The Chemical Bond Revisited N. Colin Baird University of Western Ontario, London, Ontarlo, Canada NBA 587
A chemical bond is a concept for which ab initio quantum mechanics provides a necessary and sufficient theory in that (for small molecules a t least) it predicts properties such as bond lengths, dipole moments, vibration frequencies, and intensities with hieh accuracv and dissociation enereies with a lower (but usefur) accurac{. I t would be very helpkd if one could "interpret". "exdain". "understand". or "analvze" the subtle, cknplicated a b initio theory in simpler t&ms, i.e., in terms of forces, energies, simple wavefunctions, electron densities, etc. This is a difficult undertaking. Some textbook discussions of the nature of the chemical bond are oversimplified to such an extent that many of the statements they make in fact are incorrect.' This deficiency is understandable, given that correct explanations of covalent bond formation are rare even in quantum chemistry texts! There are two fundamental difficulties associated with the rectification of this state of affairs. First, there is no one unique, "correct" way of discussing the chemical bondfor example, one can analyze either forces or energies, wavefunctions or electron densities. etc. with eaual validitv but a difference in point of view as to which are the more fundamental quantities. Second, the actual situation is rather more subtle and complicated than one might have hoped and certainly more so than is apparent from the simple but often misleading explanations in most texts. One complete and interesting analysis of the chemical bond has been provided by Ruedenberg and co- worker^,^ particularly for the fundamental systems Hz+ and Hz. The pivotal role of changes in the kinetic energy of the electron(s) is emphasized in this analysis. In the discussion that follows, I have attempted t o provide a simplified and accessible, but still quantum-mechanically correct, version of Ruedenberg's theory. As we shall see, the factors whose interplay determine the energy and size of free atoms are also those which operate (though in modified form) in molecules and which are responsible for covalent bond formation. Thus, t o understand Ruedenberg's theory of the chemical bond, we must first learn something about the energetics of simple atoms. ~~~~~~
~
~~~
Free Atoms Accordine to classical as well as wave mechanics.. there are two significant contril~utionsto the internal energy of a free atom: thr kinetic energy of the electrons due to their motim about the nuclrus and the potential enrrgy of the electrons dueto their attraction to the nucieusand their mutual repul~
~
' An example is the implication that the potential energy decreases
substantially in the molecule because the attractive potential is more negative at the midpoint between the nuclei than it is close to either nucleus, and the transfer of density to this region is favorable for that reason. In fact,as a simple summation of -l/r, and -1la indicates, the potential close to either nucleus is superior to that at midbond. Ruedenberg, K. Rev. Mod. Phys. 1962,34, 326. Feinberg, M. J.; Ruedenberg, K. J. Chern. Phys. 1971, 54, 1495. Feinberg, M. J.; Ruedenberg, K.; Mehier, E. L. Adv. Ouanturn Chem. 1970, 5, 27. The inverse of ll/rl.. is rouahlv related to the averaae - seoaration . althougn not ioent~c.sl10%. For tieburposes of this qua ilative discussion. I sna I referto(lIr).,-' as the 'average raol~s",a though it is not In fact equal to . 660
Journal of Chemical Education
sion. We shall concentrate our attention upon the hydrogen atom, which, since i t contains only one electron, has no potential energy contributions from electron-electron repulsions. s the ~ o t e n t i aenerw l Recall from elementarv ~ h v s i e that of attraction between two charged is proportioc& to the product of their charees and is inverselv-~.r o ~ o r t i o n a l t o the; separation r. In atomic units of charge, distance, and energy, the potential energy P of attraction at any instant between an electron and a nucleus of atomic number Z is given by .
&
-
A
where r is the electron-nucleus distance. The potential energy Pa, hetween the electron and nucleus is therefore given by -Z times the average value of l l r , i.e., the average of the inverse separation between them
For convenience, we redefine the average of the inverse distance as X ,
and thus P, = -Zx. Clearly the potential energy of attraction improves (i.e., becomes more negative) as the average electron-to-nucleus distance decreases. We now turnour attention t o the kineticenergy associated with the electron's motion about the nucleus. Although we know nothing about the trajectory of the electron's movements, we do possess information concerning the probability that the electron is to be found a t anv t soace. From . ~. o i nin such information, we can infer theeffective size of the sphere to which the electron is restricted most of the time (for a 1s orbital). I t is well known for aparticle in a (three-dimensional) box that the kinetic energy of motion is inversely proportional to the square of the box's dimensions; this result applies also to a sphere. Thus it is not unexpected to find that the kinetic energy of the electron in a hydrogen atom is proportional to the square of the average inverse of the electron-nucleus separation. In atomic units of energy i t is found to be:
Clearly, the kinetic average is always positive and thus "opposes" electron-nucleus binding; the kinetic energy becomes worse (i.e., more positive) as the average electron-to-nucleus distance decreases, since the electron is thereby confined t o a smaller and smaller volume. The notential enerw and kinetic enerw -,of the electron in a hydrogen atom are plotted against the average electronnucleu~seoaration.~ lartuallv Il/r.,)-lI . in Fieure 1. The total energy of the hydrogen atom is the sum of these two terms, and is also shown for the region of interest (Fig. 1). The best (lowest) energy occurs for a value of one atomic unit. The same result can be obtained by differentiating the expression for the total energy E
Figure 2. Probability of finding electron in atomic hydrogen at varlous dbtames rfrom the nucleus.
Figure 1. Kinetic (0,potential (0.and tdal energy versus average electronnucleus separation in atomic hydrogen. \
E,=K,+P,
with respect to x and setting the result equal to zero,
Thus, a t the best separation, dK, -=-
-@,
dx
dx
Now K., = 0.5x2 and Pa, = - Z X Thus,
and dx
By setting the derivatives equal and opposite, we obtain X
=z
at the optimum. For Z = 1(i.e., hydrogen), x = 1,and thus
The "avera e radius" of one atomic unit (i.e., one Bohr radius, 0.529 )represents the optimum compromise for the total enerev. If the size of the atom were to be further reduced, theidestabi~izin~) increase in kinetic energy would more than offset anv advantage to the potential energy. T o restate thisconclusibn, the hyhrogen atom is prevented from contracting to a smaller size (even though this would improve the potential energy component of the total energy) due to the large increase in kinetic energy that would result from the increase in the electron's velocity that accompanies any reduction in the volume of space accessible to it. I t is well known that the electron in a hydrogen atom has a finite probability that i t will be found a t any point in space,
i.e., it is not restricted to any particular finite volume. However, the total probability of finding the electron far from its nucleus is very small, and for our purposes i t can be ignored. The familiar plot for the probability of finding the electron a t any particular distance r from the nucleus is shown in Figure 2. The function peaks a t r = 1atomic unit, which also is the "average radius" obtained above from energy considerations. Indeed the actual potentialenergy is identical to an atom in which the electron is restricted to travel on the surface of a sphere of radius 1.0 atomic unit. The actual kineticenergyis identical t o that of an electronin a spherical box of constant ~otential,and with a wall a t a radius of 1.57 , the kinetic and potential energy atomic units. ~ h u s both functions are consistent with a hydrogen atom in which the electron travels a t a distance of about 1atomic unit from the nucleus, i.e., on the surface of a sphere of radius F. 1 a.u., venturine - - ~ ~ outward and inward. This model for electron binding can now be extended to bond formation. Clearlv. - . the kinetic enerm of the electron in a hydrogen atom could be improved, i.e.,reduced in positive value. bv increasine the effective radius of the sphere in which i t i s enclosed,or alternatively by extending the sphere in some one direction. Such an extention does not occur in atomic hydrogen because it would place, for a substantial fraction of the time, the electron in a region in which the potential is poor. T o he more precise, let us for convenience consider a cross section of the above spherical surface for the hydrogen atom; we choose a plane passing through the nucleus. This cross section is a circle (of radius 1.0 a.u.); in a sense we have returned to the Bohr model for the atom, but this causes no real difficulty for the 1s state if we are concerned solely with potential energy. ~
1
Now consider the potential energy of an electron located a t some point along the hcrirontal line that touches tangentially the circle (see diagram abo5.e). The variation in instantsnrous potential energy P with position d along this line (d = Volume 63
Number 8
August 1986
661
Since the potential experienced by the electron is relatively constant between (and ahove or below) the nuclei in Hz+, the probability distribution G2 for the electron also is relatively smooth in this region in the groundstate of thesystem. J.2is larger in the midbond region than is the average of the probabilities for the two atomic I s orbitals, 41and b,, one centered on each atom. Indeed, $ is found to he well represented hv the normalized sum of d.. l d. . with the conm~~~. ... quence that d 'runrains the crosnterm Zwlw,, which is large in thr midhond rrgion. i.e.. the volume in which thr sum o? + 4r2is smaller than near either nucleus
+
*
~
~~
~
= N(4, + 4,)
Thus, Figure 3. Potential energy (0versus distance d along a nucleus in Hand H Z . Internuclear separation is 2 a,".
line 1 a,",
above
0 represents the point of intersection with the circle) is shown as the lower curve in Figure 3. Clearly P peaks (in a negative sense) a t d = 0. I t becomes substantially smaller in magnitude whenever the electron is located away from this point in either direction to any appreciable extent. Molecule Formatlon In order to formulate a model for the chemical bond in Hz+, we bring a second nucleus close to the hydrogen atom; in particular we place it 2.0 atomic units from the first nucleus (since this corresponds to the internuclear separation in HZ+).Let us now reconsider the potential energy of the electron along the horizontal line which lies 1.0 a.u. above the line joining the nuclei.
Due to the presence of the second nucleus, the potential felt by the electron is almost 0.5 units more negative a t points along this line than in atomic hydrogen; however, this advantage to the energy is cancelled by an approximately equal but opposite term which is due to the nucleus-nucleus repulsion. Murh more important is the fact that the potential en erg^. function I S r s s r n r i o l l ~flat, ie.. P is almost constant. n,hen the electron travels aiokg the line segment between the nuclei but 1a.u. ahove them-see the upper curve in Fiaure 3. Thus the space accessible to the eleciron has been greatly increased4 in one direction in space-i.e., toward the second nucleus, where "accessible space" is defined as the region in which the potential is essentially identical with that found for the 1 a.u. radius around the nucleus. Thus in 3-D. ~.~the space accessible to the electron in Hz+, and in other diatomic molecules, is not a sphere but a cylinder with rounded ends:
The kinetic energy function in Hzf is related to the value of (Ilr)., associated with the atomic orbital function +I (or 4,) by the same inverse square dependence as in atomic hydrogen, hut with a proportionality consrant which is only a t m t 516 as large. The reduction of 116 orcurs hecause J extends well bevond the t.ffcctive limit of m, alone" the hnnd ~~-~~~ direction; as a result, the component of kinetic energy in this direction is only about half that in the isolated atom. (A further increase in the length of the "box" in the direction along the internuclear axis makes little difference to the kinetic energy as the box is long in that direction anyway.) Thus, for Hz+, ~
The total potential energy for Hz+ includes two terms: the attraction of the electron to the two nuclei and the mutual repulsion of the nuclei. The attraction term is approximately equal to that of -(l/r), of the hydrogen atom plus about -1lR for the attraction of the electron to the second nucleus. The repulsion between nuclei is exactly 11R. Thus, for Hz+,
In other words, the potential energy function is essentially identical to that in the isolated atom! If no change were to occur in the average electron-nucleus distance, then for HZ+we would have
since
i.e., E = -0.583 a.u., an improvement of -0.083 a.u., or about 52 kcal mol-' compared to that of a hydrogen atom (plus bare proton). However, there is amechanism by which the energy in Hz+ can he improved even further. Let us discover whether 1.0 (Extension of the sphere in other directions is achieved in polyatomic molecules.) This is the essence of chemical bond formation-the extension in space of the electron t o volumes having advantageous potential energy beyond those which are available to free atoms and which results in improvement in the kinetic energy value. 662
Journal of Chemical Education
'Note that there is no advantage to the kinetic (or potential)energy in allowing the electron to jump back and forth between two infinitely separated hydrogen atoms; the accessible volume at any instant is just that of the one hydrogen atom at which the electron is located, not double this value.
a.u. is the optimum radius for the electron in Hz+ as well as in atomic H. In functional form, the energy in HZ+is given by
where
Em = K,, + , p
AP = P d - patom, and AK = K,, - '%to,,
+
Differentiating and setting dE,ldx = 0, we find that to optimize E, x = 615 = 1.20. With (llr), = 1.20,
Given that E = P K, i t also follows that the change AE in total energy upon molecule formation AE=AK+e = -m
K , = +0.60 a.u.
P ,
= -1.20
E, = -0.60 am.
i.e., an improvement to E., over the hydrogen atom of -0.10 a.u., about 60 kcal mol-I. (The experimental value is 64 kcal mol-' for Hz+.) As a result of this re-optimization, the average electronnucleus distance in Hzf becomes substantially smaller than i t was in atomic hydrogen (0.83 versus 1.00 atomic units). This contraction in size of the "atoms" in Hz+ allows a 20% improvement (i.e., -0.20 a.u.) to the potential energy since the electron comes much closer on average to a nucleus. The "penalty" in terms of the increase in kinetic energy amounts to only 0.183 a.u. In contrast, a decrease in radius to this value in the original H atom would have resulted in an energy increase of 0.22 a x . I t is the relief of some of the kinetic energy "pressure" to expand the radius which allows the atom to contract, and the potential energy to improve, in HZ+.Similar conclusions appiy also to Hz and to other covalent bonds, although the analysis is more complicated in these multielectron~molecules. In terms of the redistribution of electron density (prohahilitv) which accomvanies formation of the chemical hond. thereis therefore a iharacteristic contraction of peak radius toward the nuclei as well as the well-known increase in density in the midhond region. Parenthetically i t should be added that the confusion apparent in some texts concerning explanations of the chemical bond are understandable, given the analysis ahove. For example, if no allowance for atomic contraction is made, it appears as if the kinetic energy decreases and that the potential energy becomes less favorable upon hond formation, and the bond is attributable to the decrease in kinetic energy. As we have seen, however, the bonding energy can be improved by about 20%by contraction. However, the accompanyingenergetic changes are so large that the net change in kinetic energy is actually positive rather than negative, and the potential energy is actually lowered. The latter conclusions are completely general for molecule formation according to the uirial theorem of quantum mechanics. According to this principle, the potential energy of any molecule a t its equilibrium geometry5 is equal to minus twice its kinetic energy,
P , d
=
=+el2
a.u.
-ur,,
The theorem applies also to the isolated atoms before bonding
Thus the change in potential energy due to molecular formation is minus twice the change in kinetic energy,
-
Since AE must be neeative if amolecule forms. it follows that AK is positive and A P is twice as large and is negative. Thus the phenomenon of contraction apparently occurs in all molecules (possibly along with other changes which produce an improved potential enerw -.a t the cost of onlv one-half the corresponding increase in kinetic energy). Finally, it is interesting to explore the changes to the energy components as a function of variations in internuclear separation R. The potential energy function never departs much from the form
In actual fact. the electron-to-electron nucleus attraction is not quite HS largeas 11R;the revidual positive 6 % repulsive) contribution from thesecond and third terms taken together becomes more serious as R becomes small, i.e., when the internuclear sevaratiou is reduced much below the 2.0 a.u. value discussedahove. P = -(l/r),,
+ small number1R
Well before this effect becomes serious as R decreases. however, the kinetic energy begins to rise rather quickly. 1n the 0. the region of constant ootential contours belimit R tween the nuclei must disappear and the kinetic energy function returns to the form
-
which is identical to that for the separated atoms! The real characteristic of a bond in a molecule is the existence of a region between the nuclei of constant potential contours that allows the potential energy to improue substantially by atomic contraction a t the expense of only a small increase in kinetic energy. Apparently the equilibrium distance Re (i.e., the hond length) in a molecule is that separation of nuclei which is small enough such that the potential in the internuclear region is relatively constant, but not so small that this constant-potential region is negligihly small. The nature of the (very common) two-electron chemical hond differs in no substantial way from the one-electron linkage discussed above. For example, in Hz the (llr) coutributions to the energyterms are doubled. The potential function also contains an electron-electron repulsion term; this positive (destabilizing) contribution to the energy also varies approximately as lIR, but is slightly smaller in magnitude than the extra electron-nucleus attraction which in turn is smaller than the ever-present nuclear-nuclear repulsion. Overall, we obtain
"he theorem applies only to optimized wavefunctions (at the theoretical optimum geometries) and to the real systems; thus, the results calculated for (llr), = 1 do not obey the theorem. Volume 63 Number 6
August 1986
683
Thus there is no special significance to two-electron bonds in this theory; they simply are multiples of the basic one-electron bond and correspond to the maximum number of electrons which according to quantum mechanics can travel according to the same optimum motion "path" as described above. (In particular, there is no special significance to the phenomenon of electron interchange between the two atomic orbitals, except as amechanism which allows them both to
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Journal of Chemical Education
travel in the region of both atoms rather than in just one and to correlate their motions so as to avoid being very close together a t any instant-i.e., their "jumps" between atoms are synchronized!) Acknowledgments
The author is grateful to K. Ruedenberg and to the referees for useful comments on the manuscript.
