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A comment on the pseudo-nuclear Zeeman effect
Journal of Magnetic Resonance 218 (2012) 11–15
Contents lists available at SciVerse ScienceDirect
Journal of Magnetic Resonance journal homepage: www.elsevier.com/locate/jmr
Communication
A comment on the pseudo-nuclear Zeeman effect Silvia Sottini, Edgar J.J. Groenen ⇑ Department of Molecular Physics, Huygens Laboratory, Leiden University, The Netherlands
a r t i c l e
i n f o
Article history: Received 6 February 2012 Revised 15 March 2012 Available online 24 March 2012 Keywords: Pseudo-nuclear Zeeman ENDOR High-spin systems
a b s t r a c t For high-spin systems whose magnetic sublevels are arranged in doublets at zero field, the electron-paramagnetic-resonance (EPR) spectra are commonly described by an effective spin Hamiltonian. We show that also in this approach, if the mixing of the electron spin states by the hyperfine interaction is negligible, a proper description of electron-nuclear double resonance (ENDOR) spectra can be obtained using a nuclear spin Hamiltonian in which the electron spin angular momentum operator is replaced by its expectation value. Appropriate values of this expectation value can be obtained from a wave function correct to first-order in the electron Zeeman interaction. In terms of perturbation theory, such a description is more logical than the conventional practice based on the inclusion of a second-order cross term, the socalled pseudo-nuclear Zeeman effect, which involves both the electron Zeeman interaction and the hyperfine interaction. We illustrate our analysis with calculations of the expectation value of the electron spin angular momentum and of the energies of the hyperfine levels for a high-spin cobalt complex, which we studied by EPR and ENDOR recently. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction For high-spin systems, electron-paramagnetic-resonance (EPR) spectroscopy is not always feasible at the commonly employed microwave frequencies like X-band. However, when the energy levels occur in doublets in the absence of a magnetic field, the EPR transitions within a doublet can be detected. If the zero-field splitting (ZFS) largely exceeds the microwave quantum, each doublet may be considered as an effective S = 1/2 system that can be described by an effective spin Hamiltonian. The terms of the effective spin Hamiltonian comprehend the same interactions as present in the spin Hamiltonian of a true S = 1/2 system, but are represented by effective tensors. These tensors are related to the corresponding true tensors of the high-spin system through the zero-field splitting. A calculation of EPR and ENDOR spectra based on perturbation theory is applicable. The electron Zeeman interaction lifts the degeneracy of each doublet in first order. In second-order the cross term that involves the electron Zeeman interaction and the hyperfine interaction was found of particular significance [1,2]. This term takes the form of an anisotropic nuclear Zeeman interaction and is hence referred to as the pseudo-nuclear Zeeman effect. A number of examples exist in the literature, where the inclusion of the pseudo-nuclear Zeeman effect was found to be important to properly analyze experiments involving transitions between nuclear sublevels [3–10]. The correction to the energy introduced by the ⇑ Corresponding author. E-mail address: [email protected] (E.J.J. Groenen). 1090-7807/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmr.2012.03.009
pseudo-nuclear Zeeman effect may easily exceed the true nuclear Zeeman effect, even for systems for which the mixing of the electron-spin states due to the hyperfine interaction is negligible. In this paper we report on the insufficiency of the effective-spin picture as regards the description of properties that depend on the eigenstates of the electron spin. We show that a perturbation treatment up to first order in the eigenstates is the logical approach to gain proper expectation values of the electron-spin operators. We discuss the pseudo-nuclear Zeeman effect and show how it partially corrects for the error made in the calculation of these expectation values by the use of zero-order wave functions. Finally, we compare and discuss the different levels of approximation in the framework of perturbation theory for a Co(II) high-spin complex that we have studied in detail by EPR and ENDOR recently. 2. Theory We consider a system with spin S > 1/2, described by a spin Hamiltonian H that includes the zero-field splitting interaction, the electron Zeeman interaction, the hyperfine interaction with the nuclear spins and the nuclear Zeeman interaction
! ! ! ! P ! ! ! ! H ¼ S D S þ lB B g S þ ðg Nm lN I m B þ S Am I m Þ: m
ð1Þ ! Here S represents the electron-spin angular momentum operator, ! I m the nuclear-spin angular momentum operator of the mth nu
cleus, and the D, g and Am tensors represent the interactions [11].
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Communication / Journal of Magnetic Resonance 218 (2012) 11–15
Our interest lies in those systems, for which the ZFS is the leading term in the spin Hamiltonian,
! ! H0 ¼ S D S
ð2Þ
and the energy levels in zero field are arranged in doublets. The degeneracy of each doublet is removed by the application of a magnetic field. First-order perturbation theory in terms of
! ! H 1 ¼ lB B g S
ð3Þ
results in a correction E1i to the energies E0i of H0 and in eigenstates w0i . The eigenstate w0i of an energy level in a doublet is a linear combination of the eigenstates in zero field of that doublet. The effect of the interactions with the nuclear spins is, in general, orders of magnitude smaller than the electron Zeeman interaction. Nonetheless, in such cases a second-order contribution has
been taken into account for the description of EPR and, in particular, ENDOR spectra [3–10]. This contribution concerns a cross term, which involves the electron Zeeman interaction and the hyperfine interaction for a certain nucleus. The correction Epn i for the mth nucleus, is given by Epn i ¼
X jRK
! ! ! ! ðhw0j jH1 jw0i ihw0i j S Am I m jw0j i þ hw0i jH1 jw0j ihw0j j S Am I m jw0i iÞ ðE0i E0j Þ ð4Þ
where K is the subspace defined by a certain doublet. The expression for Epn can be rewritten in the form of an anisotropic nuclei ar-Zeeman interaction, hence the name pseudo-nuclear Zeeman effect. In other words, the correction can be obtained by replacing the nuclear factor gNm by
Fig. 1. The expectation values of the electron-spin operators (left) and the energies (right) as a function of the relative magnitude of the electron Zeeman interaction and the ZFS for the Co S = 3/2 system. The spin-Hamiltonian parameters D = 386.73 GHz and E = 125.57 GHz, and gx = 2.284, gy = 2.402 and gz = 2.335 have been used in the exact calculations (solid lines) and for the calculations based on perturbation theory (dotted lines) up to first-order in the energy and up to zero-order in the eigenstates (Eq. (7)).
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Communication / Journal of Magnetic Resonance 218 (2012) 11–15
g eff Nm ¼ g Nm ð1 rm Þ:
ð5Þ
The pseudo-nuclear Zeeman effect may easily outweigh the true nuclear Zeeman effect. Because the interactions between the electron and the nuclear spin are a small perturbation compared to H1, a substantial correction through this cross term at second-order is not satisfactory in terms of perturbation theory. When the mixing of the electron-spin states within the doublet by the hyperfine interaction is negligible, the energy of the hyperfine levels should in fact be describable by a nuclear-spin Hamiltonian
! ! ! ! Hn ¼ g Nm lN I m B þ h S i Am I m ;
ð6Þ
in which as usual the electron-spin operators in the hyperfine-interaction term of the spin Hamiltonian are replaced by the corresponding expectation values. Up to zero-order, the expectation values of the electron-spin operators in Eq. (6) are
hS0k i ¼ w0i Sk w0i :
ð7Þ
A proper description of the expectation values requires inclusion of the first-order correction w1i to the eigenstates, which is given by
D w0j H1 w0i E 1 X w ¼ w0j i jRK ðE0i E0j Þ
3. An example Recently we studied a complex with CoIIS4 coordination, (Co[(SPPh2)(SPiPr2)N]2, by EPR and ENDOR spectroscopy [15,16]. This high-spin cobalt (S = 3/2) system is characterized by a rhombic ZFS of magnitude 29.6 cm1 and E/D = 0.33. A magnetically diluted single crystal of this compound has been investigated at
X-band as well as at W-band. The effective g 0 matrix and the
hyperfine tensors of the cobalt nucleus ACo and of the four nearby phosphorus nuclei APi were derived. For this system, the ZFS is the leading term in the spin Hamiltonian, both at X-band and at W-band frequencies. The energy levels are arranged in Kramers doublets, and only transitions within the lowest doublet are excited at these microwave frequencies and low temperatures. Moreover, the mixing of the electron-spin states by the hyperfine interaction is negligible for the nuclei studied and a nuclear-spin Hamiltonian (Eq. (6)) was used to describe the cobalt hyperfine splitting of the cw EPR spectra and the phosphorus ENDOR spectra. We will use the cobalt complex as an example to illustrate the analysis outlined in the previous section. Since we have all spin-Hamiltonian parameters for this complex at our disposal
ð8Þ
where K is the subspace defined by a certain doublet. The expectation value hS0k i (k = x, y, z) of the spin operators is then calculated over the eigenstates w0i , with w0i ¼ w0i þ w1i
hS0k i ¼ hwi jSk jwi i ¼ w0i þ w1i jSk jw0i þ w1i ¼ hS0k i þ w1i jSk jw1i þ w0i jSk jw1i þ w1i jSk jw0i :
ð9Þ
hS0k i
and hS0k i includes w0i
A significant difference between indicates that a description at the level of H1 that only does not suffice. Substitution of the expectation value in Eq. (9) into Eq. (6) and diagonalization of Hn in the basis of nuclear spin states yields the energy shift of the nuclear spin sublevels with respect to E0i . A related description of hyperfine interaction for a ZFS that is small compared to the electron Zeeman interaction, opposite to the case treated here, has been reported [12]. The last two terms in the expectation values of Eq. (9) give rise to contributions to the energies of the nuclear sublevels equivalent to Epn i . Consequently, the energies of the nuclear sublevels, obtained from Eq. (6) using the expectation values hS0k i will be numerically virtually identical to those obtained from the pseudo-nuclear Zeeman correction. The correct calculation using Eqs. (6) and (9) provides the insight that the pseudo-nuclear Zeeman effect lacks. While the latter description seems to imply a correction of the nuclear Zeeman interaction, the problem concerns the inadequate description of the expectation values of the electron-spin operators in the hyperfine coupling term. The significance of the first-order correction w1i , which represents the mixing of the electron-spin states from different doublets by H1, implies that the effective S = 1/2 picture breaks down. For the two electron-spin states of an S = 1/2 doublet the expectation values of the electron spin are equal in magnitude and opposite in sign, but this is not necessarily true for a doublet that derives from a system with S > 1/2 [13,14]. For example, the observation, for a nuclear-spin I = 1/2, of ENDOR lines asymmetrically shifted with respect to the nuclear Zeeman frequency does not derive from a shift of that frequency, but is symptomatic of the insufficiency of the effective-spin picture.
Fig. 2. The expectation values of the electron-spin operators for the four eigenstates of the Co S = 3/2 system as a function of the relative magnitude of the electron Zeeman interaction and the ZFS. The same spin-Hamiltonian parameters as for Fig. 1. have been used in the spin Hamiltonian for the exact calculation (solid lines) and for the first-order calculation (Eq. (9), dotted lines).
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Communication / Journal of Magnetic Resonance 218 (2012) 11–15
Fig. 3. The hyperfine energies corresponding to mI = ±1/2 and the lowest-energy ms level (left) and their energy differences (right) as a function of the relative magnitude of the electron Zeeman interaction and the ZFS for the Co S = 3/2 system. In addition to the spin-Hamiltonian parameters for Fig. 1. Values of Axx = 116 MHz Ayy = 45 MHz Axz = 5.1 MHz Azz = 110 MHz Axy = Ayz = 0 have been used for the exact calculation, for the calculation of the pseudo-nuclear Zeeman correction, and for the calculations based on the nuclear spin Hamiltonian (Eq. (6)) with electron-spin expectation values up to zero order (Eq. (7)) and up to first order (Eq. (9)).
(D = 386.73 GHz and E = 125.57 GHz, and gx = 2.284, gy = 2.402 and gz = 2.335), we can perform a full numerical diagonalization of the matrix representation of the spin Hamiltonian to find eigenvalues and eigenstates, and calculate the expectation values of the spin operators hSki, with k = x, y, z. We compare the exact values for the energies and the expectation values of the electron-spin operators with the result of calculations at the different levels of approximation described in the theory section. In Fig. 1, the values of hSki are represented together with those of hS0k i Eq. (7)) for each magnetic substate, as a function of the ratio bepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tween the electron Zeeman energy and the ZFS ½D0 ¼ D2 þ 3 E2 . In the same figure, the energy Ei of each level is compared with its first-order approximation E0i ¼ E0i þ E1i . The first-order perturbation approach reasonably well approximates the energies up to perturbations of 20–30%, but the zero-order eigenstates do not give an adequate description of the expectation values of the spin operators. In order to obtain appropriate values of the expectation values of the electron-spin, it is necessary to include higher-order effects and consider the contribution of the excited states. In Fig. 2 the values of hSki are represented together with those of hS0k i (Eq. (9)), the latter
being calculated for the eigenstates w0i which are correct up to firstorder. These values represent a better approximation to the exact expectation values, similar to the quality of the approximation of the exact energies by E0i . Then we consider the nuclear-spin Hamiltonian Hn (Eq. (6)) for the cobalt nucleus (I = 7/2) and investigate the description of the energy of the hyperfine levels at different levels of approximation of the expectation values hSki, Eqs. (7) and (9), and by the pseudonuclear Zeeman correction, Eq. (5). From the experiment, the absolute values of the principal values and the direction cosines of the cobalt-hyperfine tensor ACo are known [15]. In the numerical calcu lation the sign of the principal values of ACo is chosen negative, as suggested by a DFT study performed for this CoIIS4 complex [17]. The energies of the mI = ±1/2 levels, corresponding to the ms level of lowest energy, are shown in Fig. 3. The energy difference DE between the hyperfine levels is represented in the same figure as a function of the ratio between the electron Zeeman energy and the ZFS. The best approximation to the exact data is obtained when the values of hS0k i calculated over the eigenstates w0i , are used in the nuclear-spin Hamiltonian Hn. When the values of hS0k i are used in Hn, together with the pseudo-nuclear Zeeman correction, the exact
Communication / Journal of Magnetic Resonance 218 (2012) 11–15
values are also reasonably well approximated. The correction Epn i to the energy, in fact, comprises the two most important contributions of the first-order corrected hS0k i to Hn. Indeed, as outlined in the theory section, the pseudo-nuclear Zeeman term partially corrects for the use of the eigenvectors w0i . This is also clear from Fig. 3 where the result of the calculation with only the hS0k i in the nuclear-spin Hamiltonian Hn is represented as well. 4. Conclusion Whenever the ZFS term is the leading term in the spin Hamiltonian of a high-spin system, and the energy levels are arranged in doublets, an effective-spin S = 1/2 description provides a good approximation of the electron-spin energies. A perturbation treatment up to first order in the eigenstates, which takes into account contributions from other doublets, is necessary to obtain eigenstates that are accurate enough to calculate properties like expectation values of the electron spin angular momentum and transition probabilities. Such expectation values allow the use of a nuclear spin Hamiltonian in the calculation of the energy of hyperfine levels, and provide an adequate description of ENDOR spectra. The common practice of adding the pseudo-nuclear Zeeman term to the effective spin Hamiltonian introduces a correction in the energy that compensates for the poor approximation of the eigenstates of the electron spin, but is theoretically ill-defined. A correction at second-order in perturbation theory with a perturbing Hamiltonian that includes the hyperfine interaction is not in line with the hypothesis underlying the concept of a nuclear spin Hamiltonian. Such an approach obscures the fact that the pseudo nuclear Zeeman term actually corrects for an inadequate treatment of the electron Zeeman interaction. Acknowledgments The authors highly appreciate a discussion concerning the present study with J.H. van der Waals. The research was supported with financial aid by The Netherlands Organization for Scientific Research (NWO), Department of Chemical Sciences (CW). References [1] J.M. Baker, B. Bleaney, Paramagnetic resonance in some lanthanon ethyl sulphates, Proc. R. Soc. A 245 (1958) 156–174. [2] A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of Transition Ions, Clarendon Press, Oxford, 1970.
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[3] W. Nijman, R. de Beer, M.J. Jordaan, EPR and ENDOR of 59Co++ in La2Mg3(NO3)1224H2O, Physica 67 (1973) 595–612. [4] C.P. Scholes, A. Lapidot, R. Mascarenhas, T. Inubushi, R.A. Isaacson, G. Feher, Electron nuclear double resonance (ENDOR) from heme and histidine nitrogens in single crystals of aquometmyoglobin, J. Am. Chem. Soc. 104 (1982) 2724–2735. [5] A.E. True, P. McLean, M.J. Nelson, W.H. Orme-Johnson, B.M. Hoffman, Comparison of wild-type and nifV mutant molybdenum–iron proteins of nitrogenase from Klebsiella pneumoniae by ENDOR spectroscopy, J. Am. Chem. Soc. 112 (1990) 651–657. [6] Y. Fann, J. Ong, J.M. Nocek, B.M. Hoffman, 19F and 1,2H ENDOR study of distalpocket N(e)H F hydrogen bonding in fluorometmyoglobin, J. Am. Chem. Soc. 117 (1995) 6109–6116. [7] J.M. Baker, G.M. Copland, B.M. Wanklyn, Nuclear moments and hyperfine anomaly for gadolinium, J. Phys. C: Solid State Phys. 2 (1969) 862–869. [8] M.L. Falin, M.V. Eremin, M.M. Zaripov, I.R. Ibragimov, A.M. Leushin, R.Y. Abdulsabirov, S.L. Korableva, The peculiarities of electron-nuclear and pseudoZeeman interactions of 19F nuclei in KZnF3:Er3+, J. Phys.: Condens. Matter. 1 (1989) 2331–2340. [9] F. Trandafir, P. Heerdt, M. Fittipaldi, E. Vink, S. Dewilde, L. Moens, S. van Dorslaer, Studying high-spin ferric heme proteins by pulsed EPR spectroscopy: analysis of the ferric form of the E7Q mutant of human neuroglobin, Appl. Magn. Reson. 31 (2007) 553–572. [10] M. Fittipaldi, I. Garcia-Rubio, F. Trandafir, I. Gromov, A. Schweiger, A. Bouwen, S. van Doorslaer, A multi-frequency pulse EPR and ENDOR approach to study strongly coupled nuclei in frozen solutions of high-spin ferric heme proteins, J. Phys. Chem. B 112 (2008) 3859–3870. [11] N.M. Atherton, Principles of Electron Spin Resonance, PTR Prentice Hall, New York, 1993. [12] R. Vardi, M. Bernardo, H. Thomann, K.G. Strohmaier, D.E.W. Vaughan, D. Goldfarb, X-band pulsed ENDOR study of 57Fe-substituted sodalite: the effect of the zero-field splitting, J. Magn. Reson. 126 (1997) 229–241. [13] G. Jeschke, A. Schweiger, Hyperfine decoupling in electron spin resonance, J. Chem. Phys. 106 (1997) 9979–9991. [14] A. Schweiger, G. Jeschke, Principles of Pulse Electron Paramagnetic Resonance, Oxford University Press, Oxford, 2001. p. 57. [15] D. Maganas, S. Milikisyants, J.M.A. Rijnbeek, S. Sottini, N. Levesanos, P. Kyritsis, E.J.J. Groenen, A multi-frequency high-field electron-paramagnetic-resonance study of CoIIS4 coordination, Inorg. Chem. 49 (2) (2010) 595–605. [16] S. Sottini, G. Mathies, P. Gast, D. Maganas, P. Kyritsis, E.J.J. Groenen, A W-band pulsed EPR/ENDOR study of CoIIS4 coordination in the Co[(SPPh2)(SPiPr2)N]2 complex PCCP 11 (2009) 6727–6732. [17] D. Maganas, S. Milikisyants, J.M.A. Rijnbeek, S. Sottini, N. Levesanos, P. Kyritsis, E.J.J. Groenen, Theoretical analysis of the spin Hamiltonian parameters in Co(II)S4 complexes using density functional theory and correlated ab initio methods, Inorg. Chem. 50 (2011) 8741–8754.
Contents lists available at SciVerse ScienceDirect
Journal of Magnetic Resonance journal homepage: www.elsevier.com/locate/jmr
Communication
A comment on the pseudo-nuclear Zeeman effect Silvia Sottini, Edgar J.J. Groenen ⇑ Department of Molecular Physics, Huygens Laboratory, Leiden University, The Netherlands
a r t i c l e
i n f o
Article history: Received 6 February 2012 Revised 15 March 2012 Available online 24 March 2012 Keywords: Pseudo-nuclear Zeeman ENDOR High-spin systems
a b s t r a c t For high-spin systems whose magnetic sublevels are arranged in doublets at zero field, the electron-paramagnetic-resonance (EPR) spectra are commonly described by an effective spin Hamiltonian. We show that also in this approach, if the mixing of the electron spin states by the hyperfine interaction is negligible, a proper description of electron-nuclear double resonance (ENDOR) spectra can be obtained using a nuclear spin Hamiltonian in which the electron spin angular momentum operator is replaced by its expectation value. Appropriate values of this expectation value can be obtained from a wave function correct to first-order in the electron Zeeman interaction. In terms of perturbation theory, such a description is more logical than the conventional practice based on the inclusion of a second-order cross term, the socalled pseudo-nuclear Zeeman effect, which involves both the electron Zeeman interaction and the hyperfine interaction. We illustrate our analysis with calculations of the expectation value of the electron spin angular momentum and of the energies of the hyperfine levels for a high-spin cobalt complex, which we studied by EPR and ENDOR recently. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction For high-spin systems, electron-paramagnetic-resonance (EPR) spectroscopy is not always feasible at the commonly employed microwave frequencies like X-band. However, when the energy levels occur in doublets in the absence of a magnetic field, the EPR transitions within a doublet can be detected. If the zero-field splitting (ZFS) largely exceeds the microwave quantum, each doublet may be considered as an effective S = 1/2 system that can be described by an effective spin Hamiltonian. The terms of the effective spin Hamiltonian comprehend the same interactions as present in the spin Hamiltonian of a true S = 1/2 system, but are represented by effective tensors. These tensors are related to the corresponding true tensors of the high-spin system through the zero-field splitting. A calculation of EPR and ENDOR spectra based on perturbation theory is applicable. The electron Zeeman interaction lifts the degeneracy of each doublet in first order. In second-order the cross term that involves the electron Zeeman interaction and the hyperfine interaction was found of particular significance [1,2]. This term takes the form of an anisotropic nuclear Zeeman interaction and is hence referred to as the pseudo-nuclear Zeeman effect. A number of examples exist in the literature, where the inclusion of the pseudo-nuclear Zeeman effect was found to be important to properly analyze experiments involving transitions between nuclear sublevels [3–10]. The correction to the energy introduced by the ⇑ Corresponding author. E-mail address: [email protected] (E.J.J. Groenen). 1090-7807/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmr.2012.03.009
pseudo-nuclear Zeeman effect may easily exceed the true nuclear Zeeman effect, even for systems for which the mixing of the electron-spin states due to the hyperfine interaction is negligible. In this paper we report on the insufficiency of the effective-spin picture as regards the description of properties that depend on the eigenstates of the electron spin. We show that a perturbation treatment up to first order in the eigenstates is the logical approach to gain proper expectation values of the electron-spin operators. We discuss the pseudo-nuclear Zeeman effect and show how it partially corrects for the error made in the calculation of these expectation values by the use of zero-order wave functions. Finally, we compare and discuss the different levels of approximation in the framework of perturbation theory for a Co(II) high-spin complex that we have studied in detail by EPR and ENDOR recently. 2. Theory We consider a system with spin S > 1/2, described by a spin Hamiltonian H that includes the zero-field splitting interaction, the electron Zeeman interaction, the hyperfine interaction with the nuclear spins and the nuclear Zeeman interaction
! ! ! ! P ! ! ! ! H ¼ S D S þ lB B g S þ ðg Nm lN I m B þ S Am I m Þ: m
ð1Þ ! Here S represents the electron-spin angular momentum operator, ! I m the nuclear-spin angular momentum operator of the mth nu
cleus, and the D, g and Am tensors represent the interactions [11].
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Communication / Journal of Magnetic Resonance 218 (2012) 11–15
Our interest lies in those systems, for which the ZFS is the leading term in the spin Hamiltonian,
! ! H0 ¼ S D S
ð2Þ
and the energy levels in zero field are arranged in doublets. The degeneracy of each doublet is removed by the application of a magnetic field. First-order perturbation theory in terms of
! ! H 1 ¼ lB B g S
ð3Þ
results in a correction E1i to the energies E0i of H0 and in eigenstates w0i . The eigenstate w0i of an energy level in a doublet is a linear combination of the eigenstates in zero field of that doublet. The effect of the interactions with the nuclear spins is, in general, orders of magnitude smaller than the electron Zeeman interaction. Nonetheless, in such cases a second-order contribution has
been taken into account for the description of EPR and, in particular, ENDOR spectra [3–10]. This contribution concerns a cross term, which involves the electron Zeeman interaction and the hyperfine interaction for a certain nucleus. The correction Epn i for the mth nucleus, is given by Epn i ¼
X jRK
! ! ! ! ðhw0j jH1 jw0i ihw0i j S Am I m jw0j i þ hw0i jH1 jw0j ihw0j j S Am I m jw0i iÞ ðE0i E0j Þ ð4Þ
where K is the subspace defined by a certain doublet. The expression for Epn can be rewritten in the form of an anisotropic nuclei ar-Zeeman interaction, hence the name pseudo-nuclear Zeeman effect. In other words, the correction can be obtained by replacing the nuclear factor gNm by
Fig. 1. The expectation values of the electron-spin operators (left) and the energies (right) as a function of the relative magnitude of the electron Zeeman interaction and the ZFS for the Co S = 3/2 system. The spin-Hamiltonian parameters D = 386.73 GHz and E = 125.57 GHz, and gx = 2.284, gy = 2.402 and gz = 2.335 have been used in the exact calculations (solid lines) and for the calculations based on perturbation theory (dotted lines) up to first-order in the energy and up to zero-order in the eigenstates (Eq. (7)).
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Communication / Journal of Magnetic Resonance 218 (2012) 11–15
g eff Nm ¼ g Nm ð1 rm Þ:
ð5Þ
The pseudo-nuclear Zeeman effect may easily outweigh the true nuclear Zeeman effect. Because the interactions between the electron and the nuclear spin are a small perturbation compared to H1, a substantial correction through this cross term at second-order is not satisfactory in terms of perturbation theory. When the mixing of the electron-spin states within the doublet by the hyperfine interaction is negligible, the energy of the hyperfine levels should in fact be describable by a nuclear-spin Hamiltonian
! ! ! ! Hn ¼ g Nm lN I m B þ h S i Am I m ;
ð6Þ
in which as usual the electron-spin operators in the hyperfine-interaction term of the spin Hamiltonian are replaced by the corresponding expectation values. Up to zero-order, the expectation values of the electron-spin operators in Eq. (6) are
hS0k i ¼ w0i Sk w0i :
ð7Þ
A proper description of the expectation values requires inclusion of the first-order correction w1i to the eigenstates, which is given by
D w0j H1 w0i E 1 X w ¼ w0j i jRK ðE0i E0j Þ
3. An example Recently we studied a complex with CoIIS4 coordination, (Co[(SPPh2)(SPiPr2)N]2, by EPR and ENDOR spectroscopy [15,16]. This high-spin cobalt (S = 3/2) system is characterized by a rhombic ZFS of magnitude 29.6 cm1 and E/D = 0.33. A magnetically diluted single crystal of this compound has been investigated at
X-band as well as at W-band. The effective g 0 matrix and the
hyperfine tensors of the cobalt nucleus ACo and of the four nearby phosphorus nuclei APi were derived. For this system, the ZFS is the leading term in the spin Hamiltonian, both at X-band and at W-band frequencies. The energy levels are arranged in Kramers doublets, and only transitions within the lowest doublet are excited at these microwave frequencies and low temperatures. Moreover, the mixing of the electron-spin states by the hyperfine interaction is negligible for the nuclei studied and a nuclear-spin Hamiltonian (Eq. (6)) was used to describe the cobalt hyperfine splitting of the cw EPR spectra and the phosphorus ENDOR spectra. We will use the cobalt complex as an example to illustrate the analysis outlined in the previous section. Since we have all spin-Hamiltonian parameters for this complex at our disposal
ð8Þ
where K is the subspace defined by a certain doublet. The expectation value hS0k i (k = x, y, z) of the spin operators is then calculated over the eigenstates w0i , with w0i ¼ w0i þ w1i
hS0k i ¼ hwi jSk jwi i ¼ w0i þ w1i jSk jw0i þ w1i ¼ hS0k i þ w1i jSk jw1i þ w0i jSk jw1i þ w1i jSk jw0i :
ð9Þ
hS0k i
and hS0k i includes w0i
A significant difference between indicates that a description at the level of H1 that only does not suffice. Substitution of the expectation value in Eq. (9) into Eq. (6) and diagonalization of Hn in the basis of nuclear spin states yields the energy shift of the nuclear spin sublevels with respect to E0i . A related description of hyperfine interaction for a ZFS that is small compared to the electron Zeeman interaction, opposite to the case treated here, has been reported [12]. The last two terms in the expectation values of Eq. (9) give rise to contributions to the energies of the nuclear sublevels equivalent to Epn i . Consequently, the energies of the nuclear sublevels, obtained from Eq. (6) using the expectation values hS0k i will be numerically virtually identical to those obtained from the pseudo-nuclear Zeeman correction. The correct calculation using Eqs. (6) and (9) provides the insight that the pseudo-nuclear Zeeman effect lacks. While the latter description seems to imply a correction of the nuclear Zeeman interaction, the problem concerns the inadequate description of the expectation values of the electron-spin operators in the hyperfine coupling term. The significance of the first-order correction w1i , which represents the mixing of the electron-spin states from different doublets by H1, implies that the effective S = 1/2 picture breaks down. For the two electron-spin states of an S = 1/2 doublet the expectation values of the electron spin are equal in magnitude and opposite in sign, but this is not necessarily true for a doublet that derives from a system with S > 1/2 [13,14]. For example, the observation, for a nuclear-spin I = 1/2, of ENDOR lines asymmetrically shifted with respect to the nuclear Zeeman frequency does not derive from a shift of that frequency, but is symptomatic of the insufficiency of the effective-spin picture.
Fig. 2. The expectation values of the electron-spin operators for the four eigenstates of the Co S = 3/2 system as a function of the relative magnitude of the electron Zeeman interaction and the ZFS. The same spin-Hamiltonian parameters as for Fig. 1. have been used in the spin Hamiltonian for the exact calculation (solid lines) and for the first-order calculation (Eq. (9), dotted lines).
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Fig. 3. The hyperfine energies corresponding to mI = ±1/2 and the lowest-energy ms level (left) and their energy differences (right) as a function of the relative magnitude of the electron Zeeman interaction and the ZFS for the Co S = 3/2 system. In addition to the spin-Hamiltonian parameters for Fig. 1. Values of Axx = 116 MHz Ayy = 45 MHz Axz = 5.1 MHz Azz = 110 MHz Axy = Ayz = 0 have been used for the exact calculation, for the calculation of the pseudo-nuclear Zeeman correction, and for the calculations based on the nuclear spin Hamiltonian (Eq. (6)) with electron-spin expectation values up to zero order (Eq. (7)) and up to first order (Eq. (9)).
(D = 386.73 GHz and E = 125.57 GHz, and gx = 2.284, gy = 2.402 and gz = 2.335), we can perform a full numerical diagonalization of the matrix representation of the spin Hamiltonian to find eigenvalues and eigenstates, and calculate the expectation values of the spin operators hSki, with k = x, y, z. We compare the exact values for the energies and the expectation values of the electron-spin operators with the result of calculations at the different levels of approximation described in the theory section. In Fig. 1, the values of hSki are represented together with those of hS0k i Eq. (7)) for each magnetic substate, as a function of the ratio bepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tween the electron Zeeman energy and the ZFS ½D0 ¼ D2 þ 3 E2 . In the same figure, the energy Ei of each level is compared with its first-order approximation E0i ¼ E0i þ E1i . The first-order perturbation approach reasonably well approximates the energies up to perturbations of 20–30%, but the zero-order eigenstates do not give an adequate description of the expectation values of the spin operators. In order to obtain appropriate values of the expectation values of the electron-spin, it is necessary to include higher-order effects and consider the contribution of the excited states. In Fig. 2 the values of hSki are represented together with those of hS0k i (Eq. (9)), the latter
being calculated for the eigenstates w0i which are correct up to firstorder. These values represent a better approximation to the exact expectation values, similar to the quality of the approximation of the exact energies by E0i . Then we consider the nuclear-spin Hamiltonian Hn (Eq. (6)) for the cobalt nucleus (I = 7/2) and investigate the description of the energy of the hyperfine levels at different levels of approximation of the expectation values hSki, Eqs. (7) and (9), and by the pseudonuclear Zeeman correction, Eq. (5). From the experiment, the absolute values of the principal values and the direction cosines of the cobalt-hyperfine tensor ACo are known [15]. In the numerical calcu lation the sign of the principal values of ACo is chosen negative, as suggested by a DFT study performed for this CoIIS4 complex [17]. The energies of the mI = ±1/2 levels, corresponding to the ms level of lowest energy, are shown in Fig. 3. The energy difference DE between the hyperfine levels is represented in the same figure as a function of the ratio between the electron Zeeman energy and the ZFS. The best approximation to the exact data is obtained when the values of hS0k i calculated over the eigenstates w0i , are used in the nuclear-spin Hamiltonian Hn. When the values of hS0k i are used in Hn, together with the pseudo-nuclear Zeeman correction, the exact
Communication / Journal of Magnetic Resonance 218 (2012) 11–15
values are also reasonably well approximated. The correction Epn i to the energy, in fact, comprises the two most important contributions of the first-order corrected hS0k i to Hn. Indeed, as outlined in the theory section, the pseudo-nuclear Zeeman term partially corrects for the use of the eigenvectors w0i . This is also clear from Fig. 3 where the result of the calculation with only the hS0k i in the nuclear-spin Hamiltonian Hn is represented as well. 4. Conclusion Whenever the ZFS term is the leading term in the spin Hamiltonian of a high-spin system, and the energy levels are arranged in doublets, an effective-spin S = 1/2 description provides a good approximation of the electron-spin energies. A perturbation treatment up to first order in the eigenstates, which takes into account contributions from other doublets, is necessary to obtain eigenstates that are accurate enough to calculate properties like expectation values of the electron spin angular momentum and transition probabilities. Such expectation values allow the use of a nuclear spin Hamiltonian in the calculation of the energy of hyperfine levels, and provide an adequate description of ENDOR spectra. The common practice of adding the pseudo-nuclear Zeeman term to the effective spin Hamiltonian introduces a correction in the energy that compensates for the poor approximation of the eigenstates of the electron spin, but is theoretically ill-defined. A correction at second-order in perturbation theory with a perturbing Hamiltonian that includes the hyperfine interaction is not in line with the hypothesis underlying the concept of a nuclear spin Hamiltonian. Such an approach obscures the fact that the pseudo nuclear Zeeman term actually corrects for an inadequate treatment of the electron Zeeman interaction. Acknowledgments The authors highly appreciate a discussion concerning the present study with J.H. van der Waals. The research was supported with financial aid by The Netherlands Organization for Scientific Research (NWO), Department of Chemical Sciences (CW). References [1] J.M. Baker, B. Bleaney, Paramagnetic resonance in some lanthanon ethyl sulphates, Proc. R. Soc. A 245 (1958) 156–174. [2] A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of Transition Ions, Clarendon Press, Oxford, 1970.
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