J. Chem. SOC., Faraday Trans. I , 1987, 83 (12), 3493-3503 MAGRES: A General Program for Electron Spin Resonance, ENDOR and ESEEM Cornelus P. Keijzers," Ed J. Reijerse, Pieter Stam, Michiel F. Dumont and Michiel C. M. Gribnau Department of Molecular Spectroscopy, Research Institute of Materials, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands The program package MAGRES (MAcnetic msonance) is able to calculate e.s.r., ENDOR (without relaxation) and ESEEM spectra for, in principle, any spin system in single crystals as well as in powders. The spin Hamiltonian may be constructed from electron and/or nuclear Zeeman interactions, exchange and hyperfine couplings, zero-field splittings and nuclear quadrupole interactions. For the calculation of eigenvalues and eigenvectors the program uses exact diagonalization, hence no assumptions have to be made about the relative magnitude of the various interactions. Comparison of the calculated with the experimental frequencies of single- crystal spectra permits the optimization of the interaction tensors.The computed tensors may be checked by comparison of the experimental and the calculated intensities of single-crystal and powder spectra. Simulation of ESEEM spectra is possible for the two- and three-pulse sequence, and the effect of the dead-time may be included. The program can also be used for the calculation of C.W. n.m.r. spectra. The systems which are studied by e.s.r. and e.s.r.-related techniques (c.w.-ENDOR, ESEEM, ESE-ENDOR) are very diverse, and include model systems for biochemical molecules, crystals which show phase transitions, Jahn-Teller systems, metal dimers and clusters.' The nature of the electronic and nuclear spins involved and the relative magnitudes of the various spin-spin and spin-field interactions depend upon the particular system.Because in our laboratory all the above mentioned techniques are applied to the study of a wide variety of solid-state problems (transition-metal complexes with d o ~ b l e t ~ - ~ and t~-iplet~-~ ground states, Jahn-Teller systems,a-'o metal clusters,11 incommensurate systems12 and low-dimensional magnetic corn pound^^^-^^) the need for a general-purpose computer program has been felt for some time. This has resulted in the development of the program package MAGRES (Mmnetic REsonance). Originally the program was set up for the treatment of single-crystal e.s.r. spectra for a general spin ~ y s t e m .~ . ~ Now it is able to handle both single-crystal and powder spectra for general spin systems for all the abovementioned techniques (with some restrictions which will be mentioned below). The program package can also be used to calculate C.W. n.m.r. spectra. In this paper we describe the design of the program package, its input requirements and its various options. The versatility is illustrated with examples. Although the input is complex, a fact which is inherent in the experimental techniques, much attention has been paid to keeping the input as simple and user-friendly as possible. This is shown in fig. 1 by an input example for the calculation of a 14N-ESEEM spectrum for a powdered sample of nickel(r1) bis(dithi0carbamate) doped with copper(r1).34933494 A General Program for E.S.R., ENDOR and ESEEM * * Ni : Cu DTC ESEEM simulation with the program MAGRES * TENSOR IS0 xx YY TENSOR IS0 TENSOR IS0 xx YY TENSOR IS0 TENSOR xx YY * G tensor electron 2.041 - 0.023 - 0.020 * nuclear g value copper * hyperfine tensor copper 1.48040 - 84.5 42.6 41.4 0.40347 0.25 0.25 * nuclear g value nitrogen * nuclear quadrupole tensor nitrogen xz - 0.065 TENSOR * hyperfine tensor nitrogen xx 0.06 YY 0.06 ELECTRON electron 1 /2 1 NUCLEUS copper 3/2 2 HYPERFINE copper electron 3 * hyperfine; tensor 3 NUCLEUS nitrogen 1 4 5 * nitrogen I = 1; tensors 4, 5 HYPERFINE nitrogen electron 6 * hyperfine; tensor 6 THRESHOLD 1 .Om7 TEMPERATURE 15.0 ESEEM GAUSSIAN * pulse profile WIDTH 100.0 * pulse width TAU 0.200 * time between pulses 1 and 2 (3-pulse DEADTIME 0.800 TIME SAVE MAGNITUDE SAVE POWDER 30 * 2250 integration points in powder calculation SHAPE GAUSSIAN DERIVATIVE CHANNELS 0.01953125 WRITE * plotting info OUTPUT LAST PLOTTING SWEEP FREQUENCY SPECTRUM 3123.0 2.5 2.5 8409.0 IS0 -0.38 xz -0.015 * electron S = 1/2; tensor 1 * copper I = 3/2; tensor 2 * sequence) * save time-domain spectrum on file * save magnitude spectrum on file 0.075* lineshape and linewidth * specification of output options : plotting * frequency swept ESEEM spectrum from 0 to * 5MHz * magnetic field : 3 123.0 G * excitation frequency : 8409.0 MHz START ~~ Fig.1. Input dataset for the calculation of the spectrum of fig.5: a 14N ESEEM spectrum for a powdered sample of nickel(I1) bis(dithi0carbamate) doped with Cu". A line starting with * or text on a line following * is handled as comment by the program.C. P. Keijzers et al. 3495 Program Characteristics The Spin Hamiltonian All applications start with the calculation of the eigenvalues and eigenvectors of the spin Hamiltonian after the necessary input operations. The calculation of resonance frequencies/fields and intensities depends upon the specific technique which is applied, and this will be discussed in the appropriate sections. In the input routines of the program the spins, interactions and interaction tensors are defined. Spins may be defined as electron or nuclear, the difference being that the g tensors must be specified in Bohr magnetons or in nuclear magnetons.All half-integer and integer spins are allowed. In the following the term ‘electron spin’ will be used for any electronic spin state S 2 f. MAGRES recognizes six types of interactions: two field- dependent couplings (the electron and nuclear Zeeman interaction) and four field- independent couplings (the exchange interaction between two electron spins or two nuclear spins, the hyperfine interaction between an electron spin and a nuclear spin, the zero-field splitting of an electron spin S > + and the nuclear quadrupole interaction of a nuclear spin I > i). All and any number of combinations of spins and interactions are allowed. Every interaction is described by a second-rank tensor which may consist of an isotropic, a traceless symmetric and/or an antisymmetric part, which contain 1, 5 and 3 elements, respectively.(In case of zero field and quadrupole splitting the isotropic and antisymmetric parts are not allowed. The program will issue a warning in case of error.) The tensors may be defined by their elements in the laboratory frame of axes. A symmetric tensor may also be defined by its three principal values and the three Euler angles which specify the principal axes relative to the laboratory system. It is also posiible to define a tensor by its three principal values and the principal axes of another tensor, which is an easy way to force coinciding principal axes. The remainder of the required input will be discussed in the relevant sections.Eigenvalues and Eigenfunctions The basis functions for the calculations are the direct products of the S, and I, eigenfunctions of all the spins. The Hamilton matrix H is set up in the laboratory frame using spin matrices S,, S, and S, and I,, Iy and I, and direct product techniques. To save computing time when calculating a series of spectra with different magnetic field orientations, the field-independent part H, and the field-dependent part HBo- (for a field strength unity) of the Hamilton-matrix are calculated and stored separately. For each magnetic field direction the field-dependent Hamilton matrix is calculated and combined with the field-independent matrix. The resulting Hamilton matrix H = Ho+B,H,o=l is diagonalized : yielding the diagonal eigenvalue matrix E and the eigenfunction matrix C.E = C+HC Transition Probabilities If transition probabilities are needed the matrix HB1 of the perturbation Hamiltonian, which includes the anisotropy in the Zeeman interaction of all electron and nuclear spins, is set up on the same basis functions as used for HBo, and thereafter it is transformed into the H representation : CtHBIC = M (3)3496 A General Program for E.S.R., ENDOR and ESEEM yielding the transition moment matrix M. This two-index transformation is optimized for a sparse HB1 matrix by avoiding multiplication with zero elements of HB1 and by selecting the transitions in the spectral range of interest. The obtained set of transition positions and intensities can be plotted with a Gaussian or a Lorentzian lineshape, either in absorption or in derivative mode.Simulation vs. Optimization For single-crystal studies MAGRES may be used in two modes : spectrum simulation and tensor optimization. The simulation mode is well suited for a systematic study of the influence of the various experimental parameters. It may also be used to check the line intensities in a spectrum after the determination of the interaction tensors via the optimization mode. The two modes may also be used in one program run with the advantage that, for instance, before and/or after tensor optimization (selected) spectra may be simulated. In the simulation mode one of the two quantities field or frequency should be designated as ‘swept’; the other is automatically set to ‘fixed’. For ENDOR and ESEEM only frequency sweep is allowed, in accordance with the experimental set-up.A wide variety of output options is available to the user to show the results of the calculation. These options comprise printing of the Hamiltonian matrices, eigenvalues, eigenvectors, transition fields or frequencies and the interaction tensors and their relative orientations, as well as the plotting of a spectrum. For plotting the spectra the calculated resonance lines are collected in an array of channels for which the width in field or frequency units (whichever is swept) must be defined. In the optimization mode MAGRES is linked as a subroutine to the minimization program MI NU IT.^^ This allows for the optimization of the interaction tensors by comparison of the experimental with the calculated transition frequencies (without calculating the transition probabilities).To this end the user can declare any number of tensor elements (or the principal values and tensor orientations) as ‘variable’ ; the remaining ones are fixed. Starting from estimated values for the variable elements, MINUIT minimizes an error function which is calculated by MAGRES according to where vFPtl and v;lcd are the experimental and calculated transition frequencies, respectively. The summation runs over all assigned lines in all spectra. The error function is always calculated in the frequency domain: this means that in case of C.W. e.s.r. for every direction of B, a new H, = 1 is constructed. For every line position for a certain direction a new Hamilton mathx H is constructed [eqn.(l)] and diagonalized. E.S.R. and N.M.R. The transition probabilities between the levels inside .the specified range of the swept parameter are calculated according to the Fermi golden rule: P,, = ~Wm,6(IE,-E,I-Irv). A ( 5 ) The influence of the temperature is taken into account according to G , ( T ) = ~ , , B , , ( T ) iC. P. Keijzers et al. 3497 r . r l l l . . I I l . I I I . l l l I ~ 0 200 400 600 800 1000 1200 1400 1600 1800 2000 h Fig. 2. Part of the experimental (a) and the calculated (b) e.s.r. X-band spectrum of a powdered sample of Re’’ in TiO,. For the experimental spectrum see ref. (22); for the calculated spectrum see ref. (23). The actual spectrum is obtained from this stick spectrum by convolution with a lineshape.In e.s.r. spectroscopy it is the standard procedure that the microwave frequency is fixed whereas the external magnetic field is swept. This means that the frequency-swept spectra should be computed for a range of magnetic fields. The field-swept spectrum may then be obtained by the intersection of the frequency-swept spectra with the experimental microwave frequency. This is a very tedious procedure. Therefore, it is common practice that in case of a small sweep range the spectrum is calculated for a fixed magnetic field. We improved this procedure by means of first-order perturbation theory. The first-order energy correction for a field differing d units from the fixed, central field is: where C is the matrix of eigenvectors of the spin-Hamilton matrix at the given central field. Resonances are found at field values for which BIG r where v is the applied microwave frequency.If the field sweep is (too) large the first-order perturbation treatment might not hold. In that case more than one central field must be used with a limited field sweep. Afterwards the calculated spectra can be joined before plotting, as was done in the spectrum of fig. 2.3498 A General Program for E.S.R., ENDOR and ESEEM A complication in calculating a field-swept spectrum can be the crossing of energy levels. If this occurs, two transitions are to be expected with the same level indices. Both transitions will be found by the program and a warning is issued. ENDOR In principle, the ENDOR (c.w. and ESE) intensities depend on e.s.r. and n.m.r.transition probabilities, and also on the balance of relaxation times (in c.w.-ENDOR) or on phase destruction relations (in ESE-ENDOR). Moreover, instrumental effects such as the impedance match of the r.f. coil may influence the observed intensities. For the ENDOR intensities an approximate procedure is followed : n.m.r. transitions are selected within the specified r.f. frequency range. The ENDOR intensity of each of these transitions is calculated as the n.m.r. transition probability multiplied by the sum of the probabilities of those e.s.r. transitions which have one energy level in common with this n.m.r. transition and which are within the excitation range of the applied microwaves: In this expression i and j label the n.m.r. transitions, whereas k and I run over all energy levels.G(w0 -aik) is the excitation lineshape function around the carrier frequency wo of the microwaves (or the microwave pulses). ESEEM The expressions which are used in MAGRES for the calculation of the ESEEM intensities are generalizations of the formulae originally derived by Mims.17. l8 For the stimulated echo sequence the modulation intensity of the n.m.r. transition i-j is computed according to (1 1) p i j = “ X i j + C 2xij,kn ‘OS (cukn ‘)I k < n where x i j , kn = Re (M& Mj*, M j k ) (13) and N is a normalization constant. z is the time between the first and the second microwave pulse. The transition moments Mik are weighed with the lineshape function associated with the excitation pulse : Mik = Mik G(w0 - Wik). (14) The indices i, j , k, n in eqn (1 1H13) run over all energy levels of the spin system within the excitation range.MAGRES recognizes three types of excitation lineshapes, G : square, Gaussian and Lorentzian. The algorithm used for the calculation of the ESEEM intensities has a complexity of N4, where N is the total spin multiplicity. The computing time may be reduced significantly by omitting the second term in eqn (1 l), which is responsible for the suppression effect. The experimental analogue of such a ‘ suppression free ’ spectrum is a two-dimensional ESEEM spectrum integrated over the z dimension. MAGRES can calculate ESEEM spectra with and without the suppression effect. In fig. 3 and 4 examples are presented of experimental and calculated 14N-ESEEM spectra of a single crystal and of a powdered sample.Furthermore ESEEM spectra for the two-pulse sequence can be calculated. The following frequencies and intensities will be computed :C. P. Keijzers et al. 3499 5.0 Fig. 3. Experimental and calculated 14N Fourier-transform-ESEEM spectra in the XY plane of a single crystal of Cu/Ni (mnt),. There is a three-pulse echo sequence with z = 210 ns.* In this case, apart from the n.m.r. frequencies within an electron spin-manifold, the sums and differences of n.m.r. frequencies belonging to different m, manifolds are observed. The abovementioned ESEEM spectra are all computed in the frequency domain. However, the experimental data are obtained in the time domain. In general, this time- domain spectrum is transformed to the frequency domain by means of a Fourier transformation, as in this domain the experimental information is shown more clearly.Note that the experimental and the calculated spectrum in the frequency domain do not contain the same information. This is a consequence of the spectrometer dead-time which results in the loss of the first few points after the last microwave pulse. To handle the dead-time the following procedure was implemented in MAGRES. We start with the calculation of a stick spectrum in the frequency domain, followed by a Fourier transformation to the time domain. Subsequently, the first few points are deleted in accordance with the experimental dead-time. Zero’s are added at the end of the spectrum 116 FAR 13500 A General Program for E.S.R., ENDOR and ESEEM 1 5.0 Fig.4. Experimental and simulated 14N Fourier-transform ESEEM spectra of a powdered sample of 63C~/Ni(et,dt~)2 for various values of the magnetic field.2 in the time domain to keep the number of points of the spectrum fixed. At this stage a relaxation function (equivalent to a lineshape in the frequency domain) is also included. Finally, another Fourier transformation is carried out (but now a magnitude spectrum is calculated because of the phase distortion), yielding a spectrum in the frequency domain. This final spectrum may be compared with the experimental one, because both are obtained by calculating the magnitude spectrum after the Fourier transformation to the frequency domain. This is necessary in order to take care of the phase distortions due to the dead-time.An example of the various stages of this calculational procedure is shown in fig. 5 . Powder Spectra Powders represent an important class of materials, as many interesting materials are only available in powder form. However, the e.s.r., ENDOR and ESEEM spectra of powders are difficult t o analyse, since peak assignment is almost impossible. Therefore, an important application of MAGRES is the simulation of powder spectra. For the computation of a powder spectrum, the direction of Bo is varied over a hemisphere. ThisI 0.0 1.0 2 .o 3.0 4.0 5.0 0.0 Id.0 2d.0 30.0 40.0 50.0 frequency/MHz (256 points) time/ps ( 5 12 points) J I I I I 1 0.0 10.0 20.0 30.0 40.0 50.0 timelvs (5 12 points) Fig. 5. The various stages in the simulation of a 14N Fourier-transform ESEEM spectrum on a powdered sample of Cu/Ni(dtc),.The input dataset is listed in fig. 1. (a) Calculated stick spectrum in the frequency domain. Noise on the low-frequency side is due to the limited number (ca. 2250) of integration points. This spectrum convoluted with a lineshape is one of those in fig. 4. (b) Time- domain spectrum after Fourier transformation of spectrum (a). (c) Spectrum (b) with an exponential decay and a dead-time of 800 ns. The spectrum is filled with 800 ns of zeros in order to keep the number of points fixed. (d) Fourier-transformed magnitude spectrum of (c). The exponential decay causes convolution with a Lorentzian lineshape ; the dead-time and magnitude calculation cause lineshape deformations. w wl E33502 A General Program for E.S.R., ENDOR and ESEEM is accomplished by dividing a number of integration points over a spiral-like curve, which starts at the ‘equator’ of the hemisphere and ends on its ‘pole’.The distance between the points is kept as constant as possible. For each orientation of Bo all directions of B, perpendicular to B, should be considered. However, in order to save computer time, a random distribution of directions of B, perpendicular to B, is used. Provided that enough integration points are used, this should be a valid approximation. For each of the integration points the spin-Hamilton matrix is set up and diagonalized and the transition probabilities are calculated. The computed stick spectra are accumulated. After the last integration point the total stick spectrum can be convoluted with a lineshape and a plot of the powder spectrum may be obtained.The total stick spectrum can be saved on file in order to permit the user to try various lineshapes without recalculating the spectrum. Examples of powder e.s.r. and ESEEM spectra are shown in fig. 2, 4 and 5. Limitations and Perspectives In its current form the program MAGRES calculates the eigenvalues and eigenfunctions by means of diagonalization. Although this method requires considerable computing time and computer memory it has the important advantage of being general and applicable to any (orbitally non-degenerate) spin system. With the current mainframe computer (NAS 9060) spin systems with a total spin multiplicity N z 300 may be studied. The time-limiting step for the spectral simulations is the calculation of the transition moments.This step has a complexity of P. The necessary computer-memory increases with P. The requirements of the ESEEM simulations are even higher as the complexity is of the order N4. The requirements for computer time and memory may be reduced by applying perturbation treatments. However, to keep the program applicable to a general spin system it is then necessary to develop an algorithm that is capable of deciding whether a perturbation treatment is allowed and, if so, for which interactions. This stage has not been reached yet. In the present stage of the program, powder spectra may only be simulated: they cannot be used for the optimization of tensors. In order to achieve optimization possibilities, the calculation of the powder spectra (i.e.the integration over the sphere) must be accelerated, because in one optimization run the spectrum must be calculated very often (depending on the accuracy of the initial guess for the tensors). A new algorithm is under development. For graphical display of the simulated spectra, the program itself does not offer facilities. It will only dump the calculated intensities on disk, from where they can be picked up for input in a standard graphical package (for instance, SAS/GRAPH” or DISSPLA~,). As far as portability is concerned: the program is used under (and partly written for) the IBM operating system VM/CMS. Among other system subroutines it uses routines from the NAG-library2’ for diagonalization and Fourier transformation.In practice this means that at this moment in time, the program is not portable. However, with the rapid development of network facilities (EARN, BITNET) we are of the opinion that using these communication facilities has advantages over the development of a portable version, which would need regular updating. This implies that input datasets are sent over to our computer centre, where the program is run, and afterwards the output is sent back to the user. An extensive manual for the program MAGRES is available (in English).C. P. Keijzers et al. 3503 References 1 See, for instance, the series Electron Spin Resonance (Specialist Periodical Reports, The Royal Society 2 E. J. Reijerse, N. A. J. M. van Aerle, C. P. Keijzers. R. Boettcher, R.Kirmse and J. Stach, J. Magn. 3 R. Boettcher, R. Kirmse, J. Stach, E. J. Reijerse and C. P. Keijzers, Chem. Phys., 1986, 107, 145. 4 E. J. Reijerse, A. H. Thiers, R. Kanters, M. C. M. Gribnau and C. P. Keijzers, Inorg. Chem., in 5 D. Snaathorst, H. M. Doesburg, J. A. A. J. Perenboom and C. P. Keijzers, Znorg. Chem., 1981, 20, 6 D. Snaathorst and C. P. Keijzers, Mol. Phys., 1984, 51, 509. 7 M. A. Hefni, N. M. McConnell, F. J. Rietmeijer, M. C. M. Gribnau and C. P. Keijzers, Mol. Phys., 8 J. S. Wood, C. P. Keijzers, E. de Boer and A. Buttafava, Inorg. Chem., 1980, 19, 2213. 9 G. van Kalkeren, C. P. Keijzers, J. S. Wood, R. Srinivasan and E. de Boer, Mol. Phys., 1983, 48, 1. 10 M. L. H. Paulissen and C. P. Keijzers, J . Mol. Struct., 1984, 113, 267. I 1 J. B. A. F. Smeulders, M. A. Hefni, A. A. K. Klaassen, E. de Boer, U. Westphal and G. Geismar, 12 C. P. Keijzers, G. Zwanenburg, J. M. Vervuurt, E. de Boer and J. C. Krupa, J . Phys. C, in press. 13 P. T. Manoharan, J. H. Noordik, E. de Boer and C. P. Keijzers, J . Chem. Phys., 1981, 74, 1980. 14 0. Takizawa, R. Srinivasan and E. de Boer, Mol. Phys., 1981, 44, 677. 15 M. C. M. Gribnau, R. Murugesan, H. van Kempen and E. de Boer, Mol. Phys., 1984, 52, 195. 16 F. James and M. Roos, MINUITS, CERN, Geneva, program no. D-506. 17 W. B. Mims, Phys. Rev. 3, 1972, 5, 2409. 18 W. B. Mims, Phys. Rev. B, 1972, 6, 3543. 19 SAS/GRAPH : Statistical Analysis System, SAS-Institute Inc., Cary, North Carolina. 20 DISSPLA : Display Integrated Software System and Plotting Language, Integrated Software Systems Cooperation, 198 1, San Diego, California. 2 1 NAG : Numerical Algorithms Group, 1984, Oxford. 22 M. Valigi, D. Cordischi, D. Gazzoli, C. P. Keijzers and A. A. K. Klaassen, J. Chem. SOC., Faraday Trans. I, 1981, 77, 1871. 23 E. J. Reijerse, P. Stam, C. P. Keijzers, M. Valigi and D. Cordischi, J. Chem. Soc., Faraday Trans. 1, 1987, 83, 3613. of Chemistry, London), for extensive literature reviews on various topics. Reson., 1986, 67, 114. press. 2526. 1986, 57, 1283. Zeolites, in press. Paper 71727; Received 22nd April, 1987