Quantum numbers nonrelativistic

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

Here, ag = h2/(m e2) is the reduced Bohr radius (m = M), n is a principal quantum number of the hydrogenic state i), and a = e2/(hc) 1/137 is the fine structure constant. In the following discussion, (2) will be the only restriction imposed on values of physical parameters. In particular, no distinction will be made as to whether the photon energy ho is less or greater than the ionization potential Jj of the state i). Note that = ftwi holds true only in the nonrelativistic approximation, whereas in general I) tkui. 

In theory, an infinite number of calculations for highly excited states is required to complete the expansion of the EP given by Eq. (24), since there are only a few occupied valence orbitals in neutral atoms. This difficulty also exists in the nonrelativistic case and is resolved by using the closure property of the projection operator with the assumption that radial parts of EPs are the same for all orbitals having higher angular momentum quantum numbers than are present in the core. The same approximation is applicable in the present... 

The correct nonrelativistic limit as far as the basis set is concerned is obtained for uncontracted basis sets, which obey the strict kinetic balance condition and where the same exponents are used for spinors to the same nonrelativistic angular momentum quantum number for examples, see Parpia and Mohanty (1995) and also Parpia et al. (1992a) and Laaksonen et al. (1988). The situation becomes more complicated for correlated methods, since usually many relativistic configuration state functions (CSFs) have to be used to represent the nonrelativistic CSF analogue. This has been discussed for LS and j j coupled atomic CSFs (Kim et al. 1998). 

In nonrelativistic quantum mechanics, the degeneracy due to spherical symmetry of the Schrodinger equation is 2i +1, where f is the orbital angular quantum number. The spin just doubles the number of states, and if there is no magnetic field, the two spin-states are energetically indistinguishable. Hence, a nonrelativistic particle with spin has a 2(2f -I- l)-fold degeneracy due to spherical symmetry. 

As there is a one to one correspondence between the relativistic eigenstates and their nonrelativistic limit, the spectroscopic notation (which uses nonrelativistic quantum numbers ) can also be used to label the relativistic states. Hence the eigenstates of h. can be denoted as follows ... 

Notice that only the nonrelativistic value of the orbital angular momentum quantum number attached to the large component, appears above. 

Equation (3.1.2) is the nonrelativistic Hamiltonian. This means that the spin-dependent part of the Hamiltonian (Hso spin-orbit and Hss spin-spin) has been neglected. The electronic angular momentum quantum numbers, which are well-defined for eigenfunctions of nonrelativistic adiabatic and diabatic potential curves, are A, E, and 5 (and redundantly, Q = A + E). 

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