The Dirac Hydrogen Atom

The simplest system of relevance for molecular sciences is the hydrogen atom. The cpiantum mechanical equation of motion for an electron in the attractive (external) potential of a positively charged atomic nucleus can be solved analytically. From these analytical results many important consequences follow, which make the description of many-electron atoms and molecules feasible (such as the choice of proper basis functions for the expansion of molecular spinors). 

Separation of Electronic Motion in a Nuclear Central Field 

In the previous chapter we derived the Dirac equation for an electron moving in an arbitrary electromagnetic potential. We next proceed step-wise toward a many-electron theory which incorporates any kind of electromagnetic interactions. The most simple example is the hydrogen atom consisting of two interacting particles the electron and the proton. 

The quantum mechanical description of these two fermions is straightforward in a formal sense considering the one-particle Dirac Hamiltonians for electron e and proton p plus the interaction operator Vep, 

Note that the derivation of the Dirac equation in chapter 5 holds for any freely moving spin-1 /2 fermion and hence also for the proton we come back to such general two-particle Hamiltonians at the beginning of chapter 8 (for an explicit solution of the corresponding two-fermion eigenvalue equation we refer the reader to the work by Marsch [112-115]). In the field-free Dirac Hamiltonian only the rest mass determines which fermion is considered. Accordingly, the total wave function of the hydrogen atom reads 

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The topics of the individual chapters are well separated and the division of the book into five major parts emphasizes this structure. Part I contains all material, which is essential for understanding the physical ideas behind the merging of classical mechanics, principles of special relativity, and quantum mechanics to the complex field of relativistic quantum chemistry. However, one or all of these three chapters may be skipped by the experienced reader. As is good practice in theoretical physics (and even in textbooks on physical chemistry), exact treatments of the relativistic theory of the electron as well as analytically solvable problems such as the Dirac electron in a central field (i.e., the Dirac hydrogen atom) are contained in part 11. 

For the solution of the Dirac hydrogen atom defined by Eq. (6.4) we first note that the space and time variables of the electron are well separated — as shown in section 4.2 for the general case — so that we may use the ansatz. 

Schwabl provides many details on how the Dirac hydrogen atom can be solved, though some essential steps are treated very briefly. However, the material in Schw-abl s book is still more extensive than in many of the new publications on the subject. Also, the reader will find a more detailed comparison to the Klein-Gordon equation in external electromagnetic fields. But note that Schwabl chooses a different sign convention for the spherical spinors with effect on almost all equations of the Dirac hydrogen atom compared to Arose derived here. 

In section 6.9 we already introduced finite-size models of the atomic nucleus and analyzed their effect on the eigenstates of the Dirac hydrogen atom. This analysis has been extended in the previous sections to the many-electron case. It turned out that neither the electron-electron interaction potential functions nor the inhomogeneities affect the short-range behavior of the shell functions already obtained for the one-electron case. Table 9.5 now provides the total electronic energies calculated for the hydrogen atom and some neutral many-electron atoms obtained for different nuclear potentials provided by Visscher and Dyall [439], who also provided a list of recommended finite-nucleus model parameters recommended for use in calculations in order to make computed results comparable. 

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