We begin with the discussion of the method for a direct 1st principles calculation of the magnon states. Since the magnons are excitations, the calculations are performed under constraints: the symmetry constraint responsible for the wave length of the magnon and the constraint responsible for the required change of the magnetization of the system. We discuss the chiral splitting of the magnon states of an altermagnet in the wave vector space that has features similar to the nonrelativistic spin splitting of the altermagnetic electron states. We demonstrate the relation of the chiral splitting of magnons to the avoided crossings in the electron band structure. Next, we introduce the 1st principles method for the calculation of the weak ferromagnetic state in an antiferromagnet. Here again, the concept of the symmetry constraint playsan important role. We obtain a peculiar nonmonotonous dependence of the weak ferromagnetic moment on the strength of the spin-orbit coupling (SOC). To gain a deeper insight into the influence of the SOC on the electron states of an altermagnet we introduce the concept of magnetic structure of an electron state (MSES). We present the MSESs obtained in nonrelativistic, quasisymmetry, and relativistic calculations. The quasisymmetry calculation takes into account only one component of the SOC. We demonstrate a dramatic change of the MSES from a simple 1D object in the nonrelativistic case to a complex 3D object in the relativistic case. Both spin and orbital atomic moments are considered. We discuss the relation between relativistic avoided crossings and the nonmonotonous dependence of the weak ferromagnetic moments on the SOC strength. The presented material is based on the results collected in two papers:

[1] L. M. Sandratskii, K. Carva, V. M. Silkin, Direct ab initio calculation of magnons in altermagnets:
Method, spin-space symmetry aspects, and application to MnTe, Physical Review B 111, 184436 (2025).

[2] L. M. Sandratskii, K. Carva, V. M. Silkin, Spin-orbit coupling effects in altermagnets: Interplay of weak spin and orbital
ferromagnetism with relativistic splitting of electron states, submitted.