The equation for determining the energy of a photon is due to Max Planck,6 who first in- troduced the idea that light was transmitted in the form of small energy packets, which he called quanta. The energy could be straightforwardly related to the frequency of the radi- ation.7 Planck’s equation, E 5 hν, has become one of the cornerstones of modern atomic physics and modern chemistry.
The idea that energy in an atom is quantized was a revolutionary one, reinforced when the Danish physicist Niels Bohr8 proposed a model of the nuclear atom in which the quan- tization of energy levels was an integral part of the model.9 From Bohr’s work first emerged the concept of quantum numbers, which have become so important in more recent theo- ries of atomic structure and bonding.
The electron had been discovered in the late 19th century, and its behavior had marked it as a particle (e.g., it possesses both mass and momentum). However, electron beams can be diffracted—behavior restricted to waves—so that electrons also exhibit some charac- teristics of a wave. This initial radical departure from the principles of classical mechanics—
that particles could exhibit wave behavior and that waves could exhibit particle-type behavior—was first suggested in 1924 by Louis de Broglie,10 who proposed the term wave-particle duality11 to describe the equivalence of matter and energy at the atomic and subatomic level.
The dual nature of the electron solved some problems in atomic physics, but more problems were added by the work of Werner Heisenberg,12 who pointed out that many of the ideas of classical physics could not be applied at the atomic level. In his uncertainty principle,13 he stated that the position and momentum of a particle cannot be known simultaneously to such precision that the product of the uncertainty in each quantity is less than h/2π. Equation 4.1, due to Kennard14 and Weyl,15 uses the standard deviations of position (σx) and momentum (σp):
x p h 2
σ •σ ≥ π (4.1)
6. Max Karl Ernst Ludwig Planck (1858-1947) was educated at Munich and Berlin. In 1889, he joined the faculty of Physics at Berlin; he received the 1918 Nobel Prize in Physics. After the war, Planck resumed the pres- idency of the Kaiser Wilhelm Society, which was renamed the Max Planck Society in his honor. For more bi- ographical details, see the web site of the Nobel Foundation.
7. Planck, M. Ann. Phys. 1901, 4, 553.
8. Niels Hendrik David Bohr (1885-1962) was educated in Copenhagen and Cambridge. He returned to Copenhagen in 1916 as professor of physics. Bohr escaped the Nazi occupation of Denmark and eventually became an important contributor to the Manhattan Project. Bohr returned to Copenhagen in 1945 and devoted the rest of his life to finding peaceful uses for atomic energy. There are numerous biographies of Bohr available, including Blaedel, N. Harmony and Unity: The Life of Niels Bohr (Science Tech.: Madison, Wisconsin, 1988).
9. Bohr, N. Phil. Mag, 1913, 26, 1; 1914, 27, 506.
10. Louis Victor Pierre Raymond, 7th duc de Broglie (1892-1987) took his PhD in theoretical physics at the Sorbonne in 1924. De Broglie was professor of physics at the Université de Paris from 1928 until his retirement in 1962. He received the Nobel Prize in Physics in 1929. In 1960, he succeeded his brother as duc de Broglie. For more biographical information, see: Abraham, A. Biogr. Mem. Fellows Roy. Soc. 1988, 32, 22.
11. De Broglie, L. Thesis, Université de Paris (Sorbonne), 1924; Ann. phys. (10) 1925, 3, 22.
12. Werner Karl Heisenberg (1901-1975) was educated at Munich. He became professor of theoretical phys- ics at Leipzig in 1928 and was Professor in Berlin and Director of the Kaiser Wilhelm Institute during World War II. Heisenberg was awarded the 1932 Nobel Prize in Physics. His work on the Nazi atomic bomb is still a source of controversy. For more biographical details, see: Rechenberg, H.; Wiemers, G. Werner Heisenberg (1901–1976), Schritte in die neue Physik (Sax-Verlag Beucha, 2001).
13. Heisenberg, W. Z. Phys. 1927, 43, 172.
14. Kennard, E.H. Z. Phys. 1927, 44, 326.
15. Weyl, H. Gruppentheorie und Quantenmechanik (Hirzel: Leipzig, 1928).
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The insight of genius required to consolidate the theories of atomic structure and bonding was supplied by the Austrian physicist Erwin Schrửdinger,16 who combined a number of principles and theories extant in the mid-1920s and unified them into what we now term wave mechanics, or quantum mechanics.17 The English physicist P. A. M. Dirac18 combined the work of de Broglie, Heisenberg, and Schrửdinger into one unified theory.19 In simple terms, Schrửdinger took the approach that electrons about an atomic nucleus will behave exactly like a standing wave in three dimensions. However, he also incorpo- rated into his theory the Heisenberg uncertainty principle, so that instead of addressing the positions and energies of electrons in atoms in classical terms, he addressed the prob- lem in terms of probability. Because the spectra of atoms can be measured in very high precision, the energy levels of the electrons in atoms can be determined with a very small uncertainty. Accordingly, the position of the electron must be highly uncertain. Although it had been shown that quantization of the energy levels of the atom led to a model that could be used to predict atomic properties, one of the major questions which Bohr’s theory of the atom had left unaddressed was why the energy levels of atoms should be quantized.
Wave mechanics answered this question.
Standing Waves and Atomic Orbitals
At the heart of quantum mechanics is the premise that one can reasonably treat the elec- trons of an atom or molecule as a standing wave in three dimensions.
The simplest standing wave is the standing wave in one dimension, exemplified by a vibrating string. The displacement, r, of any part of a vibrating string at time t can be cal- culated by using the standard equation for a simple harmonic oscillator if one knows two fundamental properties of the vibration: the amplitude of the vibration, A, which is de- fined as the maximum displacement of the string from its equilibrium position, and its frequency, v
r = A sin (2nπωt) (4.2)
Equation 4.2 is the wave function for a standing wave in one dimension and is simply an algebraic expression that one can use to describe a vibrating string. The variable n in this equation is an integer that tells if one is dealing with the fundamental vibration (n = 1), or an overtone or harmonic (n = 2, 3, 4, . . .). This number is strictly analogous to the quantum numbers of atomic physics. The wave functions for the fundamental and first three harmonic vibrations of a standing wave in one dimension are plotted in Figure 4.1.
In Figure 4.1, there are several important points marked as nodes. A node is defined as a point within the standing wave where the displacement from the equilibrium position is always zero. A standing wave in one dimension must have a minimum of two nodes, one at each
16. Erwin Schrửdinger (1887-1961) was educated in Vienna and remained there until 1920. In 1921, he went to Zürich, and in 1927, he joined the faculty at Berlin but left Germany in 1933 over the Nazi treatment of the Jews. From 1940 to 1956, he taught at Dublin before returning to Vienna. He shared the 1933 Nobel Prize in Physics. For more detail, see: Moore, W.J. A Life of Erwin Schrửdinger (Canto ed.) (Cambridge University Press:
Cambridge, 2003).
17. Schrửdinger, E. Ann. Phys. 1926, 79, 361, 489, 734; 1926, 80, 437; 1926, 81, 109.
18. Paul Adrien Maurice Dirac (1902-1984) was educated at Bristol and Cambridge, where he was appointed Lucasian Professor of Mathematics at the age of 30; he was Lucasian Professor until 1969. Dirac shared the 1933 Nobel Prize in Physics with Heisenberg. Following his retirement from Cambridge, Dirac became Professor of Physics at Florida State University, a post he held until his death. For more biographical detail, see: Dalitz, R.H.;
Peierls, R. Biogr. Mem. Fellows Roy. Soc. 1986, 32, 138.
19. Dirac, P.A.M. Proc. Roy. Soc. A 1927, 113, 621; 1927, 114, 243; 1928, 117, 610.
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Figure 4.1 A vibrating string provides an example of a standing wave in one dimension
end of the wave. For convenience, we shall call these nodes terminal nodes. For the funda- mental vibration, these are the only nodes; for the harmonics, however (n = 2, 3, 4, . . . in Equation 1.6), there are (n – 1) internal nodes in addition to the two terminal nodes. In a one-dimensional standing wave, all nodes are point nodes, or nodal points.
The wave equation of a standing wave in two dimensions is more complex than the wave equation for a vibrating string because the vibration is occurring in more dimen- sions. When reduced to its simplest form, the wave equation for a standing wave in two dimensions requires two, rather than one, “quantum number”—one for each dimension—
to describe the vibration completely.
The best analogy for a standing wave in two dimensions is probably a vibrating drum- head. Some typical vibrations of a drumhead are illustrated in Figure 4.2. In two dimen- sions, the nodes are no longer nodal points, but are, instead, nodal lines; there are two major types of nodes in two dimensions—circular nodes and linear nodes—which divide the vibrations into two classes (symmetrical vibrations and antisymmetrical vibrations).
The symmetrical vibrations of a drumhead have only circular nodes; they are shown as the top line of drumhead vibrations. The antisymmetrical vibrations of a drumhead have at least one linear node, so that half of the drumhead is above the average plane, while half of the drumhead is below it. The antisymmetrical vibrations are the lower ones in Figure 4.2. Exactly what type of vibration one is examining will be determined by the values of the two “quantum numbers” in the wave equation.
Because of the additional dimension, the picture in three dimensions becomes even more complex, although the same rules apply. In three dimensions, the nodes now become surfaces, rather than lines or points, and three “quantum numbers,” rather than two, are required to describe the vibration. Just as there were two types of nodal lines in standing waves in two dimensions (a vibrating drumhead), there are two types of nodal surfaces in three dimensions: spherical nodes, which are analogous to the circular nodes in two dimensions, and planar nodes, which are analogous to the linear nodes in two dimensions.
Schrửdinger’s equation for the energy of an electron in an atom is a second-order differential equation for a standing wave in three dimensions. As with the string and the drumhead, the fact that this equation describes a standing wave means that only certain solutions are allowed (there may not be a nonintegral number of waves within the atom). The energy of the atom can have only certain values—it must be quantized.
Here, for the first time, the origin of the quantization of atomic energy levels was explained.
node node node node node
node node node node node node node node node
n=1 n=2
n=3 n=4
amplitude
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Figure 4.2 Standing waves in two dimensions. The dark areas and light areas are moving in oppo- site directions with respect to the viewer. Standing waves in two dimensions have nodal lines (cir- cles or straight lines) rather than nodal points. The displacement of the drumhead with respect to the dashed lines drawn on each diagram is shown above the diagram for the top set of vibrations and below the diagram for the lower set of vibrations.
Each of the solutions to the Schrửdinger equation is in the form of an algebraic function—the wave function in three dimensions. Each of these wave functions (usually designated by the Greek letter c for atomic orbitals, and by the Greek letter f for molecular orbitals) is properly called the probability amplitude of an electron of a given energy in any region of space around the nucleus of the atom. The square of the wave function (f2 or c2) gives the probability of finding an electron within a given volume around the nucleus of the atom. One of the important things to note about wave functions is that the sum of the squares of all the orbital wave functions in a given subshell is the equation of a sphere.
Because this is the simplest set of solutions, we usually describe the atom in terms of orbitals that are mathematically orthogonal (i.e., the product of the wave functions of any pair of orbitals is zero). The individual orbitals of an atom are described in terms of three integer quantum numbers. These quantum numbers are the principal quantum number, n, which defines the electron shell in which the orbital occurs and specifies the total number of nodes in that orbital; the azimuthal quantum number, l, which defines the subshell to which the orbital belongs and specifies the type of orbital by specifying the number of planar nodes (there are l – 1); and the magnetic quantum number, ml, which defines the orientation of that particular orbital in space. There are a total of n2 orbitals in an energy level for which the principal quantum number has the value n.
We can describe the shapes of orbitals by specifying their symmetry. This is a useful way of describing orbitals because it allows us to identify a directional component of the atomic orbital that is important in the formation of molecular orbitals. The electron prob- abilities for a 2p and a 4s orbital are given in Figure 4.3.
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All s orbitals are spherically symmetrical; they have no directional characteristics. In contrast, p orbitals have a unique axis perpendicular to a single planar node passing through the nucleus, so unlike s orbitals, p orbitals have a directional character. When a p orbital is viewed from a perspective perpendicular to the nodal plane (i.e., along the axis), it always appears circular in cross-section—it is cylindrically symmetrical.ả
The fourth quantum number for electrons is the spin quantum number, ms, which takes one of only two permitted values, ± 1/2. Because of this restriction on the possible values for the spin quantum number of an electron, the capacity of an orbital is restricted to two electrons. The origins of this restriction on the electron capacity of an orbital were first set forth in 1925 by Austrian-born physicist Wolfgang Pauli,20 in the form of the prin- ciple that bears his name, the Pauli Exclusion principle. This principle states that no two electrons in any atom may simultaneously have the same values for all four quantum numbers.21 In other words, because the electrons in any single orbital must have the same values of n, l, and ml, they must differ in the value of ms. Because there are only two possi- ble values of m2, the orbital can hold only two electrons. The Pauli Exclusion principle ap- plies not only to electrons in the orbitals of isolated atoms but also to electrons in the orbitals of molecules. Two electrons occupying the same orbital are said to be paired.
20. Wolfgang Pauli (1900-1958) was educated at Munich. He taught physics at Gửttingen, Copenhagen, and Hamburg before joining the ETH in Zürich. He received the 1945 Nobel Prize in Physics for his work in atomic physics. In 1940, Pauli joined the Institute for Advanced Study at Princeton University, and in 1946 was natu- ralized a US citizen. After World War II, he returned to Zürich, where he spent the remainder of his life. Biog- raphies of Pauli abound; see, for example, Enz, C.P. No Time to be Brief, A scientific biography of Wolfgang Pauli.
(Oxford University Press: Oxford, 2002).
21. Pauli, W. Z. Phys. 1925, 31, 765.
Figure 4.3 Projections of the electron probability distributions for 2p (left), 3p (center), and 4s (right) orbitals. Note that the 4s orbital has only spherical nodal surfaces, that the 2p and 3p orbit- als have a single planar nodal surface, and that there are additional nodes in the 3p.
ả. Spatial Properties of Wave Functions
One of the important things to note about wave functions is that the sum of the squares of the orbital wave functions in a given subshell is the equation of a sphere. Let us take the wave functions for the orbitals of the 2p subshell, where ρ is the radial distribution function, which is independent of orientation:
Let us take the wave functions for the orbitals of the 2p subshell, where ρ is the radial distribution function, which is independent of orientation:
2px: ρ(3/4π)1/2sinθcosf = ρ(3/4)1/2 x 2py: ρ(3/4π)1/2sinθsinf = ρ(3/4)1/2y 2pz: ρ(3/4π)1/2cosθ = ρ(3/4)1/2z
When squared and summed, this becomes cx2 + cy2 + cz2 = (3/4π)ρ2(x2 + y2 + z2), which is the equation for a sphere. A similar situation holds true for both subsets of the five orbitals of a d subshell (dxy, dyz, dxz and dx2-y2, dz2).
orbital axis
nodal plane nodal plane nodal surfaces
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The filling of the orbitals around the nucleus is carried out according to the Aufbau principle (from the German Aufbauprinzip, “principle of building up”). This principle, named by Bohr in 1922,22 derives its name from the German word to build up. It states that when filling orbitals around a nucleus of an atom, the lowest energy orbitals are filled first.
Like the Pauli Exclusion principle, the Aufbau principle also holds true for orbitals in molecules.
When the application of the Pauli Exclusion and Aufbau principles leads to more than one possible electron configuration for an atom or molecule (i.e., when the highest energy orbital containing electrons is part of a degenerate set of orbitals), the distinction between the possible configurations is made possible by Hund’s rule,23 first promulgated by Fried- rich Hund.24 This rule states that the lowest energy electron configuration of an atom or molecule where the highest energy electrons partially fill a set of degenerate orbitals is the one in which there is the maximum possible number of unpaired electrons.
Worked Problem
4-1 The molecular orbitals of cyclobutadiene (C4H4) fall into the pattern below, where there are three sets of degenerate bonding orbitals (C–H σ, C–C σ, and C–C π ), with their antibonding counterparts, as well as a degenerate set of nonbonding (n) orbitals. There are 20 valence electrons in the cyclobutadiene molecule. Com- plete the diagram by distributing the electrons to the appropriate orbitals.
Energy
σ σ∗
π π∗
n
§Answer on next page.