Helium Is orbital

Figure 5.8 Application of the general spreadsheet, fig5-7.xls, to reproduce the Huzi-naga results for the sto-4g) representation of the helium Is orbital and equation 5.2. Note, the setting of the Slater exponent to 1.0.

Another example of the difficulty is offered in figure B3.1.5. Flere we display on the ordinate, for helium s (Is ) state, the probability of finding an electron whose distance from the Fie nucleus is 0.13 A (tlie peak of the Is orbital s density) and whose angular coordinate relative to that of the other electron is plotted on the abscissa. The Fie nucleus is at the origin and the second electron also has a radial coordinate of 0.13 A. As the relative angular coordinate varies away from 0°, the electrons move apart near 0°, the electrons approach one another. Since both electrons have opposite spin in this state, their mutual Coulomb repulsion alone acts to keep them apart. 

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. 

A hydrogen atom (Z = 1) has one electron a helium atom (Z = 2) has two The single electron of hydrogen occupies a Is orbital as do the two electrons of helium We write their electron configurations as... 

The splitting of triplet terms of helium is unusual in two respects. First, multiplets may be inverted and, second, the splittings of the multiplet components do not obey the splitting rule of Equation (7.20). For this reason we shall discuss fine stmcture due to spin-orbit coupling in the context of the alkaline earth atomic spectra where multiplets are usually normal and... 

With these two assumptions, we can propose the electronic arrangement of lowest energy for each atom. We do so by mentally placing electrons successively in the empty orbitals of lowest energy. The electron orbital of lowest energy is the Is orbital. The single electron of the hydrogen atom can occupy this orbital. In the helium... 

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. ... 

Each helium atom does have, of course, vacant 2s and 2p orbitals which extend farther out than the filled Is orbital. The electrons of the second helium atom can "overlap with these vacant orbitals. Since this overlap is at great distance, the resulting attractions are extremely small. This type of interaction presumably accounts for the attractions that cause helium to condense at very low temperatures. 

In the ground state of helium, according to this model, the two electrons are in the Is orbital with opposing spins. The ground-state wave function is... 

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) ... 

Electronic excitation can promote one of the electrons in the Is orbital to an orbital of higher energy so that there is one electron in the Is orbital and one electron in a higher-energy orbital. Such excitation results in the formation of an excited-state helium atom. 

Since low energy usually means stability, it is reasonable that the most stable energy state (ground state) for any atom is one in which its electrons are in the lowest possible energy level. As you know, this is the Is orbital (n = 1). However, the electrons of most atoms are not packed into this orhital. In fact, experimental evidence shows that only two atoms have all their electrons in the Is orhital hydrogen and helium. To explain how and why this is the case, you must consider another property of the electron. 

With oxygen, as with helium s Is orbital and beryllium s 2s orbital, the last-added (eighth) electron is paired with a 2p electron of opposite spin. In other words, the Pauli exclusion principle applies. 

The possible states of electrons are called orbitals. These are indicated by what is known as the principal quantum number and by a letter—s, p, or d. The orbitals are filled one by one as the number of electrons increases. Each orbital can hold a maximum of two electrons, which must have oppositely directed spins. Fig. A shows the distribution of the electrons among the orbitals for each of the elements. For example, the six electrons of carbon (B1) occupy the Is orbital, the 2s orbital, and two 2p orbitals. A filled Is orbital has the same electron configuration as the noble gas helium (He). This region of the electron shell of carbon is therefore abbreviated as He in Fig. A. Below this, the numbers of electrons in each of the other filled orbitals (2s and 2p in the case of carbon) are shown on the right margin. For example, the electron shell of chlorine (B2) consists of that of neon (Ne) and seven additional electrons in 3s and 3p orbitals. In iron (B3), a transition metal of the first series, electrons occupy the 4s orbital even though the 3d orbitals are still partly empty. Many reactions of the transition metals involve empty d orbitals—e.g., redox reactions or the formation of complexes with bases. 

Particularly stable electron arrangements arise when the outermost shell is fully occupied with eight electrons (the octet rule ). This applies, for example, to the noble gases, as well as to ions such as Cl (3s 3p ) and Na"" (2s 2p ). It is only in the cases of hydrogen and helium that two electrons are already suf dent to fill the outermost Is orbital. 

If we imagine the nuclei to be forced together to = 0, the wave function Is A + Iss will approach, as a limit, a charge distribution around the united atom that has neither radial nor angular nodal planes. This limiting charge distribution has the same symmetry as the Is orbital on the united atom, Helium. On the other hand, the combination Isa Iss has a nodal plane perpendicular to the molecular axis at all intemuclear separations. Hence its limit in the united atom has the symmetry properties of a 2p orbital. A simple correlation diagram for this case is ... 

The Pauli exclusion principle states that no more than two eleetrons ean occupy each orbital, and if two electrons are present, their spins must be paired. For example, the two eleetrons of a helium atom must oeeupy the Is orbital in opposite spins. 

The layout of the periodic table (Fig. 2.5) reflects the shell structure of the electrons. Hydrogen and helium have only -shell electrons. The elements in row two have and L-shell electrons, with the Is orbitals always filled and the 2s and 2p orbitals filled in succession. Those in row three have and L-shell electrons, with Is, 2s, and 2p orbitals filled, and the 3 s and 3p orbitals are filled in succession. Elements in the fourth row have K, L, and M-shell electrons, with the Is, 2s, 3s, 2p, and 3p orbitals completely filled. After the 4s orbitals are filled, the 3d orbitals are filled, giving the transition metals. Then come the 4p orbitals. Row five is filled in an analogous fashion. In row six, the lanthanides, which fit between lanthanum and hafnium, reflect the appearance of the N-shell electrons, which fill the f orbitals. Row seven, which contains the actinides, also has K, L, M, and N-shell electrons. 

Let us first discuss the helium atom. The most stable orbital in the helium atom is the Is orbital, with n 1, l = 0, = 0. There are... 

The Helium Molecule-Ion.—The simplest molecule in which the three-electron bond can occur is the helium molecule-ion, HeJ, consisting of two nuclei, each with one stable Is orbital, and three electrons. The theoretical treatment7 of this system has shown that the bond is strong, with bond energy about 55 kcal/mole and with equilibrium internuclear distance about 1.09 A. The experimental values for these qualities, determined from spectroscopic data for excited states of the helium molecule, are a bout 58 kcal/mole and 1.080 A, respectively, which agree well with the theoretical values. It is seen that the bond energy in He He4 is about the same as that in H H+, and a little more than half as great as that of the electron-pair bpnd in H H. 

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... 

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