Download NMR Shifts in Paramagnetic Systems: A Nonmultipole Expansion

JOURNAL
NMR
OF
MAGNETIC
33, 627-641(1979)
RESONANCE
Shifts in Paramagnetic Systems: A Nonmultipole
Method
R.M. GOLDING
DepartmentofPhysical
Chemistry,
Received
UniversityofNew
February
AND L.C.
South
Australia
24, 1978;
Wales,
revised
Expansion
STUBBS
Box 1, P.O.,
June
Kensington,
2033,
N.S. W.,
19, 1978
NMR shifts arising from the electron
orbital angular momentum
and the electron
spin
dipolar-nuclear
spin angular momentum
interactions
using a nonmultipole
expansion
method are examined
for a d electron
in a crystal field of octahedral
symmetry
and with a
tetragonal
component.
Bonding
with the paramagnetic
center is included.
The NMR
shifts are illustrated
as isoshielding
contour
maps, and the results are compared
with the
multipole
expansion
method.
It is shown
that even when experimental
data are
successfully
fitted by considering
the NMR shifts as arising from the Fermi contact and
the dipolar interactions,
such an interpretation
may lead to incorrect
conclusions
about
the interactions
within the paramagnetic
system.
INTRODUCTION
Over the last few years the interpretation
of the NMR shifts in d” and f”
paramagnetic systems has been based upon the electron-nuclear interactions arising
from the Fermi contact interaction, the electron orbital angular momentum, and the
electron spin dipolar-nuclear spin angular momentum interactions. The latter two
terms have been treated as a multipole expansion in R, where R is the distance
between the NMR nucleus and the electron-bearing atom. In particular, the dipolar
term in the expansion which is related to the magnetic susceptibility anisotropy (I)
has been used almost exclusively, although more recently the higher multipole terms
(2-5) have been evaluated for some specific examples. In a recent paper (6) we have
shown that the use of such an expression may lead to serious errors in the
interpretation
of the NMR shift through this mechanism, especially if only the
dipolar term is considered.
In some cases the Fermi contact interaction has been used to interpret the NMR
shifts; see, for example, the NMR shifts in a series of transition metal ion dithiocarbamate complexes (7) and the 170 NMR shifts of the aqueous trivalent lanthanide
ions (8,9), where the NMR shift, Al?, is given by
AB = a(Sz)lgnrCLt+
r11
Here a is the hyperfine interaction constant. In other cases the dipolar term in the
multipole expansion has been used; see, for example, the proton NMR shifts insome
627
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Copyright 0 1979 by Academic Press. Inc.
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GOLDING
628
lanthanide complexes
components (1) xoo/
(IO),
where
AND
STUBBS
in terms of the magnetic
ABIB = -(/..d477N,~,, -&xx + xyyIH3 ccd 0 - 1)
+ %xXx-xy,,) sin’ 0 cos 2@]/3R3.
susceptibility
1.21
The polar coordinates of the NMR nucleus in relation to the paramagnetic center are
(R, 0, @), where the z-axis is the principal axis of the complex. However, both the
Fermi contact contribution given by [l] and the dipolar contribution given by [2] may
be important (II). Recently (12) a method has been developed whereby, from a set
of experimental results, an estimate may be made of the various contributions to the
NMR shifts.
However, few publications have been devoted to the determination of the region
where the dipolar contribution given by [2] may yield a good approximation. This is
of special importance when there is delocalization of the unpaired electrons from the
paramagnetic center. For example, it has been shown/6) that the 14N NMR shift in
the Fe(CN)63- ion may be interpreted as arising dominantly from the interaction with
the nitrogen nucleus of the unpaired electrons in the nitrogen p orbital of the 2T2
molecular orbitals.
In this paper we examine the NMR shifts arising from the electron orbital angular
momentum and the electron spin dipolar-nuclear
spin angular momentum interactions using the nonmultipole expansion method (13, 14) (i.e., applicable for any R)
for a number of specific cases. We include the effect of bonding with the paramagnetic center. The results are compared with the multipole expansion method. Finally,
we show that even when the multipole expansion method gives errors, experimental
shifts may still be fitted successfully using a combination of [l] and [2] but will lead to
incorrect conclusions about the interactions within the paramagnetic system.
THEORY
The principal values a,,, cr,+,, and cZZ of the NMR screening tensor u are
determined by considering the magnetic field interaction as parallel to the X, y, and z
directions and averaged assuming a Boltzmann distribution.
The contribution to the NMR shift, AB, is given by
AB=~B(u,,+a,,+cr,,),
[31
where
(X) refers to the Boltzmann average of the hyperfine interaction represented by the
Hamiltonian
21N ’ 1
3(& * s)rN ’ 1
x = E gNpSpN
---T-+gs
[41
rh
rN
NMR
SHIFTS
IN
PARAMAGNETIC
629
SYSTEMS
.
electron
f
ZrV r
/
tl
electron
- bearing
nucleus
FIG.
1. The
coordinate
system.
In this paper we take gS to be equal to 2 exactly. Here rN is the radius vector of the
electron about the nucleus with nuclear spin angular momentum I, as shown in Fig. 1.
The system we consider is a d’ system in a strong crystal field of octahedral
symmetry, where bonding effects have been incorporated using a molecular orbital
approach. The paramagnetic site is surrounded by six ligands lying at a distance *Z?L
along each of the three axes. It was assumed that the ligands could be treated as single
atoms with three p orbitals available for bonding. The p orbitals parallel to the X, y,
and z axes for the jth ligand are written as Ixi), Iyj), and IZj), respectively. The
numbering of the ligands is as shown in Fig. 2.
The appropriate molecular orbitals are (1.5)
&>+
W2)[lY5)-
Iyd+ Id-
aId +@/ml) - lzd+
l&l,
1%) - Ml,
~15~+~~/~~~1~,~-I~~>+lY*>-lY3~1,
NMR
FIG.
2. The
numbering
of the ligands
[a
630
GOLDING
AND
STUBBS
where the d orbitals are taken to be Slater-type 3d orbitals and are defined as
l-5= (Wh)“2~z exp(-kbrh
14 = (2~:/3dI’=
exp(-O2rL
lbl
It>=CW%~)“2x~
exp(-ML
and similarly the p orbitals are taken to be Slater-type 2p orbitals defined as
Ix) = @V~)1’2x
exd-khr),
IY>= (P:l.~)“~yexd-kbr),
II71
12) = (/~:/T)~‘~z exp(-&r).
The coefficients a and b in the molecular orbitals are defined from
k’=a2+$b2+3ab(djp),
PI
1 =a2+b2+4ab(djp),
where k’ is the orbital reduction factor and (dip) gives the amount of overlap. The
‘T2 ground state of the d’ system is split by spin-orbit coupling into two levels with
respective eigenvalues of -11;’ and f’, where
5’ = (2k’where fh and &, are the spin-orbit
respectively.
The NMR shift is given by
AB
-=--B
341~ kT
l)& + (1 -k’)&,
coupling constants for the d and p orbitals,
E + F exp(3[‘/2kT)
+ G{l -exp(35’/2kT)}kT/[’
1 + 2 exp(3[‘/2kT)
[91
where
E = $(2k’+ l)(D -2L),
F = $(l - k’)(D + 12L),
G = &(k’+ 2)(0 +4L),
where
D=u~Do(R)+:
,g Di(Ri),
I
L=a’Lo(R)+:
1
.i Li(Ri),
I
1
where Ri is the vector pointing from the NMR nucleus to the ith ligand. As an
example, Rs is shown in Fig. 2. The functions Do(R) and Lo(R) arising from the d
NMR SHIFTS IN PARAMAGNETIC
631
SYSTEMS
orbitals are
D,(R)= -
1
+y
Y&O,
0) S(t,)
+fg
7r1’*[Yoo(o,
[loal
@)]iv(to)
and
Lo(R) = “(~)1’2[~(~)1’2Y~~(B,
P;R~ 21
40,+;(;)1’2Y40(8,
+; (i) 1’2Y4-4(@, @)]K(to)+~
where to = 2&R
?r”*[Y00(0,
@)
@)]M(to),
[lob1
and
s(t)=[l-e-‘n~o$]’
~(t,=[l-e-‘j~~+~~,~)],
N(t)=[e-(-~+~+;+t+l)],
.
.
K(r)=[l-emtn~o$].
M(t)=[e-f;+;+t+l)].
.
.
The functions Di(Ri) and Li(Ri) (i = 1, . . . , 6) arising from the p orbitals on the ith
ligand are given by
Dl(Rd=
1 5 II2
Y42(01,
@l)
j 2
[ 0
1 5 l/2
Y4-2(@1,
@l)
H(t1)
+T 02
I
72?*
-m
+)‘“[;(?)l’*
1 3 112
y2-2(@*,
+z 0 2
Y*2(01,
01)
1
wl)+~
+ f
@l)
Y40(@1,
-$
Y20(@1,
4p:
~
Qjl)
w
"2 [YOO(@l,@dlJ(td,
LIlaI
63’
GOLDING
Ll(Rl)=~(~)1f2~~(~)“2
Y22t0,r
+i
3
STUBBS
(PI)-;
Y2&91,
@I)
lf2
( )
2 2
yz-Z(@l, @l,i T(fl)
[Yd@l,
D2(R2)=
AND
72~“~
1 5 1’2
PTR;
j
[
0
rllbl
@l)la~l)~
2
[llcl
L2(R2)=-~(~)1’2[~(~)1’2Y22(~2,~2)+~Y2,(B2,~2)
+I-
2 0 32
4
-3
Rs
If2
Y2-2(02,
1
@2) T(t2)+5g
7P2[YOO(@2?
@2)1C(f2L
7Tlf2
0-5
LdR5) =$(;)
[lldl
[llel
1’2[Y20(@5,
G'dlT(fs)+~
~'~"P'-oo~& @5)1C(t5),
5
and
D3(R3)
= D1CR3);
L3(R3)
= L(R31,
DA&)
= D2W;
LdW
=
DdR6) = D&b);
L2NA
LtdRd = LdRd,
[llfl
NMR SHIFTS IN PARAMAGNETIC
SYSTEMS
633
where ri = 2PlRi and
Hw=[l-e-‘n~o$],
z(c)=[l-e-r(~~5+~~,~)],
J(t)+‘( -;+,+I)],
T(t)=[l-e-‘.~o$].
C(t) = [e-‘(t+
l)].
When bonding effects are neglected (that is, taking a2 = 1 and b’= O), the results
reduce to those given by Golding and Stubbs (14).
In arriving at this expression for the NMR shift many molecular hyperfine integrals
had to be evaluated. The full set of integrals for the 2p orbitals is given in the
Appendix.
The results given by Golding and Stubbs (14,16) for the components of the
hyperfine interaction tensors for the *T2E” and *TziJ’ levels may be extended to
incorporate bonding effects by replacing, in the appropriate formulas for the tensor
components,
(5lLsk),
(5lTalakL
b#‘ahL
GtlTasld,
(dTapk>,
~~I%&),
(77I&k&415% b$cdih),
where
Tap = (3rNnrNP - rkLp)lrk
by
respectively.
(hJ&lC>,
634
GOLDING
AND
STUBBS
DISCUSSION
Initially we examine the case for a d’ transition metal ion in a strong crystal
field of
octahedral symmetry where the transition metal ion is surrounded by six ligands all
0.2 nm from the central atom along the three axes. We choose o1 = 2.275/au
and
p2 = 2.2/ao, the temperature as T = 300 K, the bonding coefficients as a = 0.8154
and b = -0.6818,
and the spin-orbit coupling constants as & = 400 cm-’ and
&, = 110 cm-‘. The overlap integral (dip) = 0.0584. The contours of equal chemical
shift AB/B in parts per million in the xy plane as determined from the expression
given in the previous section are shown in Fig. 3. The contour map is very complex
and is markedly different for the case when no bonding is considered; see Fig. 4. In
FIG. 3. An isoshielding
diagram
for the case of the NMR nucleus
in the xy plane for a d’ transition
metal ion in a strong crystal field of octahedral
symmetry,
where bonding
effects have been considered.
Ligands
are located at kO.2 nm from the central
metal ion along the three axes. The contours
of equal
chemical
shift AB/B
are in parts per million. (A bar over a number
indicates
a negative
value.)
NMR
FIG. 4. An isoshielding
metal ion in a strong crystal
contours
of equal chemical
SHIFTS
IN
PARAMAGNETIC
SYSTEMS
635
diagram
for the case of the NMR nucleus
in the xy plane for a d’ transition
field of octahedral
symmetry,
for the case when bonding is not considered.
The
shift AB/B
are in parts per million.
Fig. 3 near the transition metal ion and the six ligands, AB/B is nearly isotropic.
There are a number of regions where AB/B is a maximum or a minimum and regions
where AB/B changes rapidly with distance. The pattern in Fig. 3 gives an insight into
the very complex manner in which the NMR shift varies throughout the molecular
system.
Next we examine the reliability of considering the NMR shift in specific regions in
space for the system given by Fig. 2 using the multipole expansion approach. A
comparison of the exact solution with all the multipolar terms and only the dipolar
term is given in Table 1 at different values of R along the (loo), (1 lo), and (111) axes.
In this particular case the dipolar term arises only from the ligand part of the
electronic wavefunction. From Table 1 we observe that consideration of the dipolar
A
TABLE
1
- 244.203
- 177.257
--820.166
145.109
- 22.274
-2.350
-6.011
- 1.124
- 0.606
-0.355
-0.221
-0.144
- 0.097
0.2
0.25
0.3
0.4
0.35
0.45
0.5
0.55
0.6
0.65
0.7
-1294.244
(a)
Exact
0.1
0.15
0.05
R (nm)
AB/B
a (100)
- 0.606
-0.355
-0.221
-0.144
- 0.097
--____.--..-
2.350
- 6.023
-1.124
-23.381
-313.601 -
- 545.230
-350.816
-57,714.116
(b)
Multipolar
along
ic)
-0.566
-0.335
-0.210
-0.138
- 0.094
-..... -_--
-2.105
-5.133
- 1.032
- 17.523
- 139.897-
-22.454
- 141.984
Dipolar
..____.~
- 13.543
axis (ppm)
0.699
1.408
0.367
2.955
11.232
6.090
28.633
23.388
-803.132
0.205
0.120
0.074
0.047
0.032
-... - .-~ ..___ -- ._
_~
(a)
Exact
AB/B
a (110)
0.205
0.120
0.074
0.047
0.032
I,__- .
0.699
1.408
0.367
2.958
12.076
6.145
708.003
43.189
89.840.682
(b)
Multipolar
along
0.155
0.095
0.060
0.039
0.027
,__ .-_
0.475
0.874
0.266
1.589
1.802
2.477
- 10.759
-3.842
- 12.953
(4
Dipolar
axis (ppm)
0.188
0.130
0.091
0.064
0.046
_--.-...
0.384
0.495
0.272
0.439
--4.703
0.429
- 134.595
-25.103
-935.934
(4
Exact
AB/B
0.188
0.130
0.091
0.064
0.046
0.384
0.495
0.272
0.438
-5.289
-0.451
-712.930
-41.367
-95,644.784
a (111)
-
0.233
0.153
0.104
0.072
0.05!
_. . .” _. -
0.571
0.904
0.361
1.364
0.743
1.691
- 9.734
- 3.492
- 12.803
Cc)
Dipolar
axis (ppmj
USING (a) THE EXACT
(b)
Multipolar
along
COMPARISON, FOR DIFFERENT VALUES OF R ALONG (100). (llO),
AND (111) AXES, OF THE RESULTS FOR AB/E
SOLUTION; (b) THE MULTIPOLAR EXPANSION; AND (C)THE POINT-DIPOLE APPROXIMATION
3
VI
3
z
8
8
Et
NMR
SHIFTS
IN
PARAMAGNETIC
637
SYSTEMS
term only gives rise to substantial errors at distances less than 0.7 nm from the
paramagnetic center. However, by considering all the multipolar terms the
agreement with the exact solution occurs at distances greater than 0.4 nm.
In many cases experimental paramagnetic NMR shifts are analyzed in terms of the
Fermi contact and the dipolar interactions given by Eqs. [l] and [2]. To explore the
possible problems of interpretation associated by such a procedure we examine the
AB/B results obtained from the previous section over a temperature range and
reconsider the data as a set of experimental results. We attempt to analyze the data
when a ligand atom is the NMR nucleus as arising from a sum of the Fermi contact
and the dipolar interactions given by Eqs. [l] and [2]. In this paper we choose (S,) in
Eq. [l] as given in Ref. (17) for a d’ ion in a crystal field of octahedral symmetry. In
the evaluation of ,YIIand ,Y~we include a small tetragonal distortion from octahedral
symmetry. The quantities ~11and xL are determined from expressions in Ref. (15),
where S is a measure of the distortion from octahedral symmetry. The S value is
adjusted to give the best fit of the theoretical values of AI? to the expression
AB = a(Sz)+b(x~~-xJ.
In this case the best fit is obtained over the temperature range 200 to 400 K by
choosing a very small distortion, and the results are summarized in Table 2. From
Table 2 it follows that the analysis of the data using Eqs. [l] and [2] would incorrectly
imply that the NMR shift arises dominantly from the Fermi contact interaction with a
small contribution from the dipolar term in the multipole expansion. Although the
case above was chosen when the ligand was the NMR nucleus, similar analyses may
be made when the NMR nucleus is at any other position within the molecular system.
TABLE
2
SUMMARY OF THE RESULTS OF FI-ITING A SET OF THEORETICAL DATA
FOR AB/B AS ARISING FROMA SUM OFTHEFERMI CONTACTANDDIPOLAR
INTERACTIONS AS GIVEN BY Eos. [l] AND [2], FOR THE CASE OF A d’
TRANSITION METAL ION IN A STRONG CRYSTAL FIELD OF OCTAHEDRAL
SYMMETRY,~HERE
BONDING EFFECTS HAVE BEEN CONSIDERED
A
AB/B fitted
T W)
ABIB (ppm)
Exact
Contribution
from [l]
200
220
240
260
280
300
320
340
360
380
400
-644.4
-706.8
-752.1
-784.2
-806.2
-820.2
-828.0
-831.0
-830.3
-826.8
-820.9
-938.8
-932.1
-923.9
-914.3
-903.5
-891.6
-878.9
-865.5
-851.7
-837.5
-823.2
to Eqs.
[l]
Contribution
from [2]
293.0
225.9
173.2
131.6
98.6
72.2
51.0
34.1
20.4
9.5
0.7
and [2]
Total
-645.8
-706.2
-750.7
-782.7
-804.9
-819.4
-827.9
-831.4
-831.3
-828.0
-822.5
63X
GOLDING
AND
STUBBS
Hence, we have demonstrated that it does not necessarily follow that a good fit ot a
combination of Eqs. [I] and [2] to a set of experimental data results in a correct
understanding of the origin of the NMR shift.
Finally, employing results given in (14), we examine the AB/B contour map for a
d’ octahedral metal ion system where the crystal field environment is of tetragonal
symmetry with the tetragonal distortion along the z axis. We neglect bonding in this
example. The tetragonal distortion component of the crystal field interaction was
chosen in the form 6(/z -2), where S, the distortion parameter, was taken as
1000 cm- I. With & = 400 cm-‘, /?2 = 2.2/ ao, and T = 300 K, the position of the
NMR nucleus relative to the d-electron-bearing
atom to give rise to a specific NMR
FIG. 5. An isoshielding
diagram
for the case of the NMR nucleus in the xy plane for a d’ transition
metal ion in a strong crystal field of tetragonal
symmetry
for S = 1000 cm-‘,
& = 400 cm-‘, p2 = 2.2/ao,
and T = 300 K. The contours
of equal chemical
shift AB/B are in parts per million. (A bar over a number
indicates
a negative value.)
NMR
SHIFTS
IN
PARAMAGNETIC
SYSTEMS
FIG. 6. An isoshielding
diagram
for the case of the NMR nucleus in the zx plane for a d’ transition
metal ion in a strong crystal field of tetragonal
symmetry
for 6 = 1000 cm-‘,
& = 400 cm-‘, p2 = 2.2/a,,
and T = 300 K. The contours
of equal chemical
shift AB/B are in parts per million. (A bar over a number
indicates
a negative
value.)
shift, ABIB, was determined as a set of isoshielding contours. The AB/B contours in
the xy and IX planes are shown in Figs. 5 and 6, respectively.
In the xy plane (see Fig. 5), the isoshielding lines form a complex pattern for R
values less than 0.2 nm, whereas at larger values of R the contours, as expected, are
independent of @. Similarly, in the zx plane (see Fig. 6), the isoshielding lines form a
complex pattern for R values less than 0.3 nm. At very large values of R the pattern
approaches the contours proportional to (3 co? 0 - 1)/R’. These diagrams illustrate clearly the complex patterns formed by the isoshielding lines for a wide range of
R, namely, R = 0 to 0.5 nm.
To investigate further the method of using [l] and/or [2] to interpret paramagnetic
NMR shifts, we choose a set of AB/B values over a temperature range for this case
641)
GOLDING
AND
‘TABLE,
A SUMMARY
OF ‘I-HE RESULTS
FOR AB/B AS AR~SINCJ FROM A
INTERACTIONS
AS GIVEN
BY
TRANSITION
METAL
ION IN A
STUBBS
3
OF FI’ITING
A SET OF THEORETIC‘AI
11~1,
SUM OF THE FERMI CONTACT
ANU DIPOI AR
Eos.
[l) AND [2], FOR THF. CASE OF A ‘1”
STRONG
CRYSTAL
FIELD
OF TETRAGONAI
SYMMETRY
AB/B fitted to Eqs. [l] and [2]
ABIB (ppm)
T W
Exact
Contribution
from [l]
Contribution
from [2]
Total
200
220
240
260
280
300
320
340
360
380
400
-35.56
-30.19
-25.37
-21.12
-17.41
-14.19
-11.41
-9.03
-6.98
-5.24
-3.74
-2.69
-2.67
-2.65
-2.62
-2.59
-2.55
-2.52
-2.48
-2.44
-2.40
-2.36
-32.91
-21.51
-22.71
-18.49
-14.82
-11.65
-8.92
-6.S7
-4.55
-2.83
-1.35
-35.60
--30.18
--25.35
-21.11
-17.41
-14.20
- 1 1.43
-9.05
-6.99
m-5.23
-3.71
when the NMR nucleus is 0.2 nm along the x axis. The analysis of fitting the
temperature dependence of AB/B for the exact solution for a tetragonal distortion
component given by 6 = 1000 cm-’ to a combination of [l] and [2] is summarized in
Table 3. Such an analysis yields a distortion parameter and a distance R from [2] of
650 cm-’ and 0.220 nm, respectively, with substantial contributions from both the
Fermi contact and dipolar interactions. The result further supports our previous
finding that care needs to be exercised in interpreting NMR results of paramagnetic
systems.
APPENDIX:
MOLECULAR
HYPERFINE
INTEGRALS
FOR
2p ORBITALS
(a) The Radial Series
The method of evaluating the molecular hyperfine
Golding and Stubbs (14). Define the radial integral
m
R:=‘(t) = 4@:(-Rf
1 r;-Lb,(R,
0
and
Cl” (t) = Rc2’(t)
”
3
V,,(r) = R;‘(t),
W,,(t) = R?‘(t),
integrals is that given by
rN) drN
NMR SHIFTS IN PARAMAGNETIC
641
SYSTEMS
where t = 2/3,R and
b,(R, TN)= (rJr41’2Zn+1,2
(2P1r4Kn+3,2
(Wlr,)
- (r.Jra)1’2Zn--1,2 (2P1r4Kn+1/2
G&r>),
where r< is the smaller of R and rN and r, is the larger of R and rN and Z, and K, are
the modified Bessel functions.
The following radial integrals for 2p orbitals are required:
U*(t)=&[;-(f+t+2+~)
e-j,
vlit)=-P:[;-(t+2+;)
e-j,
VAtI=
-p:[(f-f)+(it+Y+Y+F+F)
e-11,
Wdt)=p:[~-(l+;)
e-j,
w*it)=#-$)+(2+y+$+$-q
e-j,
w4(t)=P’[(f-fy+y)-(3+T+~+~+~+~)e
8
32
200
760
1680
1680
(6) The Integrals (Pi) Tao / Pii>
Employing
the notation of Griffith
(IS), define
-G”(@, a) = YLO(@, @),
ZE4(0,
@) = (1/21’2)[YL-M(0,
Z%(O,
@) = (i/2”‘)[
@)+ YL.f(@,
YL-M(O, @)- Y&(0,
CD)],
@)I.
For the radial parts of the integrals define
f,(a)=
U2+aV1+(2-a)V3+
W,
The integrals ( PiI Tea) Pi) are
(x~T~,~x)=Z(~)“~~~(O)~~~~~,
oi-~(~)“;,(o)zk;‘(e,
@)
-
r].
642
GOLDING
AND
STUBBS
NMR SHIFTS IN PARAMAGNETIC
SYSTEMS
643
644
GOLDING
AND
STUBBS
645
646
GOLDING
(c) The Integrals ( ?Pijldrf,)
The integrals (‘Pi/I&&I
iblhdr~
Iv) = -(E)
i(xIINylritly)=
“‘(VI
= --i(3”‘(VI+
i(ylldr;tilz~
= -(fi)1’2(VI+
= - (fi)
+ W2)Z:“: (0, @I,
W2)Z:“:(@,@),
i<xLlr~ly>
iW.dr~l~)
Pi}
Pi) are given by
-(&)“i(Vl+
+Y
AND STUBBS
W2)Z2,(0,
O)+q(V,+
O)+~(~)1’2(VI+
W2)Z:“:(@,
1’2( VI + WZE
-(fj)“2(VI+
@)
(@, @),
@I,
W2)Z:“:(O,@),
v1+
+T
cv1+
i(zllNz/rilx>=
W2)Z2,(0,
(V1+ Wo)Zoo(@, @),
~(yL/&lz>= -(fi)1’21V~+W2)Z:“:(@,
i<zI&.J&lx>=
Wo)Zoo(O, @),
-(E)1’2(VI+
@)+f(;y2(v,+
w,)z:‘:(@,
Wo)Zoo(@,
W2)Z:“:(0,
W2)Z*o(O,
@)
@I,
@).
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
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IN
PARAMAGNETIC
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