7277
An Empirical Evaluation of the Individual Elements in the Nuclear Diamagnetic Shielding Tensor by the Atom Dipole Method T.D. Gierke and W . H . Flygare* Contribution from the Noyes Chemical Laboratory, University of Illinois, Urbana, Illinois 61801. Received February 5 , 1972
Abstract: Following earlier work on evaluating the average diamagnetic susceptibility and electric multipole
moments by the atom dipole method we now extend the atom dipole model to evaluate the individual elements in the nuclear diamagnetic shieldingtensor. The successful evaluation of the diamagnetic shielding tensor elements provides the link between the experimental determination of the spin-rotation constants and the total magnetic shielding tensor elements (or the shielding anisotropy).
I
t is now well known that the elements in the nuclear magnetic shielding tensor, uzz,can be separated into two dominant terms which are the diamagnetic (uZzd) and the paramagnetic (u,,P) terms.’ Qxx
= uzzd
+
uzzp
(3) uxzp =
Flygare and Goodisman3 have proposed a reliable method of evaluating uaVdwhich has been helpful in evaluating aav from the spin-rotation constants, the molecular structure, and eq 3. This equation is334
e and m are the electron charge and mass, c is the speed of light,, (0 1 indicates the ground-state wave function, the sum over i is over all electrons (the nuclear origin is appropriate for all properties), the sum over k is over all excited electronic states, cc indicates complex conjugate, and eo indicates the energy of the 0th or ground state of the system. u z z p is intimately related to the spin-rotation interaction constant, M,,, which is defined through a X = -(l/fi2)I.M. J interaction in a rotatingmolecule(1 and J are the nuclear and rotational angular momenta, respectively); M,, also contains a sum over excited electronic states. M,, and uzzp are directly related
the nuclear magneton, gI is the nuclear g value, I,, is the moment of inertia, 2, is the atomic number of the nth nucleus, r, and y n are the coordinates from the nucleus in question to the nth nucleus, and the prime on the summation over n indicates that the shielded nucleus is omitted. Thus if the molecular structure is known, u,,p can be obtained from M,, or vice versa. The major share of experimental data involving the parameters in eq 1 and 2 is through the average shielding obtained primarily from chemical shifts in liquids.
uavd
u8vd(freeatom)
+2,’ ~ 3 em2 c 2 T
(4)
Values of uavd(free atom) are well known. The error involved in using eq 4 for the diamagnetic shielding was shown, by using the Hellman-Feynman theorem, to be equal to the ratio of the bonding energy of an atom to the total electronic energy of the atom.3 Equation 4 has also been used along with eq 3 to evaluate the previously unknown signs of the spin-rotation constants in a number of linear and spherical top molecules.3J In spite of the success of using eq 4 to evaluate daydfor a nucleus in a molecule, attempts by Flygare and Goodisman3 to extend the method to evaluate the individual elements in the diamagnetic shielding tensor, uZzd,were not as successful. The equation analogous t o eq 4 to evaluate the individual elements in the diamagnetic shielding tensor is
p o is
(1) N. F. Ramsey, Phys. Rev.,78,699 (1950). (2) W. H. Flygare, J. Chem. Phys., 41, 793 (1964).
Because of the many methods now available6*’to measure the individual elements (or anisotropies) in the (3) W. H Flygare and J. Goodisman, ibid., 49, 3122 (1968). (4) See also Figure 1 in N. F. Ramsey, Amer. Sci., 49, 509 (1961). (5) H. L. Tigelaar and W. H. Flygare, Chem. Phys. Lett., 7, 254 (1970); see also a discussion of other attempts at estimating uBVdin this
reference. (6) For example, see a new pulsed nmr method in M. Mehring, R. G. Griffin, and J. S. Waugh, J. Chem. Phys., 55, 746 (1971), and references cited therein which also describe the orientated molecule methods. (7) The molecular Zeeman effect can also be used to directly measure the magnetic shielding anistropy; see ref 4; W. Hiittner and W. H. Flygare, ibid., 47, 4137 (1967); and F. H. DeLeeuw and A. Dynamus, Chem. Phys. Lett., 15, 288 (1970).
Gierke, Flygare / Individual Elements in the Nuclear Diamagnetic Shielding Tensor
7278
total shielding, it is desirable to develop an easy method to evaluate uzzd. In this paper we propose an evaluation of uzzd by using an atom dipole model.s This atom dipole model has been used quite successfully to evaluate molecular electric dipole and quadrupole moments and the molecular diamagnetic susceptibilities. This model assumes that bonded atoms retain a major share of their free atom electron densities and that the change in density associated with bonding can be represented by localized empirical atom dipoles. Comparison of our calculated values of uzzd (with this model) with the known values is quite good. We will also show from this viewpoint why Flygare and Goodisman were successful in the use of eq 4 and not as successful in the use of eq 5 for the calculation of diamagnetic shielding. Theory Equation 3 for the average diamagnetic shielding may be rewritten in the following way
and we have defined
(10)
with similar definitions for (x), and (x2),. Values for the quantities defined in eq 10 (i.e., the atom dipoles and atom electronic second moments) for several atoms in different bonding situations are given in GTF.8 To a good approximations the charge distribution on the nth atom will be nearly spherical so that (x2),
(y’),
= (z2), = (P2/3?n
(1 1)
and
=
(xy), E (xz), (yz), E 0 (12) Substituting the results of eq 7, 9, 11, and 12 into eq 6 , we obtain the average diamagnetic shielding in terms of the atom dipoles. Uavd(A> = u a t o m d ( 4
+ 3mc pe22 X ’Z, r, -
1
where we have merely partitioned the sum of a l l j electrons into the k electrons “on” nucleus A and the i, electrons “on” the other n nuclei. If the bonded atom retains a major share of the free atom electron distribution we have
I1
111 The result for the individual diagonal tensor elements is
Accurate values for the diamagnetic shielding of many free atoms have been calculated by Malli and F r ~ e s e . ~ To evaluate the second term in eq 6 , we use the coordinate transformation discussed in detail by Gierke, Tigelaar, and Flygare (GTF).8
ri,
=
rn
+ et.
(8)
The second term in eq 6 now becomes
IV This equation can be expanded in a Taylor series about the nth nucleus. Retaining the first three terms of the expansion gives Z’(OI(~~J~IO =)
ZtL2(Z2),)
n
n
+
~~r,>-3(r,~(p~.)
+ (3/~>nb)(XnY7dXY)n+ xnzn(xz)?l + Y.Z.(YZ?.)]
(9)
where we have used the Born-Oppenheimer approximation, we have assumed the molecule is a rigid rotor, (8) T. D. Gierke, H. L. Tigelaar, and W. H. Flygare, J . Amer. Chem. from this point to be referred to as GTF. (9) G. Malli and C. Froese, Inr. J . Quanrum Chem., Suppl., 1, 9 5 (1967). SOC.,94, 330 (1972);
(14) The diamagnetic shielding tensors for a large number of molecules have been calculated with eq 13 and 14, and some representative results are listed in the first column of Table I. The necessary ( p ) , and ( p 2 ) parameters are from ref 8. The known calculated (ab initio) results are listed in the second column of Table I. Discussion In eq 13 and 14 there are four types of contributions to the diamagnetic shielding tensor. The first two terms are identical with those used by Flygare and Goodisman,3the free atom contribution (I) and the contribution of the electronic point charges centered at the other n nuclei 11. The third term, 111, arises if the point charges are not centered on the nth nucleus but are displaced by a distance (p),. This term (the dipole term) is in general quite small (on the order of a few
Journal of the American Chemical Society / 94:21 1 October 18, 1972
7279 Table I. Magnetic Shielding in Several Molecules" Molecule Y
Molecule Y
t
Atom
UaYd UYY urz:\b
#e
4 X Atom UZCd 31.9 32.0n 0 0 530.0 531.23 553.0 553.9 34.7 \ 451.3 452.4 30.4 588.6 583.6 30.4 1151.8j 0 S 1152.8 529.Y 530.5 F-F F 1160.5 1163.0 480.5 1095.7 1100.9 555.4 1199.4 1194.4 555.4 0 483.8k 0 482.7 108. 9d 109.2 H H-F 508.5 505.8 54.3 \ 423.8 423.1 136.6 519.2 513.0 136.6 /N 446.3k 0 N 448.5 481.6d 484.4 F 471.2 472.9 479.8 379.3 369.8 486.7 497.8 493.5 486.7 H H 111.59 106.4 142.3' 142 6 H-Cl H 92.6 87.3 45.3 54.4 \ 94.7 94.0 190.1 186.7 147.1 137.8 190.1 186.7 /c=o 338.59 H C 338.0 1150.3* 1151.7 c1 294.2 288.5 1148.6 1146.0 355.7 355.3 1152.2 1154.6 364.2 370.6 1152.2 1154.6 452, O g 0 452.0 384.0 380.9,f 384 5d NEN N 415.8 418.3 338.4 348.9 467.6 465.2 407.1 401.6 475.1 470.0 407 1 401.6 H H 95.0,' 96. Oh 95.4 326.1 326.49 CfO C 110.7, 112.1h 110.6 271.2 280.4 / N--H 60.4 354.1 349.0 187.7 115.1 \ 354.1 349.0 N 355.0' 360.1 H 444.50 443.7 0 354.5 364.5 418.7 410.5 355.2 358.0 461.5 456.2 355.2 358.0 456.2 461.5 H 127.8~ 127.2 H 103.4 102. 8h H H 135.7 136.6 102.7 102.8 \ / P---H 79.2 75.3 80.4 78.4 168.0 164.8 130.2 \ 129.1 /O P 981 .Oe H 983.8 H 0 415.6' 420.9 979.8 984.7 415.5 420.3 981.4 983.3 415.0 416.7 981.4 983.3 416.3 425.7 121.9h H H 121.5 H H 136.1e 136.1 114.1 112.3 124.0 120.5 \ \ H---C-F 93.7 96.8 109.3 109.0 158.0 155.3 187.0 178.6 1 H H F 527. Om S 1066.7 526.4 1065 .Os 494.0 1064.5 489.9 1063.4 543.0 1063.9 544.6 1064.7 1071.9 1066.7 544.6 543.0 a Units are ppm. The first column gives the results of this paper compared with known results in the second column. * Calculated with eq 13 and 14; the known structures and the parameters are in ref 8. Theoretical ab initio calculations. d C. W. Kern and W. N. Lipscomb, J. Chem. Phys., 37, 260 (1962). S.Rothenberg, R. H. Young, and H. F. Schaefer, J. Amer. Chem. Soc., 92, 3243 (1970). 1 E. A. Laws, R. M. Stevens, and W. N. Lipscomb, J. Chem. Phys., 54,4269 (1971). 0 D. B. Neumann and J . W. Moscowitz, ibid., 50,2216 (1969). G. P. Arrighini, M. Maestro, and R. Moccia, ibid., 52, 6111 (1970). G. P. Arrighini and C. Guidotti, Chem. Phys. Leu.,6,436 (1970). i S. Rothenberg and H. F. Schaefer, J. Chem. Phys., 63, 3015 (1970). S. Rothenberg and H. F. Schaefer, Mol. Phys., 21, 317 (1971). J. F. Harrison, J. Chem. Phys., 47, 2990 (1967). m C. W. Kern and M. Karplus, unpublished, 1964; see S. C. Wofsey, J. S. Muenter, and W. Klemperer, ibid., 55, 2014 (1971). W. Kolos and L. Wolniewicz, ibid., 41, 3674 (1964).
H-H
H
I
I
I
lfE:y
Table 11. Magnetic Shielding Anisotropies (ppm) in Several Molecules MoleShielding anisotropy, cule Atom calcd I' IIb IIIC CHlF F 2uzzd - u ~ V ' - ~ $ 8-109.0 ~ -147.0 -98.0 H 2~5zd- u,vd - u s z d -27.5 -4.0 -23.5 HzO H uZsd- uzzd - uyVd -52.0 -17.7 -47.9 HzCO H u,,' - ~ ' - u,,*,d -43. . 5 -17 7 -40_ 2__ - u _HQS H uzzd - uzzd - ~ y y d -51.2 -17.7 -46.1 Calcd with eq 13 and 14. Calcd from ref 3 or eq 14 with term IV omitted. Theoretical ab initio calculations. References and axes are in Table I. I S
parts per million) relative to the preceding terms. The quadrupole term (IV) which appears only in uZZd in eq 14 arises because the electronic charge distribution on the nth nucleus is not a point charge but is spatially extended. Ignoring this term may result in significant errors in uzzd. This explains why eq 4 gives accurate estimates for the average diamagnetic shielding (no quadrupole term), but eq 5 gives only fair estimates for the individual tensor elements (quadrupole term is nonzero). This is particularly evident if we consider
Gierke, Flygare 1 Individual Elements in the Nuclear Diamagnetic Shielding Tensor
7280 Table III. Magnetic Shielding Computed with Eq 13 and 14 and the Spin-Rotation Constants"
H-H
H-F
'H
'H
'OF
H-Cl
'H
asci
H-Br
'H
78Br
H-I
'H
1271
F-F
H-C=N
19F
I4N
'H
Cl-CSN
3
~
1
'N
C1-F
IgF
3
N=N
'6N
~
1
35.7 30.4 30.4 32.9 54 137 137 109 480 487 487 484 54 187 187 143 1146 1155 1155 1152 54 350 350 251 3124 3132 3132 3129 56 465 465 329 5503 5511 551 1 5508 480 555 555 530 348 398 398 381 48 121 121 97 1152 1223 1223 1199 350 475 47 5 434 488 609 609 569 1149 1218 1218 1195 349 402 402 384
0.ou
-8.4 -8.4 -5.6 Od
-119 -119 -80 Od
-94 -94 -63
08
- 166 - 166 -111
0e
- 300 - 300 -200
OJ - 321 -321 -214 00
-768 -768 -512 OJ -424 -424 -283 00
- 1496
- 1496 -998
Oi
-1124 -1124 -750
Oi
-627 -627 -418
OJ - 109 - 109 -11 Oi
-930 -930 -620
Of -1110 -1110 -740 01
197 197 132 0 -2458j -2458 1639 0" -729 -729 -486
-
Journal of the American Chemical Society
c=o
34.7 22.0 22.0 26.3 54 18 18 29 480 393 393 421 54 21 21 32 1146 854 854 952 54 29 29 31 3124 2364 2364 2617 56 41 41 46 5503 4015 4015 4510 480 - 569 - 569 -219 348 229 - 229 - 37 49 12 12 26 1152 293 293 579 350 -635 -635 - 306 488 806 806 700 1149 - 1240 -1240 -444 349 - 327 - 327 - 102
26.6~ 170
29. 2c
bC=S
'3C
419 170
31.9
3
H-CEC-H
35h
H
H H
H H
0 'H
S
/ 'H
\ H
'H
/ \
44h
as
'H
\
C=O
/ 170
.210h
-
1 94:21
'3C
Ph-F
'OF
CH3F
'H
28k 'OF
'H
I4N 667'
-408'
pH,
L
'H
3
- 101h
I
October 18, 1972
'P
0" -484 -484 - 323
-99 -66 -71q - 37 - 108 -72
280 -135 -135 +3 419 -234 -234 - 16 299 -96 -96 35 421 -440 -440 -153 1060 331 331 574 50 21 21 31 31 41 21 31
30.2h
120 109 179 136
-88' -71 - 154 - 104
32 39 24 32
30.8h
87 94 137 106 418 468 470 452 519 649 619 616 112 97
-698 - 70
18 24 17 20 -1182 -402 460 - 375 455 216 264 332 21
280 349 349 326 419 456 456 444 299 483 483 421 421 538 538 499 1060 1141 1141 1114 50 120 120 97 103 78 129 103
155 121 490 545 544 526 111 60 115 95 364 358 358 360 80 165 136 127 983 983 985 984
00
-690 -690
-460 OP
- 579 - 579 - 386
Oi
-978 -978 -652
Oi -810 -810 540
-
OJ
-99
-120 -86 - 1600' -870 - 10 - 821 -64' - 373 -415 - 284 -85'
'i
1
)
-98 -91 -63' - 52 - 52 - 56 - 88" - 16 - 104 -61 -117w -78 -78 -91 -88"
- 108 -98 - 370" - 370 -421 - 387
29.4h
21.31
333h
1
i
29 30 426 493 493 47 1 23 44 11 28 241 280 280
26. 9c 423, 5c 489.5 489.5 467.5
30.8
