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10

Thermodynamic Properties

10.1 Microscopic–macroscopic Connection

In the very recent past, it has become possible under certain circumstances to observe single molecules in the laboratory. Nevertheless, the vast majority of chemical research concerns itself not with individual molecules, but instead with macroscopic quantities of matter that are made up of unimaginably large numbers of molecules. The behavior of such ensembles of molecules is governed by the empirically determined laws of thermodynamics, and most chemical reactions and many chemical properties are defined in terms of some of the fundamental variables of thermodynamics, such as enthalpy, entropy, free energy, and others.

Until now, we have for the most part concerned ourselves only with the potential and kinetic energies of individual electrons and nuclei in single molecules, and one of our rare connections to thermodynamics has been in some sense a misleading one, namely that we have often converted atomic units into other units more typically associated with macroscopic quantities, e.g., kcal mol−1 or kJ mol−1. This sometimes leads newcomers to the field to think of the atomic unit of energy, the hartree, as being enormously large, since 1 Eh is equal to 627.51 kcal mol−1. In reality, however, the hartree is a tremendously tiny unit, since ‘kcal mol−1’, as its name makes clear, refers to the energy associated with one mole (i.e., 6.0221 × 1023) of molecules, not with the single molecule that is the typical subject of an electronic-structure calculation. Moreover, when we make comparisons between two different calculations, say to determine the relative energies of two isomers, and we carry out such simple unit conversions, we tacitly (and often incorrectly) assume that the potential energy difference determined from the calculation is all that matters when comparing to a measured energy difference, that is almost always in the form of some thermodynamic quantity, most typically enthalpy or free energy.

In this chapter, the most common procedures for augmenting electronic-structure calculations in order to convert single-molecule potential energies to ensemble thermodynamic variables will be detailed, and key potential ambiguities and pitfalls described. Within the context of certain assumptions, this connection can be established in a rigorous way.

Note that the situation is less clear-cut for molecular mechanics calculations. As already discussed in Chapter 2, the ‘strain energy’ from a typical MM calculation must be thought

Essentials of Computational Chemistry, 2nd Edition Christopher J. Cramer

2004 John Wiley & Sons, Ltd ISBNs: 0-470-09181-9 (cased); 0-470-09182-7 (pbk)

of as a potential energy with a rather particular zero of energy, namely, one uniquely defined by all of the atom types found in the molecule. To make a connection with thermodynamics, the most common approach is to associate with each atom type a heat of formation, and then assume that the strain energies may be thought of as enthalpies (see Figure 2.8). In this instance, the connection between the microscopic and macroscopic regimes is hidden in the fitting procedure by which the heats of formation of the atomic types are assigned, and thermodynamic aspects must be considered to be wholly empirical. As such, we will not consider this point further.

10.2 Zero-point Vibrational Energy

The first step in moving from the microscopic regime to the macroscopic is to recognize that the Born–Oppenheimer potential energy surface is fundamentally a classical construct (although the energies of various points whose coordinates are defined by the fixed nuclear positions are determined from quantum mechanical calculations of the electronic energy). As already discussed in Chapter 9, when the motion of the nuclei on this surface is also accounted for in a quantum mechanical way, energy is ‘tied up’ in molecular vibrations. This is true even at a temperature arbitrarily close to absolute zero, since the lowest vibrational energy level for any bound vibration is not zero.

Within the harmonic oscillator approximation, the energy of the lowest vibrational level can be determined from Eq. (9.47) as hω/2 where h is Planck’s constant (6.6261 × 10−34 J s) and ω is the vibrational frequency. The sum of all of these energies over all molecular vibrations defines the zero-point vibrational energy (ZPVE). We may then define the internal energy at 0 K for a molecule as

where Eelec is the energy for the stationary point on the Born–Oppenheimer PES. Note that U0 is also often written as E0 in thermochemical literature.

One may legitimately ask what errors may be implicit in Eq. (10.1), introduced by invocation of the harmonic oscillator approximation. To answer that question it is instructive to consider which modes will be likely to be least harmonic in character. In general, we expect these modes to be the softest ones, e.g., weakly hindered torsions. However, such weak modes will also tend to have very small vibrational frequencies associated with them, and thus they will not contribute much to the ZPVE in any case (since ZPVE is linear in the frequencies). As a rule, then, the harmonic approximation does fairly well in computing ZPVE. Note, of course, that the frequencies themselves must be computed at a level of electronic-structure theory that ensures their acceptable accuracy. Thus, if one uses HF theory with some basis set that is known in general to require a scaling factor of 0.9 to bring computed frequencies into line with experiment, that same scaling factor should be used to compute the ZPVE (or, equivalently, the ZPVE should be computed using the scaled frequencies).

where i runs over all possible energy states of the system having energy Ei and kB is Boltzmann’s constant (1.3806 × 10−23 J K−1).

Within the canonical ensemble, and using established thermodynamic definitions, all of the following are true

where the notation associated with the partial derivatives in Eqs. (10.3) and (10.5) implies N and V held constant during differentiation with respect to T , H is enthalpy, P is pressure, S is entropy, and G is (Gibbs) free energy.

Of course, the elegance of Eqs. (10.3)–(10.6) is somewhat muted by the daunting prospect of finding an explicit representation of Q that permits the necessary partial differentiations in Eqs. (10.3) and (10.5) to be carried out. For a true ensemble, Q must be some fantastically complex many-body function involving a staggeringly enormous number of energy levels. So, in order to make progress, we indulge in a number of simplifying assumptions.

10.3.1Ideal Gas Assumption

We begin by assuming that our ensemble is an ideal gas. The first consequence of this assumption, since ideal gas molecules do not interact with one another, is that we may rewrite the partition function as

where the factor of 1/N ! derives from the quantum mechanical indistinguishability of the particles, ε is the total energy of an individual molecule, and the change on going from the first to the second line derives from expressing the exponential of all possible sums of energies as a product of all possible sums of exponentials of individual energies. On going

from the second to the third line, the sum has been changed so that it goes over discrete energy levels, rather than individual states, and gk is the degeneracy of level k. The quantity in brackets on the third line defines the molecular partition function q.

A second consequence of the ideal gas assumption is that P V in Eq. (10.4) may be replaced by N kBT . In the special case where we are working with one mole of molecules, in which case N = NA (Avogadro’s number), we may replace P V with RT , where R is the universal gas constant (8.3145 J mol−1 K−1).

10.3.2Separability of Energy Components

We have thus reduced the problem from finding the ensemble partition function Q to finding the molecular partition function q. In order to make further progress, we assume that the molecular energy ε can be expressed as a separable sum of electronic, translational, rotational, and vibrational terms, i.e.,

where again advantage is taken of the ability to express an exponential of sums as a product of sums of exponentials, and the separate lines make clear that the degeneracy of a total molecular energy level is simply the product of the degeneracies of each of its contributing components.

Note in Eqs. (10.3) and (10.5), Q always appears as the argument of the natural logarithm function. Using Eqs. (10.7) and (10.8), our assumptions to this point allow us to write

where going from the second to the third equality makes use of Stirling’s approximation for ln(N !) when N is large. This separation of terms by the logarithm function makes evident

that we may speak of individual components – electronic, translational, rotational, and vibrational – of thermodynamic functions within our approximate treatment. All that remains is to express the various components of the molecular partition function in some useful, preferably analytic, form.

10.3.3Molecular Electronic Partition Function

The electronic partition function is usually the simplest to compute. For a typical, closedshell singlet molecule, the degeneracy of the ground state is unity, and the various excited states are so high in energy that, at least at temperatures below thousands of degrees, they make no significant contribution to the partition function, so that we might effectively take

If we evaluate the electronic component of U using Eq. (10.3), we determine that it is, independent of temperature, simply Eelec.

In practice, it proves more convenient to work within a convention where we define the ground state for each energy component to have an energy of zero. Thus, we view Uelec as the internal energy that must be added to U0, which already includes Eelec (see Eq. (10.1)), as the result of additional available electronic levels. One obvious simplification deriving from this convention is that the electronic partition function for the case just described is simply qelec = 1. Inspection of Eq. (10.5) then reveals that the electronic component of the entropy will be zero (ln of 1 is zero, and the constant 1 obviously has no temperature dependence, so both terms involving qelec are individually zero).

Another commonly encountered situation involves a ground state of higher spin multiplicity than singlet, but with excited states still sufficiently high in energy that they play no role in the electronic partition function. In that case, the definition of Eelec as the zero of energy continues to make the exponential part of the partition function equal to 1, but the degeneracy is now 2S + 1, where S is the spin multiplicity ( 12 for doublet, 1 for triplet, etc.) Thus, the partition function is also 2S + 1. This still has no temperature dependence, so it makes no contribution to the internal energy, but it is no longer unity, so it makes a contribution to the entropy. In general, then, for the electronic components of U and S we simply use

where it should be emphasized to avoid confusion that the S on the l.h.s. of Eq. (10.12) refers to entropy, while that on the r.h.s. refers to spin. The usual choice that is made is to compute molar quantities of the thermodynamic functions, so that N is NA, in which case Eq. (10.12) becomes

We will assume molar quantities from this point forward.

In particular instances, the above approximation is insufficiently accurate because one or more excited electronic states lie close in energy to the ground state. A typical example occurs for heavy halogen atoms, where spin–orbit coupling creates 2P1/2 and 2P3/2 states having only a narrow energy separation. In such cases, explicit formation of the partition function cannot be avoided, but even then one typically need only include a small number of terms in the total sum, and evaluation is not unduly taxing.

10.3.4Molecular Translational Partition Function

To evaluate qtrans, we assume that the molecule acts as a particle in a three-dimensional cubic box of dimension a3 where a is the length of one side of the cube. The energy levels for this elementary quantum mechanical system are given by

where M is the molecular mass, and each energy level has associated with it the three unique quantum numbers nx , ny , and nz. Because the energy levels for the particle in a box are very, very closely spaced (at least for a box of macroscopic dimensions), the partition function sum may be replaced by an indefinite integral, and this integral can be evaluated analytically as

where the volume of the box is now written V as opposed to a3. Note that it is only the translational partition function that depends on volume, and it does so because particle-in- a-box wave functions cannot be normalized without choice of a specific, finite, non-zero volume (it can be shown that it is only the volume that matters, and not the shape of the box). It is this term, then, that dictates the necessity of choosing a ‘standard state’ volume to ensure comparison of thermodynamic values in a consistent fashion. We will have more to say about standard states below, but here we simply note that, because we have chosen to model our substance as an ideal gas, we may replace V by RT /P and specify a standardstate pressure instead. This is typically the language that is used in electronic structure calculations, and the usual choice for standard state is 1 atm pressure (corresponding to a standard-state molar volume of 24.5 L at 298 K).

Evaluating Eqs. (10.3) and (10.5) for a molar quantity of particles using Eq. (10.15) for the translational partition function gives

(The superscript ‘o’ indicates that a ‘standard state’ is being referred to (see below).) Conventionally, however, the last two terms in the final line of Eq. (10.9), i.e., those deriving from Stirling’s approximation, are typically assigned to the translational partition function as well. As they have no temperature dependence, this has no impact on Utrans; however, the entropy of translation becomes

Note that, because we are working under the assumption of an ideal gas, V o/NA in the term in brackets can be replaced by kBT /P o.

A noteworthy aspect of Eqs. (10.16) and (10.18) is that they are altogether free of any requirement to carry out an electronic structure calculation. Equation (10.16) is well known for an ideal gas and is entirely independent of the molecule in question, and Eq. (10.18) can be computed trivially as soon as the molecular weight is specified. Note, however, that the units chosen for the various quantities must be such that the argument of the logarithm in Eq. (10.18) (i.e., the partition function), is unitless.

The translational partition function is a function of both temperature and volume. However, none of the other partition functions have a volume dependence. It is thus convenient to eliminate the volume dependence of Strans by agreeing to report values that use exclusively some volume that has been agreed upon by convention. The choices of the numerical value of V and its associated units define a ‘standard state’ (or, more accurately, they contribute to an overall definition that may be considerably more detailed, as described further below). The most typical standard state used in theoretical calculations of entropies of translation is the volume occupied by one mole of ideal gas at 298 K and 1 atm pressure, namely, V o = 24.5 L.

10.3.5Molecular Rotational Partition Function

In Section 9.3.1, the approach that is taken to solving the rigid-rotor nuclear Schrodinger¨ equation in order to compute rotational wave functions and energy levels was outlined, and the particular cases of diatomic molecules and polyatomic prolate tops were explicitly presented. The solution for the diatomic case is general for any linear molecule, so long as the molecular moment of inertia I is computed in the appropriate fashion for more than 2 atoms [Eq. (9.40)]. When the energy levels from Eq. (9.39) are used in the rotational partition function with their appropriate degeneracies, as usual taking the lowest energy rotational eigenvalue as the zero of energy, the sum can again be well approximated as an indefinite integral at ‘normal’ temperatures, and solving that integral we find for a linear molecule that

where σ is 1 for asymmetric linear molecules and 2 for symmetric linear molecules (i.e., belonging to the C∞v and D∞h point groups, respectively). To be more precise, the validity