K. A. VENKATACHALAM AND M. B. KABADI
740
VOl. 59
THE DIAMAGNETIC SUSCEPTIBILITIES OF METHANE AND THE AMMONIUM ION BY K. A. VENKATACHALAM AND M. B. KABADI Institute of Science, Bombay, India Received February $ 1 , 1066
Coulson's calculations for the diamagnetic susceptibility of methane,s based on electron pair method, have now been modified by including the ionic part of the wave function. The value 0.40 X 10-18 e.s.u. has been used for the C-H bond moment. The value so obtained has been shown t o be in better agreement with experiment, than any previously calculated value. Similar calculations have been made for the ammonium ion also, and the result obtained agrees fairly well with the experimental value.
A few attempts have been made to calculate the diamagnetic susceptibilities of molecules from purely theoretical considerations, in the case of spherically symmetrical systems such as methane and the ammonium ion, where the "temperature independent term" of Van Vleck's equation1 can be neglected. Buckingham, Massey and Tibbs2 and Coulson3 have calculated the diamagnetic susceptibility of methane, while Hartmann4 has calculated the susceptibilities of both methane and the ammonium ion. Buckingham, Massey and Tibbs have used the self consistent field method of Hartree. Coulson has employed the molecular orbital and electron pair wave functions. Hartmann has treated methane and NH4+ quantum mechanically as pseudo neon atoms using a method similar t o the Mater method for atoms. Their results are given below. Method
Value X ~ O I X 108, 0.m.u.
Self consistent field method Molecular orbital method Electron pair method Pseudo neon approximation Experimental (Bitterb) Pseudo neon approximation Experimental (Trewa)
-33.2 -26.6 -27.7 -27.4 -12 2 17 9 -13 3
-
C-H bond, on the electron pair method, requires a knowledge of the relative importance of the covalent and ionic parts of the wave function. I n this case we have to consider the structures +C-H, 3-c i;[ and
+c' fi.
Since the wave function
+ -
corresponding to the structure + C H, has energy
-+
much greater than either the covalent or the 3 C H wave functions,' we may, t o a first approxima-
+-
tion, neglect the structure 3- C H, and write for the unnormalized wave function of the bond ~(c-H)
where *cov
=
=
Icov
~?i,[*(C:l)*(H:2)
+ +
Xqion *(C:2)\k(EI:1)1
= P(c:I) I'cc:~)
*ion
and #(c)is the sp3 hybrid wave function of carbon,
uix.
Substance
Methane Methane Methane Methane Methane
KH, + "1
+
It is seen that none of the methods gives a result anywhere approaching the experimental value either in the case of methane or the ammonium ion. The chief reasons are that the self consistent field method gives a too diffuse charge cloud, the m.0. method allows for ionicity but neglects exchange, the e.p. method allows for exchange but neglects ionicity, while the pseudo-neon atom approximation does not preserve the well known individuality of the C-H and N-H bonds in methane and ammonium ion, respectively. I n this paper an attempt has been made to modify the calculations based on the electron pair method, because the description of the chemical bond based on the electron pair treatment, resembles closely the picture of a bond, visualized as formed by the sharing of the electrons of the atoms between which the bond is formed. A full description of a chemical bond such as the (1) J. H.Van Vleck, "The Theory of Electric and Magnetic Susceptibilities," Clarendon Press, Oxford, 1932, p. 275. (2) R. A. Buckinghem, H. S. W. Massey and 9. R. Tibbs, Proc. Roy. Soc. (London), A178, 119 (1941). (3) C. A. Coulson, Proc. Phys. Soc. ( L o n d o n ) ,A64, 51 (1942). (4) H.Hartmann, 2. N G ~ u I .IIA, . , 480 (1947). ( 5 ) F. Bitter, Phus. Reu., 33, 389 (1929). (G) V. C. G . Tt,ew, T r a n s . Faraday Soc., 37,470 (1941).
Nois a normalizing constant so chosen that f *2cov d71 d72
=
1
and 1 and 2 refer to the electrons forming the bond C-H. Coulson3has neglected the ionic part of the wave function, in his calculations of the diamagnetic susceptibility of methane using the electron pair method. A knowledge of the value of X is required, if this is to be included. The most direct way to determine the value of X is to use the dipole moment of the bond. If we denote by x the coordinate of any electron in the C-H bond measured from C in the direction C 4 H, and if p is the internuclear distance, the centroid of the negative charge is a t a distance 2 from the carbon atom, where =
S~(I)*~(C-H)~~I~~Z f '?'(c-~)drid72
&ov
-
=
zci Zion Sci
f X(1) * 2 c o ~ d7l d72 f ~(i)IcoyS'iondrid~2
= Sx(1)*~iond~ld~2 = f *ionWcovd~ld72
The centroid of the positive charge is at the geometric center of the bond, ie., a t a distance p / 2 from the carbon nucleus and so the resultant moment of the bond is nom p =
2e(X - p / 2 )
(7) C. A. Coulson, ibid., 38, 433 (1942).
August, 1955
DIAMAGNETIC SUSCEPTIBILITIES OF METHANE AND AMMONIUM ION
74 1
If now p is known, 5 and so X can be calculated, provided fc0,,etc., are known. Calculations based on the molecular orbital method by Coulson7 show that F for the C-H bond is 0.40 X 10-18 e.s.u. and that the direction
we get = 3.314, where Yl2 is the mean square radius of any electron forming the C-H bond, from carbon. + The mean square radius of each of the two IS of polarity is C-H. A review of the polarity of the electrons is 0.092 (Coulson3). C-H bond by Gent8 shows that most of the recent Substituting these values in the Van Vleck's evidence is in favor of a polarity directed from C + equation for the diamagnetic susceptibility of a + H, vi~., C-H of magnitude 0.40 X lo-'* e.s.u. molecule, and equating the second term to zero, Accepting this value, we get Z = 1.109 ( p is positive) because of spherical symmetry of the system, it is using p = 2.06 atomic units. The values of ZoOv, seen that xmol = -21.11 x 10-6 e.m.u. etc., can be obtained using p = 2.06 and the table of integrals supplied by C o u l ~ o n . ~They are It may be pointed out that if the polarity of the Z,,,
= 1.473 = 0.6903
Zci
- +
= 0.8529
SCi= 0.7602 Z,,, These values are slightly different from those reported by C o ~ l s o nwho , ~ uses for p the value 2.0 atomic units. I n this communication the value p = 2.06 atomic units has been employed. /3 has been taken to be 1.1 and (Y = 1.625 The value of X thus obtained is 0.9672. The value of is generally considered to indicate the percentage ionic character of the bond, the percentage ionic character being given by the expression % ionic character = 1OOh2/(1 + hz) The C-H bond thus appears to be equally as ionic as it is covalent. Such a conclusion regarding the percentage ionic character of the C-H bond does not seem to be strictly valid, as can be shown below. If we define % ionic character = MC(C-H) /P*(c-H) X 100 ( 1 ) where
C-H bond is assumed to be in the direction C-H, in accordance with the electronegativity difference between C and H, and the earlier views on the C-H bond polarity, the value of X comes out to be 2.04, but the value of xlnol =
-18.66 X
e.m.u.
Both these values are distinct improvements over those calculated so far. Although the value calculated assuming the direction of the C-H bond
- +
polarity to be C-H, i.e., in the direction expected from the electronegativity difference between carbon and hydrogen, agrees more closely with the experimental value than the one where the polarity
+ -
is taken to be in the direction C-H, the data reviewed by Gent8 regarding the direction of the C-H bond polarity does not permit us to draw a con-
- +
clusion in favor of C-H. Diamagnetic Susceptibility of the Ammonium Ion.-Similar calculations were carried out for NH4+ also. The hydrogen atoms in NH4+ are ~ ( c - H ) = 2e(S - p / 2 ) arranged tetrahedrally around the nitrogen atom P*(c-H) = e p and the N-H bond is formed by the overlapping of equation 1 reduces to yo ionic character = 100 the sp3 orbital of N and the 1s orbital of hydrogen. X2/(1 Xz) provided For purposes of the calculations it is assumed that the positive charge is situated a t the nitrogen S Z C I ) \ ~ IG) * ( H : I ) ~ ~ I = 0 atom, so that N + with 6 electrons combines with I : C ( b ' ' ~=' S = S \ k ( c : ~ ) \ k ( ~ : z ) d=7 20 4H atoms, each having one electron to form thc arid ammonium ion. .fz(1)*2(C: l)d71 = 0 A calculation similar to that carried out by C o u l ~ o n ,neglecting ~ the ionic part of the wave (8 is the overlap integral, and J X ( I ) ~ ~ ( C dT1 : ~ ) is the atomic dipole term.') The values of these terms in function, and using the values /3 = 1, a = 2.125 the present case are 0.9854, 0.6378 and 0.6993, and p = 1.889 atomic units, gives respectively; values which cannot be neglected in Xmol = -24.89 X e.m.u. comparison with X = 1.109. So, in such cases, the When the ionic part of the wave function is percentage ionic character can be interpreted only introduced into the calculations, the results obas P ( C - H ) / ~ & H ) X 100 and not as 100 Xz/(l h2). tained are as follows. N + is much more electroIndeed, as Coulson points out, the ionic part of the negative than carbon, and hence the ionic part of wave function has here actually a covalent char- the wave function is \ T ~ ( N ,1)+(~:2 ) . According to acter i n that the centroid of the negative charge is Pauling,10 the percentage ionic character of the shifted by this function toward the atom using the N-H bond is twenty, the polarity being in the - + hybrid orbital. The wave function of the C-H bond may, there- direction N-H. Using these values, the value of X is found to be 2.77, and that of fore, be written as x",+ = -13.4 X e.m.u. ~ ( c - H ) = N'[*cov 0.9672*,,,,] This value appears to be in fairly good agreement where N' is the new normalizing constant. Using with those of Trew.6a11 this expression for ~ ( c - H ) in the equation
+
) I
. E (
' € '
)
+
+
(8) W. L. G. Gent, Quart. Rev., 2, 383 (1948). (9) C. A. Coulson, Proc. Cambridoe Phil. Boc., 88, 210 (1042),
(IO) L. Pauling, "Na%ureof 'the Chemmal Bond," 2d ed., Cornell XJa'iversity Press,Ithsca, N. Y . , 1940,p. 7'2. (11) V. C . G. Trew. T r a n s . Faraday Soc., 46, 217 (1949).
