Thermochemistry in Gaussian
Joseph W. Ochterski, Ph.D.
April 19, 2000
Get PDF file of this paper (you may need
to Right-Click this link to download it).
The purpose of this paper is to explain how various
thermochemical values are computed in Gaussian. The paper documents
what equations are used to calculate the quantities, but doesn't explain
them in great detail, so a basic understanding of statistical mechanics
concepts, such as partition functions, is assumed. Gaussian thermochemistry
output is explained, and a couple of examples, including calculating the
enthalpy and Gibbs free energy for a reaction, the heat of formation of
a molecule and absolute rates of reaction are worked out.
The equations used for computing thermochemical data in Gaussian are equivalent to those given in standard texts on thermodynamics.
Much of what is discussed below is covered in detail in ``Molecular Thermodynamics''
by McQuarrie and Simon (1999). I've cross-referenced several of the equations
in this paper with the same equations in the book, to make it easier to
determine what assumptions were made in deriving each equation. These
cross-references have the form [McQuarrie, §7-6, Eq. 7.27] which
refers to equation 7.27 in section 7-6.
One of the most important approximations to be aware of throughout this
analysis is that all the equations assume non-interacting particles and
therefore apply only to an ideal gas. This limitation will introduce
some error, depending on the extent that any system being studied is non-ideal.
Further, for the electronic contributions, it is assumed that the first
and higher excited states are entirely inaccessible. This approximation
is generally not troublesome, but can introduce some error for systems
with low lying electronic excited states.
The examples in this paper are typically carried out at the HF/STO-3G
level of theory. The intent is to provide illustrative examples, rather
than research grade results.
The first section of the paper is this introduction. The next section
of the paper, I give the equations used to calculate the contributions
from translational motion, electronic motion, rotational motion and vibrational
motion. Then I describe a sample output in the third section, to show
how each section relates to the equations. The fourth section consists
of several worked out examples, where I calculate the heat of reaction
and Gibbs free energy of reaction for a simple bimolecular reaction, and
absoloute reaction rates for another. Finally, an appendix gives a list
of the all symbols used, their meanings and values for constants I've
of components for thermodynamic quantities
In each of the next four subsections of this paper, I will give the
equations used to calculate the contributions to entropy, energy, and
heat capacity resulting from translational, electronic, rotational and
vibrational motion. The starting point in each case is the partition function q(V,T) for the corresponding component of the total
partition function. In this section, I'll give an overview of how entropy,
energy, and heat capacity are calculated from the partition function.
The partition function from any component can be used to determine the
entropy contribution S from that component, using the relation
[McQuarrie, §7-6, Eq. 7.27]:
The form used in Gaussian is a special case. First, molar values
are given, so we can divide by , and substitute . We can also move the first term into
the logarithm (as e), which leaves (with N=1):
The internal thermal energy E can also be obtained from the partition
function [McQuarrie, §3-8, Eq. 3.41]:
and ultimately, the energy can be used to obtain the heat capacity [McQuarrie,
§3.4, Eq. 3.25]:
These three equations will be used to derive the final expressions used
to calculate the different components of the thermodynamic quantities
printed out by Gaussian.
The equation given in McQuarrie and other texts for the translational
partition function is [McQuarrie, §4-1, Eq. 4.6]:
The partial derivative of with respect to T is:
which will be used to calculate both the internal energy and the third term in Equation 1.
The second term in Equation 1 is a little trickier, since
we don't know V. However, for an ideal gas, , and . Therefore,
which is what is used to calculate in Gaussian. Note that we didn't
have to make this substitution to derive the third term, since the partial
derivative has V held constant.
The translational partition function is used to calculate the translational
entropy (which includes the factor of e which comes from Stirling's
The contribution to the internal thermal energy due to translation is:
Finally, the constant volume heat capacity is given by:
from electronic motion
The usual electronic partition function is [McQuarrie, §4-2, Eq. 4.8]:
where is the degeneracy of the energy level, is the energy of the n-th level.
Gaussian assumes that the first electronic excitation energy
is much greater than . Therefore, the first and higher
excited states are assumed to be inaccessible at any temperature.
Further, the energy of the ground state is set to zero. These assumptions
simplify the electronic partition function to:
which is simply the electronic spin multiplicity of the molecule.
The entropy due to electronic motion is:
Since there are no temperature dependent terms in the partition function,
the electronic heat capacity and the internal thermal energy due to electronic
motion are both zero.
from rotational motion
The discussion for molecular rotation can be divided into several cases:
single atoms, linear polyatomic molecules, and general non-linear polyatomic
molecules. I'll cover each in order.
For a single atom, . Since does not depend on temperature, the contribution
of rotation to the internal thermal energy, its contribution to the heat
capacity and its contribution to the entropy are all identically zero.
For a linear molecule, the rotational partition function is [McQuarrie,
§4-6, Eq. 4.38]:
where . I is the moment of inertia.
The rotational contribution to the entropy is
The contribution of rotation to the internal thermal energy is
and the contribution to the heat capacity is
For the general case for a nonlinear polyatomic molecule, the rotational
partition function is [McQuarrie, §4-8, Eq. 4.56]:
Now we have , so the entropy for this partition function
Finally, the contribution to the internal thermal energy is
and the contribution to the heat capacity is
The average contribution to the internal thermal energy from each rotational
degree of freedom is RT/2, while it's contribution to is R/2.
from vibrational motion
The contributions to the partition function, entropy, internal energy
and constant volume heat capacity from vibrational motions are composed
of a sum (or product) of the contributions from each vibrational mode, K. Only the real modes are considered; modes with imaginary frequencies
(i.e. those flagged with a minus sign in the output) are ignored. Each
of the (or for linear molecules) modes has a characteristic
vibrational temperature, .
There are two ways to calculate the partition function, depending on
where you choose the zero of energy to be: either the bottom of the internuclear
potential energy well, or the first vibrational level. Which you choose
depends on whether or not the contributions arising from zero-point energy
will be coputed separately or not. If they are computed separately, then
you should use the bottom of the well as your reference point, otherwise
the first vibrational energy level is the appropriate choice.
If you choose the zero reference point to be the bottom of the well
( BOT), then the contribution to the partition function from
a given vibrational mode is [McQuarrie, §4-4, Eq. 4.24]:
and the overall vibrational partition function is[McQuarrie, §4-7,
On the other hand, if you choose the first vibrational energy level
to be the zero of energy (V=0), then the partition function for
each vibrational level is
and the overall vibrational partition function is:
Gaussian uses the bottom of the well as the zero of energy (BOT)
to determine the other thermodynamic quantities, but also prints out the V=0 partition function. Ultimately, the only difference between
the two references is the additional factor of , (which is the zero point vibrational
energy) in the equation for the internal energy . In the expressions for heat capacity
and entropy, this factor disappears since you differentiate with respect
to temperature (T).
The total entropy contribution from the vibrational partition function
To get from the fourth line to the fifth line in the equation above,
you have to multiply by .
The contribution to internal thermal energy resulting from molecular
Finally, the contribution to constant volume heat capacity is
Low frequency modes (defined below) are included in the computations
described above. Some of these modes may be internal rotations, and so
may need to be treated separately, depending on the temperatures and barriers
involved. In order to make it easier to correct for these modes, their
contributions are printed out separately, so that they may be subtracted
out. A low frequency mode in Gaussian is defined as one for which
more than five percent of an assembly of molecules are likely to exist
in excited vibrational states at room temperature. In other units, this
corresponds to about 625 cm , Hz, or a vibrational temperature
of 900 K.
It is possible to use Gaussian to automatically perform some
of this analysis for you, via the Freq=HindRot keyword. This
part of the code is still undergoing some improvements, so I will not
go into it in detail. See P. Y. Ayala and H. B. Schlegel, J. Chem. Phys. 108 2314 (1998) and references therein for more detail about the
hindered rotor analysis in Gaussian and methods for correcting
the partition functions due to these effects.
This section describes most of the Gaussian thermochemistry
output, and how it relates to the equations I've given above.
a frequency calculation
In this section, I intentionally used a non-optimized structure, to
show more output. For production runs, it is very important to
use structures for which the first derivatives are zero -- in other words,
for minima, transition states and higher order saddle points. It is occasionally
possible to use structures where one of the modes has non-zero first derivatives,
such as along an IRC. For more information about why it is important to
be at a stationary point on the potential energy surface, see my white
paper on ``Vibrational Analysis in Gaussian ''.
Much of the output is self-explanatory. I'll only comment on some of
the output which may not be immediately clear. Some of the output is also
described in ``Exploring Chemistry with Electronic Structure Methods,
Second Edition'' by James B. Foresman and Æleen Frisch.
- Thermochemistry -
Temperature 298.150 Kelvin. Pressure 1.00000 Atm.
Atom 1 has atomic number 6 and mass 12.00000
Atom 2 has atomic number 6 and mass 12.00000
Atom 3 has atomic number 1 and mass 1.00783
Atom 4 has atomic number 1 and mass 1.00783
Atom 5 has atomic number 1 and mass 1.00783
Atom 6 has atomic number 1 and mass 1.00783
Atom 7 has atomic number 1 and mass 1.00783
Atom 8 has atomic number 1 and mass 1.00783
Molecular mass: 30.04695 amu.
The next section gives some characteristics of the molecule based on
the moments of inertia, including the rotational temperature and constants.
The zero-point energy is calculated using only the non-imaginary frequencies.
Principal axes and moments of inertia in atomic units:
1 2 3
EIGENVALUES -- 23.57594 88.34097 88.34208
X 1.00000 0.00000 0.00000
Y 0.00000 1.00000 0.00001
Z 0.00000 -0.00001 1.00000
THIS MOLECULE IS AN ASYMMETRIC TOP.
ROTATIONAL SYMMETRY NUMBER 1.
ROTATIONAL TEMPERATURES (KELVIN) 3.67381 0.98044 0.98043
ROTATIONAL CONSTANTS (GHZ) 76.55013 20.42926 20.42901
Zero-point vibrational energy 204885.0 (Joules/Mol)
If you see the following warning, it can be a sign that one of two things
is happening. First, it often shows up if your structure is not a minimum
with respect to all non-imaginary modes. You should go back and re-optimize
your structure, since all the thermochemistry based on this structure
is likely to be wrong. Second, it may indicate that there are internal
rotations in your system. You should correct for errors caused by this
WARNING-- EXPLICIT CONSIDERATION OF 1 DEGREES OF FREEDOM AS
VIBRATIONS MAY CAUSE SIGNIFICANT ERROR
Then the vibrational temperatures and zero-point energy (ZPE):
VIBRATIONAL TEMPERATURES: 602.31 1607.07 1607.45 1683.83 1978.85
(KELVIN) 1978.87 2303.03 2389.95 2389.96 2404.55
2417.29 2417.30 4202.52 4227.44 4244.32
4244.93 4291.74 4292.31
Zero-point correction= 0.078037 (Hartree/Particle)
Each of the next few lines warrants some explanation. All of them include
the zero-point energy. The first line gives the correction to the internal
thermal energy, .
Thermal correction to Energy= 0.081258
The next two lines, respectively, are
Thermal correction to Enthalpy= 0.082202
Thermal correction to Gibbs Free Energy= 0.055064
The Gibbs free energy includes , so when it's applied to calculating for a reaction, is already included. This means that will be computed correctly when the number
of moles of gas changes during the course of a reaction.
The next four lines are estimates of the total energy of the molecule,
after various corrections are applied. Since I've already used E to represent internal thermal energy, I'll use for the total electronic energy.
Sum of electronic and zero-point energies=
Sum of electronic and thermal energies=
Sum of electronic and thermal enthalpies=
Sum of electronic and thermal free energies=
Sum of electronic and zero-point Energies= -79.140431
Sum of electronic and thermal Energies= -79.137210
Sum of electronic and thermal Enthalpies= -79.136266
Sum of electronic and thermal Free Energies= -79.163404
The next section is a table listing the individual contributions to
the internal thermal energy ( ), constant volume heat capacity ( ) and entropy ( ). For every low frequency mode, there
will be a line similar to the last one in this table (labeled VIBRATION 1).
That line gives the contribution of that particular mode to , and . This allows you to subtract out these
values if you believe they are a source of error.
E (Thermal) CV S
KCAL/MOL CAL/MOL-KELVIN CAL/MOL-KELVIN
TOTAL 50.990 8.636 57.118
ELECTRONIC 0.000 0.000 0.000
TRANSLATIONAL 0.889 2.981 36.134
ROTATIONAL 0.889 2.981 19.848
VIBRATIONAL 49.213 2.674 1.136
VIBRATION 1 0.781 1.430 0.897
Finally, there is a table listing the individual contributions to the
partition function. The lines labeled BOT are for the vibrational
partition function computed with the zero of energy being the bottom of
the well, while those labeled with (V=0) are computed with the
zero of energy being the first vibrational level. Again, special lines
are printed out for the low frequency modes.
Q LOG10(Q) LN(Q)
TOTAL BOT 0.470577D-25 -25.327369 -58.318422
TOTAL V=0 0.368746D+11 10.566728 24.330790
VIB (BOT) 0.149700D-35 -35.824779 -82.489602
VIB (BOT) 1 0.419879D+00 -0.376876 -0.867789
VIB (V=0) 0.117305D+01 0.069318 0.159610
VIB (V=0) 1 0.115292D+01 0.061797 0.142294
ELECTRONIC 0.100000D+01 0.000000 0.000000
TRANSLATIONAL 0.647383D+07 6.811161 15.683278
ROTATIONAL 0.485567D+04 3.686249 8.487901
compound model chemistries
This section explains what the various thermochemical quantities in
the summary out of a compound model chemistry, such as CBS-QB3 or G2,
I'll use a CBS-QB3 calculation on water, but the discussion is directly
applicable to all the other compound models available in Gaussian. The
two lines of interest from the output look like:
CBS-QB3 (0 K)= -76.337451 CBS-QB3 Energy= -76.334615
CBS-QB3 Enthalpy= -76.333671 CBS-QB3 Free Energy= -76.355097
Here are the meanings of each of those quantities.
- CBS-QB3 (0 K): This is the total electronic energy as defined by the
compound model, including the zero-point energy scaled by the factor
defined in the model.
- CBS-QB3 Energy: This is the total electronic energy plus the internal
thermal energy, which comes directly from the thermochemistry output
in the frequency part of the output. The scaled zero-point energy is
- CBS-QB3 Enthalpy: This is the total electronic energy plus , as it is described above with the
unscaled zero-point energy removed. This sum is appropriate for calculating
enthalpies of reaction, as described below.
- CBS-QB3 Free Energy: This is the total electronic energy plus , as it is described above, again with
the unscaled zero-point energy removed. This sum is appropriate for
calculating Gibbs free energies of reaction, also as described below.
In this section I will show how to use these results to generate various
I've run calculations for each of the reactants and products in the
reaction where ethyl radical abstracts a hydrogen atom from molecular
as well as for the transition state (all at 1.0 atmospheres and 298.15K).
The thermochemistry output from Gaussian is summarized in Table 1.
Table 1: Calculated thermochemistry values from Gaussian for the reaction . All values are in Hartrees.
Once you have the data for all the relevant species, you can calculate
the quantities you are interested in. Unless otherwise specified, all
enthalpies are at 298.15K. I'll use to shorten the equations.
and Free Energies of Reaction
The usual way to calculate enthalpies of reaction is to calculate heats
of formation, and take the appropriate sums and difference.
However, since Gaussian provides the sum of electronic and thermal enthalpies,
there is a short cut: namely, to simply take the difference of the sums
of these values for the reactants and the products. This works since the
number of atoms of each element is the same on both sides of the reaction,
therefore all the atomic information cancels out, and you need only the
For example, using the information in Table 1, the enthalpy of reaction can
be calculated simply by
The same short cut can be used to calculate Gibbs free energies of reaction:
In this section I'll show how to compute rates of reaction using the
output from Gaussian. I'll be using results derived from transition
state theory in section 28-8 of ``Physical Chemistry, A Molecular Approach''
by D. A. McQuarrie and J. D. Simon. The key equation
(number 28.72, in that text) for calculating reaction rates is
I'll use for the concentration. For simple reactions,
the rest is simply plugging in the numbers. First, of course, we need
to get the numbers. I've run HF/STO-3G frequency calculation for reaction
FH + Cl F + HCl and also for the reaction
with deuterium substituted for hydrogen. The results are summarized in
Table 2. I put the total electronic energies
of each compound into the table as well to illustrate the point that the
final geometry and electronic energy are independent of the masses of
the atoms. Indeed, the cartesian force constants themselves are independent
of the masses. Only the vibrational analysis, and quantities derived from
it, are mass dependent.
Table 2: Total HF/STO-3G electronic energies and sum
of electronic and thermal free energies for atoms molecules and transition
state complex of the reaction FH + Cl F + HCl and the deuterium substituted analog.
The first step in calculating the rates of these reactions is to compute
the free energy of activation, ((H) is for the hydrogen reaction, (D)
is for deuterium reaction):
Then we can calculate the reaction rates. The values for the constants
are listed in the appendix. I've taken .
So we see that the deuterium reaction is indeed slower, as we would
expect. Again, these calculations were carried out at the HF/STO-3G level,
for illustration purposes, not for research grade results. More complex
reactions will need more sophisticated analyses, perhaps including careful
determination of the effects of low frequency modes on the transition
state, and tunneling effects.
and Free Energies of Formation
Calculating enthalpies of formation is a straight-forward, albeit somewhat
tedious task, which can be split into a couple of steps. The first step
is to calculate the enthalpies of formation ( ) of the species involved in the reaction.
The second step is to calculate the enthalpies of formation of the species
Calculating the Gibbs free energy of reaction is similar, except we
have to add in the entropy term:
To calculate these quantities, we need a few component pieces first.
In the descriptions below, I will use M to stand for the molecule,
and X to represent each element which makes up M, and x will be the number of atoms of X in M.
- Atomization energy of the molecule, :
These are readily calculated from the total energies of the molecule
( ), the zero-point energy of the
molecule ( ) and the constituent atoms:
- Heats of formation of the atoms at 0K, :
I have tabulated recommended values for the heats of formation of
the first and second row atomic elements at 0K in Table 3. There are (at least) two
schools of thought with respect to the which atomic heat of formation
data is most appropriate. Some authors prefer to use purely experimental
data for the heats of formation (Curtiss, et. al., J.\ Chem.
Phys. 106, 1063 (1997)). Others (Ochterski, et. al., J.\
Am. Chem. Soc. 117, 11299 (1995)), prefer to use a combination
of experiment and computation to obtain more accurate values. The
elements of concern are boron, beryllium and silicon. I have arranged
Table 3 so that either may be
used; the experimental results are on the top, while the computational
results for the three elements with large experimental uncertainty
are listed below that.
- Enthalpy corrections of the atomic elements, :
The enthalpy corrections of the first and second row atomic elements
are also included in Table 3.
These are used to convert the atomic heats of formation at 0K to those
at 298.15K, and are given for the elements in their standard states.
Since they do not depend on the accuracy of the heat of formation
of the atom, they are the same for both the calculated and experimental
This is generally not the same as the output from Gaussian for a calculation on an isolated gas-phase atom. These values
are referenced to the standard states of the elements. For example,
the value for hydrogen atom is 1.01 kcal/mol. This is , and not , which is what Gaussian calculates.
Table: Experimental and calculated enthalpies of
formation of elements (kcal mol ) and entropies of atoms (cal mol K ) . Experimental enthalpy values
taken from J. Chem. Phys. 106, 1063 (1997). Calculated enthalpy values
taken from J. Am.\ Chem. Soc. 117, 11299 (1995). is the same for both calculated and
experimental, and is taken for elements in their standard states. Entropy values taken from JANAF Thermochemical
Tables: M. W. Chase, Jr., C. A. Davies, J. R. Downey,
Jr., D. J. Frurip, R. A. McDonald, and A. N. Syverud,
J. Phys. Ref. Data 14 Suppl. No. 1 (1985). No error bars are given for Mg in
- Enthalpy correction for the molecule, :
For a molecule, this is , where is the value printed out in the
line labeled `` Thermal correction to Enthalpy'' in Gaussian output. Remember, though, that the output is in Hartrees/particle,
and needs to be converted to kcal/mol. The conversion factor is 1
- Entropy for the atoms, :
These are summarized in Table 3. The values were taken from
the JANAF tables (M. W. Chase, Jr., C. A. Davies, J. R.
Downey, Jr., D. J. Frurip, R. A. McDonald, and A. N. Syverud, J. Phys.
Ref. Data 14 Suppl. No. 1 (1985)). Again, these values are
referenced to the standard states of the elements, and will disagree
with values Gaussian calculates for the gas-phase isolated
- Entropy for the molecule, :
Gaussian calculates G=H - TS, and prints
it out on the line labeled ``Thermal correction to Gibbs Free
Energy''. The entropy can be teased out of this using S=(H - G)/T.
Putting all these pieces together, we can finally take the steps necessary
to calculate and :
- Calculate for each molecule:
- Calculate for each molecule:
- Calculate for each molecule:
Here is a worked out example, where I've calculated for the reactants and products in the
reaction I described above.
First, I'll calculate for one of the species:
The next step is to calculate the :
To calculate the Gibbs free energy of formation, we have (the factors
of 1000 are to convert kcal to or from cal) :
The essential message is this: the basic equations used to calculate
thermochemical quantities in Gaussian are based on those used
in standard texts. Since the vibrational partition function depends on
the frequencies, you must use a structure that is either a minimum or
a saddle point. For electronic contributions to the partition function,
it is assumed that the first and all higher states are inaccessible at
the temperature the calculation is done at. The data generated by Gaussian can be used to calculate heats and free energies of reactions as
well as absolute rate information.
- =contribution to the heat capacity due to electronic motion
- =contribution to the heat capacity due to rotational motion
- =total heat capacity ( )
- =contribution to the heat capacity due to translation
- =contribution to the heat capacity due to vibrational motion
- =internal energy due to electronic motion
- =internal energy due to rotational motion
- =total internal energy ( )
- =internal energy due to translation
- =internal energy due to vibrational motion
- =correction to the Gibbs free energy due to internal energy
- =correction to the enthalpy due to internal energy
- =moment of inertia
- =index of vibrational modes
- =number of moles
- =Avogadro's number
- =pressure (default is 1 atmosphere)
- =gas constant
- =entropy due to electronic motion
- =entropy due to rotational motion
- =total entropy ( )
- =entropy due to translation
- =entropy due to vibrational motion
- =temperature (default is 298.15)
- =characteristic temperature for rotation (in the x, y or z plane)
- =characteristic temperature for vibration K
- =Boltzmann constant
- =the energy of the n-th energy level
- =the degeneracy of the n-th energy level
- =symmetry number for rotation
- =Planck's constant
- =mass of the molecule
- =number of particles (always 1)
- =electronic partition function
- =rotational partition function
- =translational partition function
- =vibrational partition function
- =zero-point energy of the molecule
- =the total electronic energy, the MP2 energy for example
- =standard enthalpy, entropy and Gibbs free energy -- each compound
is in its standard state at the given temperature
- =Gibbs free energy of activation
- =concentration (taken to be 1)
- =reaction rate at temperature T
Thermochemistry in Gaussian
This document was generated using the LaTeX2HTML translator Version
96.1 (Feb 5, 1996) Copyright © 1993, 1994, 1995, 1996, Nikos Drakos,
Computer Based Learning Unit, University of Leeds.
Copyright © 2000, Gaussian, Inc.
Author: Joseph Ochterski
Wed Apr 19 16:42:08 EDT 2000
Last update: 27 February 2014