Units, reaction energies and errors
Guide 5 · the arithmetic behind every experiment
From total energies to a reaction energy
Q-Chem gives total electronic energies in hartree (Eh), the atomic unit of energy. For any reaction, the reaction energy is a single sum over all species i taking part in it:
\[ \Delta E = \sum_i \nu_i\,E_i \]
where Ei is the total energy of species i and νi is its stoichiometric coefficient.
Each stoichiometric coefficient includes its sign: νi > 0 for products and νi < 0 for reactants. With this convention you never write “products minus reactants” separately. The minus sign is already inside the reactants’ coefficients, so ΔE is always just Σ νi Ei.
For example, for 2 O3 → 3 O2 the coefficients are ν(O3) = −2 and ν(O2) = +3, so
\[ \Delta E = (+3)\,E(\mathrm{O_2}) + (-2)\,E(\mathrm{O_3}). \]
The same convention is used throughout this site: the results table checks your ΔE with exactly this sum.
Then convert to kcal/mol:
\[ 1\ E_\text{h} = 627.5095\ \text{kcal/mol}. \]
For 2 O3 → 3 O2 (ν = +3 for O2, −2 for O3): ΔE = (+3) E(O2) + (−2) E(O3). If E(O2) = −150.300000 Eh and E(O3) = −225.350000 Eh (made-up numbers), then ΔE = −450.900000 + 450.700000 = −0.200000 Eh = −125.50 kcal/mol.
Keep all the digits of the total energies until the very end. Total energies are hundreds of hartree, while reaction energies are often a few kcal/mol (thousandths of a hartree).
Errors
For each method we compare the computed reaction energy with a high-level reference value:
- Signed error: ε = ΔEcomp − ΔEref. Positive means the method overestimates the reaction energy.
- Absolute error: |ε|. It tells you how far off the method is, regardless of direction.
Over a set of N reactions:
\[ \text{MSE} = \frac{1}{N}\sum_k \varepsilon_k \]
\[ \text{MAE} = \frac{1}{N}\sum_k |\varepsilon_k| \]
The mean signed error (MSE) reveals systematic bias: does the method always overshoot? The mean absolute error (MAE) measures overall accuracy. A method can have a small MSE but a large MAE, with big errors in both directions that cancel on average. That is why we look at both.
An error of about 1 kcal/mol is traditionally called “chemical accuracy”: roughly the precision needed to predict reaction rates and equilibria in a useful way. Keep it in mind as a yardstick when you judge your results.
How the website checks your arithmetic
When you enter a reaction energy, the page recomputes it from the total energies you entered and compares the two (allowing for rounding). Green means they agree; red means something is off (sign, stoichiometry, or unit conversion). The page never shows you the correct value. It only tells you whether your own numbers are consistent.