J. Phys. Chem. 1993,97, 7516-7529
7516
Sharing of Electrons in Molecules Robert L. Fulton Department of Chemistry, The Florida State University, Tallahassee, Florida 32306-3006 Received: March 5, I993
An index which gives a quantitative measure of the degree of sharing of an electron between two points in space in systems containing many electrons is introduced. This sharing index, denoted by is defined as the absolute value squared of the matrix element of a sharing amplitude, which in turn is the square root of the first-order density matrix. These quantities are invariant under transformations of the orbitals in terms of which the wavefunction is typically expressed and are independent of the basis set provided it is sufficiently complete. The sharing amplitude has many of the characteristics of a wavefunction. By integration of the sharing index over volumes assigned to atoms, indices which measure the degree of sharing of an electron between atoms in molecules are found. Bond indices (numbers) are shown to be twice the value of the atomic sharing indices. On the basis of this, prototype double and triple bonds have bond indices of 2 and 3. That the sharing index automatically accounts for the interference and/or the localization of the electron is illustrated by the values of the bond indices for He2+ and for He2, calculated using either delocalized orbitals or using localized orbitals. One consequence of interference is that the contributions of antibonding orbitals to the sharing indices tend to cancel the contributions of bonding orbitals. To further illustrate the present definition, state of H2 and for the ?r-electrons in benzene the values of the bond indices are found for the first excited lZg+ and in 1,3-butadiene in the Huckel approximation. The present bond indices for the ?r-electrons differ from the covalent bond indices recently defined by Cioslowski and Mixon. In the case of benzene, the procedure of these authors leads to an infinity of sets of equivalent localized orbitals, each set giving different values for the covalent bond indices and each set breaking the symmetry of the benzene wavefunction. In contrast, the bond indices arising from the sharing indices retain the underlying symmetry of the wavefunction. The covalent bond indices for 1,3-butadiene, a case in which there is no broken symmetry, all differ from those obtained from the sharing indices. The relation between bond indices and the bond orders of Coulson, in the Huckel approximation, is found to be similar to that between the sharing index and the sharing amplitude. The addition of correlation to a simple molecular orbital wavefunction for the ground lZg+ state of H2 is shown to decrease the interatomic sharing at the equilibrium internuclear distance. At large internuclear separations the addition of correlation results in the expected value of zero for the interatomic sharing, in contrast to the nonzero value for a single determinant wavefunction. A volumepoint sharing index is defined by integrating the point-point sharing index over but one index. A simplified description of the electronic structure of benzene, including a-electron contributions, demonstrates how this volume-point sharing index can be used to discuss in quantitative detail the geometry of the sharing of electrons which are associated with an atom. This particular sharing index therefore gives a microscopic picture of the shape of the valence of an atom in a molecule. Again using a simplified description of benzene, we show that the two point sharing amplitude itself gives a clear indication of the degree to which electrons are localized at various points in a molecule, e.g., in the regions traditional associated with a- or ?r-bonds, in the core region surrounding a nucleus, etc. These amplitudes can then be interpreted in terms of such familiar concepts as s-p hybridization, localized orbitals, delocalized orbitals of which an example is the ?r-orbital contribution in benzene, and so on. Unlike orbitals, however, the sharing amplitudes and sharing indices have the virtue that they depend only on the complete many electron wavefunction. They therefore describe, at the one electron level, the electronic structure of a many electron system in a fashion which is invariant to orbital transformations. The present indices are quite general and are not limited in applications to electrons in molecules.
(fir),
I. Introduction The idea that bonds in molecules are associated with the sharing of electrons was put forth by G. N. Lewis's2 in 1916. The qualitative ideas of Lewis have been superseded by the results of quantum theory. However the idea that there is some number of electronsshared by atoms making up molecules has permeated our concept of bonding. We mention the concept of bond number advocated by L. C. Pauling? in particular the fractional bond numbers of aromatic hydrocarbons, and theconcept of bond order introduced by C o ~ l s o n .Since ~ that time numerous elaborations of the concept of bond order have appeared in the literature.5 The problems associated with current definitionsof bond orders and indices are well-known.- Briefly, most of the definitions depend explicitlyupon the assignment of orbitals to atomic centers and not just on the complete many electron wavefunction. As 0022-365419312097-7516$04.00/0
Z(fir),
a result, different values of the bond orders can be obtained from different basis sets, any of which, if sufficiently complete, give equally good many electron wavefunctions. The covalent bond order introduced by the authors of ref 5 differs from these in that, by assigning the atoms to volumes which are based on properties of the electron density and then defining the bond order in terms of a particular set (or sets) of localized orbitals found from the natural spin-orbitals, the bond order is independent of basis set. Although such bond orders are independent of basis set once the localization procedure has been carried through, they still have drawbacks. One is that the definitiondepends on an arbitrary procedure which localizes the molecular orbitals. A second is that for certain molecules, e.g., benzene, the localization procedure gives rise to bond orders which break the symmetry of the many electron wavefunction describing the electrons.5 The 0 1993 American Chemical Society
Sharing of Electrons in Molecules bond orders between adjacent carbon atoms are not unique, there being a second set related to the first by a rotation of 60’. (In fact, in the Hdckel approximation of the ?r-electron structure of benzene there is an infinity of sets of orbitals, each of which satisfies the criterion used for the localization procedure and each of which gives differing values of the covalent bond indices, uide infra.) In ref 5 symmetry is restored to the set of bond orders by averaging over sets of equivalent bond orders, a procedure which is artificial. Better would be a procedure which does not break symmetry. A third drawback is that the bond orders of ref 5 have no connection to any more fundamental concept than bond order itself. Other definitions of bond orders not depending on the basis set but rather on bond lengths and stretching frequencies or on properties of electron densities also have their drawbacks, as noted in ref 5 , and are not detailed here. A more detailed accounting of the behavior of electrons in a molecule than that given by bond orders is provided by the ‘Fermi” hole.9-12 The culmination of this approach is in the work of Ruedenberg13 who, with the introduction of the term “self-pair density” for the Fermi hole, made explicit a microscopic concept of sharing. Ruedenberg interprets the Fermi hole as‘ ...a picture of how an electron keeps others away from the place it occupies itself...”.I4 Includedisthenotion that anelectronissharedbetween two points in space and that this sharing is determined by the behavior of the two particle distribution function. Aside from some integral sum rules, no indication is given of how this concept is to be implemented for other than single determinant wavefunctions. An important feature of the Fermi hole not shared by individual molecular orbitals is that it is independent of the particular set of orbitals used to construct the wavefunction. In spite of this, however, there are at least three fundamental problems with the approach of Ruedenberg to the sharing of electrons in a many electron system. The first is that it is based on an analysis of the contribution of the Coulomb interaction to the total energy. In theabsenceof any further specification,e.g., that the wavefunction be an eigenfunction of the Hamiltonian, there need be no connection between the wavefunction and the Hamiltonian. The analysis of the behavior of a single particle in a many particle system should be based solely on the properties of wavefunction and not on those of the Hamiltonian. The second is that the natureof the self-pairing,or sharing, in the papers of Ruedenberg and of others is inferred from what is fundamentally a two particle property, yet it should be equally applicable to a one particle system such as H2+for which the two particle density vanishes. It should be derivable solely from single particleconcepts and not rely on any two particle concepts. The same criticism applies to any analysis of the two particle distribution function which presumes to give rise to a single particle property. The third is that the analyses are given explicitly for single determinant wavefunctions only. The appealing nature of molecular orbitals, in spite of their non-uniqueness, is that one can think in terms of the behaviors of single electrons. On the other hand Fermi holes, although independent of the particular set of orbitals used to construct the wavefunction, are fundamentally derived from the two particle distribution function and cannot in general give the behavior of a single electron in a many electron system. However the Fermi holes do have a semblance of localized orbitals in both atoms” and in molecule^.^^ This raises the question: is there a way of describing,in a quantitative way, the behavior of a singleelectron in a many-electron system in a manner which is independent of the set of molecular orbitals used to construct the wavefunction and yet which retains many of the concepts which chemists have developed of bonding? In this paper we introduce an index which measures quantitatively the amount of sharing of a single electron between any two points in space in any many electron system. (We include the spin variable of the electron as a component of the position of an electron. The position of an electron is therefore described
The Journal of Physical Chemistry, Vol. 97,No. 29, 1993 7517 by the spatial coordinates x, y, z,and the spin coordinate u.) This sharing index depends only on the first-order density matrix of the system, hence, it is independent of any set of orbitals in terms of which the wavefunction is expressed. Likewise, it is a single particle property in the many electron system. We show that this sharing index is closely related to the chemists’ concept of the valence of an atom and number of bonding electrons between two atoms in a molecule. It is also closely related to the Fermi hole discussed by manysls when the wavefunction is a single determinant. We find a sum rule relating the self-sharing index of a volume to the average number of electrons in a volume and the valence. Applications to simple systems are made in order to illustrate some of the features of sharing. The paper is organized as follows. We first consider a classical measure of the sharing of a single object between points in space. After ascertaining some properties of this sharing index, we generalize it to many particle quantal systems. The quantal sharing index, denoted the point-point sharing index, gives a quantitative measure of the amount of sharing of an electron between two points in space. (In fact the sharing index can be interpreted as the probability that an electron is shared between the two points.) It takes into account, in quite a simple and automatic way, the effects due to the localization and/or the interference of electrons. The latter, which is a result of the wave nature of the electrons, is discussed in the paper. We note that the sharing index is the absolute value squared of a sharing amplitude which itself is the square root of the first-order density matrix, the relationship between the amplitude and the index being similar to that between amplitudes and probabilities in quantum theory. We then follow Bade+ in dividing space into volumes associated with nuclei (or in some instances without nuclei), although other subdivisions of space and variables other than position may be used. (It is important to note that the fundamental sharing indices are independent of this subdivision.) Sharing indices between these volumes are defined by means of integrals over the volumes. The simplerule “the sum of all sharing indices which connect a given volume to all volumes is equal to the average number of electrons in the given volume” follows from a sum rule which holds for the sharing index involving points in space, namely, ‘the integral of the sharing index over one of its variables gives the probability of finding the electron at the position of the other variable”. In order to calibrate the sharing index against the bond number, we consider the simple molecules, H2+ and H2, showing that, for these molecules, using a molecular orbital approximation for the ground state, the bond number is twice the sharing index. We take this to be the case in general, defining the bond index (number) to be twice the sharing index. In addition to giving the connection between the sharing index and bond numbers, we define the valence of a volume in space, the volume typically being associated with an atom. To show the role antibonding orbitals and/or localizationplays in the determination of sharing, we analyze the sharing index for the molecules He*+and He2. When delocalized orbitals are used, these illustrate the fact that our definition of bond index encompasses the well-known rule of the bond number being given by half the difference between the number of bonding and the number of antibonding “electrons”. What is important here is the interferencebetween the bonding and the antibondingorbitals. These latter molecules are also considered from the point of view of the use of localizedorbitals,the transformation from delocalized to localized orbitals mattering not in the calculation of the sharing index, Idealized double and triple bonds, using molecular orbital approximations, are then considered and shown to result in bond indices of 2 and of 3. General expressions are given for the sharing index in terms of the natural spin-orbitals and occupation numbers. It is at this point that we make a first comparison between the present bond indices and those recently introduced by Cioslowski and Mixonns These authors recognized the importance of defining bond indices in terms of what the electrons actually do, giving a definition
7518 The Journal of Physical Chemistry, Vol. 97,No. 29, 19‘93 based on the topology of the electron distribution and not on an interpretationbased on a particular orbital basis set. This feature is kept by the present definition. In addition,however, the present bond indices retain the symmetry dictated by the underlying wavefunction, unlike those of Cioslowski and Mixon in the case of molecules such as benzene (vide infra). In addition, our bond indices allow for the important phenomenon of interference, whereas that possibility is excluded by the indices of Cioslowski and Mixon, and our indices do not rely on the localization of orbitals as does the procedure of ref 5 . In order to illustrate the difference between our bond index and that of Cioslowski and Muon when the wavefunction cannot be expressed as a single determinant, we determine the bond indices for the lowest excited 12,+state of the hydrogen molecule. The present definition gives (at the level of approximation used) 1 for the bond index, while the index defined by Cioslowski and Mixon is 1 / ~ . Then we consider the *-electrons of two molecules, benzene and 1,3-butadiene,in the simplest Hiickel approximation. For benzene we find that the r-bond index for each of the ortho carbons is 4/9, for meta carbons is 0, and for para carbons is l/g. The procedure of Cioslowski and Mixon is shown to lead to an infinite number of sets of orbitals, each of which satisfy their localization criterion. (The sets are parametrizedby a continuous index, and, at this level of approximation, they are the local orbitals discussed by Edmiston and Ruedenberg.1’) None of the sets, all of which result in differing values of the covalent bond indices, give values which agree with the sharing indices introduced in this paper. It is at this point in the paper that we make contact with the definition of bond order originally introduced by C o ~ l s o n .For ~ benzene, at this low level of approximation, the bond index is the absolute value squared of a generalized Coulson bond order, the Coulson bond order playing the role of a sharing amplitude. This connection is also true in the case of planar 1,3-butadiene. The r-bond index for the C1-G bond in 1,3-butadieneis found to be 0.8, that for the c1-C~bond is 0.0, that for the C1-C4 bond is 0.2, and that for the C& bond is 0.2. These indices for 1,3butadiene are all shown to differ from the values obtained by using the localization procedure of Cioslowski and Mixon. In order to demonstrate how correlation affects the values of the sharing indices and to illustrate the difference between the sharing index and the two particle distribution function, we consider a set of wavefunctions for Hz encompassing a simple molecular orbital function and a simple valence bond function, as well as functionsintermediate between these two. We find for the simple model that the addition of correlationto the molecular orbital wavefunction decreasesthe amount of sharing of a single electron between the two atoms when the protons are at their equilibrium internuclear separation. When the molecule is dissociated into two neutral atoms in their ground states, the sharing between the atoms is found to vanish. This latter example points out both the similarity of and the differencebetween thesharing index and the use of the two particle distribution function to describe the sharingof an electron between two points, as proposed by Ruedenberg and others. In the particular case that the wavefunction describing the system is given by a single determinant, the point-point sharing index is closely related to the Fermi hole which has been used previously to investigate the behavior of electrons in atoms and mo1ecules.l 1 ~ L 5 For these wavefunctions, therefore, the geometrical aspects of the sharing index are similar to those of the Fermi holes. In general this is not the case, however, because the Fermi hole is defined in terms of a two particle distribution function,9JO while the sharing index is a one particle property. An extreme case of the difference between the two measures is given by dissociated H2 in the ground singlet state. The Fermi hole, as determined by the two particle distribution function, extends over the entire molecule, while the sharing index is localized in the region of one of the nuclei. This latter is in accord with the chemists’ concept
Fulton that there is no bonding between dissociated hydrogen atoms in their ground states; the former is in disagreement with this. In section IV we integrate one of the indices of the point-point sharing index over a volume Vto define a volume topoint sharing index. We then illustrate the usefulness of this index by applying it to an extension of *-electron Hiickel benzene by including electrons in the core 1s orbitals on the carbon atoms, in the u-framework, as well as in the r-orbitals. By associating the volume Vwith one hydrogen atom, we show that the sharing of the ‘hydrogen” electrons with the other regions of the molecule can be mapped in considerable detail. We then consider the sharing by the carbon electrons. It is here we find a distinction between the contributions of localized a-orbitals and the delocalized r-orbitals. The usefulness of invoking the symmetry of the molecule is illustrated by separating the contributionsof the u- and of the r-orbitals to the sharing indices. In the fifth section of the paper, we turn to the point to point amplitudes, recognizing that the corresponding sharing indices are the absolute values squared of these amplitudes. By tracing thebehavioroftheamplitudesbetweenafixtedpointanda movable point which is constrained in several instances to lie on a line, we show that the sharing behavior of an electron in a many electron system can be visualized much as a one electron wavefunction can be used to visualize the behavior of the electron in a one electron system. Important, however, is the fact that the sharing amplitudes are independent of the choice of orbitals. The amplitudes depend only on the complete wavefunction which describes all the electrons. Although we consider only the simplest possible models in this paper, we emphasize that the definition we introduce is quite general and not limited to the models used to illustrate the index. The strength of the definition of covalent bond indices defined by Cioslowski and Muon is that it is the electron density which determines the volume of space allotted to an atom, and not the center upon which an orbital is based. Our definition shares that strength. In addition, however, the inclusion of quantal interferencegivesour index a robustness against orbital transformations that is lacking in other procedures. Likewise the satisfaction of an important sum rule gives the present index a consistency which is lacking in other definitions. Lastly, the basin-basii andvolumepoint sharingindices are all derivable from the underlying concepts of the point-point sharing amplitudes and indices.
II. Sharing Index A. General Considerations. Suppose that we have some quantity (object) which is distributed between two centers A and B. We want some measure of the sharing of the distribution of that quantity between the centers. Lef f A be the fraction of the quantity on A, and fB be the fraction on B. The sum of the fractions is 1. Then, as a function of fA, the product fAfr, takes on values between 0 and and is a measure of the evenness of the distribution. The maximum value l / 4 indicates an even distribution, the fraction on each center being 1 / ~ . The minimum value, 0, indicates that the quantity is entirely on center A or on B. Which center has the greater fraction is indicated by the values of the self-termsfA2 and f2. If we have a number of centers labeled 1,2,3, ...,with a fraction of the quantity being fA on center A, a measure of the sharing between the two centers A and Cisf&. Let us call the product of the fractions the sharing index ZAc, ‘AC E f f C
We note that the sum of ZAC over all centers C is the fraction of the distribution on center A:
Sharing of Electrons in Molecules
The Journal of Physical Chemistry, Vol. 97,No. 29, 1993 7519
The double sum of ZAcover all centers is 1:
by a wavefunction but rather by the density matrix
(3)
P(S';r)
Jdf2 di-3
*.e
df,v*(r&,...,r,v)
**(r&
,...,f,v)
(4)
The diagonal term These latter are two properties that we wish to retain in a quantal definition of sharing. Now consider the hydrogen molecule positive ion H2+. Suppose that the wavefunction is lug, which, when normalized to 1, is denoted by q(f). {stands for the three spatial coordinates r and the spin coordinate u. The probability that the electron be in the volume d{ (sdr) about f is the product P(f) d f = l(P(t)l2 d r = (P*(f)(P(f) d f
For a one particle system with a continuous index, this quantity takes the place of the fraction fA. The analogue of the sharing indexZAcfor the distribution between a volumedfcentered about f and a volume d r centered about' j is
P(cn dr simply gives the probability of finding a particle in volume df about the point f. Note that the integral of p(fif) over all space is one if the wavefunction is normalized to one,
Jdfp(S';r) = 1 Again, space may be divided up into volumes by following the procedure of Bader using the probability ~ ( f i f summed ) over the spinvariables. If we proceed as above,defining the sharing indices I, as products of the probabilities PAa n d p for ~ the volumes V, and VB, 1
PA
i(fir)is symmetric with respect to interchange of the indices {
r.
r
and We note that the integral of i(fir)over is just the probability ~ ( f )with , the consequence that the double integral of i over f and is 1. For a continuous index the integration takes the place of the discrete sum above. To get the sharing of the electron between the protons, we need to ascribe volumes as belonging to each nuclei. In this case it is easy. Simply pass a plane, perpendicular to a line drawn between the nuclei, through the midpoint of that line. Call the left nucleus 1 and the right 2. The volumes belonging to the nuclei are denoted by VI and V2, respectively. When defined more generally in terms of the locations of zero-flux surfaces, Bader16 calls these volumes basins. The sharing index between these volumes is
r
J,drP(cf)
we immediately run into a problem. Consider the molecule He2, for example. The sharing of electrons between the atoms is extremely small when the overlap between the wavefunctions of the two helium atoms is small, so we want the index to be small. Using the fact that, aside from a multiplicative factor, p(fif) is the electron density and recognizing that the electron density of the pair of atoms is the sum of the electron densities of the two individually, the probabilities p A and p~ are found to be each equal to l / 2 and the product of the two is a maximum, not a minimum! What is missing is a means of allowingfor a description of the localization of the electrons on the helium atoms. (This is also related to the interferencephenomenon allowed in quantum theory.) A possible way out of this dilemma is to note that the density matrix for a one electron molecule is P ( S ' ; r ) = (P(!3cc*(r) and that the sharing index for the electron in a one electron system may equally well be written as the product of density matrices
where the volume to which an integral is restricted is shown by the subscript and where we allow for self-sharing when the subscripts are the same. Implicit in the use of the integral sign is a summation over the spin index. From the symmetry of the wavefunction, the integral of Iq(f)12over a single basin is l / ~ The . inter-atomic sharing index 1 1 2 is therefore l/4, as are the selfindices, and 122. The double sum of Z, over the volumes is 1.
If the nuclei do not have the same charge (but still only one electron), a decision must be made as to which nuclei a given volume element df belongs. A natural choice is supplied by Bader'sl6 atomic basins, namely, divide space up into basins depending upon the nucleus (or other point) to which a path which follows the gradient of the electron density (given here by &,,lq({)l2) leads. This is akin to dividing up a mountain range into mountains according to the peak climbed by following paths of steepest ascents. Basins need not be associated with nuclei, and we are not wedded to their use in all cases. Once these volumes are defined, the sharing indices are given in terms of the fractions as above. The probability of finding the electron in the volume associated with a basin A is
P(cr)P(r;r) rather than as a product of probabilities. In a many electron system we make theprovisional identificationof the sharing index iW):
i(cr)= P(S';r)P(r;f) Except in particular cases, however, there is a problem with this definition of sharing. We want to retain the property that the integral of the sharing index over be equal to the probability p({). To see what the provisional index gives when integrated, we expand the density matrix in terms of its eigenfunctions and eigenvalues. Let the orthonormal set of eigenfunctions of the Hermitian matrixp(fir) bedenoted by qn(l)(these are the natural orbitalsls used by many) and the eigenvalues by pn,
r
J d r P(eY)qn(r) = qn(f)Pn
(5)
The normalization of the density matrix gives
C P n= 1 n
The expansion of the density matrix in terms of its eigenfunctions and eigenvalues is and the sharing index between the two basins A and B is the product p ~ pthis ~ , also being the value of I , as given by the integral over This procedure works quite nicely for one electron in the presence of any number of nuclei. When we turn to many electron systems, however, the behavior of a singleelectron is not described
i(fir).
r) =
*
P (s';
q n ( t)Pnvn ( n
giving, for the provisional sharing index,
r)
Fulton
7520 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 The result of integrating this over
r is
{f = (p',~). The sharing amplitude in terms of the momenta is
m
which, in general, is not proportional to the probability p( {), pm2 appearing in place of pm. (It is proportional to p ( f i { ) if all of the nonzero pm are the same. This is the case for wavefunctions which are single determinants.) What we want for the sharing index is a quantity which, when integrated over one of the indices, gives pm in place of pmZ. Such a quantity can be constructed from the matrix which is the square root of the density matrix. This matrix may be defined by19 m
the positive square roots of the (positive) pmbeing chosen, ensuring that thenew matrixis positivesemidefiniteas is thedensitymatrix itself. [This specification means that the interference is due to the behavior of the wavefunction and not to the signs of the square roots of pm. It also means that p I / z ( f i r ) has the same invariance properties as p ( f i r ) . ] It follows that
The sharing index fir) is defined to be
&(fir) = P1/2(cr) P1/2(r;t)
r.
Thus this sharing index has these two properties in common with the sharing index for a single particle considered a t the outset. Further, for a one electron system the index does reduce to the product of wavefunctions given above. That this index correctly allows for the localization of electrons on the atoms in He2 is depends only on the first-order shown below. The index density matrix which, in turn, can be determined from the many electron wavefunction. It is therefore independent of the basis set@) in terms of which the wavefunction may be written. In addition the sharing index, being derived from the first-order density matrix, is a single particle property. The physical interpretation of d r d r is that it is the probability that a single electron be found in the volumes d{ and d r about the points { and We note that the sharing index fir) is the absolute value squared of the possibly complex quantity ~ l / ~ ( fIfi rwe ). want to use a set of variables different from position for the description of the electron, e.g., the momentum, we simply apply the usual rules for transforming quantal amplitudes from one set of variables to the other to the square root matrix, once for each set of variables, rand Thus, in its transformation properties, the square root of the density matrix p1l2(fir) behaves like an amplitude for going from the point to the point {in quantum theory. In fact the sharing amplitude has the same relation to the sharing index that ordinary amplitudes have to probabilities in that it is the absolute value squared of the first that determines the second quantity. We will refer to the square root matrix as the sharing amplitude. As one example of carrying out the transformation of the sharing amplitudes, we consider the transformation to momentum variables. Let fP = (p,.), wherepis themomentum, and, similarly,
&(fir)
&(fir)
It is also possible to transform just one set of variables to momentum, leaving the other in terms of the position, with a correspondinginterpretation in terms of sharing between a position and a momentum. In order to get the auerage number of electrons per unit volume in the neighborhood of some point {, we multiply the diagonal element of the density matrix by the total number, N, of electrons in the molecule. This quantity we denote by N ( { ) :
N ( l ) NPW)
(10) There are N electrons which can be shared so that, in similar fashion, we define a sharing index I(fij") (capital Greek iota) which is normalized to N:
I(fir)
(7)
This index (Greek iota) is real, positive semidefinite (it is the absolute value squared of a complex number), and symmetric with respect to the interchangeof the indices {and The integral of the sharing index over one of its indices, say is the probability p ( f i { ) at the other index, and the integral over both of its indices is 1:
r,
Just as there is sharing between points in space, there is sharing between different momenta. The corresponding sharing index, representing the sharing of the electron between the momenta p and p', is
NGr) (11) These two definitions are consistent in the sense that the integral of I(fir)over one of its indices gives the number density N ( { ) of the electrons at the other index. We also introduce a related amplitude by multiplying the (small) sharing amplitude by the square root of the number of electrons in the molecule,20
(fir)
( e n = N'/zP'/2(cr) (12) for then the sharing index I(fir) is the product of the (large) sharing amplitudes (fir)and (r;{), I(cr)= (cr)(r;C) (13) It is important to note that thus far we have not invoked a definition of what constitutes an atom. The definitions of the point-point sharing amplitude and sharing index are completely independent of the Hamiltonian, depending only on the wavefunction of the system being described. We now consider a division of space into subvolumes each of which is associated with an atom. The average number of electrons in basin A is NA,given by (14)
while the sharing index between basins A and B (the indices may be the same) is
r.
r.
r
the basins being conveniently chosen according to the Bader criterion using either p ( f i { ) or N ( { ) . Two results of the normalization to the total number of electrons in the molecule are that (1) the sum of IM over the basins labeled by B is equal to the average number of electrons in basin A ,
and (2) the double sum of I A is~ the number of electrons, N . It should be noted that these sum rules depend only on the fundamental sum rules obeyed by i(S;r) and not on any specific properties assumed by Baderl6 in the definition of the atomic basins. Before continuing to some additional properties of the sharing index, we consider two applications in order to establish a tentative connection with bond number^.^ For the hydrogen molecule ion,
The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 7521
Sharing of Electrons in Molecules the sharing index 112 between basins 1 and 2 is, as given above, l / 4 , as are the self-indices. The bond number is 1 / 2 , or twice the sharing index. (In fact it is the sum 112 + I z l . ) Now consider the hydrogen molecule H2 in the ground '2: state. We use a single Slater determinant to describe the electronic structure of the molecule,
with a and p the usual spin functions, normalized to 1 as is the spatial wavefunction v(r). The density matrix is
determination of bond number. That theuse of localized orbitals gives the same indices as the delocalized orbitals give is an illustration of orbital independence of the present bond indices. The wavefunctionfor Hq+is taken to be the single determinant, abbreviated by its first term,
with the overbar indicating spin down and the lack thereof indicating spin up. The density matrix arising from this wavefunction is
P(cr)
P e r ) = (1/2)v(r)v*(r')L+
+
(recognizing that a(.) a * ( d ) p(u) o*(d)= &,d) from which the square root matrix is found to be
by using the prescription given above. The sharing index I(S,r) is 24S,r),or
I(c 7) = vWv*(r') v(r')v* (4,,a
= (1/Jvg(r)vg*(r')~u,4 + vu(r)vu*(r').(a).*(d)J with pg,"(r) being the spatial parts of the lug,uorbitals. The square root matrix has 3 1 / 2in place of the 3 in the denominator. Let the integrals involving the two spatial orbitals q l ( r ) and R(r) over basin A be denoted by
From the sum rule that
- (vuO,,qu)B
- N A = N A - V'
(19)
These are general relations which hold independently of the form of the wavefunction. The sizes and shapes of the basins depend, of course, on the wavefunction, as do the values of the bond numbers. One way to think of the bond self-sharing index BM is as follows. The average number of electrons in basin A which do not participate in a bond is equal to the average number of electrons in that basin minus the valence of the basin, (NA- VA). If there were only that average number of electrons in the basin, the bond self-sharing index would be twice that value, 2 ( N A - VA).To get the total bond self-sharing index, we need to add to this number the average number of electrons from basin A participating in the bonding, VA. This gives the relation given above between the self-sharing index BM, the average number of electrons in the basin N A , and the valence VAof the basin. B. Role of Antibonding Orbitals and Localization. We now consider the ground electronic states of the two species Hez+ and He2 from delocalized and from localized molecular orbital points of view. The role antibonding orbitals play in the determination of the bond number when delocalized orbitals are used is clearly brought out, this also illustrating the role of interference in the
(vg,qu)l
-(vg,qu)Z
-
'*'
hold. As a result, the sharing index between basins is [noting that (vg,cpe)A = l/21 1
= ( q g , v g ) A ( v g , ( p , ) B = /4 a number of terms in the sum canceling. The self-indices are 5 / 4 . The bond indices are I12
Bl2
we get the connection between the self-sharing index for a basin and the valence of that basin, BAA
+ ((p8,qu)A('Pu,'pB)B + ((Pu,vg)A((Pg,'Pu)B + ( ~ u , ( P u ) A ( ( P , , ( P , ) B )
The symmetry of the orbitals is such that, provided the nuclei are not too close together, relations such as (we choose q8to be positive and fi to be positive in basin 1) (vg8'qg)A
(17) = IAB + IBA The valence of an atom may be defined as the number of bonds the atom participates in within the molecule.2c When using the basins as defined by Bader, it is convenient to think in terms of the valence of a basin. Let us denote the valence of basin A by VA. We define the valence of basin A by the sum of the bond numbers relating basin A to all basins other than A, BAB
(20)
The sharing index is then IAB = {2(vg9(Og)A((PB,(PB)B
Aside from the spin factor, this is like the index for the hydrogen molecule ion. By using the symmetry of the wavefunction the sharing index 112 between the atomic basins is found to be l / 2 , as are the self-indices. Again, the bond number for hydrogen is twice this value, or 1. In general we define the bond number BAB between two basins A and B to be twice the sharing index,
= JAdr v1*(rMr)
('P1,'PZ)"
= '12
The origin of the well-known rule that the bond number is half the differencebetween the number of bonding (2) and antibonding (1) electrons is clear in this approach. The change in sign of the antibonding orbitals (u) relative to the bonding orbitals (8) upon going from basin 1 to basin 2 gives rise to contributions to the sharing indices which tend to cancel. The characteristic interference of wavefunctions is of paramount importance in the determination of the sharing index. It is also possible to rotate the orbitals 1ugand 1uu associated with the a(.) spin function in order to produce orbitals 91.2 which are localized within the basins 1 and 2: simply use
1522 The Journal of Physical Chemistry, Vol. 97, No. 29, 199'3 becauseoneor theother is small. As a result weget theexpression
Fulton Np,. By using this relation and multiplying by 2, we get
1, = (Vg,Pg)A(Pg,(Pg)B + ('Pl,P1)A('PI,'PI)B + ((Pz,(Dz)A((P~,vJB
which gives the samevalues for the sharing indices as those found above, although now the role of the localized orbitals is emphasized. (Note that for sharing between the basins, the localized orbitals make little contribution, while for the seuterms one or the other makes the major contribution.) We note that the sharing index is completely insensitive to the transformation from delocalized orbitals to localized orbitals. Similar considerations hold for the"molecule" - Hez; the configuration for the ground state is 1uel ugluulu,. In terms of delocalized orbitals, the sharing index is IAB
= 2{(pg,@A((p,,(p,)B + ((Pg,(P,)A((P,,Pg)B + ((Pu,(Pg)A((ps,(p,)B
+ ((P,,(C,)A((P,,(P,)B~
By using the relations given in connection with He2+, we find that the sharing index between basins is small, while the self-sharing indices are IA,, = 2, the number of electrons on each atom separately. Again the signs of the antibonding orbitals relative to those of the bonding orbitals are important, the role played by interference in the calculation of the sharing indices again being apparent. As in the case of the helium molecule positive ion, localized orbitals may be used. The result is the same. This illustrates the important feature that the present index is independent of the choice of orbitals, provided that the many electron wavefunction is unchanged by the choice. C. Double and TripleBonds. As a prototype of a double bond, we consider the four electron bond between the carbon atoms in ethylene, ignoring the carbon-hydrogen bonds. The configuration is ugZ8rU?i;. The density matrix is P(r;Y) = ('/4)([9,(r)co,*(r')
+ cp,(r)cp,*(r')lbl
The square root matrix has a 1/2 in place of the l/4. The average number of electrons in each basin is 2. Using the property that the u- and r-orbitals are orthogonalto each other when integrated over any plane which is perpendicular to the internuclear axis, the sharing indices work out to be (spatial parts of the orbitals only in the integrals)
I,
= 2(((P,,(P~)A((P.g,(P,)B+ ((P,,(P,)A((P,,P~)Bl
The sharing index between basins is 112 = 1 as are the self-indices, 111 and IZZ.The bond index B12is 2. Triple bonds can be worked out in similar fashion. All that needs be done is to add the r-orbital which is orthogonal to the *-orbital in the above expression for the density matrix. The result for the interbasin sharing index is 112 = 3/2. The corresponding bond index is 3. Bond numbers given by twice the sharing index agree with the chemists' intuition in all of these cases. D. Useof NahulllSpinOrbitals. In termsof theeigenfunctions and eigenvalues of the density matrix, the sharing index I(S;r) is given by
for the bond indices defined in this paper. The bond indices of Cioslowski and Mixon5 (denoted by in the following) are defined as follows: form the provisional index by retaining only the diagonal terms in the basin integrals (use only the terms with m = n) and by replacing um by its square, vm2,
e
BA; = 2 C u , 2 ( ( P m , c p , ) * " ( ( p , , ~ m ) s B m
Then form the sum over the provisional interbasin indices
Minimize Nd by means of isopycnic transformations and use the resulting localized orbitals and transformed occupation numbers to define the covalent bond indices by the same expression as used for the provisional index. (In ref 5 Cioslowski and Mixon denote their covalent bond indices by PM.) There are three things to note about this pr,ocedure. The first is that the overall normalization of the indices (e.g.,the sums of the bond indices) may differ from 2N if the wavefunction is not a single Slater determinant or if it lacks some appropriate symmetry. This does nor affect the minimization procedure, but it does destroy a sum rule. The second is that in the provisional index, leading to the sum Nd given above, only the diagonalbasinbasin terms are retained. The result is that this sum is dependent on the set of functions used. This leads to the third item. The sum is minimized to produce a set (or sets) of localized orbitals which are rhen used to definethe bond indices. It seems as though this localization procedure in practice tends to minimize (but in general not to make zero, vide infra) the differencesbetween the two definitions of bond indices, those defined by us and those used by Cioslowski and Mixon. The procedure of Cioslowski and Mixon also has the consequence that the bond indices break the symmetry inherent in the wavefunctions of molecules such as benzene. The symmetry of the wavefunction is then forced on the bond indices by averaging the indices over a set of equivalent structures. We emphasize that our bond indices (or sharing indices) retain the full symmetry implied by the many electron wavefunction. It should also be noted that in the definition of the bond indices by Cioslowski and Mixon, each of the terms in the sum is intrinsically positive; hence interference between different parts of the orbitals is excluded. In the definition proposed here, cross terms which need not be positive (nor real) are allowed. These are the very terms which allow for the interference phenomenon which is so characteristic of quantum theory and which makes the present definition invariant to the choice of orbitals used in the calculations. If the wavefunction for N electrons can be written as a single determinant,thevalues that the pntakeon are 1/Nfor theoccupied orbitals and 0 for the unoccupied. The sharing indices are then simply given by
I(Cr) = N C r p , ( n p m ' " ~ m * ( Y ) ~ n n ( r ) P n112(Pa* (53 (21) m,n
The basin sharing indices are (using the notation for the integrals over the individual basins introduced above and indicating that the spin variable is included in the integration by the superscript s)
where the sums are over the occupied spin orbitals only. The further specification that the spin orbitals be paired such that each spatial orbital is associated with a spin up and with a spin down orbital leads to the result that the sharing indices are
=2 It is at this point that we can compare these indices with the bond indices introduced by Cioslowski and Mixon.5 Recall that the bond indices are twice the basin sharing indices. The occupation numbers are denoted by Y,, in ref 5. These are simply
((Pm,(Pn)A((Pn,(Pm)B m>n SPaWY occupl~
with the sum over the occupied spatial orbitals only. As before, the definition of Cioslowski and Mixon uses only the terms with
Sharing of Electrons in Molecules
m and n equal, the orbitals being chosen so as to minimize the sum of the provisional interbasin indices. III. Examples A. Hz,Excited ' E: State. We consider the lowest excited Hz. Because the wavefunction cannot be expressed as a single determinant, this gives a simple comparison between our definition of bond indices and that of Cioslowski and Mixon when the occupation numbers of the spin orbitals differ from one and it illustrates a case in which a simple extension of the selfpairing term considered by Ruedenberg13 for HartreeFock wavefunctionsis invalid. The configuration of this state is 1a,2ug with a singlet spin function. Let ql(r) and cpz(r) be the spatial parts of the lag and 2ag functions. We choose these to be normalized to 1 and to be orthogonal to each other. The wave function is 1 Z i state of
+ 472(rl)cpl*(r2)1 x [ . ( 0 3 * ( 4 - B(+*(a2)1
Wi-pi-2) = ('/2)[471(r1)472*(r2)
which we note cannot be expressed as a single determinant. The occupation numbers of the spin orbitals are l/2. Now the symmetry of theorbitals is such that for integrals over each basin, (pm,pn)A
e '/2&mn
As a result, the sharing indices are all equal, IM = l/z. The corresponding bond indices are all 1. When we work out the bond indices of Cioslowski and Mixon? we get, as a result of the square of the occupation number appearing in the definition of the general bond order, the value of for each of the bond indices.21 In this case, there is a significant difference between our sharing indices and those of Cioslowski and Mixon. The bond in the excited state of the hydrogen molecule differs, admittedly, from that in the ground statein that it is weaker. The index we have introduced, however, gives the number of electrons which aresharedbetween the atomic basins, and this number is the same in this excited state as it is in the ground state (the number of electrons shared is twice the bond index). The sharing index by itself gives no indication of the energy associated with the bonding. There is no reason that thebond index of Cioslowski and Mixon should reflect the energy either, The main difference between the wavefunction for this state and the earlier ground-state wavefunction is that this wavefunction, unlike the others, cannot be written as a single determinant. Both wavefunctions aregerade (in fact the orbitals are all gerade), and the number of electrons shared between the atoms is the same for both states, namely, two. B. Benzene. In this section we consider the simplest treatment of the 7-electrons in benzene, the Huckel treatment, ignoring any overlap of the atomic functions. The delocalized orbitals (ck are expressed in terms of the p-orbitals, 4a,localized on the carbon nuclei labeled by the index a which takes on the values 0-5: 5
qk
=c4acak a=O
The Journal of Physical Chemistry, Vol. 97,No. 29, 1993 7523 are determined. (To a large extent, the basins in benzene are determined by the symmetry of the molecule.) In line with the assumption of no overlap of the atomic p-orbitals, we shall ignore this complication and assign a p-orbital on a carbon atom entirely to the basin assigned to that carbon atom. As a result, the sharing indices, written in terms of the coefficientsof the atomic orbitals, are
The necessary sums are readily worked out. The 7-electron sharing indices are found to be: Io0 = 112, bl = 2/9, bz = 0, and b3 = l/18, with the bond indices being twice the value of the corresponding sharing index: Bo1 = 4/9, BOZ= 0, and Bo3 = l/9. Thegreatest amount of sharing is between adjacent basins. There is no sharing between basins meta to each other, and the para sharing index is a quarter of the ortho sharing index. It is amusing to compare these estimates with the "orders" of the linkages given by P a ~ l i n glong ~ ~ ago. On the basis of the superposition of canonical structures, Pauling gives an order of 1.46 for the ortho linkage and 0.0734 for the para linkage. To compare to the present results, we add to our ortho index 1.OOfor the contribution of the a-bond to get 1.444. Our para index is 0.1 11. In order to compare the results of our procedure with those of Cioslowski and Mixon, it is necessary to determine the set@) of localized orbitals. The localized orbitals are found as follows. A set of real orbitals, h,is defined in terms of the occupied orbitals by
The coefficients b k h satisfy the conditions h l A
= blA*
bo, = bo,* resulting from the condition that the transformed orbitals be real, as well as the conditions
arising from the orthonormality. By also using the connection between the occupied orbitals Cpk and the atomic orbitals @a, the sum of the self-indices
is written in terms of the coefficients b k , which are then chosen to maximize the sum. (This is equivalent to the minimization of the sum of the interbasin indices.) The result is that the localized orbitals are given by the sets of orbitals discussed by Edmiston and Ruedenberg:"
with
We choose k to take on the values 0, k l , k2,and 3. The basin sharing indices are
the integrals involving only the spatial parts of the orbitals. The atomic basins are determined by the total electron density; hence, the carbon-hydrogen bonds should be included when the basins
The parameter x is undetermined by the procedure, the sums of the indices used to determine the localized orbitals being independent of x. The set of localized orbitals has a period of
Fulton
7524 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 TABLE I: r-Electron Covalent Bond Indices for Benzene Bkh BOISB23r 8 4 5 B12, B34,Bso 802, Bu, BUI 4 3 , B35, BO39 825
113 0.4444 0.4444 0.2963 0.0000 0.1111
-14 0.6259 0.2630 0.2529 0.0434 0.1111
-16 0.7010 0.1878 0.1481 0.1481 0.1111
1/12 0.6259 0.2630 0.0434 0.2529 0.1111
0 0.4444 0.4444 O.oo00 0.2963 0.1111
r/12 0.2630 0.6254 0.0434 0.2529 0.1111
*I6 0.1878 0.7010 0.1481 0.1481 0.1111
4 4 0.2630 0.6259 0.2529 0.0434 0.1111
4 3 0.4444 0.4444 0.2963 O.oo00
0.1111
Twice the r-Electron Sharing Indices for Benzene 0.4444 O.oo00
0.1111
2 ~ 1 3 .There is an infinity of orbitals satisfying the criterion of Cioslowski and Mixon at the Hiickel level. As illustrated below, each set gives rise to a different set of covalent bond indices. (Cioslowski and MixonZ2indicate that the wavefunctionsresulting fromHF/6-31G1 and HF/6-31++GS* calculations for benzene give but two identical structures of the covalent bond indices. I suspectthat if the calculated wavefunctionwere truly of symmetry Dbh, an infinity of structures would be found, just as in this approximation.) To get the covalent bond indices, we need the elements of the atomic overlap matrices. These work out to be given by
[
2 cos 2( x+A?:A)]} The covalent bond indices are simply found from these elements. The bond indices generally depend on the parameter x , having the same period of 2a/3 as has the set of wavefunctions. The symmetry exhibited by the covalent bond indices is D3h rather than the Dah of the complete *-electron wavefunction. The symmetry breaking exhibited by thecovalent bond indices together with their non-uniquenessstands in sharp contrast to the retention, by the sharing indices, of the symmetry of the underlying wavefunction. This alone is a strong argument in favor of the use of the sharing index as a measure of covalent bonding rather than the proposal of Cioslowski and Mixon. Table I contains values of the covalent bond indices at selected values of x in the interval l r / 3 to +a/3. Although some subsets of the covalent bond indices are equal to twice the values of the corresponding sharing indices, for no angle is the entire set of covalent bond indices equal to the entire set of twice the values of the sharing indices. We call attention to the following. The covalent bond indices for para carbons are equal to twice the sharing indices for all angles, so that these are in agreement. For angles of x = b r / 3 and x = 0, the covalent bond indices for ortho carbons are equal to twice the sharing indices, while for other angles they are not. The subset of covalent bond indices for meta carbons is never equal to that of twice the sharing indices, the covalentbond indicesaveraging to 0.148 1while the sharing indices all vanish. In fact, the average of the covalent bond indices for the meta carbons is larger than the covalent bond indices for the para carbons, in stark contrast to the values of twice the sharing indices. It is appropriate to note at this point a connection between our sharing indices and the bond orders introduced by C o ~ l s o nFor .~ benzene at this level of approximation (and also for 1,3-butadiene considered below) the sharing indices may be written as the absolute value squared of a sum 'AB
=
21
TABLE Ik r-Electron Sharing Iadicea and r-Electron Covalent Bond Indica for 1.3-Butadiene Cioslowski-Mixon present Cioslowski-Mixon
0.05
= 1.00 = 0.80 B13 0.00 El4 = 0.20
0.45 0.05
822 823
0.45 0.45 0.05
The partial bond orders for nearest neighbors defined by Coulson are the real parts of the possibly complex productsps = CAk*CBk. The products here are a generalization of the Coulson partial
1.00 0.20
0.90 0.90 0.10 0.10 0.90 0.10
bond orders to non-nearest neighbors. The total bond order is just the sum over the occupied orbitals of the partial bond orders, so in a sense our sharing indices, in this approximation, are the squares of the Coulson bond orders. The generalized Coulson bond orders are playing the role of a sharing amplitude. This is also true for the molecule 1,3-butadiene considered next. C. 1,fButadiene. We consider the r-electrons in the ground state of planar 1,3-butadiene using the Hiickel approximation, this molecule affording a simple comparison of the present treatment of bond indices and that of Cioslowski and Mixon.5 The carbon atoms are labeled by the index a, with a taking on the values 1-4 sequentially from one end of the carbon chain to the other. Only the a-electrons are considered. The mth molecular orbital (pm is written in terms of the atomic orbitals tpa in a fashion similar to that for benzene,
The coefficients Cam are given by
m taking on the values 1-4. The r-electron sharing indices, with the assignment of an orbital on a carbon atom entirely to that carbon atom, are given by 'AB
= 2 F=~I c A -1 k * c A f B l * c B k
The a-electron sharing indices (exact for the model) are given in the left hand column of Table I1 with the a-bond indices in the third column. Note that the 2-3 and 1-4 basins have the same sharing index while the 1-3 and 2-4 sharing indices vanish in this approximation. To get the total sharing or the total bond indices, the cr-electron contributions must be added to the 1-2, the 2-3, and the 3-4 indices. As in the case of benzene, the sharing indices can be written as twice the absolute value squared of generalized Coulson bond orders: 'AB
k=-1 i'Ak*'Bi
El1 E12
=
21
2
2
&cAk*cBkl
It is quite easy to work out the corresponding CioslowskiMixon indices. When the localization procedure is carried out, the localized wavefunctions are found to be just (1/21/2)((plf e).Thevaluesoftheindices (again exact for themodel) obtained from this localization procedure are given in the second and fourth
Sharing of Electrons in Molecules
The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 7525
columns of Table 11. Although their values are fairly close to those from the present procedure, none are identical. Note, in are equal particular, that the covalent bond indices gy and as are and g, quite unlike the present values for which only B14 and B23 are equal. We note also that our basin sharing index 8 1 3 vanishes, i.e., there is no *-bond sharing in this primitive approximation between a terminal carbon and the carbon twice removed, while the Cioslowski-Mixon gives a bond index of 0.10. D. Effects of Correlation in the Ground State of Ht. Aside from the excited state of H2, the wavefunctions used above have all been single determinant functions. Using a simple example we illustrate here some of the effects of including correlation on the basin-basin sharing indices. We also show that the pointpoint sharing index gives a very different picture of the sharing of an electron than does the two particle distribution function when the wavefunction is not a single determinant wavefunction. One of the difficulties in using the basins defined by Bade+ is that the surfaces of the basins, being determined by the electron density, are in general curved. The integrals giving the atomic overlap matrices are therefore not straightforward, with analytic results being nigh impossible to obtain. To avoid this problem and to keep the argument simple, we consider a simplified model of the bonding in the hydrogen molecule, for which the basins and the molecular orbitals are determined solely from the symmetryof the molecule. The molecular orbitals are constructed from real, normalized 1s orbitals with effective charges of p! on the two centers, 41(r)and ~&(r). The bonding and antibonding orbitals are the combinations
e, e
S being the value of the overlap integral. The spatial part of the
single determinant singlet wavefunction is pb(rl) ‘Pb(r2) As indicated above, this leads to an interbasin sharing index of 112. A simpletreatment ofcorrelation is toinclude theantibonding orbitals pa@)in the wavefunction, so that the spatial part of the wavefunction is cos(X) vb(r1) ‘Pb(r2) - sin(X) pa(rl) pa(r2) The spatial part of the density matrix p(r;r‘) is simply P h ’ ) = cos2(X) (cb(r) (ob(r’)
+ sin2(X) va(r)
pa(r‘)
giving the occupation numbers V b = cos2(X) and va = sin2(X). By using the symmetry of the wavefunction and of the basins, the interbasin sharing index is found to be
All that needs to be evaluated is (Cpb,cpO)j’fI. When expressed in terms of the atomic orbitals, this element is
There are two integrals we need, S = (41,42) and ( 4 1 , 4 1 ) ~The ~. first is given by23
s = e”R[l + (YR+ ( 1 / s ) ( a ~ ) 2 ] while the second is
(41,41)HI = 1 - 1/2e-aR[1 + ( 1 / 2 ) a ~ I The effective nuclear charge a has a slight dependence on the value of A. a is 1.197 for the molecular orbital wavefunction and is 1.166 for the Heitler-London-Wang w a v e f ~ n c t i o n(valence ~~ bond with variable effective nuclear charge). For the simple configuration interaction wavefunction, Weinbaum25 gives the value of 1.193 for a. For simplicity, we choose a = 1.18 for all
TABLE III: Effect of Correlation on the Sharing Indices of
HZ wavefunction molecular orbital valence bond Weinbaum
x
I11
Iiz
Bii
4 2
O.OO0
0.500
0.500
0.187 0.111
0.645 0.587
0.355 0.413
1.000 1.290 1.176
1.OOO 0.710 0.826
values of A. Use of this together with an internuclear distance of 1.403 au gives S = 0.682 and ( 4 1 , 4 1 ) ~=~0.825. The values of X are found from the various ratios of the coefficient of the valence bond wavefunction to that of the ionic contributions. Table I11 gives the values of X and of the sharing and bond indices for the three wavefunctions, the simple molecular orbital, thevalence bond, and the Weinbaum functions at the internuclear distance of 1.403 au. There is a 17% decrease in the sharing index upon going from the molecular orbital wavefunction to that of Weinbaum and a further decrease of 12% in going to the pure valence bond wavefunction. The change is indicative of some of the changes which occur when correlation is included in a wavefunction, but it is not a general proof that inclusion of correlation always decreases the interbasin sharing indices. Indeed, our experience is that there are molecules for which some interbasin sharing indices increase rather than decrease when correlation is included. As the protons aremoved apart, thevalence bond wavefunction approaches theground-state wavefunction of the isolated hydrogen atoms. The value of the effective nuclear charge a 1 while the overlap matrix S 0. The basin overlap matrix element (p6,$0a)j’f1 l / ~when r OJ. As a result, X s/4 and the interbasin sharing vanishes, in accord with no bond formation. A strict molecular orbital wavefunction retains its interbasin sharing index of 1/2 in this dissociated limit. These results are in accord with the well-known dissociation limits of the valence bond and molecular orbital wavefunctions. This dissociated singlet state of H2 also illustrates an important difference between the sharing index and the use of the two particle distribution function in an attempt to describe the behavior of a single electron in a many electron system. The two particle distribution function R(S;r) (normalized to 2) is written a d 2
--
-
-
-
{,r).
thereby defining thecorrelation function C( [Thegeneralized exchange term II,(S;r)used by Ruedenberg13 is the negative of the correlation function.] If we choose the spin parts of the coordinates { and to have the same value, the two particle distribution function vanishes, R(S;r)= 0, because of the Pauli principle. This gives the correlation function as the negative of the product of the number densities a t the two points. Choose t to be in the vicinity of one of the protons, say atom 1, and to be in thevicinityof theother, say atom 2. Clearly thecorrelation function does not vanish. It extends over both atoms even when the atoms are not bonded. The sharing index, however, is localized in the vicinity of atom 1, vanishing when is in the vicinity of atom 2. This corresponds to the chemists’ notion that there is nobond between thetwoatoms. It isquiteclearfromthisexample that the sharing index can be obtained neither from C ( { , r )nor from the generalized exchange term. One additional point is to be noted. For all the wavefunctions considered here, the valence bond wavefunction, the Weinbaum wavefunction, and the molecular orbital wavefunction, the average number of electrons in each basin is 1, being determined solely by symmetry. It is not sensitive to other details inherent in the wavefunction. The sharing index, which in the present casevaries as the value of X is changed, is sensitive to other details. The two quantities together therefore give more information about the wavefunction than either alone.
r
r
r
Fulton
1526 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 IV. Volume to Point Sharing Indices Above we integrated one index of the sharing index I(kj+) over the volume associated with atom A and the other index over the volume associated with atom B, giving the basin sharing index IAB which is a quantitative assessment of the sharing of electrons between the two atoms. An index more sensitive to the finer details of sharing is given by integrating only one of the indices, say in I(fir)over a volume V, producing the volume-point sharing index Id{),
r,
By choosing the volume V to coincide with that of an atom in a molecule, say atom A , we can map out the details of the electron sharing associated with that atom. The integral of I"({) over the volume of atom B gives the atom-atom sharing index ZAB. The valence of an atom in a molecule may be defined as the number of bonds associated with that atom.2c As a function of C, the bond indices BAc= 21Acdescribe how the valence of atom A is divided amongst the other atoms in the molecule. The volume-point sharing index generalizes this concept of the partitioning of the valence to a microscopic level. It gives quite a detailed, and quantitative, description of the valence structure of an atom in a molecule; that is, how the electrons associated with a given atom are shared throughout the molecule. We also note another connection. When applied to an isolated atom and the volume Vis chosen to include all space, the volume-point sharing index coincides with the electron density of the atom. In a sense, then, the volume-point sharing index generalizes the concept of the electron density of an isolated atom to that of an atom in a molecule. In terms of the overlap integrals ( c p m , c p n ) ~ and the occupation numbers vn, the volume-point sharing indices are
carbon-hydrogen bonding orbital between carbon a and hydrogen a is the linear combination
with the coefficients CH and cc being such that the orbital is normalized to 1. The bonding orbital between carbon atoms a and a + 1 is with the coefficient c chosen so that the orbital is normalized to 1. The *-orbitals are formed from the p orbitals which are perpendicular to the plane of the ring. The canonical orbitals are combinations of the form
with X any of the labels used above and with $ being 4 for the carbon core 1s orbitals and 2p orbitals perpendicular to the plane of the ring and cp for the others. k takes on the values 0, f 1, f 2 , and 3. Overlaps between different localized bonding orbitals have been, and will be, ignored. The many electron wavefunction is a single determinant of the filled orbitals. There are 12 electrons in the core orbitals, 12 in the carbon4arbon bonding orbitals, 12 in the carbon-hydrogen bonding orbitals, and 6 in the *-orbitals, those having k = 0 and f l . The sharing amplitude (CY) is naturally constructed in terms of the canonical orbitals. For our illustrative purposes, however, it is convenient to express the amplitude using localized orbitals for all but the *-electron orbitals by using the transformation which is inverse to the one above. We get
(fir)=
5
[4ls;co(r)$*1s;ca(f)
+ pcH;(r(r) q*CH;o(f) +
a=O
m,n
When the wavefunction can be written as a single determinant, the occupation numbers are 1 for occupied spin-orbitals and 0 for unoccupied. We note that these volume-point sharing indices are positive semidefinite and that they are not the squares of a wavefunction. As indicated below, when the volume V is associated with the volume of an atom, the sharing index, for some types of bonds, is localized in the neighborhood of the atom. For other types of bonds the sharing index is not so localized. This index is therefore useful as an indicator of the extent of delocalization of the bonds emanating from a given basin. In order to demonstrate the details inherent in this sharing index, we extend the crude *-electron model of the structure of benzene to include the a-electrons. Looking a t the ring from above, the carbon atoms are labeled from 0 to 5 in a counterclockwise fashion. The hydrogen atoms are given the same label as the adjacent carbon atom. Let the 1s orbitals of carbon atom a be denoted by 4lS;caand the 1s orbitals of hydrogen atom a by &,a. The in-plane sp2hybrid orbitals formed from the 2s orbitals and 2p orbitals on carbon atom a are denoted by C#JR;~,4ka,and 40a, the subscripts R, L, and 0 having the following meaning. View carbon atom a from outside the molecule in such fashion that the center of the molecule is occluded by the carbon atom and the ring is horizontal. The hybrid orbital lying to the Right of the carbon atom is denoted by R, to the Left by L, and that toward theviewer by 0 (standing for Out). The carbon 2s orbitals are chosen to be orthogonal to the carbon 1s orbitals. The carbon 2p orbitals perpendicular to the plane of the ring are denoted by Note that the atomic orbitals have all been denoted by 4. Combinationsof these atomic orbitals used below are denoted by cp.
The 1s orbitals on the carbon atoms are core orbitals. The localized a-bonding orbitals are constructed as follows. The
The *-orbitals have been left in their canonical forms. We emphasize that the choice of basis set is immaterial in this construction of the sharing amplitude. The same amplitude is obtained using the canonical orbitals. It is the complete many electron wavefunction which is important here. Thevolume-point sharing indices are readily constructed from this sharing amplitude. We consider first the sharing index associated with the 0th hydrogen atom. The volume is chosen to be the basin, in the sense of Bader,16 associated with the hydrogen atom. We suppose that the core orbitals of the carbon atoms, the orbitals localized between the carbon atoms, and the *-orbitals have no significant extension to the region of the hydrogen atom. (This is a bit of fiction because wavefunctions do not have abrupt cutoffs and there must be some contribution of these orbitals to the hydrogen volume-point sharing index. We shall ignore these contributions in this paper.) As a result, the sharing index is given by IH;O({)
%
('PCH;OI'PCH;O)HO'PCH;O(~)'P*CH;O(~)
Note that there is no dependence on the spin of the electron. The sharing index in this case is simply the basin overlap integral, which is a constant, times the probability of finding the electron at position r when it is in the localized CH bonding orbital. Near the hydrogen nucleus, the probability mimics that of a Is orbital of a hydrogen atom. Near the carbon nucleus, the behavior is that of a 2s-2p hybrid associated with the carbon atom. As viewed from the hydrogen nucleus, this hybrid has a nodal surface which cuts the line running between the carbon and hydrogen nuclei. As the line (and its extensions) connecting the two nuclei is traversed, beginning outside the hydrogen nucleus and going to inside the carbon nucleus toward the center of the molecule.
Sharing of Electrons in Molecules the sharing index increases in a roughly exponential fashion, with a cusp at the hydrogen nucleus; thence it decreases until the node resulting from the 2s-2p hybrid is reached and then another exponential increase with a second cusp at the carbon nucleus, after which the sharing decreases. This sharing index is localized in the region of the carbon hydrogen bond. The sharing index IC;&) associated with the 0th carbon atom is more complicated because of the larger number of bonds originating at the nucleus. The basin overlap integrals between the *-orbitals and the u-orbitals vanish because of the different (symmetry) representations of the orbitals. We suppose that the localized u-orbitals have negligible overlap with one another. As a result, the volumepoint sharing index is the sum
ICO(f1 = ('PCHO,'PCHO)CO'PCHO(~)'P*CHO(~) + ('PO,l,'PO,l)COx ~'PO,l(r)~*O,l(r) + 'Ps,0(r)'P*5,0(r)) + ~ls;Co(r)~*ls;Co(r)+ (1/36)[34p;o(r) + 24p;i(r) + 24p;5(r) - '&3(r)l [34*~;0(r)+ 2#*~;l(r)+ 24*~;5(r) - 4*pdr)1 The core 1s orbital on carbon 0 is quite contracted and gives a fullcontribution the index, while the last termis thecontribution of the *-orbitals, the supposition being that the p orbitals can be entirely assigned to separate atoms. The qualitative nature of the sharing index can be read off simply from this expression although the precise details depend on the values of the overlap integrals. In the vicinity of the 0th carbon nucleus, the sharing looks like that due to the 1s core orbitals together with contributions from the 2s-2p hybrid orbitals and the p orbital perpendicular to the plane of the molecule. As we move along a straight line between the adjacent carbon nuclei, the contribution of the 1s orbital on the 0th carbon nucleus decreases, as do the contributions of the hybrids other than the one pointing along the bond. The r-orbitals make no contribution to the sharing in the plane of the ring. As carbon nucleus 1 or 5 is approached, the sharing index has the form of the 2s-2p hybrid centered on the appropriate nucleus, with a nodal surface%cutting the line between carbon nuclei on the side toward the 0th carbon nucleus. The sharing index along a path from the carbon nucleus to a hydrogen nucleus is similar, the difference being that the hydrogen nucleus is surrounded by a 1s orbital. These are the contributions of the localized orbitals to the sharing index. The contributions of the r-orbitals extend over the entire carbon framework, being above and below the plane of the ring. The main contribution is in the vicinity of the 0th carbon atom. As we move around the ring, the contributions from the adjacent carbon atoms is less, that from the meta carbons is zero, while the para carbon gives a small contribution to the index. Somewhere between carbon atoms 2 or 5 and atom 3 there is a nodal surface which cuts the plane of the ring.Z6 The symmetry of the benzene molecule can be used toseparate the contributions of the u and the a-orbitals. Let Pc be the operator, acting on the tcoordinates, which reflects thecoordinate perpendicular to the plane of the ring. The u- and the r-contributions to the sharing amplitudes are found from
the plus sign being used to pick out the u-contributions, the minus for the *-contributions. Because the integral involving the product of the two amplitudes vanishes, these can be used to form separate volume-point sharing indices. This division is independentofthechoiceoforbitals. Thecontributionofthe u-orbitals is the sum of the first three terms in the expressionfor the sharing index,
(fir)*
The Journal of Physical Chemistry, Vol. 97,No. 29, 1993 7527 while the *-contribution is the last term, Ico-(f) = (1/36)[34p;o(r) + 24p;i(r) 24p;s(r) - 4p;dr)1 [34*p;0(r)+ 24*p;l(r) + 24*p;5(r) - 4*p;Ar)l The localized nature of the u-contributions is apparent in the first partial index; the extended nature of the u-contributions in the second. The nodal structure of the a-contributions results in IC&) vanishing in the plane of the ring. In this case, the total volume-point sharing index is the sum of the partial indices.
v,
point to point Amplitudes
The volumepoint sharing amplitudes used above give a much more detailed picture of the behavior of electrons in molecules than that given by the basin sharing indices used in ref 1. In particular the details of the sharing of electrons become apparent in the same sense that probabilities give some indication of the behavior of one electroi wavefunctiok. On the other hand, the volumepoint sharing index gives little information about the amount of sharing between the u-bond lying between CO and C1 and that betweenCO and C5, nor does it give precise information about the nodal structure of the sharing. The point to point sharing index or, more akin to wavefunctions, the point to point sharing amplitude can be used to probe the localization and/or delocalization of the electrons in bonds. In this section we use the sharing amplitude given above to fix our qualitative notions about the sharing of electronsbetween different regions in benzene. The procedure we use is the following. The sharing amplitude After separating out the spin depends on two points, t and part of the amplitude, this being trivial for the chosen wavefunction, we fix the point [. This will be called the fixed point. The other point' j is a movable point. Typically it will be moved along a line which need not include the fixed point. There are four situations we consider. These are given below. The sharing amplitude for singlet wavefunctionscan be written as a simple product of a function of the spatial coordinates and a function of the spin coordinates,
r.
Below we consider only the spatial part of the sharing amplitude. We first choose the fixed point to be at the center of the bond between carbon 0 and hydrogen 0. The movable point is on the line connecting the carbon and hydrogen nuclei. The relevant part of the sharing amplitude is
with r fixed. As the other index is moved from the midpoint of the bond toward the hydrogen nucleus, the amplitude looks like a hydrogenic 1s orbital. Moving in the other direction, -we pick up the features of a hybrid between a 2s orbital and a 2p orbital, with the characteristic node and change in sign of the amplitude before reaching the carbon nucleus. Note that because the fixed point is beyond the extent of the 1s core orbital of the carbon atom, there is no (or little) contribution to this amplitude from the core orbitals. The cusps of the wavefunction at the nuclei are reflected in the sharing amplitude, the one at the hydrogen nucleus being positive, the one at the carbon nucleus being negative. Beyond the nuclei, the sharing amplitude diminishes. The sharing index is localizedin the traditional carbon-hydrogen bond region. For the second case, we choose the fixed point to be midway between carbon nuclei 0 and 1. Because the a-orbitals have a node in the plane of the ring and because the 1s orbitals of the carbon atoms are quite compact, there is no contribution of these to the sharing amplitude. The amplitude is The movable point is chosen to be along a line containing the two nuclei. The cp0.1 orbitals are a combination of sp2 hybrids, one from carbon 0 and one from carbon 1. Therefore as the movable
7528 The Journal of Physical Chemistry, Vol. 97, No. 29, 19’93
point is moved from the midpoint to either of the nuclei, the amplitude mimics the behavior of these hybrid orbitals. About the midpoint there is a region of roughly constant amplitude. As the point goes toward a nucleus, the amplitude decreases until it vanishes at the node from the sp2 hybrid. The amplitude then becomes negative, forming a cusp a t the nucleus and then decreasing in magnitude as the point is moved beyond the nucleus. Again the amplitude is localized in the region of the traditional carbon+arbon a bond. In order to pick up the contributions of the ?r-orbitals,we move the fixed point of the second case in a direction perpendicular to the ring so that it sits above the midpoint of the line connecting the nuclei and outside the range of the core 1s orbitals. Recall that this point is between carbon nuclei 0 and 1. The sharing amplitude is
[At the fixed point, the contribution of 4p;l(r) is the same as that of ~$~;o(r).]The first term gives the contribution of the bonding carbon4arbon a-orbital to the amplitude. As above, this has the characteristic nodes of the 2s-2p hybrids and is localized in the region between the carbon nuclei. The second term is the contribution of the r-orbitals to the sharing amplitude. Quite clearly, this contribution extends over the entire carbon framework of the ring, with a nodal surface somewhere between carbon nuclei 2 and 3 and between carbon nuclei 4 and 5 in addition to the nodal plane containing the nuclei forming the ring. As we move around the ring from the fixed point, always remaining above the planeof thering, the magnitudeof this contribution to the sharing amplitude decreases until the node is reached and then increases to a local maximum in the region of carbon nuclei 3 and 4. The reverse of this occurs as we continue around the ring, returning to the fixed point. A characteristic of sharing which is picked up quite nicely by the sharing amplitude (and reflected less strongly in the sharing index which is positive semidefinite) is the nodal structure and the relative signs of the amplitudes. For example, in the case just considered, as long as the movable point is on the same side of the ring as is the fixed point and is in the vicinity of carbon nuclei 5,0,1,and 2, the amplitude is positive. When the movable point is in the vicinity of carbon nuclei 3 and 4, the amplitude is negative. The contributions of the Q- and ?r-orbitals to the sharing amplitude may be separated by the use of the symmetry operator used in section IV. We simply form the quantities (1/2)(I f pr)(r;r’)as done before. The plus sign give the a-contributions and the minus sign gives the ?r-contributions. As is typical with amplitudes in the absence of special circumstances (e.g., the volume-point indices in section IV), these contributions must be combined before forming the sharing index. The last case considered is that with the fixed point directly above carbon nucleus 0 and outside the range of the 1s core orbital. This point lies on the nodeal planes of the p orbitals forming the sp2 hybrids. Therefore the p orbitals having their axes in the plane of the ring make no contribution to the sharing amplitude. The movable point is chosen to be on a line containing the carbon nucleus and the fixed point. The main contribution of the ?r-orbitals is from the p orbital centered on carbon nucleus 0. As a result, the amplitude is
This has the familiar characteristics of an amplitude arising from a hybrid between 2s and 2p orbitals. The precise details of the hybrid depend on the coefficient cc arising from the bonding orbital between the hydrogen and carbon atoms as well as the position of the fixed point r. The general description is that as
Fulton the movable point approaches the carbon nucleus, the node of the hybrid is reached with the amplitude becoming negative further toward the nucleus. The cusp due to the s orbitals is found at the nucleus and then a tailing off of the amplitude as we go beyond the nucleus. The amplitude is localized on the side of the nucleus containing the fixed point. Indeed, this remains true even when the fixed point and the line are moved to a location above the midpoint between two carbon nuclei with the line remaining perpendicular to the ring. It should be noted that, unlike the previous cases, this hybrid type behavior was not fed explicitly into the wavefunction. Rather it emerges from the other items which were put into the wavefunction, namely, the p orbitals on the carbon atoms and the 2s orbitals involved in the localized bonding between the carbon atoms and between the carbon atom and its adjacent hydrogen atom. This is an illustration of our thesis that the sharing amplitudes and sharing indices pick up the details of the complete wavefunction independently of the basis set of orbitals used in the construction of the wavefunction. Our use of a primitive picture of the electronic structure of benzene means that we get only the coarse features of the sharing indices and amplitudes. Fine details, such as how much sharing there is between the electrons at/or near the positions of different nuclei, are not picked up in such a treatment. The corresponding sharing amplitudes will show the characteristic cusps found in wavefunctions at the position of nuclei; however, the magnitude of the amplitude at the internuclear cusps will be much smaller than the self-cusp. Also we have not discussed the precise nature of the sharing when both indices are in and/or near the region of the core electrons surrounding a nucleus. In such a region the shell structure of the atom will be reflected in the sharing amplitudes.
VI. Conclusions The sharing index, defined as the absolute value squared of the sharing amplitude which itself is the square root of the first-order density matrix, gives a quantitative measure of the sharing of electrons between different points in space. The index is invariant under transformations of the orbitals in terms of which the wavefunction is expressed; it accounts for the locality of electrons and for the interference of the wavefunction in different parts of space. Bond indices (numbers) are found to be simply related to the sharing index when it is integrated over volumes assigned to the atoms which make up a molecule. The sum rules relating the average numbers of electrons in an atom to the bond indices is a simple consequence of one simple relation which states that the integral of the sharing index over one variable is the probability of finding an electron at the position of the other variable. Application is made to H2+and H2 for the purpose of establishing the connection between bond indices and the sharing index, to primitive models of double and triple bonds in order to demonstrate that the bond index agrees with the chemists’ intuition regarding bond numbers, and to He*+ and He2 so as to illustrate the role interference between different spatial parts of the wavefunction plays in determining the bond number when using delocalized orbitals. These latter are also considered using localized orbitals, illustrating that the bond indices are invariant to the transformation from delocalized to localized orbitals. The bond indices of excited H2 are considered in order to illustrate the difference between the bond indices defined in this paper and those defined by Cioslowski and Mixon5 in cases when the wavefunction is not a single determinant. In addition, we consider the application to the r-electrons in benzene and in 1,3-butadiene using the simple Huckel approximation. For the former, we give the results of using the localization procedure of Cioslowski and Mixon to determine thecovalent bond indices. This illustrates thesymmetry breaking inherent in their procedure when applied to benzene and the differences between the covalent bond indices and twice the sharing indices. For the latter, the differences between the
Sharing of Electrons in Molecules
The Journal of Physical Chemistry, Vol. 97, No. 29, I993 7529
two definitions are illustrated when the wavefunction can be written as a single determinant and there is no symmetry breaking. We have also demonstrated that when effects of correlation are added to the simple molecular wavefunction of Hz, the sharing of an electron between the two atoms is decreased. By integrating the point-point sharing index I(fir)over a volume V, we obtain a volume-point sharing index Id{) which, when the volume is associated with that of an atom, may be considered to be a generalization of the valence of an atom. (We use valence in the sense used by Lewis.2c Unlike the qualitative concept of valence, what we have is a quantitative, microscopic measure of valence.) In the case of the simplified picture of the electronic structure of benzene, the sharing of the electrons associated with one hydrogen atom is localized in the traditional C H bond region. The electrons associated with one carbon atom show a morecomplicated behavior. Thereis thelocalized sharing with the three adjacent atoms which can be interpreted in terms of the traditional a-orbitals, while the delocalized sharing above and below the plane of the ring can be interpreted by means of traditional n-orbitals. The symmetry of the molecule can be used to make a separation of the u- and the 7-contributions in a manner which is independent of the orbitals used in the construction of the wavefunction. The sharing amplitude can be used to probe the sharing behavior within the bonds themselves. For example, the sharing between the midpoint of adjacent carbon and hydrogen nuclei and a movable point reveals that this quantitativemeasure of sharing shows the locality of the electrons in the CH bond. Similarly, by choosing the fixed point to be midway between adjacent carbon nuclei, the locality of the u-contribution to the CC bond is found. The *-contribution is found by raising the fixed point above the plane of the ring. It is delocalized. As is the case for the volume-point sharing index, these a- and n-contributions to the sharing amplitude can be separated in an invariant fashion by using the symmetry of the molecule. Although in a sense we have built in a lot of the various contributions to the sharing indices by choosing to work with some crude descriptions of the wavefunctions, it is apparent that the power of these indices lies in the fact that one can begin with exceedingly complicated wavefunctions found from calculations including all kinds of configuration interaction or from calculations using coupled clusters and extract the essential one electron behavior of the molecule without invoking a particular orbital picture. Indicative of this is that, in the last case considered in section V, the sharing amplitude for benzene between a fixed point above a carbon nucleus and another point on a line joining the fixed point and the carbon nucleus is mimicked by a 2s-2p hybrid even though that particular hybrid is not explicitly fed into the wavefunction. The results obtained by the use of the sharing amplitudes can then be interpreted in terms of bonding concepts with which we are familiar, e.g., the hybridization of orbitals, the delocalization of orbitals, and so on, in spite of the fact that these may not be immediately apparent from the wavefunction. It should also be clear from this that these sharing indices and sharing amplitudes have uses beyond those described here. An example is in the description of chemical reactions where we can get away from expressing the electronic structure of molecules in terms of particular sets of orbitals, although we may interpret the results in terms of familiar constructs. It is also important to recognize that the sharing amplitude and the sharing index are independent of the orbitals used in the construction of the wavefunction provided that the basis set is sufficiently complete. Thus any accurate wavefunction can be used to construct these quantities which can then be interpreted
(fir)
in terms of the one electron constructs such as molecular orbitals or valence bond structures familiar to us as chemists. The sharing index presented here is of general utility. It is not limited in application to organic molecules, nor even to molecules. It is a quantitative measure of the sharing of a particle in a many particle system which automatically includes the essential quantal aspects of wavefunctions.
Acknowledgment. I am indebted to Dr. Stacey Mixon and Professor Jerzy Cioslowski for bringing the problems with the current definitions of bond indices to my attention and for discussions involving bond indices. I have also benefited from discussions with Professors Stephen Foster, Edwin Hilinski, and Ralph Dougherty. References and Notes (1) Lewis, G. N. J. Am. Chem. SOC.1916, 38, 762. (2) Lewis, G. N. Valence and the Structure of Atoms and Molecules; Chemical Catalog Co.: New York, 1923; (a) p 79; (b), p 83; (c) p 67. (3) (a) Pauling, L. C. J . Chem. Phys. 1933, 1, 280. (b) Pauling, L. C. J. Am. Chem. SOC.1935, 57, 2705. (4) Coulson, C. A. Proc. R . Soc. 1938-9, A169,413. (5) Cioslowski, J.; Mixon,S. T. J. Am. Chem. SOC.1991,113,4142. A
list of many of the papers containing definitions of bond orders is given in the introduction of this paper. (6) Steiner,E. The Determination and Interpreiation ofMolecular Wave Functions; Cambridge University Press: Cambridge, U.K., 1976; pp 154ff. 17) Jug. K. Croat. Chem. Acta 1984.57. 941. (8j Re-&, A. W.; Schleyer, P. v. R. J.'Am. Chem. SOC.1990,112, 1434. (9) Wigner, E.; Seitz, F. Phys. Rev. 1933, 43, 804. (10) Slater, J. C. Rev. Mod. Phys. 1934,6,209. See . pages - 222 and 267 for the interpretation referred to. (11) Maslen, V. W. Proc. Phys. SOC.(London) 1956, A69, 734. (12) McWeeny, R. Rev. Mod. Phys. 1960, 32, 335. Bader, R. F. W.; Stephens, M. E. J. Am. Chem. Soc. 1975, 97, 7391. (13) Ruedenberg, K. Rev. Mod. Phys. 1962,34, 326. (14) Reference 13, pp 332-337. (151 Luken, W. L.; Beratan, D. N. Theor. Chim. Acta 1982, 61. 265. Luken, W. L. Croat. Chem. Acta 1984,57, 1283. (16) Bader, R. F. W.; Tal, Y.; Anderson, S. G.; Nguyen-Dang, T. T. Isr. J. a e m . 1980, 19, 8 . (17) Edmiston, C.; Ruedenberg, K. In Quantum Theory of Atoms, Molecules, and the Solid State; L6wdin. P. O., Ed.; Academic Press: New York, 1966; p 263. (18) LBwdin, P. 0. Phys. Rev. 1955, 97, 1474. (19) The procedure for finding the square root of a matrix is essentially that given by: Wigner, E. P. Group Theory and Its Applications to the Quantum Mechanics of Aiomic Spectra; Academic Press: New York, 1959; pp 78,79. The diagonal elements of the matrix d in the reference correspond to the eigenvalues pm of the density matrix. Then &I/* is simply replaced in eq 9.25 by d1/2.The matrix elements of the unitary transformations are the spin orbitals q&). See also: Perlis, S . Theory of Matrices; Addison-Wesley Press, Inc.: Reading, MA, 1952; p 203. Halmos, P. R. Finite-Dimensional Vector Spaces, 2nd ed.;D. Van Nostrand Co., Inc.: New York, 1958; p 167. (20) We choose this notation over the more obvious one of Pl/2(6r)JP being capital rho) in order to emphasize the similarity of these sharing (the use of the (and)) and amplitudes to the transformation functions yet retain the difference from the transformation functions (the use of ; in place of the vertical bar I as the separator of the arguments). The amplitudes differ from the transformation functionsbecause they describedifferent physics. Note that (fly) = a({-Y) when the tare eigenvaluesof commuting Hermitian is generally nonlocal. operators while (21) The value is also l/2 if the sharing index is defined as p(fir)p(j';fl as it is by Ruedenberg's considerations of the two particle density for single determinant wavefunctions. This, together with the lack of the sum rule valid for arbitrary wavefunctions, is a strong argument against such an identification. (22) Reference 5, page 4145 and Figure 1. (23) Levine, Ira N. Quantum Chemistry,2nded.;Allyn and Bacon: Boston, 1974; p 303. (24) Wang, S. C. Phys. Rev. 1928, 31, 579. (25) Weinbaum, S. J. Chem. Phys. 1933,1,593. (26) In general the volume to point sharing index does not precisely vanish on a surface. The amplitudes ( do have a rich nodal structure as a function of the variables land and so the sharing index I ( f f ) has surfaces on which it vanishes. For fixed I(fir),considered as a function of {, has nodal surfaces, the location of which depend on When forming the volume to point sharing index Id(), is integrated over and so only the remnants of the nodal surfaces remain.
(nr)
(fir)
er)
r, r,
r
r.
