Basic theory of LCGTO-LSD method

The local density approximation [25] is used to solve the spin-polarized Kohn and Sham equations [26], The exact ground state energy for a spin-polarized system of N electrons in the field of a nuclear charge Z can be expressed in terms of the spinup (n+(r)) and spin-down (n (r)) densities 

EXc[n+,n ] is the exchange-correlation functional. Now, the (non-relativistic) ground state energy is  

For an inhomogeneous system of interacting electrons the functional Exc[n+,n ] is unknown. In this method the spin-polarized generalization LSD (Local Spin Density) version of LDA (Local Density Approximation) approximation is used. n(r) is supposed to vary slowly and the energy can be expressed as 

The corresponding exchange-correlation contribution to the effective potential is 

The Vosko, Wilk and Nuisar (VWN) parametrization [27] for the correlation and Xa exchange potentials is used 

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