## I.D.2. (VII.C.6.) |

**Karl K. Irikura**

Physical and Chemical Properties Division,

National Institute of Standards and Technology, Gaithersburg, MD 20899

Some computational methods, particularly *ab initio* techniques,
produce detailed molecular information but no thermodynamic information directly.
Further calculations are needed to generate familiar,
ideal-gas quantities such as the standard molar entropy (*S*°),
heat capacity (*C _{p}*°),
and enthalpy change [

Statistical thermodynamics calculations are necessary to compute properties as functions of temperature.
In some computations, such as *ab initio* electronic calculations of molecular energy,
the raw results do not even correspond to properties at absolute zero temperature
and must always be corrected.
All the corrections are based upon molecular spectroscopy,
with temperature-dependence implicit in the molecular partition function, *Q*.
The partition function is used not only for theoretical predictions,
but also to generate most published thermochemical tables.
Many data compilations include descriptions of calculational procedures (*1-3*).

**Corrections Unique to Ab Initio Predictions**

By convention, energies from *ab initio* calculations are reported in hartrees,
the atomic unit of energy (1 hartree = 2625.5 kJ/mol = 627.51 kcal/mol = 219474.6 cm^{-1})
(*4*).
These energies are negative, with the defined zero of energy being the fully-dissociated limit
(free electrons and bare nuclei).
*Ab initio* models also invoke the approximation that the atomic nuclei are stationary,
with the electrons swarming about them.
This is a good approximation because nuclei are much heavier than electrons.
Consequently, the resulting energies are for a hypothetical, non-vibrating molecule.
Although oscillators may be at rest in classical mechanics,
real (quantum-mechanical) oscillators are always in motion.
The small residual motion at absolute zero temperature is the *zero-point vibrational energy*,
abbreviated ZPVE or ZPE.
For a simple harmonic oscillator, the ZPE equals one-half the vibrational frequency.
Although all real molecular vibrations are at least slightly anharmonic,
they are usually approximated as harmonic.
Thus, the molecule's ZPE may be taken as one-half the sum of the vibrational frequencies.

Eq. 1 |

In equation 1, *N* is the number of atoms in the molecule
and the v* _{i}* are the fundamental vibrational frequencies.
There are 3

In practice, the ZPE correction is slightly complicated
by the observation that *ab initio* vibrational frequencies are often in error by +5% to +10%.
To compensate for this error, the computed frequencies are usually multiplied by empirical scaling factors.
The most recent recommendations are those of Scott and Radom (*5*).
For example, they suggest scaling HF/6-31G* frequencies
by 0.8953 to predict vibrational spectra (i.e., fundamental frequencies),
by 0.9135 for the computation of ZPEs,
by 0.8905 to predict enthalpy differences *H° *(298.15) - *H*° (0),
and by 0.8978 to predict *S*°(298.15).
The methods for computing these quantities are described below.
Common abbreviations and acronyms of the *ab initio* literature are defined
in the glossary (Appendix D) of this book.
In this Appendix, the degree sign (°) that indicates ideality and standard pressure (1 bar)
is omitted except where the thermal electron convention for ions is being emphasized (see below).

Enthalpies of formation depend upon the thermodynamic conventions for reference states of the elements.
Since this information is not intrinsic to an isolated molecule,
an *ab initio* reaction energy (i.e., energies for at least two molecules)
must be combined with experimental data to compute an enthalpy of formation,
Δ_{f}*H*°.

**Example: Δ _{f}H°_{0}
of hydrogen fluoride.**

There are many levels of approximation in

Eq. 2 |

Note that the ideal-gas energy and enthalpy are equal at 0 K,
since H = E + PV = E + *n*RT = E.
The optimized B3LYP/6-31G(d) bond lengths are 0.743, 1.404, and 0.934 Å
for H_{2}, F_{2}, and HF respectively,
in reasonably good agreement with the experimentally derived values
*r _{e}* = 0.741, 1.412, and 0.917 Å respectively (

**General Relationships of Statistical Thermodynamics**

In the present context, statistical thermodynamics is meant to include the methods used to convert molecular energy levels into macroscopic properties, especially enthalpies, entropies, and heat capacities. Molecular energy levels arise from molecular translation (i.e., motion through space), rotation, vibration, and electronic excitation. This information constitutes the spectroscopy of the molecule of interest and can be obtained experimentally or from calculations.

**Partition Function.**
The molecular energy levels ε* _{i}*
are used to compute the

Eq. 3 |

One typically chooses the lowest energy level to be the zero of energy,
so that no levels lie at negative energies.
From equation 3 it follows that the largest contributions to
*Q* are from the lowest energy levels.
Conversely, levels that lie far above *kT* (207 cm^{-1}
at room temperature) have only a minor effect on
*Q* and its derivative thermodynamic quantities.

**Thermodynamic Functions.**
Given the partition function,
the usual molar thermodynamic functions can be calculated based upon the following general equations.

Eq. 4 | |

Eq. 5 | |

Eq. 6 | |

Eq. 7 | |

Eq. 8 | |

Eq. 9 | |

Eq. 10 | |

Eq. 11 |

**Practical Calculations**

A complete set of molecular energy levels is almost never available.
To simplify the problem, one usually adopts a model in which translation,
rotation, vibration, and electronic excitation are uncoupled.
In other words, one makes the approximation that the different types of motion
are unaffected by each other and do not mix together.
This leads to a separability of *Q* into four factors
that correspond to separate partition functions
for translation, rotation, vibration, and electronic excitation.
This is shown in equation 12,
where the explicit dependence upon temperature has been dropped for simplicity.

Eq. 12 |

Different energy units are used conventionally in the fields of molecular spectroscopy,
quantum chemistry, and thermochemistry.
To provide some feeling for magnitudes, the values of the thermal energy *kT*,
at "room temperature" (298.15 K) and at 1000 K,
are listed in Table I in several units.
In this Appendix, all units are of the SI (*Système International*: kg, m, s, Pa, K)
unless otherwise indicated.

**Table I. Thermal energy ( kT) at two temperatures, expressed in various units**

Unit | Room temperature | 1000 K |
---|---|---|

kelvin (K) | 298.15 | 1000 |

wavenumber (cm^{-1}) |
207.2 | 695.0 |

Hertz (s^{-1}) |
6.212 × 10^{12} |
2.084 × 10^{13} |

kJ/mol | 2.479 | 8.314 |

kcal/mol | 0.592 | 1.987 |

electron volt (eV) | 0.0257 | 0.0862 |

hartree (atomic unit) | 0.000944 | 0.003167 |

**Translational Partition Function.**

Rigorously, *Q _{trans}* must be calculated from a sum over
all the translational energy levels that are available to a
molecule confined to a cubic box of volume

Eq. 13 | |

Eq. 14 | |

Eq. 15 | |

Eq. 16 |

As an example, we can calculate the standard entropy for neon ideal gas at *T* = 298.15 K.
The atomic mass is converted to SI units using the equivalence
*N _{A}* amu = 0.001 kg,
where

**Rotational Partition Function.**

The free rotation of a rigid molecule is also quantized
(the angular momentum and its projection are integer multiples of
*h*/2π),
so the rotational energy is restricted to certain discrete levels.
Rotational spectra are characterized by the constants
*A*,* B*,* *and *C*,
where *A* ≡ *h*/(8π^{2}*I _{A}*)
and likewise for

Linear molecules (

Fortunately, at high enough temperatures
(*kT* >> *hA*),
the sum can be replaced by an integral as it is for translation.
In the general case, the rotational partition function is given by equation 17.

Eq. 17 |

For linear molecules, equation 18 should be used instead.

Eq. 18 |

In these and subsequent equations, the symbol σ denotes the "rotational symmetry number" or "external symmetry number" for the molecule. This is the number of unique orientations of the rigid molecule that only interchange identical atoms. It preserves parity restrictions on the interchange of identical nuclei when summation is replaced by integration. Identifying the correct symmetry number is a common point of difficulty; it is discussed further below.

For the typical case (equation 17),
the thermodynamic functions are given by equations 19-21.

Eq. 19 | |

Eq. 20 | |

Eq. 21 |

Eq. 22 | |

Eq. 23 | |

Eq. 24 |

**External Symmetry Number.**

Some computer programs, such as many *ab initio* packages,
determine the molecular symmetry and external symmetry number (σ) automatically.
If such a program is unavailable, σ may be determined by hand.
With practice, this becomes very fast.

If you are familiar enough with group theory to identify the molecule's point group
(*10*), then σ can be determined from Table II
(*11*).
Without identifying the point group,
one can count manually the number of orientations of the rigid molecule
that interchange only identical atoms.

**Table II. Symmetry numbers corresponding to symmetry point groups**

Group | σ | Group | σ | Group | σ | Group | σ |
---|---|---|---|---|---|---|---|

C_{1}, C, _{i}C, _{s}C_{∞v} |
1 | D_{∞h} |
2 | T, T_{d} |
12 | O_{h} |
24 |

C, _{n}C, _{nv}C_{nh} |
n |
D, _{n}D, _{nh}D_{nd} |
2n |
S_{n} |
n/2 |
I_{h} |
60 |

For example, the benzene molecule (C_{6}H_{6})
belongs to the *D*_{6h} point group.
From Table II, σ = 12.
Alternatively, one can draw the molecule as a hexagon with numbered vertices.
Rotating the drawing by *n* × 60°,
where *n* runs from 0 to 5, generates six different orientations
that are distinguished only by the artificial numbering of the vertices.
Each of these six orientations can be flipped over to generate another orientation,
for a total of 12 unique orientations, σ = 12.

Another example is methyl chloride, CH_{3}Cl.
This belongs to the *C*_{3v} point group,
so σ = 3.
Alternatively, one can artificially number the hydrogen atoms
and see that there are three unique orientations,
related by rotations of *n* × 120°
(*n* = 0-2) around the C-Cl bond axis.

Chlorobenzene, C_{6}H_{5}Cl,
belongs to the *C*_{2v} point group,
so σ = 2.
Alternatively, one can again number the hydrogen atoms
and see that there are two unique orientations,
related by rotations of *n* × 180°
(*n* = 0-1) around the C-Cl bond axis.
In contrast, toluene (C_{6}H_{5}CH_{3})
belongs to the *C _{s}* point group, so σ = 1.
There are no ways to rotate or flip the molecule

**Vibrational Partition Function.**

To complete the simple rigid-rotator/harmonic oscillator (RRHO) model,
one must consider the molecular vibrations.
As indicated in the discussion of ZPE (equation 1),
a molecule that contains *N* atoms
has 3*N*-6 vibrational frequencies (3*N*-5 for linear molecules).
The partition function is given in equation 25,
where the product runs over all vibrational frequencies v* _{i}*.
The corresponding thermodynamic functions are given by equations 26-28.

Eq. 25 | |

Eq. 26 | |

Eq. 27 | |

Eq. 28 |

**Example: Hydrogen Fluoride**.
Earlier we used the results of *ab initio* calculations
to obtain a value for Δ_{f}*H*°_{0}(HF).
The other equations above permit us to compute *ab initio* thermodynamic functions,
which will provide an enthalpy of formation at the more useful temperature of 298.15 K.
Results are summarized in Table III.
For simplicity, we will neglect the naturally occurring heavy isotopes of hydrogen.
The molecular weight of ^{1}H^{19}F is 20.006 amu.
Using equation 14, as done above for neon,
leads to *S _{trans}* = 146.22 J mol

For enthalpy and heat capacity,
the B3LYP/6-31G(d) frequency is scaled by 0.9989 (*5*) to obtain ν = 3983 cm^{-1}.
The heat capacity *C _{p}*(HF) is calculated using equations 15, 23, and 27,
leading to

**Table III. Results for Hydrogen Fluoride Example**

Contribution | S, J/(mol K) |
C, J/(mol K)_{p} |
[H(298.15)-H(0)], kJ/mol |
---|---|---|---|

Translation | 146.22 | 20.79 | 6.20 |

Rotation | 27.67 | 8.31 | 2.48 |

Vibration | 7 × 10^{-7} |
1 × 10^{-5} |
2 × 10^{-4} |

Total |
173.89 |
29.10 |
8.68 |

**Electronic Partition Function.**
Although they may not have low-lying electronic excited states,
some molecules have degenerate electronic ground states.
Free radicals are a common example.
They may have unpaired electrons in their electronic ground states
and a net electron spin of *S* = *n _{unpaired}*/2,
where

Eq. 29 | |

Eq. 30 | |

Eq. 31 | |

Eq. 32 |

**Example: Entropy of Methyl Radical.**
For simplicity, we again neglect minor isotopes.
Results are summarized in Table IV.
The molecular weight of CH_{3} is then *m* = 15.023 amu,
so *S _{trans}* = 142.65 J mol

**Example: Entropy of Hydroxyl Radical.**
For simplicity, we again neglect minor isotopes.
Results are summarized in Table IV.
The molecular weight of OH is then *m* = 17.003 amu,
so *S _{trans}* = 144.19 J mol

**Table IV. Methyl and Hydroxyl Examples,
S°_{298}, J/(mol K)**

Contribution | Methyl | Hydroxyl^{a} |
Hydroxyl^{b} |
---|---|---|---|

Translation | 142.65 | 144.19 | 144.19 |

Rotation |
43.50 | 28.22 | 28.22 |

Vibration | 1.99 | 5 × 10^{-6} |
5 × 10^{-6} |

Electronic | 5.76 | 11.53 | 11.08 |

Total |
193.9 |
183.9 |
183.5 |

**Internal Rotation.**
This refers to torsional motion, most commonly involving methyl groups.
There are three ways to treat such a rotor, depending upon its barrier to rotation.
The free and hindered rotor models require that an internal symmetry number,
σ* _{int}*, be included.
σ

**Free Rotor.**
If the barrier to rotation is much less than *kT*,
then the rotor may be considered freely rotating.
For a symmetric rotor such as a methyl group, the partition function is given by equation 33,
where *I _{int}* is the reduced moment of inertia for the internal rotation
and is given by equation 34 (

Eq. 33 | |

Eq. 34 |

Eq. 35 | |

Eq. 36 | |

Eq. 37 |

**Harmonic Oscillator.**
If the barrier to internal rotation is much greater than *kT*,
one can consider the torsion to be a non-rotating, harmonic oscillator.
Treatment is the same as for other vibrations.

**Hindered Rotor.**
This is the common, intermediate case, when the torsional barrier *V* is comparable to *kT*.
If the torsional potential is assumed to have the simple form
,
then the tables of Pitzer and Gwinn are usually used to compute the contribution of the hindered rotor
to the thermodynamic functions (*16,17*).
Their tables are in terms of the dimensionless variables *x* and *y*,
where and
and *I _{int}* is defined as for a free rotor (see above).

**Example: Entropy of Ethane at T = 184 K **[adapted from ref (

If we consider the torsion to be a free, unhindered rotor,
then we require the corresponding reduced moment of inertia.
In this case, in the rotor axis is aligned with the *A* axis of the molecule,
so that α = 1
and β = γ= 0 in equation 34.
This gives *I _{int}* =

To apply the hindered-rotor model we need a value for the torsional barrier height.
This can be estimated from the observed torsional vibrational frequency ν
(in s^{-1}) as *V*
≈ 8π^{2}*I _{int}*ν

**Table V. Ethane Example, S°_{184} (J mol^{-1} K^{-1})**

Contribution | Harmonic Rotor | Free Rotor | Hindered Rotor |
---|---|---|---|

Translation | 141.26 | 141.26 | 141.26 |

Rotation | 62.17 | 62.17 | 62.17 |

Vibration | 0.25 | 0.25 | 0.25 |

Torsion | 3.11 | 10.09 | 3.99 |

Total |
206.8 |
213.8 |
207.7 |

**Charged Molecules: Two Conventions.**
The balanced chemical equation describing an ionization process involves at least one free electron.
There are two major conventions for the thermodynamic properties of the electron.
Most compilations of thermochemical data adopt the *thermal electron convention*.
In this convention, the free electron is treated as a chemical element,
so that its ideal-gas enthalpy of formation is zero at all temperatures.
In contrast, most of the literature in mass spectrometry and ion chemistry adopts the *ion convention*,
sometimes also called the *stationary electron convention*.
In this convention, the enthalpy content of the electron is ignored.
At absolute zero temperature there is no difference between the two conventions,
but in general enthalpies of formation under the two conventions are related by equation 38,
where *q* is the (signed) charge on the ion in question (±1 in most cases).

Eq. 38 |

**Chemical Kinetics**

The equilibrium constant for a reaction is
*K _{eq}* = exp(-Δ

Eq. 39 |

Experimental, temperature-dependent rate constants are often presented as
an *Arrhenius plot* of *r*(*T*) vs. 1/*T*.
This is motivated by the observation that such plots are nearly linear,
*r*(*T*) = *A* exp(-*E _{a}*/

Eq. 40 | |

Eq. 41 |

Eq. 42 | |

Eq. 43 | |

Eq. 44 |

*Ab initio* energies are now precise enough that it is becoming common
to use kinetic theories more sophisticated than simple transition-state theory.
When the reaction coordinate is dominated by motion of a hydrogen atom,
corrections for quantum-mechanical tunneling are often made (*21*).
The simplest is the Wigner correction,
which requires only the imaginary vibrational frequency
ν^{‡}*i*
associated with the reaction coordinate.
To apply this correction, the calculated rate is multiplied by *F _{tunnel}* (equation 45).

Eq. 45 |

**Units and Constants**

In actual calculations, many practical difficulties involve incompatible units. In addition to the standard units of the SI, many others are in use, usually for historical reasons. Conversion factors among selected units are provided in Table VI. For convenience, the values of commonly-used constants are collected in Table VII. Detailed information is available on-line at http://www.physics.nist.gov/PhysRefData/contents.html.

**Table VI. Unit Conversions**

Quantity | Unit | Conversion^{a} |
SI Unit |
---|---|---|---|

energy | hartree (atomic unit) | 2625.500 | kJ/mol |

energy | cal | 4.184 | J |

energy | cm^{-1} (wavenumber) |
0.01196266 | kJ/mol |

energy | eV | 96.48531 | kJ/mol |

energy | K (temperature) | 8.314511 × 10^{-3} |
kJ/mol |

distance | Å | 10^{-10} |
m |

distance | bohr (atomic unit) | 5.291772 × 10^{-11} |
m |

mass | amu or u |
1.660540 × 10^{-27} |
kg |

pressure | bar | 10^{5} |
Pa |

pressure | atm | 101325 | Pa |

pressure | Torr or mm-Hg |
133.32237 | Pa |

pressure (density) | cm^{-3} (at 298.15 K; ideal gas) |
4.16643 × 10^{-15} |
Pa |

pressure (density) | cm^{-3} (arb. temp.; ideal gas) |
10^{6} kT |
Pa |

pressure (density) | M or mol/L (ideal gas) |
10^{3} RT |
Pa |

dipole moment | atomic unit | 8.478358 × 10^{-30} |
C m |

dipole moment | D (debye) | 3.335641 × 10^{-30} |
C m |

**Table VII. Physical Constants**

Quantity | Value |
---|---|

k |
1.38066 × 10^{-23} J K^{-1} |

N_{A} |
6.022137 × 10^{23} mol^{-1} |

R = kN_{A} |
8.314510 J mol^{-1} K^{-1} |

h |
6.626076 × 10^{-34} J s |

c |
299792458 m s^{-1} |

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