Helium atom, perturbation method

In this section we examine the ground-state energy of the helium atom by means of both perturbation theory and the variation method. We may then compare the accuracy of the two procedures. 

Although we cannot solve the wave equation for the helium atom exactly, the approaches described provide some insight in regard to how we might proceed in cases where approximations must be made. The two major approximation methods are known as the variation and perturbation methods. For details of these methods as applied to the wave equation for the helium atom, see the quantum... 

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. 

In the helium atom two electrons revolve about the nucleus (nuclear charge 2e) we have therefore 6 co-ordinates to deal with instead of 3, with the result that an exact solution is no longer possible. For the purpose of obtaining a general idea of the possible states, an exact solution is, however, not at all necessary following Bohr, we can in the first place neglect the mutual interaction of the electrons, and for a first approximation treat the problem as if the two electrons moved undisturbed in the field of the nucleus. Afterwards, the interaction can be taken into account by the methods of the theory of perturbations. 

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... 

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. 

We find, then, that a systematic application of the theory of perturbations does not lead to a satisfactory model of the normal helium atom. It might be supposed that the failure of our method was due to the fact that we are dealing here with the normal state, where several electrons move in equivalent orbits, and that a better result would be anticipated in the case of the excited states, where the main characteristics of the spectra are reproduced by the quantum theory in the form used here. We shall now show that this again is not the case. 

The Perturbation Method and Its Application to the Ground State of the Helium Atom... 

We now apply the perturbation method to the ground state of the helium atom. The zero-order Hamiltonian is a sum of two hydrogen-like Hamiltonians ... 

The first-order correction to the wave function and the second-order correction to the energy eigenvalue are more complicated than the first-order correction to the energy eigenvalue. Appendix E contains the formulas for these quantities. No exact calculation of the second-order correction to the energy of the helium atom has been made, but a calculation made by a combination of the perturbation and variation methods gives an accurate upper bound ... 

The perturbation method as described in the previous section does not apply if several wave functions correspond to the same zero-order energy (the degenerate case). For example, the zero-order orbital energies of the 2s and 2p hydrogen-like orbitals are all equal, so that all of the states of the (1 X2 ) and (U)(2/ ) helium configurations have the same energy in zero order. A version of the perturbation method has been developed to handle this case. We will describe this method only briefly and present some results for some excited states of the helium atom. There is additional information in Appendix G. 

In the perturbation method the Hamiltonian is written as + H, where corresponds to a Schrodinger equation that can be solved. The perturbation term H is arbitrarily multiplied by a fictitious parameter k, so that A. = 1 corresponds to the actual case. The method is based on representations of energy eigenvalues and energy eigenfunctions as power series in A. and approximation of the series by partial sums. The method can be applied to excited states. In the helium atom treatment the electron-electron repulsive potential energy was treated as the perturbation term in the Hamiltonian operator. 

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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