Helium atom description

The density of He I at the boiling point at 1 atm is 125 kg m 3 and the viscosity is 3 x 10 6 Pa s. As we would anticipate, cooling increases the viscosity until He II is formed. Cooling this form reduces the viscosity so that close to 0 K a liquid with zero viscosity is produced. The vibrational motion of the helium atoms is about the same or a little larger than the mean interatomic spacing and the flow properties cannot be considered in classical terms. Only a quantum mechanical description is satisfactory. We can consider this condition to give the limit of De-+ 0 because we have difficulty in defining a relaxation when we have the positional uncertainty for the structural components. 

Atoms and ions with noble-gaS electron configurations have usually been described as having spherical symmetiy. For some considerations this description is satisfactory for others, however, it is advantageous to consider the atoms or ions to have a shape other than spherical—the helium atom can be described as deformed to a prolate ellipsoid of revolution, and the neon atom and other noble-gas atoms as deformed to a shape with cubic symmetry. 

Description Helium nucleus (not helium atom). Same properties as an electron but was ejected from the nucleus. Not a particle at all. Gamma rays are high-energy radiation. 

There are three energy contributions that must be considered in the description of the helium atom (1) the kinetic energy of the electrons as they move around the nucleus, (2) the potential energy of attraction between the nucleus and the electrons, and (3) the potential energy of repulsion between the two electrons. 

In contrast to the Schrodinger equation for the hydrogen atom, the Schrodinger equation for a polyelectronic atom cannot be solved exactly. For example, although the hydrogen and helium atoms are similar in many respects, the mathematical descriptions of these atoms are fundamentally... 

QED can be considered to be one of the most precisely tested theories in physics at present. It provides an extremely accurate description of systems such as hydrogen and helium atoms, as well as for bound-leptonic systems, for example, positronium and muonium. Remarkable agreement between theory and experiment has been achieved with respect to the determination of the hyperfine structure and the Lamb shift. The same holds true for the electronic and muonic g-factors. The free-electron g-factor is determined at present as... 

Truncate this operator to third order with CCSD(T) and it still reproduces an estimated 97% of the correlation description [9]. (It is worth noting that methods exist which explicitly include the interelectronic potential. Recent calculations on the helium atom using Hylleraas-type r12 methods were able to match the exact non-relativistic energy to an astounding 10 12 kcal/mol [10].)... 

Even when it is not possible to remove the coupling interaction, it may be convenient to use this description. Thus in our discussion of the helium atom we found certain stationary states to be approximately represented by wave functions formed by linear combination of the wave functions ls(l) 2s(2) and 2s(l) ls(2). These we identify with states A and B above, saying that each electron resonates between a Is and a 2s orbit, the two electrons changing places with the frequency 1/h times... 

Exercise 5.5. Modify fig5-9.xls to provide for distinct Slater exponents for each component of the split-basis. Repeat the calculation of the energy of the helium atom to investigate whether a better description of the ground state can be obtained using the electronic energy as the test parameter. 

Thus, for example, in the HF description of the helium atom as two electrons of opposite spin with the same spatial distribution, the two electrons may occupy the same point in space notwithstanding their mutual repulsion. Attempts at the physical interpretation of these phenomena are important in the density-functional description of electronic structure which we discuss in Chapter 33. 

Table 3.1. Contributions of various physical effects (non-relativistic, Bieit, QED, and beyond QED, distinct physical contributions shown in bold) to the ionization energy and the dipole polarizability a of the helium atom, as well as comparison with the experimental values (all quantities are expressed in atomic units i.e.. e = 1. fi = 1, mo = 1- where iiiq denotes the rest mass of the electron). The first column gives the symbol of the term in the Breit-Pauli Hamiltonian [Eq. (3.72)] as well as of the QED corrections given order by order (first corresponding to the electron-positron vacuum polarization (QED), then, beyond quantum electrodynamics, to other particle-antiparticle pairs (non-QED) li,7T,. ..) split into several separate effects. The second column contains a short description of the effect. The estimated error (third and fourth columns) is given in parentheses in the units of the last figure reported. 

Now if we look at the perturbation theory description of this system using wavefunctions which are localized on the individual atoms we find that the second-order correlation energy for two helium atoms is just twice that of a single helitun atom and in general... 

Notice that O5 and Og are two-electron functions, which cannot be factorized into one-electron functions. By calculating all matrix elements and solving the 6 x 6 eigenvalue problem, Hylleraas, in 1928, obtained, without comparison, the best description of the helium atom with the energy -2.903329 H, compared to the earlier best value of -2.86 H. With the help of modern computers, it was recently possible to determine the ground state energy with more than accurate 20 decimal places (-2.903724 H) using essentially the Hylleraas method. 

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