Coulomb function

For the purposes of this review it is convenient to focus attention on that class of molecules in which the valence electrons are easily distinguished from the core electrons (e.g., -n electron systems) and which have a large number of vibrational degrees of freedom. There have been several studies of the photoionization of aromatic molecules.206-209 In the earliest calculations either a free electron model, or a molecule-centered expansion in plane waves, or coulomb functions, has been used. Only the recent calculation by Johnson and Rice210 explicitly considered the interference effects which must accompany any process in a system with interatomic spacings and electron wavelength of comparable magnitude. The importance of atomic interference effects in the representation of molecular continuum states has been emphasized by Cohen and Fano,211 but, as far as we know, only the Johnson-Rice calculation incorporates this phenomenon in a detailed analysis. 

Johnson and Rice used an LCAO continuum orbital constructed of atomic phase-shifted coulomb functions. Such an orbital displays all of the aforementioned properties, and has only one obvious deficiency— because of large interatomic overlap, the wavefunction does not vanish at each of the nuclei of the molecule. Use of the LCAO representation of the wavefunction is equivalent to picturing the molecule as composed of individual atoms which act as independent scattering centers. However, all the overall molecular symmetry properties are accounted for, and interference effects are explicitly treated. Correlation effects appear through an assigned effective nuclear charge and corresponding quantum defects of the atomic functions. 

There are two parameters in the atomic coulomb functions, the effective nuclear charge and the quantum defect. The values of these were taken by Johnson and Rice from available spectral data. The effective atomic charge was adjusted to give the correct ionization potential of the molecule, 9.25 eV, requiring thereby z = 0.8243. The quantum defects of carbon were taken from the appropriate atomic series and were 1.04 for the 5-state and 0.73 for the p-states. It is interesting to compare the calculated molecular quantum defects (i.e., those corresponding to the Johnson-Rice LCAO function) with those which can be obtained from the various benzene Rydberg series.218 The asymptotic form of the elu orbital constructed from s atomic functions is... 

The / and g functions are commonly termed the regular and irregular coulomb functions.4,5 In the classically allowed region the / and g functions are real... 

Although the properties of the/and g functions are outlined in chapter 2, it is worth summarizing their properties here.8 The / and g coulomb functions are termed regular and irregular since asr— 0,/< rt+ and g oc r (. Due to the r = 0 behavior of the g function, in H only the/wave exists. As r — for Wt > 0 the/ and g waves are sine and cosine functions, and if Wt > 0, jiv simply specifies the phase of the wavefunction relative to the hydrogenic/wave. If Wt < 0 the/and g waves both have exponentially increasing and decreasing parts, and, as we have seen in Chapter 2, only if... 

The simplest analytical radial orbitals may be found by solving the radial Schrodinger equation for a one-electron hydrogen-like atom with arbitrary Z. They are usually called Coulomb functions and are expressed... 

J. Thompson, A.R. Barnett, COULCC A continued-fraction algorithm for Coulomb functions of complex order with complex arguments, Comput. Phys. Comm. 36 (1985) 363. 

It is not necessary for the physicist to know how to compute the Coulomb functions. They are found in subroutine libraries, for example Barnett et al. (1974). A sufficient idea of their form is obtained by putting j = L = 0 in (4.62), when they are seen to be sinp and cosp respectively. The potential terms dilate or compress the sine and cosine waves, resulting in an overall phase shift at long range. 

The forms of the Coulomb functions for very small and very large r are important... 

If the potential V(r) is a pure Coulomb potential the asymptotic partial wave is given by the regular Coulomb function (4.64), apart from a constant phase factor. We strictly have no incident plane wave since the Coulomb potential modifies the wave function everywhere. We make the normalisation of the Coulomb distorted wave t/j,j(k,r) analogous to that of (4.83) by choosing the phase factor to be the Coulomb phase shift [Pg.95]

For certain mathematical functions and operations it is necessary for the physicist to know their context, definition and mathematical properties, which we treat in the book. He does not need to know how to calculate them or to control their calculation. Numerical values of functions such as sinx have traditionally been taken from table books or slide rules. Modern computational facilities have enabled us to extend this concept, for example, to Coulomb functions, associated Legendre polynomials, Clebsch—Gordan and related coefficients, matrix inversion and diagonali-sation and Gaussian quadratures. The subroutine library has replaced the table book. We give references to suitable library subroutines. 

The 7/ / terms are the Coulomb functions introduced in eq.(104), P are the relaxed (i.e. polarized by the solvent in a self-consistent manner) elements of the density matrix P = CnC, and F are the elements of the in vacuo Fock matrix, in the form used in the semiempirical approach, but referred to the relaxed P matrix. The charges qk are drawn from P, according to the Mulliken s rules, or according to a new definition given by... 

Static model (Langmuir 1979), which assumes metal cation-OH bonding is purely electrostatic and, therefore, proportional to the coulombic function. Metal-OH complexing is stronger and becomes increasingly covalent (inner sphere) as + l) values increase for species plotting above the... 

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