T H E QUANTUM THEORY O F T‘ALEXCE’ SAUL DUSHMAN Research Laboratory, General Electric C o m p a n y , Schenectady, N e w Y o r k AND
FREDERICK SEITZ Department of Physics, Uninersity of Rochester, Rochester, New York Receit)ed September 50, 19S6 INTRODL-CTIOS
“T’alency,” according to N. Y.Sidgwick ( 5 ) , “is a general term used to describe the power which atoms possess of combining with one another to form molecules.” The concept of valence is thus one of the most important, if not the most significant, concepts in the correlation of those observations on the interactions of atoms and of molecules that constitute the science of chemistry. Our understanding of the nature of valence has undergone, during the past three decades, a profound change, which has been largely the result of discoveries in the field of atomic structure. The naive picture of two little hard balls bound together by a “hook and eye” has been displaced, through the work of Kossel, G. N. Lewis, and Langmuir, by a classification of chemical bonds into covalent and electrovalent, or homopolar and heteropolar. We designate as covalent or homopolar those bonds in which we regard the two atoms as sharing pairs of electrons, as, for instance, in hydrogen, methane, etc., and as electrovalent or heteropolar those bonds in which the tn-o atoms posseqs opposite electric charges, as, for instance, in sodium chloride and in ionic compounds in general. While such a classification has proven extremely useful in the investigation of molecular structures and in the interpretation of their chemical properties, it is obvious that it lacks any quantitative aspect. By this we mean that the Lewis-Langmuir concept of the shared-electron bond does not enable us to derive the energy relations which govern the possible modes of interaction of atoms to form stable molecules. K h a t we are interested in fundanientally is the solution of this problem: 1 Presented a t the Symposium on Molecular Structure, held a t Princeton University, Princeton, Yew Jersey, December 31, 1936 t o January 2, 1937, under the auspices of the Division of Physical and Inorganic Chemistry of the American Chemical Society. 233
234
SAUL DUSHMAN A S D FREDERICK SEITZ
Given t n o or more atonis for each of 11 hich we know the nuclear charge and electron configurations, what will be the r e d t of their interaction? Fundamentally, the problem involves a determination of the total energy of the system of nuclei and electrons as a function of internuclear distance. For any two atoms A and B the energy as a function of internuclear distance may be of the form I or I1 shown in figure 1. In the first case we conclude that the repulsive force increases as the atoms are made to approach each other and that, consequently, no ~noleculeformation will occur; in the second case, the attractive energy increases with decrease in distance of separation and passes through a minimum, which represents the energy of binding, to form a stable molecule. While classical dynamics and the Bohr theory proved incapable of dealing with this problem, the advent of quantum mechanics has given us, a t least theoretically, a method by which for the first time it is possible to make such calculations as are of interest in solving the problems of molecule formation. I n 1927 Heitler and London published their now well-known discussion of the hydrogen molecule problem. While the actual results of
FIG.1
their calculation gave a value for the energy of dissociation of hydrogen of 72.38 kg-cal. per mole, as compared with the observed value, 108.80 kgcal. per mole,? even this partial success has encouraged subsequent investigators to apply the methods of quantum mechanics to this and similar problems of molecule formation and interaction. The methods used and the results of these investigations have been reviewed recently in two extremely interesting publications, one by J. H. Tan T’leck and -4. Sherman (6) and the other by W. G. Penney (4). I t is therefore not necessary in the present paper to enter into any detailed discussion of the mathematical technique involved in the calculation. To such aspects we shall refer only briefly. I t is rather the logic and physical significance of these calculations that we shall attempt to consider in the following remarks. FUNDAMENTAL CONCEPTS
As the most fundamental idea of quantum mechanics we must recognize the validity of Heisenberg’s principle of indeterminism. For our present
* This includes the “zero point”
energy, which is 6.12 kg-cal. per mole.
QCANTCM THEORY O F VALENCE
235
purposes it may be stated in the form that precise determination of the kinetic energy of a particle at any instant is incompatible with a precise determination of the position of the particle at that instant. Since the potential energy of any system ( V ) is a function of the coordinates of the particles, and the total energy ( E )is a constant for an atomic or molecular system, this means that it is impossible to associate instantaneous values of the kinetic energy (T) with simultaneous values of the coordinates. In other words, we cannot derive any conclusions with regard to definite orbits, and consequently the only magnitude that we can calculate is the relative density distribution as a function of the coordinates. This is the physical significance of the function $$, which is derived from the solution of the Schrodinger equation for the system. Let us consider, as an illustration, the hydrogen-like atom problem. This consists of an electron of charge -e, moving in the field due to a nucleus of charge +Ze. The potential energy function for the system is given by the relation
V
=
-Ze2/r
(1)
where r is the distance of the electron from the nucleus. Schrodinger equation has the form
The corresponding
where V2 is the Laplacian operator, p is the mass of the electron, and h is Planck's constant. We can write this in the form
H$ = E# where H is an operator defined by the relation
+v \Then written in this form, H is said to be the total energy operator while h2 -- Y2 = T and V are the kinetic and potential operators, respectively. 8T'p
S o n . it can be shown that if we consider any arbitrary function 4 of the coordinates which is everywhere finite, then the so-called mean value of the energy integral
I
=
1
(5)
6H4d.r
is always greater (more positive) than the value of Eo, the lowest energy level for the system. The more closely the form of approaches the exact
+
236
SAUL DUSHMAN AND FREDERICK SEITZ
solution for $ in equations 2 and 3, the lover the resulting value of I, and when 4 is identical with $0, the eigenfunction for the system corresponding t o the lowest energy level, the value of I becomes equal to Eo, the corresponding eigenvalue. Consequently, we can express the solution in the form
EO =
/
&H$odr
(6)
where E Ois a minimum obtained by varying @ (or $) in all possible ways. Since the mean value of H for given 4 is equal to the sum of the mean values of T and Ti, that is,
B=F+P and F is always positive while 7is always negative, it is interesting t o note the way in which the two “competing” terms modify the charge distribution going with the 4 corresponding to the minimum energy. I n the first place, one could minimize T to zero by setting @ = constant, but this would not give its minimum value, since it would place practically all of the charge in regions away from r = 0. On the other hand, if one attempts to minimize P by localizing the charge distribution entirely a t the nucleus, F becomes infinite, since localization of a particle implies infinite kinetic energy on the basis of the uncertainty principle, and consequently ?i becomes infinite and positive. What actually happens in the case of the hydrogen-like system is a compromise between these opposing tendencies, and the minimum value of E is that for which T = -$V, and E = T + V = +V. The corresponding eigenfunction for the normal state has the form
where p = r/ao and a0 = radius of normal Bohr orbit. If we plot $* as a function of p , it is observed that while this distribution function has a maximum value a t p = 0, as might be expected from the fact that in this region V approaches a minimum value (of - a),there is a considerable probability for the occurrence of the electron in the region for which r > ao,-a fact which is the result of the tendency to have T as small as possible. This display of comproniise between two opposing tendencies by the wave function is characteristic of the quantum-mechanical description of atomic systems, and recurs constantly. Individual terms of the energy operator make certain demands upon the wave function in order that they may best contribute toward minimizing the integral (equation 5). The wave function does not favor any one of these demands to the limit, how-
23 7
QUaNTUM THEORY O F VALENCE
ever, but effects a compromise between all. It is in fact just this state of affairs that makes quantitative investigations of atomic problems difficult, because too rough a treatment of any one term will lead to considerable error in the description. THE HYDROGEN MOLECULE
The Schrodinger equation for this system (consisting of nuclei A and B and electrons 1 and 2 ) has the form v2+
+ cry23 - V ) + = 0
(8)
where = 8ir2p/h2
and
-e2+ - e2+ - +e2-
‘A1
‘A2
‘81
(9) ‘BZ
Thus, we find in the potential energy function (equation 9) three different types of terms corresponding to the following : ( 1 ) nuclear repulsion, represented by the first term; ( 2 ) electronic repulsion, represented by the 2nd term; and (3) electron-nucleus attraction, represented by the four terms in the bracket. It follows that V as an energy operator contains, in addition to those of the type occurring in the hydrogen atom, an electronelectron interaction term. There are two methods by which this system and all molecular systems are usually treated t o a first approximation. The first is by the use of so-called atomic orbitals. This is the method of Heitler and London. The second, which is designated the Hund-Mulliken method, involves the use of molecular orbitals. Let us first consider the Heitler-London method. I n this treatment the eigenfunction which corresponds to attraction of the two atoms and molecule formation is represented by the linear sum of two eigenfunctions in the form 1
where U A ( 1 ) represents the single electron eigenfunction for the condition in which electron 1 is associated with nucleus A, and corresponding interpretations are to be given to the other three single electron functions inside the parentheses in equation 10. That is,
and similar expressions are to be used for u ~ ( 2 )U ,B ( l ) ,and
UB
(2).
238
SAUL DUSHMAX AND FREDERICK SEITZ
The quantity S2 in equation 10 is defined by tthe relation (12) and it is found that when this integral is eraluated, the result is S2 =
E-’’{
1
+ D + (1/3)D2I2
(13)
That is, S2is a function of D = R / a , with a value varying from 1 for D = OtoOforD-+
=.
According to Heitler and London, the binding energy for the formation of Hz is given by
W8 = E11 1
+
+
E12 s 2
where E11 and El% are integrals over the configuration space. By evaluating the integral for E,, i t is found that this has a minimum value El1 = -0.4877 v.e. for D = 1.90. Since the corresponding value of S2is 0.3466, it follows that Ell/(l + 82) = 0.3622 v.e. = 8.349 kg-cal. per mole. The evaluation of the integral E12was carried out by Sugiura, and he deduced the value W s = 3.2 v.e., so that the term Ell/(l + S2) constitutes only 11.32 per cent of the total theoretical binding energy. Comparing 0.3622 v.e. with the observed value WTS= 4.72 v.e., it is seen that the integral E11 corresponds to only 7.67 per cent of the total experimental binding energy. Now, if there were no possibility of interchange of electrons between the two atoms, the total binding energy would be given by the term Ell alone, which corresponds to the interaction energy of the nuclei and electron charge distributions for each electron about their respective nuclei. However, as the atoms are made to approach there is an increase in the frequency of interchange of electrons, and this is taken into account by the term Elz. Hence, Ell has been designated the Coulomb integral and E12 the exchange integral. X great deal has been written about the non-classical nature of the term E12,and since this term accounts, as shown abore, for a large part of the energy of formation of H2, a distinction has been drawn between the types of forces involved in the two energy terms. Evidently such a distinction is oidy the result of the mathematical computation, for, as a matter of fact, the quantum-mechanical treatment recognizes that the only forces involved in the binding of two hydrogen atoms are those which arise from electrostatic attraction and repulsion between the four particles which constitute the system. The exchange integral is merely an expression of the physical requirement that the electrons in Hz cannot be regarded as localized about the nuclei with which they were associated in the separated atoms.
239
QUANTUM THEORY O F VALESCE
The fact that the Heitler-London method lead. to a binding energy which is only 3.2 v.e. as compared rvith the observed value of 4 . i 2 v.e., however, shows that this treatment is incomplete. In accordance with the remarks made at the end of the previous sectioI1, the existence of this diqcrepancy suggests at once that the effect of at least one of the energy terms ha. been slighted in this approximate treatment of the problem. Actually, tJvo very important effects, which n-e shall now enumerate, have been neglected. In the first place the kinetic energy term will demand that the complete function be spread smoothly throughout the entire ,space, while the nuclear potential terms will require localization in the line joining the two protons, since the nuclear field is strongest there. The Heitler-London treatment doe5 not give the best distribution for the purpose of these t n o term.., and by altering the atomic functions used in the previous computations in such a way that there is a large amount of charge spread smoothly throughout the internuclear region, Rosen succeeded in increasing the theoretical binding energy to 4 volts. The final distribution shows the expected coniproniise in the tn o terms. Thig appears to be about the best that one can do with an energy that is divided into only exchange and Coulomb terms, and since three-quarters of a volt is left unaccounted for, we see that it is not at all correct to describe the valence bond as due t o the action of socalled exchange and Coulomb forces. This is even more striking in caheq in which the binding energy per electron is smaller than in the case of H2. The remaining discrepancy is to be associated, principally, with the second neglected effect, that is, the effect of the presence in the expression for Ti of the electron repulsion term. In order to satisfy this term alone, we would expect a charge distribution in which the two electrons are as far apart as possible. Since this implies that one electron be a t zc and the other at - w , which would not be the most suitable for the other energy terms, a compromise results. The electrons will move around in a region the size of molecular dimensions, but will do so in such a way that they are seldom near each other. -4 part of this effect is contained in the previous approximation and appears in the fact that the electrons spend more time near different atonis than on the same one. The exchange energy is essentially the term arising from this effect, and, as remarked previously, constitutes a large percentage of the binding energy. This is not entirely sufficient, however, and the remaining discrepancy would be almost completely removed if the repulsion effect were included in more detail. -4ctually it proves very difficult to do this in an approximation scheme that is based on the Heitler-London first approximation, and has not been carried through for H2. I t ha5 been treated in a way that bears comparison with the Hund-1\Iulliken scheme, however, so we shall consider this next. I n the method of molecular orbitals we start with electronic wave functions which represent the behavior of each electron in the field resulting
+
THE J O U R N A L OF PHYSICAL CHEMISTRY, YOL. 41, N O 2
240
SAUL DUSHMAN AKD FREDERICK SEITZ
from the presence of the two nuclei and the other electron. While niolecular orbitals may be constructed out of atomic orbitals, this is not a t all necessary. Having chosen some function involving one or more arbitrary parameters, n-e then apply the variational method to determine the energy of the system in a manner analogous to that already described. But this method, like the Heitler-London method, also does not take adequate cognizance ( 7 ) of the fact that the electrons n-ill tend to avoid each other. This may be illustrated by reference to figure 2 (a and b). If we use the simple Hund-3lulliken view, the probability function for each electron as a function of interelectronic distance ( ~ ~ 2is) given by figure 2a, and the calculation yields a value for the energy of binding which is about one-half the observed value. On the other hand, when account is taken of electronic repulsion which represents another stage of approximation, we obtain the probability function shown in figure 2b, and we find, as mentioned previously, that this gives an additional binding energy, which is sufficient to raise the total calculated binding energy to approximately
-r2 P
FIG.2
the same value as that observed. This term which thus arises in the computation as a result of taking into account the correlation of the electrons more fully than the one-electron function scheme allows is known as correlation energy. The exchange energies in both the Heitler-London and Hund-Mulliken schemes may be regarded as correlation energies which are included in a manner that is incidental to the use of an antisymmetric function constructed of one-electron functions. It is more convenient, however, to reserve the latter expression for the additional term arising from a higher approximation. It is of interest to compare the two points of view. As an illustration let us consider still further the case of the hydrogen molecule. We can express the molecular orbital for each electron in terms of atomic orbitals and thus obtain for the wai-e function of the system an expression of the form
QUANTUM THEORY O F VALENCE
24 1
The first two terms are the same as in the Heitler-London theory; the last two terms are ionic in nature, since they indicate that the two electrms are associated with either nucleus A or nucleus B. Thus the use of molecular orbitals implies that we regard the bonds as partly homopolar and partly heteropolar, and if necessary we can replace the right-hand side of equation 15 by an expression of the form
where corresponds to the sum of the first two terms on the right-hand side of equation 15 and GM to the sum of the second two terms. Furthermore, we may use for a more general expression of the form
+
If a = b, this becomes identical with equation 15, while if a is very much less than b, so that both ab and a2 are small compared to b2, then we can speak of the molecule as of the polar type. The differences between the different states corresponding to these eigenfunctions may be represented symbolically thus : Shared electron bond (pure Heitler-London eigenfunction) A :B Ionic bond A : B or A-B+ A :B or ii+BEigenfunction in expression 15 (A:B) + (A: B) + (A :B) The introduction of ionic terms along with Heitler-London terms in the function IJ represents an attempt to take care of electronic repulsion without having to introduce the so-called r12 term. This however leads in many cases to an overemphasis of ionic terms, which implies excessive simultaneous localization of the electrons a t the two nuclei and therefore a smaller value for the binding energy than the true value. It is evident from this discussion that in quantum mechanics the distinction between homopolar and ionic compounds is not nearly as sharp as chemists have usually considered it to be. As Van T’leck and Sherman point out, “There are elements of truth in the old-fashioned chemistry that HC1 has the structure H+Cl-, as the true wave function of HC1 is expressible as a linear combination of various ionized types, and certainly H+Cl- must be given some representation. . . . One great service of quantum mechanics is t o show very explicitly that all gradations of polarity are possible, so that in a certain sense it is meaningless to talk of such idealizations as homopolar bond, heteropolar bond, covalent bond, dative bond, etc.” Pauling has presented the same ideas in an interesting paper (2) in which he uses as illustrations of such transitions from one bond type to another the alkali and hydrogen halides.
242
SAUL DCSHMAPLT AXD FREDERICK SEITZ DIRECTED V I L E N C E S
For the chemist the concept of directed valence bonds is a logical consequence of the properties and behaTior of many compounds. While such ideas could not be deduced from the siniple Heitler-London theory, it has been shown by both L. Pauling and J. C. Slater, working independently, that it is possible to construct on the basis of quantum mechanics single electron eigenfunctions which have directional properties and may thus be used to interpret directed valence bonds. “It has been found,” Pauling w i t e s , “ t h a t the strength and direction of an electron-pair bond formed by an atom are determined essentially by one electronic eigenf u n c t i o n . The bond tends to be formed in the direction in \yhich the eigenfunction has its maximum value, and the greater the concentration of the eigenfunction in the bond direction, the stronger the bond will be. The spherically symmetrical s eigenfunction can form a bond in any direction of strength 1 according to the semiquantitative treatment, and a p eigenfunction a bond of strength 1.732 in either of two opposite directions. But in most atoms which form f o u r or more bonds the s and p eigenfunctions do not r e t a i n their Identity, being instead combined to f o r m n e w eigenf u n c t i o n s , better suited t o bond j o r m a t i o n . The best bond eigenfunction which can be formed from the one s and three p eigenfunctions in a given shell has the strength 2.000. Moreover, three other equivalent bond eigenfunctions can also be formed, and the j o u r bonds are directed toxard the corners of a regular tetrahedron. This result immediately gives the quantum-mechanical justification of the chemist’s tetrahedral carbon atom, with all its properties, such as free rotation about a single bond (except when restricted by steric effects) and lack of it about a double bond, and shows t h a t many other atoms direct their bonds toward tetrahedron corners.” 4 s is evident from a plot of one of these tetrahedral eigenfunctions we must not consider that in the C-H bond the electrons are actually localized in a “cigar-like” region betn-een the two nuclei. Rather, what we infer from Pauling’s deductions is that in methane the electronic charge density will tend to be greatest along those lines which are directed from the carbon nucleus towards the corners of a regular tetrahedron. While Pauling has not carried out any computation of the binding energy by use of his suggested eigenfunctions, it is evident that such a calculation would yield too low a value of the binding energy because of the failure to take into account correlation energy. For, while a part of the binding energy undoubtedly arises from the tendency for the electrons to localize bet ween the carbon nucleus and each of the four hydrogen nuclei, the rest of the binding energy must be ascribed to the fact that the electrons will also tend t o avoid each other. Slater has derived expressions for the energy of a system consisting of two, three, or four univalent atoms in each of which the valence electron is in the Cstate. The resulting expression for the binding energy involves both Coulomb and exchange integrals which are similar in nature to those occurring in the Heitler-London treatment of hydrogen. Such computa-
QTAXTLW THEORY OF T‘ALENCE
243
tions must however be regarded as only a first approxiniation to the true values, since they omit any conqideration of correlation energy terms. That the inclusion in the calculation of these latter ternis offers grave mathematical difficulties ib, of course, the main reason for the failure on the part of “theoreticians” to attempt this task. However, it should be realized that all calculations of binding energies, and energies of activation n hich are based on thii artificial device of dividing the energy into Coulomb arid exchange energy are liable to be only the roughest kind of approxiniations to the true value^.^ The present status of this whole problem has been very well described by TTanSleck and Sherman at the beginning of their comprehensive review. K e cannot do better than quote their remarks. “ T h e subject of valence is really concerned with energy relations. If n e knew the energies of all the possible different kinds of electron orbits in molecules, and also in the atoms out of which the molecule is formed, the rules of valence would automatically follow. “ S o n the principles of quantum mechanics enable one to m i t e dov-n an equation for any system of nuclei and electrons, the solution of which would provide us with complete information concerning the stability of the system, spatial arrangements of the nuclei, etc. . . . “ T h e complexities of the n-body problem are, alas, so great t h a t only for the very simplest molecule, namely Hl, has i t proved possible to integrate the Schroedinger wave equation with any real quantitative accuracy. Hence to date, anyone is doomed to disappointment n ho is looking in Diogenes-like fashion for honest, straightforn ard calculations of heats of dissociation from the basic postulates of quantum mechanics. HOK, then, can i t be said t h a t n e have a quantuni theory of valence? The ansner is t h a t to be satisfied one must adopt the mental attitude and procedure of an optimist rather than a pessimist. The latter demands a rigorous postulational theory, and calculations devoid of any questionable approximations or of empirical appeals to knonn facts. The optimist, on the other hand, is satisfied n i t h approximate solutions of the n a v e equation. If they favor, say, tetrahedral and plane hexagonal models of methane and benzene, respectively, or a certain order of sequence among activation energies, or a paramagnetic oxygen molecule he is content t h a t these same properties will be possessed by more accurate solutions. He appeals freely to experiment to determine constants, the direct calculation of vhich nould be too difficult. The pessimist, on the other hand, is eternally worried because the omitted terms in the approximations are usually rather large, so t h a t any pretense of rigor should be lacking. The optimist replies t h a t the approximate calculations do nevertheless give one an excellent ‘steer’ and a 1 ery good iden of ‘ hon things go,’ permitting the systematization and understanding of n hat would othernise be a maze of experimental d a t a codified by purely empirical valence rules I n particular, he finds t h a t a mechanism is really provided by quantum mechanirs for the Lewis electron pair bond, and for the stereochemistry of complicated oiganic compounds. I t is, of course, futile to argue nhether the optimist or pessimist is right . . . One thing is clear. I n the absence of rigorous computaSee the criticism of such calculations by A . S. Coolidge and H . 31. James (J. Chem. Physics 2 , 811 (1934)).
244
SAUL DUSHMAN A 9 D FREDERICK SEITZ
tions, it is obviously advantageous t o use as many methods of approximation as possible. If they agree in predicting some property (for instance, the tetrahedral structure of methane) we can feel some confidence t h a t the same property would be exhibited by a more rigorous solution,-otherwise none. . .
.”
IONIC COMPOUNDS
I n quantum mechanics there is no rigid distinction, as already mentioned, between homopolar and ionic compounds. Even in the case of such molecules as hydrogen and methane in which the vale ce bonds are ordinarily regarded as of the shared electron or homopolar type, it is found that the most satisfactory electron eigenfunctions are obtained as a linear combination of Heitler-London and ionic termi:. When we consider the case of a typically ionic compound, such as sodium chloride, we find that from the point of view of quantum mechanics the electronic charge distribution is not localized to nearly the extent demanded by such a model as that of Born. iiccording to the prevalent view the sodium atom in sodium chloride has lost an electron which the chlorine atom has gained. I n terms of quantum mechanics this means that there is an excess charge distribution in the neighborhood of the chlorine nucleus which corresponds to that of one electron, and that there is corresponding deficiency around the sodium atom. However, some recent calculations have shown that even in this case the bond eigenfunction is best represented by a linear combination of an ionic function (#[) and a Heitler-London function (9s) of the following form
ic
=
a+I+
v‘1
-
a2+,
(18)
where a > > d1 - a2. On the whole then, we must conclude that the strictly classical picture of ionic substances is not wholly correct, and that an appreciable fraction of the binding energy arises from electronic distributions which are the same as those met with in valence compounds. This occurs as a result of the same need for compromise between competing energy terms which we have discussed in previous sections. Additional computations on solid ionic crystals, based on a Hund-Mulliken type of approximation, also confirm the significance of this point of view. QUANTUM-MECHANICAL RESONAXCE
One of the most significant deductions which has appeared as a result of the application of quantum mechanics to atomic and molecular interactions is the existence of the resonance effect. If n-e have a system consisting of three univalent atoms in which the atoms are at approximately equal distances from each other, there are three possible ways by which a bond may be established between any pair. Since only two of these three bond-structures are independent, the eigenfunction for the qystem is
represented by a linear combination of t n o ternis, one for each of the so-called canonzcal qstrucfu/cs,and it iq found that the energy of the complex of three atoms is l o ~ \ e than r that for either of the independent structures by itself. This is generally described as being due to a sort of coupling action between the two poasible structures. From the standpoint we hare emphasized above, it may be regarded as arising from the fact that the linear combination of functions leaves room for a more intricate description of electronic motions than any single one of the functions, and hence one which is nearer the true description, so that the mean values of all energy terms are mutually lon-ered. The most interesting illustration of this resonance energy is furnished by the benzene ring and similar aromatic compounds. As is well knon.n, it was not possible by means of the classical concepts of valence to determine the correct method of assigning bonds to the carbon atoms in the benzene ring. KekulB, Claus, Ladenburg, Dewar, and others each postulated a certain structure. Now according to L. Pauling and G. 77’. Wheland (3) the actual state of the molecule is apparently to be regarded as a mixture of these different possible structures. There are five independent bond structures or canonical structures. If we represent these by $I to \Lv, it is found that the eigenfunction which represents the state of the molecule most adequately is of the form $ = a(h
+
$11)
+
b(+III
+ +IY +
$VI
where I and I1 refer to the two possible Kekul6 structures and 111, IT.-,and 5-refer to the Dewar structure. Because of the constant interaction of these five structures, the binding energy is greater than that which would be obtained for any of the strucLures alone. The difference constitutes the so-called resonance energy, and it may be regarded as contributing towards an increased stability of the system. VAX DER WAALS FORCES
The dewlopipent of an adequate interpretation of the nature of van der IT-aals forces must be regarded as another iniportant achievement of quantum mechanics. According to London there exists between any pair of atoms or molecules an attractive energy which T aries in]-ersely as R6, wherc R is the intermolecular distance. I n order to understand the origin of this attraction w e form a model of the atom in which, owing to the 1110tion of t h e electron, we niay consider the system of electron and nucleus as a dipole, of wliich both the moiiient and direction of orientation are varying continuously. This fluctuating dipole gives rise to a field, and when two atonis are made to approach, there is a tendency for the electrons in each atom to move in phaqe owing to interaction of the two fields, thus resulting
246
SSCL DUSHMAS AND FREDERICK SEITZ
in a n energy of attraction. This niotioii is, of course, closely connected with the motions which give rise t o correlation energy. From the fact that latent heats of evaporation are considerably smaller than molecular heats of formation, it is evident that the magnitude of the van der Waals energy must in general be of a different order from that of the energy exhibited in bond formation. However, from the point of view of quantum niechanics there must also occur cases in which the difference between the so-called physical type of cohesion and the chemical form of cohesion ic not a t all sharply defined, for both these forms of attraction energy are essentially electrostatic in origin. Only, in the case of bond formation the electric charge.. are so near each other that the inverse square law of attraction ic of prinie significance, while in the case of van der V7aals attraction, the individual atoms are so far apart that their interaction is more suitably represented as that between dipoles, which, as already mentioned, are not stationary, but fluctuating rapidly. Obviously, as two hydrogen atonis are made to approach each other from a n infinite distance there should be a region of internuclear separation in which the energy of attraction changes gradually from the more loose form to the valence form. So far no theory has been developed to take into account such a transition in the nature of the energy. But it i 3 possible, in the caqe of atomic hydrogen, to derive some idea of the extent of the region in which the energy changes over by gradual stages from the van der Waals to the valence form. For the van der Kaals attraction energy 1)etween tn-o hydrogen atoniz E the most accurate calculations lead to the relation >$-,
=
- 13T$'o(ao R)6
where Wo = ionization energy of H = 13.53 v.e. This relation is valid only for values ( K / a o ) ? > > 9, that i., I2 ' a o > > 2.08. Using the Morse form of potential energy function for the hydrogen niolecule and the value for the dissociation energy, D o = 4.72 \-.e. (which iiicludes zero point energy), we obtain for the valence energy the relation
E, where
p =
4.723(ep4p - 2 6 ~ ' ~Y.C. )
R - 0.74. Hence, E,
For R
=
=
- 21.23~"(2 - 4 . 3 9 3 6 ~ ~ ' ~ )
> 4 A.U., we have the relation E
=
- 4 2 . 4 6 ~ ~V.P. '~
QVANTVM THEORY O F VALESCE
247
If iiow n-e plot E and also E each as a function of R, it is found that for R 5 7.0 A.U. (approximately) E > E w , while for R > 7.0 A"., E < E w . This value of R is uiicloubtedly too great, since actual observation on the equation of state for hydrogen leads t o a maximum van der Kaals energy of 2.5 X v.e. a t an interniolecular distance of about 3.5 X.U. Furthermore, it is probably not justifiable to extrapolate the Morse curve to much larger values of R than this. But qualitatively it is evident that for values of R greater than a certain definite value, E +; must he greater than E v. That van der Waals energies of cohesion may be of the same order of magnitude as bond energies is evident froni the observation that a very large number of organic and some inorganic compouiids deconipose long before the temperature attains a value that even approaches the theoretically calculated value for boiling or sublimation at atmospheric prez-ure. Even an approximate calculation shows that the total energy of evaporation of such compounds niay be accounted for on the baqis of the quantum niechaiiically deduced relations. Recently A. Muller (1) has shown that a more rigorous calculation for the case of a normal CHz-chain molecule in a paraffin crystal actually yields values in very good agreement with t ho-e deduced from measurements of vapor pressure. For a hoiiiologou> ieries of hydrocarbons it niay be s1io1v-n that the latent heat of evaporation is proportional to the total surface of t h e molecule. I t follows that the value of the latent heat of evaporation for very large molecules must I~ecomeof the same order of iiiagnitude as the energy of some of the neaker hoiicl+, and lieiice when the liquid is heated the tendency t o break iuch bonds may exceed that for the molecules to become separated a4 indi1-idual qyqteiiiq. METALS
V-hile there is not a.niple space for a thorough dismission of the cohesivc properties of metals, it is worth mcntioniiig that these substances, n-hich were generally omitted from chemists' classifications of boiid types, may be described very satisfactorily b y the methods of quantum niochaiiics. I t turiis out that the biiidiiig energy arises from aliiiost exactly the sanic sources as the binding energy of valence compounds. The iiiiportant difference between metals aiicl valenrc coiiipouiids, from the staiitlpoiiit of cohesion, lies in the fact that directioiisl propertics of the wave fuiictiow are not stressed nearly as much in iiietal.. I n the case of the alkalies, computations have been carried to ai1 extent in which the correlation cnergy is included, and many of the remarks relating to this quantity, n-hich wcre niacle in previou? sections, are based upon an exaniiiiation of thcse cases.
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SAKL DUSHMAS I N D FREDERICK SEITZ
coscLusIos About twenty years ago it began to dawn upon chemists that there is no fundamental distinction between so-called physical and chemical reactions. Both may be treated by the same therniodyiiamical method, and such a classification is merely of historical and pedagogical interest. Similarly, a t the present time, we must realize that in all forms of cohesive energy, whether between atonis in non-polar molecules, between ions in polar compounds, betn-een the carbon atoms in diamond or the sodium atoms in metallic and liquid sodium or between the molecules of methane in liquid methane, the cohesive energy arises from electrostatic forces. There are no other mysterious forms of forces. It is true that it is possible to calculate the magnitude of this energy in only a relatively few simple cases. This arises largely because of the difficulties in calculating correlation energies for the binding of the electron
