Gaussian Thermochemistry

Originally by Prof. Hendrik Zipse, this page has been modified by John Keller, University of Alaska Fairbanks (2026). Most of HZ's text is used, but here the δEvib equation graphic is corrected, and a somewhat more complex example is shown: water dimer (H2O)2. Unlike Zipse's H2 molecule, the water dimer has several low frequency vibrations that affect the thermochemical properties.


A detailed account of how thermochemical values are calculated in Gaussian by Joseph W. Ochterski (Gaussian Inc.) is available here as a pdf file.

A number of constants and conversion factors helpful in thermochemical calculations can be found here.

The thermochemical analysis in Gaussian is based on the harmonic vibrational frequencies calculated in OPT FREQ or FREQ jobs. The thermochemical values are calculated at 298.15K and 1.0 atm unless otherwise stated in the input file.
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The Gaussian input file for this calculation is as follows:

#N B3LYP/6-311++G(2d,p) OPT=vtight FREQ EmpiricalDispersion=GD3 NOSYMMETRY

2wat-6311ppG2dp-b3lyp-d3-opfr

0 1
O  0.00000000  0.00000000  0.00000000
H -0.43754800 -0.39439800  0.76183900
H -0.43000000  0.85290100 -0.12257500
O  2.83628100  0.06678700  0.14314900
H  1.86727000  0.03697100  0.08186300
 H  3.15093    -0.47229    -0.5878

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The complete Gaussian output file
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The key thermochemical properties are as follows:
Etot        the total electronic energy Etot as calculated by a given theoretical model. This is the energy of the molecular system under study relative to separate nuclei and electrons. Remember that semiempirical methods such as PM7 use a different point of reference and produce heats of formation. For the water dimer (H2O)2 calculated at the B3LYP level with a medium- sized basis set 6-311++G(2d,p), and with Grimme's D3 dispersion correction, the total energy of the optimized system appears at line 3804 in the output before any other thermochemical data. The atomic unit (a.u.) of energy is the Hartree, which = 627.5095 kcal/mol. 

SCF Done:  E(RB3LYP) =  -152.929043813     A.U. after    1 cycles
ZPVE        The zero point vibrational energy ZPVE (or ZPE) results from the vibrational motion of molecular systems even at 0 K and is calculated for a harmonic oscillator model as a sum of contributions from all i vibrational modes of the system:


 


For the water dimer there are 12 vibrational modes. With 0.5hc = 2.27817*10-6 cm-hartree, we obtain a zero point vibrational energy of 0.046225 hartrees after adding up the 12 values (see column 2 in the table below). This result is given on line 4012 of the Gaussian output file as:


Zero-point correction=                    0.046225 (Hartree/Particle)
Conversion to Joules or kilocalories gives using 627.5095 Kcal Mol-1 hartree-1:

Zero-point vibrational energy      121364.3 (Joules/Mol)
                                   29.00675 (Kcal/Mol)
E0        The zero point corrected total energy E0 is the sum of the total electronic energy Etot and the zero point vibrational energy ZPVE:

E0 = Etot + ZPVE

This result is listed in the output file (line 4016) as:

Sum of electronic and zero-point Energies=             -152.882819
E(0-298)        is the thermal correction to the internal energy at 298.15K and is given as a sum of four components:  electronic, vibrational, rotational and translational:

E(0-298) = δEel + δEvib + δErot + δEtrans

The first of these terms δEel describes the contribution of electronically excited states to the internal energy of the system. With excitation energies even to the first electronically excited state being much higher than kBT at room temperature and with the zero point of energy taken as the electronic energy of the ground electronic state, there is usually no contribution to the internal energy from occupation of electronically excited states at room temperature. Therefore, δEel = 0.

By far the largest contribution to the internal energy at room temperature stems from vibrational degrees of freedom, the zero point vibrational energy being one important component. The occupation of higher vibrational levels gives rise to an additional contribution δEvib which can be calculated according to:



The vibrational temperature of each mode i is not only a helpful quantity for evaluation of E(0->298) using the above equation, but also serves as a qualitative indicator for the extent of thermal excitation of a vibrational mode. In the example of (H2O)2, five vibrational modes have low frequencies and vibrational temperatures, which will slightly increase δEvib over the zero point energy. Adding them up as follows


Then δEvib = 0.00198722 kcal mol-1 K-1* 15513.46242 K = 30.829 kcal mol-1 .  Gaussian does this automatically; there is no need to calculate it manually. But this is where "30.829" in column 2 of the table below (line 4027 in the log file) comes from.  ( E (thermal) column as "Vibrational").

The spacing of rotational energy levels is much narrower than that of vibrational energy levels. An approximate formula for the contribution of rotational energy levels to the internal energy at room temperature (or above) for an asymmetric top is:

δErot = 3/2 RT

And so the contribution of rotational motion to the internal energy at 298.15K is 0.8887 kcal/mol (or 3.7185 kJ/mol).

The translational energy of an ideal gas δEtrans at temperature T is given (in molar quantities) as

δEtrans = 3/2 RT

implying that at 0 K there is no contribution to the internal energy from translational motion, but that the translational energy increases linearly with increasing absolute temperature. At 298.15 K this amounts to 0.8887 kcal/mol (or 3.7185 kJ/mol).

Thus the total thermal correction = 30.829 + 0.8887 + 0.8887 = 32.606 kcal mol-1 = 0.051961 hartree

The overall value for E(0-298) appears together with the ZPVE as "Thermal correction to Energy":


Zero-point correction=                           0.046225 (Hartree/Particle)
Thermal correction to Energy=                    0.051961
To repeat what was said above: The individual components of the internal energy at 298.15K are listed a few lines below in the following format (together with contributions to heat capacities cv and entropies S): (For displaying this table, Gaussian rounds Translational and Rotational values to 3 decimal places, however the Total values are correct to 3 decimal places because in the background, the program's arithmetic steps do not round off.)

                     E (Thermal)             CV                S
                      KCAL/MOL        CAL/MOL-KELVIN    CAL/MOL-KELVIN
 TOTAL                   32.606             15.956             69.028
 ELECTRONIC               0.000              0.000              0.000
 TRANSLATIONAL            0.889              2.981             36.674
 ROTATIONAL               0.889              2.981             21.110
 VIBRATIONAL             30.829              9.994             11.244
 Vibration 1              0.616              1.910              2.759
 Vibration 2              0.623              1.886              2.500          
 Vibration 3              0.624              1.882              2.462
 Vibration 4              0.636              1.846              2.172
 Vibration 5              0.759              1.489              0.995
E298        is the sum of E0 and E(0-298). For (H2O)2 this appears in the output file as:

Sum of electronic and thermal Energies=                -152.877083
H298        ..is the enthalpy at 298.15K H298 is based on the equation:

H298 = E298 + PV = E298 + RT

the latter equality being valid for molar quantities of an ideal gas. (kBT is used for one particle.) At 298.15K, RT = 0.592490 kcal mol-1 or 0.0009442 hartree. This is the difference between the thermal energies and enthalpies listed in the output as: 
Sum of electronic and thermal Enthalpies=              -152.876139
G298        ..is the Gibbs free energy at 298.15K and is equal to H298 -T*S298 , where S298 is 69.028 cal/mol-K from the table above. The total appears in the output files as: 

Sum of electronic and thermal Free Energies=           -152.908936

The S298 factor is calculated in the background using standard formulas. For example, Strans is calculated by the Sackur-Tetrode equation. Svib is calculated as shown below: (from the Q-Chem website 2026)



 

cm-1 K
ni qi q/298.15K Ai Bi Ai-Bi
143.335 206.23 0.692 0.6937 -0.6946 1.38832
164.295 236.38 0.793 0.6554 -0.6025 1.25793
167.658 241.22 0.809 0.6494 -0.5893 1.23872
195.832 281.76 0.945 0.6008 -0.4921 1.09295
391.367 563.09 1.889 0.3366 -0.1640 0.50067
669.716 963.57 3.232 0.1329 -0.0403 0.17314
1627.074 2340.98 7.852 0.0031 -0.0004 0.00345
1639.448 2358.79 7.911 0.0029 -0.0004 0.00327
3694.455 5315.47 17.828 3.224E-07 -1.809E-08 3.405E-07
3806.269 5476.35 18.368 1.937E-07 -1.054E-08 2.042E-07
3885.638 5590.54 18.751 1.348E-07 -7.189E-09 1.420E-07
3905.433 5619.02 18.846 1.231E-07 -6.534E-09 1.297E-07
sum = 5.6585
R*[] 11.24
cal/K-mol