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Atoms in Molecules Acc. Chem. Res. 1985,18, 9-15 9 Atoms in Molecules R. F. W. Bader Department of Chemistry, McMaster University, Hamilton, Ontario, Canada L8S 4M1 Received May 25, 1984 (Revised Manuscript Received September 28, 1984) Approximate quantum mechanical state functions of atoms linked by a network of bonds is the operational have recently been obtained for the norbornyl cation.1 principle underlying our classification and present un- These calculations have yielded the energy of its derstanding of chemical behavior. Therefore, it would equilibrium geometry and the relative energies of appear that to find chemistry within the framework of neighboring geometries. It is the purpose of this ac- quantum mechanics one must find a way of determining count to illustrate that more chemical information than the values of observables for pieces, that is, subsystems, simply energies and their associated geometries can be of a total system. But how is one to choose the pieces? obtained directly from a state function. A state func- Is there only one or are there many ways of dividing a tion contains the necessary information to both define molecule into atoms and its properties into atomic the atoms in a molecule and determine their average contributions? If there is an answer to this problem properties.2 It also enables one to assign a structure, then the necessary information must be contained in that is, determine the network of bonds linking the the state function, for 'k tells us everthing we can know atoms in a molecule and determine whether or not the about a system. structure is stable.3 The state function further de- Thus the question “Are there atoms in molecules?” termines where electronic charge is locally concentrated is equivalent to asking two equally necessary questions and depleted. Quantum mechanics can be used to re- of quantum mechanics: (a) Does the state function late these properties to local energy contributions,4 predict a unique partitioning of a molecule into sub- thereby providing an understanding of the geometry systems? (b) Does quantum mechanics provide a com- and reactivity of a molecule.5 plete description of the subsystems so defined? To The norbornyl cation provides a clear illustration of answer questions a and b one must turn to a principle how quantum mechanics can be used to obtain this kind that directly determines the observables, their values, of information. The protracted discussion concerning and equations of motion for a quantum system. Such the possible structures of this ion and their relationship is Schwinger’s principle of stationary action,7 the power to its chemistry have been the subject of a recent issue and elegance of which derives from its unified of Accounts.6 The possibility of unambiguously as- approach—a single principle providing a complete de- signing a structure to the norbornyl cation demonstrates velopment of quantum mechanics. The approach is also the usefulness of the theory. a very general one, so general that one can, within its seek answers to a and b.8 It is Atoms in Molecules framework, questions found that the same principle does apply to a subsys- It is a postulate of quantum mechanics that every- tem of a total system if the subsystem is bounded by thing that can be known about a system is contained a surface which satisfies a particular physical condition. in the state function 'k. The value of a physical quan- The condition is stated in terms of a property of the tity is obtained from the state function through the elctronic charge density, as determined by the state action of a corresponding operator on Thus quan- function of the total system. Thus the state function tum mechanics is concerned with observables, that is, does predict that a system can be uniquely partitioned with operators corresponding to physical properties. into subsystems. Since the principle of stationary ac- Downloaded via SAINT EDWARD'S UNIV on November 12, 2018 at 19:54:44 (UTC). Depending on the nature of the state function some tion applies to these particular subsystems, the values observables yield “sharp” values, others yield “average” and corresponding subsystem equations of motion for See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. values. The manner in which the value of each ob- each observable are also uniquely determined. Thus servable changes with time is given by a corresponding J. J. Am. Chem. Soc. of called a of (1) Goddard, D.; Osamura, Y.; Schaefer, F., III, equation motion, Heisenberg equation 1982,104, 3258. motion. The theorems of quantum mechanics that yield (2) Bader, R. F. W.; Nguyen-Dang, T. T. Adv. Quantum Chem. 1981, relationships between various average values, such as 14, 63. (3) Bader, R. F. W.; Nguyen Dang, T. T.; Tal, Y. Rep. Prog. Phys. are the virial theorem and Ehrenfest relations, derived 1981, 44, 893. from such equations. Questions we have about a (4) Bader, R. F. W.; Essen, H. J. Chem. Phys. 1984, 80, 1943. R. F. P. C. D. H. J. Am. system are therefore, answered in terms of the (5) Bader, W.; MacDougall, J.; Lau, Chem. quantum Soc. 1984, 106, 1594. values and equations of motion for the relevant physical (6) Grob, C. A. Acc. Chem. Res. 1983 16,426. Brown, H. C. Acc. Chem. observables. These values and relationships refer of Res. 1983,16,432. Olah, G. A.; Surya Prakash, G. K.; Saunders, M. Acc. to the total An of Chem. Res. 1983, 16, 440. Walling, C. Acc. Chem. Res. 1983, 16 448. course, system. understanding (7) Schwinger, J. Phys. Rev. 1951, 82, 914. chemistry requires answers of a more regional nature. (8) In fact, the search for a quantum definition of an atom in a mol- The notion that a molecule can be viewed as a collection ecule leads in a natural way to the principle of stationary action. This principle is obtained through a general variation of the quantum action integral. In varying the action integral for a subsystem one must nec- Richard F. W. Bader was born in 1931. He received his M.Sc. in 1955 essarily retain the variations in 4' on the boundaries of the subsystem and from McMaster University and his Ph.D. in 1957 from the Massachusetts In- this leads to their inclusion at the time end points of the action integral. stitute of Technology. He did postdoctoral work at MIT in 1957-1958 and in This is precisely the step that transforms the usual action principle into 1958-1959 at Cambridge University. He began his scientific career as a Shwinger’s more general principle of stationary action. It is the variations physical organic chemist. He was at the University of Ottawa in 1960-1963 in 4' at the time end points that are identified with and determine the and then moved to McMaster University, where he remains today. properties of the observables of a quantum system. 0001-4842/85/0118-0009$01.50/0 © 1985 American Chemical Society 10 Bader Accounts of Chemical Research one has a quantum mechanics for a subsystem. The same theorems derivable from the Heisenberg equations of motion for the total system may be derived for a subsystem. The virial theorem and Ehrenfest rela- tionships for example, apply to these subsystems. The virial theorem yields a definition of the electronic en- ergy of a subsystem. It has been demonstrated that this energy faithfully reflects the near constancy or change in the distribution of charge over the subsystem when it is observed in different molecules.9 At this point one has a quantum definition of a sub- system stated in terms of a particular property of the charge distribution. One has next to determine whether or not a molecular charge distribution exhibits the properties demanded by this definition and whether the resulting subsystems, if they are so defined, correspond to the chemical atoms. Thus one is led to study the topology of a molecular charge distribution. Topology of Molecular Charge Distributions The condition defining the quantum subsystem is stated in terms of a property of the gradient vector of the charge density p. This is most fortunate, for the gradient vectors of p (collectively called the gradient vector field of p) are precisely what one studies to obtain both a qualitative and quantitative understanding of the topology of p. Indeed, if one assumes that a mo- lecular charge density contains the information neces- sary for the definition of molecular structure, then there already exists a complete mathematical theory that defines both structure and the stability of this structure in terms of the properties of the gradient vector field associated with a scalar field such as p.10,11 Figure 1, A portion of a contour map of a charge distribution The quantum condition of the subsystem is simply showing its intersection with two partitioning surfaces. The line stated: the surface the shall not of intersection for surface a is crossed by the vectors Vp. The bounding subsystem vector is to the line of intersection of surface b at each be crossed vectors of la shows Vp tangent by any gradient p. Figure point on the surface. The two aims of surface b meet at a point the intersection of an arbitrarily defined surface with where Vp = 0, a critical point. a contour map of p. Since the gradient vector of p, denoted by Vp always points in the direction of greatest properties, a point where Vp = 0, called a critical point. increase in p, it must always be perpendicular to lines Each of the arms indicating the intersection of the of constant density.
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