SPECIAL ISSUE PREFACE pubs.acs.org/JPCA
Autobiography of Richard F. W. Bader
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am indebted to the people who have contributed to my “festschrift”, as I am to all of those who find the quantum theory of atoms in molecules (QTAIM) useful in predicting and understanding the behavior of matter at the atomic level. To have others find one’s work useful in their own scientific endeavors is the ultimate personal accolade for a scientist. I am confident that with the ongoing efforts of many, as amply illustrated in this issue of the Journal, QTAIM will continue to grow in its breadth and depth of application long after my 80th birthday has passed. I have been exceedingly fortunate in having led a research group whose work has culminated in a new branch of physics— the physics of an open system derived from the principle of least action. Throughout, I was steadfast in insisting that our research to garner an understanding of the behavior of matter at the atomic level hue to the principle of basing one’s models and theories on observation followed by an appeal to physics. The sign over my office door stated: “if you cannot measure it or define it using physics, I do not want to discuss it”. It was my grounding in experimental chemistry that convinced me to switch from the then prevalent orbital and valence bond models to the observable density in my search for a physical basis of the molecular structure hypothesis and the attendant concept of a functional group. My interests in chemistry began in high school in my basement laboratory. I began my professional life as an experimental chemist taking a Master’s degree in physical organic chemistry under Professor A. N. Bourns at McMaster University using isotope effects in the study of reaction mechanisms, a field I continued to follow in my doctoral studies at MIT under Professor C. G. Swain. My Ph.D. thesis proposed a model for solvent isotope effects in light and heavy water by modeling the structure of liquid water in terms of the librational motions (the hindered rotations) of tetrahedrally coordinated water molecules. The librational frequency was found to provide a measurable indicator of the “structure” exhibited by liquid water. Ion solvation changed water’s structure as measured by the changes in the libration frequency, whose isotopic shift is the source of the solvent isotope effect. The effect on heats of hydration, entropy, and free energy (and ultimately on reaction rates) were accounted for and predicted using the measured shifts in the librational frequencies incurred by ionic solvation. It became clear from the many discussions I had with Professor Swain that chemists operated under a considerable handicap. At that time, chemists used the notions of atoms, bonds, and structure but everyone had their own understanding of these terms. How could we discuss the making and breaking of bonds between atoms when we had differing ideas of what constituted an atom and a bond? Chemists possessed a common language, but everyone had their personal dictionary. In fact, there was no theory underlying the language of chemistry, one that related the observations of chemistry that were ensconced in the molecular structure hypothesis, to the underlying physics. We had in chemistry an understanding based on a classification scheme that was both powerful and at the same time, because of its empirical nature, limited. It was after completing my degree at MIT in 1958 that I left for Cambridge University to study theoretical r 2011 Richard F. W. Bader
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chemistry under Professor Longuet-Higgins with the express purpose of developing a theoretical understanding of the language of chemistry by providing everyone with a common dictionary. In what follows I will not mention the names of group members associated with particular advances in the theory. With just a few exceptions, every member of the group contributed to its development and every observation was important. Naming some but not all of the members would be unfair. In switching to theory, I was embarking on what was to be for me a voyage of remarkable discovery. As a physical organic chemist my knowledge of quantum mechanics was limited to the usual undergraduate courses of the early 1950s, providing a smattering of molecular orbital and valence bond theory. While at MIT I sat in on Professor Slater’s course on quantum mechanics, concentrating mostly on his treatment of chemical bonding. The observational work on the electron density beginning in the 1960s in collaboration with LMSS in Chicago gave me the opportunity to use Feynman’s electrostatic theory of chemical bonding that I first learned of in Professor Slater’s lectures. In 1972, the studies of the electron density led to the proposal that a virial theorem should hold for a bounded piece of a system— for an atom in a molecule. With the arrival of a post doctoral fellow from the Technion, I began my journey into the world of quantum mechanics starting with Schr€odinger’s first four papers, necessitating my learning the calculus of variations. With the dawning of the realization that we were proceeding down a path that had its origin in the action principle, I turned to a reading of Dirac’s book and its beautiful discussions of the concepts that are common to and that link the quantum and classical worlds, a step that required the reading of Goldstein’s book on classical mechanics. Paramount in the bridges established by Dirac was the laying of the foundations for the quantum analogue of the classical Lagrangian approach underlying the principle of “least action”. It was this 1933 paper of Dirac that gave Feynman the clue he needed to develop his path integral technique, followed two years later by Schwinger’s postulation of the “quantum action principle” whose incorporation into the theory of atoms in molecules required my learning field theory. What a journey! It was accomplished with the unstinting help of the members of my group. As a result of my having to learn the necessary physics as needed to account for our observations, tethered in my belief that chemistry is grounded in physics, I was able to fully appreciate the beauty of the path that began with Schr€odinger and culminated with Schwinger. I have in the notes I took (and still possess) in Professor Slater’s course, his statement regarding the importance of the virial and Feynman theorems. Slater later referred to these theorems as “two of the most powerful theorems applicable to molecules and solids.” Nothing better illustrates the state of theoretical chemistry Special Issue: Richard F. W. Bader Festschrift Published: November 10, 2011 dx.doi.org/10.1021/jp208874a | J. Phys. Chem. A 2011, 115, 12432–12435
The Journal of Physical Chemistry A in the 1960s and 1970s than the observation that both of these theorems were strongly criticized in the chemical literature, because they were adjudged by some to be at variance with the models then in vogue. When I arrived at Cambridge in 1958, I found a series of papers by A. C. Hurley who had been a visitor at the laboratory before my arrival, then under the tutelage of Professor Lennard-Jones. They offered a beautiful, clear, and physical understanding of chemical bonding in terms of the forces exerted on the nuclei by the electron density, as determined by the Feynman electrostatic theorem. Although the theorem used by Feynman had been derived earlier, it had never been used in the discussion of chemical bonding. He was the first to set the external parameter of the theorem equal to a nuclear coordinate, thus determining the force on a nucleus in a molecule, demonstrating that the forces exerted on the nuclei are understood and calculable using classical electrostatics, with the force of attraction being determined by the electron density. Once quantum mechanics is used to determine the density of electrons, the discussion of the forces experienced by the nuclei is rigorously couched in the language of classical electrostatics. Better still, one deals with the real, space-filling, measurable electron density rather than the illusory orbitals that exist only in some mathematical space. I read as well the paper by Berlin wherein he defined the binding and antibinding regions of a diatomic molecule building on Feynman’s theorem. Feynman’s electrostatic theorem is the cornerstone of our understanding of chemical bonding. I never looked back and after my return to Canada, eventually to McMaster University, my research interests remained centered on the electron density. My first theoretical paper arose from a Cambridge tea time discussion between the Professor and Leslie Orgel. They proposed the use of the HerzbergTeller Hamiltonian in conjunction with second-order perturbation theory in the determination of the transition density leading to the most facile vibrational displacement of the nuclei to predict the signs of potential interaction constants in valence force potential functions. This caused me to read Mulliken’s Journal of Chemical Physics papers to find the ordering of molecular excited states, my first rigorous introduction to molecular orbital theory. The model yielded complete agreement with the observed signs (1960). In 1962 I realized that I could use the same method to predict the favored course of a chemical reaction, resulting in chemistry’s first symmetry rule. The paper presents an orbital correlation diagram from reactants to transition state to products that demonstrates how in an allowed reaction, a new low-lying excited state is generated as a consequence of the noncrossing rule for orbitals of the same symmetry, creating a transition density that facilitates the motion of the system over the energy barrier. It is a simple extension to the case of a forbidden reaction where such a state would not be generated. The model was likened to a secondorder JahnTeller effect. I should add that on my departure from Cambridge, “the” Professor strongly advised against the study of the electron density, stating that it would never lead to anything useful. In all fairness to him, I should add that this was the prevailing view at the time—as I was to discover. The history of the early studies of electron density has been reviewed in a 2010 Journal of Molecular Structure: THEOCHEM article. I was most fortunate in my efforts in this endeavor by collaborating with Professor Mulliken’s laboratory for molecular structure and spectra (LMSS) at the University of Chicago. Professor Roothaan’s group was in the process of obtaining wave
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functions of near HartreeFock quality for hundreds of diatomic molecules, both homonuclear and heteronucler, in ground and in excited states. I was given access to these functions in the late 1960s, work that resulted in the first comprehensive study of the properties of molecular electron density distributions, published as a series of papers in the Journal of Chemical Physics. Electron densities of crystals are now routinely obtained from X-ray scattering studies, but in our work, the LMSS wave functions were the source of the densities on which we made our “experimental observations”. In 1971 in a JACS paper we proposed a partitioning of diatomic densities by a line passing through the point of minimum density. In a footnote to this paper we drew attention to the fact that a “natural partitioning surface” could be defined in terms of the paths of steepest descent from the same density minimum—paths defined by the gradient vector field of the density. A coauthoring graduate student of that paper played an essential role in the further development of the theory by writing the first program for the determination and display of the gradient vector field of the electron density. I pressed him so hard to obtain a display of the “zero-flux surface” that he developed a rash, which his doctor advised him would disappear if he switched research supervisors. These displays followed hard on our 1969 study of the kinetic energy density and to our sudden realization that the zero-flux surface yielded a unique definition of the kinetic energy of an atom in a molecule. During these studies of the density, the group was conducting computational studies of potential energy surfaces, particularly those generated by the reaction of O(3P,1D) with H2(1Σg+) to account for the differing behavior of singlet and triplet species toward insertion and abstraction. The differences in chemistry of O(3P) and O(1D) were related to a polarization of the spin density, a concept that was shown to be useful in the interpretation of systems with unpaired spins. An extensive investigation of the surfaces was undertaken, with particular emphasis on the insertion of O(1D) into H2 to yield water and its abstraction of H to yield OH and H. This study resulted in the first theoretical discovery of a “conical intersection”, as predicted and studied by Teller. A study of the zero-flux surfaces generated by the LMSS densities and some triatomics left no doubt that we had found the atoms of chemistry, and in 1972 we proposed that the topology of the electron density (hereafter labeled F(r)) provided a unique and “natural” partitioning of the space of a molecule or a crystal into mononuclear regions that I shall refer to as atoms. Equally important, it soon became apparent that the resulting regions were transferable to varying extents between molecules. Indeed, if one insisted that the pieces transferred had to exhaust the system, they had the property of maximizing any transferability that was present. I wish to emphasize the initial excitement that was incurred by the realization that the mononuclear regions so defined recover the chemistry of an atom in a molecule. This natural definition satisfies the two essential requirements imposed on any possible definition of an atom in a molecule that stem from the concept of a functional group: (1) it maximizes the transferability of the atom’s distribution of charge from one system to another and (2) this transferability in the density is accompanied by a paralleling transferability in the other properties of the atom including its energy. This finding is but the satisfaction of the dictum that two atoms that look the same must necessarily make the same contributions to all properties. Point (2) was a result of a parallel 12433
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The Journal of Physical Chemistry A study in 1969 that we were conducting on the kinetic energy density of the electrons, a function that, like the electron density F(r), is defined in real space. As reported in 1972, I made what was to be the most important of all my observations of the electron density: that the transferability of the kinetic energy density parallels the transferability of F(r). Two atoms that “looked” the same in real space would possess the same electronic kinetic energy. This was amazing, because just suppose the virial theorem of quantum mechanics applied to such a topological atom, then one could define the total energy of atom in a molecule! This follows from the statement of the virial theorem that equates the total energy E to the negative of the kinetic energy T, for a system governed by Coulombic forces and in the absence of Feynman forces. For an atom Ω in a molecule, one could equate E(Ω) to T(Ω), where E(Ω) is the total electronic energy and T(Ω) its electronic kinetic energy. The same identification enables one to relate the total energy of the molecule E to the sum of the atomic contributions, that is, E = ΣΩE(Ω) = ΣΩT(Ω). We had in our hands a definition of an atom in a molecule that accounted for the obvious requirements that two atoms that look the same, possess the same energy and, when summed over all the atoms, yield the total energy. Our original derivation of the atomic statement of the virial theorem in 1975 that required satisfaction of the “zero-flux boundary condition”, was achieved by imposing a spatial boundary on Schr€ odinger’s energy functional. The variation of this functional, as I was to learn, corresponds to a constrained variation of the action integral for an instant of time. We soon discovered that we were following in the footsteps of Schwinger’s generalization of the “stationarity of the action” that he achieved by a variation of both the time-like and space-like surfaces bounding the action integral to obtain the principle of stationary action. I have been accused of unnecessarily invoking Schwinger, which is most unfortunate. We limited the spatial extent of the action integral, by placing an “Ω” on Schr€odinger’s energy functional out of necessity to define an open system. Schwinger on the other hand, realized that in varying both the time-like and space-like surfaces bounding the action integral he could obtain all of physics from a single principle. If we had not referred to Schwinger, then we would have been accused of plagiarism. By 1978 the theory was linked to Schwinger’s principle, and via Dirac, to the principle of least action. By 1975 we were actively studying the gradient vector field of F(r), and the student responsible for the calculation of the trajectories in polyatomic molecules, being an ardent skier, christened it SCHUSS. In that year we attempted to publish a paper in JACS showing that nuclei bonded in the chemical sense were linked by a line of maximum electron density—by a bond path. The paper was rejected by the then editor Professor Parr as being too simplistic and more suitable for the Journal of Chemical Education. In any event, I at least got the definition of a bond path published in 1975 and the full paper in 1977. By 1979 we were embarked on the development of the theory of molecular structure and structure stability, as made manifest in the dynamics of the gradient vector field of the density, the dynamics in our case being a result of the motions of the nuclei. Once again we were following a path laid out by others, this time by the mathematicians Thom, Palis, and Smale among others, who demonstrated how one could make quantitative the notion of structure and predict the mechanisms of structural change using qualitative dynamics. It was most exciting to read Thom’s book on catastrophe theory and find in his examples of “elementary
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catastrophes” all of the structural changes we had encountered in our study of the gradient vector field of the electron density. So now we had not only a theory of the quantum mechanics of an atom in a molecule but also a complete theory of structure and structural stability, both components of the theory being reviewed in 1981. In 1982 an applied mathematician, who decided to join my group upon the completion of his Ph.D. studies in Germany, wrote the first efficient and transportable algorithm for the calculation of the “properties of atoms in molecules”, PROAIM. At last count it had close to 1000 citations. It has since been superseded by much faster and more efficient programs, but it played an indispensible role in the acceptance and use of the theory. The beauty of the theory is how everything follows from the initial identification of the quantum boundary condition for an open system with the bounding surface of zero-flux in the gradient vector field of the electron density. This identification leads not only to the physics of an open system but also, through the topological properties of a bond critical point, to the physically unique association of a molecular structure with the lines of maximum electron density linking the nuclei of bonded atoms. No one presently denies that every molecular structure is but a mapping of “the bonds of chemistry” onto lines of maximum electron density, the bond paths of QTAIM. Fine, but where is the Lewis model in all of this? 1975 was a good year. We published the variational derivation of the atomic virial theorem and in addition the physical basis of the concept of electron localization by following in the footsteps of Lennard-Jones who emphasized the roles of the exclusion principle and the same-spin pair density. We demonstrated that the quantum mechanical requirement for the spatial localization of an electron in a many-electron system is that the density of the Fermi correlation hole, obtained by the double integration of the exchange density, be totally contained within the same spatial region. Correspondingly, the extent to which this requirement is not met provides a quantitative measure of its delocalization over the remaining space of the system. This definition of electron localization/delocalization serves as the basis for all such subsequent discussions. One now had a physical handle on electron localization, but its spatial properties had yet to be determined. The Laplacian of the electron density plays a dominant role throughout the theory: the stationarity in its variation over an atomic volume yielding the zero-flux boundary condition and its appearance in the local statement of the virial theorem, for example. But there was much more to come, a result of the Laplacian of a scalar function determining where it is locally concentrated and locally depleted. In 1984 we studied the Laplacian along with other properties at a bond critical point to develop a bonding classification scheme. In this study we noted that displays of the Laplacian distribution exhibited regions where r2F(r) < 0, regions of charge concentration, and r2F(r) > 0, regions of charge depletion. When we obtained the display for ClF3, we realized that we had discovered the physical basis of Gillespie’s valence shell electron pair repulsion (VSEPR) model, the topology of the Laplacian faithfully recovering all of the nuances of his model. We tacked the diagram for the Laplacian of ClF3 onto his office door and from that point on, he became a convert to the study of the electron density. It was the reporting of the spatial properties of the Laplacian at a Gordon research conference in 1984 that finally convinced workers in the field of accurate determination of the electron density by X-ray diffraction methods to switch from density difference maps to displays 12434
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The Journal of Physical Chemistry A of the total density, to obtain the “bond paths” to define structure along with the Laplacian to recover the “lone” and “bonded” pairs of electrons so dear to their hearts. In 1999 we demonstrated the existence of a mapping of the Laplacian of the conditional pair density onto the Laplacian of the density, demonstrating that the Laplacian recovered the essential pairing information in real space determined by the six-dimensional pair density. The Laplacian plays the dominant role in determining intermolecular interactions in condensed matter by the alignment of local charge concentrations with local charge depletions. Still to come were the extensions of the theory to measurable properties long studied in terms of their “atomic” contributions: dipole moments, electric polarizabilities, magnetic susceptibilities, atomic polar tensors, and contributions to the intensities of transitions induced by light. Any physical model of these properties revealed that they all had a property in common, an apparent origin dependence. Because atomic and group contributions to the electric and magnetic susceptibilities were well established experimentally, the origin dependence was clearly nothing more than a sign that previous approaches were missing some essential physics. In 1992 we began a study of the topology of the current density, a vector field, necessitating the development of a new method for its calculation, because there were no existing methods that yielded representations of the quality required for its topological analysis, a consequence of the “gauge origin” problem. Following our usual path we went straight for the calculation of the measurable current. After all, we had a theory of atoms in molecules and one could calculate the current induced by a magnetic field in an atom by placing the origin at the nucleus. Hence the use of individual gauges for atoms in molecules, the IGAIM method for calculating the induced current. The results were phenomenal compared to previous attempts to calculate an induced current. We were the first to tabulate values of the divergence of the current as a test of the quality of the calculated current, a quantity that must necessarily vanish in a stationary state. But the best was still to come. The graduate student who wrote the IGAIM program asked the question “why not go to the obvious limit and use a new gauge origin for every point in space?”, giving rise in 1993 to the method “continuous set of gauge transformations”, the CSGT method. We now produced what were divergence-free representations of the induced current (IGAIM gave difficulties at the boundaries between the atoms). The resulting portrayals of the induced current are beautiful, the map for benzene clearly showing the presence of a “ringcurrent”. Now the road was clear for the first definitive study, in 1993, of the topology of the induced current using the mathematical theorems governing the critical points and flow of a vector field. It became clear that the atomic contributions to all of the properties exhibiting apparent origin dependence were all describable in terms of a single “master equation”. One simply inserted the proper field, internal polarization, electric, magnetic, or electromagnetic field and all of the atomic contributions were not only calculable but also found to be in agreement with the experimental values. “Glasstone”, my second year physical chemistry textbook, described the experimental determination of the additive atomic and group contributions to heats of formation, and to the electric and magnetic susceptibilities, and now we had a theory that recovered all of these values. New applications of the theory continue to appear, some in the field of molecular biology, for example. A relatively recent paper challenging existing models is the reporting in 2003 of
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hydrogenhydrogen bonding and thence of other bonding between what were classified under the misnomer “nonbonded interactions”. I am constantly amazed by the refusal of some to accept the presence of any bonding that is not predicted by the Lewis model. Why should a bond path, which recovers all Lewis structures, not be associated with bonding in other cases? Why refuse to accept the findings of a theory of bonding that transcends all models? The electrostatic force of attraction of the nuclei for the electron density is the sole attractive force responsible for chemical bonding, intra- or intermolecular and its presence is made manifest through the presence of a bond path. The theorem of Hohenberg and Kohn, that the electron density is a unique functional of the external potential, applies to a closed system with a fixed number of electrons. Transferability of the electron density of an atom between differing systems, necessary to account for the fundamental role of a functional group with characteristic properties in chemistry is, on the other hand, necessarily a problem in the physics of an open system. Finding the chemical basis for the manifestation of the “nearsightedness of electronic matter” resulted in the development of the quantum theory of atoms in molecules. Transferability in chemistry requires a new theorem stated in terms of the density: that the electron density of an atom in a molecule or crystal determines its additive contribution to all properties of the total system, its transferability being determined by a paralleling degree of transferability in the atom’s virial field, the virial of the Ehrenfest force exerted on its electron density. Transferability of the virial field is found in spite of the large, unavoidable changes in the external potential that occur on transfer. QTAIM has always progressed by following the path determined by observation and its relation to theory, a path that now points to the search for the theoretical basis in place of the empirical one presently relating the virial field to the near sightedness of electronic matter. I close with some words for young people. Do not go through life without reading Dirac’s book or Feynman’s 1949 Physical Reviews paper. Chemistry courses will eventually include the Lagrangian approach and the principle of least action, one that, as Planck pointed out, is the guiding principle of everything that occurs around us. If you are fortunate enough to discover something new, do not expect universal acceptance. Be instead prepared for unbridled criticisms from those whose ideas you are challenging. Out of the many struggles that I faced, and still do, one stands out. My submission of the 1972 paper to the Journal of Chemical Physics, which laid the observational basis for the theory of atoms in molecules, required the efforts of Professor Polanyi, an associate editor of the Journal, to force the editor of the Journal to obtain a report from a reluctant referee who made no useful comments. Nearly all of my most cited papers were published with difficulty. One must never lose faith in the belief that science will in the end win out. Richard F. W. Bader
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