The helium atom ground state

Now we turn to the helium atom, whose coordinate system is shown in Fig. 2.2.1. The potential energy of this system (in a.u.) is 

It is obvious that both terms on the left side of eq. (2.2.8) must each be equal to a constant, corresponding to Ea and Eb, respectively, and 

So the simplified Schrodinger equation eq. (2.2.7) can be separated into two (familiar) equations, each involving only the coordinates of one electron  

We have already solved eqs. (2.2.10) and (2.2.11) in the treatment of the hydrogen atom the only difference is that now Z = 2. If we take the ground state wavefunction and energy (in a.u.), 

To improve on this result, we need to understand what causes the error in the solution given by eq. (2.2.14), aside from the fact that electronic repulsion has been ignored. From i 2(2), and ir (1,2) given by eqs. (2.2.12) to 

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For the helium atom ground state, which we shall later generalize to many election atoms and molecules. 

Abstract. With an eye on the high accuracy ( 10 MHz) evaluation of the ionization energy from the helium atom ground state, a complete set of order ma6 operators is built. This set is gauge and regularization scheme independent and can be used for an immediate calculation with a wave function of the helium ground state. 

Here we overview our applications to the helium atom ground state. In the Hyller-aas coordinate defined by... 

PERTURBATION TREATMENT OF THE HELIUM-ATOM GROUND STATE... 

Section 9,3 Perturbation Treatment of the Helium-Atom Ground State 253... 

EXAMPLE Do an SCF calculation for the helium-atom ground state using a basis set of two Is STOs with orbital exponents = 1.45 and 2 = 2.91. [By trial and error, these have been found to be the optimum s to use for this basis set see C. Roetti and E. Clementi, J. Chem. Phys., 60, 4725 (1974).]... 

Now consider variation functions for the helium-atom ground state. If we used 4° [Eq. (9.49)] as the trial function, we would get the first-order perturbation result, -74.82 eV. To improve on this result, we introduce a variational parameter into (9.49). We try the normalized function... 

An application of the variation method to the lithium atom ground state uses an orbital wave function containing hydrogen-like orbitals with variable orbital exponents (effective nuclear charges) similar to that used with helium, except that different effective... 

Mainly for considerations of space, it has seemed desirable to limit the framework of the present review to the standard methods for treating correlation effects, namely the method of superposition of configurations, the method with correlated wave functions containing rij and the method using different orbitals for different spins. Historically these methods were developed together as different branches of the same tree, and, as useful tools for actual applications, they can all be traced back to the pioneering work of Hylleraas carried out in 1928-30 in connection with his study of the ground state of the helium atom. 

To test the accuracy and convenience of the method of superposition of configurations, the problem of the ground state of the helium atom has recently been reexamined by several authors. According to Hylleraas (1928), the total wave function may be expressed in the form... 

TABLE VI. Ground State of the Helium Atom Obtained by Superposition of Configurations ... 

Kinoshita, T., Phys. Rev. 105, 1490, Ground state of the helium atom. ... 

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. 

To arrive at the correct formulation of the ground state of the helium atom it is necessary to also take into account the effect of spin, represented by the functions a and / . There are four possibilities, according to the electrons having the same spin, either up or down ... 

The linear combination is used instead of the unsyinmetrical states / (l)cv(2) and j3 2)a ). It is reasonable to expect that each of these spin states could occur in combination with the ground-state function ij> r) to yield four different levels at the ground state. However, for the helium atom only one ground-state function can be identified experimentally and it is significant to note that only one of the spin functions is anti-symmetrical, i.e. 

Fig. 13.4 Logarithm of error(Eh) in the configuration interaction energy for the ground state of the helium atom as a function of maximum orbital quantum number, L, of the one-electron basis functions. The data were obtained in an...

To expose the essence of the R12 method of Kutzelnigg [19], consider the simplest two-electron system, the helium atom in its ground state. The exact wave function in the vicinity of an electron-electron coalescence point r can be expressed [13] as... 

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