Helium atom probability density

Although it is strictly a probability distribution, it has become common to call p(f) the electron density. This terminology is obviously more realistic for the Uranium atom than for the helium atom but the meaning is clear. 

If the interaction between two ground-state He atoms were strictly repulsive (as predicted by MO theory), the atoms in He gas would not attract one another at all and the gas would never liquefy. Of course, helium gas can be liquefied. Configuration-interaction calculations and direct experimental evidence from scattering experiments show that as two He atoms approach each other there is an initial weak attraction, with the potential energy reaching a minimum at 3.0 A of 0.0009 eV below the separated-atoms energy. At distances less than 3.0 A, the force becomes increasingly repulsive because of overlap of the electron probability densities. The initial attraction (called a London or dispersion force) results from instantaneous correlation between the motions of the electrons in one atom and the motions of the electrons in the second atom. Therefore, a calculation that includes electron correlation is needed to deal with dispersion attractions. 

The mutual avoidance of electrons in the helium atom or in the hydrogen molecule is caused by Coulombic repulsion of electrons (described in the previous subsection). As we have shown in this chapter, in the Haitree-Fock method the Coulomb hole is absent, whereas methods that account for electron correlation generate sueh a hole. However, electrons avoid each other also for reasons other than their charge. The Pauli principle is another reason this occurs. One of the consequences is the fact that electrons with the same spin coordinate cannot reside in the same place see p. 34. The continuity of the wave function implies that the probability density of them staying in the vicinity of each other is small i.e.. 

Fig. 10.1. Absence of electronic correlation in the helium atom as seen by the Hartree-Fock method. Visualization of the cross-section of the square of the wave function (probability density distribution) describing electron 2 within the plane xy provided electron 1 is located in a certain point in space a) at (—1,0,0) b) at (1,0,0). Note, that in both cases the conditional probability density distributions of electron 2 are identical. This means electron 2 does not react to the motion of electron 1, i.e. there is no correlation whatsoever of the electronic motions (when the total wave function is the Hartree-Fock one).

Hydrogen [54] and helium [55] atoms are known to exhibit regular/chaotic dynamics in the presence of external field of different colors and intensities. Chaotic ionization from the Rydberg states of the atoms [54,55] has been very intriguing for the experimentalists. Both QFD [56,57] and quantum theory of motion (QTM) [58,59] have been able to explain the quantum domain behavior of the classically chaotic systems [60], In QFD, the quantum dynamics is mapped onto that of a probability fluid of density and current density p(r, t) and j(r, t) respectively obtainable as solutions to the QFD equations. The fluid moves under the influence of the classical Coulomb potential augmented by a quantum potential defined as... 

As described in Ref. [25], the Hartree approach has been applied to get energies and density probability distributions of Br2(X) 4He clusters. The lowest energies were obtained for the value A = 0 of the projection of the orbital angular momentum onto the molecular axis, and the symmetric /V-boson wavefunction, i.e. the Eg state in which all the He atoms occupy the same orbital (in contrast to the case of fermions). It stressed that both energetics and helium distributions on small clusters (N < 18) showed very good agreement with those obtained in exact DMC computations [24],... 

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