Matta, Boyd. The quantum theory of atoms in molecules

52 2 The Lagrangian Approach to Chemistry

lists the atomic theorems for molecules and crystals obtained for several important generators [30]. One notes the presence in these theorems of the electron

$

density rðrÞ, the current density j(r) and the quantum stress tensor sðrÞ, quantities introduced by Schro¨dinger in 1926 and held by him to be essential to understanding the mechanical, electrical, and magnetic properties of matter. For example, the atomic virial theorem is required for definition of the energy of an open system and of the pressure acting on its surface. The atomic current theorem is required to determine the atomic contribution to the molecular diamagnetic susceptibility arising from the flux in the position-weighted flux in the induced current through the atomic surface. The use of more of these theorems will be illustrated in the contributions to this book.

2.5

Molecular Structure and Structural Stability

2.5.1

Definition of Molecular Structure

The remarkable encompassing physical aspect of the theory is that the same topology that defines an atom in a molecule, leads to a theory of molecular structure and structural stability. The response of the electron density to the interaction between two atoms is ubiquitous, resulting in the formation of a ð3; 1Þ critical point whose associated trajectories define not only the presence of the zero-flux interatomic surface but also delineate a line of maximum electron density that links the nuclei of neighboring atoms – the ‘‘bond path’’ [3, 31–33]. The network of bond paths generate a molecular graph that defines a system’s structure. The topological structures have been shown to recover the ‘‘chemical structures’’ in a multitude of systems, in terms of densities obtained from both theory and experiment, structures that were previously inferred from classical models of bonding in conjunction with observed physical and chemical properties [34].

A bond path meets all the physical requirements set by the Ehrenfest, Feynman and virial theorems that the atoms be bonded to one another [34]; the two atoms experience an attractive Ehrenfest force drawing their atomic basins together – no Feynman force, either attractive or repulsive, acts on the nuclei, because of the balancing of the repulsive and attractive forces by accumulation of electron density in the binding region. This same accumulation leads to a reduction of the electron–nuclear potential energy, the magnitude of which exceeds the increases in the electron and nuclear repulsion energies, resulting in a decrease in the potential energy equal to twice the decrease in the total energy, all as demanded by the virial theorem. Thus a bond path is indicative of the accumulation of density between the nuclei that is necessary for the presence of attractive Ehrenfest forces, for balancing of the Feynman forces on the nuclei, and for the decrease

2.6 Reflections and the Future 53

in energy. Its presence is both necessary and su cient for two atoms to be bonded to one another [35, 36].

2.5.2

Prediction of Structural Stability

A central concept in the analysis of the stability of gradient vector field is the equivalence relationship. Two vector fields are said to be equivalent if every trajectory of one field can be mapped on to a corresponding trajectory of the other. By application of this equivalence relationship to the gradient vector field of the electron density in behavior space, one arrives at a partitioning of nuclear configuration space RQ, the control space, into a finite number of disjoint regions, the structural regions, each of which is characterized by a unique molecular graph. The structural regions form a dense open subset of RQ and a point belonging to such a region is a stable structure and is called a regular point. The catastrophe set C is the collection of all structurally unstable points in RQ and serves as the union of the boundaries of all the structural regions. The result is a structure diagram, a diagram that defines all possible structures and all structural changes linking the structures for a given system [3, 37]. The Palis–Smale theorem of structural stability [38] shows that a change in structure can occur by only one of two possible mechanisms – the bifurcation mechanism arising from the formation of a degenerate critical point in the density or through the conflict mechanism wherein the manifolds of two critical points intersect in what is a manifestly unstable manner.

The topological theory of molecular structure and structural stability leads to several important observations [3]. A molecular geometry, a point in RQ, should be distinguished from a molecular structure, which represents an open region of RQ, that is, structure is generic. Motion in RQ changes the geometry, but leaves the structure unchanged for motion within the open region associated with a given structural region. A change in structure is an abrupt and discontinuous process and occurs when a system point crosses a boundary at a nuclear configuration in RQ belonging to the catastrophe set separating two structural regions. One finds these ideas being increasingly applied to a wide range of problems.

2.6

Reflections and the Future

2.6.1

Reflections

It is indeed a pleasure to write this chapter for a book that demonstrates the remarkable progress that has been made in the development and application of QTAIM. All problems at the atomic level are subject to study by QTAIM and the

54 2 The Lagrangian Approach to Chemistry

scope of its application is forever widening. The use of the atomic theorems derived from the Heisenberg equations of motion Eq. (19), some of which are displayed in Table 2.1 [30], extends beyond the asking of chemical questions of bonding, structure, and reactivity. The theorems apply to all questions about the behavior of matter at the atomic level. Indeed, there are many problems that specifically require the physics of an open system for their statement and solution. Examples are the operation of the atomic force and electron tunneling microscopes [39], defects in solids [40], the quantum definition of pressure [41], and the polarization of a dielectric [42]. The atomic statement of the Ehrenfest force is the equation of motion for an open system. It is capable of describing the motion of an adsorbed atom on the surface of a substrate or of the forces required for manipulation of individual atoms or molecules, thereby providing a basis for nanotechnology. The physical understanding obtainable from the use of the theorems for an open system is only beginning to be explored.

Molecular orbital theory is the theory for the understanding and prediction of the electronic structure of molecules, predicting the ordering and classification of many-electron states in terms of one-electron states, and it is indispensable to all chemists for the understanding of the properties of many-electron systems. Orbital ordering and state classification forms the basis for the application of the ‘‘second-order Jahn–Teller’’ symmetry rule [43] that underlies Fukui’s frontier orbital theory [44] and orbital conservation [45].

Unlike QTAIM, which builds upon the chemical concept of a functional group with characteristic properties, molecular orbital theory, as Libit and Ho mann point out [46], is incapable of recovering this concept, because each molecular orbital extends over the entire molecule. Recent papers illustrate the complementary roles of molecular orbital theory and QTAIM and how QTAIM provides the possibility of assessing the viability of orbital models [47, 48].

To argue, as some do, that QTAIM overlooks explanations of bonding a orded by simple orbital models is wrong. What QTAIM does do is enable one to go beyond the models. To state that QTAIM, by finding a bond path in Ar2 for example, fails to predict the absence of electron pair bonding between closed-shell molecules is based on improper use of the orbital model [49]. The orbital model when properly applied and understood using QTAIM, predicts weak bonding between closed shell systems. It is a travesty to claim that molecular orbital theory cannot account for the bonding between closed-shell systems by insisting that one terminates the theory at the single determinant level. Electron correlation is known to be responsible for the bonding in such cases and is readily accounted for by inclusion of the interaction of excited configurations with the ground state. Thus, for example, a CI calculation predicts both bonding and the presence of a bond path in rare gas dimers [50], a result in accord with the experimental detection of bound He2 [36].

In summary, there is no conflict of QTAIM with molecular orbital theory. The conflict is with those who introduce nonphysical concepts, postulating the presence of repulsive forces in systems wherein no definable repulsive forces act on the density or on the nuclei [49] (appropriate responses being given in [51, 52]).

2.6 Reflections and the Future 55

One can hope that younger scientists, not having been exposed to models that have outlived any usefulness they might have once enjoyed, will place their trust in physics.

2.6.2

The Future

A necessary next step in the development of the theory is its extension beyond the fixed nucleus approximation of the Born–Oppenheimer procedure. The topology of the electron density is a consequence of, and summarizes the physics that underlies, the form of matter. Whatever new topological features the charge density may be found to exhibit as a consequence of its averaging over nuclear motions, they may be incorporated into an expanded theory to provide a still deeper understanding of the behavior of matter at the atomic level.

Relativistic e ects cause no problem, as the mathematical formalism of Schwinger’s and Feynman’s approaches is manifestly co-variant with regard to Lorentz transformations, if one adheres to a relativistically invariant Lagrangian [7], as is done in the development of QTAIM.

James Anderson, a student of Dr. Paul Ayers, has recently extended QTAIM to the relativistic domain using the ZORA Hamiltonian, as a result of attending my graduate course. Schwinger’s theory is, of course, relativistically invariant, but there are a number of crucial steps involved in its extension to an open system, not the least of which is the zero-flux boundary condition. He finds QTAIM to be ‘robust’, the entire theory comes through unchanged and can be applied with the same zero-flux boundary condition across the periodic table – including the actinides. It is beautiful the way in which all of the important properties of the relativistic Lagrangian mimic the essential properties of the non-relativistic case, all of the derived relativistic expressions reducing to their non-relativistic forms in the limit of infinite c. I can see no reason for anyone doubting that the zero-flux boundary is a fundamental property of matter, providing the basis for the generalization of physics to its atomic constituents.

In many problems, the system of interest is an open system embedded in a much larger one. For example, in the development of molecular devices in nanotechnology one wishes to determine the conductivity of an organic molecule linking two conductors. In biological systems one’s interests may focus on just the active or binding site of an enzyme. Can one define and study the open system of interest, rather than resorting to existing ‘‘embedding’’ methods. The properties of an open system are totally determined by its bounding surface [53]. The determination of the open system in a simpler environment, then its transfer to the system of interest, is a possibility that has already been successfully explored in the construction of biological molecules, in which the groups of interest have high transferability and close matching of the interatomic surfaces is possible.

The open system variation principle, Eq. (11), o ers the possibility of obtaining the wave function for the entire system by performing the variation over just the open system of interest, including its surface, and requiring that the variations

56 2 The Lagrangian Approach to Chemistry

equal the surface flux in the infinitesimal current generated by the variations in c. The variation is, of course, subject to the constraint that at every stage the system be bounded by a zero-flux surface. Baranger [54] has shown that at Hartree– Fock, minimization of the energy (of the total system) is equivalent to satisfaction

h ½ ^ ; ^ & i ^

of the vanishing of c H G c , Eq. (15), for all one-body operators G, and has given a procedure for doing so. One could apply the same procedure to a open system by noting that satisfaction of Eq. (11) is equivalent to satisfying Eq. (16) for a proper open system, and thus requires that the commutator average over the open system be given by the surface flux in the corresponding currents for

^

all one-body operators G.

What is, to me, among the most important of the results derived from the development of the theory of atoms in molecules is the paralleling behavior in the form and properties found for a proper open system. This observation is valid at all levels of transferability of the density – from near transferability, as found for the Li atom in its hydride, oxide, and fluoride molecules in the original 1972 paper (the observation that sparked the development of the theory) to the essentially perfect transferability observed, for example, for the transferable methyl and methylene groups in linear hydrocarbons or amino acid residues in a polypeptide [55]. Although it is an obvious physical necessity that form determine properties, it is only through the atoms of QTAIM that this condition is realized, and it is striking. One need only view the previously illustrated [56] remarkable degree of paralleling transferability of the electron density, the kinetic energy density, and the virial field (the potential energy density) of the methylene groups in butane and pentane as an example. The virial field is a real-space representation of the average e ective potential experienced by a single electron in a many-electron system. It is the most short-range description possible of this interaction potential [57] and its integral over all space yields the total potential energy of the molecule in the fixed nucleus approximation [58]. Add to this the observation that it is structurally homeomorphic with the electron density [59] and it would seem to be a promising starting point for investigation of the energy $ density relationship. The local statement of the atomic virial theorem, by relating the Laplacian of the density to the kinetic energy density and the virial field, provides a link between a property of the density with energy [3, 16].

It is clear from the tabulations of experimentally derived group properties that group additivity must often be only apparent, as is found to apply when a group is unavoidably perturbed by its environment. The additivity in these cases is a result of compensatory transferability, the change in a property, energy for example, and in the charge that is experienced by one group being equal and opposite to the changes experienced by the other group [60]. Charge is, of course, necessarily conserved, but for this conservation to be paralleled by all properties is remarkable, one that can be illustrated by the data in the tables of Benson et al. [61, 62] and in theoretical calculations [53, 60]. There seems to be a Le Chatellier principle at work – one that states that two open systems brought into contact respond in such a way as to minimize the overall changes in their form and properties [63]. Is it possible to formulate an extremization principle that minimizes the sum of the energy changes of the two open system when brought into contact?

References 57

Chemistry is a consequence of the short-range nature of the one-electron density matrix that determines all the mechanical properties of an atom in a molecule [57] with the additional important proviso that all of the necessary physical information is obtained in its expansion up to second-order with regard to both the diagonal and o -diagonal terms [3]. The diagonal terms yield the density rðrÞ, its gradient vector field ‘rðrÞ, and its dyadic ‘‘rðrÞ whose trace yields the Laplacian ‘2rðrÞ and determines the critical points in ‘rðrÞ and hence in rðrÞ.

$

The o -diagonal terms yield the current density j(r), the stress tensor sðrÞ and the tensor ‘jðrÞ whose properties determine the critical points in j(r) [64]. There is much to be studied. Whereas the topologies of rðrÞ [37] and of j(r) [64] have been completely characterized and related to the physical properties of the sys-

$ $

tem, the same is not true of the stress tensor sðrÞ. The topology of sðrÞ, whose properties determine the local mechanics of the density, its trace equaling the virial field and the local statement of the virial theorem, is largely unstudied. Its eigenvalues and eigenvectors at a degenerate critical point in the density that is indicative of a change in structure could summarize the mechanical consequences. Similarly, the topology of the Ehrenfest force field, a vector field defined

$

by ‘ sðrÞ, could prove invaluable in understanding the mechanics of the density.

A surface of zero-flux in this field would demark a region of space for which the Ehrenfest force vanishes. The final relationship between the density and the energy will have to account for their paralleling behavior as evinced by the atoms of QTAIM, so it seems reasonable that the observations of QTAIM regarding the role of the density in the determination of atomic properties could serve as a starting point in the search for such a relationship [57, 58].

References and Notes

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9	M. Gell-Mann, Physics To-Day 1989,		variational principle states that the
	42 50.		energy obtained by averaging the

58 2 The Lagrangian Approach to Chemistry

Hamiltonian operator over a trial function, which usually includes variational parameters in the minimization of the ‘‘variational integral’’, is an upper bound to the true energy E.

14The Lagrangian approach is used throughout mechanics and in every case, from di usion to

electromagnetics to general relativity to Boltzmann’s law, to cite just a few examples, the expression for the Lagrangian contains a squared gradient term of the form representing the kinetic energy as in Eq. (4). Frieden (Frieden, R. B. Physics from Fisher Information, Cambridge University Press, 1998) has argued that the presence of such a term is natural for all fields and provides a measure of the field’s intrinsic information called the ‘‘Fisher information’’.

15R. F. W. Bader, H. J. T. Preston, Int. J. Quantum Chem. 1969, 3 327–347.

16R. F. W. Bader, T. T. Nguyen-Dang,

Ad. Quantum Chem. 1981, 14 63–124.

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16–21.

18R. F. W. Bader, Chem. Phys. Lett.

2006, 426, 226–228.

19P. A. M. Dirac, Physik. Zeits. Sowjetunion 1933, 3 64.

20An account of Dirac’s proposal using the multiplicative law of transformation functions and its consequences for Feynman’s path integral is given in Section 8.6 of my book. Schwinger’s use of his di erential characterization of the transformation function as the basis for his quantum action principle is given in Section 8.2.

21R. F. W. Bader, S. Srebrenik, T. T. Nguyen Dang, J. Chem. Phys. 1978, 68 3680–3591.

22R. F. W. Bader, Theor. Chem. Acc.

2001, 105 276–283.

23R. F. W. Bader, Theor. Chem. Acc.

2002, 107 381–382.

24The expression for the total time derivative of the expectation value for an open system contains in addition

to the terms shown in Eq. (19), a surface integral of the change in the surface with time weighted by the property density of the observable G.

25R. F. W. Bader, Phys. Rev. 1994, B 49 13348–13356.

26H. Bethe, New York Times Obituary

2005, March 18.

27R. L. A. Haiduke, R. E. Bruns, J. Phys. Chem. A 2005, 109 2680–2688.

28P. H. Ce´sar, S. H. D. Faria, J. V. da Silva, R. L. A. Haiduke, R. E. Bruns,

Chem. Phys. 2005, 317 35–42.

29C. F. Matta, R. F. W. Bader, J. Phys. Chem. A 2006, 110 6365–6371.

30R. F. W. Bader, P. L. A. Popelier, Int.

J.Quantum Chem. 1993, 45 189–207.

31G. R. Runtz, R. F. W. Bader, R. R. Messer, Can. J. Chem. 1977, 55

3040–3045.

32R. F. W. Bader, S. G. Anderson, A. J. Duke, J. Am. Chem. Soc. 1979, 101

1389–1395.

33R. F. W. Bader, in M. T. P. International Series of Science, Theoretical Chemistry, Vol. II, (Eds.:

C.A. Coulson, E. D. A. Buckingham), Butterworths, 1975, pp. 43–79.

34R. F. W. Bader, J. Phys. Chem. A 1998, 102 7314–7323.

35For a system in a stationary state, the wave function and the electron density it determines are such as to minimize the total energy. Thus the formation of an atomic interaction line

– a line of maximum density linking a pair of nuclei – is associated with the minimization of the energy of interaction between the atoms it links. In the case where the attractive restoring forces return a system to its equilibrium geometry it is termed a bond path. All bond paths are transformed into atomic interaction lines when the separation between the atoms is compressed to the point where repulsive forces dominate the interaction, as determined by the kinetic energy T exceeding the negative of the total energy E, T > E. One tires of reading the criticism of a bond path being present in the case of ‘‘repulsive interactions’’ such as He2. At its equilibrium separation,

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	compressed and the bond path is		6253–6263.
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61

3

Atomic Response Properties

Todd A. Keith

3.1 Introduction

This chapter describes some applications of the quantum theory of atoms in molecules (QTAIM) [1] with the objective of an atomic description of molecular response properties. For the purposes of this chapter, molecular response properties are observable measures of how a molecule changes as a result of interactions with external sources, such as applied electric and/or magnetic fields, nuclear magnetic moments, nuclear displacements from equilibrium, etc. For example, electric dipole polarizability and hyperpolarizability tensors are measures of how a molecule’s electric dipole moment changes in response to an external electric field. Molecular response properties of interest are typically origin-independent. However, in many cases a corresponding response property density is origindependent and a simple definition of an atomic response property as the integral of such a density over the space of the atoms results in origin-dependent atomic contributions. This origin-dependence arises from a ‘‘null’’ molecular property that vanishes for the whole molecule but not for an atom in a molecule. The classic example is the electric dipole moment (a first-order response to an external electric field) for neutral molecules, in which case the ‘‘null’’ molecular property is the net charge. Because property densities are not unique (any function which integrates to zero for the molecule can be added to a ‘‘basic’’ property density), one may try and circumvent this problem by defining a property density that is origin-independent or whose atomic integrals are origin-independent. This is not usually possible, however.

A physically meaningful method for defining origin-independent atomic contributions to electric dipoles, electric polarizabilities, magnetizabilities and other response properties was first introduced by Bader et al. [2–10]. The essence of this method is to express each atomic contribution as the sum of an atomic ‘‘polarization’’ contribution and a set of ‘‘transfer’’ contributions associated with each group to which the atom is bonded. The expression and interpretation of an atomic polarization contribution, as the polarization of a ‘‘null’’ property density