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General description of nuclear interaction terms in a spin Hamiltonian including a system of ligands
JOURNAL
OF MAGNETIC
RESONANCE
59,7
l-78
(1984)
General Description of Nuclear Interaction Terms in a Spin Hamiltonian Including a System of Ligands* S. J. SFERCO,-~ M. C. G. PASSEGGI,$
AND E. A. ALBANEsIt
INTEC.9 Casilla de Correo 91, 3000-Santa Received
November
Fe, Argentina
16, 1983
Following a procedure described by R. Calvo (J. Magn. Reson. 26 (1977)) to construct the ligand hyperline terms contributing to the spin Hamiltonian of isolated paramagnetic centers, the second-rank terms in nuclear spin operators are derived with reference to a system of C,, symmetry. It is shown that these describe appropriately quadrupolar as well as coupling effects between different nuclei. The symmetry is automatically reflected in fixing the number of independent parameters demanded by the addition of such interactions, without new requirements from the group theory point of view. The matrices representing these effects in a six-atom cluster of C,. symmetry are explicitly given as an example.
In a paper by Calve (I) it was shown how the symmetry of an atomic cluster can be advantageously used in deriving a spin Hamihonian to describe the ligand hyperhne structure of isolated paramagnetic centers. A general recipe based on group theory was given in order to construct a set of invariants under symmetry operations in terms of electronic and nuclear spin operators in the presence of a magnetic field. An application to C,, symmetry was also presented with particular reference to the EPR and ENDOR spectra of heme proteins including up to linear terms in the components of nuclear spin operators. In the present paper we apply the same formalism to derive second-order terms in these operators and to describe appropriately the effective quadrupole interactions as well as the coupling terms between nuclear spins, which could add important contributions to the spin Hamiltonian, depending on the nature of the bonding scheme and the nuclear spins of the atoms involved. In particular, it will be shown that the procedure used by Calvo is particularly suitable to this purpose since no new symmetry-adapted combinations of spin operators are necessary, with the result that the new coupling terms automatically incorporate the symmetry in fixing the number of new independent parameters. To illustrate the procedure we have applied the formalism to a system of C,, symmetry. Although very recent ENDOR results by Scholes et al. (2) show that for * t $ 5
Work financed through the BID-CONICET program. Supported by CONICET. Member of the Carrera de1 Investigador Cientifico y Tecnolbgico de1 CONICET. Instituto de Desarrollo Tecnologico para la Industria Quimica, Universidad National and Consejo National de Investigaciones Cientificas y T&micas (CONICET). 71
0022-2364184
de1 Litoral
(UNL)
$3.00
Copyright Q 1984 by Academic Pres.s. Inc. All ri@s of reproduction in any form reserved.
72
SFERCO,
PASSEGGI,
AND
ALBANESI
the system described by Calvo (I) the four nitrogens of the heme ring cannot be considered as equivalent, we will assume that the Cd0 point group can still be used as an approximate symmetry to describe the geometrical configuration of that cluster. SUMMARY
OF
THE
Following T. Ray (3), the spin Hamiltonian
THEORY
is presented (I) under the general form
111 where
Pkr;)= 2 (r’l”)lr,,rk,)(r’,“‘)l r,rjD)P( r~~))fPq)( @))I(‘)(rg))
PI
where ,@‘)(I’$~‘) and l(fl(rpJ’) represent irreducible tensor operators of range p and r transforming as the (Y and y component of the I’i, rk irreducible representations of the point group, respectively. In Eq. [2] indexes ns and nr distinguish between cases where a given irreducible representation appears more than once in the process of classifying tensor operators of a given range adapted. to the representations of the group. Similar considerations are valid with reference to indexes n and n’. The Htq)( I$“‘) are functions of range q written in terms of powers of the components of the magnetic field adapted in symmetry with the same prescription we have just described for the operators. To apply the same formalism to a system where the effective electronic spin and/or the magnetic field interact with several ligand nuclei, the same procedure can be applied with the prescription of classifying the nuclei into “families” of ligands. Nuclei which belong to the same family are those corresponding
Nw
/’
/
/
/ ‘I.1
Fe’+i+,,y I#*(/2 --
FIG. family.
I. Description
T,-------,p
of the spin system
t ,/ -------
analyzed
in the text.
zJ
Subscripts
refer
to ligands
belonging
to each
NUCLEAR
INTERACTION
IN
SPIN
HAMILTONIAN
73
to equivalent (and identical) ligands: those whose positions interchange among themselves under the symmetry operations of the group. Taking into account the different components of nuclear spin operators of ligands belonging to a family, linear combinations among them can be constructed so as to transform irreducibly. In this sense the procedure is similar to the construction of symmetry-adapted nuclear displacement coordinates so commonly used in the analysis of vibrational normal modes of a cluster; the differences are that in the present case the equivalent combinations corresponding to translational as well as rigid body rotational modes should be retained, and also that nuclear spin components behave as classical pseudovectors under inversion and reflections instead of the vectorial properties of nuclear displacements. Consequently, tables of symmetry vibrational coordinates for a given cluster can be easily adapted and used advantageously to classify the combinations of nuclear spin operators. The operators thus constructed for a given family will obviously be of first range since the nuclear operators will be contained linearly in each expansion. Higherorder operators can then be easily generated by performing the direct products and projecting them into the desired irreducible representations. Considering the cluster of Fig. 1 with C,, symmetry, the different J#‘$‘) have
TABLE SYMMETRY
LINEAR
1 COMBINATIONS
OF
NUCLEAR SPIN COMPONENTS FOR THE SECDND FAMILY OF LIGANDS”
1 = -2 (12.2.x
-
14.2.x
- z1,Z.y+
J2(r:)
= ; (z1,2.x-
z3,2,x
+
J2(r:)
(zl.2.r
12.2,
J2(r3)
(12.2.x
k2.x
z3.2,x
z2,2~
12.2.~
13.2,~
52trl)
z2,2,
13.2.y)
- &y,
=; + + + =1 - +ZL2Y - -+ 1 - +- 1 J2(r:)
1 = -
(f1.2,~
J2u3
1 = -
(I,.~.~
J2(%)
2
2
=
13.2,
Z3.2,)
(1/2”2&2,x
+
J2(r:+.) = (1/21flxz 23~ 52(rgx) = (1/23z
+
2.2.x
+
z3,2$) z4,2~,
14.2.x)
m:‘y)
=
(i/2’nx~l,2~
+
13.2,)
~2u-%
=
(1/2”2xz1,2,z
-
z3,2,z)
J2(r:y)
=
( w2xz2,2s
-
z4.2.z)
’ Taken
from
Ref. (I).
h2.A
z4,2#)
14.2,~
74
SFERCO,
PASSEGGI,
AND
ALBANESI
been already given by Calvo (I) and are reproduced here in Table 1 for completeness. It is worthwhile mentioning that for that problem with an iron atom surrounded by five nitrogens three “families” of ligand nuclei are obtained. While the iron and the nitrogen (labeled as ligand 3) atoms (see Fig. 1) each form a family by themselves, with Z, and Zr3 transforming as I’* and Z,I,, ZXJZ,,sas basis for the two dimensional representation rs, respectively, the four nitrogen ligands of the heme ring form another family (2) whose operators transform as I’r + 2I’* + I3 + 2r4 + 31Y5(see Table 1). SECOND-RANK
LIGAND
NUCLEI
CONTRIBUTIONS
Restricting ourselves to second-rank contributions we can write explicitly ~~!~~i,, =
C
in the nuclear spin operators,
A$?(n, i(mm’))@(n,
i(mm’))
[31
f>f-n,i(mm’)
where Jj&n,
i(mm’)) = 2 Jf(rlP)J~(r~~“)(r~‘:
I’irI’ir,).
[41
Y-r’ In the present case (I$“‘: I’iJ’jr,) = 6,,@?‘, EiJir). Concerning Eq. [4] and the number of independent parameters A&z, i(mm’)) to be considered, there is one additional point which deserves some examination. Whenever terms with f # f’ are considered, all nuclear spin operators contained in Jf(I’!y’) commute for they belong to different ligands. However, this will not be necessarily the case forf = f’. Generally .$‘(n, i(mm’)) # .$‘(n, i(m’m)) since it can happen that [J,(I-‘,“‘), Jf (I-‘,““)] # 0.
[51
As can be easily verified, this is precisely the case of the operators involved within family 2 and for [JAJ%, J#‘$)l; [JAri), JD:)l, [JG%d, J~U%d. Since these commutators will give rise to isolated linear terms in nuclear spin operators as part of the spin Hamiltonian, a requirement should be imposed to keep the overall operator invariant under time reversal. This can be achieved very easily by requiring that &‘(n,
i(mm’))
= A$‘(n, i(m’m))
[61
which is equivalent to asking that only symmetric products in different components of the spin operators of a given ligand be retained. The 17 different operators J(222) for f = f’ = 2 are shown in Table 2. These 17 operators will not give rise to 17 independent parameters since the requirement imposed by time reversal invariance through Eq. [6] imply that only 14 can be selected as independent. By expanding each of the J2(I’G) explicitly we can write the spin Hamiltonian contributions of these terms as
NUCLEAR
INTERACTION
IN SPIN HAMILTONIAN
15
TABLE 2 SEC~NDORDERINVARIANTPRODUCTSINNUCLEAR SPINOPERATORSOFLIGANDSBELONGINGTO FAMILY 2 INC&SYMMETRY
The explicit form of the second-order tensors are given in Table 3 (note that no effects quadratic in the spin of iron nuclei (family 1) are included, since 1, = l/2), where the Q:” (i = 1, 14) are the 14 independent constants for family 2 and there are two more parameters corresponding to family 3. only four independent parameters are involved in the effective quadrupole interaction terms for family 2 and the remaining 10 correspond to interactions between different nuclei of the same family (six from ligands connected by C, rotations, and four from the ligands connected by Cz operations).
76
SFERCO,
PASSEGGI, TABLE
AND
ALBANESI
3
QUADRUP~LE TENSORMATRICESIN&,SYMMETRY ACCORDINGTOFIG. 1
0:: =r.Q:’ .
Q:’
L
Q:' . Q:' Q:'
a:: = [
a:: = [
32
Q:'
*I -1 L -1
h:’ Q:’
Q? -Qi’ Q:’
Q:’ QG’ Q:’ Q:’ Qf’ Q:’ Q:’ -Qf’ -% Q:’ Qi’ Q;’ Q::
1 Q:' Q;; =1-Q:' 1 Q::e:: a:: =c-e:: -1 22
a:: =
g
[
-Q;’
Q::
a:: =
Q:' .
a%,:
Q:'
[:
0:: =
Q:' -QG’ _ 8:’ Q8 Qi2 L Qf’ 22
-Q:;
22
32
Qg $2 Q;:
Q::
=
.
!-
In addition, terms corresponding to coupling between ligand nuclei belonging to different families will add another set of 12 more parameters describing these interactions. The corresponding matrix representation of F&, is given in Table 4. CONCLUSIONS
In the present paper we have shown how simply nuclear interaction terms as well as effective quadrupole coupling can be incorporated in the ligand spin Hamiltonian and how the elements of the corresponding coupling tensors adopt automatically the correct form according with the symmetry of the system. With regard to the system under consideration we have been able to show that these terms will add a set of 28 new parameters which should be considered as contributing to the general structure of the spin Hamiltonian. It is obvious that only some of them can be resolved by EPR or ENDOR experiments but the majority will not be accessible to experimental determination. However, since they could have relative strength sufficient to influence the ENDOR spectra, it is important to mention the possible microscopic sources of these parameters. Concerning the quadrupole terms, most of the data on these systems are interpreted in terms of the Townes-Dailey model (4) which assumes that the field gradients at the nitrogen nuclei are due to different populations of the p orbitals. This corresponds to contributions which can be called direct, since they arise from the coupling between the nuclear quadrupole moments and the field gradients produced by the charge
NUCLEAR
INTERACTION
IN TABLE
MATRICES
SPIN
HAMILTONIAN
77
4
FOR NUCLEAR INTERACTION ACCORDING TO FIG. 1
TERMS
IN C.,,,
distribution in the molecule. However, the microscopic interaction between electrons and nuclear spins will result in additional contributions (usually called pseudoquadrupolar) (5) to the effective quadrupole parameters, as well as an indirect effective coupling between different nuclei, when considered up to second order in perturbation theory. Unfortunately, estimates for these second-order corrections involve the knowledge of the excited states and their excitation energies. Molecular orbital calculations on the system under examination have been performed using the extended Hiickel method (6), but to our knowledge calculations on the effects we are referring to remain to be made and are beyond the scope of the present paper. As a final remark it is to be noted that second-order splitting corrections caused
78
SFERCO, PASSEGGI, AND ALBANESI
by the hyperfme terms of the spin Hamiltonian have been examined by Scholes et reference to cross terms between different nitrogens of the heme ring, concluding that such corrections are of an order of magnitude less than the observed splittings. However, internuclear coupling terms were not included in their Hamiltonian, even if these terms are allowed from a purely phenomenological point of view, and they could produce a first-order correction to these splittings.
al. (2) with particular
ACKNOWLEDGMENT Helpful suggestions made by Dr. R. Cnlvo are gratefully acknowledged. REFERENCES 1. R. CALVO, J. Magn. Reson. 26, 445 (1977). 2. C. P. *HOLES, A. LAPIDOT, R. MASCARENHAS, T. INUBUSHI, R. A. ISAACSON, AND G. FEHER, J. Am. Chem. Sot. 104,2724 (1982). 3. T. RAY, Proc. Roy. Sot. London Ser. A 277, 76 (1963). 4. A. SCHWEIGER,“Electron Nuclear Double Resonance of Transition Metal Complexes with Organic Ligands,” Springer-Verlag, New York, 1982. 5. A. ABRAGAM AND B. BLEANEY, “Electron Paramagnetic Resonance of Transition Ions,” Oxford Univ. Press (Clarendon), Oxford, 1970. 6. S. K. MON, J. C. CHANG, AND T. P. DAS, J. Am. Chem. Sot. 101, 5562 (1979).
OF MAGNETIC
RESONANCE
59,7
l-78
(1984)
General Description of Nuclear Interaction Terms in a Spin Hamiltonian Including a System of Ligands* S. J. SFERCO,-~ M. C. G. PASSEGGI,$
AND E. A. ALBANEsIt
INTEC.9 Casilla de Correo 91, 3000-Santa Received
November
Fe, Argentina
16, 1983
Following a procedure described by R. Calvo (J. Magn. Reson. 26 (1977)) to construct the ligand hyperline terms contributing to the spin Hamiltonian of isolated paramagnetic centers, the second-rank terms in nuclear spin operators are derived with reference to a system of C,, symmetry. It is shown that these describe appropriately quadrupolar as well as coupling effects between different nuclei. The symmetry is automatically reflected in fixing the number of independent parameters demanded by the addition of such interactions, without new requirements from the group theory point of view. The matrices representing these effects in a six-atom cluster of C,. symmetry are explicitly given as an example.
In a paper by Calve (I) it was shown how the symmetry of an atomic cluster can be advantageously used in deriving a spin Hamihonian to describe the ligand hyperhne structure of isolated paramagnetic centers. A general recipe based on group theory was given in order to construct a set of invariants under symmetry operations in terms of electronic and nuclear spin operators in the presence of a magnetic field. An application to C,, symmetry was also presented with particular reference to the EPR and ENDOR spectra of heme proteins including up to linear terms in the components of nuclear spin operators. In the present paper we apply the same formalism to derive second-order terms in these operators and to describe appropriately the effective quadrupole interactions as well as the coupling terms between nuclear spins, which could add important contributions to the spin Hamiltonian, depending on the nature of the bonding scheme and the nuclear spins of the atoms involved. In particular, it will be shown that the procedure used by Calvo is particularly suitable to this purpose since no new symmetry-adapted combinations of spin operators are necessary, with the result that the new coupling terms automatically incorporate the symmetry in fixing the number of new independent parameters. To illustrate the procedure we have applied the formalism to a system of C,, symmetry. Although very recent ENDOR results by Scholes et al. (2) show that for * t $ 5
Work financed through the BID-CONICET program. Supported by CONICET. Member of the Carrera de1 Investigador Cientifico y Tecnolbgico de1 CONICET. Instituto de Desarrollo Tecnologico para la Industria Quimica, Universidad National and Consejo National de Investigaciones Cientificas y T&micas (CONICET). 71
0022-2364184
de1 Litoral
(UNL)
$3.00
Copyright Q 1984 by Academic Pres.s. Inc. All ri@s of reproduction in any form reserved.
72
SFERCO,
PASSEGGI,
AND
ALBANESI
the system described by Calvo (I) the four nitrogens of the heme ring cannot be considered as equivalent, we will assume that the Cd0 point group can still be used as an approximate symmetry to describe the geometrical configuration of that cluster. SUMMARY
OF
THE
Following T. Ray (3), the spin Hamiltonian
THEORY
is presented (I) under the general form
111 where
Pkr;)= 2 (r’l”)lr,,rk,)(r’,“‘)l r,rjD)P( r~~))fPq)( @))I(‘)(rg))
PI
where ,@‘)(I’$~‘) and l(fl(rpJ’) represent irreducible tensor operators of range p and r transforming as the (Y and y component of the I’i, rk irreducible representations of the point group, respectively. In Eq. [2] indexes ns and nr distinguish between cases where a given irreducible representation appears more than once in the process of classifying tensor operators of a given range adapted. to the representations of the group. Similar considerations are valid with reference to indexes n and n’. The Htq)( I$“‘) are functions of range q written in terms of powers of the components of the magnetic field adapted in symmetry with the same prescription we have just described for the operators. To apply the same formalism to a system where the effective electronic spin and/or the magnetic field interact with several ligand nuclei, the same procedure can be applied with the prescription of classifying the nuclei into “families” of ligands. Nuclei which belong to the same family are those corresponding
Nw
/’
/
/
/ ‘I.1
Fe’+i+,,y I#*(/2 --
FIG. family.
I. Description
T,-------,p
of the spin system
t ,/ -------
analyzed
in the text.
zJ
Subscripts
refer
to ligands
belonging
to each
NUCLEAR
INTERACTION
IN
SPIN
HAMILTONIAN
73
to equivalent (and identical) ligands: those whose positions interchange among themselves under the symmetry operations of the group. Taking into account the different components of nuclear spin operators of ligands belonging to a family, linear combinations among them can be constructed so as to transform irreducibly. In this sense the procedure is similar to the construction of symmetry-adapted nuclear displacement coordinates so commonly used in the analysis of vibrational normal modes of a cluster; the differences are that in the present case the equivalent combinations corresponding to translational as well as rigid body rotational modes should be retained, and also that nuclear spin components behave as classical pseudovectors under inversion and reflections instead of the vectorial properties of nuclear displacements. Consequently, tables of symmetry vibrational coordinates for a given cluster can be easily adapted and used advantageously to classify the combinations of nuclear spin operators. The operators thus constructed for a given family will obviously be of first range since the nuclear operators will be contained linearly in each expansion. Higherorder operators can then be easily generated by performing the direct products and projecting them into the desired irreducible representations. Considering the cluster of Fig. 1 with C,, symmetry, the different J#‘$‘) have
TABLE SYMMETRY
LINEAR
1 COMBINATIONS
OF
NUCLEAR SPIN COMPONENTS FOR THE SECDND FAMILY OF LIGANDS”
1 = -2 (12.2.x
-
14.2.x
- z1,Z.y+
J2(r:)
= ; (z1,2.x-
z3,2,x
+
J2(r:)
(zl.2.r
12.2,
J2(r3)
(12.2.x
k2.x
z3.2,x
z2,2~
12.2.~
13.2,~
52trl)
z2,2,
13.2.y)
- &y,
=; + + + =1 - +ZL2Y - -+ 1 - +- 1 J2(r:)
1 = -
(f1.2,~
J2u3
1 = -
(I,.~.~
J2(%)
2
2
=
13.2,
Z3.2,)
(1/2”2&2,x
+
J2(r:+.) = (1/21flxz 23~ 52(rgx) = (1/23z
+
2.2.x
+
z3,2$) z4,2~,
14.2.x)
m:‘y)
=
(i/2’nx~l,2~
+
13.2,)
~2u-%
=
(1/2”2xz1,2,z
-
z3,2,z)
J2(r:y)
=
( w2xz2,2s
-
z4.2.z)
’ Taken
from
Ref. (I).
h2.A
z4,2#)
14.2,~
74
SFERCO,
PASSEGGI,
AND
ALBANESI
been already given by Calvo (I) and are reproduced here in Table 1 for completeness. It is worthwhile mentioning that for that problem with an iron atom surrounded by five nitrogens three “families” of ligand nuclei are obtained. While the iron and the nitrogen (labeled as ligand 3) atoms (see Fig. 1) each form a family by themselves, with Z, and Zr3 transforming as I’* and Z,I,, ZXJZ,,sas basis for the two dimensional representation rs, respectively, the four nitrogen ligands of the heme ring form another family (2) whose operators transform as I’r + 2I’* + I3 + 2r4 + 31Y5(see Table 1). SECOND-RANK
LIGAND
NUCLEI
CONTRIBUTIONS
Restricting ourselves to second-rank contributions we can write explicitly ~~!~~i,, =
C
in the nuclear spin operators,
A$?(n, i(mm’))@(n,
i(mm’))
[31
f>f-n,i(mm’)
where Jj&n,
i(mm’)) = 2 Jf(rlP)J~(r~~“)(r~‘:
I’irI’ir,).
[41
Y-r’ In the present case (I$“‘: I’iJ’jr,) = 6,,@?‘, EiJir). Concerning Eq. [4] and the number of independent parameters A&z, i(mm’)) to be considered, there is one additional point which deserves some examination. Whenever terms with f # f’ are considered, all nuclear spin operators contained in Jf(I’!y’) commute for they belong to different ligands. However, this will not be necessarily the case forf = f’. Generally .$‘(n, i(mm’)) # .$‘(n, i(m’m)) since it can happen that [J,(I-‘,“‘), Jf (I-‘,““)] # 0.
[51
As can be easily verified, this is precisely the case of the operators involved within family 2 and for [JAJ%, J#‘$)l; [JAri), JD:)l, [JG%d, J~U%d. Since these commutators will give rise to isolated linear terms in nuclear spin operators as part of the spin Hamiltonian, a requirement should be imposed to keep the overall operator invariant under time reversal. This can be achieved very easily by requiring that &‘(n,
i(mm’))
= A$‘(n, i(m’m))
[61
which is equivalent to asking that only symmetric products in different components of the spin operators of a given ligand be retained. The 17 different operators J(222) for f = f’ = 2 are shown in Table 2. These 17 operators will not give rise to 17 independent parameters since the requirement imposed by time reversal invariance through Eq. [6] imply that only 14 can be selected as independent. By expanding each of the J2(I’G) explicitly we can write the spin Hamiltonian contributions of these terms as
NUCLEAR
INTERACTION
IN SPIN HAMILTONIAN
15
TABLE 2 SEC~NDORDERINVARIANTPRODUCTSINNUCLEAR SPINOPERATORSOFLIGANDSBELONGINGTO FAMILY 2 INC&SYMMETRY
The explicit form of the second-order tensors are given in Table 3 (note that no effects quadratic in the spin of iron nuclei (family 1) are included, since 1, = l/2), where the Q:” (i = 1, 14) are the 14 independent constants for family 2 and there are two more parameters corresponding to family 3. only four independent parameters are involved in the effective quadrupole interaction terms for family 2 and the remaining 10 correspond to interactions between different nuclei of the same family (six from ligands connected by C, rotations, and four from the ligands connected by Cz operations).
76
SFERCO,
PASSEGGI, TABLE
AND
ALBANESI
3
QUADRUP~LE TENSORMATRICESIN&,SYMMETRY ACCORDINGTOFIG. 1
0:: =r.Q:’ .
Q:’
L
Q:' . Q:' Q:'
a:: = [
a:: = [
32
Q:'
*I -1 L -1
h:’ Q:’
Q? -Qi’ Q:’
Q:’ QG’ Q:’ Q:’ Qf’ Q:’ Q:’ -Qf’ -% Q:’ Qi’ Q;’ Q::
1 Q:' Q;; =1-Q:' 1 Q::e:: a:: =c-e:: -1 22
a:: =
g
[
-Q;’
Q::
a:: =
Q:' .
a%,:
Q:'
[:
0:: =
Q:' -QG’ _ 8:’ Q8 Qi2 L Qf’ 22
-Q:;
22
32
Qg $2 Q;:
Q::
=
.
!-
In addition, terms corresponding to coupling between ligand nuclei belonging to different families will add another set of 12 more parameters describing these interactions. The corresponding matrix representation of F&, is given in Table 4. CONCLUSIONS
In the present paper we have shown how simply nuclear interaction terms as well as effective quadrupole coupling can be incorporated in the ligand spin Hamiltonian and how the elements of the corresponding coupling tensors adopt automatically the correct form according with the symmetry of the system. With regard to the system under consideration we have been able to show that these terms will add a set of 28 new parameters which should be considered as contributing to the general structure of the spin Hamiltonian. It is obvious that only some of them can be resolved by EPR or ENDOR experiments but the majority will not be accessible to experimental determination. However, since they could have relative strength sufficient to influence the ENDOR spectra, it is important to mention the possible microscopic sources of these parameters. Concerning the quadrupole terms, most of the data on these systems are interpreted in terms of the Townes-Dailey model (4) which assumes that the field gradients at the nitrogen nuclei are due to different populations of the p orbitals. This corresponds to contributions which can be called direct, since they arise from the coupling between the nuclear quadrupole moments and the field gradients produced by the charge
NUCLEAR
INTERACTION
IN TABLE
MATRICES
SPIN
HAMILTONIAN
77
4
FOR NUCLEAR INTERACTION ACCORDING TO FIG. 1
TERMS
IN C.,,,
distribution in the molecule. However, the microscopic interaction between electrons and nuclear spins will result in additional contributions (usually called pseudoquadrupolar) (5) to the effective quadrupole parameters, as well as an indirect effective coupling between different nuclei, when considered up to second order in perturbation theory. Unfortunately, estimates for these second-order corrections involve the knowledge of the excited states and their excitation energies. Molecular orbital calculations on the system under examination have been performed using the extended Hiickel method (6), but to our knowledge calculations on the effects we are referring to remain to be made and are beyond the scope of the present paper. As a final remark it is to be noted that second-order splitting corrections caused
78
SFERCO, PASSEGGI, AND ALBANESI
by the hyperfme terms of the spin Hamiltonian have been examined by Scholes et reference to cross terms between different nitrogens of the heme ring, concluding that such corrections are of an order of magnitude less than the observed splittings. However, internuclear coupling terms were not included in their Hamiltonian, even if these terms are allowed from a purely phenomenological point of view, and they could produce a first-order correction to these splittings.
al. (2) with particular
ACKNOWLEDGMENT Helpful suggestions made by Dr. R. Cnlvo are gratefully acknowledged. REFERENCES 1. R. CALVO, J. Magn. Reson. 26, 445 (1977). 2. C. P. *HOLES, A. LAPIDOT, R. MASCARENHAS, T. INUBUSHI, R. A. ISAACSON, AND G. FEHER, J. Am. Chem. Sot. 104,2724 (1982). 3. T. RAY, Proc. Roy. Sot. London Ser. A 277, 76 (1963). 4. A. SCHWEIGER,“Electron Nuclear Double Resonance of Transition Metal Complexes with Organic Ligands,” Springer-Verlag, New York, 1982. 5. A. ABRAGAM AND B. BLEANEY, “Electron Paramagnetic Resonance of Transition Ions,” Oxford Univ. Press (Clarendon), Oxford, 1970. 6. S. K. MON, J. C. CHANG, AND T. P. DAS, J. Am. Chem. Sot. 101, 5562 (1979).