Theoretical Investigation of wave functions and fine structure of ground state configuration 5d36s2 of Tantalum atom.*

Hydrogen is the only element for which Schrodinger’s equation gives an exact result. For transition metal with large atomic number and rare-earth metals Russel Saunders coupling is still suitable. When a number of electron is increased, different type of perturbation added in Hamiltonian, and it becomes complicated to calculate exact hamiltonian for multielectronic system. In this study, coefficient of fractional parentage method is used to calculate the wavefunctions of Tantalum. The coefficient of fractional parentage method shows the contribution of parent atom in making final terms.

Tantalum is a transition metal and symbolized as Ta. It is a disputed element due to its use in making weapons. Due to its temperature stabiltiy and immunological respnse, tantalum is a suitable element  in the field of oil and gas exploration and exposed to human body. In 2016, L Windholz et al discussed the possibility of unknown energy levels of neutral Ta, Pr and La by using high-resolution Fourier-transform and LASER spectroscopy. In 2016, some new energy levels of Ta II were determined with accuracy better than 0.10 cm^-1. In 2018, the energy levels of neutral Tantalum i.e. 216 Ta of even parity and 29 of odd parity have been reported with uncertainties below 0.010 cm^-1.

The ground state wave function of quantum many-body system is a complicated task to calculate. In this research, fine structure and wavefunctions of configuration 5d³6s²2 is calculated which is a ground state electronic configuration of Ta I. In the configuration of 5d³6s², d³ is an equivalent electron and s2 has completely filled orbitals, so it will not participate in generating final terms that is why only d³ is contributing in making final terms of 5d³6s². In this study, we have reported 21 term symbols with different values oftotal angular momentum (J) ranging from 1/2 to 11/2, 120 possible MLand MS combination of allowed terms by considering Pauli’sexclusion principle, a complete matrix table, 55 number of energylevels and forty-coupled wave function along with their coefficientof fractional parentage. The term 4F3/2 is verified as the groundstate term of Ta I using Hund’s rule. The forty wave functions areorthonormalized using Gram Schmidt technique. These wavefunctions can used to calculate energy, momentum and other various spectroscopic quantities of Ta I.