Thermochemistry in GaussianJoseph W. Ochterski, Ph.D. April 19, 2000 Get PDF file of this paper (you may need to Right-Click this link to download it). Abstract: 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 used.
Sources 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
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.
Contributions from translation The equation given in McQuarrie and other texts for the translational partition function is [McQuarrie, §4-1, Eq. 4.6]:
The partial derivative of which will be used to calculate both the internal energy The second term in Equation 1 is a little trickier, since
we don't know V. However, for an ideal gas, which is what is used to calculate The translational partition function is used to calculate the translational entropy (which includes the factor of e which comes from Stirling's approximation):
The contribution to the internal thermal energy due to translation is:
Finally, the constant volume heat capacity is given by: Contributions from electronic motion The usual electronic partition function is [McQuarrie, §4-2, Eq. 4.8]: where Gaussian assumes that the first electronic excitation energy
is much greater than 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.
Contributions 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, For a linear molecule, the rotational partition function is [McQuarrie, §4-6, Eq. 4.38]: where
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
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
Contributions 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 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, Eq. 4.46]:
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 The total entropy contribution from the vibrational partition function is: 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 vibration is
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 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.
Thermochemistry output from GaussianThis section describes most of the Gaussian thermochemistry output, and how it relates to the equations I've given above.
Output from 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)
48.96870 (Kcal/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 situation.
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 and
where Thermal correction to Enthalpy= 0.082202 Thermal correction to Gibbs Free Energy= 0.055064 The Gibbs free energy includes 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
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 ( 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
Output from 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, means. 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.
Worked-out ExamplesIn this section I will show how to use these results to generate various thermochemical information. I've run calculations for each of the reactants and products in the reaction where ethyl radical abstracts a hydrogen atom from molecular hydrogen: 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.
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
Enthalpies 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 molecular data. 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
The first step in calculating the rates of these reactions is to compute
the free energy of activation,
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.
Enthalpies 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 ( 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.
Putting all these pieces together, we can finally take the steps necessary
to calculate
Here is a worked out example, where I've calculated First, I'll calculate
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) : SummaryThe 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.
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.
Last update: 23 September 2011 |
, and substitute 
. We can also move the first term into
the logarithm (as e), which leaves (with N=1): 
with respect to T is: 
and the third term in Equation 
, and 
. Therefore, 
is the degeneracy of the energy level, 
is the energy of the n-th level. 
. 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: 
. 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. 
. I is the moment of inertia.
The rotational contribution to the entropy is 
, so the entropy for this partition function
is 
is R/2. 
(or 
for linear molecules) modes has a characteristic
vibrational temperature, 
. 
, (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). 
. 
, 
Hz, or a vibrational temperature
of 900 K. 
. 
. 
, so when it's applied to calculating 
for a reaction, 
is already included. This means that 
for the total electronic 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 
, as it is described above with the
unscaled zero-point energy removed. This sum is appropriate for calculating
enthalpies of reaction, as described below.
, 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.
. All values are in Hartrees. 
to shorten the equations. 
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 
((H) is for the hydrogen reaction, (D)
is for deuterium reaction): 
) of the species involved in the reaction.
The second step is to calculate the enthalpies of formation of the species
at 298K. 
:
), the zero-point energy of the
molecule ( 
) and the constituent atoms: 
:
:
, and not 
, which is what Gaussian calculates. 
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
JANAF.
:
, where 
:
:
and 
: 
for each molecule:
for each molecule:
for each molecule:
) 
) 
) 
