Cramer C.J. Essentials of Computational Chemistry Theories and Models

of Eq. (10.19) requires that the temperature be such that qrot 1, but this is almost always the case unless one is dealing with very low temperatures (say, 10 K and below) or very light molecules (like diatomics) or both.

Evaluation of the rotational components of the internal energy and entropy using the partition function of Eq. (10.19) gives

As was also previously noted in Section 9.3.1, the completely general rigid-rotor Schrodinger¨ equation for a molecule characterized by three unique axes and associated moments of inertia does not lend itself to easy solution. However, by pursuing a generalization of the classical mechanical rigid-rotor problem, one can derive a quantum mechanical approximation that is typically quite good. Within that approximation, the rotational partition function becomes

where IA, IB, and IC are the principal moments of inertia, and σ is again a symmetry number. In this case, σ is the number of pure rotations that carry the molecule into itself. Table 10.1 lists the specific values of σ for all chemically relevant symmetry point groups. Note that

Table 10.1 Rotational symmetry numbers for molecular point groups

a See Appendix B for explanations of point groups.

the presence of symmetry in a molecule may also require a structural degeneracy correction to the molecular partition function for certain conformers, as described in more detail in Appendix B.

Evaluation of the rotational components of the internal energy and entropy using the partition function of Eq. (10.22) for more typically encountered non-linear molecules gives

Again, it must be noted that evaluating the rotational components of U and S requires relatively little in the way of molecular information. All that is required is the principal moments of inertia, which derive only from the molecular structure. Thus, any methodology capable of predicting accurate geometries should be useful in the construction of rotational partition functions and the thermodynamic variables computed therefrom. Also again, the units chosen for quantities appearing in the partition function must be consistent so as to render q dimensionless.

10.3.6Molecular Vibrational Partition Function

In a polyatomic molecule with many vibrations, we simplify the vibrational partition function much as the original molecular partition function was simplified: we assume that the total vibrational energy can be expressed as a sum of individual energies associated with each mode, in which case, for a non-linear molecule, we have

where the energies εk are the vibrational energy levels associated with each mode k, and there are 3N − 6 such modes in a non-linear molecule (3N − 5 in a linear molecule) where N is the number of atoms.

To evaluate the sums associated with each mode, we assume that the modes can be approximated as quantum mechanical harmonic oscillators (QMHOs), in which case the energy levels are given by Eqs. (9.47) and (9.48). In this case, we are offered a choice with respect to convention. We may either take the zero of energy as the bottom of the potential energy well on the PES, in which case the zeroth vibrational level has energy 12 hω, or we may take the zero of energy as the energy of the equilibrium structure plus the ZPVE, in which case the energy of the zeroth vibrational energy level is zero for every mode. Both

conventions are used routinely, and one must simply be careful to ensure consistency – the entropy is independent of the choice of convention, and the internal energy varies by the ZPVE as a function of which convention is chosen.

Here, we will adopt the convention of including the ZPVE in the zero of energy [Eq. (10.1)], so that each zeroth vibrational level has an energy of zero. In that case, any individual mode’s partition function can be written as

k=0

The sum in Eq. (10.26) is well known as a convergent geometric series, so that we may write

This is a serendipitous result, insofar as the energy level spacing for most molecular vibrations is sufficiently large that significant errors would be introduced by replacing the sum by the corresponding indefinite integral as we did successfully for translation and rotation (such a replacement actually would amount to assuming a classical harmonic oscillator, for which qvib = kBT / hω; by expanding the exponential in Eq. (10.27) as its corresponding power series, one can see that the classical and quantum partition functions agree only when kBT hω).

Using Eq. (10.27) for each mode, the full vibrational partition function of Eq. (10.25) can be expressed as

=

where implies a product series (the multiplicative analogy of a sum), and the upper limit would be 3N − 5 for a linear molecule. Evaluation of the vibrational components of the internal energy and entropy using the partition function of Eq. (10.28) provides

=

Note that Eqs. (10.29) and (10.30) take the vibrational frequencies as independent variables, and as such cannot be calculated ab initio without first optimizing a structure at some level of theory and then computing the second derivatives in order to obtain the frequencies within the harmonic oscillator approximation. (Of course, one could avoid the harmonic oscillator approximation (see, for example, Barone 2004), but the necessary calculations and

the less tractable vibrational partition functions restrict this choice to only the most ambitious of calculations.)

In practice, then, it is fairly straightforward to convert the potential energy determined from an electronic structure calculation into a wealth of thermodynamic data – all that is required is an optimized structure with its associated vibrational frequencies. Given the many levels of electronic structure theory for which analytic second derivatives are available, it is usually worth the effort required to compute the frequencies and then the thermodynamic variables, especially since experimental data are typically measured in this form. For one such quantity, the absolute entropy So, which is computed as the sum of Eqs. (10.13), (10.18), (10.24) (for non-linear molecules), and (10.30), theory and experiment are directly comparable. Hout, Levi, and Hehre (1982) computed absolute entropies at 300 K for a large number of small molecules at the MP2/6-31G(d) level and obtained agreement with experiment within 0.1 e.u. for many cases. Absolute heat capacities at constant volume can also be computed using the thermodynamic definition

and the various equations for components of U above.

Absolute internal energies, enthalpies, and free energies, on the other hand, are somewhat less straightforward. From a theoretical standpoint, using the electronic energy as something to which thermodynamic components are added is equivalent to setting the absolute zero of energy as corresponding to all nuclei and electrons infinitely separated one from another and at rest. In the laboratory, this is a very inconvenient zero, since the relevant elementary particles are not easily handled. The alternative conventions in common use for reporting H and G as determined from experiment, and the steps which must be taken so that theory and experiment may be consistently compared, are addressed next.

10.4Standard-state Heats and Free Energies of Formation and Reaction

The experimental convention for assigning a zero to an enthalpy or free-energy scale is that this is the value that corresponds to the heat or free energy of formation associated with every element in its most stable, pure form under standard conditions (273 K, 1 atm). Thus, for instance, the elemental standard states for the first few elements are hydrogen gas (diatomic), helium gas (monatomic), solid lithium, solid beryllium, solid boron, solid carbon as its graphite allostere, nitrogen gas (diatomic), oxygen gas (diatomic), fluorine gas (diatomic), and neon gas (monatomic). Following this convention, the meaning of an experimental heat of formation for a molecule is that it is the (molar) enthalpy change associated with removing each of the atoms in the molecule from its elemental standard state and assembling them into the molecule.

Put in this manner, it is easy to imagine this as a two-step procedure. There is first an enthalpy cost to pull each atom out of its elemental standard state – always a non-negative quantity, since the elemental standard states are chosen to be the most stable forms. This

is followed by the enthalpy change for combining them into the molecular structure, which is the negative of the enthalpy of atomization. As an example, the 0 K heat of formation of 2-butanone (methyl ethyl ketone, a widely used industrial solvent) is tabulated as −51.9 kcal mol−1. The molecule is composed of eight atoms of hydrogen, four of carbon, and one of oxygen. The enthalpy cost to split 4 moles of hydrogen gas to create 8 moles of hydrogen atoms at 0 K is 413.5 kcal, the cost to strip 4 moles of carbon atoms from an infinite graphite block at 0 K is 1066.8 kcal, and the cost to split one-half mole of oxygen gas to create 1 mole of oxygen atoms at 0 K is 59.0 kcal. The 0 K atomization enthalpy of 1 mole of 2-butanone is 1591.2 kcal. Thus, the tabulated 0 K heat of formation cited above is determined as (413.5 + 1066.8 + 59.0 − 1591.2).

An important technical point that must be mentioned here is that some attention must be paid to the states of the atoms to ensure that the difference between the molecular atomization enthalpy and the enthalpies of formation of the atoms is carried out consistently. When one atomizes a species like 2-butanone experimentally, each resulting atom will typically contain a number of spin-unpaired electrons equal to its number of formal bonds in the molecule (because each bond is being ruptured into two unpaired electrons, one on each atom formerly involved in the bond). Thus, for instance, each carbon atom will have four unpaired electrons, corresponding to the quintet S (5S) term of the atom. However, this is not the ground state of the carbon atom (the ground state is 3P), so the value of 1066.8 kcal noted above for the enthalpy change associated with stripping 4 moles of carbon atoms from a graphite block may, under some experimental conditions, be measured as 680.6 kcal, the cost to generate the 4 moles of C atoms in their 3P ground state, plus 4 × 96.5 kcal mol−1, where the latter energy is the molar enthalpy cost to excite an atom of C from 3P to 5S.

When one speaks of a computational atomization energy for a molecule, it should be carefully specified whether the energies of the product atoms are being computed in their ground states or in excited states that may be more convenient to work with for one reason or another. This specification is also critical to determining a computed heat or free energy of formation, as described next.

10.4.1Direct Computation

Direct computation of a molecular heat or free energy of formation is something of a misnomer, since it would imply computing the difference in H or G for some molecule compared to the reference elemental standard states. Such a calculation might readily be imagined for a molecule like HF, because the standard states of H and F are gaseous diatomics. However, carrying out a high-level quantum mechanical calculation on an infinite block of graphite is another matter altogether. As a result, almost all so-called direct computations of heats of formation are carried out as illustrated in Figure 10.1. All quantities in the large inset region are computed relative to the theoretical zero of energy (all nuclei and electrons infinitely separated and at rest). To determine a standard-state molecular thermodynamic quantity, the computed energy difference between the molecule and its constituent atoms is added to the experimental thermodynamic value determined for the identical atoms. For instance, if we wanted to predict the 298 K heat of formation above

370 10 THERMODYNAMIC PROPERTIES

the elemental standard states are not, and thus the combinations of the last two terms on the r.h.s.s of Eqs. (10.32) and (10.33) are not equal. It is probably simplest to see this by considering the example of molecular hydrogen. Its translational enthalpy is given by Eq. (10.16) as ( 32 )RT . So, at 0 K it has no translational enthalpy and at 298 K it has roughly 0.9 kcal mol−1 of such enthalpy. Analogous changes are associated with rotation and vibration. However, molecular hydrogen is the elemental standard state, so it is defined experimentally to have a zero heat of formation at whatever temperature. Thus, when we compute the 298 K thermal contributions to the enthalpy of two H atoms, we determine from theory an absolute translational contribution of 3RT (again from Eq. (10.16) now applied to two separate particles), but experimentally we would only obtain 32 RT for this term, since the reference elemental standard state also has increased absolute enthalpy at 298 K.

Having discussed in detail how to go about computing heats and free energies of formation, we should now consider how useful typical electronic-structure methods are for that purpose. The somewhat disappointing answer is that most single levels of theory are disastrously bad, with the problem lying primarily in the computation of E between the molecule and its constituent atoms (the leftmost vertical line in Figure 10.1). As there is vastly more correlation energy in a molecule, with its collection of bonded pairs of electrons, than there is in a collection of atoms, and as practically affordable correlated electronic-structure methods capture at best perhaps 70–90% of the correlation energy, the differential error can be very large. Only with very, very small molecules is it possible to apply a single sufficiently high level of theory to accurately compute heats and free energies of formation ab initio. However, a number of different approaches employing varying degrees of semiempiricism have been promulgated to improve on this situation.

10.4.2Parametric Improvement

In Section 7.7, parametric methods for improving the quality of correlated electronicstructure calculations were discussed in detail. Similarly, in Section 8.4.3, the mild parameterization of density functional methods to give maximal accuracy was described. Given that background, and the substantial data presented in those earlier chapters, this section will only recapitulate in a rough categorical fashion the various approaches whose development was motivated by a desire to compute more accurate thermochemical quantities.

Most attention has been focused on the computation of Eelec, because even fairly modest levels of theory can compute molecular geometries and vibrational frequencies sufficiently accurately to give good ZPVEs and thermal contributions, particularly if the frequencies are scaled by an appropriate factor (see Section 9.3). The simplest approach to improved Eelec estimation is to scale it as a raw value as well, and this is the formalism implicit in the PCI-80 and SAC methods described in Section 7.7.1.

At a higher level of complexity, correlation energies are computed assuming that effects associated with basis-set incompleteness and, say, truncated levels of perturbation theory,

can be corrected for in a piecewise fashion. Such extrapolation schemes are described in Section 7.7.2, and specific recipes for extrapolation are at the heart of the G2 and G3 methods and the various CBS methods. Of course, scaling and extrapolation need not be mutually exclusive, and the combination of the two is the hallmark of the multicoefficient methods described in Section 7.7.3.

The G2 and G3 methods go beyond extrapolation to include small and entirely general empirical corrections associated with the total numbers of paired and unpaired electrons. When sufficient experimental data are available to permit more constrained parameterizations, such empirical corrections can be associated with more specific properties, e.g., with individual bonds. Such bond-specific corrections are employed by the BAC method described in Section 7.7.3. Note that this approach is different from those above insofar as the fundamentally modified quantity is not Eelec, but rather H . That is, the goal of the method is to predict improved heats of formation, not to compute more accurate electronic energies, per se. Irikura (2002) has expanded upon this idea by proposing correction schemes that depend not only on types of bonds, but also on their lengths and their electron densities at their midpoints. Such detailed correction schemes can offer very high accuracy, but require extensive sets of high quality experimental data for their formulation.

Finally, hybrid DFT methods have a somewhat murky status with respect to their parameters, with some being founded on theoretical arguments while others are unabashedly empirical in their design to give improved agreement with experiment. From a practical standpoint, the hybrid DFT methods tend to offer the lowest overhead with respect to bookkeeping: all computed quantities in Figure 10.1 can usually be determined conveniently from a single level of theory. As noted in Chapter 8, however, the best DFT results are still somewhat less reliably accurate than the best multilevel models, although it must be borne in mind that the latter tend to be considerably more expensive than the former. As might be expected given their success in the context of MO theoretical methods, the use of bond additivity correction schemes to improve DFT performance has begun to be explored (see, for instance, Cloud and Schwartz 2003 and Winget and Clark 2004), as has the use of multicoefficient models (Zhao, Lynch, and Truhlar 2004).

One point meriting additional discussion concerns dispersion. Most of the databases used to validate the predictive ability of different theoretical models for heats of formation have been restricted to fairly small molecules. As such, there are few examples of molecules having different portions that interact with one another through London dispersion forces (as might be expected for a coiled long-chain alkane in the gas-phase, for instance). While the highly correlated MO-theory based models should perform acceptably for such cases (if cost is not prohibitive), DFT models would be expected to do less well, since they do poorly in general in predicting weak non-bonded interaction energies. This is also true for NDDO models, but these are already sufficiently inaccurate on average that failure to account for dispersion may not necessarily lead to substantially increased error. In any case, the magnitude of the error for DFT and NDDO models would be expected to increase with molecular size, so this is a source of some concern. Resolution of this issue will require greater attention to large molecules for which accurate data are available (see, for example, Winget and Clark 2004).

10.4.3Isodesmic Equations

An alternative method for computing heats (or free energies) of formation involves consideration of a balanced chemical equation, e.g.,

where A, B, C, and D are molecules and m, n, r, and s indicate the number of moles of each in the balanced equation. The heat of reaction for a chemical transformation is defined as the difference between the heats of formation of the products and those of the reactants when these are defined relative to consistent standard states. For the reaction of Eq. (10.34), we would have

Hrxno ,298 = [r Hfo,298(C) + s Hfo,298(D)] − [m Hfo,298(A) + n Hfo,298(B)] (10.35)

where we have arbitrarily selected 298 K as the temperature of interest. Note that the standard-state symbol on the heat of reaction (as opposed to the heats of formation) does not imply the use of elemental standard states to assign a zero of enthalpy. Because the reaction is balanced, the standards used to define the zeroes for the heats of formation must cancel out on the two sides of the equation. So it is equally valid to write

where H298 is the quantity typically addressed theoretically, i.e., the enthalpy relative to all nuclei and electrons infinitely separated and at rest.

Insofar as the r.h.s. of Eq. (10.35) must then be equal to the r.h.s. of Eq. (10.36), if the experimental heats of formation for all but one of the species in Eq. (10.34) are known (say B), we may rearrange our equality to determine this quantity as

This technique at first seems rather cumbersome, since we must perforce compute H298 for four different species in this example, but it has one great advantage over the apparently simpler a priori calculation of a single heat of formation, and that is that the difficulty in computing heats of atomization can be avoided. As noted above, computed heats of atomization tend to be highly inaccurate unless heroic levels of theory are employed, because the correlation energies for the electrons in the atoms and in the molecule are so enormously different. However, assuming experimental data are available, we may select our balanced chemical Eq. (10.34) in such a way that the various bonds on the leftand right-hand sides are essentially identical. That being the case, we would expect bond-by-bond errors in correlation energy to largely cancel in the computed heat of reaction (the top line on the