Thermochemistry in Gaussian

1 Thermochemistry inGaussianJoseph W. Ochterski, 2000, Gaussian , 2, 2000 AbstractThe purpose of this paper is to explain how various thermochemical values arecomputed inGaussian. The paper documents what equations are used to calculatethe quantities, but doesn t explain them in great detail, so a basic understandingof statistical mechanics concepts, such as partition functions, is output is explained, and a couple of examples, including calculatingthe enthalpy and Gibbs free energy for a reaction, the heat of formation of a moleculeand absolute rates of reaction are worked Introduction22 Sources of components for thermodynamic Contributions from translation.

2 Contributions from electronic motion .. Contributions from rotational motion .. Contributions from vibrational motion ..63 Thermochemistry output Output from a frequency calculation .. Output from compound model chemistries ..114 Worked-out Enthalpies and Free Energies of Reaction .. Rates of Reaction .. Enthalpies and Free Energies of Formation ..145 Summary1711 IntroductionThe equations used for computing thermochemical data inGaussianare equivalent to thosegiven in standard texts on thermodynamics. Much of what is discussed below is coveredin detail in Molecular Thermodynamics by McQuarrie and Simon (1999).

3 I ve cross-referenced several of the equations in this paper with the same equations in the book, tomake it easier to determine what assumptions were made in deriving each equation. Thesecross-references have the form [McQuarrie, 7-6, Eq. ] which refers to equation insection of the most important approximations to be aware of throughout this analysis isthat all the equations assume non-interacting particles and therefore applyonlyto an idealgas. This limitation will introduce some error, depending on the extent that any systembeing studied is non-ideal.

4 Further, for the electronic contributions, it is assumed that thefirst and higher excited states are entirely inaccessible. This approximation is generally nottroublesome, but can introduce some error for systems with low lying electronic examples in this paper are typically carried out at the HF/STO-3G level of intent is to provide illustrative examples, rather than research grade first section of the paper is this introduction . The next section of the paper, I give theequations used to calculate the contributions from translational motion, electronic motion,rotational motion and vibrational motion.

5 Then I describe a sample output in the thirdsection, to show how each section relates to the equations. The fourth section consists ofseveral worked out examples, where I calculate the heat of reaction and Gibbs free energy ofreaction 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 Sources of components for thermodynamic quanti-tiesIn each of the next four subsections of this paper, I will give the equations used to calculatethe contributions to entropy, energy, and heat capacity resulting from translational, elec-tronic, rotational and vibrational motion.

6 The starting point in each case is the partitionfunctionq(V,T) for the corresponding component of the total partition function. In thissection, I ll give an overview of how entropy, energy, and heat capacity are calculated fromthe partition partition function from any component can be used to determine the entropy con-tributionSfrom that component, using the relation [McQuarrie, 7-6, Eq. ]:S=NkB+NkBln(q(V,T)N)+NkBT( lnq T)VThe form used inGaussianis a special case. First, molar values are given, so we candivide byn=N/NA, and substituteNAkB=R. We can also move the first term into the2logarithm (ase), which leaves (withN= 1):S=R+Rln (q(V,T)) +RT( lnq T)V=Rln (q(V,T)e) +RT( lnq T)V=R(ln(qtqeqrqve) +T( lnq T)V)(1)The internal thermal energyEcan also be obtained from the partition function [Mc-Quarrie, 3-8, Eq.]

7 ]:E=NkBT2( lnq T)V,(2)and ultimately, the energy can be used to obtain the heat capacity [McQuarrie, ,Eq. ]:CV=( E T)N,V(3)These three equations will be used to derive the final expressions used to calculate thedifferent components of the thermodynamic quantities printed out Contributions from translationThe equation given in McQuarrie and other texts for the translational partition function is[McQuarrie, 4-1, Eq. ]:qt=(2 mkBTh2)3 partial derivative ofqtwith respect toTis:( lnqt T)V=32 Twhich will be used to calculate both the internal energyEtand the third term in Equation second term in Equation 1 is a little trickier, since we don t knowV.

8 However, foran ideal gas,PV=NRT=(nNA)NAkBT, andV=kBTP. Therefore,qt=(2 mkBTh2)3 is what is used to calculateqtinGaussian. Note that we didn t have to make thissubstitution to derive the third term, since the partial derivative hasVheld translational partition function is used to calculate the translational entropy (whichincludes the factor ofewhich comes from Stirling s approximation):St=R(ln(qte) +T(32T))=R(lnqt+ 1 + 3/2).3 The contribution to the internal thermal energy due to translation is:Et=NAkBT2( lnq T)V=RT2(32T)=32 RTFinally, the constant volume heat capacity is given by:Ct= Et T= Contributions from electronic motionThe usual electronic partition function is [McQuarrie, 4-2, Eq.]

9 ]:qe= 0e 0/kBT+ 1e 1/kBT+ 2e 2/kBT+ where is the degeneracy of the energy level, nis the energy of then-th that the first electronic excitation energy is much greater , thefirst and higher excited states are assumed to be inaccessibleat any tempera-ture. Further, the energy of the ground state is set to zero. These assumptions simplify theelectronic partition function to:qe= 0,which is simply the electronic spin multiplicity of the entropy due to electronic motion is:Se=R(lnqe+T( lnqe T)V)=R(lnqe+ 0).Since there are no temperature dependent terms in the partition function, the electronicheat capacity and the internal thermal energy due to electronic motion are both Contributions from rotational motionThe discussion for molecular rotation can be divided into several cases: single atoms, linearpolyatomic molecules, and general non-linear polyatomic molecules.

10 I ll cover each in a single atom,qr= 1. Sinceqrdoes not depend on temperature, the contributionof rotation to the internal thermal energy, its contribution to the heat capacity and itscontribution to the entropy are all identically a linear molecule, the rotational partition function is [McQuarrie, 4-6, Eq. ]:qr=1 r(T r)where r=h2/8 the moment of inertia. The rotational contribution to theentropy isSr=R(lnqr+T( lnqr T)V)=R(lnqr+ 1).The contribution of rotation to the internal thermal energy isEr=RT2( lnqr T)V=RT2(1T)=RTand the contribution to the heat capacity isCr=( Er T)V= the general case for a nonlinear polyatomic molecule, the rotational partition functionis [McQuarrie, 4-8, Eq.]