Boltzmann Distribution    Partition Functions    Computing Equilibria    Bosons & Fermions   

Example from Statistical Equilibrium

We start with a screen shot from program STATEQU
It shows how chemical equilibria are established microscopically. The package contains much more refined, real problems, e.g. the calculation of equilibrium constants involving molecules with dozens of atoms, internal torsional modes etc. See the downloads.

Example from Partition Functions


Several MathCad programs (and their exact replicates in Pascal, callable under DOS/WIN) treat the origin of the Partition Functions for translational, rotational, vibrational, and electronic degrees of freedom. The rotational specific heat of hydrogen is generated here.

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Where do you go from here?

Many semiempirical or ab initio Quantum Chemical electronic structure systems (ESS) offer to determine the thermodynamic functions of the computed molecule. For this to work they have to compute a stable molecule (in a well of the Born-Oppenheimer potential energy hypersurface). They then explore the forces connected to the movements of the nuclei on that surface, and obtain the characteristic vibrational frequencies of (the normal) harmonic oscillators, and hence the vibrational partition function. Since the minimum energy structure allows to calculate the moments of inertia, the rotational partition function is easily obtained. Thus (almost) everything is known, to get all the thermodynamic functions.

Mopac7 is such a program. It is very easy to use and gives good results. However, it can treat only molecules without hindered internal rotations.

Another one is Gaussian 03, a popular ab initio program which can - in principle - compute any molecular property. It is capable of treating some cases of hindered rotators but the algorithms are not generally applicable, yet. Whereas the hindered rotator eigenfunctions and eigenvalues as well as their thermal properties are correctly evaluated e.g. for ethane and dimethylether, the same treatments for methanol and methylamine produce useless data (-nan with Gaussian98, Rev. A.7, keyword 'freq=hindered'). Gaussian Inc. has been informed about this deficiency (end 1999) and has confirmed it. Whether the treatment has been corrected by now (April 2002) is not known since the keyword 'hindered' has not been described or released officially! (It is only documented in tests 440 and 441 of the Gaussian suite of tests.) For the underlying theory see P.Y. Ayala & H.B. Schlegel, JCP 108,2314(1998) and references therein. The authors also give a discussion of several other approximations in use to account for hindered rotators. It is still a good idea to use the semiempirical corrections of Pitzer & Gwinn, e.g. with my MathCad programs, mentioned above, to compute entropies and free enthalpies of molecules with hindered rotators. If this is felt to be cumbersome the usual klu(d)ge is to treat inner rotors as harmonic torsional vibrations or as free rotors. Depending on temperature and barrier height the approximations are often tolerable.

GAMESS produces thermal properties, if runtyp=hessian is chosen; the molecules are treated in the usual harmonic oscillator/rigid rotator approximation.

Except for Gaussian the programs are free of charge.

Both Gaussian and Gamess are capable of computing anharmonic vibrations and vibrational couplings with good basis functions and much more computer time.

All ESS's come with excellent manuals. Gaussian, as G03W, and Gamess as PC Gamess and WinGamess are available also for Win9x/NT4.0/Win2000,XP, see main page.

Notice that for most ab initio model chemistries and methods, computed vibrational frequencies have to be empirically corrected by as much as -12% (see J.B. Forseman & Ae. Frisch, 'Exploring Chemistry with Electronic Structure Methods", 2nd ed., table p.64, Gaussian Inc., 1996), to give satisfactory zero point and thermal energies.-
If you download Babel (e.g. Babelw16) you may convert input files among all those and other ESS's, force field methods, and chemical drawing programs.
With a good molecular structure (e.g. X-ray structure) and the vibration spectrum you can use the MathCad programs without an ESS. With large molecules there are often no vibration frequencies available. However, there exist methods to estimate them described e.g. in O.A. Hougen & K.M. Watson, Chemical Process Principles, John Wiley (1947 and newer editions!), Part 2: Thermodynamics, or in the literature. 

Java animated real time demo

for 'The Second Law' from Christopher J. Grayce