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4 M Any-Electron Atoms pArt i PART II PART III atomic Structure MOLECULAR MOLECULAR STRUCTURE INTERACTIONS 1 classical and quantum mechanics 2 the Schrödinger equation 3 one-electron atoms 4 Many-Electron Atoms many-electron 4 atoms learning objectiVeS goal Why Are We Here? After this chapter, you will be able to Th e principal goal of this chapter is to introduce the Schrödinger equation do the following: for the many-electron atom, together with some new physics related to ➊ Write the Hamiltonian for spin, and show how we can solve that Schrödinger equation by approxi- any atom. mations to the energies and wavefunctions. We need the energies in order ➋ Find a zero-order wavefunction (the electron to predict the energies of chemical bonds; we need the wavefunctions confi guration) and zero-order because the distribution of the electrons will determine the geometry of energy of any atom. the molecules we build. ➌ Use the Hartree-Fock orbital energies to estimate the context Where Are We Now? effective atomic number of Chapter 3 describes—in considerable detail—a two-body problem: fi nding the orbital. the energies and distributions of one electron bound to an atomic nucleus. ➍ Determine the permutation Chemistry is largely determined by where electrons go, and if we’re to get symmetry of a many-particle function. closer to real chemistry (where most electrons come in pairs), we’ll need ➎ Determine the term symbols more than one electron. In this chapter, we add the other electrons. Th at and energy ordering of the may sound simple enough, but the many -body problem cannot be solved term states resulting from any exactly. Much of our eff ort in this chapter must be directed toward fi nding electron confi guration. approximations that help us approach the right answer, while maintaining some intuitive understanding of the structure of the atom. We can also expect new intricacies, both in the electric fi eld interactions as we add more charges and in the magnetic fi eld interactions because each new electron introduces at least one new magnetic moment. But once we have built a comprehensive model of the many-electron atom, we’ll have the entire periodic table at our disposal as we move on to assembling molecules. Supporting text How Did We Get Here? Th e main qualitative concept we draw on for this chapter is that the electron in an atom is suffi ciently wavelike that quantum mechanics restricts its energy and other parameters to specifi c values. For a more 155 M04_COOK4169_SE_01_C04.indd 155 10/29/12 11:54 AM 156 chapter 4 Many-Electron Atoms detailed background, we will draw on the following equations and sections of text to support the ideas developed in this chapter: • Section 2.1 explained how we can use operators, such as a combination of multiplication and differentiation, to extract information from a function. n In particular, the Hamiltonian H operates on a wavefunction c to return the energy E times the wavefunction. We call this the Schrödinger equation (Eq. 2.1): n Hc = E c. • However, the Schrödinger equation works as written only if the wavefunction is an eigenstate of the Hamiltonian. We will find in this chapter that one useful approximation uses a time-independent wavefunction c that is not an n eigenstate of H, and in that case we can use the average value theorem to find an expectation value of the energy (Eq. 2.10): n A = c*(Ac)dt. all Lspace • Section 2.2 introduced the8 particle9 in a one-dimensional box, a fundamental problem in quantum mechanics in which the Schrödinger equation can be solved analytically to give the wavefunctions (Eq. 2.33) 2 npx c (x) sin n = a a and the energies (Eq. 2.31) A a b n2p2U2 En = . 2ma2 • The one-dimensional box depends only on a single Cartesian coordinate. The atom, however, has a potential energy function that can be more naturally described in spherical coordinates, because the Coulomb attraction between the electron and the nucleus depends only on r, the distance between them. As a result, the solutions to the Schrödinger equation are also easier to express in spherical coordinates. To convert between the two coordinate systems, we use Eqs. A.6: x = r sin u cos f r = x2 + y2 + z2 1/2 z y r sin u sin f u a1rccos 2 = = r a yb z r cos u f arc tan . = = x These relationships also allow us to convert the Cartesiana b volume element dx dy dz to the volume element in spherical coordinates, r2dr sin u dudf, which is an essential component of any integrals we may need to carry out. • The Schrödinger equation for the one-electron atom can also be solved analytically, yielding the energies (Eq. 3.30) 2 4 2 Z mee Z En = - 2 2 2 K - 2 Eh. 2(4pe0) n U 2n We also obtain the wavefunctions as products of a radial wavefunction Rn,l (r) ml (given in Table 3.2) and an angular wavefunction Yl (u, f) (given in Table 3.1): ml c u f n,l,ml = Rn, l(r)Y l ( , ). M04_COOK4169_SE_01_C04.indd 156 10/29/12 11:54 AM 4.1 Many-Electron Spatial Wavefunctions 157 • One tool we will turn to for a critical step is the Taylor expansion (Eq. A.29), ϱ n n (x - x0) d f (x) f(x) = a n , n=0 n! dx x0 which lets us rewrite any algebraic expressiona asb a` power series. The trick with power series is to set them up so that each higher-order term is smaller than the last term, so that you can ignore all but the first few terms. In this chapter, we use a Taylor series expansion of the Schrödinger equation itself to find approximate solutions to the energies and wavefunctions. 4.1 Many-Electron Spatial Wavefunctions Neglecting some relatively small effects,1 we can solve the Schrödinger equation of the hydrogen atom to get algebraic expressions for the energies and wavefunc- tions. Once we add a second electron to an atom, that becomes impossible. Although many numerical methods work well, the exact solution cannot be written in general form. We will go as far as we can in the space of one chapter, which is pretty far. Checkpoint as in Chapter 3, Electron Configurations we are color-coding the math related to the Schrödinger equation to help distinguish First, let’s write the Hamiltonian for a two-electron atom as an extension of the terms in the Hamiltonian operator Hamiltonian we used for the one-electron atom: (blue) from contributions to the 2 2 2 2 2 scalar energy (red), and to help n U 2 U 2 Ze Ze e illustrate the transformation of , (H ٌ(1) ٌ(2 = - - - - + (4.1) one to the other. 2me 2me 4pe0r1 4pe0r2 4pe0r12 where 2 2 2 2 0 0 0 2 2 2 1 2 ; ( i) = 2 + 2 + 2 , ri = (xi + yi + zi)ٌ 0xi 0yi 0zi > a b xi, yi, and zi are the Cartesian coordinates for electron i; and r12 is the distance between electrons 1 and 2. These coordinates are defined in Fig. 4.1. The potential z energy terms are now written relative to the energy of the completely ionized atom; 1 in other words, the potential energy is zero when r1, r2, and r12 are all infinite. For –e the more general case of the atom with N electrons, we sum up the kinetic energies of all the electrons and their potential energies of Coulomb attraction to the nucleus. Then, foreach pair of electrons we add the repulsion energy: r N 2 2 i- 1 2 1 n U 2 Ze e r12 (H = a - ٌ(i) - + a . (4.2 i=1 2me 4pe0ri j =1 4pe0rij y c d What’s new and disturbing in the many-electron atom Hamiltonians is the +Ze x 2 electron–electron repulsion term, e (4pe0r12), which prevents the electron 1 r2 –e and electron 2 terms of the Hamiltonian in Eq. 4.1 from being separable. 2 > ▲ FigurE 4.1 Coordinate definitions for the two- 1Magnetic interactions, finite mass of the nucleus, special relativity, to name a few. electron atom. M04_COOK4169_SE_01_C04.indd 157 10/29/12 11:54 AM 158 chapter 4 Many-Electron Atoms example 4.1 Many-Electron Atom Hamiltonians context The element beryllium doubles as one of the most useful and frightening metals available to us. It is the lightest of all the structurally useful metals (neglecting lithium, which is soft and highly reactive), and also among the most hazardous. It has a strength comparable to steel, but at a fraction of the mass. Beryllium-based products were widely produced in the United States and Europe in the decades prior to World War II, but then it became gradually clear that the beryllium in factory dust was sickening and killing workers, sometimes years after exposure. Chronic beryllium disease resembles pneumonia, and can be contracted by repeated exposure to 0.5 mg/m3 or more of beryllium in the air. The mechanism for its biological activity is only partly understood, but it apparently combines with a protein and then deposits itself primarily in lung tissue, triggering an over-reaction by the body’s immune system—a response called hypersensitivity that leads to destructive inflammation of the lungs. The high toxicity has led to severe restrictions on its use in manufacturing, but its properties are still tempting. By accurately simulating the electronic properties of beryllium on a computer, we may identify the most valuable applications of this element at minimal risk.
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