The total energy is obtained from a model in which the nuclei are frozen in place. Therefore, the total energy fails to include the kinetic energy of the nuclei and cannot be compared directly to the standard enthalpy, H, at 298 K, 1 atm. This post describes the procedure for obtaining the missing kinetic energy (and the standard gas-phase enthalpy) with SPARTAN'10 using the EDF2/6-31G* method.
Calculations for methanol, CH3OH, are used to illustrate these method. To follow this example on your own, build methanol and calculate its Equilibrium Geometry at Ground State with Density Functional EDF2 6-31G* in Vacuum. Set Total Charge: Neutral and Multiplicity: Singlet and check the boxes next to IR and Thermodynamics.
The missing kinetic energy is routinely broken down into several components:
- ZPE or zero-point energy. This is kinetic energy contributed by the nuclei at 0 K and comes from the vibrational motions of the nuclei.
- Increase in vibration energy from 0 to 298 K. As the temperature rises, excited vibration states get populated and the kinetic energy due to vibration rises.
- Rotation energy at 298 K. The molecule is treated as a rigid body that rotates about its center-of-mass. If we assume that the molecule behaves like an ideal gas, this term is 1.5RT (or just RT if the molecule is linear).
- Translation energy at 298 K. The molecule is treated as a point mass concentrated at the center-of-mass. If we assume that the molecule behaves like an ideal gas, this term is 1.5RT.
- Pressure-volume work at 298 K. If we assume that the molecule behaves like an ideal gas, this term is nRT (or just RT for 1 mole of gas).
Looking back at these components, we see that the first 2 require information about vibration frequencies, the last 4depend on temperature, and the very last depends on the gas pressure (or quantity of gas). In terms of magnitude, ZPE is usually much larger than any of the others. For example, for the EDF2/6-31G* model of methanol, we find:
- ZPE 135.27 (all kinetic energies in kJ/mol)
- Increase in vibration energy 1.21
- Rotation 3.72
- Translation 3.72
- PV work 2.48
The total energy and standard gas-phase enthalpy are related by the following equation (values for methanol): enthalpy (-115.582297 au) = total energy (-115.638056 au) + contributions (146.3953 = 135.27 + 1.21 + 3.72 + 3.72 + 2.48 kJ/mol).
Finding SPARTAN's energy values. The total energy, ZPE, and enthalpy, are reported in the Molecular Properties window and are easily found. Click Display: Properties to open the window. The total energy is reported in the Molecule tab under Energy. The zero-point energy and enthalpy are reported in the Thermodynamics tab under ZPE and Ho, respectively.
The individual contributions to the kinetic energy are harder to locate. They are all reported in the Output window and you must search through the output to find them. First, click Display: Output to open the window. Next, scroll down to the section headed by Standard Thermodynamic quantities at 298.15 K and 1.00 atm. The four temperature-dependent contributions are reported in a table under the column labeled Enthalpy.
Manual calculation of enthalpy. Although SPARTAN reports the enthalpy in the Molecular Properties window, it can also be obtained by manual calculation. Simply add the total energy (-115.638056 au) to the five kinetic energy contributions (135.27+1.21+3.72+3.72+2.48 kJ/mol) to obtain the enthalpy (-115.582297 au). Take care to combine au and kJ/mol correctly (1 au = 2625.5 kJ/mol). Instead of locating and combining the five kinetic energies, you can find and use their sum (146.3953 kJ/mol) in two places in the Output window: scroll down to the same section and look under the Enthalpy column (value is labeled Totals) or look in a second table titled, Vibrational (v) Corrections, under Temp. Correction Hv.
Frequency scaling. Unfortunately, most methods for computing ZPE (and any other vibration energies) slightly overestimate the magnitude of these energies. This is partly due to the fact that computational models assume harmonic vibrations whereas real molecular vibrations are anharmonic. Therefore, it is routine practice to multiply the vibration energies, ZPE + the increase in vibration energy, by a scaling factor before adding them to the total energy. In the case of EDF2/6-31G* models, a scaling factor of 0.9595 has been recommended by Merrick, Moran and Radom (2007).
This factor can be applied to SPARTAN'10 models in several ways.
The easiest method is to type FREQSCALE=0.9595 in the Options text box in the Setup: Calculations dialog window. If this is done, the Molecule Properties window displays all of the appropriate quantities and no further work is required:
- the correct total energy (identical to the unscaled value) -115.638056 au
- the scaled ZPE 129.79 kJ/mol (= 135.27 * 0.9595)
- the scaled enthalpy of -115.584348 au ( = total energy + [135.27+1.21]*0.9595 + 3.72 + 3.72 + 2.48]
- A dismal alternative is to take the contributions reported by SPARTAN'10, scale the two that derive from vibrations, and then combine all five with the total energy as shown above.