# Types of thermochemical data

(For brevity, this discussion is limited to gas-phase species)

## Output Data

A thermochemical (a.k.a. thermodynamical) table or database provides
basic thermodynamical data for a broad range of chemical species of
interest. The thermochemical data consists of standard thermodynamical
functions (enthalpy of formation, heat capacity, entropy, enthalpy
increment, Gibbs function, etc.) These functions are species-dependent
and temperature-dependent. Two particular temperatures, 298.17 K
(standard temperature, favored by general chemists) and 0 K (favored by
spectroscopists), have a special meaning, since they are common
reference points.

Thermochemical tables differ vastly in what they offer as an output. On
the meager end, for every considered species the table simply tabulates
the enthalpy of formation (sometimes supplemented by entropy and Gibbs
energy) at 298 K (CRC Tables). Some tables list in addition the 0 K
enthalpy (NBS Tables). On the other extreme, the thermochemical table
provides for every considered species a full listing of all standard
functions at a number of temperatures differing by some convenient
increment (JANAF, Russian compilation of Gurvich et al.)

The thermochemical tables also differ vastly in what pedigree
information they provide. Some tables are unannotated (NBS) or provide
only a generic list of sources (CRC Tables). Some give for each species
a single reference to the source for the selected enthalpy of formation.
The most valuable tables provides for every species references to all
sources of data that were considered, provide a description that
explains how and why the particular enthalpy of formation was selected,
and list the molecular constants used to calculate the partition
function (JANAF, Russian compilation). Such tables are often referred to
as "critical data evaluations".

The primary *raison d'être* for a thermochemical table is to allow the
user to retrieve the value of a particular function at a particular
temperature for one, or more frequently, several species. If the desired
temperature is not listed, the user derives the values by interpolation.
The extracted values are then typically used to calculate some
measurable thermochemical quantity, such as a reaction enthalpy, free
energy, kinetic equilibrium constant, etc.

## Experimental Input Data

The input data from which these thermochemical functions are derived can
be conveniently partitioned into two sets.

*Species-specific data*. A change in this data affects primarily only
one species. This subset is typically of spectroscopic origin (symmetry
and multiplicity of the ground electronic state, vibrational frequencies
and anharmonicities, rotational constants/moments of inertia/geometry,
term values for excited electronic states, etc). This data is used to
construct the partition function for the species of interest, through
which the thermal dependence of the thermochemical functions can be
obtained.

*Species-interconnecting data*. The nature of this data transcends a
single species, since its links together in some fashion thermochemical
properties of several species. A change in this data typically affects
more than one species. These are thermochemical quantities that pertain
to a chemical reaction, such as bond dissociation energies, reaction
enthalpies, kinetic equilibrium constants, etc. They come from a variety
of measurements (kinetics, photoionization, photodissociation,
calorimetry, etc.) The data is primarily used to obtain the enthalpies
of formation of one species relative to other species. The enthalpies
are anchored to a firm scale by reference to elements in their standard
states, which are by convention defined as having a zero enthalpy of
formation at all temperatures.

## Theoretical Input Data

*Spectroscopic theory.* This theory can be described as an empirical
superstructure sitting on a quantum mechanical foundation. Its use is
implicit in the experimental input data of spectroscopic origin, since
spectroscopic constants such as vibrational frequencies, rotational
constants, and even some dissociation energies have been obtained by
interpreting observed spectroscopic transitions within the
quantum-mechanical model of a vibrating rotator.

*Electronic structure theory.* With very few exceptions, quantities
obtained from electronic structure theory have been largely ignored by
thermochemists. However, at a sufficiently high level of theory, and for
sufficiently small species, calculated atomization energies and reaction
enthalpies start approaching chemical accuracy. This is also true for
other properties, such as vibrational frequencies, geometries, etc.
Hence, when experimental data is dodgy or entirely missing, good
calculations can provide a very valuable input. The main problem of
using theoretical data is the general lack of error bars, although for
some types of calculations (the most notorious being those of the Gn
series, such as G2, G3, and variations such as G3(MP2) or G3//B3LYP)
have at least "generic" error bars developed by comparing calculated and
experimental properties for a number of species.

# How are standard thermochemical tables constructed.

For every species involved, the enthalpy of formation is derived at a
single temperature from species-interconnecting data.
Temperature-dependent heat capacity, entropy, and enthalpy increments
are calculated from species-specific data, and together with the
selected enthalpy of formation, used to develop the temperature
dependence of the enthalpy of formation and the remaining functions. In
some cases the temperature dependence obtained from species-specific
data may be used to re-reference the species-interconnecting data to the
desired temperature.

Since species-interconnecting data produce enthalpies of formation only
relative to other species involved in the chemical reaction, the
standard way in which tables of enthalpies of formation are developed is
through a manual and sequential "aufbau" process, i.e. a tabulation of
enthalpies is developed in incremental steps. The procedure starts from
the elements in their standard states and follows a particular sequence
of elements. Each of these steps adds the enthalpy of formation of
another species to the table, using experimental measurements that
relate its thermochemical properties to those that have been determined
in previous steps. Once a step is completed, the enthalpy of that
species is considered fixed, and is used in following steps as a
constant. Since during any step all but one enthalpy of formation have
to be already known and fixed, the procedure becomes a stack of highly
localized sequential steps

The sequential building process has dear consequences. The localized
nature of the steps prevents the derived values and the attached error
bars to properly reflect global relationships implicit in the input
data. Every enthalpy of formation has a large number of dependencies on
other entries in the table, which are difficult to disentangle. As a
consequence, it is extremely difficult to update the table with new
measurements. Since nobody is willing to sequentially redo the whole
table in order to incorporate her new measurement, such updates are
again highly local, and while they may improve the enthalpy of one
specific species, they generally introduce even larger global
inconsistencies.

# The approach of an active thermochemical table.

The most important difference is in how the species-interdependent data
is treated. The active table approaches the information content hidden
in the species-interdependent data from the viewpoint of a
thermochemical network. Every vertex (node) of this network represents
the enthalpy of formation of one species. The species-interdependent
data define the topology of the network graph by providing the edges
(links) in the network. Competing measurements provide multiple
(parallel) links in the network. The relational information defining the
topology maps onto a set of equations, with the enthalpies of the
involved species as unknowns, the stoichiometry of the reactions
defining the coefficients, and the measurements and their error bars
providing the free elements. Since the number of equations regularly
exceeds the number of unknowns, the system is overdetermined. The best
solution to the network is obtained in two steps. The first step
statistically analyzes the network with the goal of checking the error
bars of individual data items for consistency. All inconsistencies are
identified and proposals for their resolutions generated. The
resolutions generally proceed through incrementing the error bars of the
offending data items to their statistically indicated values, and the
analysis is repeated. During subsequent steps the error bars of the
offending data items usually oscillate up and down until
self-consistency across the whole network is reached. The analysis is
carried out both in unweighted and error-weighted space. The second step
occurs once the values and error bars of data items achieve consistency
across the whole network. This step consists of finding simultaneously
the optimal solution to all nodes by χ^2^ minimization (in
error-weighted space).

The species-specific data is used to prepare the network for analysis by
re-expressing all data items as reaction enthalpies at one common
temperature, and to calculate the partition functions and hence develop
the temperature dependence of the standard thermochemical functions.

It should be noted that, as opposed to the simultaneous solution
described above, the traditional sequential approach would correspond to
a procedure where the nodes are solved one at a time in a prescribed
sequence. Each of the steps corresponds to selecting a particular path
in the network leading from solved nodes to the next node, and ignoring
all other possible paths. Both the fact that an overdetermined system of
coupled equations is solved for one unknown at the time and that some of
the equations are ignored contribute to the lack of global consistency
in traditional tables.

# Benefits of the active table approach

Unlike the quantities found in a traditional table, the thermochemical
quantities (and their error bars) obtained from the active table
properly reflect the globalism of relationships implied by the
underlying thermochemical network. All values/error bars are consistent
in a global sense.

An active table allows rapid update with new measurements (and possibly
calculations) by globally propagating the new information through the
table. Quality and integrity of the table is protected throughout the
updates by error analysis, which runs in both directions: The error bar
of the new experiment may shrink error bars in the table; however, the
error bars of other experiments in the table might challenge, via the
statistical analysis discussed above, the error bar assigned to the new
experiment.

An active table allows \"what if\" tests. Such tests provide a critical
evaluation of the tested data, and its impact on prior thermochemical
knowledge, or, if the test corresponds to a gedanken experiment,
provides feedback on sensitivity of the network to various measurements.
The approach has also potential to become an interesting learning and
education tool, etc.

An active table can provide a ranked list of links that are missing or
weak from a statistical viewpoint, i.e. it can provide pointers to what
would be the most useful new experiments or calculations.