A broken-(spin) symmetry (BS) method is now widely used for systems that involve (quasi) degenerated frontier orbitals because of their lower cost of computation. The BS method splits up-spin and down-spin electrons into two different special orbitals, so that a singlet spin state of the degenerate system is expressed as a singlet biradical. In the BS solution, therefore, the spin symmetry is no longer retained. Due to such spin-symmetry breaking, the BS method often suffers from a serious

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... em called a spin contamination error, so that one must eliminate the error by some kind of projection method. An approximate spin projection (AP) method, which is one of the spin projection procedures, can eliminate the error from the BS solutions by assuming the Heisenberg model and can recover the spin symmetry. In this chapter, we illustrate a theoretical background of the BS and AP methods, followed by some examples of their applications, especially for calculations of the exchange interaction and for the geometry optimizations.