The orbital concept and the Pauli exclusion principle allow us to understand the periodic table of the elements. An orbital is a one-electron spatial wave function. We have used orbitals to obtain approximate wave functions for many-electron atoms, writing the wave function as a Slater determinant of one-electron spin-orbitals. In the crudest approximation, we neglect all interelectronic repulsions and obtain hydrogenlike orbitals. The best possible orbitals are the Hartree-Fock SCF functions. We build up the periodic table by feeding electrons into these orbitals, each of which can hold a pair of electrons with opposite spin.
Latter [R. Latter, Phys. Rev., 99, 510 (1955)] calculated approximate orbital energies for the atoms of the periodic table by replacing the complicated expression for the Hartree-Fock potential energy in the Hartree-Fock radial equations by a much simpler function obtained from the Thomas-Fermi-Dirac method, which uses ideas of statistical mechanics to get approximations to the effective potential-energy function for an electron and the electron-density function in an atom (Bethe and Jackiw, Chapter 5). Figure 11.2 shows Latter's resulting orbital energies for neutral ground-state atoms. These AO energies are in pretty good agreement with both Hartree-Fock and experimentally found orbital energies (see J. C. Slater, Quantum Theory of Matter, 2nd ed., McGraw-Hill, 1968, pp. 146, 147, 325, 326).
Orbital energies change with changing atomic number $Z$. As $Z$ increases, the orbital energies decrease because of the increased attraction between the nucleus and the electrons. This decrease is most rapid for the inner orbitals, which are less well-shielded from the nucleus.
For $Z>1$, orbitals with the same value of $n$ but different $l$ have different energies. For example, for the $n=3$ orbital energies, we have $\varepsilon{3 s}<\varepsilon{3 p}<\varepsilon_{3 d}$ for $Z>1$. The splitting of these levels, which are degenerate in the hydrogen atom, arises from the interelectronic repulsions. (Recall the perturbation treatment of helium in Section 9.7.) In the limit $Z \rightarrow \infty$, orbitals with the same value of $n$ are again degenerate, because the interelectronic repulsions become insignificant in comparison with the electron-nucleus attractions.
The relative positions of certain orbitals change with changing $Z$. Thus in hydrogen the $3 d$ orbital lies below the $4 s$ orbital, but for $Z$ in the range from 7 through 20 the $4 s$ is below the $3 d$. For large values of $Z$, the $3 d$ is again lower. At $Z=19$, the $4 s$ is lower; hence ${ }_{19} \mathrm{~K}$ has the ground-state configuration $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s$. Recall that $s$ orbitals are more
FIGURE 11.2 Atomicorbital energies as a function of atomic number for neutral atoms, as calculated by Latter. [Reprinted figure with permission from R. Latter, redrawn by M. Kasha, Physical Review Series II, 90, 510, 1955. Copyright 1955 by the American Physical Society.] Note the logarithmic scales. $E_{H}$ is the ground-state hydro-gen-atom energy, -13.6 eV .
penetrating than $p$ or $d$ orbitals; this allows the $4 s$ orbital to lie below the $3 d$ orbital for some values of $Z$. Note the sudden drop in the $3 d$ energy, which starts at $Z=21$, when filling of the $3 d$ orbital begins. The electrons of the $3 d$ orbital do not shield each other very well; hence the sudden drop in $3 d$ energy. Similar drops occur for other orbitals.
To help explain the observed electron configurations of the transition elements and their ions, Vanquickenborne and co-workers calculated Hartree-Fock $3 d$ and $4 s$ orbital energies for atoms and ions for $Z=1$ to $Z=29$ [L. G. Vanquickenborne et al., Inorg. Chem., 28, 1805 (1989); J. Chem. Educ., 71, 469 (1994)].
One complication is that a given electron configuration may give rise to many states. [For example, recall the several states of the $\mathrm{He} 1 s 2 s$ and $1 s 2 p$ configurations (Sections 9.7 and 10.4).] To avoid this complication, Vanquickenborne and co-workers calculated Hartree-Fock orbitals and orbital energies by minimizing the average energy $E_{\text {av }}$ of the
states of a given electron configuration, instead of by minimizing the energy of each individual state of the configuration. The average orbitals obtained differ only slightly from the true Hartree-Fock orbitals for a given state of the configuration.
For each of the atoms ${ }{1} \mathrm{H}$ to ${ }{19} \mathrm{~K}$, Vanquickenborne and co-workers calculated the $3 d$ average orbital energy $\varepsilon{3 d}$ for the electron configuration in which one electron is removed from the highest-occupied orbital of the ground-state electron configuration and put in the $3 d$ orbital; they calculated $\varepsilon{4 s}$ for these atoms in a similar manner. In agreement with Fig. 11.2, they found $\varepsilon{3 d}<\varepsilon{4 s}$ for atomic numbers $Z<6$ and $\varepsilon{4 s}<\varepsilon{3 d}$ for $Z=7$ to 19 for neutral atoms.
For discussion of the transition elements with $Z$ from 21 to 29, Fig. 11.2 is inadequate because it gives only a single value for $\varepsilon{3 d}$ for each element, whereas $\varepsilon{3 d}$ (and $\varepsilon{4 s}$ ) for a given atom depend on which orbitals are occupied. This is because the electric field experienced by an electron depends on which orbitals are occupied. Vanquickenborne and co-workers calculated $\varepsilon{3 d}$ and $\varepsilon{4 s}$ for each of the valence-electron configurations $3 d^{n} 4 s^{2}, 3 d^{n+1} 4 s^{1}$, and $3 d^{n+2} 4 s^{0}$ and found $\varepsilon{3 d}<\varepsilon{4 s}$ in each of these configurations of the neutral atoms and the +1 and +2 ions of the transition elements ${ }{21} \mathrm{Sc}$ through ${ }_{29} \mathrm{Cu}$ (which is the order shown in Fig. 11.2).
Since $3 d$ lies below $4 s$ for $Z$ above 20, one might wonder why the ground-state configuration of, say, ${ }{21} \mathrm{Sc}$ is $3 d^{1} 4 s^{2}$, rather than $3 d^{3}$. Although $\varepsilon{3 d}<\varepsilon{4 s}$ for each of these configurations, this does not mean that the $3 d^{3}$ configuration has the lower sum of orbital energies. When an electron is moved from $4 s$ into $3 d, \varepsilon{4 s}$ and $\varepsilon{3 d}$ are increased. An orbital energy is found by solving a one-electron Hartree-Fock equation that contains potential-energy terms for the average repulsions between the electron in orbital $i$ and the other electrons in the atom, so $\varepsilon{i}$ depends on the values of these repulsions and hence on which orbitals are occupied. For the first series of transition elements, the $4 s$ orbital is much larger than the $3 d$ orbital. For example, Vanquickenborne and co-workers found the following $\langle r\rangle$ values in Sc: $\langle r\rangle{3 d}=0.89 \AA$ and $\langle r\rangle{4 s}=2.09 \AA$ for $3 d^{1} 4 s^{2} ;\langle r\rangle{3 d}=1.11 \AA$ and $\langle r\rangle{4 s}=2.29 \AA$ for $3 d^{2} 4 s^{1}$. Because of this size difference, repulsions involving $4 s$ electrons are substantially less than repulsions involving $3 d$ electrons, and we have $(4 s, 4 s)<(4 s, 3 d)<(3 d, 3 d)$, where $(4 s, 3 d)$ denotes the average repulsion between an electron distributed over the $3 d$ orbitals and an electron in a $4 s$ orbital. (These repulsions are expressed in terms of Coulomb and exchange integrals.) When an electron is moved from $4 s$ into $3 d$, the increase in interelectronic repulsion that is a consequence of the preceding inequalities raises the orbital energies $\varepsilon{3 d}$ and $\varepsilon{4 s}$. For example, for ${ }{21} \mathrm{Sc}$, the $3 d^{1} 4 s^{2}$ configuration has $\varepsilon{3 d}=-9.35 \mathrm{eV}$ and $\varepsilon{4 s}=-5.72 \mathrm{eV}$, whereas the $3 d^{2} 4 s^{1}$ configuration has $\varepsilon{3 d}=-5.23 \mathrm{eV}$ and $\varepsilon{4 s}=-5.06 \mathrm{eV}$. For the $3 d^{1} 4 s^{2}$ configuration, the sum of valence-electron orbital energies is $-9.35 \mathrm{eV}+2(-5.72 \mathrm{eV})=-20.79 \mathrm{eV}$, whereas for the $3 d^{2} 4 s^{1}$ configuration, this sum is $2(-5.23 \mathrm{eV})-5.06 \mathrm{eV}=-15.52 \mathrm{eV}$. Thus, despite the fact that $\varepsilon{3 d}<\varepsilon_{4 s}$ for each configuration, transfer of an electron from $4 s$ to $3 d$ raises the sum of valence-electron orbital energies in Sc. [As we saw in Eq. (11.10) for the Hartree method and will see in Section 14.3 for the Hartree-Fock method, the Hartree and Hartree-Fock expressions for the energy of an atom contain terms in addition to the sum of orbital energies, so we must look at more than the sum of orbital energies to see which configuration is most stable.]
For the +2 ions of the transition metals, the reduction in screening makes the valence $3 d$ and $4 s$ electrons feel a larger effective nuclear charge $\mathrm{Z}{\text {eff }}$ than in the neutral atoms. By analogy to the H-atom equation $E=-\left(Z^{2} / n^{2}\right)\left(e^{2} / 8 \pi \varepsilon{0} a\right)$ [Eq. (6.94)], the orbital energies $\varepsilon{3 d}$ and $\varepsilon{4 s}$ are each roughly proportional to $Z{\text {eff }}^{2}$ and the energy difference $\varepsilon{4 s}-\varepsilon{3 d}$ is roughly proportional to $Z{\text {eff }}^{2}$. The difference $\varepsilon{4 s}-\varepsilon{3 d}$ is thus much larger in the transitionmetal ions than in the neutral atoms; the increase in valence-electron repulsion is no longer
enough to make the $4 s$ to $3 d$ transfer energetically unfavorable; and the +2 ions have ground-state configurations with no $4 s$ electrons.
For further discussion of electron configurations, see W. H. E. Schwarz, J. Chem. Educ., 87, 444 (2010); Schwarz and R. L. Rich, ibid., 87, 435.
Figure 11.2 shows that the separation between $n s$ and $n p$ orbitals is much less than that between $n p$ and $n d$ orbitals, giving the familiar $n s^{2} n p^{6}$ stable octet.
The orbital concept is the basis for most qualitative discussions of the chemistry of atoms and molecules. The use of orbitals, however, is an approximation. To reach the true wave function, we must go beyond a Slater determinant of spin-orbitals.