1. Introduction
This chapter aims at understanding what is at stake when thermodynamics is used in chemical quantum calculations. Quantum chemistry and thermodynamics seem to be two incommensurable scientific worlds the assumptions and the statements of which are thoroughly different. So my questions are: How was thermodynamics been integrated into the chemical quantum background at the very beginning of quantum chemistry? What are its role and status in current
I refer both to history and epistemology to grasp this entanglement of scientific approaches. First of all, I propose to analyze how thermodynamics became involved in chemistry. In this respect, I will point out how the concept of energy provides the old chemical affinity with a quantitative tool to understand chemical transformations. The birth of thermochemistry aroused opposition between two old rival conceptions of matter that framed the history of chemistry, that is to say the aggregate and the ‘mixt’ stances A ‘mixt’ is a chemical combination composed of elements but not bearing the same properties as the constitutive elements. Conversely, an aggregate is a mere additive combinations of elements and their properties.
I will then highlight that this duality of conceptions was still at stake when Mulliken and Pauling created two different quantum chemical approaches. In this context, thermodynamics was not just used as a mere tool to calibrate methods; it also guided the contrivance of new quantum concepts or parameters from the outset. Following this line of reasoning, I will query how the concept of ‘state’, be it electronic or thermodynamic, allows us to bridge thermodynamics to quantum chemistry in a different way. I will indicate why and how the second law of thermodynamics is reflexively of importance to understand molecular calculations and to better grasp the relation between a molecular “whole” and its respective parts.
These investigations are widened by a global overview of the ways thermodynamic parameters are currently involved in workaday quantum methods in order to describe molecular reactivity.
To conclude, the paper will query the status of thermodynamics in predictive quantum methods. I will insist on the status of the concept of energy and the heuristic power of the second law of thermodynamics on quantum grounds.
2. The integration of thermodynamics into chemical grounds: From a qualitative to a quantitative affinity
The new rules of the French Royal Academy of sciences (1699), Wilhelm Homberg’s work on the interchangeability of ‘average’ -now called ‘neutral’- salts, the mechanist philosophy influences at the end of the seventeenth century, and as well Paracelsus and the alchemists’ traditions, paved the way for the empirical production of affinity tables during the 18th century. From Etienne-François Geoffroy (1718) to Bergman (1775), these tables were multiplied; some chemists, such as Guyton de Morveau (1773), developed the first experimental devices to quantify these affinities (Mi Gyung, 2003; Partington, 1962).
A shift of the explanatory function of the principles – Aristotelian, Paracelsian, or other, which previously accounted for qualities and chemical transmutations, towards the state of union between two chemical substances and the concept of process which implies union and disunion, gradually occurred (Bensaude-Vincent & Stengers, 1996). This major epistemological upheaval led to the attraction between chemical bodies being operationally redefined within the context of salts chemistry. The key question of the force or power which governed the chemical combinations remained rather unclear and mysterious according to Henri Sainte Claire Deville (Deville, 1864) until the chemists integrated knowledge of calorific and thermodynamics into their own practices.
Using a new calorimeter with mercury, J.T. Silbermann and P.A Favre showed for the first time in 1852 that a chemical decomposition could involve a release of heat. At the same time, Julius Thomsen published a paper entitled The French original sentence is : ‘Lorsque la combinaison se produit, il se dégage une quantité de chaleur proportionnelle à l’affinité des deux corps.‘
As Thermochemistry began to develop, chemists paid attention to other facts which first appeared foreign from each other. In 1852, Edmond Fremy and Henri Becquerel showed that the production of ozone was an incomplete reaction, a conclusion that Berthelot and Pan de Saint Gilles also reached for the esterification reaction ten years later. The chemical reaction appeared limited and dependent on the time factor, Sainte Claire Deville and his collaborators widened and strengthened those findings thanks to many experiments (Daumas, 1946). After many attempts, Maximilian Güldberg and Peter Waage asserted in 1861 that they were able to "find for each element and each chemical combination, numbers which express their relative affinity" The French original sentence is : " (…) trouver pour chaque élément et pour chaque combinaison chimique, des nombres qui expriment leur affinité relative"
The development of the energy approach in chemistry was the result of a fortuitous combination of independent works proposed by Wilhelm Hortsmann in Germany, by Josiah Willard Gibbs in America and by Bakhuis Roozeboom and J.H. Van't Hoff in Holland. Hortsmann integrated Rudolf Clausius’ considerations on isolated systems into chemistry. In so doing, he rediscovered in 1873 the law of mass action by means of calculation without having any idea that it had already been found on other grounds. The same year, Gibbs, published a paper entitled ‘
Applications to experimental chemistry by the Dutch school, for example, Roozeboom had to cope with difficulties in interpreting hydrobromic acid decomposition in the presence of water in the gas phase. His colleague physicist J.D. Van der Waals suggested to him to use Gibbs’s work and helped him to put forward the so-called
At the beginning of the twentieth century, chemists attempted to know not loner why, but how matter is transformed. Chemical kinetics studied the process of transformation of matter. Swante Arrhenius introduced the concept of energy activation, researches gradually turned to focus on the question of the energy transfer and the direction of collisions between chemical bodies. Wilhelm Ostwald succeeded in describing chemical equilibrium without making any reference to atoms (Ostwald, 1919). Two antagonistic approaches of matter were at stake. Thermochemistry revolved around energy and denied any reality to atoms whereas chemical kinetics was based on the atomic assumption. Thomsen, for instance, used structural theory to assign heats of formation to specific bond types found in organic molecules. In this respect, he tried to reduce chemical properties to a mere juxtaposition of atomic properties. Others, like F.W. Clarke tried to connect the heat of formation with the one and only number of atomic linkages within the molecule. By doing so, he tried to connect valence with affinity (Weininger, 2001). All the attempts that tried to understand affinity thanks to additive and reductive descriptions failed.
To sum up this first part, I would like to emphasize that the integration of thermodynamics within the frameworks of chemistry was made possible because chemists were looking for a quantitative measure of affinity. The way thermodynamics became thermochemistry depended on the instrumentation and the practices that chemists contrived to tackle the challenge of affinity. As the philosopher Joseph Rouse points out: ‘Practices are not just pattern of action, but the meaningful configurations of the world within which actions can take place intelligibly, and thus practices incorporate the objects that they are enacted with and on and the settings in which they are enacted’. (Rouse, 1996, p.135). Thermodynamics was thus integrated into chemical projects and then transformed by such integration because it made chemists goals achievable and intelligible within such new practical backgrounds.
I suggest we should take more distance and consider the whole history of chemistry to analyze the way this integration actually took place. Let us widen the circle to grasp what is at stake behind this integration and how the duel between different conceptions of matter will remain active at the very beginning of quantum chemistry. This study will enable us to understand the role of thermodynamics in the first chemical quantum calculations.
3. The integration of thermodynamics into first quantum methods: The reviving of the aggregate/’mixt’ duel
3.1. Two conceptions of matter and the thermodynamics embodiment within chemical practices
First and foremost, I would like to develop the opposition of conceptions of matter we previously stressed. Duhem’s claim for an energy description of molecules that need not rely on any atomic assumption reminds us of other historical oppositions.
In the seventeenth century for instance, Nicolas Lemery in his famous
Not only did thermodynamics enable chemists to construe a quantitative version of affinity but it also fitted very well into the cultural background that had been framing chemists’ activities for a long time. Thermodynamics embodiment within chemical practices was thus at least twofold; it provided chemists with quantitative tools for understanding chemical reaction while recasting old oppositions of matter representations. Along with this perspective, thermodynamics could easily be integrated into the usual chemical way of thinking about matter while reconfiguring it. As Rouse claims (1996, p.157): ’In order to understand how scientific knowledge is situated within practices, we need to take account of how practices are connected to one another, for knowledge will be established only through these interconnections. Scientific knowing is not located in some privileged type of practice, whether it be experimental manipulation, theoretical modeling, or reasoning from evidence, but in the ways these practices and others become intelligible together.’
Duhem focused his work on the dichotomy between the ‘mixt’ and the aggregate referring to Aristotle’s philosophy (Needham, 1996). Like Sainte-Claire Deville and Berthellot, but not because of the same positivist reasons, he rejected atomism then deeply rooted in structural organic chemistry. According to the structural molecular paradigm, the physical arrangement of the constituent elements accounted for the properties of the whole compound. Since Lavoisier, chemists have been explaining the properties of compounds by reference to the nature, the proportion and, more recently, the bonds of its constitutive parts, be they atoms or elements: a logic that runs from simple to complex frameworks in post-Lavoisian chemistry (Bensaude-Vincent & Simon, 2008). Conversely, the holistic energy approach used compounds to explain the properties of the elements. In this respect, atomism had a weak explanatory power because it could not completely illuminate chemical processes. According to Duhem, chemical formula could make chemists believe that substances remained unchanged when they entered into combinations whereas they only existed potentially within them (Duhem, 1902). Joseph Earley has recently proposed an argument on the same lines. He uses the example of sea water in which salt and water cease to exist in their actual states–because for instance of solvatation- but they can be reproduced by distillation (Earley, 2007). When the ‘mixt’ ceases to exist, it is made to reproduce its separate constituents as Venel might have asserted. In this respect, water and salt potentially exist in sea water but do not actually exist within it. Duhem then undertook to retranslate Aristotle’s concept of power into that of the thermodynamic potential (Duhem, 1902). Measurable properties and mathematics allowed him to describe chemical reaction within the context of thermochemistry.
Duhem rejected both the idea of valence taken as an intrinsic atomic property and the concept of atomicity. According to him, the whole components could only give rise to valence information but not the contrary. The opposition between a holistic approach of chemical bodies on the one hand and the aggregative atomic description on the other hand will appear of primary importance at the very beginning of quantum chemistry. I propose to study how Linus Pauling and Robert Sanderson Mulliken created the first chemical quantum approaches in the context described before and how they integrated thermodynamics and quantum mechanics into chemistry.
3.2. The ‘mixt’ and the aggregate: A framework for the embodiment of thermodynamics into quantum chemistry?
Both standardization and precision were required if thermodynamic bond measurements were to play a significant role in calibrating innovative methods and stabilizing new theories about affinity as well as about valence or the chemical bond (Servos, 1990). The Russian-Polish Wojciech Swietolawski played a leading role in this challenge (Médoire & Tachoire, 1994). His work provided chemists with more accurate average bond energies that legitimized heat of reactions calculations. Weininger clearly shows how those thermodynamic data made researchers get to grips with valence within the atomist conception. He points out for instance how Morris Kharash used the Niels Bohr’s orbit model to propose a physical picture of thermodynamic quantities. This heuristic approach validated by Swientoslawski’s data enabled him to derive heats of combustion for hydrocarbons in quite good agreement with experiment (Weininger, 2001). But it was Linus Pauling who succeeded in bridging valence, atomic theory and thermochemistry.
Pauling’s work constitutively entangled thermodynamics with the Pauli Exclusion Principle, Heisenberg and Dirac’s approach of resonance, structural chemistry and Born’s probabilistic description (Pauling, 1928). We should bear in mind that he was first trained as a crystallographer to understand the way he shaped his experimental and theoretical crowded network that was the
To understand Pauling’s molecular description, one needs: (1) to connect the molecular structure to its constitutive atoms; (2) to study how those atoms interact from within the molecule. This model retains the integrity of the atoms inside the molecule, a molecule is considered as an
The study of diatomic molecule enabled Pauling to propose the concept of ‘normal
Mulliken proposed a very different quantum approach based on molecular spectroscopy. With regard to the concept of valence considered as an intrinsic property of the atom, Mulliken opposed the notion of ‘energy state’ deduced from molecular spectra on the basis of an
The semantic shift from the concept of molecular
At the very beginning of his investigations, Mulliken mainly used molecular spectroscopy data. He seldom referred to thermochemistry except for necessary calibration requirements. It is important to notice nevertheless that thermodynamics was influential when he envisaged the study of larger molecules by using group theory. I think it is important not only to check if his holistic molecular conception changed the way thermodynamics became involved in chemical quantum works; but also to compare it to Pauling’s own use of thermal data.
Mulliken’s studies of hyperconjugation are a relevant case study to grasp the role and the status of thermodynamics in such a chemical quantum background (Mulliken et al., 1941). Mulliken’s calculations taken in connection with thermal and bond distance data indicated the conjugating power of chemical groups such as the landmark methyl group. With respect to strength and stability, he could then label the single or the multiple bonds of a conjugated system as acceptor and donor bonds, respectively. The thermal data allowed him to postulate that the hyperconjugation energy of saturated hydrocarbons was to a good approximation a function only of the numbers of different types of bonds. Using localized and non-localized molecular orbitals, he described the conjugation or resonance energy as the energy of delocalisation. In order to approximate quantitative calculations, he wrote the molecular orbital as a Linear Combination of Atomic Orbitals –LCAO- within the Hartree-Fock self-consistent field approach –labelled LCAO MO SCF-.
Unlike Pauling, he systematically used heats of combustion rather than bond energies referring to Karash and W.G. Brown’s corrected tables mainly construed by using hydrogenation heats data. Mulliken and al. wrote: ‘Our procedure for deriving conjugation energy from thermal data is similar to that of Pauling and Sherman who, assuming additivity of bond energies (with corrections for special groups), compute energies of formation and interpret deviations therefrom as resonance energies. However, we shall work with heats of combustion.’ (Mulliken et al., 1941).
Heats of combustion enabled Mulliken to put forward formula to calculate conjugation energies from heats of combustion that fitted the available consistent data for gaseous saturated hydrocarbons - except methane - with considerable accuracy – mostly better than 1 kcal. The current practice of research then involved a rich set of corrections within which quantum formalism, approximations, chemical knowledge and thermochemistry were deeply intertwined in order to create a stabilized composite knowledge of conjugation energy for particular types of molecules. For instance, Mulliken tailored Lennard-Jones’s curves to make them fit the empirical data, he then determined wave function coefficients by defining and substituting new parameters in the secular determinant, and finally extracted from the computed conjugation energies some energy quantities - the third-order conjugation energy - to make a direct comparison with observed conjugation energy. By trial and error, a host of other corrections and readjustments enabled him to determine the total conjugation energy and to compare it to thermodynamic outcomes. Mulliken and al. wrote (p.56): ‘Perhaps the most uncertain feature of our analysis is the derivation from thermal data. (...). Our empirical parameters, our bond order curve, and our numerical conclusions would then be so strongly altered, since they are decidedly sensitive to variations in the empirical conjugation energies to which they are fitted. Nevertheless, their self-consistency gives a distinct support to our numerical results, since we have found that such self-consistency is not easy to attain.’ (Mulliken et al., 1941). The authors called for more accurate thermal and bond distances data, those researches got into an endless and open circle of refinements that linked calculations with empirical data. It is of importance to notice that this work led the authors to provide Hückel’s resonance parameter ‘β’ with a new interpretation that allowed a more satisfactory understanding of energy interactions within unsaturated molecules. This theoretical accommodation was then confirmed by spectroscopic data. Thermodynamics not only took part in a motley complex of scientific practices that made it possible for a quantum chemist to calculate molecular properties and to predict chemical reactivity, but it also partly altered the meaning of the theoretical quantum background. I wish to emphasize that thermodynamics was not a mere tool for calibrating a semi-empirical method but a constitutive active part of a techno scientific network that Mulliken and others shaped to study a molecule understood as a ‘mixt’.
In addition to this conclusion, there are other interesting facts we should take a look at. Mulliken and Parr studied the decrease in ‘π’ electron energy for the change from a Kekulé to a proper benzene structure by using a
I claim that this difference of practice from Pauling to Mulliken was in a way a consequence of the two conceptual schemes at stake. On the one hand, the aggregative Pauling’s approach focused on a reified chemical bond that resulted in valence electrons share. Pauling was indeed interested by the formation energy of a molecule from its parts. On the other hand, Mulliken used chemical reaction combustion data because he considered the way the ‘whole’ molecule reacted and released energy by thermal transfer in the presence of other chemical reactants and their surroundings. Pauling’s bottom-up analysis collapsed Mulliken’s holistic way of thinking. I think that my statement is to be qualified insofar as we should wonder if pragmatic reasons were also at stake concerning this choice of data. Heats of combustion corrected tables probably were more useful for Mulliken than others.
At that time, chemical affinity turned out to play no role in the integration of thermodynamics into quantum methods simply because researchers’ presumptions did not consider it as a challenge to face anymore. On the contrary, the duality of the two conceptions of matter were still at work and underpinned the way Mulliken and Pauling were using thermochemistry while doing quantum chemistry. So I emphasize that the way thermodynamics became involved in quantum chemistry partly depended on different human stories and skills -Pauling was first a chemist and crystallographer whereas Mulliken was trained as a chemist and a spectroscopist. Others were mathematicians, organic chemists, and so on. But it also depended on different representations of matter – the aggregate and the ‘mixt’. Practices of research, human skills and goals, human and non human agency, time, concepts and representations interactively took part in the integration of thermodynamics into the earlier quantum realm.
Before I move on to modern quantum chemistry, I would like to further examine the relation between earlier quantum methods and thermodynamics by querying the concept of ‘state’, be it electronic, quantum or thermodynamic.
3.3. The concept of ‘state’ and the relation between quantum chemical methods and thermodynamics
Quantum chemistry is the result of a deep entanglement of scientific and human practices within which thermodynamics was an active generator of concepts and a tool for method calibration. If we want to query the role and status of thermodynamics in quantum chemistry, it is necessary to consider the practices of research from which they originate,
It is of interest to point out that quantum formalism gives rise to miscellaneous chemical quantum approaches depending on both chemical cultural resources and practical scientific backgrounds. It is astonishing however to notice that an atomic approach such as that of Pauling could have successfully developed on quantum grounds. The notion of atomic parts within a molecule is indeed deprived of meaning in quantum mechanics. The holistic approach of Mulliken seems much more understandable in a holistic, contextual and non-representionalist quantum theory. The final results reached by those methods are not pure quantum physics applications. This is a crucial point to bear in mind.
Let us deepen our study of Mulliken’s molecular orbital framework to illuminate his fine-grained relation with thermodynamics. Mulliken first worked on the couplings between orbital kinetic moments and of spin suggested by Friedrich Hund. In 1927, Hund developed an approach radically different from the work developed by Walter Heitler and Fritz London and generalized the study of Oyvind Burrau to diatomic molecules. Rather than built a molecular wave function from those describing isolated atoms, he proposed to describe each electron in the total molecular electric field of the nuclei and other electrons. Hund focused on the evolution of electronic energy during the transfer of an orbit around the joined nuclei to an orbit around the separate atoms isolated from each other. On the basis of works developed by Erwin Schrödinger, Pascual Jordan and Max Born, Hund was able to describe the exact stationary states of the two subsystems knowing those of the system by using linear combination. He wrote: ’We investigate a system with one degree of freedom as an analogous for a molecule with several atoms, using quantum mechanics. Its potential energy has several minima. We can relate the stationary states of such a system to those of partial systems that result when the separation between the minima becomes infinite or when the potential energy separating them becomes infinite. In agreement with this (and in opposition to the classical theory) we obtain an adiabatic relation between the states of two separated atoms or ions, the states of a two-atomic molecule and the states of the atom that would result when the nuclei are united. This relation allows for a qualitatively valid term system of the molecule and for an explanation of the terms ‘polar molecule’ and ‘ion lattice
Within the framework of thermodynamics, a system is involved in an adiabatic process if it does not exchange any thermal energy – any heat - with the outside. It can exchange only work. In mechanics, an adiabatic process is characterized by the fact that within infinitely slow changes of external parameters, the system evolves through successive states of equilibrium. In this kind of process, some quantities remain invariant, physicists call them adiabatic invariants. The adiabatic hypothesis, which was originally developed by Paul Ehrenfest, considers that the quantum conditions must always be such that the adiabatic invariants of classical mechanics are equal to an integer multiple of the quantum of action. You can infer the values of the states of a system from quantum states of another system that can be reached by an adiabatic transformation. The difficulty related to the conservation of quantities when changing orbits, evoked by Hund, disappears when the problem is studied within the framework of quantum theory. We realize that beyond semantic diversity of words such as ‘state’ or ‘adiabatic’, what is at stake is the way quantum physics can encompass classics physics as a limited case in precise contexts. Researchers were inventing a new quantum chemical scheme, while using general scientific and linguistic devices to link it with different previous theories. The notion of ‘state’ related to that of the ‘equilibrium state’ involved in thermodynamics is not tantamount to that of a ‘quantum state’ that only provides scientists with the calculation of the probability of each set of ‘observables’ from within a precise experiment context (Bitbol, 1998). The quantum state is related to a predictive symbolism that enables scientists to study holistic systems constitutively entangled with apparatus, that is to say the study of which cannot be separated from the context of measurement. Thermodynamics and quantum chemistry are nevertheless holistic, the former is descriptive at a macroscopic level, the later is predictive at a microscopic one. In this respect, it is not surprising that scientists tried and try to bridge those approaches in what we call different levels of our universe. What may the link between the two levels be? What are the necessary pre-conditions for tuning them? What may be the link between an energy quantum study of a molecule understood as a ‘whole’ at a microscopic level, and the energy of a set of molecules at a level described by thermodynamics?
Dealing with relations between a molecule and it parts, G.K. Vemulapalli noticed that: ‘While properties of the whole are not the sums or products of the properties of parts, the states of the system can be obtained by adding the states of parts. Because properties in turn can be derived from the states, it appears that we have shown that properties of wholes are completely determined by parts. But there are two problems here. (1) It is true that the states of the system are composed of states of the parts, but there are also weighting factors in the composition. There are the constants λ in the linear combination. What factors determine these constants? (2) Just as in the molecular wave function, an atomic wave function may also be represented by a sum of an arbitrary set of functions. Thus one may claim that an atomic function is a linear combination of molecular functions or atomic states (parts) reduced to molecular states (wholes!).’ (Vemulapalli, 2003). If we set apart that the notion of properties as open to criticism in quantum contexts and the linguistic traps related to it, the author’s insight is relevant to query the interrelation between levels of description studied by quantum chemistry.
The arbitrary character of the relation between the whole and its parts is highlighted. It remains more than ever present in current semi-empirical or
Vemulapalli referred to the second law of thermodynamics to explain why the studied molecular system continuously eliminates its excess energy by interactions with its environment. An energy transformation into local entropy returns legitimates the use of the Variational Principle. Vemulapalli added: ‘Thus we are led to conclude that it doesn’t matter what the states of the parts are, but it does matter the surroundings soak up the excess energy of the molecule, increasing entropy, and make the molecule settle down into the lowest energy state. It is that part of the universe coupled to the system, and the varieties of interactions between the system (molecules) and the surroundings that determines the structure of the molecule. Holism thus appears as the root of the apparent reduction of properties of a molecule to its parts through coupling states. We are able to follow a reductionist program in calculating molecular properties, but what we are able to do is a gift of holism
To sum up, we have focused our work on the way thermodynamics was used from within the earlier quantum chemical methods. We have shown that the opposition between the ‘aggregate’ and the ‘mixt’ was still at stake when explaining the integration of thermodynamics into quantum chemistry. Taking distance from linguistic traps concerning words such as ‘state’ or ‘adiabatic’, and by reflecting upon the relations between the levels of scientific description – a molecule to its alleged constitutive atoms or the macroscopic and microscopic scales -, we confirm that epistemology can provide us with another kind of understanding of the interrelations between thermodynamics and quantum chemistry. I would like to turn now to modern quantum methods and to examine how they involved thermodynamics. I choose to develop the example of the density functional theory - DFT -which has been widely used for twenty years in research laboratories.
4. The role and status of thermodynamics in modern quantum chemistry
Kohn–Sham density functional theory has become one of the most popular tools in electronic-structure theory due to its excellent performance-cost ratio as compared with correlated wave function theory, WFT. Within this theory, the molecular space is divided into grids of cubes; researchers define an electronic density for each point of this space. It is a holistic approach that enables quantum chemists to calculate molecular geometry or total energy exhaustively thanks to its electronic density – ‘ρ(r)’ -, provided that its Ground-State is not degenerate. The total energy is in consequence a
The first generation of functionals is called the local spin density approximation – LSDA -, in which density functionals depend only on local spin densities. Although LSDA gives accurate predictions for solid-state physics, it is not a useful model for chemistry due to its severe overbinding of chemical bonds and underestimation of barrier heights. The second generation of density functionals is called the generalized gradient approximation – GGA -, in which functionals depend both on the electronic density and its gradient. GGA functionals have been shown to give more accurate predictions for thermochemistry than LSDA ones, but they still underestimate barrier heights (Trulhar & Zhao, 2008a). In third-generation functionals, a Laplacian term density is added in the functional form; such functionals are called meta-GGAs. LSDAs, GGAs, and meta-GGAs are “local” functionals because the electronic energy density at a single spatial point depends only on the behavior of the electronic density and kinetic energy at and near that point. Local functionals can be mixed with nonlocal Hartree–Fock – HF - exchange as justified by the adiabatic connection theory (Becke, 1993). Functionals containing HF exchange are usually called hybrid functionals, and they are often more accurate than local functionals for main group thermochemistry (Trulhar & Zhao, 2008a, 2008b). This field of research aims at creating new density functionals with broader applicability to chemistry by including, for instance, non-covalent interactions. The crucial step is the calibration of new functionals against benchmark databases or best theoretical estimates (Goerigk & Grimme, 2010). Let us consider a case study developed by Truhlar and Zhao in order to understand the role and the status of thermochemistry in such a current context.
The most popular density functional, ‘B3LYP’, an hybrid GGA, has some serious shortcomings among which is its underestimation of barrier heights by an average of 4.4 kcal/mol for a database of 76 barrier heights. This underestimation is usually ascribed to the self-interaction error (unphysical interaction of an electron with itself) in local DFT (Trulhar & Zhao, 2008a). Truhlar and Zhao change parameters and include new ones while shaping a new mathematical functional form that takes physical phenomena into account. In so doing, they design a new functional by trial and error. They then use databases to appraise the reliability of a new functional within a defined purpose. Two databases gather all the thermodynamic quantities: (1) the data base ‘TC177’ is a composite database consisting of 177 data for main-group thermochemistry including atomization energies, ionization potentials, electron affinities, proton affinities of conjugated polyenes, and hydrocarbon thermochemistry among others data; (2) ‘DBH76’ is database of 76 diverse barrier heights concerning for instance nucleophilic substitution and hydrogen transfer. Truhlar and Zhao then discuss the performance of new functionals for these databases, they conclude that functionals labeled ‘MO6-2X’ and ‘MO5-2X’ are the ‘
Choosing a functional of electron density depends upon: (1) the necessary accuracy; (2) the chemical system; (3) the time of calculation. It also requires choosing a set of functions called a basis to achieve calculations for each atom. The basis change according to the type of atoms and different effects such as diffusion, polarization, pseudo potentials for chore electrons, and the size of functions -double, triple zeta-. The functional and its relative basis set define a level of calculation, the process of which requires choosing a computer program such as Gaussian type or Turbomole to be processed. If calculations are not convergent, researchers can change the functional, the size of the grids and convergence thresholds in order to optimize geometry or to calculate molecular energy. Each step reveals know-how, chemical culture and pragmatic compromises. Notwithstanding their basic differences, the ways thermochemistry is involved within molecular orbital approximation or DFT approach are quite similar. Modeling includes thermochemistry as a tool for calibration but also as a heuristic guide for theoretical parameters adjustments inside functionals or wavefunctions or for the design of new quantum methods (Grimme et al., 2007). The structure, within which calculations are made, is well framed by the Variational Principle. We thus realize that thermodynamic quantities partly shape current quantum practices of optimization of geometry and calibration. Calculations help researchers to find out the energy surface associated with a particular chemical reaction. The knowledge of the minimum points on an energy surface makes it possible for a chemist to interpret thermodynamic data. Besides, thermodynamics can retroactively justify minimization of energy as we have already explained. Thermodynamics and energy surface are thus interconnected to determine transition structure and reaction pathways. Modelling structural configurations is of importance in this context and the quantum calculations of entropy play a leading role in such descriptions and predictions.
Before I conclude, I would like to focus on a last case study to widen and deepen my enquiry. Let us consider how thermodynamic quantities are used to model solvatation effects and to scrutinize a chemical reaction mechanism within the DFT calculation background. I will refer to a study about zinc-thiolate complexes reactivity depending on the zinc ligands (Picot et al., 2008). Some calculations are shaped by thermodynamic quantities especially designed for quantum context, that is to say that do not exist in classic thermodynamics. It is typically the case of the zero-point vibrational energy labeled ‘ZPVE’. The molecular vibration energy is not equal to zero at absolute zero –O K-, it is a quantum mechanical effect which is a consequence of the Uncertainty Principle. Once a stationary point is localized, be it an energy minimum or a transition state, its energy turns out to be less important than the experimental energy of the molecule. For comparison with experimentally obtained thermochemical data, zero‐point vibrational energy is required to convert total electronic energies obtained from
ΔEelec, ΔZPVE, ΔET and ΔS are the differences of electronic energy, zero-point vibrational energy, thermal energy and entropy between the products and the reactants, respectively (Picot et al., 2008).
The solvatation free energy of each compound is determined by calculations depending on a model. This quantity is always defined as the required amount of energy necessary to transfer a molecule of gaseous solute into the solvent. The crucial step is to appraise how the solvent gets involved in a chemical reaction. Its action can be direct if some molecules of solvent take part in the chemical process or indirect if the solvent –then labeled the ‘bulk medium’- only modifies reactants reactivity compared with that of the same molecules in the gas phase. Whatever the context may be, the solvatation free energy is calculated from the equation (Leach, 2001):
ΔGelec quantifies the interaction between the solvent and the solute, it is all the more important as the iconicity or polarity is great. ΔGvdw takes into account Van der Waals interactions between the two. To finish, ΔGcav quantifies the cavity occupied by the solute while counting solvent reorganization around the cavity and the necessary work to fight against solvent pressure when the cavity is created. It is possible to encompass the two last terms within the equation:
a and b are constants, and S is the area of contact between the solute and the solvent. The different models that enable chemists to calculate ΔGsolv mostly differs by the way they appraise ΔGelec. From earlier models developed by Born (1920) and Onsager (1936) to the PCM model –Polarisable Continuum Method-, the form of the cavity and the study of polarization between the solvent and the solute were continuously modified and improved (Barone et al., 2004; Cossi et al., 2002). The surface of the cavity was divided into fine-grained fragments labeled ‘tesserae’, the wavefunction of solute is determined by Self-Consistent Field iteration. Two others models were performed, the COSMO theory –Conductor-Like Screening Model- and C-PCM approach –Conductor-Like PCM-. Modeling the interactions between the solute and the solvent is a challenge for current quantum chemists. In this context, thermodynamic quantities are the heuristic framework that shapes quantum investigations for achieving better models. The calculation of such thermodynamic quantities stir up: (1) new polarization descriptions and understanding; (2) the creation of new algorithms and cavity topological models (Barone et al., 2004); (3) the continuous recasting of levels of description and software to optimize geometry or to calculate energy quantities (Takano & Houk, 2005); (4) the modelling of the electronic density of the solute especially outside the cavity.
It is then easy to express the free energy of chemical reaction in water using the following classic thermodynamic cycle (Picot et al., 2008):
This cycle in turn implied the following formula:
Let us analyze how those thermodynamic quantities guide Picot et al. during their investigation of zinc-thiolate complexes alkylation. This short study will allow us to grasp thermodynamics role and status in workaday chemical quantum practices of research.
They first need biomimetic models that are appropriate for both structural and mechanistic studies. Based on the experimental data, they search for a consistent series of zinc complexes in which the ligands, the electric charge, and the availability of hydrogen bonding to the atom of sulfur can be varied. They choose the Gaussian 03 software and a level of calculation for the geometry optimizations using basis especially designed for each atom or physical contraction, diffusion or polarization. For each possible mechanistic pathway -see figure 2 below-, they scrutinize each stationary point by using frequency analysis. Each transition state –labeled TS1-3 in the mechanisms presented below- was verified by stepping along the reaction coordinate and confirming that the transformation occurred.
They then calculate the gas phase Gibbs free energy, and use C-PCM model to calculate the solvatation free energy within a precise set of levels of calculations. They can finally work out the react free energy in aqueous phase. They assess the adequacy of the chemical modeling and of the level of computation against observed databases of zinc complexes. They thus propose all the necessary thermodynamic quantities to analyze the chemical reaction - table 1 below.
Those thermodynamic quantities guide the authors along their line of enquiry. They compared energy barriers required to reach transition states in order to elucidate all the influencing parameters such as the global charge of the complex, the hydrogen bond, the role of zinc ligands and that of the solvent. In doing so, they confirm that their
computational outcomes are in agreement with several experimental studies. They for instance show that the net electronic charge of the complex plays a significant role not only on its reactivity, but especially on the mechanism of thiolate alkylation. They finally discuss the nature of the pathways depending on all those energy considerations. Once again, geometry and molecular configurations of the transition state are modeled and assumed to make those predictions become achievable. The entropic contribution is thus of primary importance to query such chemical potential mechanisms.
Thermodynamics is thus a tool for calibrating levels of computation (Curtis et al., 1997; Trulhar & Zhao, 2008b), but it also shapes solvatation modeling and the basic reasoning of mechanistic investigation (Takano & Houk, 2005). In a way, thermodynamics embeds a wide class of quantum activities of seeking and predicting. It provides quantum chemical methods with necessary conditions for reasoning and inventing new methods for calculations (Grimme et al., 2010).
5. Conclusion
The study of both earlier and recent quantum chemical methods highlights the way that thermodynamics is intertwined with quantum methods within a large network of scientific practices that includes computation, chemistry, spectroscopy, crystallography, physics, and so on. As Rouse claims concerning scientific practices (1996, p. 177): ‘What results is not a systematic unification of the achievements of different scientific disciplines but a complex and partial overlap and interaction among the ways those disciplines develop over time.’ Chemists connect ways of doing science and transform them within ongoing open-ended processes of research. As we have pointed out, thermodynamics was transmuted into thermochemistry through chemical practices, and conversely chemical instrumentation and ways of modeling were transformed by thermochemistry.
The role of thermodynamics is undoubtedly to validate models and methods while stirring up techno scientific creativity. The status of thermodynamics within quantum chemical methods is that of a reference framework that enables chemists to carry out their semi-empirical calculations or to create new
This study also points out that alleged incommensurable scientific worlds such as thermodynamics and quantum mechanics, the assumptions, the formalisms and the natures – descriptive or predictive - of which are completely different, can constitutively interact to form the composite field of quantum chemistry. Epistemological queries thus arise concerning inter-levels description of what we call ‘reality’ and the way scientific fields and knowledge can be mutually stabilized. To this extent, this study also stresses the importance of an epistemology that focuses its attention on scientific practices while including historical insights.
It is interesting to notice that chemical affinities reappear in the latest quantum chemical background. Truhlar and Zhao, among others, refer to affinities –electron affinities, proton affinities of different molecules- in their benchmark databases. Thermodynamics was first introduced in chemistry, we have shown, because it provided chemists with a notion of quantitative affinity. This concept went astray in earlier chemical quantum works and then reappeared from within databases or concepts that help current quantum chemists to shape their functionals according to thermochemistry and to investigate chemical reactivity. Further epistemological investigations are considered necessary to open up the reviving role of the concept of affinity to scrutiny in modern chemistry.
Acknowledgments
I would like to thank Rom Harré for his second reading of this paper, his advice and his generosity. I also would like to thank Miss Zgela, the Publishing Process Manager in charge for the book, for her help during the different steps of the publishing process.
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Notes
- A ‘mixt’ is a chemical combination composed of elements but not bearing the same properties as the constitutive elements. Conversely, an aggregate is a mere additive combinations of elements and their properties.
- ‘The foundations of a thermodynamic system’, my translation.
- The French original sentence is : ‘Lorsque la combinaison se produit, il se dégage une quantité de chaleur proportionnelle à l’affinité des deux corps.‘
- The French original sentence is : " (…) trouver pour chaque élément et pour chaque combinaison chimique, des nombres qui expriment leur affinité relative"