The chapter introduces the quantum mechanics of many-electron systems, focusing on the development of approximate practical methods to explain complex atomic systems without extensive computation. It begins by discussing the limitations of the old orbit theory, which assumed individual orbits for each electron, and the introduction of spin and Pauli's exclusion principle to explain multiplet structure. The exchange interaction between electrons, arising from their indistinguishability, is highlighted as a key factor in understanding multiplet structure. The chapter then delves into the mathematical treatment of permutations as dynamical variables, showing how they can be used to describe the exclusive sets of states in a system. It proves that for each stationary state of an atom, there is a definite numerical value for the total spin vector, which is crucial for multiplet structure. The chapter also applies these concepts to electrons, demonstrating that the perturbation energy can be expressed as a linear function of the permutation variables, with coefficients representing exchange energies. Finally, it discusses the application of these results to the calculation of energy levels in atomic systems, providing a framework for understanding the degeneracy and multiplicity of energy levels.The chapter introduces the quantum mechanics of many-electron systems, focusing on the development of approximate practical methods to explain complex atomic systems without extensive computation. It begins by discussing the limitations of the old orbit theory, which assumed individual orbits for each electron, and the introduction of spin and Pauli's exclusion principle to explain multiplet structure. The exchange interaction between electrons, arising from their indistinguishability, is highlighted as a key factor in understanding multiplet structure. The chapter then delves into the mathematical treatment of permutations as dynamical variables, showing how they can be used to describe the exclusive sets of states in a system. It proves that for each stationary state of an atom, there is a definite numerical value for the total spin vector, which is crucial for multiplet structure. The chapter also applies these concepts to electrons, demonstrating that the perturbation energy can be expressed as a linear function of the permutation variables, with coefficients representing exchange energies. Finally, it discusses the application of these results to the calculation of energy levels in atomic systems, providing a framework for understanding the degeneracy and multiplicity of energy levels.