6.Energies & Thermochemistry

To be improved further

Let's say we want to calculate the reaction:

A + B -> C

where A, B and C can be atoms, molecules or molecular fragments. The first thing to make sure is that the equation is balanced, the same number of atoms must be present on both sides and the charge must be the same on both sides. N₂ +H₂ -> NH₃ is not a balanced equation for example while N₂ + 3H₂ -> 2NH₃ is.

Note that the discussion here is valid for any reaction. If you are calculating a conformational energy or isomerization energy instead you would be calculating the reaction A -> C instead (and you can then leave out all terms involving B in equations below). If you have A + B -> C + D then you must add terms for D.

Calculating relative electronic energies and ZPE-corrected energies

We start out by solving the electronic problem for each species (A, B or C; we'll use X as a general label ) separately.

This gives us the ∆E^el reaction energy (reaction energy at 0 Kelvin but neglects zero-point motion):

∆E^el = E^el_C - E^el_A - E^el_B

The E^el_X energies you get from the ORCA output in the line: FINAL SINGLE POINT ENERGY -XX.XXXXX

This energy is called total energy, potential energy or electronic energy (even though includes nuclear repulsion also).

For example: grep FINAL outputfile

The energy will be in hartrees. It's common practice to add/subtract the total energies in hartree units but then convert the energy difference (here ∆E^el ) immediately to kcal/mol or eV by multiplying by the appropriate constant.

If you ran a geometry optimization you want to make sure that the job finished properly and then since the line above is printed for each step, to take the energy from the last line

This would typically be called an electronic energy difference or potential energy difference. Even though it's strictly not comparable to any experiment (as it neglects zero-point motion and it's at 0 Kelvin) it's still often reported as it is the most important term and usually a very good approximation to the enthalpy difference (less for free energy because of entropy).

Next we calculate the Zero-point energy (ZPE) for each species (by harmonic vibrational frequencies) and get the ZPE correction:

The ZPE energies you get from the ORCA frequency output. There will be a line that in the ORCA frequency output:

Zero point energy ... => ZPE in hartree units

∆ZPE_corr = ZPE_C - ZPE_A - ZPE_B

We can then add : ∆E^el + ∆ZPE_corr = ∆E⁰ (zero-point corrected reaction energy at 0 K)

We could also do it in a different way: E^el_A + ZPE_A = E⁰_A and E^el_B + ZPE_B = E⁰_B and E^el_C + ZPE_C = E⁰_C

Then: ∆E⁰ = E⁰_C - E⁰_A - E⁰_B

In the PES exercises you are asked to calculate electronic energies and zero-point energies for different isomers and then compare ∆E⁰ to ∆E^el values.

Calculating Enthalpies and Free energies at room temperature

Then we calculate the thermal energy correction at e.g. room temperature (E²⁹⁸_corr) for each species and calculate the ZPE correction:

∆E²⁹⁸_corr = E²⁹⁸_C,corr - E²⁹⁸_A,corr - E²⁹⁸_B,corr (E²⁹⁸_X,corr = E²⁹⁸_{X,vibrational +} E²⁹⁸_{X,rotational +} E²⁹⁸_{X,translational )}

and then add : ∆E^el + ∆ZPE_corr + ∆E²⁹⁸_corr = ∆E²⁹⁸ (zero-point corrected reaction energy at 298 K)

Add a k_BT term to get enthalpy:

∆E^el + ∆ZPE_corr + ∆E²⁹⁸_corr + k_BT = ∆H²⁹⁸ (enthalpy difference at 298 K)

Calculate entropic term TS = T( S_el + S_vib + S_rot + S_trans )

and calculate Gibbs free energy difference:

∆E^el + ∆ZPE_corr + ∆E²⁹⁸_corr + k_BT - TS= ∆G²⁹⁸ (free energy difference at 298 K)

In the ORCA frequency output you can find all of these terms:

"Final Gibbs free enthalpy ..." => G²⁹⁸

"Final entropy term ..." => TS term at 298 K

"Total Enthalpy ..." => H²⁹⁸ term

G-E(el) ... X.XXXXX Eh => (G²⁹⁸ - E^el) Handy free-energy correction term.

So all the thermochemical quantities here you can get from a frequency calculation in ORCA.

Most of these do not come from quantum mechanics but basic thermodynamics and statistical mechanics but are included in the output for convenience.

These quantities are approximate as we assume that the molecular vibrations can be described well by the harmonic approximation and that the rotations can be described well by the rigid-rotor approximation.