Abstract
Atomic structure of N-electron atoms is often determined by solving the Hartree-Fock equations, which are a set of integro-differential equations. The integral part of the Hartree-Fock equations treats electron exchange, but the Hartree-Fock equations are not often treated as an integro-differential equation. The exchange term is often approximated as an inhomogeneous or an effective potential so that the Hartree-Fock equations become a set of ordinary differential equations (which can be solved using the usual shooting methods). Because the Hartree-Fock equations are an iterative-refinement method, the inhomogeneous term relies on the previous guess of the wavefunction. In addition, there are numerical complications associated with solving inhomogeneous differential equations. This work uses matrix methods to solve the Hartree-Fock equations as an integro-differential equation. It is well known that a derivative operator can be expressed as a matrix made of finite-difference coefficients; energy eigenvalues and eigenvectors can be obtained by using linear-algebra packages. The integral (exchange) part of the Hartree-Fock equation can be approximated as a sum and written as a matrix. The Hartree-Fock equations can be solved as a matrix that is the sum of the differential and integral matrices. We compare calculations using this method against experiment and standard atomic structure calculations. This matrix method can also be used to solve for free-electron wavefunctions, thus improving how the atoms and free electrons interact. This technique is important for spectral line broadening in two ways: it improves the atomic structure calculations, and it improves the motion of the plasma electrons that collide with the atom.
Highlights
Calculations of spectral line broadening are important for a variety of applications, including diagnosing laboratory plasma conditions [1,2], determining temperatures and gravities of white dwarf stars [3], and calculations of stellar opacities [4]
The work here has an impact on spectral line broadening in two ways: the atomic structure and the behavior of plasma electrons when they impact the radiator
We present a different technique for evaluating exchange effects in multi-electron atoms and in collision processes
Summary
Calculations of spectral line broadening are important for a variety of applications, including diagnosing laboratory plasma conditions [1,2], determining temperatures and gravities of white dwarf stars [3], and calculations of stellar opacities [4]. Line-broadening calculations do not often consider exchange between the radiator and the plasma electrons (though it is included for the calculation of the internal atomic states). Many calculations are performed in a high-temperature regime where exchange effects are assumed to be negligible [13,14] Some systems, such as lithium-like ions, are known to be dominated by close-range interactions [15]1. We begin by looking at changes in the wavefunctions and energy of neutral helium using a matrix method for solving the Hartree-Fock equations with exact exchange This is a significant improvement since most calculations treat exchange as an inhomogeneous term [21] or treat it using a local-density approximation [22]. We discuss some implications on how collisions of plasma electrons with Li-like ions could change when using our new wavefunctions
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