Coulomb wavefunction

The fitted value of 8p 8rf is in good agreement with the number calculated from the quantum defects of the atom and the phases of the Coulomb wavefunctions. 

Using the procedures outlined above we may calculate bound and continuum wavefunctions as well as matrix elements of r°, for cr>0. These wavefunctions are often called coulomb wavefunctions, and properties calculated using them are said to be obtained in the coulomb approximation. In addition, we can calculate matrix elements of inverse powers of r for H. We cannot calculate with confidence matrix elements of inverse powers of r for anything but H since the inverse r matrix elements weight r 0 heavily and the results can be highly dependent on the radius at which we truncate the sums of Eq. (2.45). 

Table 17.3. Ba+ Dipole and quadrupole polarizabilities, ad and aq, determined from an analysis of measured 6s n intervals and calculated using coulomb wavefunctions.a...

The wavefunctions of the normal scattering modes are the standing waves produced by a linear combination of incoming coulomb wavefunctions which is reflected from the ionic core with only a phase shift. The composition of the linear combination is not altered by scattering from the ionic core. These normal modes are usually called the a channels, and have wavefunctions in the region rc[Pg.418]

If the energy is raised above the second limit there are two open channels. In Fig. 20.2 at an energy WB for r > rB the wavefunction is composed of a linear combination of 0, and 02. If we put a radial box of radius rB around the ionic core we can again ask, What are the normal modes for electron scattering from the contents of the box In other words, what linear combinations of incoming coulomb wavefunctions will suffer at most a phase shift when scattering from the contents of the box There are two wavefunctions, labelled by p = 1,2. They are linear combinations of 0, and 02, given by... 

Gasaneo G, Colavecchia FD (2003) Two-body Coulomb wavefunctions as kernel for alternative integral transformations. J Phys A Math Gen 36(31) 8443... 

Fig. 3.1. Recapitulation of the inner nodes for radial Coulomb wavefunctions also shown are the near-threshold continuum function with delta function normalisation (dotted curve) and two continnum functions above the threshold (adapted from H. Friedrich [112]).

The shifting dominance of centrifugal and Coulomb energies is thus substantially unaffected in equation (3) by the diamagnetic term. This dominance can accordingly be illustrated through the behavior of Coulomb wavefunctions which we cast in the Milne form [10] ... 

This is efficiently performed by block inverse iteration [4]. The continuum states so obtained are then normalized, fitting the asymptotic part to the regular and irregular Coulomb wavefunctions. 

Corrections to the mean-field model are needed to describe the instantaneous Coulombic interactions among the electrons. This is achieved by including more than one Slater determinant in the wavefunction. 

Imagine a model hydrogen molecule with non-interacting electrons, such that their Coulomb repulsion is zero. Each electron in our model still has kinetic energy and is still attracted to both nuclei, but the electron motions are completely independent of each other because the electron-electron interaction term is zero. We would, therefore, expect that the electronic wavefunction for the pair of electrons would be a product of the wavefunctions for two independent electrons in H2+ (Figure 4.1), which I will write X(rO and F(r2). Thus X(ri) and T(r2) are molecular orbitals which describe independently the two electrons in our non-interacting electron model. 

In the limit of infinite atom separations, or if we switch off the Coulomb repui. sion between two electrons, all four wavefunctions have the same energy. But they correspond to different eigenvalues of the electron spin operator the first combination describes the singlet electronic ground state, and the other three combinations give an approximate description of the components of the first triplet excited state. 

We assume that standard Coulomb-correlated models for luminescent polymers [11] properly described the intrachain electronic structure of m-LPPP. In this case intrachain photoexcitation generate singlet excitons with odd parity wavefunctions (Bu), which are responsible for the spontaneous and stimulated emission. Since the pump energy in our experiments is about 0.5 eV larger than the optical ran... 

Term wavefunctions describe the behaviour of several electrons in a free ion coupled together by the electrostatic Coulomb interactions. The angular parts of term wavefunctions are determined by the theory of angular momentum as are the angular parts of one-electron wavefunctions. In particular, the angular distributions of the electron densities of many-electron wavefunctions are intimately related to those for orbitals with the same orbital angular momentum quantum number that is. 

In the foregoing, U is the interaction potential, M is the reduced mass of the colliding system, ftk and ftk are respectively the momentum of the projectile before and after the collision, ig and in are respectively the wavefunctions of the atom (or molecule) in the ground and nth excited states, and the volume element dt includes the atomic electron and the projectile. Since U for charged-particle impact may be represented by a sum of coulombic terms in most cases, Eq. (4.11) can be written as (Bethe, 1930 Inokuti, 1971)... 

We consider an /V-electron system where the electrons experience the mutual Coulomb interaction along with an external potential vv( r) due to the nuclei. The system is then subjected to an additional TD scalar potential (r, t) and a TD vector potential A(r, t). The many-electron wavefunction [Pg.74]

The density functional theory (DFT) [32] represents the major alternative to methods based on the Hartree-Fock formalism. In DFT, the focus is not in the wavefunction, but in the electron density. The total energy of an n-electron system can in all generality be expressed as a summation of four terms (equation 4). The first three terms, making reference to the noninteracting kinetic energy, the electron-nucleus Coulomb attraction and the electron-electron Coulomb repulsion, can be computed in a straightforward way. The practical problem of this method is the calculation of the fourth term Exc, the exchange-correlation term, for which the exact expression is not known. 

In the first term, Uc, usually called the Coulombic term, the initially excited electron on D returns to the ground state orbital while an electron on A is simultaneously promoted to the excited state. In the second term, called the exchange term, Liex, there is an exchange of two electrons on D and A. The exchange interaction is a quantum-mechanical effect arising from the symmetry properties of the wavefunctions with respect to exchange of spin and space coordinates of two electrons. 

To illustrate the convergence of the FCI principal expansion with respect to short-range electron correlation, we have in Fig. 1.1 plotted the ground-state He wavefunction with both electrons fixed at a distance of 0.5 ao from the nucleus, as a function of the angle 0i2 between the position vectors ri and r2 of the two electrons. The thick grey lines correspond to the exact nonrelativistic wavefunction, whereas the FCI wavefunctions are plotted using black lines. Clearly, the description of the Coulomb cusp and more generally the Coulomb hole is poor in the orbital approximation. In particular, no matter how many terms we include in the FCI wavefunction, we will not be able to describe the nondifferentiability of the wavefunction at the point of coalescence. 

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