_u^2 1\pi_g^1$. We neglect the $\sigma$$\sigma$'s orbitals since they're full. Possible configurations are:

So that we have for $\pi _{u}$$\pi_u$ the terms (following the order of the image): $^{3}\Sigma _{g}^{+}\quad ^{1}\Sigma _{g}^{+}\quad ^{1}\Delta _{g}$$^3\Sigma_g^+ \quad ^1\Sigma_g^+ \quad ^1\Delta_g$. For $\pi _{g}^{*}$$\pi_g^*$ there is just one term that is $^{2}\Pi _{g}$$^2\Pi_g$. Now we add each term corresponding to $\pi _{u}$$\pi_u$ possible configurations to the term related to $\pi _{g}^{*}$$\pi_g^*$, and we obtain:

$${\begin{aligned}^{2}\Pi _{g}+^{3}\Sigma _{g}^{+}\rightarrow \underbrace {^{4}\Pi _{g}} _{8{\mbox{states}}}\,\underbrace {^{2}\Pi _{g}} _{4{\mbox{states}}}\\^{2}\Pi _{g}+^{1}\Sigma _{g}^{+}\rightarrow \underbrace {^{2}\Pi _{g}} _{4{\mbox{states}}}\\^{2}\Pi _{g}+^{1}\Delta _{g}\rightarrow \underbrace {^{2}\Phi _{g}} _{4{\mbox{states}}},\underbrace {^{2}\Pi _{g}} _{4{\mbox{states}}}\end{aligned}}$$

$${\begin{aligned}^{2}\Pi _{g}+^{3}\Sigma _{g}^{+}\rightarrow \underbrace {^{4}\Pi _{g}} _{8{\mbox{states}}}\,\underbrace {^{2}\Pi _{g}} _{4{\mbox{states}}}\\^{2}\Pi _{g}+^{1}\Sigma _{g}^{+}\rightarrow \underbrace {^{2}\Pi _{g}} _{4{\mbox{states}}}\\^{2}\Pi _{g}+^{1}\Delta _{g}\rightarrow \underbrace {^{2}\Phi _{g}} _{4{\mbox{states}}},\underbrace {^{2}\Pi _{g}} _{4{\mbox{states}}}\end{aligned}}$$

The number of states is easily found considering that each term (except $\Sigma$$\Sigma$'s) respresents 2 possible values of $M$$M$ so that the total states are $2\cdot (2s+1)$$2\cdot(2s+1)$ for each non-$\Sigma$$\Sigma$ term and $2s+1$$2s+1$ for $\Sigma$$\Sigma$'s terms Let us verify our result. The configuration $1\pi _{u}^{2}1\pi _{g}^{1}$$1\pi_u^2 1\pi_g^1$ corresponds to ${\binom {4}{2}}\cdot {\binom {4}{1}}=6\cdot 4=24$$\binom{4}{2}\cdot\binom{4}{1}=6\cdot 4 = 24$ states, which is just the number found.

Non-homonuclear molecules

A molecule is said to be non-homonuclear if it's made of two atoms of different species. For this kind of molecules there are less symmetries than in homonuclear molecules:

• There isn't symmetry with respect to the center : $\left[H,P\right]\neq 0$$\left[H,P\right]\neq 0$
• The symmetry with respect to the bound axis, which we choose to be the z axis,is still present: $\left[H,L_{z}\right]=0$$\left[H,L_{z}\right]=0$
• The symmetry with respect to every plane orthogonal to the bound axis is still present and the same considerations hold.

The procedure for finding molecular orbitals and molecular terms is almost the same sketched for homonuclear molecules. The main difference is:

• Since symmetry with respect to the center has been broken there is no supscript $g/u$$g/u$ neither in molecular orbitals nor in molecular terms. Furthermore we can combine also atomic orbitals with $\mathbf {l} \neq \mathbf {l'}$$\mathbf{l}\neq\mathbf{l'}$ if they have $\mathbf {m} =\mathbf {m'}$$\mathbf{m}=\mathbf{m'}$ For example $2s$$2s$ and $2p_{z}$$2p_z$ can mix together. Because of the energy difference between different n shells and in order to simplify calculations we will not assume $2s-2p$$2s - 2p$ (and other analogue pairs) mixing to be possible if one of these orbitals is completely full.