**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 (Médard &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 *Valence Bond Theory*. The use of both accurate thermodynamic and crystallographic data enabled Pauling to notice that the covalent radii sum of the bonded atoms approximated bond lengths very well. He then linked bond energies with experimental heats of formation of gaseous molecules (Pauling, 1932). The key step was to choose a set of molecules that could supply the data necessary for extracting those bond energies (Weininger, 2001). This approach allowed him to express the total energy of formation of the molecule as a mere sum of energy terms characteristic of the different bonds assuming that the molecule was obtained from separate atoms (Pauling, 1932). The referent molecules only had to have a single Lewis electronic structure (Pauling & Sherman, 1933a, 1933b). Atoms are the basic units of Pauling's system, this atomic standpoint shaped the way he used thermodynamic data.

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 *aggregate* of atoms. Each atom has stable atomic orbits - 2s, 2p for instancethat will be used to form stable bonds inside a molecule or to induce ad hoc directed valence (Pauling, 1931; Slater, 1931). He stated that bonds resulted from the overlapping of two atomic eigenfunctions, the larger the overlap is, the stronger the bond gets.

The study of diatomic molecule enabled Pauling to propose the concept of 'normal*'* covalent bond and to express what he called the 'normal' covalent molecular wave function as a mere sum of covalent and ionic terms so as to provide his electronegativity concept with a quantum counterpart (Pauling, 1932). Thermochemistry was once again a touchstone for the

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

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 (Médard &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

thermodynamics and quantum mechanics into chemistry.

**thermodynamics into quantum chemistry?** 

the way he used thermodynamic data.

**3.2 The 'mixt' and the aggregate: A framework for the embodiment of** 

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 *Valence Bond Theory*. The use of both accurate thermodynamic and crystallographic data enabled Pauling to notice that the covalent radii sum of the bonded atoms approximated bond lengths very well. He then linked bond energies with experimental heats of formation of gaseous molecules (Pauling, 1932). The key step was to choose a set of molecules that could supply the data necessary for extracting those bond energies (Weininger, 2001). This approach allowed him to express the total energy of formation of the molecule as a mere sum of energy terms characteristic of the different bonds assuming that the molecule was obtained from separate atoms (Pauling, 1932). The referent molecules only had to have a single Lewis electronic structure (Pauling & Sherman, 1933a, 1933b). Atoms are the basic units of Pauling's system, this atomic standpoint shaped

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 *aggregate* of atoms. Each atom has stable atomic orbits - 2s, 2p for instancethat will be used to form stable bonds inside a molecule or to induce ad hoc directed valence (Pauling, 1931; Slater, 1931). He stated that bonds resulted from the overlapping of two

The study of diatomic molecule enabled Pauling to propose the concept of 'normal*'* covalent bond and to express what he called the 'normal' covalent molecular wave function as a mere sum of covalent and ionic terms so as to provide his electronegativity concept with a quantum counterpart (Pauling, 1932). Thermochemistry was once again a touchstone for the

atomic eigenfunctions, the larger the overlap is, the stronger the bond gets.

validity of this quantum mechanical treatment of chemical bonding; it was as not just a mere tool to calibrate methods. Empirical data really aroused Pauling's creativity and guided him to adapt his quantum work. By applying the rules for the electron-pair bond, Pauling removed the apparent incompatibility between chemistry and quantum theory (Gavroglu & Simões, 1994). Pauling answered more directly the concerns of the chemists by stressing the three-dimensional structure of molecules, the electrons being the bonding officers of the atoms. The valence bond approach which he developed with Slater was more quickly acknowledged by chemists because resonance corresponded to their usual representations and structural formula (Llored & Bitbol, 2010).

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 *electronic configuration*, *i.e.*, of a distribution of the molecular electrons in different orbits. In this description, each orbit is delocalized over all the nuclei and can contribute, depending on each specific case, a stabilizing or destabilizing energy contribution to the total energy of the molecule (Llored, 2010). The sum of the energy contributions of each electron in its orbit determined whether the electronic configuration allowed for the existence of a stable molecule, *i.e.*, whether its energy was stabilizing overall. For Mulliken, the atom did not exist as a component in a molecule. His concept of molecular state suggested molecular variability of energy and geometry that could not even be considered within the approaches of Lewis and Irving Langmuir. Mulliken proved that the spectral states of the molecules could be obtained from that of their molecular ions by the mere addition of an electron without changing the quantum numbers and, thus, worked out his molecular *Aufbauprinzip* (Llored, 2010). This close connection between the quantum theory and the spectral studies gave birth to the correlation diagrams of 1932 (Mulliken, 1932b). Those diagrams made it possible to consider the degree of likeness between a molecule and its separated atoms or its united atom - a fictitious atom obtained by the coalescence of the two atoms such as helium He for two hydrogen H atoms - thanks, in particular, to empirical knowledge of the inter-nuclear distances, energy dissociation and of the charges of the nuclei. The molecule from then on was considered as a composite, *i.e.*, a new entity rather than a mere aggregate of individualized atoms. He wrote: 'In the 'molecular' point of view advanced here, the existence of the molecule as a distinct individual built up of nuclei and electrons is emphasized, whereas according to the usual atomic point of view the molecule is regarded as composed of atoms or of ions held together by valence bonds. From the molecular point of view, it is a matter of secondary importance to determine through what intermediate mechanism (union of atoms or ions) the finished molecule is most conveniently reached. It is really not necessary to think of valence bonds as existing in the molecule (Mulliken, 1931). Despite their irreducible differences, Duhem's thermodynamic potential echoed the electronic states developed by Mulliken insofar as both considered a molecule from an energy standpoint as a 'mixt' not as an 'aggregate'. The '*electronic state'*, the '*binding capacity'*, the '*promotion*' of an electron, '*the energy-bonding-power',* are among the many concepts Mulliken built to explain the capacity of the electrons to be linked to nuclei to form a molecule seen as a whole (Harré & Llored, 2011).

The semantic shift from the concept of molecular *orbit* to that of molecular *orbital* –MOoccurred in 1932. The concept of orbital took all its significance from Max Born's probabilistic interpretation that the square of a molecular orbital corresponded to the

The Role and the Status of Thermodynamics in Quantum Chemistry Calculations 477

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 *completely theoretical method* (Mulliken & Parr, 1951). In order to make a comparison with the ordinary empirical resonance energy, they had to make several corrections that involved: (1) the 'compressive energy' needed to adjust the lengths of the single and double Kekulé's bonds to those of the proper benzene; (2) hyperconjugation and related effects. They discussed the corrections and estimated their magnitudes before concluding that a reliable value could only be obtained for the compression energy. Following this line of reasoning, they determined that the computer net resonance energy was 36.5 kcal. This outcome agreed, with the uncertainties due to the omitted correction terms, with the value 41.8 kcal of the empirical resonance energy 'Δ' based on thermodynamic data. They then used 'Δ' as the point of departure of the calculation of the actual heat of formation 'ΔH°f' of benzene from the value given by a standard formula for nonresonating hydrocarbons. They proposed a new standard formula containing corrections for the mutual effects of neighboring carbon-carbon bonds while discussing its significance. This analysis allowed them to clarify what was meant by 'resonance energy' and to query the significance of 'nonresonating' structures and repulsion terms in their own theory. They always sought to identify the conditions that made it possible for a chemist to make a clean-cut comparison between theory and experiment. In quite that light, thermodynamic data guided the way they wrote equations relating theoretical energy quantities to a sum of empirically based terms. This work allowed them to define new useful concepts such as 'standard hydrocarbon' – held with Δ = 0 kcal - that fostered calculations and comparisons. To sum up, they continually queried their model and its meaning. Thermochemistry, quantum chemical methods, chemical practices and culture, computers, instruments were *constitutively intertwined*, and they were *interactively stabilized*. Modelling is an *open-ended process* that includes thermochemistry as a foundation to create a new quantum account of a molecular 'mixt'. As Andrew Pickering asserts: 'Existing culture constitutes the surface of emergence for the intentional structure of scientific practice, and such practices consists in the reciprocal tuning of human and material, tuning that can itself

probability density of finding this electron at a certain location in space. Mulliken wrote: 'By an atomic orbital is meant an orbital corresponding to the motion of an electron in the field of a single nucleus plus other electrons, while a molecular orbital corresponds to the motion of an electron in the field of two or more nuclei plus other electrons. Both atomic and molecular orbitals may be thought of as defined in accordance with the Hartree method of the self-consistent field, in order to allow so far as possible for the effects of other electrons than the one whose orbital is under consideration*.'* (Mulliken, 1932a).

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.

probability density of finding this electron at a certain location in space. Mulliken wrote: 'By an atomic orbital is meant an orbital corresponding to the motion of an electron in the field of a single nucleus plus other electrons, while a molecular orbital corresponds to the motion of an electron in the field of two or more nuclei plus other electrons. Both atomic and molecular orbitals may be thought of as defined in accordance with the Hartree method of the self-consistent field, in order to allow so far as possible for the effects of other electrons

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

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-

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

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.

than the one whose orbital is under consideration*.'* (Mulliken, 1932a).

Fock self-consistent field approach –labelled LCAO MO SCF-.

work with heats of combustion.' (Mulliken et al., 1941).

data.

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 *completely theoretical method* (Mulliken & Parr, 1951). In order to make a comparison with the ordinary empirical resonance energy, they had to make several corrections that involved: (1) the 'compressive energy' needed to adjust the lengths of the single and double Kekulé's bonds to those of the proper benzene; (2) hyperconjugation and related effects. They discussed the corrections and estimated their magnitudes before concluding that a reliable value could only be obtained for the compression energy. Following this line of reasoning, they determined that the computer net resonance energy was 36.5 kcal. This outcome agreed, with the uncertainties due to the omitted correction terms, with the value 41.8 kcal of the empirical resonance energy 'Δ' based on thermodynamic data. They then used 'Δ' as the point of departure of the calculation of the actual heat of formation 'ΔH°f' of benzene from the value given by a standard formula for nonresonating hydrocarbons. They proposed a new standard formula containing corrections for the mutual effects of neighboring carbon-carbon bonds while discussing its significance. This analysis allowed them to clarify what was meant by 'resonance energy' and to query the significance of 'nonresonating' structures and repulsion terms in their own theory. They always sought to identify the conditions that made it possible for a chemist to make a clean-cut comparison between theory and experiment. In quite that light, thermodynamic data guided the way they wrote equations relating theoretical energy quantities to a sum of empirically based terms. This work allowed them to define new useful concepts such as 'standard hydrocarbon' – held with Δ = 0 kcal - that fostered calculations and comparisons. To sum up, they continually queried their model and its meaning. Thermochemistry, quantum chemical methods, chemical practices and culture, computers, instruments were *constitutively intertwined*, and they were *interactively stabilized*. Modelling is an *open-ended process* that includes thermochemistry as a foundation to create a new quantum account of a molecular 'mixt'. As Andrew Pickering asserts: 'Existing culture constitutes the surface of emergence for the intentional structure of scientific practice, and such practices consists in the reciprocal tuning of human and material, tuning that can itself

The Role and the Status of Thermodynamics in Quantum Chemistry Calculations 479

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 nonrepresentionalist quantum theory. The final results reached by those methods are not pure

Let us deepen our study of Mulliken's molecular orbital framework to illuminate his finegrained 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*'.'* (Hund, 1927). The new quantum theory thus allowed him to explain the adiabatic passage between two stationary states of the same system. Hund made this result suitable for the study of molecules and proposed an interpolation between the quantum states of the isolated atoms, the united atom and the molecule. Hund further added: 'The complete transition from the case of nuclei separated by a large distance to the case of a small separation cannot be done adiabatically in the classical model. If we start in the case of nuclei separated by a large distance with some given quantum numbers, then we first arrive at orbit type II, but for a certain internuclear distance this type is no longer possible. The classical motion becomes a limiting motion. The same occurs when we approach from the other side, with nuclei placed close together; for a certain distance between the nuclei, orbit type I becomes impossible and the motion becomes a limit. An adiabatic transition going over the limiting case is not possible because of the vanishing

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

quantum physics applications. This is a crucial point to bear in mind.

frequency*.' (*Hund*,* 1927*).* 

reconfigure human intentions. The upshot is, on occasion, the reconfiguration and extension of scientific culture.' (Pickering, 1995). The *dialectics of resistances and accommodations* between thermochemistry and the quantum chemical model made Mulliken continuously recast his approach so as to stabilize a great amount of tables and concepts about molecular properties. He produced a great number of tables throughout his academic life. From spectroscopic to conjugation energy tables as well as from correlation diagrams to Mulliken-Walsh ones, he knitted a network of data thanks to a constitutive interaction between theory and experiment.

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, *i.e.*, the techno-scientific closure which combines quantum mechanics, approximations, instrumental and algorithmic techniques, chemical know-how, and the use of Principles which do not belong to quantum theory such as the Pauli Principle. The predictive capacity of these chemical quantum approaches does not only rely on the molecular wave function but also on a host of approximations and compromises that make it possible for numerical properties and molecular landscapes to be calculated (Llored, 2010, 2012).

reconfigure human intentions. The upshot is, on occasion, the reconfiguration and extension of scientific culture.' (Pickering, 1995). The *dialectics of resistances and accommodations* between thermochemistry and the quantum chemical model made Mulliken continuously recast his approach so as to stabilize a great amount of tables and concepts about molecular properties. He produced a great number of tables throughout his academic life. From spectroscopic to conjugation energy tables as well as from correlation diagrams to Mulliken-Walsh ones, he knitted a network of data thanks to a constitutive interaction between theory

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.

of thermodynamics into the earlier quantum realm.

be it electronic, quantum or thermodynamic.

**thermodynamics** 

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

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',

**3.3 The concept of 'state' and the relation between quantum chemical methods and** 

properties and molecular landscapes to be calculated (Llored, 2010, 2012).

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, *i.e.*, the techno-scientific closure which combines quantum mechanics, approximations, instrumental and algorithmic techniques, chemical know-how, and the use of Principles which do not belong to quantum theory such as the Pauli Principle. The predictive capacity of these chemical quantum approaches does not only rely on the molecular wave function but also on a host of approximations and compromises that make it possible for numerical

and experiment.

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 nonrepresentionalist 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 finegrained 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*'.'* (Hund, 1927). The new quantum theory thus allowed him to explain the adiabatic passage between two stationary states of the same system. Hund made this result suitable for the study of molecules and proposed an interpolation between the quantum states of the isolated atoms, the united atom and the molecule. Hund further added: 'The complete transition from the case of nuclei separated by a large distance to the case of a small separation cannot be done adiabatically in the classical model. If we start in the case of nuclei separated by a large distance with some given quantum numbers, then we first arrive at orbit type II, but for a certain internuclear distance this type is no longer possible. The classical motion becomes a limiting motion. The same occurs when we approach from the other side, with nuclei placed close together; for a certain distance between the nuclei, orbit type I becomes impossible and the motion becomes a limit. An adiabatic transition going over the limiting case is not possible because of the vanishing frequency*.' (*Hund*,* 1927*).* 

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

The Role and the Status of Thermodynamics in Quantum Chemistry Calculations 481

order of energy as well as its correlation diagram thanks to those of the two fragments. In doing so, he included all the characteristics of the molecular orbital diagram of the ethylene molecule and checked it using molecular spectroscopy. Mulliken could just as easily have considered a fragment "C2" and another "H4" of adapted symmetries. The relation on whole "C2H2" with its parts was of secondary interest. The fundamental choice relates to the nature and the extent of the basis sets of the calculation. Vemulapalli threw light on the role of the weighting coefficients appearing in front of the orbitals of the key basis sets. These coefficients determined by the Variational Principle are those which minimize the molecular potential energy. How is this minimum of energy justified? What can explain the use of the

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*.*' (Vemulapalli, 2003). A molecule is always in relation with its surroundings, it can at least emit a photon even in a strong vacuum. So the study at a molecular level requires a study of interactions at an upper level while microscopic descriptions require quantum predictions. Levels of description need one another, they are co-stabilized. The Variational Principle that underpins Mulliken's work at a molecular level can find a justification within the context of thermodynamics. It is an *a posteriori* analysis that allows us to widen our understanding of the possible links between thermodynamics and quantum chemistry from

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 -

Variational Principal? A quantum principle?

another point of view, that of inter-levels relations.

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

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 *ab initio* methods of molecular orbital calculation that depend on the choice of atomic or molecular orbital used. Mulliken developed *the fragment method* in 1933, two fragments could interact provided they had the same kind of symmetry and that the energy gap, measured by spectroscopy, was not too high. For the ethylene molecule 'C2H4', Mulliken considered two fragments 'CH2' and determined a suitable molecular orbital by using the irreducible representations of ethylene. He could thus propose a representation of a molecular orbital of ethylene by increasing

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

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

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 *ab initio* methods of molecular orbital calculation that depend on the choice of atomic or molecular orbital used. Mulliken developed *the fragment method* in 1933, two fragments could interact provided they had the same kind of symmetry and that the energy gap, measured by spectroscopy, was not too high. For the ethylene molecule 'C2H4', Mulliken considered two fragments 'CH2' and determined a suitable molecular orbital by using the irreducible representations of ethylene. He could thus propose a representation of a molecular orbital of ethylene by increasing

between levels of description studied by quantum chemistry.

thermodynamics?

order of energy as well as its correlation diagram thanks to those of the two fragments. In doing so, he included all the characteristics of the molecular orbital diagram of the ethylene molecule and checked it using molecular spectroscopy. Mulliken could just as easily have considered a fragment "C2" and another "H4" of adapted symmetries. The relation on whole "C2H2" with its parts was of secondary interest. The fundamental choice relates to the nature and the extent of the basis sets of the calculation. Vemulapalli threw light on the role of the weighting coefficients appearing in front of the orbitals of the key basis sets. These coefficients determined by the Variational Principle are those which minimize the molecular potential energy. How is this minimum of energy justified? What can explain the use of the Variational Principal? A quantum principle?

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*.*' (Vemulapalli, 2003). A molecule is always in relation with its surroundings, it can at least emit a photon even in a strong vacuum. So the study at a molecular level requires a study of interactions at an upper level while microscopic descriptions require quantum predictions. Levels of description need one another, they are co-stabilized. The Variational Principle that underpins Mulliken's work at a molecular level can find a justification within the context of thermodynamics. It is an *a posteriori* analysis that allows us to widen our understanding of the possible links between thermodynamics and quantum chemistry from another point of view, that of inter-levels relations.

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.
