Example the helium atom

We first consider the helium-like atom, with a nucleus of charge Z fixed at the origin. The electronic Hamiltonian defined in Section 1.1 takes the form 

The solutions of the eigenvalue equation (1.2.1) cannot be obtained in simple closed form owing to the presence of the term g(l, 2). To obtain a very crude description of the system, we may negect this term altogether, i.e. consider a model system with non-interacting electrons and the Hamiltonian 

Now the two terms on the left are entirely independent of each other, being functions of different sets of variables, and their sum can be a constant, for all values of the variables, only if each term is itself a constant. Thus, denoting the two constants by and tz. the product (1.2.5) will satisfy the eigenvalue equation (1.2.4) provided that 

The names of the variables in (1.2.7) are irrelevant (pi and 2 are functions of position of a point in space and the operator h works on the variables (e.g. cartesian coordinates) defining the point. It follows that 0i and 02 roay be any solutions of the one-electron eigenvalue problem  

The fact that the total energy is the sum of the energies of the separate electrons is not imexpected in view of the omission of the interaction term g(l, 2). 

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For example, the helium atom electron wave function can be written as... 

For example the helium atom Levine IN (2000) Quantum chemistry, 5th edn. Prentice Hall, Engelwood Cliffs, NJ, pp 256-259... 

We have already dealt with the calculation of the wave functions of the hydrogen atom. We now proceed to consider many-electron atoms, first dealing with the simplest such example, the helium atom which possesses two electrons. The Hamiltonian for a helium-like atom with an infinitely heavy nucleus can be obtained by selecting the appropriate terms from the master equation in chapter 3. The Hamiltonian we use is... 

BernardiJ Application of the full basis set 2 when calculating Ea results in the wave function of A containing not only its own atomic orbitals, but also the atomic orbitals of the ( absenf ) partner B, the ghost orbitals (see Fig. 13.3b). As a by-product, the charge density of A would exhibit broken symmetry with respect to the symmetry of A itself (if any) for example, the helium atom would have a small dipole moment, etc. 

Example The electron configuration for Be is Is lsfi but we write [He]2s where [He] is equivalent to all the electron orbitals in the helium atom. The Letters, s, p, d, and f designate the shape of the orbitals and the superscript gives the number of electrons in that orbital. 

So far, no exact application of this extended HF scheme has been carried out, but, by using the helium atom as a typical example, one can get an idea of the possibilities and limitations of this ap-... 

Note that the other electrons do not block the influence of the nucleus they simply provide additional repulsive coulombic interactions that partly counteract the pull of the nucleus. For example, the pull of the nucleus on an electron in the helium atom is less than its charge of +2e would exert but greater than the net charge of +e that we would expect if each electron balanced one positive charge exactly. 

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. 

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... 

In previous sections, the basis for applying quantum mechanical principles has been illustrated. Although it is possible to solve exactly several types of problems, it should not be inferred that this is always the case. For example, it is easy to formulate wave equations for numerous systems, but generally they cannot be solved exactly. Consider the case of the helium atom, which is illustrated in Figure 2.7 to show the coordinates of the parts of the system. 

Continuing to work in the helium atom example (realize that this could be any two electron system) pick ( , j) = (1, 2) and look at that one term. 

The concept of resonance was introduced into quantum mechanics by Heisenberg16 in connection with the discussion of the quantum states of the helium atom. He pointed out that a quantum-mechanical treatment somewhat analogous to the classical treatment of a system of resonating coupled harmonic oscillators can be applied to many systems. The resonance phenomenon of classical mechanics is observed, for example, for a system of two tuning forks with the same characteristic frequency of oscillation and attached to a common base, hich... 

Even in atoms in molecules which have no permanent dipole, instantaneous dipoles will arise as a result of momentary imbalances in electron distribution. Consider the helium atom, for example. It is extremely improbable that the two electrons in the Is orbital of helium will be diametrically opposite each other at all times. Hence there will be instantaneous dipoles capable of inducing dipoles in adjacent atoms or molecules. AnothCT way of looking at this phenomenon is to consider the electrons in two or more "nonpolar" molecules as synchronizing their movements (at least partially) to minimize electron-electron repulsion and maximize electron-nucleus attraction. Such attractions are extremely short ranged and weak, as are dipole-induced dipole forces. The energy of such interactions may be expressed as... 

The number of protons in the nucleus of an atom is called the proton number (or atomic number) and is given the symbol Z. Hence in the diagram shown in Figure 3.3, the helium atom has a proton number of 2, since it has two protons in its nucleus. Each element has its own proton number and no two different elements have the same proton number. For example, a different element, lithium, has a proton number of 3, since it has three protons in its nucleus. 

Hence, in the example shown in Figure 3.3 the helium atom has a nucleon number of 4, since it has two protons and two neutrons in its nucleus. If we consider the metallic element lithium, it has three protons and four neutrons in its nucleus. It therefore has a nucleon number of 7. 

Figure 4.4 shows another example, also from close-coupling calculations by Burke et al. [43], for and 2P electron scattering by the helium atom near... 

Now suppose the two colliding partners have quite different masses. For example, suppose a helium atom (mass 6.64 x 10-27 kg) is traveling perpendicular to the flat wall of a container (mass 1 kg) at a speed of 1000 ms-1. Choose the z-axis as the initial direction of motion of the helium atom. The motion after the collision will also be along the z-axis, since the atom is moving perpendicular to the wall—no force is... 

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