Spin orbitals construction algorithms

The arguments just described for the construction of the 2-RDM were extended without difficulty to the higher-order RDMs. The algorithms for these high-order RDMs were originally reported by Colmenero et al. [20] in a spin-free basis and Valdemoro et al. [46] obtained later on the algorithms in a spin-orbital basis. For the 3-RDM, the algorithm in a spin-orbital basis is... 

The decision of whether to work with 2-spinors or a scalar spin-orbital basis must be made at an early stage of computer program construction because it affects all stages of the SCF process evaluation of the integrals, construction of the Fock matrix, and solution of the SCF equations. However, at each stage, the scalar spin-orbital basis can be transformed to the 2-spinor basis. Transformation of the integrals to a 2-spinor basis is not particularly difficult it is similar in principle to the transformation from Cartesians to spherical harmonics. Some efforts have been made to develop new algorithms in which these transformations are incorporated, and RKB is implemented from the start in the 2-spinor basis (Quiney et al. 1999, 2002, Yanai et al. 2002). 

A complete set of spin eigenfunctions, e.g. oo i l = 1, 2,. .., 5) in the case of a six-electron singlet, can be constructed by means of one of several available algorithms. The most commonly used ones are those due to Kotani, Rumer and Serber [13]. Once the set of optimized values of the coefficients detining a spin-coupling pattern is available [see in EQ- (2)], it can be transformed easily [14] to a different spin basis, or to a modified set reflecting a change to the order in which the active orbitals appear in the SC wavefunction [see Eq. (1)]. 

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