CALCULATION OF THE GEMINAL COUPLING CONSTANT
1761
A Calculation of the Geminal Coupling Constant Based upon the Dirac-Van Vleck Vector Model1
by Harry G. Hecht University of California, Los Alamos Scientific Laboratory, Los Alamos, New Mexico (Received October 88,1966)
8764.6
The geminal proton-proton coupling constant is calculated using the Voge formulation of the methane problem. This is more general than the previous calculations in that valence-state promotion is considered explicitly by configuration interaction with the s2p2and p4 atomic states of carbon. The energy matrix is solved for four different sets of empirical integrals. It is found that although the calculated coupling constant is still positive, it does not depend so critically on the precise values of the integrals used as was the case in former calculations.
I. Introduction The early calculations of nuclear spin-spin coupling constants by the valence bond method appeared remarkably but at present these results are regarded by most workers as somewhat fortuitous. Although the valence bond approach continues to be used for the correlation of some effects,2fit is generally viewed with much skepticism, since it has been shown to predict positive constants for both geminal and vicinal couplings,2bvCwhereas the recent experiments show that they are of opposite relative sign,s with the geminal coupling presumably the negative one. It is supposed that there is no fundamental difficulty with the methodJ4but that the approximate nature of the molecular wave functions is responsible for the discrepancy. More specifically, Karplus5 has pointed out that a near cancellation of the exchange integrals, p and y (see notation of Section 111), leads to a geminal coupling constant nearly proportional to 6.
J H H‘V~ X(26
- fl - 7)
(1)
6 is not known with certainty2‘ and it has been shown
that the calculated geminal coupling constant can be changed by more than f30 cps by varying the exchange integrals within reasonable limits.0 Thus it would appear that the valence bond method is severely limited by the inaccuracies inherent in the evaluation of the necessary exchange integrals. The molecular orbital calculation of coupling con-
stants had a much different genesis. The initial calculations did not look very promising.’ Here again only positive contact couplings were calculated and it seemed that the difficulty in determining the extent of configuration interaction would seriously limit the method. The delocalized molecular orbital method of Pople and Santry* does allow for both signs, but again cancellations occur which preclude any quantitative calculations.
(1) Work performed under the auspices of the U.5. Atomic Energy Commission. (2) (a) E. Aihara, J. Chem. Phys., 26, 1347 (1957); (b) M. Karplus and D. H. Anderson, ibid., 30, 6 (1959); (c) M. Karplus, ibid., 30, 11 (1959); (d) H. 9. Gutowsky, M. Karplus, and D. M. Grant, ibid., 31, 1278 (1959); (e) S. Alexander, ibid., 34, 106 (1961); (f) M. Barfield and D. M. Grant, J. Am. Chem. SOC.,83, 4726 (1961); 85, 1899 (1963); J . Chem. Phys., 36, 2054 (1962). (3) R. R. Fraser, R. U. Lemieux, and J. D. Stevens, J . Am. Chem. SOC.,83, 3901 (1961); R. R. Fraser, Can. J. Chem., 40, 1483 (1962); F. Kaplan and J. D. Roberts, J . Am. Chem. Soc., 83, 4666 (1961); C. A. Reilly and J. D. Swalen, J. Chem. Phys., 35, 1522 (1961); R. Freeman and K. Pachler, Mol. Phys., 5, 85 (1962); H . S. Gutowsky and C. Juan, J . Chem. Phys., 37, 120 (1962); F. A. L. Anet, J . Am. Chem. SOC.,84, 1053, 3767 (1962); P. C. Lauterbur and R. J. Kurland, ibid., 84, 3405 (1962). (4) M. Barfield and D. M. Grant in “Advances in Magnetic Resonance Spectroscopy,” Vol. I, J. S. Waugh, Ed., Academic Press Inc.. New York, N. Y., 1965. (5) M. Karplus, J. Am. Chem. SOC.,84, 2458 (1962). (6) M. Barfield and D. M. Grant, Abstracts, 144th National Meeting of the American Chemical Society, Los Angeles, Calif., April 1963. (7) H. M. McConnell, J . C h m . Phys., 24, 460 (1956). (8) J. A. Pople and D. P. Santry, Mol. Phys., 7,269 (1964); 9, 301, 311 (1965).
Volume 71, Number 6 May 1067
HARRYG. HECHT
1762
It is probably fair to say that on the whole the valence bond method has been favored for the interpretation of coupling constants, because of the intimate way in which the pairwise coupling of spins is incorporated. The same can of course be said for the Dirac-Van Vleck vector modeL9 Because of the conceptual simplicity of this approach, it seems that calculations based upon a more complex model, in an attempt to better understand the nature of its limitations, can be justified.
vector model need not be repeated here; Voge12 has solved the problem using the elegant method of treating permutation degeneracy developed by Serber. l 4 It is found that the 7 X 7 problem involves three states of spin-paired configuration (A), t a t b t c t d , as previously mentioned, together with three states with one orbital occupied by two electrons (B), t a ' t b t c , and one state with two filled orbitals (C), ta2tb2. Voge shows that this is tantamount to consideration of the carbon atom configurations, sp3,s2p2,and p4.
11. The Model It might be supposed that it is better to evaluate the
111. Calculations and Results
required molecular integrals theoretically, because of the inherent difficulties in estimating them empirically with sufficient accuracy to make meaningful calculations. A problem is encountered here, however. Although wave functions have been tabulated for most of the atomic states of carbon,'O any quantitative calculation would require the wave function for the socalled valence state.g-l 1 Thus, even Slater orbitals may be mort: accurate for the calculation of the molecular integrals than more exact solutions, because of the way in which an empirical account is taken of overlap and valencestate promotion. Because of such considerations, empirical integrals probably offer the best approach and other factors must be sought in order to achieve the necessary improvement in the molecular wave function. It seems that although some account is taken of valence-state promotion by use of empirical integrals, a model which treats it explicitly is to be preferred. In the usual valence bond treatment, only the quadrivalent sp3 configuration is considered. A more exact calculation involves interaction with other configurations, particularly the divalent s2p2ground state of the carbon atom. This problem has been considered by Voge,12 who shows that the AI states of Eyring, Frost, and Turkevich13 can be regarded as providing the three spin-paired basis functions out of a set of seven functions for a complete calculation. The calculation of the geminal proton coupling in methane by Karplus and Anderson2b used just these three symmetry-adapted functions. As Voge points out,12 the energy is not greatly changed by resonance with the other atomic states, but the promotional energy of the valence state and the wave function are altered considerably. Of course, just such changes are of utmost significance in the nuclear spin-spin coupling problem and one might hope that the critical dependence upon 6 might be minimized, although there is probably no a priori reason for optimism on this point. The fornulation in terms of the Dirac-Van Vleck The Journal of'Physical Chemistry
Following Voge,12 we use the following notation for the molecular integrals in terms of which the energy matrix is expressed
HI**
=
(AIHIA)
HI^*
HIBB= (BIHIB) QI
= (taha; hntn)
P
=
(hahb; htha)
7 =
(tntb;
6
(tatb; hbta)
=
HICC = (CIHIC)
tbta)
= (BIHIA)
e
=
K
= (tahc; hob)
(tatb;
tctJ
= (tatb;
tatc)
(tatb;
tctd)
P =
(2)
17 = (taha; hatb)
where (tahn; hat,)
E
SSta(l)ha(2)Hha(l)ta(2)d?ld?2 (3)
etc. Some of these integrals are purely atomic quantities (7, e, A, p ) and can be expressed in terms of the Slater parameters. l5 The remaining integrals require a knowledge of the molecular properties. Our calculations were performed for the following four sets of integrals. Set I uses Beardsley's formulas16 for F0(2s, as), F0(2s, 2p), and F0(2p, 2p), together with the values of GI, Fz, and I(2p) - I(2s) derived by Van Vleck" by fitting to the sp3 atomic energy levels. The molecular integrals assume N , , = 1.91 ev, N,, = 2.0 ev, N,, = 2.2 ev, N , , = -0.6 ev (see Van Vleck" for the definition of these integrals in terms of which a, 6, 7, and K (9) J. H . Van Vleck and A . Sherman, Rev. Mod. Phys., 7, 167 (1935). (10) A. Tubis, Phys. Rev., 102, 1049 (1956). (11) J. H. Van Vleck, J . Chem. Phys., 2, 20 (1934). (12) H. H. Voge, ihid., 4, 581 (1936). (13) H. Eyring, A. A. Frost, and J. Turkevich, ihid., 1, 777 (1933). (14) R.Serber, ibid., 2, 697 (1934); 3, 81 (1935). (15) J . C . Slater, Phys. Rev., 34, 1293 (1929). (16) N. F. Beardsley, ibid., 39, 913 (1932). (17) J. H. Van Vleck, J . Chem. Phys., 2, 297 (1934).
CALCULATION OF THE GEMINAL COUPLING CONSTANT
~
~~~~~~~~~~~
1763
~
Table I: Results of Calculations of Geminal Contact Spin-Spin Coupling Interaction of Protons in Methane, Using the Four Seta of Integrals Discussed in the Text (All Integrals are in Electron Volts and J E H ‘is in Cycles per Second) HI”*
Set
I
I1 I11 IV
HxBB
0,745-+Z 0.7744-z 9.774-tz 9.774 3-2
Hicc
12.489fz 12.8444-2 12.844fz 12.844fz
15.233-b 15.5144-2 15.5144-2 15.514+~
HxBA -2.487 -2.472 -2.472 -2.472
B
a -3.804 -3.804 -3.091 -3.989
-1,080 -1.060 -1.060 -1,080
are expressed), which are the values found by Voge18 by fitting to the heat of sublimation of graphite, 170 kcal.l9 These values closely parallel those used by Karplus and Anderson,2b but they do not contain the correction for zero-point vibrations. Set I1 uses Ufford’s atomic states20 for F0(2s, 2s), F0(2s, 2p), and F0(2p, 2p). They are perhaps better than those derived by Beardsley,l6since they are based on SCF functions for carbon given by Torrance.21 G1,Fz, and I(2p) - I(2s) were taken from Voge’s re examination18of the methane problem and the same N values were used as in set I. Set I11 is the same as set I1 for the atomic integrals, but the set NSa = 2.0 ev, N,, = 2.3 ev, N,, = 1.0 ev, N , , = -0.6 ev, derived by Van Vleck,” is used for the molecular integrals. Set IV is the same as set I1 for the atomic integrals, but the set NBa = 2.0 ev, Nu, = 2.2 ev, N , , = 2.1 ev, N , , = -0.6 ev, derived by Penney22by fitting to the bending vibration of methane, is used for the molecular integrals. In each case, 6 = -1.06 ev is used, which is 0.9 of the Morse function for H2 for the proton separation appropriate to methane. The term =
Mm
- ‘/&fur - 2/aMrr
(4)
also enters the energy matrix, where the M values are defined by
These integrals were evaluated using the screening constants for the carbon 2s and 2p orbitals given by Hartree23 and the tables of Kotani and Amemi~a.~‘ It was found that 2 = 0.064 ev. Since the average energy approximation has been well justified for this case,25we used the following equation for the contact coupling constant.2b JHH~=
&(&)( l6n@6 7) X yH2
Here
PKgt
(
+ 1)I )
8(rKH)b(fKH’)(2PKK’
*O
IK,Kt
*O
@)
is an electron transposition operator and
r 0.901 0.336 0.338 0.338
d 0.268 0.288 -0.003 0,323
P
-0.471 -0.490 -0.499 -0.499
7
-0.501 -0,501 -0.214 -0.556
e -0.067 -0.007 -0.097 -0.097
x
-0.332 -0.332 -0,803 -0.277
JHH’ 10.8 15.7 18.8 15.8
@O is the ground-state electronic wave function. The assumptions involved in the actual calculations are similar to those used by Karplus and Anderson.2b The necessary spin functions for the 14 coupling schemes are not given by Voge,12but they can easily be written using Clebsch-Gordon coefficients. The results of the calculations are summarized in Table I.
IV. Conclusions The techniques used t o evaluate the integrals mentioned in Section I11 are those that have been frequently employed by various workers and it is felt that the ranges of values for these integrals listed in Table I are fairly representative of the uncertainties inherent in an empirical evaluation of them. It will be observed that the lack of precision in many cases is quite considerable. The calculated proton spin-spin coupling constants compare favorably in magnitude with the experimental value of 12.4 and are quite constant considering the magnitude of the changes in several of the integrals. By comparison, eq 1 (with X = 30 cps) leads to values 20.9, 41.9, 21.5, and 41.1 cps for sets I, 11, 111, and IV, respectively. Thus, the critical dependence on 6, 6, and y implied by eq 1 is a t least in part an artifact of the simplified perturbation approach2d and it does appear that more complete calculations based on empirical estimates of the integrals can lead to more stable results. It will be noted that the geminal coupling constants calculated here are still of what appears to be the (18) H. H . Voge, J . Chem. Phys., 16, 984 (1948). (19) W. A. Chupka and M. G. Ingram, J. Phys. Chem., 59, 100 (1955); F. H. Field and J. L. Franklin, “Electron Impact Phenomena,” Academic Press Inc., New York, N. Y., 1957. (20) C. W. Ufford, Phys. Rev.,53, 569 (1938). (21) C. C. Torrance, ibid., 46, 388 (1934). (22) W. G. Penney, Trans. Faraday SOC.,31, 734 (1935). (23) D. R. Hartree, “The Calculation of Atomic Structures,” John Wiley and Sons, Inc., New York, N. Y., 1957. (24) M. Kotani and A. Amemiya, PTOC.Phys-Math. SOC.Japan. 22, 1 (1940). (25) M. Karplus, J. Chem. Phys., 33, 941 (1960). (26) M. Karplus, D. H. Anderson, T. C. Farrar, and H. S. Gutowsky, ibid., 27, 597 (1957).
Volume 71,Number 6 M a y 1067
WISHVENDER K. BEHLAND JAMESJ. EGAN
1764
wrong sign. Although we still cannot rule out the possibility that integral variations are responsible for this discrepancy, the improved stability of the above calculations with respect to such variations suggests that this factor is not as significant as previously thought. Thus, the neglect of other factors, such as ionic terms and multiple exchange integrals, may be quite important. Ionic terms are not easily included in the usual formulation of the theory, but could be incorporated in a manner similar to that reported by Craig.*' It is very unlikely that this would be fruitful, however, in view of the relatively small changes in
couplings calculated by Hiroike28 and Ranft,2Qwhere ionic character was introduced in a more empirical way. Multiple-exchange interactions are almost always excluded] apparently in many cases for no better reason than that they are difficult to calculate. They probably are worthy of consideration, even if they could only bc dealt with in a very approximate way. ~~
~
~
(27) D. P. Craig, Proc. Roy. SOC.(London), A200, 272, 390, 401 (1950). (28) E. Hiroike, J . Phys. SOC.Japan, 15, 270 (1960); Progr. Theoret. Phys. (Kyoto), 26, 283 (1961). (29) J. Ranft, Ann. Phy8ik., 8, 322 (1961); 9, 124 (1962).
Transference Numbers and Ionic Mobilities from Electromotive Force Measurements on Molten Salt Mixtures'
by Wishvender K. Behl and James J. Egan Brookhaven National Laboratory, Upton, New York
(Received November 4, 1966)
Transference numbers and ionic mobilities of the cations relative to the chloride ion in the mixtures of molten chlorides LiC1-PbC12, KCl-PbClZ, KC1-CaCl2, KCI-MgC12, and KCI-NaC1 were determined over the entire composition range from electromotive force measurements. The experimental arrangement of the cell employs a special junction using alumina powder to join the two salt compartments. The relative ionic mobilities of the cations were equal in the system KC1-NaCl over the whole concentration range, while in other systems the alkali ion had a greater mobility than the divalent cation. The transference numbers and ionic mobilities obtained in the present measurements for the systems KC1-PbC12 and LiC1-PbC12 agree very well with literature values obtained by moving-boundary and Hittorf-type measurements. The systems KC1-CaC12, KC1-MgC12, and NaCl-KCl have not been previously studied.
Introduction Transference numbers in molten salt mixtures are in general determined by one of three methods-Hittorftype measurements, moving-boundary measurements, or studies of emf of cells with transference. Examples of Hittorf-type measurements may be found in the works of Azia and Wetmore, Duke, Laity, and coWorkers.2d The moving-boundav n ~ t h o dhas been The Journal of Physical Chemistry
used by Klemm and co-worker~.~JAlso, Berlin, et al.,* have studied trace amounts of cations in nitrate melts by countercurrent electromigration experiments. Emf (1) This work was performed under the auspices of the U. S. Atomic (2) P. M. Aziz and F. E. W. Wetmore, Can. J . Chem., 30, 779 (1952). (3) F. R . Duke and G.Victor, J . Am. Chem. SOC.,83, 3337 (1961).
