Bound state solutions of the two-electron Dirac-Coulomb equation
We present a variational method for solving the two-electron Dirac-Coulomb equation. When the expectation value of the Dirac-Coulomb Hamiltonian is made stationary for all possible variations of the different components of a well-behaved trial function one obtains solutions representative of the physical bound state wave functions. The ground state wave function is derived from the application of a minimax principle. Since the trial function remains well-behaved, the method remains safe from the twin demons of variational collapse and continuum dissolution.
The ground state wave function thus derived can be interpreted as a linear combination of different configurations. In particular, the admixing of intermediate states having one (two) electron(s) deexcited to a negative-energy orbital (orbitals) contributes a second-order level shiftE0−(2) which can be identified with the second-order shift due to the Pauli blocking of the production of one (or two) virtual electron-positron pair(s). Thus the minimax solution corresponds to the renormalized ground state in quantum electrodynamics, with deexcitations to negative-energy orbitals taking the place of the avoidance of virtual pairs.
If one extends the relativistic configuration interaction (RCI) treatment by additionally including negative-energy and mixed-energyeigenvectors of the Dirac-Hartree-Fock hamiltonian matrix in the two-electron basis, the calculated energy will be shifted from the conventional RCI value by an amount that is much smaller thanE0−(2). For two-electron atoms, we have derived expressions for the all-spinor limit (δE) and thes-spinor limit (δEs) of this shift in leading orders. The all-spinor limit (δE) is of orderα4Z4 1/3 whereas thes-spinor limit (δEs) is of orderα4Z3 2/3. leading components are related to the 1-pair component ofE0−(2) in a simple way, and the relationships offer the possibility of computing energy due to virtual pairs. Numerical results are discussed.
Volume 94, 2020
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