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Second-order perturbation solutions of the spin hamiltonian Ws=~BS. g. B+S. D. S+I. A. s+I. P. I are presented for the case of most general symmetry conditions, when the principal axes of the different tensors may be non-collinear and the tensors g and A may be asymmetric. In order to obtain a concise formalism we introduce projectors on the quantization directions of the electron and nuclear spins, which make the determination of complete Euler transformation matrices unnecessary. The angular dependence of the intensities are discussed both for the allowed and different first-order forbidden transitions for randomly oriented samples. INTRODUCTIONIn his classical work [1] Bleaney gave a second-order perturbation solution to the eigenvalue problem of the spin hamiltonian for axial symmetry. For general symmetry conditions the problem is discussed in the book of Abragam and Bleaney [2] but only first-order expressions are given. The recently developed more accurate computer simulations of electron spin resonance spectra, however, require the inclusion of higher-order terms. This explains the simultaneous appearance of papers on this topic [3][4][5][6][7][8][9][10][11]. For rhombic symmetry, second-order solutions were given by Azarbayejani [3], McClung [4], Golding [5], Sakaguchi et al. [6] and Lin [7] when the principal axes of the tensors in the spin hamiltonian coincide ; by Kirmse et al. [8], Pilbrow and Winfield [9] and Golding and Tennant [10] when one or more principal axes of the symmetric tensors g and A do not coincide ; by Rockenbauer and Simon [11] when the tensors g and A are not symmetric. For accurate simulation the angular dependence of transition probabilities should be also considered. The formula for axial symmetry was given by Bleaney [12], for rhombic symmetry by Kneubfihl and Natterer [13], Johnston and Hecht [14], Pilbrow [15], Golding et al. [16], Abragam and Bleaney [2], Sakaguchi et al. [6] and Rockenbauer and Simon [11]. Since the large number of papers published on this topic concern only part of the problem, a systematic study of the perturbation solution of the spin hamiltonian seems to be justified. Therefore we give, up to second order, a perturbation solution to the eigenvalue problem of a relatively complex spin hamiltonian consisting of a Zeeman term, a zero-field term, a hyperfine term and a quadrupole term. We obtain formulae for allowed and different kinds of forbidden transitions.