THE ISOTROPIC HYPERFINE INTERACTION
possibility of using endor to analyze for different nuclear spin transition selection rules. An analysis of the relevant experimental results showed that they are consistent with both the END mechanism as the dominant “endor-active” process
51
and with exchange processes acting to reduce the endor signals. Acknowledgment. We wish to thank Dr. James S. Hyde for communication of his experimental results prior to publication.
The Isotropic Hyperfine Interaction’
by B. R. McGarvey Department of Chemistry, Polytechnic Institute of Brooklm, Brooklyn, New York (Received September 27, 1966)
The isotropic contribution to the hyperfine interaction resulting from exchange polarization of inner s electrons has been computed for first-, second-, and third-transition metal ions from electron spin resonance data. Various trends with respect to bonding and the periodic table are noted. A perturbation approach to the calculation of the isotropic hyperfine term is developed and used to explain some of the observed trends.
Introduction The isotropic contact term in the hyperfine interaction of paramagnetic ions has been the subject of many studies.2 Abragam, Horowitz, and Pryce3 have observed that the quantity x, defined as
is negative and of nearly constant magnitude for ions in the first transition series. Since unpaired electrons in d orbitals cannot contribute to x, they proposed that the finite value of x results from a polarization of the inner filled s orbitals by the unpaired d electrons. Their attempt to calculate x by using configuration interaction with the excited state resulting from the promotion of a 3s electron to a 4s orbital was, however, unsuccessful. More successful calculations4 of x have been made using unrestricted Hartree-Fock methods. These calculations revealed that the polarization of both 2s and 3s orbitals is important and that the negative contribution of the 2s shell dominated the positive contribution of the 3s shell.
Matamuras and Title6 have noted that x for d5 ions decreases as the electronegativity difference between the anion and cation of the host lattice decreases. A similar behavior has been assumed for x in copper complexes by several investigators’ when calculating molecular orbital coefficients from spin-Hamiltonian parameters, but Kuska and Rogers8 have recently reported some measurements on substituted copper(I1) acetylacetonates which contradict such an assumption. (1) Work supported by NSF Grant GP-4215. (2) For a complete review see A. J. Freeman and R. E. Watson, “Magnetism,” Vol. IIA, G. T. Rad0 and H. Suhl, Ed., Academic Press, Inc., New York, N. Y., 1965, p 167. (3) A. Abragam, J. Horowitz, and M. H. L. Pryce, Proc. Roy. SOC. (London), A230, 169 (1955). (4) J. H. Wood and G. W. Pratt, Jr., Phys. Rev., 107, 995 (1957); V. Heine, ibid., 107, 1002 (1957); R. E. Watson and A. J. Freeman, ibid., 120, 1134 (1960); A. J. Freeman and R. E. Watson, ibid.. 123, 2027 (1961). ( 5 ) 0. Matamura, J . Phys. SOC.Japan, 14, 108 (1959). (6) R. S. Title, Phys. Rev., 131, 623 (1963). (7) D. Kivelson and R. Neiman, J . Chem. Phys., 35, 149 (1961); H. R. Gersman and J. D. Swalen, ibid., 36, 3221 (1962). (8) H. A. Kuska and M. T. Rogers, ibid., 43, 1744 (1965).
Volume 71, Number 1
January 1967
B. R. MCGARVEY
52
Geschwind2p9 has observed that isoelectronic ions in the same host lattice have similar values of x, indicating that x is not strongly dependent on the charge of the ion. In this paper are presented the results of calculating x for ions of most configurations with a variety of ligands. Various correlations and trends will be noted and an attempt to understand these results, theoretically, will be reported upon.
are largest in this case. Also certain conclusions reached in this case will be found useful for the treatment of other configurations. For Cu2+ complexes the crystal field is most often octahedral with a large tetragonal distortion. In this case the pertinent molecular orbitals for the positive hole have the form u = adzi-yz -
Calculation of x from Experimental Data In the following calculations it is necessary to have values of
P = 2.0023g~p&~(r-~)~~
(2)
for the various ions. p, and /3N are the Bohr and nuclear magnetons, respectively, and gN is the nuclear g factor. The values of P used in the following computations are listed in Table I in units of cm-I. , , obtained from the HartreeThe values of ( F ~ )were Fock calculations of Freeman and Watson.2 The numbers followed by an asterisk were obtained by either interpolation or extrapolation.
cm-1
47149Ti3-I-
-25.7* 85.7 128
59C~+ 50CoZ+
WNb4+
56Mn2 + 6Wn3+ %/In4 +
-45.0 -50.2* 187 211 235,
56MnK + 67Fe3+
282* 32.9
+
a Asterisks indicate or extrapolation.
Ian
06MoS +
+
WMO3
D6Mo5 f
97M05+ OOTcZ + 107AgZ + 109AgZ +
228 254 112
192* -54.4 -55.6 -66.7* -68.2* 200 -62.5 -73.6
values obtained by interpolation
d1 and d9. These two configurations are treated in similar fashion since the d9 configuration can be treated as a single-hole configuration. They are also the most difficult ones for which good values of x can be obtained, owing to the large dipolar and indirect dipolar contributions to the hyperfine coupling. The d9 case will be considered first because the indirect dipolar terms The JOUTnd of Phyaical Chemistry
y2)
- P’~$L(w)
T
= 8dzy
=
Pldzz,vz - Pl’+L(ZZ,YZ)
(3)
= gIIPeH,Sz
+ gLPe[H,SZ + H,S,I + AIzS, + B[IzSz + IYSVl
(4) and hyperfine terms, A and B, can be related to the molecular orbitals in eq 3 by the equations’*’O
- 4a2P + (gll 7
- 2.0023)PZ11
+
104~, om-1
1 0 4 ~ ~
Ian
51\79
-
where the ligand atoms are along the z and y axes. For Cu2+the a-antibonding orbital is the ground-state orbital and T and sl are excited states which are connected to the u orbital through the spin-orbit interaction. The esr of the Cu2f complex can be fitted to the spin-Hamiltonian
A = -K
Table I : Values of P for Transition Ions”
61V
Tl
a’i$L(~’
B
=
11 + 72-a2P + -(g1 14
-K
T(n) = n - ( 1
- 2.0023)PZL
- n2)”zR8(ZpZ,)”/” X (2s
- ZP)/(Z,
(5)
+ zp)5a0
Zp and Z, are the effective changes for p and s electrons on the ligand atom, R is the distance between the metal ion and the ligand atom, and n2 is the fraction of p hybridization in the ligand orbitals making up @L(z2y2). Equations 5 were obtained assuming no overlap for the T orbitals. K in eq 5 is the isotropic contact term and x can be calculated from it using
x=
--
(9) S. Geschwind, J. A p p l . Phys., 36, 920 (1965); P. R. Locher and S. Geschwind, Phys. Rev. Letters, 11, 333 (1963). (10) A. H.Maki and B. R. McGarvey, J . Chem. Phys., 29, 31, 35
(1958).
THEISOTROPIC HYPERFINE INTERACTION
Table XI : Comparison of Approximate Method of Calculation of Complete Calculation of Several Copper(11) Complexes
53
and x with
( ~ 2
-Approximate----Host crystal
Pd[(CHaCO)zCH]2. Ni[0C6H4CHNH]p Zn [SZCN(CZH&*
Zl I
1.08 1.14 1.01
Complet-
ZL
a,
x , au
at
x , au
1.05 1.11 1.26
0.74 0.74 0.51
-3.45 -3.45 -2.57
0.77 0.78 0.50
-3.64 -3.67 -2.64
' Reference 10. '' T. R. Reddy and R. Srinivasan, J. Chem. Phys., 43, 1404 (1965).
I n eq 6, K is in units of cm-l and x is given in atomic units. The second term in the equations for A and B is the direct dipolar term and the terms involving gib and gl are indirect dipolar terms resulting from both the nuclear and electron spins being coupled to the orbital motion of the electron. The indirect dipolar terms are particularly large for Cu2+owing to the large value of the spin-orbit coupling parameter. 211and 2, can be obtained from a knowledge of 911, g,l S, the spin-orbit coupling parameter, and the energy difference between the c orbital and the excited R and r1 orbitals. Not all of these quantities are available for many complexes, so it was assumed for the calculations that 21, = 2, = 1.0 in every case. A comparison of values for cy2 and x obtained with this assumption and from a more complete calculation is made for three complexes in Table 11. It is seen that the values of x differ by only 0.1 au and a2 by 0.03. Thus we see that good estimates of the indirect dipolar terms can be made by using the experimental g values and the P value of the free ion, and this is how these terms will be estimated in the calculations for other configurations. In these calculations P = 360 X cm-l was used instead of the value given in Table I because this is the value used by most authors' for Cu2+ and was the value used in the complete calculz+ tions. In Table I11 are given calculated values of x and a2 for the d9 configurations of Cu2+, Ag2+, and Au2+ along with the values of 911, gl, A , and B from which they were calculated. The signs for A and B in parentheses are assumed. For those cases where the z and y values of g and A are different, the average has been taken and reported as g, and B. For copper the value of P = 360 X cm-l was used so that the values of a2would be comparable to what others have reported. The value of x is not very sensitive to the exact choice of P . For Au2+, (r3),, was estimated to be 9.6 au giving P = 28 X cm-'. This was not a critical estimate since the closeness of g to 2.0023 made x nearly independent of the value used for P . For the isotropic
solution spectrum of Au2+ as well as the octahedral cases of MgO, CaO, and high-temperature hexahydrates, the value of x was obtained from the equation 1 (A) = -(A 3
+ 2B) = - K + (8,"
- 2.0023)P (7)
No value of a2 is reported for Cs2ZnClr because Cu2+ is in a tetrahedral site and in tetrahedral symmetry the ground-state 3d orbital can mix with the 4p orbitals which contribute dipolar terms of opposite sign to those from the 3d orbital. Sharnoff" has shown that the small dipolar terms in this case are due primarily to an admixture of 4p orbitals into the ground state rather than a large covalency of the bonds. It should be mentioned that only a few results from frozen solution studies of copper complexes have been included in Table I11 because they are generally of lower accuracy, particularly in the determination of B. In some instances values reported in the literature are obviously incorrect.12 For d1 configurations the unpaired electron is found in either the d,, or the d,, orbital. For the d,, orbital the equations used for A and B were A
=
-K
- 47-B2P + (gll - 2.0023)P + ?(gl
7 B
=
-K
+ 72-p2P + g(,:
- 2.0023)P
(8)
- 2.0023)P
while for the d,, orbital the equations were
B = -K - -p2p 2 f 15 -(gL 7 14
- 2.0023)P
(9)
(11) M. Sharnoff, J . Phys. Chem., 41, 2203 (1964); 42,3383 (1965). (12) J. A. McMillan and B. Smaller, ibid., 35, 763 (1961).
Volume 71, Number 1 January 1967
B. R. MCGARVEY
54
Table 111: Values of x and
cyz
for do Ions 1 0 4 ~ ~
Host lattice
Ql I
cm -1
Qlbb
104~,bb
X.
cm -1
au
Ref
a’
Cu2+ (3dQ) MgO CaO MgaBi~(N0&~.24HzO(90°K) Mg3Biz(NO&. 24H20 (20’K) Mg~Laz(N03)12.24H~0 (90’K)
Ag( C1O4)2in HClOr AgSOc in HzO Ag( NO3)2 in HN03 Ag[(CaH?)2NCSzlz Ag deuterioporphyrin I X Cd(CsH6N)aSzOe
i-
2.190 2.221 2.219 2.454
( 119 (-)21.5 j27
2.096
(-)110
2.470 2.344 2.2661 2.2004 2.086 2.1085 2.179 2.162 2.171 2.172 2.287 2.273 2.264 2.087 2.083 2.25 2.169 2.339 2.446
2.097 2.099 2.0535 2.0448 2.024 2.023 2.050 2.047 2.020 2.042 2.066 2.062 2.042 2.023 2.024 2.05 2.062 2.070 2.092
-113 -88 160 185 (-)162 (-)142.4 ( - )202 (-)215 (-)183 ( -)162 (-)171 (-)171 (-)169 154 (-)171 (-)199 ( - )188 (-)113 ( - 125
2.337 2.265 2.341 2.035 2.104 2.18
2.071 2.065 2.069 2.011 2.029 2.04
2.219
( -129
-
-
Agz+ (4dQ) ( )51 (+I51 ( )49 ( 137 (+)71 ( )34
+
-2.72 -3.12 -3.28 -3.18 -3.34 -3.31 -3.40 -3.43 -3.43 -2.96 -2.57 -3.53 -3.83 -3.53 -3.15 -3.97 -3.90 -3.34 -2.81 -3.25 -3.68 -3.84 -2.96 -3.62 -9.16 -9.10 -9.16 -5.82 -9.44 -6.30
+ + +
... ...
... 0.90
... 0.92 0.57 0.74 0.74 0.49 0.51 0.78 0.77 0.65 0.63 0.76 0.70 0.79 0.48 0.48 0.87 0.65 0.73
... 0.76 0.63 0.71 0.26 0.68 0.40
a,
b
a C C
c, d C
e
f f 9
h i j, k
1 1 m n 0
P Q
r S
t U
21 W V
P, x Y z, aa
A u ~ +(5d9) 2.040
(
- 128
-15.1
...
X
* P. Auzins, J. W. Orton, and J. E. Wertz, “Paramagnetic Resource,’’ Vol. 1, W. Low, Ed., Academic Press Inc., New York, N. Y., 1963, p 90. W. Hayes, see J. W. Orton, Rept. Progr. Phys., 22,204 (1959). B. Bleaney, K. D. Bowers, and R. S. Trenam, Proc. D. Bijl and A. C. Rose-Innes, Proc. Phys. SOC.(London), A66,954 (1953). H. J. Gerritsen Roy. SOC.(London), A228,157 (1955). and A. Stan, Arkiv Fysik, 25, 13 (1963). Reference 10. A. H. Maki, N. Edelstein, A. Davidson, and R. H. Holm, J . Am. Chem. SOC.,86, 4580 (1964). Footnote b, Table 11. S. E. Harrison and J. M. Assour, J . Chem. Phys., 40, 365 (1964). J. E. Bennett and D. J. E. Ingram, Nature, 175,130 (1955). R. M. Deal, D. J. E. Ingram, and R. Srinivasen, Proc. X I I , Colloque Ampke Bordeaux, Reference 7. G. F. Kokoszka, H. C. Allen, and G. Gordon, J . Chem. Phys., 42, 3730 (1965). * H. C. Allen, 1963, 239 (1964). G. F. Kokoszka, and R. G. Inskeep, J . Am. Chem. SOC.,86, 102 (1964). H. G. Hecht and J. P. Frazier, 111, J. Chem. Phys., 44, 1718 (1966). * R. Petterson and T. Viinnghrd, Arkiv Kemi, 17, 249 (1961). * E. Billig, R. Williams, I. Bernal, J. H. Waters, and H. B. Gray, Inorg. Chem., 3,663 (1964). ‘H. Abe and M. Ohtsuka, J . Phys. SOC.Japan, 11,896 (1956). * E. M. Roberts and W. S. Koski, J . Am. Chem. Soc., 82, 3006 (1960). J. H. M. Thornley, B. W. Mangum, J. H. E. Griffiths, and J. Owen, Proc. Phys. SOC. Reference 11. ’ N. S. Garifyanov, B. M. Kozyrev, and E. I. Semenova, Dokl. Akad. Nauk SSSR, (London), 78, 1263 (1961). J. A. McMillan and B. Smaller, J . Chem. Phys., 35, 1698 (1961). ‘T. Vanngard and S. Akerstrom, Nature, 184, 147, 365 (1962). F. K. Kneubuhl, W. S. Koski, and W. S. Caughey, J. Am. Chem. Soc., 83, 1607 (1961). ’ H. M. Gijsman, H. J. Ger183 (1959). ritsen, and J. van der Handel, Physica, 20, 15 (1954). T. Buch, J. Chem. Phys., 43, 761 (1965). bb When different x and y values were reported, the average was taken.
’
’
’
@ 2 in eq 9 is the fraction of time spent by the electron in the d,, orbital of the metal ion. The same symbol is used as in d,, because in those cases where d,, is
the ground state it forms vantibonding molecular orbitals just as d,, does. Equation 9 is obtained on the assumption that the spin-orbit parameter X is
THEISOTROPIC HYPERFINE INTERACTION
55
small compared to the energy separation of d,, and d,,,,,. For some cases this appears not to be correct and in these cast.s the following equations were used y
B = -K
“ I
- --sin2y 7
- 7-PP + [-7 sin2 y + 14 2
15
4
L “
gll = 2.0023 cos 2y
gL = 3.0023 cos2 y
tan2y =
-- “7 1 “k
’/27
; rl=
k sin 2y
x Ex,,$, - E,,
= (dill+ldo)
1
sin 2r p
+ 2k sin2 y
+d
P
(10)
ligand rather than on the metal ion. The signs assumed for the corresponding chromium complexes were chosen to give values of x similar to those of vanadium, since the evidence is strong that x changes very little for first transition series ions in similar environments. d2. For the d2 configuration the only reported esr results are for the ion in a tetrahedral crystal field. In this case both g and A are isotropic and A is given by
A
=
-K
+ (9 - 2.0023)P
(11)
Table V has the values of x computed for the few ions reported. d3. The d3 ions are always found in octahedral sites for which the g and A terms are isotropic and A is given by eq 11. The computed values of x are given in Table VI. For Tc4+ the value of P for Tc2+ was used since any reasonable choice of P will give a correct x owing to closeness of g to 2.0023. For Re4+ an estimated value of P = 241 X cm-l was used in calculating x. If we had taken P = 0.0, x would be -14.5 au, so we can estimate that, at the greatest, the error in x is probably less than 1 au. d4. The esr spectrum of &In3+ in TiO, has been reported by Gerritsen and Sabisky.16 If we assume the unoccupied orbital is d,2-,2, then A and B are given by
I n the case of isotropic solution spectra, eq 7 was used to find x. I n Table IV are given the calculated values of x and p2 for d’ ions. The value of P used for V02+ions was that listed for the V2+ ion in Table I. The value of p2 is observed to be constant for the various complexes13 and this is taken to indicate that the ‘IT bond to ligands in the zy plane is essentially ionic and therefore 02 should be near unity. It is found that p2 is -0.9 if we use P for the V z + ion. Thus the effective 1 charge on vanadium in the V 0 2 + ion is 2+ owing to A = -K -e2P 7 charge transfer from the oxygen atom. The same value (12) 1 was also used for V4+. Since the effective charge on vanadium is 2 less than its oxidation number, it is assumed a similar effect occurs for other ions, and thus where e is the molecular orbital coefficient for d,,, which values of P for S b 2 + , Cr3+, and Xo3+ were used for forms a o-antibonding orbital. I n obtaining eq 12, Xb4+, Cr5+, and lI05+. For Mn6+ the value of P it was assumed that the d,,, d,,, and d,, orbitals all for Xn2f was used, and for Ti3+the Ti3+value was had the same coefficients and that the second-order used. For the third transition series ions, x was computed assuming f12 = 1.0 since no values for ( T - ~ ) ~ , , dipolar contributions could be neglected, since the g values art! close to 2.00 (911 = 1.99, g1 = 2.00). were available. It should be emphasized that the Using the reported values of A and B,15A = (-)52.8 choice of P has only a slight influence on the value comX cm-l and B = (-)82.6 X cm-I, we obputed for x because, in most instances, the g values are tain x = -2.45 au and cy2 = 0.71. close to 2.0023. d5. For the weak crystal field, the d5ion is in an 6S The ions V [S2C2(CPi)2]32- and 1/10[SZCZ(CF&]~state for which there are no dipolar or second-order present a special problem. They should both have the dipolar contributions to the hypefine interactions. symmetry D3 and we would expect the ground-state Therefore orbital to be dz2. I n the case of V[SZCZ(CK)Z]~~-, we know from the isotropic solution hyperfine splitting A = -K (13) that A and I3 have the same sign and the only reasonValues of x for d5ions are given in Table VII. able assignment for a d,, ground state is‘to take both A In Table VI11 are given values of x and p2 for some and B as positive, which gives the positive value of x cases of d5 in a strong crystal field. For the nitrosyl listed in Table IV. As we shall see later, a positive value of x is not unlikely for this case. A similar situa(13) D. Kivelson and S.Lee, J . Chem. Phys., 41, 1896 (1964). tion pertains to 310[S2C2(CF3)2]3-. The small values (14) A. Davidson, N. Edelstein, R. H. Holm, and A. H. Maki, of p2 for both ions support the contention of DavidJ . Am. Chem. SOC.,8 6 , 2799 (1964). et al., that the unpaired electron is mainly on the (15) H. J. Gerritsen and F. S.Sabisky, Phys. Rev.,132, 1507 (1963).
+
Volume 7 1 , Number 1 January 1967
B. R. MCGARVEY
56
Table IV: Values of x and B* for d1 Ions Host lattice A1f(CHaC0)zCHls TiF2(CH30H)4+ TiClz(CH30H)4+ ( C~Hs)pTicl~Al( CzH5)z
Qll
OlPP
2.00
1.921 1.9465 1.951 1.976
lO'A, crn -1
1 0 4 ~ ~ cm -1
Ti3+ (3d1) (f16.3 (+)17.5 (+)16.7 (+)IS. 3 (+ D0.2
x.
au
Ref
-2.15 -2.29 -2.54 -1.42
0.59
-3.09 -3.34 -2.86 -2.79 -2.76 -3.17 -3.12 -3.37 -3.31 -3.31 -2.78 -2.70 -2.74 -3.27 -3.24
0.93 0.93 0.87 0.89 0.91 0.90 0.93 0.92 0.78 0.78 0.91 0.92 0.88
-2.01 -1.87 -1.96 +1.94 -2.08
0.92 0.95 0.87 0.43
-2.68 -2.64 -2.56
0.79
...
...
a b Cl
d
e1
f
VO2+ (3d1) ( NH4)~InC16HzO Zn( NH4)4SO& 6Hz0 Kz[TiO( CZO& '2Hz0 HzPc ZnPc GeOz VO[( CH~CO)ZCH]Z VOClz in 30% HCl VOC~Z in C&OH VOSOd in CZH~OH VO tetraphenylporphyrin VO etioporphyrin I VO etioporphyrin I1 VOFz in 20% H F VOClz in HzO 9
-
1.9450 1.9847 1.9331 1.9807 1.972 1.940 1.989 1.966 1.987 1.966 1.976 1.929 1.944 1.996 1.93 2.00 1.98 1.92 1.98 1.92 1.964 1.989 1.948 1.987 1.947 1.989 1.968 1.962 1.955 1.943 1.963 1.974
1.913 1.921 1.921 1.974 1.99
K&rOCh CrOC14- in CH~COZH CrOC14- in 20% HCl
2.008
1.977 1.9877 1.986
( -)173.0 (-)182.81 (-)163 (-)158 (-)158 ( -)175.5 ( -)173.5 (-)182 (-)171 (-)171 (-)159.1 (-)159 (-)158
(-)63.8 (-)71.82 ( - 160 ( )56 (-)54 (-)68.2 (-)63.5 ( 171 ( 176 ( - 176 (-)54.2 ( -152 ( - )54
-
-
(-)lo6 (-)lo6 V4 + (3d*) ( -)141.5 (-)140.3 (-)134.36 (+)92.4 (-)66.1
(-)37.5 )31.5 (-)37.11 (+141 (
Cr08+ (3d') (+)36.1
-
(+)9.7 (+)18.2 (+)17-.7
9
h i jl
k 1 m n 0
0
P P T
0
...
5
t-v W
1 21
Y
z
...
aa bb
cc
Cld+ (3dl) 1.973 1.9941 1.996 KzCrO4
...
2
...
ad
- 129
-1.91
0.60
ee
(+)32.6 (+)38.5 (+)34.5
-5.61 -6.97 -7.60 -6.26 -5.44
0.88 0.61 0.97 0.83 0.66
(+)41.18 (+)27.57 (+)39.1 (-)8.96
-3.54 -4.01 -3.91 +1.48
0.55 0.79 0.52 0.16
-5.05 -3.07
0.80 0.80
(-)14.8 (-)17.8
1.938
Mn8+ (3d') 1.968 (-)135
1.9632 1.965 1.874 1.951 1.928
1.9400 1.940 1.918 1.939 1.944
1.987 1.9125 1.9981 2.011
1.887 1.8000 1.9889 2.009
1.965 1.910
1.809 1.844
M003f (4d') (+)74.7 (+)76.1 (+ 199.8 ( 179 (+)68. 4
+
cc
-2.47 $2.26 f2.69
(+ )17.6
(
+
( P7.0 (+ )48.2
9
ff BB hh
ii
Mol+ (4d')
Nb( 0CHs)Cls'SnOz
The Journal of Physical Ch?nktry
(+)8.39 (+)65.85 (+)14 (-)16.3
Nb02+ (4d') ( - )232 (-)186
(-)135 ( -)TO
jj
kk 11 2
mm
nn
THEISOTROPIC HYPERFINE INTERACTION
Table IV
57
(Continued) Host lattice
104~~ cm -1
Qll
QlPP
1.9819
1.9677
1.90
1.77
W6+
ReOCl4 in H2S04
(5d1) (-)18.7
Reo4+( 5d1) ( -)425
104~,pp
cm-1
(-)68.5 (-)331
XI
au
P2
-10.4 -12.7
Ref
11
...
00
a B. R. McGarvey, J . Chem. Phys., 38,388 (1963). E. L. Waters and A. H. Maki, Phys. Rev., 125,233 (1962). N. S. Garifyanov, Ai V. Danilova, and R. D. Shagidullin, Opt. i Spektroskopiya, 13,212 (1962). N. S. Garifyanov, E. I. Semenova, and N. F. Usacheva, Zh. Strukt. Khim., 3,596 (1962). ‘A. H. Maki and E. W. Randall, J. Am. Chem. SOC., 82,4109 (1960). P. E. M. Allen, J. K. Brown, and R. M. S. Obaid, Trans. Faraday SOC.,59,1808 (1963). K. De Armond, B. B. Garrett, and H. S. Gutowsky, J . Chem. Phys., 42, 1019 (1965). * R. H. Borcherts and C. Kikuchi, ibid., 40,2270 (1964). R. M. Golding, Mol. Phys., 5, 369 (1962); Trans. Faraday SOC.,59, 1513 (1964). D. J. E. Ingram and J. E. Bennett, Discussions Faraday SOC.,19, 140 (1955). J. M. Assour, J. Goldmacher, and S. E. Harrison, J . Chem. Phys., 43, 159 (1965). I. Siegel, Phys. Rev., 134, A193 (1964). Reference 7. N. S. Garifyanov and S. E. Kamenev, Zh. Eksperim. i Teor. Fiz., 46, 501 (1964). ’ N. S. Garifyanov and N. F. Usacheva, Zh. Fiz. Khim., 38, 1367 (1964). p D. Kivelson and S. Lee, J . Chem. Phys., 41, 1896 (1964). D. E. O’Reilly, ibid., 29,1188 (1958). ‘ E. M. Roberts, W. S. Koski, and W. S. Caughey, ibid., 34, 591 (1961). * N. S.Garifyanov and B. M. Kozyrev, Dokl. Akad. Nauk SSSR, 98, 929 (1954). G. M. Zverev and A. M. Prokhorov, Zh. Eksperim. i. Teor. Fiz., 39,222 (1960). ” H. J. Gerritsen and H. R. Lewis, Phys. Rev., 119, 1010 (1960). ’ E. Yamaka and R. G. Barnes, ibid., 135, A144 (1964). C. Kikuchi and I. Chen, J . Chem. Phys., 42, 181 (1965). A. Davidson, N. Edelstein, R. H. Holm, and A. K. Maki, J. Am. Chem. SOC.,86,2799 (1964). ” A. Davidson, N. Edelstein, R. H. Holm, and A. H. Maki, Znorg. Chem., 4, 55 (1965). ’ J. C. W. Chien and C. R. Boss, J . Am. Chem. SOC.,83,3767 (1961). aa H. Kon and N. E. Sharpless, J . Chem. Phys., 42,906 (1965). H. Kon, Bull. Chem. SOC.Japan, 35, 2054 (1962); J . Znorg. Nucl. Chem., 25, 933 (1963). cc N. 8. Garifyanov, Dokl. Akad. Nauk SSSR, 155,385 (1964). dd J. H. Waters, R. Williams, H. B. Gray, G. N. Schrauaer, and H. W. Finck, J . Am. Chem. SOC.,86,4198 (1964). A. Carrington, D. J. E. Ingram, K. A. K. Lott, D. S. Schonland, and M. C. R. Symons, Proc. Roy. SOC.(London), A254,lOl (1960). ” N. S. Garifyanov and V. N. Fedotov, Zh. Eksperim. i Toer. Fiz., 43, 376 N. S.Garifyanov, V. N. Fedotov, and N. S. Kucheryavenko, Zzv. Akad. Nauk SSSR, Ser. Khim., 743 (1964). hh J. Owen (1962). and I. M. Ward, Phys. Rev., 102, 591 (1956). N. S.Garifyanov, B. M. Kozyrev, and 17. N. Fedotov, Dokl. Akad. Nauk SSSR, 156, 641 (1964). G. H. Azarbayejani and A. L. Merlo, Phys. Rev., A137, 489 (1965). kk R. Kyi, ibid., 128, 151 (19623. “ B. R. McGarvey, Inorg. Chem., 5, 467 (1966). m m P. G. Rasmussen, H. A. Kuska, and C. H. Brubaker, Jr., ibid., 4, 343 (1965). nn W. H. From, P. B. Dorain, and D. R. Locker, Bull. Am. Phys. SOC.,11,220 (1966). I o N. S.Garifyanov, Zh. Eksperim. i Teor. Fiz., 45,1819 (1963). p p When different z and y values were reported, the average was taken.
’
’
‘
@
’’
Table V : Values of Host lattice
x for d2 Ions 0
104~,
XI
cm -1
au
Ref
CdS
V a + (3d2) 1.933 (-)65
-1.77
a
Si
Cr4+( 3d2) 1.9962 (+)12.54
-1.84
u
Si NaaVO4-12HzO
Mn6+ (3d2) 2,0259 -63.09 2 (-)62.5
-2.34 -2.13
a b, c
a F. S. Ham and G. W. Ludwig, “Paramagnetic Resonance,” Vol. 1, W. Low, Ed., Academic Press Inc., New York, N. Y., 1963, p 130. Footnote ee, Table IV. A. Carrington, D. J. E. Ingram, D. Schonland, and M. C. R. Symons, J. Chem. SOC., 4710 (1956).
complexes, eq 8 was used since the unpaired electron is in a d,, orbital, and for the solution resonances eq 11 was used. d7. When a ti7 ion is in a tetrahedral or cubic field,
it is similar t,o a d3ion in an octahedral field and both g and A are isotropic. Thus we can use eq 11 for d7 in a tetrahedral field. The problem is more complex for an octahedral field since, in this case, the orbital motion is not quenched by the crystal field. For an undistorted octahedral field, the isotropic hyperfine interaction has been shown by Low16to be given by 5 A = --K 3
+ 63-(I2
-159)+ ~
where r 2 is a mixing coefficient for the excited *P state. The analysis of g and A for a small distortion in an octahedral field has been done by Abragam and Pryce17 and it has been found that the behavior is too complex to allow good va,luesof x to be computed. I n Table IX are given values of x computed for Co2+ (16)W.Low,Phys. Rev., 109, 256 (1958). (17)A. Abragam and M. H. L. Pryce, Proc. Roy. soc. (London), A206, 173 (1951).
Volume 71,Number 1
January 1967
B. R. MCGARVEY
58
Table VI: Values of x for d3 Ions Host lattice
lO‘A, cm-1
x , au
-75.1 (-)76.0 (-)70.7 (-)86.2 (-)93.0 -82.5 -83.9 (-)76.7 -55.5 ( )26 (-)88.0
-2.32 -2.30 -2.18 -2.64 -2.73 -2.53 -2.58 -2.32 -1.74 -0.84 -2.67
(+)16.0 (4-116.8 (+)16.2 (+)l5.3 (+)18.5 (+)17.0 (+ )16.7 (+)16.7 (+)16.2 (+)16.5 (+ )14.7 (+)16.8 (+)I7 ($117 (+)14.9
-2.27 -2.35 -2.29 -2.20 -2.62 -2.41 -2.38 -2.38 -2.33 -2.32 -2.14 -2.39 -2.4 -2.3 -2.12
(-)70.8 (-)69.4 (-)70.0 ( -)71.8 (-)72.4 (-)72.0
-2.35 -2.30 -2.16 -2.36 -2.36 -2.45
ff
+)39
-4.50
11
( -)137.8
-5.00
mm
0
Ref
V i + (3d3)
KMgF3 CaFz (NH~)zZ~(SO 6HzO ~)Z. ZnSiF6 CdCL K4Fe(CN)6.3H20 \r(C&)Z VCL in HzO
1.9800 1.9683 1.9896 1.9720 1.939 1.9728 1.974 1.9690 1,9920 2.00 1.965
MgO CaO SrTi03 MgS KAl(SeO&. 12H20 AlC13 6Hz0 A1 [(CHsCO)zCH]a CO[(CH~CO)ZCHI~ Co(en)lCla * NaCl .6Hz0 t-[CoClp(en)z]Cl.HCl .2Hz0 K3Col CN)6 A1203 TiOz ZnW04 ( NH4)zInC16. HzO
1.9800 1.9734 1.9780 1.9874 1.976 1.977 1.9820 1.9802 1.9874 1.9765 1.991 1.980 1.97 1.963 1,9842
MgO
CaO LlgS
-
a, b C
d e
f g, h, i 9 k 1, m n, 0
P
Cr3+ (3d3) bl
d, 4, r
c, d S
d t U
V V W W
m, 00 Yl
2, Y ,
aa, bb
cc, dd
ee
Mn4+ (3d3) MgO SrTiOa A1203 Ti02 SnOz CsnGeF6
1.994 1.994 1.9737 1,9919 1.9885 N2
Moa+ ( 4 d 3 ) K31nC16.2H20
1.93
KzPtC&
1,9896
KzPtCla
1.815
(
99
Y, hh, ii jj kk
Tc4+ (4d3) Re4+(5d3) (
- )389
-12.8
nn
a W. Low, Phys. Rev., 101, 1827 (1956). * J. S. van Wieringen and J. E. Rensen, “Paramagnetic Resonance,” Vol. 1, W. Low, Ed., Academic Press Inc., New York, N. Y., 1963, p 105. W. Low and R. S. Rubins, “Paramagnetic Resonance,” Vol. 1, W. Low, Ed., Academic Press Inc., New York, N. Y., 1963, p 79. Footnote a, Table 111. e T. P. P. Hall, W. Hayes, R. W. H. Stevenson, and J. Wilkens, J . Chem. Phys., 38, 1977 (1963); 39, 35 (1963). V. T. Hochli, Bull. Am. Phys. Soc., 11, 203 (1966). Footnote h, Table IT‘. * C. Kikuchi, H. M. Sirveta, and V. W. Cohen, Phys. Rev., 92, 109 (1953). B. Bleaney, D. J. E. Ingram, and H. E. V. Scovil, Proc. Phys. Soc. (London), A64,601 (1951). J. M. Baker, see J. W. Orton, Rept. Progr. Phys., 22,204 (1959). I. Y. Chan, D. C. Doetschman, C. A. Hutchison Jr., B. E. Kohler, and J. W. Stout, J . Chem. Phys., 42, 1048 (1965). J. M. Baker and B. Bleaney, Proc. Phys. SOC.(London), A65,952 (1952). 7h J. M. Baker, B. Bleaney, and K. D. Bowers, ibid., A69, 1205 (1956). R. E. Robertson and H. M. McConnell, J. Phys. Chem., 64, 70 (1960). ’ H. M. McConnell, W. W. Porterfield, and R. E. Robertson, J . Chem. Phys., 30, 442 (1959). * N. S. Garifyanov, Dokl. Akad. Nauk SSSR, 109, 725 (1956); I z v . Akad. Nauk SSSR, Ser. Fiz., 21, 824 (1957); Dokl.Akad. Nauk SSSR, 138, 612 (1961). W. Low, Phys. Rev., 105, 801 (1957). ‘ J. E. Wertz and P. Auzins, ibid., 106, 484 (1957). a K. A. Muller, “Paramagnetic Resonance,” Vol. 1, W. Low, Ed., Academic Press Inc., New York, N. Y., 1963, B. Bleaney and K. D. Bowers, Proc. Phys. Soc. (London), A64,.1135 (1951). p 17. E. Y. Wong, J. Chem. Phys., 32, 598 (1960).
’
ff
’
The Journal of Physical Chemistry
THEISOTROPIC HYPERFINE INTERACTION
59
B. R. McGarvey, ibid., 41, 3743 (1964). ' H. J. Gerritsen, S. E. Harrison, H. R. Lewis, B. R. McGarvey, ibid., 40, 809 (1964). and J. P. Wittke, Phys. Rev. Letters, 2, 153 (1959). 'J. Sierro and R. Lacrok, Helv. Phys. Acta, 32, 286 (1959). * S. Geschwind, P. Kisliuk, M. P. Klein, J. P. Remeika, and D. L. Wood, "Paramagnetic Resonance," Vol. 1, W. Low, Ed., Academic Press Inc., New York, N. Y., 1963, p 113. H. J. Gerritsen, S. E. Harrison, and H. R. Lewis, J . Appl. Phys., 31, 1566 (1960). " J. Sierro, K. A. Muller, and R. Lacroix, Arch. Sci. (Geneva), 12,122 (1959). '' V. A. Atsarkin, E. A. Gerasimova, I. G. Matveeva, and A. V. Frautsesson, Zh. Ehperim. i Teor. Fiz., 43, 1272 (1962). dd E. N. Emel'yanov, N. U. Karlov, A. A. Manenkov, V. A. Milyaeu, A. M. Prokhorov, S. P. Smirnov, and A. V. Shirokov, ibid., 44,868 (1963). B. B. Garrett, K. DeArmond, and H. S. Gutowsky, J . Chem. Phys., 44,3393 (1966). 'I M. Nakada, K. Awaeu, S. Ibuki, T. Miyako, and M. Date, J . Phys. SOC. Japan, 19,781 (1964). w K. A. Muller, Phys. Rev. Letters, 2, 341 (1959); Helv. Phys. A'cta, 33, 497 (1960). H. G. Andersen, J . Chem. Phys., 35, 1090 (1961); Phys. Rev., 120, 1606 (1960). l i E. Yamaka and R. G. Barnes, ibid., 135, A144 (1964). i i W. H. From, P. B. Dorain, and C. Kikuchi, ibid., 135, J. Owen and I. M. Ward, see K. D. A710 (1964). kk L. Helmholz, A. V. Gueeo, and R. N. Sanders, J . Chem. Phys., 35,1349 (1961). Bowers and J. Owen, Rept. Progr. Phys., 18,304 (1955). nn W. Low and P. M. Llewellyn, Phys. Rev., 110,842 (1958). nn P. B. Dorain and R. Rahn, J . Chem. Phys., 36,561 (1962); 41,3249 (1964). O0 M. W. Walsh, Jr., Phys. Rev., 114, 1485 (1959).
in weak tetrahedral, cubic, and octahedral fields. For octahedral fields, values of T~ for MgO and CaO were taken from Low,16 and for the hexahydrate complexes a value of T~ = 0.039 was assumed. For the tetrahedral and cubic crystal fields, the sign for A was chosen so as to give a dependence of x upon the ionic character of the bond similar to that observed for d3 and d5 ions (this will be discussed later in the article). The values of x resulting from the opposite choice of sign are given in parentheses. The Go2+ ion is also found in strong crystal fields in which there is only one unpaired electron. For the case of the phthalocyanines, the evidence18 suggests that the unpaired electron is in a d,, orbital and that eq 10 should be used to obtain x, except that P2 should be replaced by azsince the dz9orbital is a a-antibonding orbital. The values obtained for x and a2 are listed in Table X. The maleonitriledithiolato complex of Co2+ has been interpreted by Maki, et aZ.,I9 to have its unpaired electron in a d,, orbital and the values reported in Table X are obtained using their interpretation. ds. In an octahedral field the d8 ion should have an isotropic g and A similar to the d3 configuration and A will be given by eq 11. I n Table X I are listed values of x for d8 ions. The values of x for W i 2 + are based on the magnetic moment for 6lNi given by Locher and Geschwindm which in turn is based on the assumption that x is the same for other d8 ions in similar lattices. Observations on the Value of x x and the Covalent Bond. One purpose of this investigation was to determine how the value of x was affected by the nature of the metal ligand bond. Matamura6 and Title6 have reported that the magnitude of x for d5 ions decreases as the ionic character of the bond decreases. I n Figure 1 are plotted the values of x for d5 ions vs. ( X A - X C ) ,the difference between the electronegativities of the anion and cation in the host lattice. The electronegativities used were those
I
I
I
I 0
Mn*2
' 1
Fe+'
+ cr+'
Q + O O
0
3.0
2.0
1.0
0.0
(X,-X,)
Figure 1. x
US.
(XA - X c ) for d6 ions.
computed by Gordy and Thomas. 21 Although there is considerable scatter in the points, a general trend can be discerned in which x varies from -3.2 for ionic bonds to about -1.6 for covalent bonds. The large scatter in the range of ( X A - X C ) = 1.0-1.5 is due to the values of x for the silver and cadmium halides which give the high points in this range. The higher values of x at ( X A - X C ) = 2.0 are for the hydrated ions which probably have more ionic bonds than the metal oxides with which they are grouped in this plot. In Figures 2 and 3 are given similar plots of x vs. ( X A - X c ) for d3 and d7 ions, respectively. These plots also show the magnitude of x decreasing as covalency increases, although the trend is less pronounced in the case of d3 ions. As will be shown further on, we can (18)J. M. Assour and K. Kahn, J . Am. Chem. Soe., 87, 207 (1965). (19) A. H.Maki, N. Edelstein, A. Davidson, and R.H. Holm, ibkl., 86, 4580 (1964). (20) P. R. Locher and S . Geschwind, Phys. Rev. Letters, 11, 333 (1963). (21) W. Gordy and W. J. 0. Thomas, J . Chem. Phys., 24, 439 (1956).
Volume 71, Number 1 January 1967
B. R. MCGARVEY
60
Table VII: Values of x for d6 Ions in Weak Field Host lattice
104~,
x,
Host
cm -1
8U
lattice
ZnS ZnSe
(+)13.4 (+)13.3
-2.00 -1.99
LiF NaF KMgFs CaFa BaFz MgFi CdFz SrFz NaMgFs KCaFs KCdFa KzMgFi KF CsCaFa MgO CaO SI-0 AlzOa ZnO CdO CdWO4 SrClz LiCl NaCl KCl AgCl MgClz CdClz
-87.7 (-)88.6 -91.0 (-)95.4 -95.0 (-)90.6 -92 ( - 193 -92.5 -93.1 -92.6 -91.5 (-)95.4 (-)90.8 -81.2 -81.7 (-)78.7 -79.6 -74.10 ( )87.3 (-p3.0 ( -)81.2 -79.7 -81.0 (-)88.6 -80.5 -82 -81.5
-2.99 -3.02 -3.10 -3.25 -3.24 -3.09 -3.14 -3.17 -3.15 -3.18 -3.16 -3.12 -3.25 -3.10 -2.77 -2.78 -2.68 -2.72 -2.53 -2.98 -2.83 -2.77 -2.72 -2.76 -3.02 -2.74 -2.80 -2.78
MgO CaO ZnO ZnWO4
-
( -)lo. 1
(-)10.5 (-)9.02 (-)9.63
Cr+ (3d') ZnTe CdTe Mn*+ (3d9 ZnS CdS MgS CaS
SrS ZnSe MgSe CaSe CdSe ZnTe CdTe CaTe MgTe (NH~)zZ~(SOI) * 6HIo ZnSiFe. 6Hz0 MgTiFs. 6Hz0 MgSO4 *7&0 Mg,Biz( 24H20 CdBrz AgBr ZnFz KNs CdIz cacos Zn( HCO&.2HzO Mg( CHsCOz)~*4Hz0 Si Ge GaAs Q
Fe*+(3d5) CaCOa ZnS Si
-2.64 -2.74 -2.36 -2.51
lO'A, om-1
X, BU
(+)12.4 (+)12.78
-1.85 -1.91
C
-63.7 -64.8 (-)71.9 (-)75.7 (-)75.0 -61.7 (-)71.2 (-)72.9 -62.7 -56.1 -57.1 (-)67.2 (-)58.1 -91.1 -94.2 ( )92 ( 188 90 (-)79.8 ( - 181 96 -83.6
-2.17 -2.21 -2.45 -2.58 -2.56 -2.10 -2.43 -2.48 -2.14 -1.92 -1.95 -2.29 -1.98 -3.11 -3.21 -3.14 -3.00 -3.07 -2.72 -2.76 -3.27 -2.85 -2.73 -2.99 -3.10 -2.97 -1.82 -1.45 -1.79
nn-w h, 1, w,rr
-
-
( -180
(-)87.8 ( 191 ( 187 -53.5 -42.5 -52.4
-
-11.23 (-)7.8 (-)7.0
Ref
d
1 1 1
rr-tt 1 1 1 SS
h, 1, mm, 88, uu
1 1 vv-YY vu, zz
xx
ww am z, bbb
kk ccc-fff BBB bbb ddd, hhh-kkk 111 111
mmm nnn 000,
-2.94 -2.04 -1.83
ttt,
PPP
uuu
b, vw
www
a J. Dieleman, R. S. Title, and W. V. Smith, Phys. Letters, 1, 334 (1962). R. S. Title, Phys. Rev., 131, 623 (1963). R. S. Title, ibid., 133, A1613 (1964). G. W. Ludwig and M. R. Lorenz, ibid., 131,601 (1963). e Footnote e, Table VI. T. T. Chang, W. H. Tanttila, and J. S. Wells, J . Chem. Phys., 39, 2453 (1963). W. Hayes and D. A. Jones, Proc. Phys. SOC.(London), 71, 503 (1958). J. S. Van Weiringen, Discwrsions Faraday SOC.,19, 118, 173 (1955). S. Ogawa and Y. Yokozawa, J . Phys. SOC.Japan, 14, 1116 (1959). S. Ogawa, ibid., 15, 1475 (1960). W. Low, Phys. Rev., 105,792, 793 (1957); Proc. Phys. SOC.(London), B69, 837 (1956). Reference 5. J. M. Baker, B. Bleaney, and W. Hayes, Proc. Roy. SOC.(London), A247,141 (1958). J. E. Drumheller, J . Chem. Phys., 38,970 (1963). ' Footnote c, Table VI. P. P. Sorokin, I. L. Gelles, and W. V. Smith, Phys. Rev., 121, 1513 (1958). A. J. Shuskus, Phys. Rev., 127, 1529 (1962). ' W. Low and R. S. Rubins, Phys. Letters, 1,316 (1962). L. V. Holroyd and J. L. Kolopus, Phys. Status Solidi, 3, K456 (1963). W. Low and J. T. SUM,Phys. Rev., 119, 132 (1960). " V. J. Folen, ibid., 125, 1581 (1962). " J. Schneider and S. R. Sircar, 2.Naturforsch., 178,570 (1962). P. B. Dorain, Phys. Rev., 112, 1058 (1958). R. E. Donovan and A. A. Vuylsteke, ibid., 127, 76 (1962). C. F. Hempstead and K. D. Bowers, ibid., 118, 131 (1960). ' H. Koga, K. Horai, and 0. Matamura, J . Phys. SOC.Japan, 15, 1340 (1960). G. L. Bir and L. S. Sochawa, F k . Tuerd. Tela, 5, 3594 (1963). bb W. Low and V. Rosenberger, Phys. Rev.,116, 561 (1959). cc H. Yoshimura, J . Phys. SOC.Japan, 15, 435 (1960). d d Y. Yokozawa and Y. Kazumata, ibid., 16, 694 (1961). " K. Fukuda, Y. Uchida, and H. Yoshimura, ibid., 13, 971 (1958). " G. D. Watkins, Phys. Rev., 113, K. Morigaki, M. Fujirnoto, and J. Itola, J . Phys. SOC.Japan, 13, 1174 (1958). IvI H. Abe, H. Nogano, M. Nagusa, 79, 91 (1959). J. Schneider and S. R. Sircar, Z.Naturand K. Oshima, J . Chem. Phys., 25,378 (1956). ii W. Low, Phys. Rev., 101, 1827 (1956). K. Fukuda, H. Matsumoto. T. Takagi, and Y. Uchida, forsch., 178, 155 (1962). kk H. Abe, J . Phys. Soc. Japan, 12,435 (1957). ibid., 16,1256 (1961). ** T. P. P. Hall, W. Hayes, and F. I . B. Williams, Proc. Phys. SOC.(London), 78,883 (1961). nn J. Schneider,
'
'
'
'
The Journal of Physical Chemistry
'
THEISOTROPIC HYPERFINE INTERACTION
61
S. R. Sircar, and A. Rauber, 2. Naturforsch., 18a, 980 (1963). L. M. Matarese and C. Kikuchi, J. Phys. Chem. Solids, 1, 117 (1956); Phys. Rev., 100, 1243 (1955). pp K. A. Muller, Helv. Phys. Acta, 28,450 (1955). aa G. D. Watkins, Phys. Rev., 110,986 (1958). " R. S. Title, ibid., 131. 2503 (1963). *' Footnote a, Table V. " B. C. Cavenett, Proc. Phys. SOC.(London), 84, 1 (1964). uI1 J. Lambe and C. Kikuchi, Phys. Rev., 117, 102 (1960). '"B. Bleaney and D. J. E. Ingram, Proc. Roy. SOC.(London), A205, 336 (1951). ww I. Hayashi and K. Ono, J. Phys. SOC.Japan, 8, 270 (1953). '' B. Bleaney, Physica, 17, 175 (1951). yy P. Locher and G. J. Gorter, T. Arakawa, J. Phys. SOC.Japan, 17,706 (1962). ibid., 28,797 (1962). R. S. Trenam, Proc. Phys. SOC.(London), 66,118 (1953). bM C. G. Windsor, J. H. E. Griffiths, and J. Owen, ibid., 81, 373 (1963). ccc D. M. S. Bagguley, B. Bleaney, J. H. E. Griffiths, R. P. Penrose, and B. I. Plumpton, ibid., 61, 551 (1948). ddd C. MacLean and G. J. W. Kor, Appl. Sci. Res., B4,425 (1955). M. Tinkham, Proc. Roy. SOC.(London), A236, 535 (1956). S. G. Salikhov, Zh. Eksperim. i Teor. Fiz., 34, 39 (1958). Oog G. J. King and B. S. Miller, J. Chem. Phys., 41, 28 (1964). lihh S. G. Salikov, Zh. Eksperim. i Teor. Fiz., 17, 1070 (1947). 'Iz B. M. Kozyrev and F. W. Lancaster and W. Gordy, J. Chem. Phys., 19, 1181 (1951); 20,740 (1952). kkk F. K. S. G. Salikov, ibid., 19, 185 (1949). Hurd, M. Sachs, and W. D. Herschberger, Phys. Rat., 93,373 (1954). D. J. E. Ingram, Proc. Phys. SOC.(London), A66, 412 (1953); Phys. Rev., 90,711 (1953). mmrn H. H. Woodbury and G. W. Ludwig, ibid., 117, 102 (1960). nnn G. D. Watkins, Bull. Am. Phys. SOC., 2, 345 (1957). Ooo R. Bleekrode, J. Dieleman, and H. J. Vegter, Phys. Letters, 2, 355 (1962). *" N. Almeleh and B. Goldstein, Phys. Rev., 128, 1568 (1962). qq9 J. Schneider, 2. Naturforsch., 17% 189 (1962). ' I r W. M. Walsh, Jr., and L. W. Rupp, Jr., Phys. Rev., 126, W. G. Nilsen and S. K. Kurtz, ibid., 136, A262 (1964). "' S. A. Marshall and A. R. Reinberg, ibid., 132, 134 (1963). 952 (1962). uuu S. A. Marshall, J. A. Hodges, and R. A. Serway, ibid., 133, A1427 (1964). '"' A. Rauber and J. Schneider, 2. Naturforsch., 17a, 266 (1962). wvIw G. W. Ludwig, R. 0. Carlson, and H. H. Woodbury, Bull. Am. Phys. SOC.,4, 22 (1959); Phys. Rev. Letters, 1,295 (1958).
"'
"'
Table VIII: Values of x and p2 for d6 Ions in Strong Field Host lattice
Cr( CN)aNOa- in KCl [Cr(C&")3ClO4 [C~(CSHENZMB~ [C~(CEHE)ZII
lo", cm-1
81
811
1 ,9722
2.0044
104~. cm - 1
8%
0.75
a
...
b
(+ W . 9
-2.59 -3.01 -2.51 -2.30
(-)77.3 (-)59.4
-2.43 -1.87
... ...
9
-2.61 -2.55
0.63 0.73
i
Cr+ (3d5) ( 130.8
+
1.9973 1.993 1.9863
X.
au
(
+P O . 3
(+)11.0
1
Ref
...
C
...
d, e
VO ( 3d6) 1.9831 1,9866
Naz[Fe(CN)bNO] [Mn(CN)5N0]2-in .HzO
1,9892 1.9873
2.0265 2.0279
Mn2+ (3d6) (-)150.5 (-)153.5
(-)36.2 (-)31.8
a, f
h
H. A. Kuska and M. T. Rogers, J . Chem. Phys., 42,3034 (1965). E. Konig, Z. Naturforsch., 19a, 1139 (1964). B. Elschner and N. N. Bubnov and V. M. Chibrikin, Zh. Fiz. Khim., 33,1891 (1959). S. Herzog, Arch. Sci. (Geneva), 11, special no. 160 (1958). 'K. H. Hausser, 2. Naturforsch., 14a, 425 (1959); Naturwissenschaften.,48,426,666 (1961). Footnote y , Table IV. K. H. Hausser, 2. Naturforsch., 16a, 1190 (1960). J. J. Fortman and R. G. Hayes, J. Chem. Phys., 43, 15 (1965). ' D. A. C. McNeil, J. B. Raynor, and M. C. R. Symons, J . Chem. SOC.,410 (1965).
'
attribute the lesser dependence of x upon (XA - XC) for d3 ions to the fact that the electrons in d3 are influenced only by T bonding, which probably changes less, while the d6 and d7 ions have electrons that are influenced by u bonding. The plots for d6and d3also illustrate rather markedly an observation made by Geschwindg that the value of x is constant for isoelectronic ions in similar environments. This similarity in x becomes even more marked when one compares x for different ions in the same lattice, for example, V2+,Cr,3+and h4n4+in MgO. Kivelson and Neiman7 in their analysis of Cu2+ complexes assumed that x is given by
x
= a2xo
(15)
which assumes a behavior for de similar to that observed for d3, d5, and d7 in Figures 1-3. I n Figure 4 are plotted values of x vs. a2 for Cu2+. The values of x for CdC12 and Cs2ZnClr are not included in Figure 4 because these have different symmetries for the crystal field. The scatter of points in Figure 4 is large, but it is clear that eq 15 will not represent the behavior of x. I n fact, it would appear that x first increases in magnitude as a2 decreases and then decreases in magnitude when cy2 < 0.7. Thus for d9 there is no s i m p l e r e b tionship between x and covalency. Volume 71, Number 1
January 1967
B. R. MCGARVEY
62
Table IX : Values of x for Coa (3d7)in a Weak Field +
Host lattice
2.3093 2,248 2.292 2.278 2.269 4.2785 4.372 4,293 4,303
CdTe ZnS ZnTe CdFz CdS MgO CaO Mg,Biz( N03)lz. MgoLadNO,),, .24H20
xla au
lO4A. cm-1
0
(-)23.4 (-)1.8 (-)17.5 ( 123 (-)4.6 (+)97.79 ( +)132.2 (+W . 0 (+)95.8
Ref
-3.61( -1.95) -2.29( -2.16) -3.24 ( -2.00) -3.32 (-1.68) -2.58 ( -2,25) -3.08 -2.91 -3.19 -3.26
-
a Numbers in parentheses are values of x obtained by taking the sign of A to be opposite that given in the table. F. S. Ham, G. W. Ludwig, G. D. Watkins, and H. H. Woodbury, Phys. Rev. Letters, 5,468 (1960). T. P. P. Hall and W. Hayes, J. Chem. Phys., 32, 1871 (1960). J. Lambe, J. Baker, and C. Kikuchi, Phys. Rev. Letters, 3, 270 (1959). W. Low, Phys. Rev., 109, 256 (1958). D. J. I. Fry and P. M. Llewellyn, Proc. Roy. SOC.(London), A266, 84 (1962). Footnote c, Table VI. Footnote r, Table VI1 Footnote aaa, Table S’II. I W. B. Gager, P. S. Jastrum, and J. G. Daunt, Phys. Rev., 111, 803 (1958).
’
Values of
Table X :
x and
a2for Co2+(3d7) in Strong Field
Host lattice
1.91 1.89 2,007
p-ZnPc P-HzPc a-ZnPc
a
Reference 18.
Table XI:
1044
cm-1
cm -1
2.90 2.94 2.422
z
[(C4H9)4N12Ni[S~C~(CN)~1~ 1.977
104~.
81
811
X
2.798
(+ )I60 ( )I60
( (
+
(+)I16
Y 2.025
2
(-)23
X
(-)50
+)265 +1280 ( + )66
x , au
a1
Ref
$2.21 $2.21 $0.44
0.83 0.83 0.82
a a
-3.4
0.79
b
a
Y
(-)28
Reference 19.
Values of x for d8 Ions
Host lattice
B
MgO
2.1728
Co+ (3d8) (-y4.0
-3.31
a
MgO A1203
2,225 2.1885
Ni2+ ( 3d8) (-)8.3 ( - )11.32
-3.15 -3.04
a-d e
A1203
2,0776
Cu3+(3d8) -61.51
-2.98
f, g
104~,
XI
cm-1
au
Ref
solutions of substituted copper(I1) acetylacetonates decreased linearly as the g value increased. This, in itself, was not surprising since the hyperfine splitting in A should be given by eq 11 which predicts that / A / should decrease with increasing g when A is negative in value, but the observed decrease was almost double that predicted by eq 11 if K were constant.
‘ J. W. Orton, P. Auzins, and J. E. Wertz, Phys. RW., 119, W. Low, Phys. Rev., 1691 (1960). Footnote a, Table VI. 109, 247 (1958). A. M. Germanier and R. Lacroix, Helv. W. E. BlumPhys. Acta, 34, 401 (1961). e Reference 20. berg, J. Eisinger, and S. Geschwind, Phys. Rev., 130, 900 (1963). W. E. Blumberg, J. Eisinger, S. Geschwind, and J. P. Remeika, tiparamagnetic Resonance,” Vol. 1, W. Low, Ed., Academic Press Inc., New York, N. Y., 1963, p 125.
’
’
Q
The fact that eq 15 might not be correct for Cu2+ was first pointed out by Kuska and Rogers,* who observed that the magnitude of the hyperfine splitting in The Journal of Physical Chemivtry
(X,-X,)
Figure 2. x vs. (XA
- X c ) for dS ions.
THEISOTROPIC HYPERFINE INTERACTION
I
I
63
I
.8
-
0
-3.0-
0
c: ?
0
Y
e
x
-
8
-2.00
Octahrdral S i t e 1
0
T r t r o h r d r a l and Cubic S i t e 8
I
I
1
covalency changes. This difference appears also in the case of N b 0 2 +us. Nb4+and Moo3+us. Mo6+. x and the Periodic Table. It was first pointed out by Abragam, Horowitz, and Pryce3 that x changed very little throughout the first transition series. I n Table XI1 are given the average values of x for each configuration of the first transition series in MgO, CaO, AlzOa, and TiOz. These oxide lattices were chosen because most configurations have been studied in one or more of these crystals, except for d2 in which the Na3V04-12H20value is given. The MgO and CaO values of x for Cu2+are not included because there are special problems involved in interpreting the d9 results
Table XI1 : Average Value of x in Oxide Lattices for Each Configuration of the First Transition Series dl
0
0
0
2
0
2-3.0-
-
0 0
I
Figure 4.
-2.1
-2.3
d4
-2.5
d6
d'
d*
do
-2.7
-3.0
-3.2
-3.4
0 0
0
-2.0
d'
I
0
0
0
?
x,au
0
I
d*
I
x vs. a? for Cu2+complexes.
Thus the ligands with the more ionic bonds and larger g values had xJs of smaller magnitude. To complicate the picture further, they found that changing the solvent produced an identical plot except that the more basic solvents (which one would expect to interact more covalently with copper ion) had the larger g values and hence the smaller magnitude for x. Thus it appears evident that a simple theory which states that the magnitude of x is determined only by the fraction of time the unpaired electron spends in the d orbital of the metal ion is not sufficient to account for the behavior of x in d9configurations. Similar discrepancies are observed in the d' configuration. For V02+,p2 is practically constant for the complexes observed, although x is noticeably smaller in magnitude for the phthalocyanines and porphyrins. The most noticeable result for vanadium is the much larger magnitude of x for V 0 2 + than for V4+ in TiOt, SnO2, and Ge02. It is difficult to associate this with
There is a gradual increase of x from -2.0 to -3.4 in the first transition series as we go from d' to ds. A similar trend is observed in the second and third transition series with x going from -4.0 in 4d' to -9.0 in 4d9 and from - 10 in 5d' to - 15 in 5d9. The change in x going down a group in the periodic table is larger
Theory of x Successful attempts to calculate x for metal ions of the first transition series have been made4 using the unrestricted Hartree-Fock approach. The difficulty with such calculations is that they throw very little light on the effect of covalent bonds on the value of x. I n the following paragraphs a perturbation approach to the calculation of x will be presented. The purpose of developing this approach is not to calculate good values of x, as better values can be expected from the Hartree-Fock method, but to see if examination of the resulting equations might reveal the reasons for the changes that occur in x with changes in the bonding of the ion. The perturbation method is ideal for this type of examination because the effect of replacing atomic orbitals by molecular orbitals can be readily evaluated. The source of x is taken to be the polarization of the inner s electron by an exchange interaction with the unpaired electrons. This polarization is accomplished by mixing into the ground state those excited states resulting from the promotion of an inner s electron into Volume 71, Number 1
January 1967
B. R. MCGARVEY
64
an empty s orbital. First, consider the case of one unpaired electron in the 3d orbital. The ground state can be written $0
= (3~+)(3~-)(3d+)
(16)
where #o is a determinental function and the other filled orbitals are understood to be present although not explicitly written down. The first excited states to be considered are the ones in which a 3s electron is promoted to a 4s orbital. This promotion produces three functions which can be written
e2
J(~,b,c,d)= J~*(l)b*(2)-~(l)d(2) d r r12
Combining eq 1, 21, and 23, the equation for x becomes
x
=
$2
= (3~+)(4~-)(3d+)
$3
=
-
J(4~,3d,3~,3d)] [2c1*c1
These functions cannot be mixed directly with the ground state because they are not eigenfunctions of S2. The function
+
2+
'$1 = a d 1 '$2
=
@2$1
+ +
+ +
b1$2 bZ$2
3/2
Ul*Ul
+
UZ*&
=
Ul*Cl
and the two
CZ$3
+ bl + c1 = -uz + bz + cz = 0
x (20)
As will become apparent later, the actuaI vaIues of the coefficients in the doublet functions are not required. The two functions and '$2 are added to the groundstate function giving, to first order in perturbation theory, the wave function $ =
$0
+
+ a242
a1Q1
(21)
where a1and a2 are given by
AE1 and AEz are the differences in energy of '$1 and d2 from $y,. Substituting eq 19 into eq 22, we obtain
J(4s,3d,3s,3d)1
+ ("*
The J o u r d of Physical Chemistry
+ (24)
bl*b,
+ b2*bz = 2 + CZ*C~ = 3
(25)
+ uz*c2 = UlCl* + u2cz* = -13
If we assume the difference between AEl and AE2 is small compared to their magnitude, we can factor out AE and using eq 25 obtain for x
C1IC.a
are doublet functions with S = l/z provided that --a1
UlCl*
where [3s(O)4s(O)] is the value of the product of the 3s and 4s orbit,als evaluated at the nucleus. From the orthogonality and normality equations plus eq 20 it can be shown that
C1*C1 is a triplet-state function with S = functions
+
+ &*C2 + a%*]J(4s,3d,3d,3s)} AEz
2c2*c2 (17)
(3s+) (4~+)(3d+)
+
[J(4s,3s,3s13s)
(4~+)(3~-)(3d+)
$1
2(Ul*Ul - bl*bl) AE1
= -4n[3s(0)4s(O)l{[
Ck*'J(4s,3d,3d,3s) (23)
=
8n[3~(0)4~(O)]J(4~,3d,3d,3s)/AE (26)
Considering all functions resulting from promotion of inner s electrons to empty s orbitals, the final equation for x becomes 3
x
m
= 8n n = l m=4
[ns(O)ms(O)l X J(ms,3d,3d,ns)/(Em - E,)
(27)
The exchange integral J(ms,3d,3d,ns) is independent of the type of d orbital. By extension of the above approach, it can be shown that eq 27 is correct for all configurations in all crystal fields. Accurate computations of x using eq 27 are not to be expected because A E for promotion to 5s and higher orbitals is not greatly larger than AE for promotion to 4s, and slow convergence of the infinite series would be expected. As an illustration of this fact, the first three terms in eq 27 (promotion of Is, 2s, and 3s electrons to a 4s orbital) were computed for the manganese atom from the wave functions of Watson.22 The resulting value of x was +0.13 au. (22)
R. E.Watson, Phys. Rev.,
119, 1934 (1960).
THEISOTROPIC HYPERFINEINTERACTION
65
The two e, orbitals are a-antibonding orbitals and the three tzg orbitals are T-antibonding orbitals. If we consider only the d orbitals to be involved in bonding and assume the s orbitals to remain atomic orbitals, then the exchange integrals in eq 27 are reduced by the factors a2and Pz. Thus for the high-spin configurations we can write
IO
-
" i 0
1
-
n* I
-
c
c "
r
-2
d1,dz,d3
x
d4
X = &az
d6
x
= 2'2
d7
x
= -@a2
d8,ds
x
= azxo
=
P2xo 1
+ 3P2)Xo
-0
In Figure 5 are plotted values of [ns(0)4s(O)]/ (E4 - E,) for n = 1, 2, and 3 for the neutral atoms of vanadium through copper. These were computed from Watson'sz2 functions by taking (E4 - E,) to be the difference in Hartree-Fock one-electron energies for the two s orbitals. Also plotted in Figure 5 is the value of J(4s,3d,3d14s) which was calculated by Watson.22 From Figure 5 it can be seen that all three values of [ns(O).ls(O)]/(E4- E,) increase gradually over the transition series, the change being about 20% in each case. At the same time the exchange integral decreases gradually, the total change being about 14%. This suggests that all the terms in eq 27 change gradually over the transition series, and, since eq 27 applies to all configurations, it seems reasonable to expect a gradual variation of x over the transition series. The observed independence of x on charge for a series of isoelectronic ions might result from chargetransfer effects by covalent interactions. For the d3 ions it has been observedz3 from the variation in g that the effective charge seen by the outer d electrons for V2+, Cr3+, and ;\In4+ in oxide lattices was about 2+ in every case. Thus the constancy of x may simply be evidence that the larger the formal charge on the ion, the more the charge on the ion is nullified by covalent charge transfer from the ligand. x and the Covalent Bond. For an ion bonded to ligand atoms, eq 27 can be considered still correct, except that the atomic orbitals are replaced by molecular orbitals. For octahedral or near-octahedral symmetry, there are two types of d orbitals which can be written
1
1 3
+ P2)xo
+ P2)xo
where xo is the value of x for the free ion. Equations 29 appear to be sufficient to account for the behavior of d3, d6, and d7 ions. Some alterations in eq 29 are necessary for ions in tetrahedral or cubic fields because of the inversion of filling of the orbitals. From eq 29 we can interpret the lesser dependence of x upon electronegativity for the d3 ions as showing that there is less ?r bonding and that it changes less with changes in electronegativity than does the u bonding. In obtaining eq 29 it was assumed that the 4s orbital was not involved in forming a bond between the metal ion and the ligand. If bonding involving the 4s orbital is present, we can expect the situation to become more complex than that represented by eq 29. Consider the case of ds in D 4 h symmetry. The unpaired electron is in the molecular orbital 1
- 2-a'(-pzl
+ py2 + pz3 - py4) +
(30)
where ligand atoms 1 and 3 are on the and - portions of the x axis and atoms 2 and 4 are on the and portions of the y axis. For simplicity, it is assumed that only the p orbital of the ligand is involved, although symmetry allows the ligand s orbital to mix in also. The s orbitals can be included by replacing the p orbital in eq 30 by an sp hybrid. If we assume the 4s orbital to be involved in the bonding, then there is a filled bonding orbital of the form shown in eq 31. (23)
+
B. R. McGarvey, J. Chem. Phys., 41, 3743 (1964).
Volume 71,Number 1 January 1967
B. R. MCGARVEY
66
rL
=
1 + 2Y(-Pz1
7'4s
- Pv2 + Pz3 + Pv4)
(31)
and an empty antibonding orbital of the form $* = 64s
1
- i6'(-pz'
+ pz3 + py4)
- pv2
(32)
I n addition to the inner-shell s orbitals, there is now another filled orbital with some s character. Polarization of $ can also contribute to x. Using eq 30 for the 3d orbital, eq 32 for the 4s oribtal, and eq 31 for an additional filled s orbital, we obtain for x the equation 3
x
= 8na2
Szc[ns(O)4s(O)]J(4s,3d,3d,ns)/ n=l
3
+ c 2 [ns(o>ms(O)lx J(ms,3d,3dlns)/(Em - E,) +
(E',* - E,)
(T')~$
n = l m=5
[4s(O)ms(O)]J(rns,3d73d,4s)/
m=5
orbitals between the bonding state and the unpaired electron, and the donation of charge by the solvent molecule to 4s might reasonably be expected to reduce the charge donation from the ligand atoms in the zy plane. Direct Contribution of s to x. I n certain instances it is possible for the d orbital to be directly mixed with an s orbital. I n such a case the contribution to x would be positive. I n octahedral, tetradehedral, and cubic symmetry all d orbitals belong to irreducible representations different from that of the s orbital and this direct contribution to x is not allowed. However, in D4h symmetry the d,, orbital belongs to the same irreducible representation as the s orbital and a direct contribution to x is allowed. This may account for the plus values of x observed for the phthalocyanines of Co2+ in which the unpaired electron is in a d,, orbital. Also, in D3 and D4d symmetries the d,, orbital can mix with the s orbital, and this could account for the low value of x in Ti[CH3CO)2CH]3and in M o (CN)83- and the positive values of x for d' complexes of vanadium, chromium, and molybdenum with SzCzRz2-. In the case of X ~ O [ S ~ C ~ ( C Fit~ should ) ~ ] ~ - be noted that it takes only a small direct admixture of s to obtain a large value of x, so that a large value of x is not inconsistent with the observation that the unpaired electron is primarily on the ligand. For D2h symmetry, the d,z-y2 orbital (z, y, and z axes are the twofold symmetry axes) can mix with the s orbital as well as the d,, orbital. Thus for d' in rhombic symmetry, it is possible for the unpaired electron to have s character and this may be the reason V4+in Ti02 and Sn02has a x smaller in magnitude than that for V 0 2 + since the rhombic distortion is quite severe in both crystal lattices. Kuska, Krigas, and Rogersz4 have noted that x is low in magnitude for some V 0 2 + unsymmetrical complexes and attributed it to s character in the d orbital of the unpaired electron. The small magnitude of x in cyclopentadienyl complexes is, also, most likely due to s character in the d orbitals. It should be pointed out that even in Deh symmetry the unpaired electron in dg cannot mix with an s orbital, so that the direct interaction makes no contribution to x in copper complexes.8
+
(33) All overlap terms were neglected in obtaining eq 33. The last term in eq 33 is the term resulting from the polarization of spin in the bonding state #, It is significant only when there is covalent bonding in both 3d and 4s orbitals and, further, it should be negative in sign because J(p,,p,,p,,p,) is much larger than J(4s,3d,3d,4s). Because J(pz,p2,p2,pz) can be large and (E,* - E,) can be one of the smaller energy differences, it s e e ~ reasonable s that the last term in eq 33 might sometimes dominate and lead to more negative values of x as covalency increases. This polarization of spin in a 4s bonding orbital by exchange interaction on the ligand may help account for the variation of x with a2 observed in Figure 4 for copper ions. It is possible to interpret the observations made by Kuska and Rogers8 on copper complexes in terms of changes in bonding involving the 4s orbital. If we were to consider that adding electron-donating groups to the acetylacetonate ring would increase the donation of charge by the ligand to the 4s orbital of CuP+, then from eq 33 we might expect x to become larger in magnitude, as has been observed. Further, the donation of charge by solvent molecules along the x axis could be expected to have the reverse effect, because there is no exchange interaction involving solvent ligand The Journal of Physical Chemistry
Discussion M. C . R. SYMONS (Leicester University, Leicester). (1) I wonder if spin Polarization of the V-0 0-bonding electrons may be an important source of isotropic hyperfine coupling from vanadium? ( 2 ) I am unhappy about your assignment of posi(24) H. A. Kuska, T. Krigas, and M. T. Rogers, 151st National Meeting of the American Chemical Society, Pittsburgh, Pa., March
1966.
