Molecular Orbital Diagrams in LaTeX: TikZ, chemfig and modiagram

A molecular orbital (MO) diagram shows how the atomic orbitals of two atoms combine into the orbitals of a molecule, and where the electrons go. It predicts bond order, bond strength and magnetism, including facts a Lewis structure gets wrong, such as O₂ being paramagnetic. The first half of this guide explains how to read and fill an MO diagram, with worked diagrams for H₂, O₂, N₂ and CO. The second half shows how to draw them in LaTeX with the modiagram package, in plain TikZ, and with chemfig and mhchem for the structures and formulas beside them.

How to Read a Molecular Orbital Diagram

Energy runs upward. The atomic orbitals (AOs) of the two atoms are drawn as short horizontal lines on the left and right; the molecular orbitals (MOs) they form sit in the middle, joined to their parent AOs by thin correlation lines. Each pair of AOs that overlaps gives two MOs:

• a bonding orbital (σ or π), lower in energy than the AOs, where the wavefunctions add and electron density builds up between the nuclei;
• an antibonding orbital (σ* or π*), higher in energy, with a node between the nuclei.

σ orbitals come from head-on overlap (s with s, or p_z with p_z along the bond axis); π orbitals come from sideways overlap of p_x and p_y, so they come in degenerate pairs. The electrons are then filled in exactly as for atoms: lowest orbital first (Aufbau), at most two per orbital with opposite spins (Pauli), and one electron in each degenerate orbital before pairing (Hund). The result is read off with one formula:

bond order = (electrons in bonding MOs − electrons in antibonding MOs) / 2

Bond order 1 is a single bond, 2 a double bond, 3 a triple bond, 0 means the molecule does not form. Fractional values are allowed and show up in ions. If any MO holds unpaired electrons the species is paramagnetic; if all are paired it is diamagnetic.

H₂: The Simplest Diagram

Each hydrogen brings one 1s electron. The two 1s AOs combine into σ_1s (bonding) and σ*_1s (antibonding). Both electrons go into σ_1s with paired spins:

H2:   (σ1s)²           bond order = (2 − 0)/2 = 1,  diamagnetic
He2:  (σ1s)² (σ*1s)²   bond order = (2 − 2)/2 = 0,  no bond
He2+: (σ1s)² (σ*1s)¹   bond order = (2 − 1)/2 = ½,  observed as a weak ion

The same diagram explains why helium is monatomic: in He₂ the two extra electrons fill the antibonding orbital and cancel the bonding ones.

Second-Period Diatomics: O₂ and N₂

For Li₂ to Ne₂ the 1s electrons are core and are left out. The 2s AOs give σ_2s and σ*_2s; the three 2p AOs give σ_2p, a degenerate π_2p pair, a degenerate π*_2p pair and σ*_2p. There are two orderings of the 2p-derived levels, and this is the point most students lose marks on:

• O₂, F₂, Ne₂: σ_2s < σ*_2s < σ_2p < π_2p < π*_2p < σ*_2p
• Li₂ to N₂: σ_2s < σ*_2s < π_2p < σ_2p < π*_2p < σ*_2p

The difference is s–p mixing. In the lighter atoms the 2s and 2p orbitals are close in energy, so σ_2s and σ_2p (which have the same symmetry) interact: σ_2s is pushed down and σ_2p is pushed up, above the π_2p pair. From oxygen onwards the 2s–2p gap is too large for the mixing to matter and σ_2p drops back below π_2p.

O₂: bond order 2 and paramagnetic

Oxygen has six valence electrons, so O₂ has twelve to place. Fill in order: σ_2s (2), σ*_2s (2), σ_2p (2), π_2p (4). Two electrons are left for the two degenerate π*_2p orbitals, and Hund's rule puts one in each with parallel spins:

O2:  (σ2s)² (σ*2s)² (σ2p)² (π2p)⁴ (π*2p)¹(π*2p)¹
     bonding = 2 + 2 + 4 = 8,  antibonding = 2 + 2 = 4
     bond order = (8 − 4)/2 = 2        two unpaired electrons → paramagnetic

The Lewis structure O=O has every electron paired and predicts a diamagnetic molecule; liquid oxygen is in fact held between the poles of a magnet. This is the classic success of MO theory. The ions follow immediately: O₂⁺ loses a π* electron (bond order 2.5, shorter bond), superoxide O₂⁻ gains one (1.5) and peroxide O₂²⁻ gains two (1.0, diamagnetic).

N₂: bond order 3

Nitrogen has five valence electrons, ten in total for N₂, and the mixed ordering applies: σ_2s (2), σ*_2s (2), π_2p (4), σ_2p (2). Nothing reaches the antibonding 2p levels:

N2:  (σ2s)² (σ*2s)² (π2p)⁴ (σ2p)²
     bond order = (8 − 2)/2 = 3        all paired → diamagnetic
     HOMO = σ2p (not π2p), confirmed by photoelectron spectroscopy

Both orderings give bond order 3 for N₂, but only the mixed one gets the highest occupied orbital right, which matters for ionisation energies and for how N₂ binds to metals. B₂ is the other tell-tale case: with the mixed ordering its two 2p electrons go one each into the π_2p pair, so B₂ is paramagnetic, as observed; the unmixed ordering would wrongly pair them in σ_2p.

Heteronuclear Diatomics: CO and NO

When the two atoms differ, their AOs start at different energies: the more electronegative atom (O in CO) has lower-lying orbitals and is drawn lower on its side. The MOs are no longer symmetric combinations. Each bonding MO lies closer in energy to the lower AO and has more of its character, each antibonding MO is closer to the higher AO. CO has ten valence electrons, the same as N₂, and the same mixed ordering, so its bond order is 3. Its bonding orbitals are mostly oxygen in character, while the HOMO (the σ orbital at the top of the filled set) is concentrated on carbon as a lone pair, which is why CO binds to metals through the carbon atom. NO has eleven electrons, one of them in π*: bond order 2.5 and paramagnetic, and losing it to form NO⁺ strengthens the bond to order 3.

Drawing MO Diagrams with the modiagram Package

modiagram (CTAN, by Clemens Niederberger) draws the whole diagram from a description of the orbitals. It loads TikZ, expl3 and chemgreek itself. Inside the modiagram environment, \atom places the AOs of the left or right atom, \molecule places the MOs, and \EnergyAxis adds the arrow. Each orbital is given as {energy;electrons}, where the energy is a height in centimetres and the electrons are up, down or pair:

\documentclass{article}
\usepackage{modiagram}

\begin{document}
\begin{modiagram}[names, labels]
  \atom[H]{left}{1s = {0;up}}
  \atom[H]{right}{1s = {0;up}}
  \molecule[H$_2$]{1sMO = {0.75;pair}}
  \EnergyAxis
\end{modiagram}
\end{document}

For a molecular orbital the number is the splitting: 1sMO = {0.75;pair} puts σ_1s 0.75 cm below the lower 1s AO and σ*_1s 0.75 cm above the higher one, and fills σ with a pair. A second electron specification after a comma fills σ*: 1sMO = {0.75;pair,pair} is He₂. The names option prints the atom and molecule names from the optional arguments, labels writes 1s, σ_s and so on under each level, and the correlation lines are drawn automatically.

O₂ with modiagram

The 2p AOs take three electron specifications (p_x, p_y, p_z). The 2pMO key takes two energies, the σ splitting and the π splitting, then six electron slots in the order σ, π_y, π_z, π*_y, π*_z, σ*. modiagram takes x as the bond axis, so its labels read 2σ_x, 2π_y and 2π_z rather than σ_2p. A larger σ splitting than π splitting puts σ_2p below π_2p, which is the oxygen ordering:

\begin{modiagram}[names, labels]
  \atom[O]{left}{2s = {0;pair}, 2p = {4;pair,up,up}}
  \atom[O]{right}{2s = {0;pair}, 2p = {4;pair,up,up}}
  \molecule[O$_2$]{
    2sMO = {0.8;pair,pair},
    2pMO = {1.8,0.9;pair,pair,pair,up,up}   % σ, πy, πz, πy*, πz*, (σ* empty)
  }
  \EnergyAxis
\end{modiagram}

Trailing empty slots may be left out, so the two up electrons land in π*_y and π*_z and σ*_2p stays empty: exactly the paramagnetic configuration above.

N₂ and the mixed ordering

To put π_2p below σ_2p while keeping σ*_2p on top, give the bonding and antibonding splittings separately with a slash:

\begin{modiagram}[names, labels]
  \atom[N]{left}{2s = {0;pair}, 2p = {4;up,up,up}}
  \atom[N]{right}{2s = {0;pair}, 2p = {4;up,up,up}}
  \molecule[N$_2$]{
    2sMO = {0.8;pair,pair},
    2pMO = {0.6/2.2, 1.4/1.6;pair,pair,pair}   % σ/σ*, π/π* splittings
  }
  \EnergyAxis
\end{modiagram}

Here σ_2p is 0.6 cm below the 2p level and π_2p 1.4 cm below, so π comes first; σ*_2p is 2.2 cm above and π*_2p 1.6 cm above, so σ* is highest. The three pairs fill σ, π_y and π_z: ten electrons, bond order 3.

A heteronuclear diagram: CO

Give the two atoms different AO energies. modiagram measures each bonding MO down from the lower of the two AOs and each antibonding MO up from the higher one, which is the correct qualitative picture:

\begin{modiagram}[names, labels]
  \atom[C]{left}{2s = {0.8;pair}, 2p = {4.8;up,up}}
  \atom[O]{right}{2s = {0;pair}, 2p = {3.8;pair,up,up}}
  \molecule[CO]{
    2sMO = {0.6;pair,pair},
    2pMO = {0.7/2.0, 1.3/1.6;pair,pair,pair}
  }
  \EnergyAxis[title]
\end{modiagram}

Options worth knowing

The orbital node names used by color and \connect are 1sleft, 2pzright, 1sigma, 1sigma*, 2sigma, 2psigma, 2piy, 2piz* and so on. The package only knows 1s, 2s and 2p; for d orbitals or a ligand field diagram you need TikZ. Older documents use \begin{MOdiagram} with a capital MO: it still works but prints a deprecation warning, and \MOsetup has become \setmodiagram.

A Pure TikZ Molecular Orbital Diagram

Drawing the diagram directly in TikZ gives full control: each level is a short thick line, each electron a \uparrow or \downarrow node on it, and the correlation lines are dashed. Styles defined once in the picture options keep it tidy. This is the O₂ diagram again, with mhchem for the formulas:

\documentclass{standalone}
\usepackage{tikz}
\usepackage[version=4]{mhchem}

\begin{document}
\begin{tikzpicture}[
  level/.style={very thick},
  corr/.style={gray, dashed, thin},
  el/.style={font=\normalsize},
  lbl/.style={font=\small}
]
  % energy axis
  \draw[->] (-0.5,-0.3) -- (-0.5,5) node[above] {$E$};

  % atomic orbitals of the two oxygen atoms
  \foreach \x/\lab in {0/O, 6/O} {
    \draw[level] (\x,1) -- ++(0.8,0) node[midway, el] {$\uparrow\downarrow$};
    \node[lbl, below] at (\x+0.4,0.9) {$2s$};
    \draw[level] (\x-0.5,3.5) -- ++(0.8,0) node[midway, el] {$\uparrow\downarrow$};
    \draw[level] (\x+0.4,3.5) -- ++(0.8,0) node[midway, el] {$\uparrow$};
    \draw[level] (\x+1.3,3.5) -- ++(0.8,0) node[midway, el] {$\uparrow$};
    \node[lbl, below] at (\x+0.8,3.4) {$2p$};
    \node[above] at (\x+0.4,5) {\ce{\lab}};
  }

  % molecular orbitals
  \draw[level] (3,0.3) -- ++(0.8,0) node[midway, el] {$\uparrow\downarrow$};
  \node[lbl, right] at (3.9,0.3) {$\sigma_{2s}$};
  \draw[level] (3,1.7) -- ++(0.8,0) node[midway, el] {$\uparrow\downarrow$};
  \node[lbl, right] at (3.9,1.7) {$\sigma^*_{2s}$};
  \draw[level] (3,2.6) -- ++(0.8,0) node[midway, el] {$\uparrow\downarrow$};
  \node[lbl, right] at (3.9,2.6) {$\sigma_{2p}$};
  \draw[level] (2.5,3.1) -- ++(0.8,0) node[midway, el] {$\uparrow\downarrow$};
  \draw[level] (3.5,3.1) -- ++(0.8,0) node[midway, el] {$\uparrow\downarrow$};
  \node[lbl, right] at (4.4,3.1) {$\pi_{2p}$};
  \draw[level] (2.5,4.1) -- ++(0.8,0) node[midway, el] {$\uparrow$};
  \draw[level] (3.5,4.1) -- ++(0.8,0) node[midway, el] {$\uparrow$};
  \node[lbl, right] at (4.4,4.1) {$\pi^*_{2p}$};
  \draw[level] (3,4.8) -- ++(0.8,0);
  \node[lbl, right] at (3.9,4.8) {$\sigma^*_{2p}$};
  \node[above] at (3.4,5) {\ce{O2}};

  % correlation lines
  \draw[corr] (0.8,1) -- (3,0.3);    \draw[corr] (6,1) -- (3.8,0.3);
  \draw[corr] (0.8,1) -- (3,1.7);    \draw[corr] (6,1) -- (3.8,1.7);
  \draw[corr] (2.1,3.5) -- (3,2.6);  \draw[corr] (5.5,3.5) -- (3.8,2.6);
  \draw[corr] (2.1,3.5) -- (3,4.8);  \draw[corr] (5.5,3.5) -- (3.8,4.8);
\end{tikzpicture}
\end{document}

Do not name a style dash: TikZ already has a key of that name and stops with The key '/tikz/dash' requires a value. To extend the drawing, add levels and arrows in the same pattern; for a ligand field splitting (t_2g/e_g) the same lines-and-arrows approach works with five d levels on one side. If you would rather not place coordinates by hand, the TikZ generator produces a starting picture you can edit.

Lewis Structures with chemfig and Formulas with mhchem

An MO diagram is often shown next to the Lewis structure it improves on. chemfig draws the structure, with \charge adding lone pairs and single electrons around an atom as a list of angle=symbol pairs: \: is a lone pair, \. a single electron and \| a pair drawn as a line, with 0 = right, 90 = top, 180 = left and 270 = bottom. (Older tutorials use \lewis{2:6:,O}; chemfig 1.6 and later no longer define it, so on a current TeX Live it fails with Undefined control sequence.) mhchem's \ce formats formulas and charges, and works inside modiagram and TikZ labels:

\usepackage{chemfig}
\usepackage[version=4]{mhchem}
\usepackage{modiagram}

\ce{O2}, \ce{N2}, \ce{O2^-}, \ce{O2^2-}, \ce{He2^+}, \ce{2 H2 + O2 -> 2 H2O}

\chemfig{H-H}                                             % H2
\chemfig{\charge{90=\:,270=\:}{O}=\charge{90=\:,270=\:}{O}}   % O=O, two lone pairs each
\chemfig{\charge{180=\:}{N}~\charge{0=\:}{N}}                % N≡N (~ is a triple bond)
\chemfig{\charge{180=\:}{C}~\charge{0=\:}{O}}                % C≡O
\chemfig{\charge{45=\.}{O}}                                  % a single (radical) electron

\begin{figure}[h]
  \centering
  \begin{minipage}{0.3\textwidth}\centering
    \chemfig{\charge{90=\:,270=\:}{O}=\charge{90=\:,270=\:}{O}}
  \end{minipage}
  \begin{minipage}{0.6\textwidth}\centering
    \begin{modiagram}[labels, distance=2.5cm]
      \atom{left}{2p = {0;pair,up,up}}
      \atom{right}{2p = {0;pair,up,up}}
      \molecule{2pMO = {1,0.5;pair,pair,pair,up,up}}
    \end{modiagram}
  \end{minipage}
  \caption{Lewis structure and 2p molecular orbitals of \ce{O2}; bond order $(8-4)/2 = 2$.}
\end{figure}

Names in \molecule[...] and \atom[...] accept \ce too, so \molecule[\ce{O2}]{...} gives a properly formatted subscript.

Exporting the Figure

For a standalone image, put the diagram in a document with \documentclass{standalone} (as in the TikZ example) and compile; the PDF is cropped to the picture and can be dropped into another document with \includegraphics or converted to SVG for a web page with pdftocairo -svg or dvisvgm --pdf. In a long thesis, the external TikZ library (\usetikzlibrary{external}, \tikzexternalize) caches each picture as a PDF so it is not redrawn on every compile; it needs shell escape.

Common Errors and Fixes

Quick Reference

Frequently Asked Questions

How do you calculate bond order from a molecular orbital diagram?

Count the electrons in bonding orbitals, subtract the electrons in antibonding orbitals and divide by two: bond order = (bonding − antibonding) / 2. For O₂ that is (8 − 4) / 2 = 2, a double bond; for N₂ it is (8 − 2) / 2 = 3; for He₂ it is (2 − 2) / 2 = 0, which is why He₂ does not exist. A higher bond order means a shorter, stronger bond.

Why is O₂ paramagnetic according to MO theory?

O₂ has 12 valence electrons. After filling σ_2s, σ*_2s, σ_2p and the two π_2p orbitals, two electrons remain for the two degenerate π*_2p orbitals. By Hund's rule they occupy one each with parallel spins, so O₂ has two unpaired electrons and is attracted into a magnetic field. A Lewis structure, with all electrons paired, cannot explain this; the MO diagram does.

Why is the MO order different for N₂ and O₂?

For B₂, C₂ and N₂ the 2s and 2p atomic orbitals are close enough in energy that the σ_2s and σ_2p molecular orbitals mix (s–p mixing). The mixing pushes σ_2p up above the π_2p pair, so the order is σ_2s < σ*_2s < π_2p < σ_2p. In O₂ and F₂ the 2s–2p gap is larger, the mixing is weak, and σ_2p lies below π_2p. The two orderings give the same bond order for O₂ and N₂ but change which orbital is the HOMO.

What LaTeX package draws MO diagrams?

modiagram is the dedicated package: \usepackage{modiagram} gives you a modiagram environment with \atom for the atomic orbitals, \molecule for the molecular orbitals and \EnergyAxis, and it draws the levels, electrons and correlation lines for 1s, 2s and 2p for you. It is built on TikZ, so for anything it does not cover (d orbitals, ligand field diagrams, more than two fragments) you draw the diagram directly in TikZ.

How do I write O₂, N₂ or O₂⁻ with proper formatting in LaTeX?

Load mhchem with \usepackage[version=4]{mhchem} and write \ce{O2}, \ce{N2}, \ce{O2^-} and \ce{CO}. \ce sets the subscripts and charges upright and in the right position, works in text and in math mode, and also handles reaction arrows such as \ce{2 H2 + O2 -> 2 H2O}. It can be used inside modiagram and TikZ labels too.

Next Steps

Diagram drawn? These guides cover the surrounding figure work:

• TikZ Diagrams in LaTeX – Coordinates, nodes, styles and \foreach, everything the TikZ version above relies on
• Plots with pgfplots – Potential energy curves and spectra to put next to the MO diagram
• Greek Letters and Math Symbols – σ, π and the arrows used for electrons
• Subscripts and Superscripts – Formulas and orbital labels without mhchem
• LaTeX Tables – Tabulate bond orders and configurations
• Learn LaTeX – Complete beginner tutorial covering all the basics

Free LaTeX Tools

• TikZ Generator – Generate a TikZ picture to start your diagram from
• LaTeX Preview – Compile modiagram and chemfig code with a live PDF preview
• LaTeX Symbols Reference – Arrows and Greek letters for labels