The contribution to the internal thermal energy due to translation is:įinally, the constant volume heat capacity is given by: The translational partition function is used to calculate the translational entropy (which includes the factor of e which comes from Stirling’s approximation): Note that we didn’t have to make this substitution to derive the third term, since the partial derivative has V held constant. Which is what is used to calculate in Gaussian. The second term in Equation 1 is a little trickier, since we don’t know V. Which will be used to calculate both the internal energy and the third term in Equation 1. The partial derivative of with respect to T is: The equation given in McQuarrie and other texts for the translational partition function is : 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 internal thermal energy E can also be obtained from the partition function :Īnd ultimately, the energy can be used to obtain the heat capacity : We can also move the first term into the logarithm (as e), which leaves (with N=1): First, molar values are given, so we can divide by, and substitute. The form used in Gaussian is a special case. The partition function from any component can be used to determine the entropy contribution S from that component, using the relation : In this section, I’ll give an overview of how entropy, energy, and heat capacity are calculated from the partition function. The starting point in each case is the partition function q( V, T) for the corresponding component of the total partition function. 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. Sources of components for thermodynamic quantities Finally, an appendix gives a list of the all symbols used, their meanings and values for constants I’ve used.
#Single point energy quantumwise free#
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. Then I describe a sample output in the third section, to show how each section relates to the equations. 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. The first section of the paper is this introduction. The intent is to provide illustrative examples, rather than research grade results. The examples in this paper are typically carried out at the HF/STO-3G level of theory. This approximation is generally not troublesome, but can introduce some error for systems with low lying electronic excited states. Further, for the electronic contributions, it is assumed that the first and higher excited states are entirely inaccessible. This limitation will introduce some error, depending on the extent that any system being studied is non-ideal. 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. These cross-references have the form which refers to equation 7.27 in section 7-6. 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. Much of what is discussed below is covered in detail in “Molecular Thermodynamics” by McQuarrie and Simon (1999). The equations used for computing thermochemical data in Gaussian are equivalent to those given in standard texts on thermodynamics.
#Single point energy quantumwise pdf#
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