M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 38
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This property establishes the connection between thesymmetry of a molecule and its wave function. The preceding statement follows from Wigner’s theorem, which says that all eigenfunctions of a molecular system belong to one of the symmetry species ofthe group [14].In the expression of the energy of a system the following type ofintegral appears:⌿i Ĥ ⌿ j dτDepending on the problem, ⌿i and ⌿j may be atomic orbitals usedto construct molecular orbitals, or they may represent two different6.1. One-Electron Wave Function241electronic states of the same atom or molecule, etc.
The energy, then,expresses the extent of interaction between the two wave functions ⌿iand ⌿j . As was shown in Chapter 4, an integral will have a nonzerovalue only if the integrand is invariant to the symmetry operations ofthe point group, i.e., it belongs to the totally symmetric irreduciblerepresentation.The above energy integral contains the Ĥ operator, which alwaysbelongs to the totally symmetric irreducible representation. Therefore,the symmetry of the whole integrand depends on the direct product of⌿i and ⌿j . As also was shown in Chapter 4, the direct product ofthe representations of ⌿i and ⌿j belongs to, or contains, the totallysymmetric irreducible representation only if ⌿i and ⌿j belong tothe same irreducible representation.
Consequently, the energy integral will be nonzero only if ⌿i and ⌿j belong to the same irreduciblerepresentation of the molecular point group.6.1. One-Electron Wave FunctionBefore discussing many-electron systems, the hydrogen atom (a oneelectron system) will be described. This is essentially the only atomicsystem for which an exact solution of the wave function is available.The spherical symmetry of the hydrogen atom makes it convenient toexpress the wave function in a polar coordinate system. Such a systemis shown in Figure 6-1 with the proton in the origin.
Ignoring the translational motion of the hydrogen atom, the Schrödinger equation canbe simplifed as follows [15]:Ĥe ⌿ = E⌿e(6-2)where Ĥe depends only on the coordinates of the electron.The electronic wave function can be represented as a product of aradial and an angular component:⌿e = R(r ) · A(⌰, ⌽)(6-3)The radial wave function R(r) depends on two quantum numbers,n and l.
The principal quantum number, n, determines the electron shell. The numbers n = 1, 2, 3, 4, ... correspond to the shellsK, L, M, N, respectively. For the hydrogen atom, n completely deter-2426 Electronic Structure of Atoms and MoleculesFigure 6-1. The relationship between Cartesian coordinates and spherical polarcoordinates, illustrated for the hydrogen atom with the proton at the origin.mines the energy of the shell, which is inversely proportional to n2 .Since this energy is negative, E is smallest for the first (K) shell, andincreases with increasing n.
The azimuthal quantum number, l, is associated with the total angular momentum of the electron and determines the shape of the orbitals. It may have integral values from 0 ton – 1. The s, p, d, f, ... orbitals correspond to the azimuthal quantumnumbers, l = 0, l, 2, 3, ..., respectively.The angular wave function A(⌰, ⌽) depends also on two quantumnumbers, l and ml . The magnetic quantum number, ml , is associatedwith the component of angular momentum along a specific axis in theatom.
Since the hydrogen atom is spherically symmetrical, it is notpossible to define a specific axis until the atom is placed in an externalelectric or magnetic field. This also means that the quantum numberml has no effect on the energy and shape of the wave function of thehydrogen atom in the absence of such an external field. Generally, mlmay have values –l, –l+1, ..., 0, ..., l–1, l, altogether 2l+1 of them, andthe orbitals are subdivided accordingly.Usually we refer to the energy of orbitals while what is reallymeant is the energy of an electron in that orbital. It was mentionedearlier that only the principal quantum number n influences the orbitalenergy in the hydrogen atom.
This means that while 1s and 2s orbitalshave different energies, the 2s and all three 2p orbitals have the same6.1. One-Electron Wave Function243energy, i.e., these four n = 2 orbitals are degenerate in the hydrogenatom.In many-electron atoms the value of l also influences the energy ofthe orbitals; thus, the 2s and 2p orbitals, the 3s, 3p, and 3d orbitals, orthe 4s, 4p, 4d, and 4f orbitals will no longer be degenerate.
However,there are always three p orbitals, five d orbitals, and seven f orbitalsin a shell, and they differ only in the quantum number ml and will bedegenerate. As there are 2l + 1 values of ml for an orbital with quantumnumber l, the p orbitals (l = 1) will always be threefold degenerate, thed orbitals (l = 2) will always be fivefold degenerate, while the f orbitals(l = 3) will always be sevenfold degenerate.Harris and Bertolucci [16] illustrated the relationship betweensymmetry and degeneracy of energy levels with a simple and attractive example. There are three parrallelepipeds in Figure 6-2. Each ofthem has six stable resting positions.
The potential energy of thesepositions depends on the height of the center of the mass above theFigure 6-2. Illustration of the interrelation of symmetry and degeneracy after Harrisand Bertolucci [17]. Used with permission. See text for details.2446 Electronic Structure of Atoms and Moleculessupporting surface. This height, in turn, is determined by the choiceof face on which the body rests. Three different positions are possiblefor the first parallelepiped (1) according to its three different kindsof faces. The potential energy of 1 will be largest when it stands onan ab face, since its center of mass is at the highest possible position. There are only two energetically different positions for 2 sinceits center of mass is at the same height when it rests on face bc oron face ac.
Parallelepiped 3 is indeed a cube, and all possible positions will be energetically equivalent. Looking at the degeneracy ofthe most stable (lowest energy) position, it is twofold degenerate for 1,four-fold degenerate for 2, and six-fold degenerate for the cube. Thus,with increasing symmetry, the degree of degeneracy increases. Theconnection between symmetry and degeneracy is strikingly obvious.The greater the degree of symmetry the smaller will be the number ofdifferent energy levels and the greater will be the degeneracy of theselevels.This correlation between symmetry and degeneracy of energylevels is fundamental to understanding the electronic structure ofatoms and molecules. This relationship is valid not only whenincreasing symmetry renders the energy levels degenerate but alsowhen energy levels are split as molecular symmetry decreases.Let us now return to the wave function description of electronicstructure.
The separation of the wave function into two parts is convenient since these two parts relate to different properties. The radialpart determines the energy of the system and is invariant to symmetryoperations. The square of the radial function is related to probability.If we fix the angular variables, ⌰ and ⌽, they define a direction fromthe nucleus. Then the square of the radial function is proportionalto the probability of finding the electron in a volume element alongthis direction. In order to determine the probability of finding theelectron anywhere in a spherical shell surrounding the nucleus at adistance r from the nucleus, integration over both angular variablesmust be performed.
The result is the radial distribution function.Consider now the angular part of the one-electron wave function. Itsays nothing about the energy of the system but it can be altered bysymmetry operations. Therefore, we shall be dealing with this function in greater detail. The function A(⌰, ⌽) may have different signs(+ and –) in different spatial regions. A change in sign indicates adrastic change in the wave function. These signs might be thought6.1.
One-Electron Wave Function245of as signs of the amplitudes of the wave function; they certainlyhave nothing to do with electric charges. The places where the wavefunction changes sign are called nodes. The number of nodes is n–1,where n is the principal quantum number. Again, the squared functionhas physical significance; it is positive everywhere. The probabilityof finding an electron at a node is zero. However, as one proceeds ineither direction from the nodes, the squared wave function has equalvalues relating to equal probabilities; to wit, the probability of findingthe electron on the “positive” or “negative” side of the wave functionis equal.It usually helps to visualize and understand a problem in a pictorial way.
However, since the wave function depends upon three variables, it can be represented only in four dimensions. To overcomethis problem, symbolic representations are used to emphasize variousproperties of the wave function.The angular wave function, A(⌰, ⌽), is shown for the H 1s and2pz orbitals in Figure 6-3a. The H 1s orbital is positive everywhere, but the 2pz orbital has one node, through which it changessign. The A2 (⌰, ⌽) function is shown for the same orbitals inFigure 6-3b. For both orbitals, the shape of this function is similarto the shape of the A(⌰, ⌽) function, but this function is positiveeverywhere.
It represents the region in space where the electroncan be found with a large probability (usually 90 % or more). Theboundary surface of this space is determined by the square of theangular function. The squared angular function does not say anything,however, about the variation of the probability density within thissurface. That information is contained in the radial distribution function. A way to illustrate the latter is shown for the 1s and 2pzorbitals in Figure 6-3c, where a cross section of the electron densityFigure 6-3.
Representations of the hydrogen 1s and 2pz orbitals; (a) Plot ofthe angular wave function, A(⌰, ⌽); (b) Plot of the squared function, A2 (⌰, ⌽);(c) Cross section of the squared total wave function, ⌿2 , representing the electrondensity. Reprinted by permission of Thomas H. Lowry [18].2466 Electronic Structure of Atoms and Moleculesdistribution is depicted. The varying amount of shading reflectsthe square of the radial function. Thus, this picture represents thesquared total wave function, ⌿2 . Rotating this picture around anyaxis for the 1s orbital and around the z axis for the 2pz orbitalwould give the three-dimensional representation of the total wavefunction.Whereas the square of the angular function has outstanding physical significance, the angular function itself contains valuable information regarding the symmetry properties of the wave function. Theseproperties are lost in the squared angular function.The well-known shapes of the one-electron orbitals are presented inFigure 6-4; these are, in fact, representations of the angular wave functions.
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