Penetration and shielding

Key points: The ground-state electron configuration is a specification of the orbital occupation of an atom in its lowest energy state. The exclusion principle forbids more than two electrons to occupy a single orbital. The nuclear charge experienced by an electron is reduced by shielding by other electrons. Trends in effective nuclear charge can be used to rationalize the trends in many properties. As a result of the combined effects of penetration and shielding, the order of energy levels in a shell of a many electron atom is s<p<d<f.

It is quite easy to account for the electronic structure of the helium atom in its ground state, its state of lowest energy. According to the orbital approximation, we suppose that both electrons occupy an atomic orbital that has the same spherical shape as a hydrogenic 1s orbital. However, the orbital will be more compact because, as the nuclear charge of helium is greater than that of hydrogen, the electrons are drawn in towards the nucleus more closely than is the one electron of an H atom. The ground-state configuration of an atom is a statement of the orbitals its electrons occupy in the ground state. For helium, with two electrons in the 1s orbital, the ground-state configuration is denoted 1s² (read ‘one s two’).

As soon as we come to the next atom in the periodic table, lithium (Z=3), we en-counter several major new features. The configuration 1s3 is forbidden by a fundamental feature of nature known as the Pauli exclusion principle:

No more than two electrons may occupy a single orbital and, if two do occupy a single orbital, then their spins must be paired. By ‘paired’ we mean that one electron spin must be ↑ and the other ↓; the pair is denoted ↑↓. Another way of expressing the principle is to note that, because an electron in an atom is described by four variable quantum numbers, n, l, ml , and ms, no two electrons can have the same four quantum numbers. The Pauli principle was introduced originally to account for the absence of certain transitions in the spectrum of atomic helium. Because the configuration 1s3 is forbidden by the Pauli exclusion principle, the third electron must occupy an orbital of the next higher shell, the shell with n=2. The question that now arises is whether the third electron occupies a 2s orbital or one of the three 2p orbitals. To answer this question, we need to examine the energies of the two subshells and the effect of the other electrons in the atom. Although 2s and 2p orbitals have the same energy in a hydrogenic atom, spectroscopic data and calculations show that this is not the case in a many-electron atom.

In the orbital approximation we treat the repulsion between electrons in an approximate manner by supposing that the electronic charge is distributed spherically around the nucleus. Then each electron moves in the attractive field of the nucleus and experiences an average repulsive charge from the other electrons. According to classical electrostatics, the field that arises from a spherical distribution of charge is equivalent to the field generated by a single point charge at the centre of the distribution (Fig. 1.17). This negative charge reduces the actual charge of the nucleus, Ze, to Z_eff e, where Zeff (more precisely, Zeff e) is called the effective nuclear charge. This effective nuclear by the charge other electrons is called shielding

The effective nuclear charge is sometimes expressed in terms of the true nuclear charge and an empirical shielding constant, σ,

by writing Z_eff=Z- σ. The shielding constant can be determined by fitting hydrogenic orbitals to those computed numerically.

The closer to the nucleus that an electron can approach, the closer is the value of Zeff to Z itself because the electron is repelled less by the other electrons present in the atom. With this point in mind, consider a 2s electron in the Li atom. There is a nonzero probability that the 2s electron can be found inside the 1s shell and experience the full nuclear charge (Fig. 1.18). The presence of an electron inside shells of other electrons is called penetration. A 2p electron does not penetrate so effectively through the core, the filled inner shells of electrons, because its wavefunction goes to zero at the nucleus. As a consequence, it is more fully shielded from the nucleus by the core electrons. We can conclude that a 2s electron has a lower energy (is bound more tightly) than a 2p electron, and therefore that the 2s orbital will be occupied before the 2p orbitals, giving a ground-state electron configuration for Li of 1s²2s¹. This configuration is commonly denoted [He]2s1, where [He] denotes the atom’s helium-like 1s² core.

Table 1.2 Effective nuclear charge, Z_eff

The pattern of energies in lithium, with 2s lower than 2p, and in general ns lower than np, is a general feature of many-electron atoms. This pattern can be seen from Table 1.2, which gives the values of Zeff for a number of valence-shell atomic orbitals in the ground state electron configuration of atoms. The typical trend in effective nuclear charge is an increase across a period, for in most cases the increase in nuclear charge in successive groups is not cancelled by the additional electron. The values in the table also confirm that an s electron in the outermost shell of the atom is generally less shielded than a p electron of that shell. So, for example, Z_eff= 5.13 for a 2s electron in an F atom, whereas for a 2p 5.10, a lower value. Similarly, the effective nuclear charge is larger for an electron in an np orbital than for one in an nd orbital. As a result of penetration and shielding, the order of energies in many-electron atoms is typically ns np nd nf because, in a given shell, s orbitals are the most penetrating and f orbitals are the least penetrating. The overall effect of penetration and shielding is depicted in the energy-level diagram for a neutral atom shown in Fig. 1.19. Figure 1.20 summarizes the energies of the orbitals through the periodic table. The effects are quite subtle, and the order of the orbitals depends strongly on the numbers of

electrons present in the atom and may change on ionization. For example, the effects of penetration are very pronounced for 4s electrons in K and Ca, and in these atoms the 4s orbitals lie lower in energy than the 3d orbitals. However, from Sc through Zn, the 3d orbitals in the neutral atoms lie close to but lower than the 4s orbitals. In atoms from Ga (Z=31) onwards, the 3d orbitals lie well below the 4s orbital in energy, and the outer most electrons are unambiguously those of the 4s and 4p subshells.

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