The helium atom two electrons

7 Many-electron atoms The helium atom two electrons 

The next simplest atom is He (Z = 2), and for its two electrons, three electrostatic interactions must be considered  

In terms of obtaining wavefunctions and energies for the atomic orbitals of He, it has not been possible to solve the Schrodinger equation exactly and only approximate solutions are available. For atoms containing more than two electrons, it is even more difficult to obtain accurate solutions to the wave equation. 

In a multi-electron atom, orbitals with the same value of n but 

The preceding sections have been devoted mainly to hydrogen-like species containing one electron, the energy of which depends on n and Z (eq. 1.16). The atomic spectra of such species contain only a few lines associated with changes in the value of n (Fig. 1.3). It is only for such species that the Schrodinger equation has been solved exactly. 

In the ground state of the He atom, two electrons with ntg = and —j occupy the I5 atomic orbital, i.e. the electronic configuration is l5. For all atoms except hydrogen-like species, orbitals of the same principal quantum number but differing / are not degenerate. If one of the l5 electrons is promoted to an orbital with n = 2, the 

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A young English scientist recently promulgated the theory that the atomic number corresponds with the electro-positive charges that form the nucleus of any atom. In other words, if the atomic number corresponds to the number of units of positive electricity that go to make up the atomic nuclei, there must be one electron or negative unit. Thus, the hydrogen atom contains one electron, the helium atom two electrons, lithium three electrons and so on through the entire list. 

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. 

GAUSSIAN BASIS SET CALCULATIONS FOR THE HELIUM ATOM -TWO-ELECTRON INTEGRALS OVER GAUSSIAN BASIS FUNCTIONS... 

In the helium atom, two electrons, each of charge —e, circle a nucleus with charge -h 2e (Fig. 1.7a). The wave function of the system must now allow for the results of simultaneous measurement of the positions of two electrons. Our wave function must tell us the probability of finding electron 1 in some... 

Figure 16-3D shows the simplified representation of the interaction of two helium atoms. This time each helium atom is crosshatched before the two atoms approach. This is to indicate there are already two electrons in the Is orbital. Our rule of orbital occupancy tells us that the Is orbital can contain only two electrons. Consequently, when the second helium atom approaches, its valence orbitals cannot overlap significantly. The helium atom valence electrons fill its valence orbitals, preventing it from approaching a second atom close enough to share electrons. The helium atom forms no chemical bonds. ... 

The helium hydride two-electron ion has been detected in mass spectra and Szabo and Ostlund (47) report it to be of interest in astrophysics, for example, as the decay product in the /6 emission of the heavy dihydrogen species HT and in proton scattering experiments involving helium atoms. There are, too, accurate calculations by Wolniewicz (84), which suggest an equilibrium bond length of 1.4632 a.u. and an electronic binding energy of 0.0749 a.u. 

The intensity of shading at any point represents the magnitude of 1, i.e. the probability of finding the electron at that point. This may also be called a spherical charge-cloud . In helium, with two electrons, the picture is the same, but the two electrons must have opposite spins. These two electrons in helium are in a definite energy level and occupy an orbital in this case an atomic orbital. 

VV e now wish to establish the general functional form of possible wavefunctions for the two electrons in this pseudo helium atom. We will do so by considering first the spatial part of the u a efunction. We will show how to derive functional forms for the wavefunction in which the i change of electrons is independent of the electron labels and does not affect the electron density. The simplest approach is to assume that each wavefunction for the helium atom is the product of the individual one-electron solutions. As we have just seen, this implies that the total energy is equal to the sum of the one-electron orbital energies, which is not correct as ii ignores electron-electron repulsion. Nevertheless, it is a useful illustrative model. The wavefunction of the lowest energy state then has each of the two electrons in a Is orbital ... 

The final wavefunction stUl contains a large proportion of the Is orbital on the helium atom, but less than was obtained without the two-electron integrals. 

Write the Hamiltonian for the helium atom, which has two electrons, one at a distance r and the other at a distance r2-... 

The helium atom is similar to the hydrogen atom with the critical difference that there are two electrons moving in the potential field of a nucleus with a double positive charge (Z = 2) (Eig. 8-1). 

The first floor of the atom can hold only two electrons. When two hydrogen atoms, each with one electron, come together, each nucleus is in effect sharing two electrons, and as far as each nucleus is concerned, its shell is filled. The helium atom already contains two electrons its first floor is filled. That is why helium is unreactive. Wlien two helium atoms collide, rather than stick together, they simply bounce apart. (.See Table 1.)... 

Since the nucleus has positive charge, it attracts electrons (each with negative charge). If a nucleus attracts the number of electrons just equal to the nuclear charge, an electrically neutral atom is formed. Consider a nucleus containing two protons, a helium nucleus. When the helium atom has two electrons as well (2— charge), an electrically neutral helium atom results ... 

As an example we may calculate the energy of the helium atom in its normal state (24). Neglecting the interaction of the two electrons, each electron is in a hydrogen-like orbit, represented by equation 6 the eigenfunction of the whole atom is then lt, (1) (2), where (1) and (2) signify the first and the second electron. 

P (2) p (2) — p (1) p (1) ip (2) (2) that is, in the normal state of the helium atom the two electrons have oppositely directed spins. Other consequences of the exclusion principle, such as that not more than two electrons can occupy the K-shell of an atom, follow directly. 

The existence of the first HK theorem is quite surprising since electron-electron repulsion is a two-electron phenomenon and the electron density depends only on one set of electronic coordinates. Unfortunately, the universal functional is unknown and a plethora of different forms have been suggested that have been inspired by model systems such as the uniform or weakly inhomogeneous electron gas, the helium atom, or simply in an ad hoc way. A recent review describes the major classes of presently used density functionals [10]. 

The helium atom serves as a simple example for the application of this construction. If the nucleus (for which Z = 2) is considered to be fixed in space, the Hamiltonian operator H for the two electrons is... 

For hydrogen, the notation Is1 is used, where the superscript denotes a single electron in the Is state. Because an electron can have a spin quantum number of +1/2 or -1/2, two electrons having opposite spins can occupy the Is state. The helium atom, having two electrons, has the configuration Is2 with the electrons having spins of +1/2 and —1/2. 

We have just explained that the wave equation for the helium atom cannot be solved exacdy because of the term involving l/r12. If the repulsion between two electrons prevents a wave equation from being solved, it should be clear that when there are more than two electrons the situation is worse. If there are three electrons present (as in the lithium atom) there will be repulsion terms involving l/r12, l/r13, and l/r23. Although there are a number of types of calculations that can be performed (particularly the self-consistent field calculations), they will not be described here. Fortunately, for some situations, it is not necessary to have an exact wave function that is obtained from the exact solution of a wave equation. In many cases, an approximate wave function is sufficient. The most commonly used approximate wave functions for one electron are those given by J. C. Slater, and they are known as Slater wave functions or Slater-type orbitals (usually referred to as STO orbitals). 

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