Relativistic cusp correction

The behavior of the local energy around the nucleus is more severe for the ZORA-QMC results, because relativistic effects cause the contraction of s and p orbitals and increase the electron density near the nucleus. Thus, an adequate relativistic cusp correction scheme is necessary to treat heavier elements where the relativistic effects are important. Recently, we proposed a cusp correction scheme for the ZORA-VMC formalism [15]. Strictly speaking, the term cusp correction may not be adequate for our orbital correction scheme because relativistic wave function does not have a cusp but divergences at electron-nucleus collision. However, we call our relativistic orbital correction scheme as a relativistic cusp correction scheme because the term cusp correction makes clearer that our scheme is a relativistic extension of the NR cusp correction scheme. 

In Fig. 10.1, the Is orbitals of the Ne atom are plotted with and without the cusp correction. GTO and STO orbitals are used in this calculation. The condition (0.08 30), where Tc = 0.08a.u. and rmn is 30% of r, is used. Note that the corrected orbitals are not normalized. The original and corrected orbitals have similar shapes except in the region of r 0. This observation in the relativistic case is the same as that seen for the case of NR cusp correction scheme of Ma et al. in the GTO case. 

Fig. 10.1 Relativistic Is orbital of Ne atom with and without the cusp correction. Both GTO and STO orbitals are obtained from the ZORA-HF calculation...

We have developed a new relativistic treatment in the QMC technique using the ZORA Hamiltonian. We derived a novel relativistic local energy using the ZORA Hamiltonian and tested its availability in the VMC calculation. In addition, we proposed a relativistic electron-nucleus cusp correction scheme for the relativistic ZORA-QMC method. The correction scheme was a relativistic extension of the MO correction method where the 1 s MO was replaced by a correction function satisfying the cusp condition. The cusp condition for the ZORA wave function in electron-nucleus collisions was derived by the expansion of the ZORA local energy and the condition required the weak divergence of the orbital itself. The proposed relativistic correction function is the same as the NR correction function of Ma et al. in the NR... 

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