ELEMENTS of the QUANTUM THEORY' X I . SLATER-PAULZNG THEORY OF VALENCE BONDS SAUL DUSHMAN Research Laboratory, General Electric Co., Schenectady, New York GENERAL REMARKS
T
HE HL theory of formation of the Hz molecule leads to the general conclusion that an electronpair (homopolar) bond is formed by the interaction of an unpaired electron on each of two atoms. The energy of the bond is largely due to the fact that the two electrons of opposite spins are constantly interchanging places, and may be calculated from a knowledge of the exchange integrals which are derived from the corresponding electron wave functions (orbitals). While Heitler and London showed that from this point of view it is possible to account for the observed valences of the elements of the first two periods,t little or no attention was paid in this early work to the problem of directed valence bonds. The first successful darts to treat this topic by the methods of quantum mechanics were those published by J. C. Slater' and, independently of the latter, by L. Pauling in a series of papers published since 193L2 Further investigations on the structures of specific molecules, such as Hz0 and CH4have also been published by J. H. Van Vleck,f as well as by W. Heitler, E. Hiickel, and G. Rumer.** On the whole, the method used by L. Pauling, especially in his first papers on this topic, is more "physical" in the sense that i t relies to a large extent on a geometrical interpretation of the significance of electron eigenfunctions in the formation of bonds. While some of the conclusions reached by Pauling may not prove to be sufficiently well founded, his method of attacking the problem of valence bonds is certainly extremely suggestive, and it is largely for this reason that the topic has been discussed in the following sections. Slater's method is more quantitative, inasmuch as it leads to approximate methods for calculating both the direction and energy of bond formation. For this reason. however, it resents meater mathematical diffi' This is the fifteenth and concluding anicle of a wries presentct ing a more detailed indextended treatmcnt of t h c ~ u h ~ ematter covered in Dr. Dushman's contribution to the svmwsium on Modernizina the Course in General Chemistry, conducted by thc Dinsicln of Chernical Education at t h r r i c h t y - w ~ h r h merung of the American Chemical Society, Clevclnnd, Ohio. Pcprembcr 12, 1934 l'hc author re,rrves thc rieht to ouhlication irt hook fonn. -~ t See discussion b y J. H. VAN-VLE& AND A. SHERMAN, Rev. Modern Pkys., 7,167 (1935). especially pp. 196-7. The abbreviation V. V. S. will be used in subsequent references. This topic has also been discussed bv the author in "Treatise on ~ h v s i c a l & ch&istrv." b y H. S. T A ~ O R . Van Nostrand Co., ~ e kork, 1930, vdl. 2, pp. 1369-72. 1 J. C. SLATER, Phys. Rev., 37, 481; 38, 1109 (1931). a L. P A ~ I N O 3.. Am. C h . Soc.. 53. 1367 (1931). ~
culties and the calculation becomes extremely tedious when dealing with polyatomic molecules. While Slater, Pauling, Eying, and other investigators have developed "short-cut" rules by which roughly quantitative results can be deduced without too much labor, it is possible, in this chapter, to touch only upon the simplest aspects of the methods of solution used by the different investigators. PAULWG'S TREATMENT OF HOMOPOLAR BONDS
In his iirst paper, Pauling introduces the following six postulates which are to be used as a guide in the determination of relative energies and directions of different bonds in molecule formation. The first three are merely a restatement of the HL theory: 1. "The electron-pair bond is formed.through the interaction of an unpaired electron on each of two atoms. 2. "The spins of the electrons are opposed when the bond is formed, so t h a t they cannot contribute to the paramagnetic susceptibility of the substance. 3. "Two electrons which form a shared pair cannot take part in forming additional pairs."
To these Pauling adds three more rules "which are justified by the qualitative considerations of the factors influencing bond energies." 4. "The main resonance (exchange) terms for a single eketron-pair bond are those involving only one dgenfunction from each atom. 5. "Of two eigenfnnctions with the same dependence on r, the one with the larger value in the bond direction will give rise to the stronger bond, and for a given eigenfunction the bond will tend to be formed in the direction with the largest value of the eigenfunction. 6. "Of two eigenfunctions with the same dependence on 0 and+,ttt the one with the smaller mean value of r, that is, the one corresponding to the lower energy level for the atom, will give rise to the strower - bond."
Let us consider the application of these rules to the determination of bond directions in such molecules as HzO and NH3. According to the Lewis-Langmuir theory of valence these molecules are represented as H :0 :H and H :N..:H, respectively. The electron in a
H
normal hydrogen atom is in the 1s (n = 1, 1 = 0) state. The electron configuration of N and 0 in the normal state are ( 2 ~ ) = ( 2 pand ) ~ ( 2 ~ ) ~ ( Z prespectively. )~, The 2s electrons are paired and therefore do not take part in bond formation (except when a change occurs in quantization of the electron eigenfunction owing to bond formation, as will be discussed subsequently). Hence the electrons which act in bond formation in the
case of N and 0 are of the type 2p (n = 2, 1 = 1). Now, as Pauling points out, s and p eigenfunctions with the same value of n, "do not differ very much in their mean values of 7, but their dependence on 0 and $ is widely different."
electrons will form stronger bonds than s-electrons and that the bonds formed by p-electrons in an atom tend b be oriented at right angles to one another." Van Vleck and Shermant state the argument for this conclusiou as follows:
For s eigenfunetians, +,(r, 8, 6 ) = S,(r).s(B, For p eigenfunctions, IL.I(r, 8, 4) = S.,(r)M8, 4).Pr+(8, 6) .pr-(8, m)
"Let us suppose there is an electron-pair bond hetween an s electron of some attached atom and the p~zel.eleclronof the central atom. Then the exchange enerm associated with this oarticular pair is greatest if thdattached atom lies an the s &is. since the exchange integrals will clearly be largest in absolute value if the wave functions of the two atoms overlap as much as possible. This requirement clearly demands that the attached atom be located on the axis of the dumb-bell associated with the particular electron of the central atom with which it is paired. "If a second atom is brought up, and if the pairing between p, and the first attached atom is not broken, then clearly the only possibility is for the second atom t o pair with one of the other wave functions, p, or p., so that i t will become located on the y or z axis. Hence in a molecule such as HSOthe angle between the two OH axes should be 90". The exoerimental value is 10fiO. The departures from 90° are to be blamed upon repulsions between the attached atoms anduponspahybridization. Similarly, if the first two atoms have preempted the z and y directions a third atom tends to become located on the z axis, so that in a molecule like NHa the three N H axes should make angles of 90" with each other. The NHI molecule is then pyramidal in structure, each axis making an angle of 51.7' with theaxis of the NHI pyramid. The experimental value is 67-, and the discrepancy is t o be attributed t o the same causes as in HnO."
6)
The s-eigenfunction is symmetrical, and from Table 2, Chapter VI,3 it is seen that the normalized function has the form s(S. +) = Xoo(@.Zo(6) = 1/d4T
There are three p-eigenfunctious:
m)
p,+(S,
PA8, Q) = Xd8)Zo(+) = .cos 8 6) = X,,(E)Z,(+) = v'3/(4r).sin 8&/.\/Z
. sin e.sCi+/.\/Z
fi-(S, 6) = XI. - I ( B ) Z _ I ( ~= ) )-
Since p,+ and h- are complex, i t is desirable to replace them by real functions which can be represented as functions of the rectangular coordinates x, y and z. Replacing c"'+ by (cos 4 * i sin $), we obtain the real functions: p,'= f l . s i n 8 . c o s 6 = fl.sine.sin 6 pz = f l cos e
h
(486)
.
~
as compared with the eigenfunction where the factor 1 / d G has been discarded, since we are interested only in relative magnitudes. It will be observed that pz is identical with p. and that 9.' Aa Ps2 = PT+' PW-' 9.'.
+ +
+
+
For $ = 0, p, = fi sin 0 ; and p, = 0. Thus, fi sin 0 represents a section of the spherical function p, in the xz plane, in which i t has its maximum value for any given value of 0. As shown in Figure 63 this function is represented by two circles in contact a t the origin and each of diameter fi units, as compared with the s-eigenfuuction which, on the same scale, is represented by a circle of unit radius.* Thus I p, 1 consists of two spheres, with the long axis in the direction of the x-axis. The similar functions 1 p, I and / p. I have their long axes in the direction of the y- and z- axes, respectively. In Figure 35, Chapter VI,I the distribution functions are shown corresponding to these three p-eigenfunctions. "From Rule 5," as Pauling observes, "we conclude that pa J. CBEM.EDUC.,12, 534 (Nov.. 1935).
*This follows from the following simple consideration: For any point on the circle, the distance from the origin is Let D denote the diameter of each given by r = .circle. Then i t follows from the properties of any triangle inscribed in the circle on D as a base, that
/sin 8 = D 4
= r/D.
J. CHEM.EDUC., 12, 535 (Nov., 1935).
63.-SFIGURE
AND P-EIGENPUNCTIONS (PAULING)
CHANGE IN QUANTIZATION OF BOND EIGENFUNCTIONS
In the case of a normal carbon atom, with the electron configuration 2s22p2 (known spectroscopically as SP state) there are only two unpaired electrons, and this would account for the double bond in C : : O . Only about 1.6 v.e. of energy (36,900 cal./mole) is required to excite one of the 2s electrons to a 2p state, and this would give three unpaired p electrons and one unpaired s electron, thus accounting for the valence of four as in CHI. But, since the s-bond is not a specially directed one, this answer cannot be sufficient. A much more satisfactory point of view is the introduction by Pauling of a new concept-that of "hybridization" of eigenfunctions to form combination functions which take the place of the single electron functions. Thus, in the case of the formation of CHI, if the energy of interaction per H-atom is greater than the difference in energy of the electron in the 2s and 2p states, then the interaction will cause the electron to be "promoted and now we must consider each bond as being formed
t V. V. S., loc. cit.. p. 199.
by the grouping together of hydrogen-like s and p eigenfunctions. As Pauliig points out this criterion is satisfied for qnadrivalent carbon, and he therefore proThe negative sign is due to the fact that for this bond ceeds to determine the zeroth-order eigenfunctions cos 6 must be equal to -1 as compared to the value "which will form the strongest bonds for the case when -I-1 for in the xz plane. the s-p quantization is broken." He assumes that the Applying the condition fpr normalization and the four bonds are each represented by one of the four com- additional requirement, bination eigenfunctions. 9;= ais
+ biA + ciP, + &A
(487)
where i = 1, 2, 3, or 4, and the coefficients are subject to the orthogonality and normalization requirements,*
and
Si
9;YiY/dr= 0 or aiax
+ bibn-+ cicx + Adr = 0
it is readily shown that This will have a maximum value M = 2 for definite values of 6 and b2, such that d%/db = 0 and d%/d6 = 0. From these conditions it follows that
(489)
where Z k, and i, k = 1, 2, 3, 4. For a single bond, we can choose the direction arbitrarily. If we take it along the x-axis, for which p? = p, = 0, the corresponding eigenfunction has the form
sin 8 cos
+=
-1/3;
+ = 180°, 0 = 109" 28'.
That is, the second bond eigenfunction makes an angle of 109' 28' with the first, "which i s just the angle between the lines d r a m from the center to two corners of a regular tetrahedron." The actual expression for the bond function has' the form,
The maximum value of this function is evidently (where the subscripts may he discarded). Also since a 2 b2 = 1
+
M = W + b f i . Introducing the condition d M / d b = 0, it follows that
By similar methods it may be shown that the third and fourth eigenfunctions are 1 1 1 ua=-s-,Pr+-?"----PI 2
243
4
1 46
(492)
and
and
This has a maximum value M = 2, which is considerably greater than the value 1.732 for a pure p-eigenfunction. Figure 64 shows a graph of this function in the xz plane.
A second bond function may be introduced in the same ulane, of the f o m
* This point is discussed in a footnote in the paper by V. V. S., p. 202.
which are directed toward the other two corners of the tetrahedron. Pauling also points out that an equivalent set of four tetrahedral eigenfunctions is
which differs from the previous set by a rotation of the atom as a whole. "This calculation," as Pauling remarks, "provides the quantum mechanical justification of the chemist's tetrahedral carbon atom, present in diamond and all aliphatic carbon compounds," as well as for a number of other tetrahedral atoms and ions. Furthermore, since "each of these tetrahedral bond eigenfnnetions is cylindrically symmetrical about its bond direction, the bond energy is independent of orientation about this direction, so that there will be free rotation about a bond." On the other hand, there can be no free rotation aboul a double bond.
The reader will be well repaid by studying further Panling's original paper upon which the discussion has been based, as well as his subsequent papers. There is a wealth of material there which is of great signilicance for the quantum mechanical interpretation of directed valence bonds.
The problem which Slater has attacked is that of calculating the electronic energy states of molecules, from a consideration of the interaction of the atomic orbitals. The calculation is essentinllv an extension to three or more atoms of the method of secular equations discussed in Chapter X, Part 11. Starting with oneelectron wave functions which involve coordinates both of position and spin, zero-order wave functions are built up which are antisymmetric in the electrons (in accordance with Pauli's principle). Let *I, *2 . denote such combinations. If H i s the Hamiltonian operator of the problem, then the secular equation to be solved for the energy is of the form, H,, - Wd,,, H,,- Wd,,, . . . HI
