M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 39
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The f orbitals will not be discussed further, since their participation in chemical bonding is limited. The representations depicted inFigure 6-4 are used commonly for illustrations because they describeaccurately the symmetry properties of the wave function. In orderto give the total wave function, however, they must be multipliedby an appropriate radial function.
Another representation, shown inFigure 6-5, is a three-dimensional computer drawing of the total function including both the radial and the angular functions. These are stillnot real “pictures” of the orbitals, since they represent a cross sectionof the wave function in one plane only. The vertical scale gives thevalue of ⌿ for each point in the xy plane. These diagrams show howthe sign and magnitude of ⌿ vary in the xy plane, and they also help usvisualize the electronic wave function as a wave.
On the other hand,they do not illustrate its symmetry properties so well as do the simplediagrams in Figure 6-4.As mentioned before, the symmetry properties of the one-electronwave function are shown by the simple plot of the angular wavefunction. But, what are the symmetry properties of an orbital and howcan they be described? We can examine the behavior of an orbitalunder the different symmetry operations of a point group.
This will beillustrated below via the inversion operation.The s and d orbitals are transformed into themselves as the inversion operation is applied to them (Figure 6-6). Both the magnitudeand the “sign” of the wave function will remain the same under theinversion operation. These orbitals are said to be symmetric withrespect to inversion. The effect of the inversion operation on the porbitals is demonstrated in Figure 6-7. Whereas their magnitude does6.1. One-Electron Wave Function247(a)(b)Figure 6-4.
Shapes of one-electron orbitals. They are representations of the angularwave function, A(⌰, ⌽): (a) s, p, and d orbitals; (b) f orbitals.not change, their “sign” changes upon inversion. These orbitals aresaid to be antisymmetric with respect to inversion. In the charactertables, this is indicated by +1 for symmetric and –1 for antisymmetricbehavior under each symmetry operation. As mentioned in Chapter 4,the atomic orbitals always belong to the same irreducible representations of the given point group as do their subscripts (x, y, z, xy, x2 – y2 ,etc.).2486 Electronic Structure of Atoms and MoleculesFigure 6-5. Three-dimensional computer drawings of the total wave function, ⌿,of the iodine atom, calculated with a 3-21G basis set [19].
They show the values of⌿ in a cross section. Courtesy of István Kolossváry. (a) 1s orbital; (b) 2px orbital;(c) 3dxy orbital.Figure 6-6. The effect of inversion on the s and d orbitals. They are symmetric tothis operation.6.2. Many-Electron Atoms249Figure 6-7. The effect of inversion on the p orbitals. They are antisymmetricto inversion, as the inversion operation changes their sign.6.2. Many-Electron AtomsThere is interaction among all the electrons in a many-electronatom.
Thus, the wave function for even one electron in a manyelectron system will, in principle, be different from the wave function for the one electron in the hydrogen atom. Since the electronsare mutually indistinguishable, it is not possible to describe rigorously the properties of a single electron in such a system. There isno exact solution to this problem, and approximate methods must beadopted.In the most commonly utilized approximation, the many-electronwave functions are written in terms of products of one-electron wavefunctions similar to the solutions obtained for the hydrogen atom.These one-electron functions used to construct the many-electronwave function are called atomic orbitals.
They are also called“hydrogen-like” orbitals since they are one-electron orbitals and alsobecause their shape is similar to that of the hydrogen atom orbitals.2506 Electronic Structure of Atoms and MoleculesCoulson referred to the atomic orbitals as “personal wave functions”to emphasize that each electron is allocated to an individual orbital inthis model [20].At this point we can, again, appreciate the possibility of separatingthe total wave function into a radial and an angular wave function.The angular wave function does not depend on n and r, so it will bethe same for every atom.
This is why the “shapes”of atomic orbitalsare always the same. Hence, symmetry operations can be applied tothe orbitals of all atoms in the same way. The differences occur in theradial part of the wave function; the radial contribution depends onboth n and r and it determines the energy of the orbital, which is, ofcourse, different for different atoms.While the energy of a one-electron orbital depends only on n, in amany-electron atom the energy of the orbital is determined by bothn and l. Thus, an electron in a 2p orbital has higher energy than anelectron in a 2s orbital.
The order of orbital energies in many-electronatoms is generally as follows:1s < 2s < 2 p < 3s < 3 p < 4 s ≈ 3d < 4 p < 5s < 4d < . . .There are some cases, however, when the order is changed somewhat. For example, the 3d orbital sometimes lies below the 4s orbital.A diagram which illustrates the order of orbital energies is shown inFigure 6-8.In addition to the three quantum numbers used to describe theone-electron wave function, the electron has also a fourth, the spinquantum number, ms . It is related to the intrinsic angular momentumof the electron, called spin. This quantum number may assume thevalues of +1/2 or –1/2. Usually the sign of ms is represented by arrows,(↑ and ↓), or by the Greek letters ␣ and .
Thus, the wave function ofan orbital is expressed as⌿e = R(r ) · A(⌰, ⌽) · S(s)(6-4)rather than as in Eq. (6-3). However, the introduction of spin does notalter any of the properties discussed previously that relate to the shapeand symmetry of the orbitals. The reason is that the spin function isindependent of the spatial coordinates.6.2. Many-Electron Atoms251Figure 6-8. The sequence of orbital energies.An important postulate in connection with the spin of the electron iscalled the Pauli principle. It states that if a system consists of identicalparticles with half-integral spins, then all acceptable wave functionsmust be antisymmetric with respect to the exchange of the coordinatesof any two particles.
In our case, the particles are electrons, and thePauli principle is formulated accordingly: No two electrons in an atomcan have the same set of values for all four quantum numbers.The electronic configuration of an atom tells us how many electronsthe atom has in its subshells. A subshell is a complete set of orbitalsthat have the same n and l. The building up of electronic configurations is governed by the Pauli principle and by Hund’s first rule,according to which, for a given electronic configuration, the state withthe greatest number of unpaired spins has the lowest energy.There is a marked periodicity in the electronic configuration of theelements and this is the underlying idea of the periodic table (seeChapter 1).
As the chemical properties of the atoms are determined bytheir electron configuration, the atoms with similar electron configurations will have similar chemical properties.2526 Electronic Structure of Atoms and Molecules6.3.
Molecules6.3.1. Constructing Molecular OrbitalsIn the discussion of the electronic structure of atoms, the Schrödingerequation could be reduced to one involving only the electrons. Thiswas achieved by separating the electronic energy of the atom from thenuclear kinetic energy, which is essentially determined by the translational motion of the atom.Such a separation is exact for atoms. For molecules, only the translational motion of the whole system can be rigorously separated, whiletheir kinetic energy includes all kinds of motion, vibration and rotationas well as translation. First, as in the case of atoms, the translationalmotion of the molecule is isolated. Then a two-step approximationcan be introduced.
The first is the separation of the rotation of themolecule as a whole, and thus the remaining equation describes onlythe internal motion of the system. The second step is the application ofthe Born–Oppenheimer approximation, in order to separate the electronic and the nuclear motion. Since the relatively heavy nuclei movemuch more slowly than the electrons, the latter can be assumed tomove about a fixed nuclear arrangement. Accordingly, not only thetranslation and rotation of the whole molecular system but also theinternal motion of the nuclei is ignored.
The molecular wave function is written as a product of the nuclear and electronic wave functions. The electronic wave function depends on the positions of bothnuclei and electrons but it is solved for the motion of the electronsonly.As was emphasized before (cf. Chapter 3), a molecule is not simplya collection of its constituting atoms. Rather, it is a system of atomicnuclei and a common electron distribution.
Nevertheless, in describingthe electronic structure of a molecule, the most convenient way is toapproximate the molecular electron distribution by the sum of atomicelectron distributions. This approach is called the linear combination of atomic orbitals (LCAO) method. The orbitals produced by theLCAO procedure are called molecular orbitals (MOs). An importantcommon property of the atomic and molecular orbitals is that bothare one-electron wave functions. Combining a certain number ofone-electron atomic orbitals yields the same number of one-electronmolecular orbitals. Finally, the total molecular wave function is the6.3.
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