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

r.h.s. of Eq. (10.37)) so that any error might be no larger than the errors associated with the experimentally measured heats. Such a reaction is called ‘isodesmic’.

As a specific example, we might seek the heat of formation of 6-methylquinoline, the size of which is such that application of a methodology like G3 would be computationally rather intensive. However, consider the isodesmic reaction

where 6-methylquinoline is now molecule ‘B’ of Eq. (10.34). So long as heats of formation for the common molecules naphthalene, quinoline, and 2-methylnaphthalene are known, we may then compute enthalpies for all four species and predict the heat of formation of 6-methylquinoline using Eq. (10.37). Note that, by construction, all of the bonds on the l.h.s. of Eq. (10.38) are essentially identical to those on the r.h.s. (they are only non-identical once one starts to define them not only according to the two atoms which are bonded, but based on their distant neighbor atoms as well). As such, we might expect a much more affordable level of theory, say a DFT calculation, to be useful in the evaluation of Eq. (10.37).

Note that the construction of an isodesmic equation is something of an art, depending on chemical intuition and available experimental data. In the above situation, if an experimental heat of formation for quinoline were not available, we might decide to resort to a reaction like

While this reaction is still balanced, it is less ideal than Eq. (10.38). For instance, on the r.h.s. of Eq. (10.39), there are two aromatic C –H bonds where the carbon atom is bonded to a nitrogen, but on the l.h.s. there is only one. As a result, we might be forced to go to higher levels of theory to ameliorate any error this might introduce. In the extreme, one can imagine balanced reactions like

N

The necessary experimental heats of formation are known to exquisite accuracy (or defined as zero, in the case of H2), and the calculations will be trivial for such small molecules, but accurately accounting for the enormous differences in the natures of the bonds on the two sides of Eq. (10.40) will require levels of electronic-structure theory nearly as high as those that would be necessary for a direct or parametric computation on 6-methylquinoline alone. The one virtue of Eq. (10.40), which is an example of a ‘bond separation reaction’, is that the total amount of unpaired electron spin on the two sides of the reaction is the same (in this case, zero); such a reaction is called ‘isogyric’. Note that atomization processes are

the molar internal energy, since in Eq. (10.29), both the numerator and the denominator of the term associated with the vanishing frequency go to zero. However, if we examine this behavior using a power series expansion for the exponential, we see that

Thus, each vanishing frequency contributes a factor of RT to the molar internal energy (and thus the enthalpy).

Equation (10.44) can also be used to indicate that the first term in brackets on the r.h.s. of Eq. (10.30) goes to 1 as the frequency vanishes. However, if we examine the second term in brackets on the r.h.s. of Eq. (10.30), we discover

hω

=lim −R ln

which is certainly not a very pleasant result, since free energies will become infinitely negative with infinitely positive entropies. A careful analysis of Eq. (10.45) also indicates that small errors in very small non-zero frequencies can lead to very large errors in entropies. Unfortunately, it is precisely for low-frequency motions that we typically expect the harmonic oscillator approximation to be most poor. Thus, when a molecule is characterized by one or more very low-frequency vibrations, it is usually best not to discuss the molecular free energy, but instead restrict oneself to enthalpy or internal energy.

Note that there is nothing ‘wrong’ with Eq. (10.45). The entropy of a quantum mechanical harmonic oscillator really does go to infinity as the frequency goes to zero. What is wrong is that one usually should not apply the harmonic oscillator approximation to describe those modes exhibiting the smallest frequencies. More typically than not, such modes are torsions about single bonds characterized by very small or vanishing barriers. Such situations are known as hindered and free rotors, respectively.

More accurately, ‘free rotor’ is used to imply any torsion having a barrier substantially below kBT . In such a situation, the contribution of the free rotor to the molar internal

energy is

i.e., only half that computed for a harmonic oscillator using Eq. (10.44). The contribution of a free rotor to the entropy is given by

where σint and Iint are the reduced symmetry numbers and moments of inertia associated with the free rotor. The definitions for these quantities may be found in the definitive work of Pitzer and Gwinn (1942) on this subject.

Pitzer and Gwinn have also provided tables to determine the thermodynamic contributions of hindered rotors (those having torsional barriers on the order of kBT ) when such rotors are well described by the torsional potential

where V is the torsional barrier height and θ is the torsion angle. For the most careful work, this is the appropriate treatment to employ.

Note that if the torsional barrier is considerably greater than kBT , then the harmonic oscillator approximation is as valid as for any vibration, and no special precautions need be taken.

10.5.3Equilibrium Populations over Multiple Minima

It is not uncommon for a single molecule to have multiple populations. At non-zero temperatures, the population of different conformations will be dictated by Boltzmann statistics. If we make the approximation that we may neglect the continuous character of conformational space and simply work with discrete potential energy minima, we can replace a statistical mechanical probability integral with a discrete sum, and the equilibrium fraction F of any given conformer A at temperature T may be computed as

e−GoA /RT

F (A) = (10.49)

e−Goi /RT

i

where i runs over all possible conformers, each characterized by its own free energy Go. In measurements on systems at equilibrium, it is rarely possible to determine the free energies of individual components of the equilibrium. Rather, one refers to the free energy of the whole equilibrium population, which may be written

i {A}

where {A} emphasizes computation over the population of all conformers of A (where this set can include structures differing only by atom labeling, as detailed further in Appendix B). Free-energy changes, then, between two species each of which exist as populations over multiple conformers, must be computed as the difference between their averages. Note that the formalism of Eq. (10.50) may also be applied to determine averaged transition state free energies provided multiple transition state structures exist all of which lead to the same product; the difference between an averaged reactant free energy and an averaged transition state free energy defines a free energy of activation.

In fortunate instances, one conformer in a population has a free energy that is much lower than that of any of the other possibilities. Inspection of Eq. (10.50) makes clear that in that instance, only the low-energy term contributes significantly to the sum, in which case that free energy may be taken as the population free energy.

10.5.4Standard-state Conversions

Two issues associated with thermodynamic standard states bear some further attention. The first is associated with the enthalpy of ions. Ion heats of formation may be defined based on the heats of ionization of neutral molecules (or electron attachments thereto). For example, one might consider a reaction like

and define the heat of formation of the radical cation A+ž as the sum of the heat of formation of A and the enthalpy change for Eq. (10.51). In that case, however, one needs to assign a heat of formation to the free electron. The thermal electron convention takes the free electron as the ‘electron standard state’, i.e., its enthalpy of formation is always zero. The so-called ion convention, on the other hand, takes the electron at rest to be the standard state (this is the usual theoretical convention as well, recall), and predominates in the mass spectrometric literature. The conversion between the two is straightforward, namely

where superscript ‘o’ represents the thermal electron standard state, superscript ‘o ’ represents the ion convention standard state, and q is the signed charge on X.

A separate issue arises in the discussion of standard-state free energies. Recall that the entropy of translation requires a concentration specification to be included as part of the standard-state conditions. Different tabulations of data often adopt different concentration conventions, and it is very important that care be taken to ensure consistent comparisons. To convert from one convention to another, we write

where Q is the reaction quotient (i.e., the ratio of concentrations that appear in the equilibrium constant) evaluated with all species at their standard-state concentrations, expressed so that the logarithm is dimensionless. As an example, consider the gas-phase condensation reaction

where we will define the ‘o’ standard state to imply all species at 1 atm pressure and the ‘o ’ standard state to imply all species in the gas phase at 1 mol L−1. If A, B, and C are ideal gases, their concentration at 1 atm may be derived from the ideal gas law as mol L−1 at 298 K. Since the reaction quotient Q is [C]/[A][B], Eq. (10.53) becomes

Additional standard-state issues can arise in condensed phases, and these will be dealt with in subsequent chapters.

10.5.5Standard-state Free Energies, Equilibrium Constants, and Concentrations

While our focus has been primarily on thermodynamic quantities, like free energy, it should be borne in mind that the ultimate motivation for computing free energy differences is usually to permit calculation of chemical concentrations in actual systems. To accomplish this for a generic equilibrium is straightforward. For example, consider the following reaction (chosen in a completely arbitrary fashion)

From the relationship between the equilibrium constant and the free energies of the reactants and the products we may write

where the standard-state symbol on the free energy change dictates the units used for the concentrations of the species. Thus, if we were carrying out all free energy calculations for gas-phase species at 1 atm pressure, we would express the reactant and product concentrations in those units. Stoichiometry then permits Eq. (10.57) to be rewritten as

where x is the concentration of E (and half the concentration of D) in units of partial pressure at equilibrium, and the initial partial pressures of reactants A, B, and C appear as constants in the denominator on the l.h.s. of Eq. (10.58). Note that the sign of the free energy change for Eq. (10.56) is only predictive of the side to which the equilibrium shifts when all species are initially present at their unit standard-state concentrations. All other situations require explicit evaluation of equations like Eq. (10.58) in order to determine the final concentrations predicted at equilibrium.

One variation on this theme that should be borne in mind when analyzing actual chemical situations is that certain species in the real system may be ‘buffered’. That is, their concentrations may be held constant by external means. A good example of this occurs in condensed phases, where solvent molecules may play explicit roles in chemical equilibria but the concentration of the free solvent is so much larger than that for any other species that it may be considered to be effectively constant. Modeling solvation phenomena in general is covered in detail in the next two chapters, but it is instructive to consider here a particular case as it relates to computing equilibria. Consider such a reaction as

That is, three monosolvates of A are in equilibrium with a disolvate of trimeric B (i.e., B = A3) and a liberated solvent molecule. A rather typical protocol for evaluating the ratio of monomer to trimer in solution would be the following: (i) compute the gas-phase free energies of A·S, B·2S, and S at the appropriate temperature and a partial pressure of 1 atm (the default in most electronic structure programs), (ii) add to these gas-phase free energies the appropriate solvation free energies (usually computed assuming no change in standardstate concentration, as described in Chapters 11 and 12), and (iii) convert the free-energy change on going from reactants to products to standard-state units of 1 M concentration following the protocol of Eq. (10.55) because this is the more conventional standard state in solution. Having accomplished this, we would then be able to write

where x is the moles of A monosolvate converted at equilibrium to x/3 moles of trimeric B disolvate and [S]0 is the concentration of the solvent (determined from its density and molecular weight). To cement this example with actual values, imagine the solvent to be water ([S]0 = 55.56 M) and, for a particular choice of A and B, the free energy change in solution (i.e., for the ‘o ’ standard state) to be −3.0 kcal mol−1. If we take the starting concentration of A monosolvate (x0,AžS) to be 0.2 M, we determine from solving the cubic Eq. (10.60) that at equilibrium x is 0.037 M, which is to say that there is about one molecule of Bž2S for every 16 molecules of AžS. The failure of the reasonably large negative free-energy change to lead to substantial trimerization seems paradoxical only if one forgets that that negative number refers specifically to all species being at their standard-state concentrations

(1 M)–actual systems may be quite far from that reference point.

10.6 Case Study: Heat of Formation of H2NOH

Synopsis of Saraf et al. (2003) ‘Theoretical Thermochemistry: ab initio Heat of Formation for Hydroxylamine’.

Hydroxylamine (H2NOH) is a highly reactive molecule. As such, handling bulk quantities poses significant safety concerns, and indeed serious accidents occurred in 1999 and 2000 with this molecule in industrial settings. A direct measurement of the 298 K gas-phase enthalpy of formation of hydroxylamine is not available. Data from the solid phase have been interpreted to suggest a value of −12.0 ± 2.4 kcal mol−1, but so large an uncertainty suggests that theory might prove useful in providing an improved estimate of this quantity, and this in turn might aid in the design of reaction conditions for reactors containing hydroxylamine. With that goal in mind, Saraf et al. surveyed a very large number of different levels of theory, including composite levels, to assess their likely utility for this task.

We consider here three different reaction protocols for predicting the enthalpy of formation of H2NOH:

The latter two equations were used by Saraf et al. since the 298 K gas-phase enthalpies of formation of hydrogen, water, and ammonia are all known to very high accuracy. Thus, the procedures outlined in Section 10.4.3 may be used to compute the unknown hydroxylamine enthalpy of formation. As isodesmic equations go, Eq. (10.62) is not particularly good. The H−H and N−O bonds appearing on the l.h.s. are replaced by new N−H and O−H bonds on the r.h.s., and there is not much reason to expect these bonds to have similar errors in computed correlation energies. Eq. (10.63) is an improvement to the extent that the only major difference in bonding from the l.h.s. to the r.h.s. is the change of an N−O bond to an O−O bond. As both bonds are heteroatom to heteroatom for first-row atoms, we may expect a much more favorable cancellation of errors. Saraf et al. did not discuss the atomization energy, Eq. (10.61), which is in some sense the worst possible isodesmic reaction (perhaps one should call it the nihildesmic reaction) However, in the limit of perfect accuracy there is no need for the systematic cancellation of errors that isodesmic reactions are designed to provide, so we will consider Eq. (10.61) here for comparison.

As can be seen in Table 10.4, AM1 semiempirical theory is poorly suited for this application. With a polarized valence-double-ζ basis set, HF theory provides surprisingly good agreement with much higher levels of theory, but this is a case of fortuitous cancellation of errors, since use of a polarized valence-quadruple-ζ basis set decreases that agreement. The B3LYP model with a good basis set provides predictions that are not overall particularly much of an improvement over HF theory. The MP2 level with a large basis set does better for the more balanced isodesmic equation (10.63), but fares poorly with Eq. (10.62). Some improvement can be had with CCSD(T), but the cost of such a

a Average ± standard deviation from G2, G2MP2, G3B3, G3, and CBS-Q.

calculation far exceeds every other entry in the table. Since the G2, G2MP2, G3B3, G3, and CBS-Q models (all discussed in Chapter 7) are cheaper than CCSD(T)/cc-pVQZ and moreover designed specifically for the purpose of computing enthalpies of formation, there is ample reason to focus more closely on their performance.

Rather than attempting to rationalize why any one of these composite levels might be more or less good than another, let us examine their joint performance. The final

row of Table 10.4 provides the means and standard deviations of the predicted Hf,o298 (H2NOH) values from these levels for Eqs. (10.61) to (10.63). The largest standard deviation is associated with Eq. (10.61), the next largest with Eq. (10.62), and the smallest, only 0.19 kcal mol−1, with Eq. (10.63). This trend is entirely consistent with the above discussion of the relative quality of the three isodesmic equations, and provides some quantitative feel for how difficult the accurate computation of an atomization energy really is. Given this analysis, it appears reasonable to take the average value from the last five methods and Eq. (10.63) as a best estimate: −11.4 kcal mol−1. Further support for this choice comes from considering a different reaction, namely

Note that this is the analog to Eq. (10.62) with hydrogen peroxide replacing hydroxylamine. In this case, all enthalpies of formation are known experimentally to high accuracy, so the performance of the various theoretical models may be directly assessed. Applying the same averaging procedure, one finds that the models predict an enthalpy of formation for H2O2 that is too negative by 0.3 kcal mol−1. Note that if one assumes that this correction may be applied to the results from Eq. (10.62) for hydroxylamine, one predicts −11.4 kcal mol−1, in perfect agreement with the uncorrected results from Eq. (10.63).

Note that the average atomization energy prediction differs from −11.4 kcal mol−1 by only 1.1 kcal mol−1, which is about the range of accuracy typically quoted for the models