16-08-2012, 02:18 PM
Thermochemistry in Gaussian
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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.
Introduction
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 crossreferenced
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, x7-6, 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
rst 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 rst 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, x7-6, Eq. 7.27]:
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, qr = 1. Since qr 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.
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
agged with a minus sign in the output) are ignored. Each
of the 3natoms