What Is the Difference Between Single-Point Energy and Dimerization Energy?

A single-point energy and a dimerization energy are not competing types of calculation. The first describes an energy evaluation at a specified geometry; the second describes an energy change between chemical species. You can calculate a dimerization energy from several single-point calculations, or from energies produced during geometry optimizations.

For the reaction $A + B \rightarrow AB$, the usual electronic dimerization energy is the energy of the dimer minus the energies of the separated monomers. The sign, monomer geometries, and treatment of basis-set error all need to be clear before comparing reported values.

What a single-point energy means

A single-point calculation evaluates the electronic energy for the nuclear coordinates supplied to the program. It does not, by itself, move the nuclei to find a minimum-energy geometry. The calculation also uses a specified electronic-structure method—such as a density functional—and a basis set.

For example, after optimizing a water molecule’s geometry, you could run a higher-level calculation on those same coordinates. The resulting energy is a single-point energy at the optimized geometry. If you instead change the coordinates and run the calculation again, that gives another single-point energy, even if the molecule and method are unchanged.

A geometry optimization typically requires many energy and gradient evaluations as it searches for a stationary point. The energy at the final geometry may be reported as the optimized energy, but a separate calculation at that geometry is also often called a single-point calculation.

What dimerization energy means

For two monomers forming a dimer, the electronic energy change is commonly defined as

$$\Delta E_{\mathrm{dim}} = E_{AB} - E_A - E_B.$$

Here, $E_{AB}$ is the energy of the dimer and $E_A$ and $E_B$ are the energies of the isolated monomers. Under this convention, a negative value means the dimer has lower electronic energy than the separated monomers, so dimer formation is favorable in electronic energy.

Some sources instead report a positive binding energy or interaction energy, defined with the reverse sign:

$$E_{\mathrm{bind}} = E_A + E_B - E_{AB} = -\Delta E_{\mathrm{dim}}.$$

Always check the definition before comparing values. The word “dimerization” can also refer to a chemical reaction that changes bonds or composition; the same energy-difference principle applies, but the stoichiometry and reference species must match the stated reaction.

A worked example

Suppose a calculation gives these electronic energies, all in Hartree and at the same method and basis set:

• Dimer $AB$: $E_{AB} = -200.010$ Hartree
• Monomer $A$: $E_A = -100.000$ Hartree
• Monomer $B$: $E_B = -100.000$ Hartree

Then

$$\begin{aligned} \Delta E_{\mathrm{dim}} &= -200.010 - (-100.000) - (-100.000) \\ &= -0.010\ \text{Hartree}. \end{aligned}$$

Using $1$ Hartree $\approx 2625.5$ kJ mol$^{-1}$, this is approximately $-26.3$ kJ mol$^{-1}$. With the positive binding-energy convention, the same result is $+26.3$ kJ mol$^{-1}$. These figures illustrate the arithmetic; meaningful results require consistent geometries and appropriate treatment of computational errors.

Which monomer geometries should you use?

There are two common questions, and they do not generally have the same answer.

Overall energy change from relaxed monomers: Optimize the isolated monomers and the dimer, then subtract the monomer energies from the dimer energy. This compares the relaxed products and reactants. The value includes any energy cost of distorting the monomers into the shapes they adopt in the dimer.

Interaction energy at the dimer geometry: Keep each monomer at the geometry it has inside the dimer, calculate its energy separately, and subtract those fragment energies from the dimer energy. This isolates the interaction between fragments at that geometry, but excludes the cost of distorting the isolated monomers.

The relationship can be written as

$$\Delta E_{\mathrm{dim}} = E_{\mathrm{interaction}} + E_{\mathrm{deformation}},$$

where the deformation term is the energy required to change the isolated monomers from their relaxed geometries to their geometries in the dimer. For flexible molecules, this contribution can matter substantially.

Basis-set superposition error

In a finite basis set, each monomer in the dimer can use basis functions centered on the other monomer. The dimer may therefore appear artificially stabilized compared with calculations on the isolated monomers in their own smaller sets of basis functions. This is called basis-set superposition error (BSSE).

The counterpoise method estimates this error by calculating each isolated fragment in the full dimer basis, treating the other fragment’s basis functions as “ghost” functions. For a dimer at fixed geometry, the counterpoise-corrected interaction energy is commonly expressed as

$$E_{AB}^{AB} - E_A^{AB} - E_B^{AB},$$

where the superscript indicates the basis used. If you want an overall dimerization energy from relaxed monomers, include the monomer deformation contribution as well and state the correction protocol. Larger, well-converged basis sets generally reduce BSSE, though they do not make geometry choices irrelevant.

Common mistakes

• Calling every energy difference a single-point energy: A single-point energy is one energy evaluation; dimerization energy is a difference between species.
• Mixing geometries without saying so: A relaxed-monomer reference and a frozen-fragment reference answer different physical questions.
• Comparing incompatible calculations: Use the same electronic-structure method, compatible basis sets, charge and spin states, and energy units for all species.
• Assuming electronic energy is a free energy: The difference above is not automatically a reaction enthalpy or Gibbs free energy. Zero-point, thermal, entropic, and solvation contributions may be needed for those quantities.
• Ignoring the sign convention: A negative dimerization energy and a positive binding energy can describe the same stabilization.

When reporting a value, specify the reaction, sign convention, geometries, method and basis set, and whether BSSE or vibrational and thermal corrections were applied. That makes clear what the number measures and allows others to reproduce or compare it.