Molecular orbitals of PtF6 and E110 F6 calculated by the self‐consistent multiple scattering Xα method

1974 ◽  
Vol 60 (11) ◽  
pp. 4466-4470 ◽  
Author(s):  
J. T. Waber ◽  
F. W. Averill
Keyword(s):  
Multiple Scattering ◽  
Molecular Orbitals ◽  
The Self ◽  
Self Consistent

scholarly journals Retrieving scattering clouds and disequilibrium chemistry in the atmosphere of HR 8799e

2020 ◽  
Vol 640 ◽  
pp. A131 ◽  
Author(s):  
P. Mollière ◽  
T. Stolker ◽  
S. Lacour ◽  
G. P. P. L. Otten ◽  
J. Shangguan ◽  
...  
Keyword(s):  
Radiative Transfer ◽  
Multiple Scattering ◽  
Functional Form ◽  
Cloud Model ◽  
The Self ◽  
Model Interpretation ◽  
Cloud Models ◽  
Cloudy Atmosphere ◽  
Self Consistent

Context. Clouds are ubiquitous in exoplanet atmospheres and they represent a challenge for the model interpretation of their spectra. When generating a large number of model spectra, complex cloud models often prove too costly numerically, whereas more efficient models may be overly simplified. Aims. We aim to constrain the atmospheric properties of the directly imaged planet HR 8799e with a free retrieval approach. Methods. We used our radiative transfer code petitRADTRANS for generating the spectra, which we coupled to the PyMultiNest tool. We added the effect of multiple scattering which is important for treating clouds. Two cloud model parameterizations are tested: the first incorporates the mixing and settling of condensates, the second simply parameterizes the functional form of the opacity. Results. In mock retrievals, using an inadequate cloud model may result in atmospheres that are more isothermal and less cloudy than the input. Applying our framework on observations of HR 8799e made with the GPI, SPHERE, and GRAVITY, we find a cloudy atmosphere governed by disequilibrium chemistry, confirming previous analyses. We retrieve that C/O = 0.60−0.08+0.07. Other models have not yet produced a well constrained C/O value for this planet. The retrieved C/O values of both cloud models are consistent, while leading to different atmospheric structures: either cloudy or more isothermal and less cloudy. Fitting the observations with the self-consistent Exo-REM model leads to comparable results, without constraining C/O. Conclusions. With data from the most sensitive instruments, retrieval analyses of directly imaged planets are possible. The inferred C/O ratio of HR 8799e is independent of the cloud model and thus appears to be a robust. This C/O is consistent with stellar, which could indicate that the HR 8799e formed outside the CO2 or CO iceline. As it is the innermost planet of the system, this constraint could apply to all HR 8799 planets.

scholarly journals EXTENDED FOLDY–LAX APPROXIMATION ON MULTIPLE SCATTERING

2014 ◽  
Vol 19 (1) ◽  
pp. 85-98 ◽  
Author(s):  
Jie Liao ◽  
Chao Ji
Keyword(s):  
Scattering Amplitudes ◽  
Multiple Scattering ◽  
Numerical Approximation ◽  
Harmonic Wave ◽  
The Self ◽  
New Approach ◽  
Consistent System ◽  
Relative Radius ◽  
Time Harmonic ◽  
Self Consistent

The Foldy–Lax self-consistent system has been widely used as an efficient numerical approximation of multiple scattering of time harmonic wave through a medium with many scatterers when the relative radius of each scatterer is small and the distribution of scatterers is sparse. In this paper, an “extended” Foldy–Lax self-consistent system including both source and dipole effects as well as corrections due to the self-interacting effects will be introduced, in which the scattering amplitudes and the corrections are determined as powers of the small scaled radius. This new approach substantially improves the accuracy of the approximation of the original Foldy–Lax approach.

scholarly journals The relativistic self-consistent field

1935 ◽  
Vol 152 (877) ◽  
pp. 625-649 ◽  
Keyword(s):  
Wave Equation ◽  
Field Equation ◽  
Wave Functions ◽  
The Self ◽  
Magnetic Interactions ◽  
Consistent Field ◽  
Exchange Effects ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Schrödinger's Equation

1—The method of the self-consistent field for determining the wave functions and energy levels of an atom with many electrons was developed by Hartree, and later derived from a variation principle and modified to take account of exchange and of Pauli’s exclusion principle by Slater* and Fock. No attempt was made to consider relativity effects, and the use of “ spin ” wave functions was purely formal. Since, in the solution of Dirac’s equation for a hydrogen-like atom of nuclear charge Z, the difference of the radial wave functions from the solutions of Schrodinger’s equation depends on the ratio Z/137, it appears that for heavy atoms the relativity correction will be of importance; in fact, it may in some cases be of more importance as a modification of Hartree’s original self-nsistent field equation than “ exchange ” effects. The relativistic self-consistent field equation neglecting “ exchange ” terms can be formed from Dirac’s equation by a method completely analogous to Hartree’s original derivation of the non-relativistic self-consistent field equation from Schrodinger’s equation. Here we are concerned with including both relativity and “ exchange ” effects and we show how Slater’s varia-tional method may be extended for this purpose. A difficulty arises in considering the relativistic theory of any problem concerning more than one electron since the correct wave equation for such a system is not known. Formulae have been given for the inter-action energy of two electrons, taking account of magnetic interactions and retardation, by Gaunt, Breit, and others. Since, however, none of these is to be regarded as exact, in the present paper the crude electrostatic expression for the potential energy will be used. The neglect of the magnetic interactions is not likely to lead to any great error for an atom consisting mainly of closed groups, since the magnetic field of a closed group vanishes. Also, since the self-consistent field type of approximation is concerned with the interaction of average distributions of electrons in one-electron wave functions, it seems probable that retardation does not play an important part. These effects are in any case likely to be of less importance than the improvement in the grouping of the wave functions which arises from using a wave equation which involves the spins implicitly.

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