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Comments on second-order spin-hamiltonian energies
JOURNAL
OF MAGNETIC
RESONANCE
18,
113-116
(1975)
Commentson Second-OrderSpin-Hamiltonian Energies JOHN
A. WEIL
Department of Chemistry and Chemical Engineering, University of Saskatchewan, Saskatoon S7N 0 WO, Canada Received June 26,1974 The second-order perturbation energy of a spin Hamiltonian, valid for low symmetry and an arbitrary number of nuclei, is reported and discussed. The nuclear Zeeman terms, as well as certain cross terms missing in previous publications, have been included. In recent papers (Z-.5), formulae have been given for spin-Hamiltonian energies correct to second order, which are valid even for low symmetry (triclinic) situations. The
following additions, comments and corrections seem required. The present work includes generalization to more than one nucleus, consideration of the nuclear Zeeman terms as well as of certain cross terms. Herein,
the spin Hamiltonian
Xs=/?S.g*B+S*D*S+
is taken to be (6) i~~(ii.Ai.S-P,i,p,.B+I,.Pi.I,),
111
where index i runs over N nuclei, N denotes transpose and B is the applied Zeeman field The resulting second-order energy is found from perturbation theory to be
vector.
. ., mN) = g/WI4 + +Tr(D) [S(S + 1) - M2]
EW,mI,m2,.
+ id, [3M2 - S(S + l)] + &j$l,
- d:)[8M2 + 1 - 4S(S + l)]M
-![Tr(D2) - 2d2 + df - 2d-, Det(D)] [2S(S + 1) - 2M2 - l]M + 8gPB +
2
(JZfW
+
hB[
3 (Tr(& *Ai) - kf}M{Zi (Zi + 1) - mf}
I - !Zj+
1
{S(S + 1) - M2}WZi + (I$ - Kf) Mm;
1
+ 2(e, - dl)Ki(3M2 - S(S + l>}mi - G$NBmi + +Tr(PJ[Zi (I i+ 1) - mf] + +pli [3mf - Zi (Zi + l)] + Copyright Q 1975 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
113
114
JOHN
1
A. WEIL
(Lij - KiKj)Mmimj
+LBiB2 ci>j
+a
x(gf
M
- Gf)m, + pe
I
[Sm: - 41, (Z, + I) + I]m,
+ Tr(P:) - 2p,i + p:i - 2~~1, Det(Pi) [2Zi(Zi + 1) - 2m: - l]m, 8Ki - y@I
PI
- Gipli) [3mF - Zi (Zi + l)]) +
Here /I = Bohr magneton, /IN = nuclear magneton, q=B/IB] gz
=
g2& =
=B/B,
q.g.g.q, t.g.D”.g.q
n = integer,
g2Kf = ~.&&.A,g.~, g2K:kf = f~~&A,&A,~g~~, g2Kfei = d*g*$(D*&*Ai
+ &*Al*D)*g+q,
g~=ii’Ei’gi”I,
gKiGi=~.~(g.Ai.gi+gi.A,.g).tl, g2K;p,, = &g&P;.Ai.g.q gKiq* =e*$(g*Ai*Pi*gi g’KiKjL,j=
$*g*$(Ai*A,*Aj*Aj
n = integer, + g,.Pi.Ai*g)*q, + Aj.Aj*Af*Ai).g.q.
[31
Here the Zeeman term in [I] has been assumed to be dominant, and thus yields the unperturbed levels. The last term in [2], varying as M-l, has been derived under the assumption that the first-order hyperfine energy KiMmi for every nucleus is larger than any corresponding nuclear Zeeman or quadrupole contribution. Thus application of [2] to states with M = 0 is invalid. It is important to recognize explicitly that the quantization axis for each nuclear spin has been chosen along the unit vector Ai.g.tl/gKi. Use of some other scheme, for example, the more general one required when the hyperfine and nuclear Zeeman Hamiltonians are of equal importance (7), would alter the energy expression and the meaning of the nuclear spin component quantum number(s) mi, as well as the selection rules for EPR transitions. In the present scheme, the convention gj?B > 0 implies that the lowest electron state is labeled with electron spin component quantum number M equal to -S. Since Ki > 0, this causes mi = +Zi to be associated with the lowest state and means that the primary
SECOND-ORDER
SPIN-HAMILTONIAN
ENERGIES
115
EPR hyperfine components are to be labeled with mi from +Ii to -Ii with increasing magnetic field. Both the electronic and nuclear quadrupole matrices are symmetric for all local symmetry situations, i.e.,D=D and P,=P, (8). Since d-, (i.e.,D-l)isundefinedif Det(D) happens to vanish, it is useful to note that the alternative expression Tr(Adj(D)) d,Tr(D) + d2 can replace h, Det(D) in [2]; an analogous expression holds for peli Det(P,). By convention, Tr(D) and Tr(Pi) are generally taken to be zero when only spectroscopic data are to be considered. Our notation, unlike that used in Ref. (1) permits d,, and pni to be negative for n odd. Indeed, it is evident that the relative signs of some of the matrix quantities are obtainable from measurement. Thus the sign of the product of the three principal values of Ai is available when the term in Det(A,) does not vanish. The nuclear Zeeman term varies linearly with Gi, rather than withg, = +[q .gi .gi .$]1/2, and thus depends on the sign of the nuclear magnetic moment. Application of Eq. [2] is limited to intermediate fields B, since for very large fields, the nuclear spin is aligned via the nuclear Zeeman term rather than via the hyperfine term in Eq. [I] so that the more general theory (7) would be required. Note that, as a symptom of this limitation, a term in PN2B2/KiM occurs. Thus relation [2] is likely to be inapplicable when considering spectra with small hyperfine splittings (for example, arising from ligand atoms). The signs of D and Pi are available from suitable measurements (6, 9), as is evident from Eq. [2]. The latter can also yield the relative sign of g and gi (from Gi or qi) if suitable transitions are observable, at appropriate values of B. It is to be noted that Refs. (Z-4) omit the mixed term (ej - d,)Ki arising from the electronic quadrupole and nuclear hyperfine effects. However, such a term has been given previously (IO) for the situation when uniaxial, coaxial matrices g, D, Ai occur; our expression reduces correctly to this special case. These publications have not treated the situation when an arbitrary number of nuclei occur. It is noteworthy that mixed terms, coupling contributions from pairs of different nuclei (i.e., proportional to mim.j), occur. Such terms have been mentioned previously (II) but are given in a more complete and elegant form herein. The cross term qi, between the nuclear Zeeman and quadrupole terms, does not seem to have been described previously. It is especially noteworthy that the energy of each state is completely expressible in terms of quantities (Tr, TrAdj, Det) invariant to spatial rotations and of quantities having the form q *Y .g, where matrices Y are products of the various matrices in the spin Hamiltonian and are symmetric or may be chosen so (12). Thus, while “tensors” g, Ai and gi may be asymmetric in low-symmetry situations (13), the observable energies [2] are not affected by these asymmetries. The relationship between projection terms of the form 9 .Y .q and crystal symmetry has been discussed in detail elsewhere (12). The generalization to higher than second-order perturbations, i.e., to the exact energy solution, will be considered elsewhere (14). It follows from the above that, typically, it is the tensors g.g, Ai .Ai and &*gi (rather than g, Ai and pi) which are involved in determining single crystal EPR line-position data. Note that special line-intensity measurements can yield Det(g) (15), i.e., the sign of g. Knowledge of measurables g and gK does not suffice to give g or A* A (in the case of one nucleus) completely. In a previous paper(l6), in contrast to the above expansion of the energy in terms of
116
JOHN A. WEIL
inverse powers of the field B, perturbation theory was applied to bring in the effects of anisotropy on the spin-Hamiltonian energy of an isotropic (Breit-Rabi) situation, for S = 3 and one nucleus. Since it is x*A which is measured, rather than A itself, it is necessary to rewrite Eq. [14] (and [5]) in Ref. (16) with absolute magnitude signs, i.e.,
where A = q*A*q (and g = q*g.q), and hv is the transition energy. Use of relationship [4], together with field and g values measured at various orientations of 11,cannot yield A, but only gives the approximation AI1 A-A 1: A0
Au + AZI A 22
A,, + A,, A23
+A,2
A 33
i
,
[51
i
where A, is the isotropic component of A, and must be large compared to the anisotropit part for [4] to hold. This then is consistent with the results discussed above. REFERENCES
1. A. ROCKENBALJERAND P. SIMON, J. Magn. Resonance 11,217 (1973). 2. U. SAKAGUCHI, T. ARATA, AND S. FUJIWARA, J. Magn. Resonance 9, 118 (1973). 3. W. C. LIN, Mol. Phvs. 25,247 (1973). 4. W. Low, “ Paramagnetic Resonance in Solids,” Solid State Physics, Suppl. 2, Chap. 2, Academic Press, New York, 1960. 5. R. M. GOLDING AND W. C. TENNANT, Mol. Phys. 25,1163 (1973); 28,167 (1974). The latter paper presents some third-order perturbation energy terms. 6. A. ABRAGAM AND B. BLEANEY, “Electron Paramagnetic Resonance of Transition Ions,” Chap. 3, Clarendon Press, Oxford, 1970. Note that nuclear spin-spin interactions are omitted in Eq. [I]. 7. J. A. WEIL AND J. H. ANDERSON, J. Chem. Phys. 35, 1410 (1961). 8. F. KNEUB~~HL, Phys. Kondens. Mater. 1,410 (1963). 9. M. E. FOGLIO, Nuouo Cimento 50,158 (1967); SlB, 570 (1967). IO. B. BLEANEY, Phil. Mug. 42,441 (1951). II. J. A. WEIL, G. L. GOODMAN, AND H. G. HECHT, “Paramagnetic Resonance” (W. Low, Ed.), Vol. 2, p. 880, Academic Press, New York, 1963. 12. J. A. WEIL, T. BUCH, AND J. E. CLAPP, Advan. Mug. Resonance 6,183 (1973). 13. F. KNEUB~HL, Phys. Kondens. Mater. 4, 50 (1965). 14. R. SKINNER AND J. A. WEIL, to be submitted for publication, 15. M. H. L. PRYCE,Phys. Rev. Lett. 3,375 (1959). 16. J. A. WEIL, J. Magn. Resonance 4, 394 (1971).
OF MAGNETIC
RESONANCE
18,
113-116
(1975)
Commentson Second-OrderSpin-Hamiltonian Energies JOHN
A. WEIL
Department of Chemistry and Chemical Engineering, University of Saskatchewan, Saskatoon S7N 0 WO, Canada Received June 26,1974 The second-order perturbation energy of a spin Hamiltonian, valid for low symmetry and an arbitrary number of nuclei, is reported and discussed. The nuclear Zeeman terms, as well as certain cross terms missing in previous publications, have been included. In recent papers (Z-.5), formulae have been given for spin-Hamiltonian energies correct to second order, which are valid even for low symmetry (triclinic) situations. The
following additions, comments and corrections seem required. The present work includes generalization to more than one nucleus, consideration of the nuclear Zeeman terms as well as of certain cross terms. Herein,
the spin Hamiltonian
Xs=/?S.g*B+S*D*S+
is taken to be (6) i~~(ii.Ai.S-P,i,p,.B+I,.Pi.I,),
111
where index i runs over N nuclei, N denotes transpose and B is the applied Zeeman field The resulting second-order energy is found from perturbation theory to be
vector.
. ., mN) = g/WI4 + +Tr(D) [S(S + 1) - M2]
EW,mI,m2,.
+ id, [3M2 - S(S + l)] + &j$l,
- d:)[8M2 + 1 - 4S(S + l)]M
-![Tr(D2) - 2d2 + df - 2d-, Det(D)] [2S(S + 1) - 2M2 - l]M + 8gPB +
2
(JZfW
+
hB[
3 (Tr(& *Ai) - kf}M{Zi (Zi + 1) - mf}
I - !Zj+
1
{S(S + 1) - M2}WZi + (I$ - Kf) Mm;
1
+ 2(e, - dl)Ki(3M2 - S(S + l>}mi - G$NBmi + +Tr(PJ[Zi (I i+ 1) - mf] + +pli [3mf - Zi (Zi + l)] + Copyright Q 1975 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
113
114
JOHN
1
A. WEIL
(Lij - KiKj)Mmimj
+LBiB2 ci>j
+a
x(gf
M
- Gf)m, + pe
I
[Sm: - 41, (Z, + I) + I]m,
+ Tr(P:) - 2p,i + p:i - 2~~1, Det(Pi) [2Zi(Zi + 1) - 2m: - l]m, 8Ki - y@I
PI
- Gipli) [3mF - Zi (Zi + l)]) +
Here /I = Bohr magneton, /IN = nuclear magneton, q=B/IB] gz
=
g2& =
=B/B,
q.g.g.q, t.g.D”.g.q
n = integer,
g2Kf = ~.&&.A,g.~, g2K:kf = f~~&A,&A,~g~~, g2Kfei = d*g*$(D*&*Ai
+ &*Al*D)*g+q,
g~=ii’Ei’gi”I,
gKiGi=~.~(g.Ai.gi+gi.A,.g).tl, g2K;p,, = &g&P;.Ai.g.q gKiq* =e*$(g*Ai*Pi*gi g’KiKjL,j=
$*g*$(Ai*A,*Aj*Aj
n = integer, + g,.Pi.Ai*g)*q, + Aj.Aj*Af*Ai).g.q.
[31
Here the Zeeman term in [I] has been assumed to be dominant, and thus yields the unperturbed levels. The last term in [2], varying as M-l, has been derived under the assumption that the first-order hyperfine energy KiMmi for every nucleus is larger than any corresponding nuclear Zeeman or quadrupole contribution. Thus application of [2] to states with M = 0 is invalid. It is important to recognize explicitly that the quantization axis for each nuclear spin has been chosen along the unit vector Ai.g.tl/gKi. Use of some other scheme, for example, the more general one required when the hyperfine and nuclear Zeeman Hamiltonians are of equal importance (7), would alter the energy expression and the meaning of the nuclear spin component quantum number(s) mi, as well as the selection rules for EPR transitions. In the present scheme, the convention gj?B > 0 implies that the lowest electron state is labeled with electron spin component quantum number M equal to -S. Since Ki > 0, this causes mi = +Zi to be associated with the lowest state and means that the primary
SECOND-ORDER
SPIN-HAMILTONIAN
ENERGIES
115
EPR hyperfine components are to be labeled with mi from +Ii to -Ii with increasing magnetic field. Both the electronic and nuclear quadrupole matrices are symmetric for all local symmetry situations, i.e.,D=D and P,=P, (8). Since d-, (i.e.,D-l)isundefinedif Det(D) happens to vanish, it is useful to note that the alternative expression Tr(Adj(D)) d,Tr(D) + d2 can replace h, Det(D) in [2]; an analogous expression holds for peli Det(P,). By convention, Tr(D) and Tr(Pi) are generally taken to be zero when only spectroscopic data are to be considered. Our notation, unlike that used in Ref. (1) permits d,, and pni to be negative for n odd. Indeed, it is evident that the relative signs of some of the matrix quantities are obtainable from measurement. Thus the sign of the product of the three principal values of Ai is available when the term in Det(A,) does not vanish. The nuclear Zeeman term varies linearly with Gi, rather than withg, = +[q .gi .gi .$]1/2, and thus depends on the sign of the nuclear magnetic moment. Application of Eq. [2] is limited to intermediate fields B, since for very large fields, the nuclear spin is aligned via the nuclear Zeeman term rather than via the hyperfine term in Eq. [I] so that the more general theory (7) would be required. Note that, as a symptom of this limitation, a term in PN2B2/KiM occurs. Thus relation [2] is likely to be inapplicable when considering spectra with small hyperfine splittings (for example, arising from ligand atoms). The signs of D and Pi are available from suitable measurements (6, 9), as is evident from Eq. [2]. The latter can also yield the relative sign of g and gi (from Gi or qi) if suitable transitions are observable, at appropriate values of B. It is to be noted that Refs. (Z-4) omit the mixed term (ej - d,)Ki arising from the electronic quadrupole and nuclear hyperfine effects. However, such a term has been given previously (IO) for the situation when uniaxial, coaxial matrices g, D, Ai occur; our expression reduces correctly to this special case. These publications have not treated the situation when an arbitrary number of nuclei occur. It is noteworthy that mixed terms, coupling contributions from pairs of different nuclei (i.e., proportional to mim.j), occur. Such terms have been mentioned previously (II) but are given in a more complete and elegant form herein. The cross term qi, between the nuclear Zeeman and quadrupole terms, does not seem to have been described previously. It is especially noteworthy that the energy of each state is completely expressible in terms of quantities (Tr, TrAdj, Det) invariant to spatial rotations and of quantities having the form q *Y .g, where matrices Y are products of the various matrices in the spin Hamiltonian and are symmetric or may be chosen so (12). Thus, while “tensors” g, Ai and gi may be asymmetric in low-symmetry situations (13), the observable energies [2] are not affected by these asymmetries. The relationship between projection terms of the form 9 .Y .q and crystal symmetry has been discussed in detail elsewhere (12). The generalization to higher than second-order perturbations, i.e., to the exact energy solution, will be considered elsewhere (14). It follows from the above that, typically, it is the tensors g.g, Ai .Ai and &*gi (rather than g, Ai and pi) which are involved in determining single crystal EPR line-position data. Note that special line-intensity measurements can yield Det(g) (15), i.e., the sign of g. Knowledge of measurables g and gK does not suffice to give g or A* A (in the case of one nucleus) completely. In a previous paper(l6), in contrast to the above expansion of the energy in terms of
116
JOHN A. WEIL
inverse powers of the field B, perturbation theory was applied to bring in the effects of anisotropy on the spin-Hamiltonian energy of an isotropic (Breit-Rabi) situation, for S = 3 and one nucleus. Since it is x*A which is measured, rather than A itself, it is necessary to rewrite Eq. [14] (and [5]) in Ref. (16) with absolute magnitude signs, i.e.,
where A = q*A*q (and g = q*g.q), and hv is the transition energy. Use of relationship [4], together with field and g values measured at various orientations of 11,cannot yield A, but only gives the approximation AI1 A-A 1: A0
Au + AZI A 22
A,, + A,, A23
+A,2
A 33
i
,
[51
i
where A, is the isotropic component of A, and must be large compared to the anisotropit part for [4] to hold. This then is consistent with the results discussed above. REFERENCES
1. A. ROCKENBALJERAND P. SIMON, J. Magn. Resonance 11,217 (1973). 2. U. SAKAGUCHI, T. ARATA, AND S. FUJIWARA, J. Magn. Resonance 9, 118 (1973). 3. W. C. LIN, Mol. Phvs. 25,247 (1973). 4. W. Low, “ Paramagnetic Resonance in Solids,” Solid State Physics, Suppl. 2, Chap. 2, Academic Press, New York, 1960. 5. R. M. GOLDING AND W. C. TENNANT, Mol. Phys. 25,1163 (1973); 28,167 (1974). The latter paper presents some third-order perturbation energy terms. 6. A. ABRAGAM AND B. BLEANEY, “Electron Paramagnetic Resonance of Transition Ions,” Chap. 3, Clarendon Press, Oxford, 1970. Note that nuclear spin-spin interactions are omitted in Eq. [I]. 7. J. A. WEIL AND J. H. ANDERSON, J. Chem. Phys. 35, 1410 (1961). 8. F. KNEUB~~HL, Phys. Kondens. Mater. 1,410 (1963). 9. M. E. FOGLIO, Nuouo Cimento 50,158 (1967); SlB, 570 (1967). IO. B. BLEANEY, Phil. Mug. 42,441 (1951). II. J. A. WEIL, G. L. GOODMAN, AND H. G. HECHT, “Paramagnetic Resonance” (W. Low, Ed.), Vol. 2, p. 880, Academic Press, New York, 1963. 12. J. A. WEIL, T. BUCH, AND J. E. CLAPP, Advan. Mug. Resonance 6,183 (1973). 13. F. KNEUB~HL, Phys. Kondens. Mater. 4, 50 (1965). 14. R. SKINNER AND J. A. WEIL, to be submitted for publication, 15. M. H. L. PRYCE,Phys. Rev. Lett. 3,375 (1959). 16. J. A. WEIL, J. Magn. Resonance 4, 394 (1971).