Helium atom quantum numbers

Liquid Helium-4. Quantum mechanics defines two fundamentally different types of particles bosons, which have no unpaired quantum spins, and fermions, which do have unpaired spins. Bosons are governed by Bose-Einstein statistics which, at sufficiently low temperatures, allow the particles to coUect into a low energy quantum level, the so-called Bose-Einstein condensation. Fermions, which include electrons, protons, and neutrons, are governed by Fermi-DHac statistics which forbid any two particles to occupy exactly the same quantum state and thus forbid any analogue of Bose-Einstein condensation. Atoms may be thought of as assembHes of fermions only, but can behave as either fermions or bosons. If the total number of electrons, protons, and neutrons is odd, the atom is a fermion if it is even, the atom is a boson. 

A neutral helium atom has two electrons. To write the ground-state electron configuration of He, we apply the aufbau principle. One unique set of quantum numbers is assigned to each electron, moving from the most stable orbital upward until all electrons have been assigned. The most stable orbital is always ly( = l,/ = 0, JW/ = 0 ). 

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. 

A ground-state helium atom has two paired electrons in the Is orbital (Is2). The electrons with paired spin occupy the lowest of the quantised orbitals shown below (the Pauli exclusion principle prohibits any two electrons within a given quantised orbital from having the same spin quantum number) ... 

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...

Another way of stating the exclusion principle is that no two electrons in an atom have the same four quantum numbers. This important idea means that each electron in an atom has its own unique set of four quantum numbers. For example, compare the quantum numbers that distinguish a ground state hydrogen atom from a helium atom. (Recall that a helium atom has two electrons. Note also that mg quantum number is given as +. It could just as easily have a value of —By convention, chemists usually use the positive value first.)... 

Refer to the sets of quantum numbers for hydrogen and helium that you saw earlier. Then use the quantum numbers for lithium to infer why a lithium atom has the ground state electron configuration that it does. 

In order to appreciate the size of the basis sets required for fully converged calculations, consider the interaction of the simplest radical, a molecule in a electronic state, with He. The helium atom, being structureless, does not contribute any angular momentum states to the coupled channel basis. If the molecule is treated as a rigid rotor and the hyperfine structure of the molecule is ignored, the uncoupled basis for the collision problem is comprised of the direct products NMf ) SMg) lnii), where N = is the quantum number... 

Following the development of quantum theory by Heisenberg [1] and Schrodinger [2] and a few further discoveries, the basic principles of the structure of atoms and molecules were described around 1930. Unfortunately, the complexity of the Schrodinger equation increases dramatically with the number of electrons involved in a system, and thus for a long time the hydrogen and helium atoms and simple molecules as H2 were the only species whose properties could really be calculated from these first principles. In 1929, Dirac [3] wrote ... 

Abstract. The antiprotonic helium,pe He2+ (= pHe+), is a peculiar metastable atom, interfacing between matter and antimatter. A series of metastable states axe composed of the He nucleus, one electron in the ground Is configuration and one antiproton orbiting with large quantum numbers (n, l), where n l e 38. They possess... 

Fig. 2. Atomic and molecular views of Antiprotonic Helium. The large (n, l) states in the atomic yrast region in the atomic model axe also assigned as the molecular states of corresponding rotational and vibrational quantum numbers (J,v) = (l,n — l — 1) in the one-dimensional potential for each J. The radiative transitions with Av = 0, as shown by arrows, are favoured because of the maximum overlapping of the radial densities. In this sense, the atomcule system has a dual character by itself...

The SAPT potential for the He-C02 complex was also used in the calculations of the rovibrational spectra of the He -CC clusters 366. High resolution experimental data were also reported in this paper. Comparison of the theoretical and experimental effective rotational constants B and other spectroscopic characteristics as functions of the cluster size N is shown on Figure 1-9. Again, the agreement between the theory and experiment is impressive showing that theory can describe with trust spectroscopic characteristics of small clusters He -CO This especially true for the effective rotational constant and the frequency shift of the C02 vibration due to the solvation by the helium atoms. One may note in passing that the clusters HeA,-C02 with the number of helium atoms N around 20 do not exhibit all the properties of the C02 molecule in the first solvation shell of the (quantum) liquid helium at very low temperatures. 

With lithium (Z = 3) there are the two electrons in a spherical cloud (as with helium) plus a third electron. In most cases, in considering the electronic configurations of the element, the last electron is the only one that need be considered, all the remaining having been present in the preceding atom in the periodic table (there are a few important exceptions, however). For the third lithium electron there are no more possible combinations of quantum numbers where w = 1 since neither l or m can exceed zero if n is only one. The last (outermost) electron of lithium has the quantum numbers ... 

A necessary condition to be used is the Pauli exclusion principle, which states that no two electrons in the same atom can have the same set of four quantum numbers. It should also be recognized that lower n values represent states of lower energy. For hydrogen, the four quantum numbers to describe the single electron can be written as n = 1, l = 0, mt = 0, ms = +1/2. For convenience, the positive values of mt and ms are used before the negative values. For the two electrons in a helium atom, the quantum numbers are as follows ... 

Because an atomic energy level can be denoted by the n value followed by a letter (s, p, d, or /to denote 1 = 0, 1, 2, or 3, respectively), the ground state for hydrogen is Is1, whereas that for helium is Is2. The two sets of quantum numbers written previously complete the first shell for which n = 1, and no other sets of quantum numbers are possible that have n = 1. 

The Pauli exclusion principle states that no two electrons in an atom can have the same set of quantum numbers. Each electron exists in a different quantum state. Consequently, none of the electrons in an atom can have the same energy. The Is orbital has the following set of allowable numbers n= 1, t=0, m =0, m=+1/2 or -1/2. All of these numbers can have only one value except for spin, which has two possible states. Thus, the exclusion principle restricts the Is orbital to two electrons with opposite spins. A third electron in the Is orbital would have to have a set of quantum numbers identical to that of one of the electrons already in the orbital. So, the third electron needed for lithium must go into the next higher energy shell, which is a 2s orbital. The question about the Bohr atom that had so vexed scientists—why two electrons completely fill the lowest energy shell in helium—was now answered. There are only two electrons in the lowest energy shell because the quantum numbers derived from Schrodinger s equation and Paulis principle mandate it. 

A helium atom has two electrons, so we need two sets of quantum numbers. To represent the atom in its lowest energy state, we want each electron to have the lowest energy possible. If we let the first electron have the value of 1 for its principal quantum number, its set of quantum numbers will be the same as one of those given previously for the one electron of hydrogen. The other electron of helium can then have the other set of quantum numbers. 

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