Part 1. Background Material In this portion of the text, most of the topics that are appropriate to an undergraduate reader are covered. Mamy of these subjects are subsequently discussed again in Chapter 5, where a broad perspective of what theoretical chemistry is about offered. They are treated again in greater detail in Chapters 6-8 where the three main disciplines of theory are covered in depth appropriate to a graduate-student reader Chapter I. The Basics of Quantum Mechanics Why Quantum Mechanics is Necessary for Describing Molecular Properties. We know that all molecules are made of atoms which in turn contain nuclei and electrons. As I discuss in this introductory section, the equations that govern the motions of electrons and of nuclei are not the familiar Newton equations F=m a but a new set of equations called Schrodinger equations. When scientists first studied the behavior of electrons and nuclei, they tried to interpret their experimental findings in terms of classical Newtonian motions, but such attempts eventually failed. They found
1 Part 1. Background Material In this portion of the text, most of the topics that are appropriate to an undergraduate reader are covered. Many of these subjects are subsequently discussed again in Chapter 5, where a broad perspective of what theoretical chemistry is about is offered. They are treated again in greater detail in Chapters 6-8 where the three main disciplines of theory are covered in depth appropriate to a graduate-student reader. Chapter 1. The Basics of Quantum Mechanics Why Quantum Mechanics is Necessary for Describing Molecular Properties. We know that all molecules are made of atoms which, in turn, contain nuclei and electrons. As I discuss in this introductory section, the equations that govern the motions of electrons and of nuclei are not the familiar Newton equations F = m a but a new set of equations called Schrödinger equations. When scientists first studied the behavior of electrons and nuclei, they tried to interpret their experimental findings in terms of classical Newtonian motions, but such attempts eventually failed. They found
that such small light particles behaved in a way that simply is not consistent with the Newton equations. Let me now illustrate some of the experimental data that gave rise to these paradoxes and show you how the scientists of those early times then used these data o suggest new equations that these particles might obey. I want to stress that the Schrodinger equation was not derived but postulated by these scientists. In fact, to date, no one has been able to derive the Schrodinger equation From the pioneering work of Bragg on diffraction of x-rays from planes of atoms or ions in crystals, it was known that peaks in the intensity of diffracted x-rays having wavelength n would occur at scattering angles 0 determined by the famous Bragg equati n入=2dsin where d is the spacing between neighboring planes of atoms or ions. These quantities are illustrated in Fig. 1. I shown below. There are may such diffraction peaks, each labeled by a different value of the integer n(n=1, 2, 3,). The Bragg formula can be derived by considering when two photons, one scattering from the second plane in the figure and the second scattering from the third plane, will undergo constructive interference. This condition is met when the extra path length"covered by the second photon (i.e, the length from points a to B to C)is an integer multiple of the wavelength of the photons
2 that such small light particles behaved in a way that simply is not consistent with the Newton equations. Let me now illustrate some of the experimental data that gave rise to these paradoxes and show you how the scientists of those early times then used these data to suggest new equations that these particles might obey. I want to stress that the Schrödinger equation was not derived but postulated by these scientists. In fact, to date, no one has been able to derive the Schrödinger equation. From the pioneering work of Bragg on diffraction of x-rays from planes of atoms or ions in crystals, it was known that peaks in the intensity of diffracted x-rays having wavelength l would occur at scattering angles q determined by the famous Bragg equation: n l = 2 d sinq, where d is the spacing between neighboring planes of atoms or ions. These quantities are illustrated in Fig. 1.1 shown below. There are may such diffraction peaks, each labeled by a different value of the integer n (n = 1, 2, 3, …). The Bragg formula can be derived by considering when two photons, one scattering from the second plane in the figure and the second scattering from the third plane, will undergo constructive interference. This condition is met when the “extra path length” covered by the second photon (i.e., the length from points A to B to C) is an integer multiple of the wavelength of the photons
A C B Figure 1. 1. Scattering of two beams at angle 0 from two planes in a crystal spaced by d The importance of these x-ray scattering experiments to electrons and nuclei appears in the experiments of Davisson and Germer in 1927 who scattered electrons of (reasonably) fixed kinetic energy E from metallic crystals. These workers found that plots of the number of scattered electrons as a function of scattering angle 0 displayed"peaks at angles 0 that obeyed a Bragg-like equation. The startling thing about this observation is that electrons are particles, yet the Bragg equation is based on the properties of waves An important observation derived from the Davisson-Germer experiments was that the scattering angles 0 observed for electrons of kinetic energy e could be fit to the bragg n n= 2d sine equation if a wavelength were ascribed to these electrons that was defined by
3 Figure 1.1. Scattering of two beams at angle q from two planes in a crystal spaced by d. The importance of these x-ray scattering experiments to electrons and nuclei appears in the experiments of Davisson and Germer in 1927 who scattered electrons of (reasonably) fixed kinetic energy E from metallic crystals. These workers found that plots of the number of scattered electrons as a function of scattering angle q displayed “peaks” at angles q that obeyed a Bragg-like equation. The startling thing about this observation is that electrons are particles, yet the Bragg equation is based on the properties of waves. An important observation derived from the Davisson-Germer experiments was that the scattering angles q observed for electrons of kinetic energy E could be fit to the Bragg n l = 2d sinq equation if a wavelength were ascribed to these electrons that was defined by
λ=h(2mE)2, where me is the mass of the electron and h is the constant introduced by Max Planck and Albert einstein in the early 1900s to relate a photon s energy e to its frequency v via e hv. These amazing findings were among the earliest to suggest that electrons, which had always been viewed as particles, might have some properties usually ascribed to waves That is, as de broglie has suggested in 1925, an electron seems to have a wavelength inversely related to its momentum, and to display wave-type diffraction. I should mention that analogous diffraction was also observed when other small light particles(e.g protons, neutrons, nuclei, and small atomic ions) were scattered from crystal planes. In all such cases, Bragg -like diffraction is observed and the Bragg equation is found to govern the scattering angles if one assigns a wavelength to the scattering particle according to 入=h(2mE where m is the mass of the scattered particle and h is Plancks constant(6.62 x10-erg sec) The observation that electrons and other small light particles display wave like behavior was important because these particles are what all atoms and molecules are made of. So, if we want to fully understand the motions and behavior of molecules, we must be sure that we can adequately describe such properties for their constituents Because the classical Newtonian equations do not contain factors that suggest wave properties for electrons or nuclei moving freely in space, the above behaviors presented significant challenges
4 l = h/(2me E)1/2 , where me is the mass of the electron and h is the constant introduced by Max Planck and Albert Einstein in the early 1900s to relate a photon’s energy E to its frequency n via E = hn. These amazing findings were among the earliest to suggest that electrons, which had always been viewed as particles, might have some properties usually ascribed to waves. That is, as de Broglie has suggested in 1925, an electron seems to have a wavelength inversely related to its momentum, and to display wave-type diffraction. I should mention that analogous diffraction was also observed when other small light particles (e.g., protons, neutrons, nuclei, and small atomic ions) were scattered from crystal planes. In all such cases, Bragg-like diffraction is observed and the Bragg equation is found to govern the scattering angles if one assigns a wavelength to the scattering particle according to l = h/(2 m E)1/2 where m is the mass of the scattered particle and h is Planck’s constant (6.62 x10-27 erg sec). The observation that electrons and other small light particles display wave like behavior was important because these particles are what all atoms and molecules are made of. So, if we want to fully understand the motions and behavior of molecules, we must be sure that we can adequately describe such properties for their constituents. Because the classical Newtonian equations do not contain factors that suggest wave properties for electrons or nuclei moving freely in space, the above behaviors presented significant challenges
Another problem that arose in early studies of atoms and molecules resulted from the study of the photons emitted from atoms and ions that had been heated or otherwise excited(e.g, by electric discharge ). It was found that each kind of atom(i.e, H or C or O)emitted photons whose frequencies v were of very characteristic values. An example of such emission spectra is shown in Fig. 1. 2 for hydrogen atoms 2/nm TotaH Aimer Lyman A即ay Brackett Figure 1. 2. Emission spectrum of atomic hydrogen with some lines repeated below to illustrate the series to which they belong In the top panel, we see all of the lines emitted with their wave lengths indicated in nano meters.The other panels show how these lines have been analyzed(by scientists whose names are associated) into patterns that relate to the specific energy levels between which transitions occur to emit the corresponding photons
5 Another problem that arose in early studies of atoms and molecules resulted from the study of the photons emitted from atoms and ions that had been heated or otherwise excited (e.g., by electric discharge). It was found that each kind of atom (i.e., H or C or O) emitted photons whose frequencies n were of very characteristic values. An example of such emission spectra is shown in Fig. 1.2 for hydrogen atoms. Figure 1.2. Emission spectrum of atomic hydrogen with some lines repeated below to illustrate the series to which they belong. In the top panel, we see all of the lines emitted with their wave lengths indicated in nanometers. The other panels show how these lines have been analyzed (by scientists whose names are associated) into patterns that relate to the specific energy levels between which transitions occur to emit the corresponding photons