## Quantized States of Atoms

The application of Schrödinger's equation to the case of an atom allows one to obtain permissible energy levels of electrons moving around the nucleus of the atom. These are the various quantized electronic energy levels of an atom. For a specific energy level that the electron occupies (in other words, an electron possessing a specific permissible energy value), the probability of finding the electron in the space around the nucleus is described by a wave function that defines an orbital. An orbital can be visualized as the region of space where the probability of finding an electron is high. The Schrödinger equation for a hydrogen atom can be solved mathematically to arrive at an exact solution. However, such a mathematical solution is not possible for a many-electron atom, due to the presence of repulsion between two electrons simultaneously moving around the nucleus. Hence, approximations such as that of self-consistent field theory are introduced. These mathematical descriptions, however, are outside the scope of this book. For such details, refer to books by Atkins and dePaula (2002) or Levine (2000). Here we only provide a flow sheet qualitatively listing the various steps to obtain the quantized energy

Figure 2.8. Shapes of the 5 and p atomic orbitals, pz not shown.

levels and the orbitals for an atom using the Schrödinger equation. These steps are listed in Table 2.5. The orbitals with modified energies, as shown in Figure 2.8, are used to derive electronic distribution (electronic configuration) in many-electron atoms using the following guiding principles:

• Aufbau (a German word meaning "building up") principle, which says the electrons fill in the orbitals of successively increasing energy, starting with the lowest-energy orbital.

• Pauli's principle, which says that each orbital can accommodate a maximum of two electrons, provided that their spins are of opposite signs (i.e., they are paired).

• Hund's rule, which says that if more than one orbital has the same energy (this is called degeneracy, as it exists for the three types of p orbitals), electrons are to be filled singly in each orbital before pairing them up.

Some examples of electronic configurations are He 1s2 and Li 1s22si. Here the superscript at the top of an orbital designation represents the number of electrons in the orbital. A single electron in any of these orbitals is characterized by four quantum numbers: n, l, and mi—derived from the orbital it is in— and ms from its spin orientation (+1/2 for up and -1/2 for down spin, in its simplest description). The quantum number l represents its orbital angular momentum, with ml defining the direction of the angular momentum.

For many electrons, the correlation effects (interactions between electrons) produce overall angular momentum quantum numbers L and ML. Similarly, the coupling of their spins produces an overall quantum number S. Another important manifestation is the spin-orbit coupling, which can be viewed as resulting from the magnetic interaction between the magnetic moment due to the spin of the electron and the magnetic field produced by the electron's

TABLE 2.5. Schematics of Quantum Mechanical Approach for an Atom

Hydrogen atom Motions of one electron and one nucleus

Transform to nuclear-based coordinate

Schrodinger equation describing relative motions of the electron

(i) Define potential energy term, V, as electrostatic attraction between the electron and the nucleus

(ii) Apply the boundary conditions y (r) = 0 at distance r = • from the nucleus

Energy states of the hydrogen atom defined by orbitals 1s; 2s, 2p; 3s, 3p, 3d ... each characterized by three quantum numbers n, 1, mi

(i) Introduce more electrons

(ii) Change the nuclear charge

Many-electron atoms

(i) Use the same orbitals as the starting point

(ii) Treat the additional potential energy of electron-electron repulsion by self-consistent field theory (SCF), which assumes any electron interacting only with the space-averaged charge distribution due to all other electrons

1s; 2s, 2p; 3s, 3p, 3d ... orbitals with modified energies orbital motion around the nucleus (just like a current in a coil produces a magnetic field). This spin-orbit coupling is dependent on the atomic number (the charge on the nucleus). Therefore, heavier atoms exhibit strong spin-orbit coupling, often called the heavy atom effect, which leads to a strong mixing of the spin and the orbital properties. The mixing leads to characterization of overall angular momentum by another quantum number J. The spin-orbit coupling plays an important role in spectroscopy, as discussed in Chapter 4.

The electron orbital and spin correlation effects and the spin-orbit coupling produce shifting and splitting of the atomic energy levels which are characterized by a term symbol (Atkins and dePaula, 2002; Levine, 2000). A term symbol is given as 2S+1[L}J. Here 2S + 1, with S as the overall spin quantum number, represents the spin multiplicity. When S = 0, 2S + 1 = 1, it represents a singlet state. The S = 1,2S + 1 = 3 case represents a triplet state. {L} is the appropriate letter form representing the L value, which represents the total orbital angular momentum; for example, 0, 1, 2, 3 values are represented by S, P, D, F. Thus, a term symbol 1S0 for the atom represents an energy level with spin S = 0, orbital quantum number L = 0, and the overall angular momentum quantum number J = 0. These term symbols are often used to designate energy states of an atom and the spectroscopic transitions between them. They are used to designate the transition between two quantized states of an atom or an ion in the production of a laser action, as described in Chapter 5.

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