Electron bound states

The active electron bound state (r-/ ) satisfies the Schrodinger equation... 

The energy eigenvalues of the hydrogen electronic bound states are inversely proportional to the square of the principal quantum number, in SI units,... 

Let the electron-acceptor interaction be described by the short-range potential UsA(f—fA) where rA is the center of acceptor coordinate. It is supposed that the wave function of the electron bound state on the acceptor with energy E, TG(F — ta, E), is an exact one, i.e. it is considered not only UA(f—rA) but the donor-electron interaction also. Then, the exact value of the matrix element of the electron donor-acceptor transfer is equal [1] ... 

Electronic bound-state levels are inversely proportional to the square of an effective quantum number, E oc — 1/n2, as shown on an arbitrary scale in the diagram. 

In atomic physics we encounter one-electron bound states with different types of boundary condition. For the first type the electron is completely confined to a spherical box, near the boundary of which the potential is negligible. The second type involves a potential which falls to zero... 

In the previous sections the potential scattering problem has been defined in terms of a Schrodinger differential equation with outgoing spherical-wave boundary conditions. The description and computational methods are analogous to those used for one-electron bound-state problems. In this section we see that the whole problem in the coordinate representation can be written in terms of a single integral equation, which in many ways is easier to understand physically than the differential equation. 

The extension of the matrix solution of section 4.3 for one-electron bound states to the Hartree—Fock problem has many advantages. It results in radial orbitals specified as linear combinations of analytic functions, usually normalised Slater-type orbitals (4.38). This is a very convenient form for the computation of potential matrix elements in reaction theory. The method has been described by Roothaan (1960) for a closed-shell or single-open-shell structure. 

The expression proportional to (Aei) in (99) is called the Darwin term. It is sometimes heuristically explained as an effect related to Zitterbewegung, but this is rather doubtful, because electronic bound states do not exhibit any Zitterbewegung according to the Dirac equation. 

We note that a negative 7 corresponds to an attractive Coulomb potential. This system describes a hydrogen-like ion. A positive 7 corresponds to a repulsive Coulomb potential. It is repulsive for electrons, but attractive for positrons (resp. negative-energy wave packets). Hence the Dirac equation will have bound states also for positive 7. It is not necessary to discuss this separately, because the positronic bound states (which exist for 7 > 0) can be obtained from the electronic bound states (which exist for 7 < 0) by a charge conjugation. 

We also note that the relativistic electron bound state energies are always below the corresponding nonrelativistic eigenvalues. 

As we have just implied, solutions to the many-electron scattering problem, like solutions to the many-electron bound-state problems of quantum chemistry, are obtained in terms of products of one-electron functions, subject to constraints of spin, exchange antisymmetry (the Pauli principle), and possibly spatial (point... 

The first step toward a practical relativistic many-electron theory in the molecular sciences is the investigation of the two-electron problem in an external field which we meet, for instance, in the helium atom. Salpeter and Bethe derived a relativistic equation for the two-electron bound-state problem [135,170-173] rooted in quantum electrod)mamics, which features two separate times for the two particles. If we assume, however, that an absolute time is a good approximation, we arrive at an equation first considered by Breit [101,174,175]. The Bethe-Salpeter equation as well as the Breit equation hold for a 16-component wave function. From a formal point of view, these 16 components arise when the two four-dimensional one-electron Hilbert spaces are joined by direct multiplication to yield the two-electron Hilbert space. 

If the electron-electron interaction is switched on, we will face either continuum or autoionizing states. In the autoionizing states, a bound state couples to the continuum, which would lead to its decay. As a consequence, the Dirac-Coulomb model is not considered a useful physical Hamiltonian. However, it is the most widely applied Hamiltonian as projection on the square-integrable one-electron bound states yields remarkably accurate results, despite their dependence on the choice of the projection operators (see the next sections for a further discussion of these issues). 

A relativistic consistent quantum-theoretical description of electronic bound states in atoms was first introduced in atomic physics as early as the late 1920s and has been pushed forward since that time. It was believed, however, that effects stemming from Einstein s theory of relativity were of little or even no importance to chemistry. This changed in the 1970s when it was recognized by Pyykkd, Pitzer, Desclaux, Grant and others that several unusual features in heavy-element chemistry and spectroscopy can only be explained in terms... 

B. Klahn, /. Chem. Phys., 83,5754 (1985). The Convergence of Cl Calculations for Atomic and Molecular Electronic Bound States in a Basis of Floating Gaussian Orbitals. 

Setting two f-electrons in the core on the central site as well as depleting all other f-character out of this site is simply a device used to obtain the appropriate potential with which to analyze the unoccupied states. As such this method is very general. This device has been used to analyze the spectrum of a two-f-electron bound state on a cerium site, but one could easily simulate other many-body excitations (two-hole bound states, for example). 

Ultrafast excitation experiments and corresponding dynamical simulations reveal that on the molecular time scale caging is sensitive both to the detailed molecular structure of the solvent surrounding the molecule and to the dynamics of the dissociation process. As an example let us take excitation of U to an electronically bound state (B) that dissociates because of a non-adiabatic transition to a repulsive state, Figure 11.10. The transition is symmetry forbidden in the isolated homonuclear molecnle bnt allowed in solntion, where coupling to the solvent breaks the symmetry of the isolated molecule. ... 

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