Worlds apart in chemistry: a personal tribute to J. C. Slater.

A reading of the book of Dirac's life entitled The Strangest Man is a most stirring experience, bringing one back to the beginnings of quantum mechanics where every attempt was made "to establish a basis for theoretical quantum mechanics founded exclusively on relationships between quantities which are in principle observable." The prime movers in this quest were Heisenberg and Dirac. One of Dirac's most important contributions in the passage from classical to quantum mechanics, a passage that consumed much of his early efforts, was unfortunately published in an obscure Russian journal where it remained largely unread until it was found by Feynman while a graduate student at Princeton. The paper posed the question, "what corresponds in the quantum theory to the Lagrangian method of classical mechanics?", a method that, as Dirac pointed out, is clearly superior in the simplicity of its structure to that of the classical Hamiltonian approach. Dirac's partial answer to this question provided the key to solving the problem of introducing the action integral into quantum mechanics that occupied Feynman's mind, leading to his formulation of the path integral technique. His contribution was followed two years later by Schwinger's independently derived statement of the quantum action principle, each contribution providing a complete formulation of quantum mechanics stated in terms of single principle. The present paper points out that the successful introduction of the action principle into quantum mechanics made possible by Dirac, enables one to proceed still further by extending Schwinger's quantum action principle to an open system, to an atom in a molecule. Thus the quantum theory of an atom in a molecule has its roots in the question posed by Dirac in 1933. The paper proposes a return to a greater use of the theorems of quantum mechanics in interpretive chemistry from that begun by Slater in 1933, a staunch advocate of theory following in the footsteps of observation.

Definition of molecular structure: by choice or by appeal to observation?

There are two schools of thought in chemistry: one derived from the valence bond and molecular orbital models of bonding, the other appealing directly to the measurable electron density and the quantum mechanical theorems that determine its behavior, an approach embodied in the quantum theory of atoms in molecules, QTAIM. No one questions the validity of the former approach, and indeed molecular orbital models and QTAIM play complementary roles, the models finding expression in the principles of physics. However, some orbital proponents step beyond the models to impose their personal stamp on their use in interpretive chemistry, by denying the possible existence of a physical basis for the concepts of chemistry. This places them at odds with QTAIM, whose very existence stems from the discovery in the observable topology of the electron density, the definitions of atoms, of the bonding between atoms and hence of molecular structure. Relating these concepts to the electron density provides the necessary link for their ultimate quantum definition. This paper explores in depth the possible causes of the difficulties some have in accepting the quantum basis of structure beginning with the arguments associated with the acceptance of a "bond path" as a criterion for bonding. This identification is based on the finding that all classical structures may be mapped onto molecular graphs consisting of bond paths linking neighboring atoms, a mapping that has no known exceptions and one that is further bolstered by the finding that there are no examples of "missing bond paths". Difficulties arise when the quantum concept is applied to systems that are not amenable to the classical models of bonding. Thus one is faced with the recurring dilemma of science, of having to escape the constraints of a model that requires a change in the existing paradigm, a process that has been in operation since the discovery of new and novel structures necessitated the extension of the Lewis model and the octet rule. The paper reviews all facets of bonding beginning with the work of Pauling and Slater in their accounting for crystal structures, taking note of Pauling's advocating possible bonding between large anions. Many examples of nonbonded or van der Waals interactions are considered from both points of view. The final section deals with the consequences of the realization that bonded quantum atoms that share an interatomic surface do not "overlap". The time has come for entering students of chemistry to be taught that the electron density can be seen, touched, and measured and that the chemical structures they learn are in fact the tracings of "bonds" onto lines of maximum density that link bonded nuclei. Matter, as we perceive it, is bound by the electrostatic force of attraction between the nuclei and the electron density.

Bond paths are not chemical bonds.

This account takes to task papers that criticize the definition of a bond path as a criterion for the bonding between the atoms it links by mistakenly identifying it with a chemical bond. It is argued that the notion of a chemical bond is too restrictive to account for the physics underlying the broad spectrum of interactions between atoms and molecules that determine the properties of matter. A bond path on the other hand, as well as being accessible to experimental verification and subject to the theorems of quantum mechanics, is applicable to any and all of the interactions that account for the properties of matter. It is shown that one may define a bond path operator as a Dirac observable, making the bond path the measurable expectation value of a quantum mechanical operator. Particular attention is given to van der Waals interactions that traditionally are assumed to represent attractive interactions that are distinct from chemical bonding. They are assumed by some to act in concert with Pauli repulsions to account for the existence of condensed states of molecules. It is such dichotomies of interpretation that are resolved by the experimental detection of bond paths and the delineation of their properties in molecular crystals. Specific criticisms of the stabilization afforded by the presence of bond paths derived from spectroscopic measurements performed on dideuteriophenanthrene are shown to be physically unsound. The concept of a bond path as a "bridge of density" linking bonded atoms was introduced by London in 1928 following the definition of the electron density by Schrödinger in 1926. These papers marked the beginning of the theory of atoms in molecules linked by bond paths.

QTAIM study on the degenerate cope rearrangements of 1,5-hexadiene and semibullvalene.

Using the homotropylium cation (1) as an archetypal example of a homoaromatic molecule, we carried out a quantum theory of atoms-in-molecules (QTAIM) computational study--at DFT (density functional theory), CCSD (coupled cluster with singles and doubles), and CASSCF (complete active space self-consistent field) levels--on 1 and the degenerate Cope rearrangements of 1,5-hexadiene (2) and semibullvalene (3) including the evaluation of delocalization indexes and a visualization of atomic basins. This study yielded new insights into the factors determining the reaction barriers and the bonding of the ground and transitions states of 2 and 3. Contrary to conclusions reached in earlier studies, we found that the transition state for the degenerate rearrangement of 2 is not aromatic and that the driving force for the very facile Cope rearrangement of semibullvalene is caused by the stabilization of individual atoms as well as electronic delocalization, not by the release of strain in the three-membered ring.

Nearsightedness of electronic matter as seen by a physicist and a chemist.

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 two systems, necessary to account for the fundamental role of a functional group with characteristic properties in chemistry, however, is a problem in the physics of an open system. 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 unavoidable changes in the external potential that occur on transfer. The properties of an atom in a molecule are determined by the Heisenberg equation of motion for a "proper open system" derived from the principle of stationary action. The theorem is grounded in the common sense observation that two atoms that "look the same", i.e., have the same charge distribution, must possess identical properties. Transferability is discussed in terms of the properties of the electron density, the one-density and the two-density matrix, the latter demonstrating that both the Coulombic and exchange contributions to the energy of a group are separately transferable. It is demonstrated that the exchange indices determined by the two-density matrix provide a readily determinable measure of the effective range of the exchange energy.

Everyman's derivation of the theory of atoms in molecules.

This paper demonstrates that it is straightforward to develop the theory of an atom in a molecule--the extension of quantum mechanics to an open system--by deriving the necessary equations of motion from Schrödinger's equation, followed by a comparison of the predicted properties with experiment to determine the correct boundary condition. Although less fundamental than the variational derivation of the quantum theory of atoms in molecules, this heuristic approach makes the quantum mechanics of an atom in a molecule accessible to "everyman" possessing a knowledge of Schrödinger's equation, aiding its general acceptance by experimental chemists.

Forces in molecules.

Chemistry is determined by the electrostatic forces acting within a collection of nuclei and electrons. The attraction of the nuclei for the electrons is the only attractive force in a molecule and is the force responsible for the bonding between atoms. This is the attractive force acting on the electrons in the Ehrenfest force and on the nuclei in the Feynman force, one that is countered by the repulsion between the electrons in the former and by the repulsion between the nuclei in the latter. The virial theorem relates these forces to the energy changes resulting from interactions between atoms. All bonding, as signified by the presence of a bond path, has a common origin in terms of the mechanics determined by the Ehrenfest, Feynman and virial theorems. This paper is concerned in particular with the mechanics of interaction encountered in what are classically described as 'nonbonded interactions'--are atoms that 'touch' bonded or repelling one another?

An experimentalist's reply to "What is an atom in a molecule?".

Parr, Ayers and Nalewajski have opined in this Journal that the concept of an atom in a molecule "is an object knowable by the mind or intellect, not by the senses." This view is countered by the two hundred years of experimental chemistry underlying the realization that the properties of some total system are the sum of its atomic contributions. This paper concludes that an experimentalist has no doubt but that he or she is measuring the properties of atoms when performing an experiment.

Pauli repulsions exist only in the eye of the beholder.

This paper presents a rebuttal to the preceding paper in this issue entitled "Hydrogen-Hydrogen Bonding in Planar Biphenyl, Predicted by Atoms-In-Molecules Theory, Does Not Exist". The arguments presented therein are based on an arbitrary partitioning of the energy into contributions from physically unrealizable states of the system. The response given here is presented in terms of the Feynman, Ehrenfest, and virial theorems of quantum mechanics and the observable properties of a system. A reader is thus free to choose between subjectivity or physics.

Properties of Atoms in Molecules: Caged Atoms and the Ehrenfest Force.

This paper uses the properties of atom X enclosed within an adamantane cage, denoted by X@C10H16, as a vehicle to introduce the Ehrenfest force into the discussion of bonding, the properties being determined by the physics of an open system. This is the force acting on an atom in a molecule and determining the potential energy appearing in Slater's molecular virial theorem. The Ehrenfest force acting across the interatomic surface of a bonded pair atoms [Formula: see text] atoms linked by a bond path [Formula: see text] is attractive, each atom being drawn toward the other, and the associated surface virial that measures the contribution to the energy arising from the formation of the surface is stabilizing. It is the Ehrenfest force that determines the adhesive properties of surfaces. The endothermicity of formation for X = He or Ne is not a result of instabilities incurred in the interaction of X with the four methine carbons to which it is bonded, interactions that are stabilizing both in terms of the changes in the atomic energies and in the surface virials. The exothermicity for X = Be(2+), B(3+), and Al(3+) is a consequence of the transfer of electron density from the hydrogen atoms to the carbon and X atoms, the exothermicity increasing with charge transfer despite an increase in the contained volume of X.

Atoms-in-molecules study of the genetically encoded amino acids. III. Bond and atomic properties and their correlations with experiment including mutation-induced changes in protein stability and genetic coding.

This article presents a study of the molecular charge distributions of the genetically encoded amino acids (AA), one that builds on the previous determination of their equilibrium geometries and the demonstrated transferability of their common geometrical parameters. The properties of the charge distributions are characterized and given quantitative expression in terms of the bond and atomic properties determined within the quantum theory of atoms-in-molecules (QTAIM) that defines atoms and bonds in terms of the observable charge density. The properties so defined are demonstrated to be remarkably transferable, a reflection of the underlying transferability of the charge distributions of the main chain and other groups common to the AA. The use of the atomic properties in obtaining an understanding of the biological functions of the AA, whether free or bound in a polypeptide, is demonstrated by the excellent statistical correlations they yield with experimental physicochemical properties. A property of the AA side chains of particular importance is the charge separation index (CSI), a quantity previously defined as the sum of the magnitudes of the atomic charges and which measures the degree of separation of positive and negative charges in the side chain of interest. The CSI values provide a correlation with the measured free energies of transfer of capped side chain analogues, from the vapor phase to aqueous solution, yielding a linear regression equation with r2 = 0.94. The atomic volume is defined by the van der Waals isodensity surface and it, together with the CSI, which accounts for the electrostriction of the solvent, yield a linear regression (r2 = 0.98) with the measured partial molar volumes of the AAs. The changes in free energies of transfer from octanol to water upon interchanging 153 pairs of AAs and from cyclohexane to water upon interchanging 190 pairs of AAs, were modeled using only three calculated parameters (representing electrostatic and volume contributions) yielding linear regressions with r2 values of 0.78 and 0.89, respectively. These results are a prelude to the single-site mutation-induced changes in the stabilities of two typical proteins: ubiquitin and staphylococcal nuclease. Strong quadratic correlations (r2 approximately 0.9) were obtained between DeltaCSI upon mutation and each of the two terms DeltaDeltaH and TDeltaDeltaS taken from recent and accurate differential scanning calorimetry experiments on ubiquitin. When the two terms are summed to yield DeltaDeltaG, the quadratic terms nearly cancel, and the result is a simple linear fit between DeltaDeltaG and DeltaCSI with r2 = 0.88. As another example, the change in the stability of staphylococcal nuclease upon mutation has been fitted linearly (r2 = 0.83) to the sum of a DeltaCSI term and a term representing the change in the van der Waals volume of the side chains upon mutation. The suggested correlation of the polarity of the side chain with the second letter of the AA triplet genetic codon is given concrete expression in a classification of the side chains in terms of their CSI values and their group dipole moments. For example, all amino acids with a pyrimidine base as their second letter in mRNA possess side-chain CSI < or = 2.8 (with the exception of Cys), whereas all those with CSI > 2.8 possess an purine base. The article concludes with two proposals for measuring and predicting molecular complementarity: van der Waals complementarity expressed in terms of the van der Waals isodensity surface and Lewis complementarity expressed in terms of the local charge concentrations and depletions defined by the topology of the Laplacian of the electron density. A display of the experimentally accessible Laplacian distribution for a folded protein would offer a clear picture of the operation of the "stereochemical code" proposed as the determinant in the folding process.

Hydrogen-hydrogen bonding: a stabilizing interaction in molecules and crystals.

Bond paths linking two bonded hydrogen atoms that bear identical or similar charges are found between the ortho-hydrogen atoms in planar biphenyl, between the hydrogen atoms bonded to the C1-C4 carbon atoms in phenanthrene and other angular polybenzenoids, and between the methyl hydrogen atoms in the cyclobutadiene, tetrahedrane and indacene molecules corseted with tertiary-tetra-butyl groups. It is shown that each such H-H interaction, rather than denoting the presence of "nonbonded steric repulsions", makes a stabilizing contribution of up to 10 kcal mol(-1) to the energy of the molecule in which it occurs. The quantum theory of atoms in molecules-the physics of an open system-demonstrates that while the approach of two bonded hydrogen atoms to a separation less than the sum of their van der Waals radii does result in an increase in the repulsive contributions to their energies, these changes are dominated by an increase in the magnitude of the attractive interaction of the protons with the electron density distribution, and the net result is a stabilizing change in the energy. The surface virial that determines the contribution to the total energy decrease resulting from the formation of the H-H interatomic surface is shown to account for the resulting stability. It is pointed out that H-H interactions must be ubiquitous, their stabilization energies contributing to the sublimation energies of hydrocarbon molecular crystals, as well as solid hydrogen. H-H bonding is shown to be distinct from "dihydrogen bonding", a form of hydrogen bonding with a hydridic hydrogen in the role of the base atom.

Atoms-in-molecules study of the genetically encoded amino acids. II. Computational study of molecular geometries.

The geometries of the 20 genetically encoded amino acids were optimized at the restricted Hartree-Fock level of theory using the 6-31+G* basis set. A detailed comparison showed the calculated geometries to be in excellent agreement with those determined by X-ray crystallography. The study demonstrated that the geometric parameters for the main-chain group and for the bonds and common functional groups of the side-chains exhibit a high degree of transferability among the members of this set of molecules. This geometric transferability is a necessary prerequisite for the corresponding transferability of their electron density distributions and hence of their bond and atomic properties. The transferability of the electron distributions will be demonstrated and exploited in the following paper of this series, which uses the topology of the electron density to define an atom within the quantum theory of atoms in molecules. Particular features of the geometries of the amino acids are discussed. It has been shown, for example, how the apparent anomaly of the Calpha-N bond length in a peptide being shorter than in the charged species Calpha-NH3+ is resolved when the charge separation is gauged by the differences in the charges of the Calpha and N atoms as opposed to the use of formal charges. A compilation of literature sources on experimental geometries covering each member of the 20 amino acids is presented. A set of rules for labeling the atoms and bonds, complementing the generally accepted IUPAC-IUB rules, is proposed to uniquely identify every atom and bond in the amino acids.

Geometry of the Fluorides, Oxofluorides, Hydrides, and Methanides of Vanadium(V), Chromium(VI), and Molybdenum(VI): Understanding the Geometry of Non-VSEPR Molecules in Terms of Core Distortion.

This paper describes a study of the topology of the electron density and its Laplacian for the molecules VF(5), VMe(5), VH(5), CrF(6), CrMe(6), CrOF(4), MoOF(4), CrO(2)F(2,) CrO(2)F(4)(2)(-) and CrOF(5)(-) all of which, except VF(5,) CrF(6), and CrOF(5)(-) have a non-VSEPR geometry. It is shown that in each case the interaction of the ligands with the metal atom core causes it to distort to a nonspherical shape. In particular, the Laplacian of the electron density reveals the formation of local concentrations of electron density in the outer shell of the core, which have a definite geometrical arrangement such as four in a tetrahedral arrangement or five in a square pyramidal or trigonal bipyramidal and six in an octahedral arrangement. Ligands that are predominately covalently bonded are found opposite regions of charge depletion between these core charge concentrations. In VH(5), VMe(5), CrOF(4), and MoOF(4), these core charge concentrations have a square pyramidal arrangement, and the regions of charge depletions have the corresponding inverse square pyramidal arrangement so that these molecules have a square pyramidal geometry rather than a trigonal prism geometry. In CrMe(6), there are five core charge concentrations with a trigonal bipyramidal arrangement so that the regions of charge depletion have a trigonal prismatic arrangement and the molecule has the corresponding trigonal prism geometry rather than an octahedral geometry. In contrast, molecules in which the only ligand is the more ionically bound fluorine are less affected by core distortion and have VSEPR-predicted structures. The unexpected bond angles in CrO(2)F(2) and the preference of CrO(2)F(4)(2)(-) for a cis structure are also discussed in terms of the pattern of core charge concentrations.