Helium perturbation theory calculations

For completeness, in fig. 5 we compare some of the more precise experimental measurements of helium-like n = 2 fine structure at higher Z with the allorders relativistic perturbation theory calculations of Plante et al. [27]. Fig. 5a... 

Korona T, Williams HL, Bukowski R, Jeziorski B, Szalewicz K (1997) Helium dimer potential from symmetry-adapted perturbation theory calculations using large Gaussian geminal and orbital basis sets. J Chem Phys 106 5109—5122... 

There are two basic approaches to the theory of atomic helium, depending on whether the nuclear charge Z is small or large. For low-Z atoms and ions, the principal challenge is the accurate calculation of nonrelativistic electron correlation effects. Relativistic corrections can then be included by perturbation theory. For high-if ions, relativistic effects become of dominant importance and must be taken into account to all orders via the one-electron Dirac equation. Corrections due to the electron-electron interaction can then be included by perturbation theory. The cross-over point between the two regimes is approximately Z = 27... 

A symmetry-adapted perturbation theory approach for the calculation of the Hartree-Fock interaction energies has been proposed by Jeziorska et al.105 for the helium dimer, and generalized to the many-electron case in Ref. (106). The authors of Refs. (105-106) developed a basis-set independent perturbation scheme to solve the Hartree-Fock equations for the dimer, and analyzed the Hartree-Fock interaction energy in terms of contributions related to many-electron SAPT reviewed in Section 7. Specifically, they proposed to replace the Hartree-Fock equations for the... 

In two previous papers [8,9] we have calculated the static polarizabilities and hyperpolarizabilities for ls3p Pj (J = 0, 2)-states of helium. The method was based on degenerate perturbation-theory expressions for these quantities. The necessary dipole matrix elements were found by using the high-precision wave function on framework of the configuration-interaction (Cl) method [10]. The perturbed wave functions are also expanded in a basis of accurate variational eigenstates [11]. These basis sets of the wave functions explicitly take account of electron correlation. To control the result we have also carried out similar calculations with Fues model potential method. 

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. 

We add a few remarks on the wave mechanics of many-body problems, Here of course we are concerned with the solution of a wave equation in many-dimensional space thus the calculation of the helium spectrum needs as many as six co-ordinates, and that of the lithium spectrum nine. It is clear that in these cases an exact solution is not to be looked for, so that we must be content with an approximate solution of the problem. The methods of a highly developed perturbation theory enable us to push this approximation as far as we please the labour involved, however, increases without limit with the order of the approximation. The lowest terms of He, Li+ and Li have already been successfully calculated by this method, with results in good agreement with experiment (Hylleraas, 1930). 

There are many problems of wave mechanics which cannot be conveniently treated either by direct solution of the wave equation or by the use of perturbation theory. The helium atom, discussed in the next chapter, is such a system. No direct method of solving the wave equation has been found for this atom, and the application of perturbation theory is unsatisfactory because the first approximation is not accurate enough while the labor of calculating the higher approximations is extremely great. 

In Section 236 we treated the normal state of the helium atom with the use of first-order perturbation theory. In this section we shall show that the calculation of the energy can be greatly increased in accuracy by considering the quantity Z which occurs in the exponent (p = 2Zr/a0) of the zeroth-order function given in Equations 23-34 and 23-37 as a parameter Z instead of as a constant equal to the atomic number. The value of Z is determined by using the variation method with given by... 

Problem 29-1. Evaluate the integrals J and K for l 2s and ls2p of helium, and calculate by the first-order perturbation theory the term values for the levels obtained from these configurations. Observed term values (relative to He+) are ls2s lS 32033, ls2s tS 38455, ls2p P 27176, and ls2p SP 29233 cm-1. 

The value of the polarizability a of an atom or molecule can be calculated by evaluating the second-order Stark effect energy — %aF2 by the methods of perturbation theory or by other approximate methods. A discussion of the hydrogen atom has been given in Sections 27a and 27e (and Problem 26-1). The helium atom has been treated by various investigators by the variation method, and an extensive approximate treatment of many-electron atoms and ions based on the use of screening constants (Sec. 33a) has also been given.3 We shall discuss the variational treatments of the helium atom in detail. 

Manning and Sanders (33) used the Z-dependent perturbation theory combined with the complex rotation method to calculate the resonance position and width for the 2s2p autoionizing states of all members of the helium isoelectronic sequence. 

This paper reviews progress in the application of atomic isotope shift measurements, together with high precision atomic theory, to the determination of nuclear radii from the nuclear volume effect. The theory involves obtaining essentially exact solutions to the nonrelativistic three- and four-body problems for helium and lithium by variational methods. The calculation of relativistic and quantum electrodynamic corrections by perturbation theory is discussed, and in particular, methods for the accurate calculation of the Bethe logarithm part of the electron self energy are presented. The results are applied to the calculation of isotope shifts for the short-... 

E.A. Hylleraas, Zeit. Phys. 54 (1929) 347. Egil Andersen Hylleraas arrived in 1926 in Gottingen, to collaborate with Max Born. His professional experience was related to crystallography and to the optical properties of quartz. When one of the employees fell ill. Born told Hylleraas to continue his work on the helium atom in the context of the newly developed quantum mechanics. The helium atom problem had already been attacked by Albrecht Unsold in 1927 using first order perturbation theory, but Unsold obtained the ionization potential equal to 20.41 eV, while the experimental value was equal to 24.59 eV. In the reported calculations (done on a recently installed calculator) Hylleraas obtained a value of 24.47 eV (cf. contemporary accuracy, p. 134). 

A typical high-harmonic spectrum is shown in Fig. 4.4 for the helium atom. The squares represent experimental data taken from [66], and the solid line was obtained from a calculation using the EXX/EXX functional [67]. The spectrum consists of a series of peaks, first decreasing in amplitude and then reaching a plateau that extends to very high frequency. The peaks are placed at the odd multiples of the external laser frequency (the even multiples are dipole forbidden by symmetry). We note that any approach based on perturbation theory would yield a harmonic spectrum that decays exponentially, i.e. such a theory could never reproduce the measured peak intensities. TDDFT, on the other hand, gives a quite satisfactory agreement with experiment. 

Moszynski et al. calculated the interaction-induced polarizability of the helium dimer for the internuclear separations 3computational approach relied on spin-adapted perturbation theory (SAPT) calculations with a large basis set of [5s4p3d2f] size. Subsequently, they determined the polarized and depolarized Raman spectrum of the dimer. The computed polarized spectrum displays fair agreement with experiment. Most important, the computed intensities of the depolarized spectrum agree quite well with the experimental data reported by Proffitt et al7 ... 

Other methods which go beyond the Hartree-Fock level of approximation include Cluster Methods and Many-Body Perturbation Theory (Wilson 1984). These approaches involve the introduction of repulsion effects due to simultaneous interactions between three, four, and even more electrons in the expansion of the wavefunction. One important drawback of cluster methods and many-body perturbation theory is that they are not variational. That is to say, the calculated energies no longer represent upper bounds and it is possible to obtain predictions in excess of 100% of the experimental values. Nevertheless, their use is capable of reducing the error in the calculation of the energy of the helium atom to something of the order of lO %. 

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