CHEMICAL PHYSICS LETTERS 1 (1967) 173-176. NORTH-HOLLAND PVBLISHING COMPANY, AMSTERDAM

A PHENOMENOLOGICAL TREATMENT OF THE ABSOLUTE SIGNS OF SPIN-SPIN COUPLING CONSTANTS

JACOB SCHAEFER Central Research Department, Xonsanto Com$arzy, St.Louis, hfksom-i

and

ROBERT YARIS *$ Defiartment of Chemis@ _ Washington Univevsi!j~, St. Louis. Missouti

Received 13 June 1967

By using a perturbation-variation treatment, and some assumptions about the first-order perturbed trial function, the absolute signs of the spin-spin coupling constants for the systems RX, H-C-X, and H-C-C-X (where X is D, B, C, N,O, OF F and all carbons are saturated) are examined.

1. INTRODUCTION

Although the measurement of spin-spin coup- ling constants in molecules has been very useful in empirical applications of NMR [ 11, it has proven tc be too difficult to obtain detailed infor- mation concerning the electronic structuse of molecules from the experimental data in a mean- ingful quantitative way**. Hence, we have set ourselves the restricted task of treating only the sign of the spin-spin coupling constant in a pheno- menological manner. We proceed as follows: we limit ourselves to the restricted class of com- pounds, HX, H-C-X, and H-C-C-X, where X is D, B, C, N, 0, or F and all carbons are satu- rated. We expect that these systems are likely to satisfy the following conditions. 1) The Fermi contact potential is the dominant interaction me- chanism. 2) Short and long range spin-spin coup- ling can be compared without interference from complicated electronic bonding around atomic centers other than those of interest (H and X). 3) Signs of coupling constants can be transferred from molecule to molecule.

* Work partially supported by the National Aeronautics and Space Administration through a multidiscipltnary grant to Washington University.

t Alfred P. Sloan Fellow. ** A discussion of this difficulty is in ref. [2].

2. AN EXPRESSION FOR THE COUPLING CON- STANT

Following Ramsey [3], it is as,cumed that in a rotationally averaged system Me i.nteractioK en- ergy is

E&V = h &NV /N*IN*,

(2) where ENN~ is the second-order interaction en- ergy between nuclei N and N’ having spins fN and IN’, respectively, and JNN~ is the spin- spin coupling constant***.

Rather than attempt to solve the perturbation equations directly we use a Hylleraas variati al procedure [2,5] to approximately solve for E

w 2 in terms of a trial first-order perturbed wave’

t

function. That is, we define the energy functional

WNNV = [2(F(l)/ VN + VNV[*~) f

where H,, E,, and q. are the unperturbed Hamil- tonian, ground state energy, and ground state eigenfunciion, respectively; VI) is the trial first-

*** When we compare coupling constants between VOF- ious systems we shall actually use the rediiced coupling constant &~NV = @r/h WN~)*! as in ref. [4]. This is done so that the sign of only the electronic contribution to the co@ing constant is consider&d.

174 J-SCHAEFER and R.YARIS

order wave function; VN contains the Fermi con- tact potential of nucleus N [2,3] summed over all electrons as. well as electron and nuclear spin operators, and the subscript NN’ indicates that only terms proportional to IN-IN’ are retained.

Within the class of systems considered, the many-electron ground state wave function can be written [6] as a Hartree-Fock term plus a corre- lation term restricted to electron pairs. Since the spin-spin coupling term measures the corre- lation of electronic spin, including correlation

- ;“ry

s in the trial function should be important. represents the two-electron correlation cor-

rection to the ground state wave function which makes the dominant contribution to IVNNV.

q/(l) = kzTjSkG,(',?), ,

where aN is a variationa.l parameter, Sk is the electron spin operator, XN is a constant contain- ing the gyromagnetic ratio of nucleus N, and &&,j) is a two-electron function which is spa- tially orthogonal to the ground state. The ortho- gonality restriction is necessary since the trial function must be triplet-like and-hence spatially orthogonal to singlet functions. tr,.j has a singlet spin part which becomes triplet upon appl-ying the spin opErator. Making JVNNr stationary with

t’o”~(%? t to +(I) yields JVNNI as an approximation

NT”

The denominator in the expression for WNNV is always positive since E. is the lowest eigenvalue of Ho. Thus, the sign of the dominant contribution to J.+$NN~ and hence to +N depends on the rela- tive pkase of +(I) at each of the two nuclei.

3. SPIN-SPIN COUPLING BETWEEN DIRECTLY BONDED NTJCLEI

For directly bonded nuclei, the dominant-con- tribution to the coupling con&& comes from the bonding tight pair (the two bonding electrons in the same molecular orbital). For the molecule (or fragment) HX we write the ground state wave function, Jlo (unnormalized), as

IC/o~~(l)~(2)+Q(2)~(1)+K~(1~(2)~~(l)~(2)-~(2)8(1)),

where a and b are atomic orbitals, or combina- tions of orbitals, on nuclei N and N’, respective- ly, and both of which have S-like character; -(N and j3 are spin functions, and K is a constant. It has been assumed, for convenience, that N’ is the more electronegative center and -has the dominant ionic contribution and that a valence bond approximation, including both covalent and ionic parts, is a sufficiently good representation of the bonding electrons. Using a ground state function of this form with the electronic corre- lation explicitly included and a sim$ar, suitably orthogonalized and parameterized q(l), a posi- tive sign was obtained for the coupling constant in HD [2J. Also, all predominantly covalent trial functions of the above form give a positive Jo.

The possibility exists, however, that G(1) should be predominantly ionic, especially if there is a substantial electronegativity difference between H and X. To investigate this possibility we choose an ionic 201~: .

w12 = 4ww{~ m(2) - Q mm3, where a’ is an orbital centered on N similar to but not identical with c. We choose a’ with the same relative phase as Q, since the Fermi contact perturb&i-n on center N should not change the relative phr ‘5 3 a function centered on N.

We now SC: r?idt-orthr,gonalize ~12 to Q. to obtain U112 :

2212 = a(l)a’(2; - a4l)b(2) + 42)b(l) +

•t K b(l)b(2)l){o (l)P(2) - o (2)fi(l)),

where

= (S’ + Kss’ +ss”)/[2(1 + s2) + 4Ks -I- K2],

and

S = (a\ b), S’ = (aI/ b} and S” = (n’j a).

Substituting the wave functions into the expression for IV -, and neglecting off-center terms such as 6(r01 N9 b(l), yields

W $Jpq’ =

= K[yS’-2c][-2c(1+ 2KS+ K2)]/(6121 Ho - Eel ti’l2).

In the akrove equation, K is a positive constant which depends on the properties of the two nuclei N and N’, on the electron density at N and N’ of (z and b, on the results of the spin integration, and on tie various normalization constants. y is a positive constant greater than or equal to one whose size depends on the electron density of a’

A PHENOMENOLOGICAL TREATMENT OF THE ABSOLUTE SIGNS 175

at N. Since TS’ > 2C > 0, the ionic contribution to WN-N~ is negative *.

Jf we had chosen our trial function q(l) - b(l)b’(2), corresponding to the ionic trial function arising from the Fermi contact pertur- bation on nucleus N, we obtain a similar result, with a different K (reflecting the inadequacies of a one-term treatment); however, the sign of the ionic contribution remains negative.

Thus, in general, we expect that covalent pair terms contribute positively to the directly bonded spin-spin coupling constant, while ionic terms contribute negatively.

4. SFIN-SPIN COUFIJNG BETWEEN INDIRECT- LY BONDED NUCLEI

The contributions to JHCX and to JHccX arise from two kinds of electron pairs - from tight pairs as in HX, and from loose pairs, the two-electron interactions between electrons not in the same molecular orbital. The details of the analysis of electron-pair contributions to indirect coupling constants are similar to those of the con- tributions to direct coupling constants in section3. Qualitatively, in an HCX fragment there are two bonding tight pairs, HC and CX. If X is more electronegative than C, then the CX pair is polar- ized toward X, and with respect to H acts as an ionic structure tight pair leading to a negative contribution to JHCX- However, the HC pair is, if anything, slightly polarized toward the carbon. Hence, with respect to the X nucleus the HC pair does not act strongly ionic and so leads to a po- sitive contribution to $.ICX. If X is about as elec- tronegative as C then both pairs act as weakly ionic structures on the near nucleus relative to the more distant one and both contribute negative- ly to JHCX.

As X becomes very electronegative, the con- tribution of the CX pair to the coupling constant at a distant nucleus goes to zero negatively since the electron density of that pair at the distant nu- cleus becomes negligible. Thus, ionic tight pair contributions to non-bonded spin-spin coupling constants decrease rapidly with nuclear separa- tion (through bonds) and with increased electro- negativity difference between C and X. Covalent tight pair contributions also decrease rapidly ~5th nuclear separation.

* The inequality@’ > 2C > 0 fails only if S’ becomes small while S zmd.5” rem&n large. This is unlikely for the orbitals in the molecular systems being con- sidered.

Table 1 Absolute signs of selected reduced * spin-spin co:@ng

cixlstants $.

X JHX JHCX JHCCX Footnotes

D + : a,b.c B f (-) d C A- (‘) e.f.g N t-1 (+) 0 (+) (+) F + f h,i.j

* See footnote in text. $ Signs of coupling constants not experimentally estab-

lished are in parentheses. a.

b.

C.

d.

e.

f.

6.

h.

1.

!.

JHD is positive as recently determined by moIecular beam resonance experiments (i_ Ozier, P-n Pi z.zd N. F. Ramsey, private communication)_ JHC and JHCH have opposite signs as determmed by double resonance NMR experiments on CH2DOH (F. .4. L. Anet, J. Am. Chem.Soc. 84 (1962) 37Gi). JHCH and JRCCR have opposite signs as determined by double resonance EihIR experiments on BrCH2CHBrC02H (R. Freeman, K. A. McLaughlan, J-1. hIusher and K. G. R. PachIer, Mol. Pbys. 5 (1962) 321) and for many other systems of the type considered here (ref. [Ii). JHB and JHBF have the same sign as determined by double resonance fTMR experiments on HEFz. (E. B_ Whipple, T. H. B::own. ‘?. C. Farrar and T. D. CoyIO; ref. [8J in text. ) AssumingJHBF/JHCF is positive. JHB is positive.

G% is positive as determined bv relaxation rates in lines of CHFC12 (E. I. RIackor and C. MacLean;

ref. [7] in text). JIIC and JHCC have opposite s!gns as determined 53 anal sis of the high resolution spectrum of H3 IY 13 C CH3 (R. hl. Lynden-Bell and N.Sheppard, Proc Roy.Soc. (London) -4369 (1962) 385). and ethyl iodide (H. Dreeskamp and E. Sackman. 2. Physik. Chem. 3-t (1962) 273).

I .

&C and ZRCC have the same signs, however, in l-1.2.2-tetrabromo ethane as determined bv double r&knce experiments (R. Freeman and iV.k. Ander- son, J. Chem. Phys. 42 (1965) 1299). Tht extremely heavy substitution of the H-C-C-X system may re- sult in a failure to meet the three conditions listed in the first paragraph in the text. JHCF is positive (E. I. Mackor and C. BracLean. op. cit. )_ JHC

-f is positive as determined by anaIysis of lH

and 9F spectra of CH3F in a nematic liquid cry;staI (R. A. Bernheim and B. J. Lxverv. J. Am. Chen;.soc. s9 (1967) 1279).

_ _

JHCF and JHC~F have the same sign in a range of compounds of the type considered here as determined by high resolution -Gd double resonance KlVR expe- riments. (See D. F. Evans, S. L. hknatt and D.D. Elleman, J._&n.Chem.Soc. 85 (1963) 238.1

:

The dominant loose pair contributions to the coupling constant can be either positive or nega- tive corresponding to whether their perturbed spatial wave function is better represented as essentially covalent or essentizlly ionic. Loose

176 J. SCHAEFER and R. YARIS

pair contributions to the nuclear coupling de- crease with internuclear separation.

5. ASSUMPTIONS REQUIRED TO RATIONALIZE EXPER.lMENTAL SIGNS OF COUPLLNG CON- STANTS

Table 1 gives the results of some experiments providing signs for the reduced coupling constants in molecular systems of the type considered. The values in parentheses can be added to the *table giving a consistent and reasonable picture for the signs of the coupling constants in terms of the above molecular picture if the tollowing additio- nal assumptions are made regarding the nature of s(l): 1) Jm is determined by the bonding tight pair contribution, wNch is essentially cova- Zent unless the electronegativit_* difference be- tween l-l and X exceeds 1.5 units. 2) Tight pair contributions to JHCX dominate loose pair con- tributions. The tight pair which makes the domi- nant contribution to JHCX can be covalent or ion- ic in nature depending on the electronegativity difference between H and X. The dominant tight pair will be covalent only if this difference ex- ceeds 1.5 units *. 3) Loose pairs determine &ICCX exclusively, and because of the moderat- ing, intervening saturated carbon system between H and X, are essentially covalent.

6. DISCUSSION

The choice of 1.5 eiectronegativity units as the dividing line between an essentially ionic and an essentially covalent tight pair contribution to JHX is rather arbitrary, but does reflect the fact that negative coupling constants between directly bonded nuclei have been reported (CF, [3] BF, [8] SiF, [9] in which such a value is indicated.

Pople and Santry [4] have made a successful attempt to explain negative coupling constants be- tween directly bonded nuclei in their molecular orbital perturbation treatment of JHF. Faced with the problems of unavailable molecular wave functions, and unknown excited state energies, they gave a plausible argument for the possibility of a negative coupling constant. Our approach is

* Then, in HCX, the CX tight pair will be quite polar- ized towards X, resulting in a rather small CX tight pair electron density at H and hence a small nenative contribution to JHCX. while the CH pair contribution will be positive and large reflecting the covalent na- ture of that pair relative to X and i%s substantial elec- tron density at X.

similar to theirs in many respects, in that using a single term variational function to represent the dominant contributions is analogous to sum- ming the dominant terms (or term) in a pertur- bational sum. A major difference is the added flexibility the variational formulation gives since in selective partial summation the form of the available terms is already fixed.

We have avoided making numerical calcula- tions in this work because, considering the pre- sent state of the art, we do not know what mean- ing to ascribe to them. A more definitive reso- lution of the importance of the various contribu-

tions in determining coupling constants can only be achieved by more difficult and elaborate cal- culations than have been attempted to date. We only wish to indicate a simple, physically plau- sible, and consistent explanation of a physical property and to point out some experimentally accessible implications of this treatment, a few of which are the following; 1) JHX and JHa are opposite in sign. 2) JHCCX is always positive.

3, JHCs 1 -- ---

‘s usually smaller in magnitude than

JHCCX + [lo].

REFERENCES

[1] -4. 4. Bothner-By, Advan. Magnetic Resonance 1 (1965) 195.

[2] J. Scha.efer and R.Yaris, J. Chem. Phys. 46 (1967) 948.

[3] N. F. Ramsey and E. N.Purcell, Phys. Rev. 85 (1952) 143; N. F. Ramsey, Phys. Rev. 91 (1953) 303.

[4] .T. A. Dople and D. P.Santry, Mol. Phys. 8 (1964) 1. [5] J. 0. Hirschfelder, S. T. Epstein and W. Byers

Brown, Advan.Quantum Chem. l(l964) 255. [S] O.Sinanoglu. Advan. Chem. Phys. 6 (1964) 315; J.

Chem. Phys. 36 (1962) 706, 3198. 1 r71 E. L. Mackor and C. MacLean. J. Chem. Phvs. 44 _ _

(1966) 64. [8] E. B Whipple, T. H. Brown, T. C. Farrar and T.D.

Coyle, J. Chem. Phys. 43 (1965) 1841. [9] S.S.Danyluk, J. Am. Chem.Soc. 86 (1964) 4504.

[lo] S. L. Stafford and J. D. Baldeschwieler, J. Am. Chem. Sot. 83 (1961) 4473.

** This is the result of the partial cancellation of the tight and loose pair contributions to JHCX and the absence of any cancellations in contributions to JHCCX. When the electronegativity difference be- tween H and X is between 1.0 and 1.5 units, JHCX is unusually small in magnitude since, in addition to possible tight and loose pair cancellations, the contributions from the two tight pairs in JHCX are comparable in size but opposite in sign. (For this reason, exceptions to the signs in table 1 are most likely to occur for JHCX. Also, see footnote g in table 1.)