Explicit calculations are made to examine whether the tetrahedral model of CH4 is really the most stable in the Hund‐Mulliken and Slater‐Pauling theories. The secular determinant involved in the H‐M procedure is too complicated (degree 8) to solve generally, but three methods of approximation are given applicable in three different limiting cases, in all of which the tetrahedral model proves to be that of least energy quite irrespective of repulsions between the H atoms. The fact that two of the methods do not assume the carbon s‐p separation to be small shows that s‐p hybridization is not a necessary condition for tetrahedral valence bonds. Calculations are also given which show that in the Heitler‐London‐Pauling‐Slater method the regular tetrahedron is superior to other models of somewhat lower symmetry. It is further calculated according to both the H‐M and H‐L‐P‐S schemes that in compounds of the form CH2X2, CHX3, and CH3X, the most stable models are tetrahedra of less symmetry than the regular tetrahedron; i.e., models with the valence angles somewhat distorted from 109.5° unless the C – H and C — X bonds are of equal intensity. The predicted directions of the deviations from 109.5° are in agreement with x‐ray diffraction data for CH2Cl2 and CHCl3. The conclusion is rather prevalent in the literature that with s‐p hybridization and electron pairing two bond axes tend to set themselves at 109.5°. This is shown incorrect; instead the angle can be anything between 90° and 180° depending on the relative intensities of the s and p bonds. If the s bonding power is not negligible, the angle between an NH axis and the pyramidal axis in NH3 should be somewhat greater than the value 54.7° which is obtained if the three NH axes are orthogonal and which is characteristic of pure p‐bonding. Actually Dennison and Uhlenbeck find 68°, and Lueg and Hedfeld 73°, from band spectra. It is shown that CH4+ should be a flattened rather than regular tetrahedron, conceivably even being plane. Also CH3 should be a flatter pyramid than NH3.

1.
J. H.
Van Vleck
, Part I,
J. Chem. Phys.
1
,
177
(
1933
).
2.
See Part I for references to the literature.
3.
Cf.
Heitler
and
Rumer
,
Zeits. f. Physik
68
,
12
(
1931
) and discussion to be given in Part III.
4.
In using (6) it must be mentioned that to specify a real π wave function completely, it is necessary to specify the phase ε in its azimuthal factor cos(φ−ε). In (6) the value of ε is zero if the apse line for measuring φ is taken in the plane x−C−i. In calculating the values of the elements r in (4), use must be made of the fact that if ψπ,ψπ denote π wave functions of C with different phases,
∫∫∫ψπψπVHidυ = −Cπcos(ε−ε′)
.
5.
The matrix elements of S−1 needed for computing S−1KS are found by noting that (8) gives nearly a unit matrix, so that
S−1 = 1−(S−1)+(S−1)2−(S−1)3+⋯
.
6.
If we set T = 0, the root sum Y which interests us when WC−WH≫Q is Z−X where Z is the spur of the complete matrix (4) and X is our previous root sum. The value of Z is independent of the angles, and hence Y has an extremum when X does. When WC−WH≫Q the tetrahedral arrangements gives a minimum of Z−X and a maximum of X rather than the reverse as previously. This is because sign changes in τ and in certain of the integrals (3), occasioned by the electronic drift being from C to H rather than from H to C, make the expressions β, γ defined in (19, 20) become negative.
7.
For the regular tetrahedron, which is a special case of the models of class (a) or (c), and which has the most symmetry of all, none of the approximations (I), (II), (III) are necessary, and an exact expression can be derived for the energy, which was given in Eq. (4), Part I. The constants employed in Part I have the following significance in terms of the integrals (3), etc.
a+3b = 4Cs+WsC, a−b = 43Cσ+83Cπ+WpC
,
c+3e = 2(−Qs+TsWsC), c−e = −(43)12(Qσ−TσWpC)
,
d+3f = −2Ts, d−f = −(43)12Tσ, g∼0, h∼0, k = WH+B
.
8.
Most commonly the atoms X will be halides, in which case they will be in P2 rather than in S2 states like hydrogen. Despite this difference, the analysis is still applicable if we suppose that in the halide’s incomplete shell ms2 mpπ4 mpσ, the axis of mpσ is directed towards the central atom C and if we neglect the bonding power of ms and mpπ. Such a choice of the axis of mpσ is clearly demanded for overlapping favorable to bonding. With these assumptions, only the 2pσ electron of the halide is involved in the secular problem (4), and the integrals (3) obviously are to be calculated for an mpσ rather than 1s state of X. Also no harm is done if the letter H similarly symbolize some atom other than hydrogen and not necessarily in an S state. Thus the conclusions of our calculations also apply to compounds such as CCl4,CCl2Br2,CCl3Br, as well as CH4,CH2X2,CH3X.
9.
The assumption x2+z2 = x′2+z′2 is, however, always needed for the simplification P = Q which we make in connection with CH4.
10.
G. N. Lewis, Valence and the Structure of Atoms and Molecules, Chem. Cat. Co., 1923.
11.
L.
Pauling
,
J. Am. Chem. Soc.
53
,
1367
(
1931
).
12.
Essentially this point has also been noted in an interesting paper by
R.
Hultgren
,
Phys. Rev.
40
,
891
(
1932
). His wave functions are more general than ours in that d states are admitted to the linear combinations, but less general in that all the functions are supposed to have the same hybridization ratio. Eqs. (33) and (34) each involve two such ratios. Four would be required for CWXYZ. Presumably the number of different ratios required is the same as the number of different kinds of atoms attached by C. If this is the case then all the C wave functions of CH4 have the same ratio and the tetrahedral model follows uniquely simply from the requirement of orthogonality. Nevertheless it seems of some interest for us to verify that introduction of two unequal ratios in place of one raises the value of (35) in CH4.
13.
Cf. for instance
J. C.
Slater
,
Phys. Rev.
34
,
1307
(
1929
). His K is our J.
(a) See, for example the table given by
Coolidge
,
Phys. Rev.
42
,
198
(
1932
).
14.
J. C.
Slater
,
Phys. Rev.
41
,
255
(
1932
).
15.
Lueg
and
Hedfeld
,
Zeits. f. Physik
75
,
559
(
1932
);
D. M.
Dennison
and
G. E.
Uhlenbeck
,
Phys. Rev.
41
,
313
(
1932
)
or
N.
Rosen
and
P. M.
Morse
,
Phys. Rev.
42
,
210
(
1932
); these determinations are presumably more accurate than older spectroscopic work which gave values of θ in the neighborhood of 60°.,
Phys. Rev.
16.
L.
Bewilogua
,
Phys. Zeits.
32
,
265
(
1931
);
P.
Debye
,
Zeits. f. Electrochemie
36
,
612
(
1930
).
17.
R.
Mecke
and
W.
Baumann
,
Phys. Zeits.
33
, (
1932
).
18.
It seems quite probable that the functions (42) and first two of (34) are the proper wave functions for C double bonds, rather than the tetrahedral sp3 functions assumed by Pauling and Slater. At any rate it is certain that the s‐p hybridization ratios for double bonds will not be the same as for CH4. We suggest this combination of (42) with (34) because it gives three directed coplanar valences, viz., those corresponding to ψaC,ψbC,ψd. The C–C bond is here to be located along the z axis, so as to bisect X–C–X. This subject of double bonds will be discussed more fully in a paper by W. G. Penney.
19.
This and shortly following formulas for the bonding energy are obtained by application of (35) and (6), with allowance for the finite s‐p separation in the same fashion as in connection with (41). The addition of −2Nss and the omission of f(ω′) are consequences of the fact that in (38) those terms containing Nσσ,Nσs,Nππ which arise from interaction of the third and fourth electrons of C with X2, are proportional to f(ω′), while the corresponding terms containing Nss are proportional to 1−f(ω′). Such terms must be dropped and new terms added because of the substitution of new wave functions in place of the third and fourth of (34).
20.
For instance when P = Q,ω = ±ω′, one finds W1+2W2<W1+2W2 only if Qσ2cos2ω>3(WH−WC)2. Here W1,W2 and W1,W2 are levels for certain spatial and plane models as given in (24), (28). The remaining roots (25) are common to both models.
21.
Only five electrons are involved in this sum because two are absorbed by the deep 2s states.
22.
Eq. (44) would also give the best value of θ for NH3 in the H‐L‐P‐S method were it not for the fact that the energy of NH3 involves the additional terms (41). The term proportional to Wp−Ws in (41) will tend to make θ smaller in NH3 than in CH3.
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