Helium atom simple calculations

Following the development of quantum theory by Heisenberg [1] and Schrodinger [2] and a few further discoveries, the basic principles of the structure of atoms and molecules were described around 1930. Unfortunately, the complexity of the Schrodinger equation increases dramatically with the number of electrons involved in a system, and thus for a long time the hydrogen and helium atoms and simple molecules as H2 were the only species whose properties could really be calculated from these first principles. In 1929, Dirac [3] wrote ... 

Within the last few years, there has been a resurgence of interest in high-accuracy calculations of simple atomic and molecular systems. For helium, such calculations have reached an extraordinary degree of precision. These achievements are only partially based on the availability of increased computational power. We review the present state of developments for such accurate calculations, with an emphasis on variational methods. Because of the central place occupied by the helium atom and its ground state, much of the discussion centers on methods developed for helium. Some of these methods have also been applied to more complex systems, and calculations on such systems now approach or even surpass a level of precision once only associated with calculations on helium. Hence, other atoms and molecules amenable to high-precision methods are also discussed. 

We have calculated the second- and fourth-order dipole susceptibilities of an excited helium atom. Numerical results have been obtained for the ls2p Pq-and ls2p f2-states of helium. For the accurate calculations of these quantities we have used the model potential method. The interaction of the helium atoms with the external electric held F is treated as a perturbation to the second- and to the fourth orders. The simple analytical expressions have been derived which can be used to estimate of the second- and higher-order matrix elements. The present set of numerical data, which is based on the Green function method, has smaller estimated uncertainties in ones than previous works. This method is developed to high-order of the perturbation theory and it is shown specihcally that the continuum contribution is surprisingly large and corresponds about 23% for the scalar part of polarizability. 

There are no analytical forms for the radial functions, / ni(r), as solutions of the radial wave equation. Hartree, in 1928, developed the standard solution procedure, the self-consistent field method for the helium atom by using the simple product forms of equation 1.10 to represent the two-electron wave function. Herman and Skillman (4) programmed a very useful approximate form of the Hartree method in the early 1960s for atomic structure calculations on all the atoms in the Periodic Table. An executable version of this program, based on their FORTRAN code, modified to output data for use on a spreadsheet is included with the material on the CDROM as hs.exe. 

Figure 5.1 Part details of the spreadsheet, fig5-lto5-3.xls, for the calculation of the energy of the helium atom, based on Hartree s analysis in Section 5.1. The assumption of simple hydrogenic behaviour does not return good energies.

It is appropriate therefore to present the mathematical details, here too, using the simple case of a double-zeta Slater basis function calculation of the electronic energy of the helium atom. This mathematics is sufficient for the calculations in Chapter 6... 

The helium atom has a metastable Si excited state with quantum numbers a = y a = 1 and a radiative lifetime of 8000 s. To carry out quantum scattering calculations on collisions of two such atoms, we need channel functions to handle the two spin quantum numbers and the mechanical angular momentum L. Once again it would be possible to use simple product functions as channel functions. 

A systematic approach which gives a simple interpretation of the spectral line shifts and of other structural properties of spectra of foreign atoms surrounded by liquid helium has been presented. Results for hydrogen and helium atoms show that the qualitative behaviour of the calculated spectral properties agrees with experimentally available data. No specific experimen-... 

The ionization potential (energy to remove one electron) of a helium atom in its ground state is 24.58 eV. a. What effective nuclear charge does this correspond to Corr5)are with the Z value from the simple variation calculation. 

There are several commonly used approximation schemes that can be applied to the electronic states of multielectron atoms. The first approximation scheme was the variation method, in which a variation trial function is chosen to minimize the approximate ground-state energy calculated with it. A simple orbital variation trial function was found to correspond to a reduced nuclear charge in the helium atom. This result was interpreted to mean that each electron in a helium atom shields the other electron from the full charge of the nucleus. A better variation trial function includes electron correlation, a dependence of the wave function on the electron lectrcm distance. ... 

The Rydberg constant is the scale factor that connects all theoretical calculations and experimental measurements of energy levels in any system involving electrons. This includes all atoms, molecules and condensed matter. In simple systems, such as hydrogen, positronium, muonium, and possibly helium, the theoretical accuracy is comparable to that of experiments. In this case, experimenters can be said to measure the Rydberg constant, if not to test theory. 

A wide variety of plasma diagnostic applications is available from the measurement of the relatively simple X-ray spectra of He-like ions [1] and references therein. The n = 2 and n = 3 X-ray spectra from many mid- and high-Z He-like ions have been studied in tokamak plasmas [2-4] and in solar flares [5,6]. The high n Rydberg series of medium Z helium-like ions have been observed from Z-pinches [7,8], laser-produced plasmas [9], exploding wires [8], the solar corona [10], tokamaks [11-13] and ion traps [14]. Always associated with X-ray emission from these two electron systems are satellite lines from lithium-like ions. Comparison of observed X-ray spectra with calculated transitions can provide tests of atomic kinetics models and structure calculations for helium- and lithium-like ions. From wavelength measurements, a systematic study of the n and Z dependence of atomic potentials may be undertaken. From the satellite line intensities, the dynamics of level population by dielectronic recombination and inner-shell excitation may be addressed. 

The simple expression or the more complicated ones can be used to describe solvent perturbations on infrared spectra. They do not explain solvent shifts, since their explanation requires a priori calculations of 17, C7",. This kind of calculation demands a detailed quantum mechanical examination of the intricate many-body interactions between the electrons and nuclei of the solvent and those of the dissolved molecule. Such calculations may be just barely possible for, say, a system of hydrogen atoms dissolved in liquid helium. They are not tractable for most solvent-solute systems unless drastic approximations are made. 

One of the problems with VMC is that it favors simple states over more complicated states. As an example, consider the liquid-solid transition in helium at zero temperature. The solid wave function is simpler than the liquid wave function because in the solid the particles are localized so that the phase space that the atoms explore is much reduced. This biases the difference between the liquid and solid variational energies for the same type of trial function, (e.g. a pair product form, see below) since the solid energy will be closer to the exact result than the liquid. Hence, the transition density will be systematically lower than the experimental value. Another illustration is the calculation of the polarization energy of liquid He. The wave function for fully polarized helium is simpler than for unpolarized helium because antisymmetry requirements are higher in the polarized phase so that the spin susceptibility computed at the pair product level has the wrong sign ... 

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