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
In the early attempts to rationalize such spectra in terms of electronic motions one described an electron as moving about the atomic nuclei in circular orbits such as shown in Fig. 1. 3 Two circular orbits of radii ri and r2 Figure 1. 3. Characterization of small and large stable orbits for an electron movin around a nucleus A circular orbit was thought to be stable when the outward centrifugal force characterized by radius r and speed v(me vi/r)on the electron perfectly counterbalanced the inward attractive Coulomb force(Ze/r)exerted by the nucleus of charge v2/r=Ze/r 6
6 In the early attempts to rationalize such spectra in terms of electronic motions, one described an electron as moving about the atomic nuclei in circular orbits such as shown in Fig. 1. 3. Figure 1. 3. Characterization of small and large stable orbits for an electron moving around a nucleus. A circular orbit was thought to be stable when the outward centrifugal force characterized by radius r and speed v (me v2 /r) on the electron perfectly counterbalanced the inward attractive Coulomb force (Ze2 /r2 ) exerted by the nucleus of charge Z: me v2 /r = Ze2 /r2 r2 Two circular orbits of radii r1 and r2. r1
This equation, in turn, allows one to relate the kinetic energy 1/2 me v to the Coulombic energy Zer, and thus to express the total energy e of an orbit in terms of the radius of the orbit E=1/2mv2-Ze2/r=-1/2Ze2/r The energy characterizing an orbit or radius r, relative to the e=0 reference of energy at r->00, becomes more and more negative (i. e, lower and lower ) as r becomes smaller. This relationship between outward and inward forces allows one to conclude that the electron should move faster as it moves closer to the nucleus since v2=Ze/(rm) However, nowhere in this model is a concept that relates to the experimental fact that each atom emits only certain kinds of photons. It was believed that photon emission occurred when an electron moving in a larger circular orbit lost energy and moved to a smaller circular orbit. However, the Newtonian dynamics that produced the above equation would allow orbits of any radius, and hence any energy, to be followed. Thus, it would appear that the electron should be able to emit photons of any energy as it moved from orbit to orbit The breakthrough that allowed scientists such as Niels Bohr to apply the circular orbit model to the observed spectral data involved first introducing the idea that the electron has a wavelength and that this wavelength 2 is related to its momentum by the de broglie equation 2 =h/p. The key step in the bohr model was to also specify that the radius of the circular orbit be such that the circumference of the circle 2t r equal an integer(n) multiple of the wavelength 2. Only in this way will the electron's wave
7 This equation, in turn, allows one to relate the kinetic energy 1/2 me v2 to the Coulombic energy Ze2 /r, and thus to express the total energy E of an orbit in terms of the radius of the orbit: E = 1/2 me v2 – Ze2 /r = -1/2 Ze2 /r. The energy characterizing an orbit or radius r, relative to the E = 0 reference of energy at r ® ¥, becomes more and more negative (i.e., lower and lower) as r becomes smaller. This relationship between outward and inward forces allows one to conclude that the electron should move faster as it moves closer to the nucleus since v2 = Ze2 /(r me ). However, nowhere in this model is a concept that relates to the experimental fact that each atom emits only certain kinds of photons. It was believed that photon emission occurred when an electron moving in a larger circular orbit lost energy and moved to a smaller circular orbit. However, the Newtonian dynamics that produced the above equation would allow orbits of any radius, and hence any energy, to be followed. Thus, it would appear that the electron should be able to emit photons of any energy as it moved from orbit to orbit. The breakthrough that allowed scientists such as Niels Bohr to apply the circularorbit model to the observed spectral data involved first introducing the idea that the electron has a wavelength and that this wavelength l is related to its momentum by the de Broglie equation l = h/p. The key step in the Bohr model was to also specify that the radius of the circular orbit be such that the circumference of the circle 2p r equal an integer (n) multiple of the wavelength l. Only in this way will the electron’s wave
experience constructive interference as the electron orbits the nucleus. Thus, the bohr relationship that is analogous to the bragg equation that determines at what angles constructive interference can occur is 2πr=nλ. Both this equation and the analogous bragg equation are illustrations of what we call boundary conditions, they are extra conditions placed on the wavelength to produce some desired character in the resultant wave (in these cases, constructive interference). of course, there remains the question of why one must impose these extra conditions when the Newton dynamics do not require them The resolution of this paradox is one of the things that quantum mechanics does Returning to the above analysis and using n=h/p=h/(mv), 2 r=na, as well as the force-balance equation me v2/r= Ze /r, one can then solve for the radi that stable Bohr orbits obey m and. in turn for the velocities of electrons in these orbits =Ze/(nh/2π) 8
8 experience constructive interference as the electron orbits the nucleus. Thus, the Bohr relationship that is analogous to the Bragg equation that determines at what angles constructive interference can occur is 2 p r = n l. Both this equation and the analogous Bragg equation are illustrations of what we call boundary conditions; they are extra conditions placed on the wavelength to produce some desired character in the resultant wave (in these cases, constructive interference). Of course, there remains the question of why one must impose these extra conditions when the Newton dynamics do not require them. The resolution of this paradox is one of the things that quantum mechanics does. Returning to the above analysis and using l = h/p = h/(mv), 2p r = nl, as well as the force-balance equation me v2 /r = Ze2 /r2 , one can then solve for the radii that stable Bohr orbits obey: r = (nh/2p) 1/(me Z e2 ) and, in turn for the velocities of electrons in these orbits v = Z e2 /(nh/2p)
These two results then allow one to express the sum of the kinetic(1/2 me v) and Coulomb potential (-Ze"/r)energies E=-1/2m。Z2e(nh/2 Just as in the Bragg diffraction result, which specified at what angles special high intensities occurred in the scattering, there are many stable Bohr orbits, each labeled by a value of the integer n. Those with small n have small radi, high velocities and more negative total energies(n b, the reference zero of energy corresponds to the electron at r oo, and withv=0). So, it is the result that only certain orbits are allowed" that causes only certain energies to occur and thus only certain energies to be observed in the emitted photons It turned out that the bohr formula for the energy levels (labeled by n)of an electron moving about a nucleus could be used to explain the discrete line emission spectra of all one-electron atoms and ions(i.e, H, He, Li, etc. )to very high precision In such an interpretation of the experimental data, one claims that a photon of energy hv=R(l/m2-1/n2) is emitted when the atom or ion undergoes a transition from an orbit having quantum number n, to a lower-energy orbit having n here the symbol r is used to denote the following collection of factors
9 These two results then allow one to express the sum of the kinetic (1/2 me v 2 ) and Coulomb potential (-Ze2 /r) energies as E = -1/2 me Z2 e4 /(nh/2p) 2 . Just as in the Bragg diffraction result, which specified at what angles special high intensities occurred in the scattering, there are many stable Bohr orbits, each labeled by a value of the integer n. Those with small n have small radii, high velocities and more negative total energies (n.b., the reference zero of energy corresponds to the electron at r = ¥ , and with v = 0). So, it is the result that only certain orbits are “allowed” that causes only certain energies to occur and thus only certain energies to be observed in the emitted photons. It turned out that the Bohr formula for the energy levels (labeled by n) of an electron moving about a nucleus could be used to explain the discrete line emission spectra of all one-electron atoms and ions (i.e., H, He+ , Li+2, etc.) to very high precision. In such an interpretation of the experimental data, one claims that a photon of energy hn = R (1/nf 2 – 1/ni 2 ) is emitted when the atom or ion undergoes a transition from an orbit having quantum number ni to a lower-energy orbit having nf . Here the symbol R is used to denote the following collection of factors:
R=1/2m2Ze(h/2x)2 The Bohr formula for energy levels did not agree as well with the observed pattern of emission spectra for species containing more than a single electron. However, it does give a reasonable fit, for example, to the Na atom spectra if one examines only transitions involving only the single valence electron. The primary reason for the breakdown of the Bohr formula is the neglect of electron-electron Coulomb repulsions in its derivation Nevertheless, the success of this model made it clear that discrete emission spectra could only be explained by introducing the concept that not all orbits were allowed". Only special orbits that obeyed a constructive- interference condition were really accessible to the electrons motions. This idea that not all energies were allowed, but only certain quantized"energies could occur was essential to achieving even a qualitative sense of agreement with the experimental fact that emission spectra were discrete In summary, two experimental observations on the behavior of electrons that were crucial to the abandonment of newtonian dynamics were the observations of electron diffraction and of discrete emission spectra. Both of these findings seem to suggest that electrons have some wave characteristics and that these waves have only certain allowed (i.e, quantized) wavelength So, now we have some idea about why Newton s equations fail to account for the dynamical motions of light and small particles such as electrons and nuclei. We see that extra conditions(e.g, the Bragg condition or constraints on the de broglie wavelength) could be imposed to achieve some degree of agreement with experimental observation
10 R = 1/2 me Z2 e4 /(h/2p) 2 . The Bohr formula for energy levels did not agree as well with the observed pattern of emission spectra for species containing more than a single electron. However, it does give a reasonable fit, for example, to the Na atom spectra if one examines only transitions involving only the single valence electron. The primary reason for the breakdown of the Bohr formula is the neglect of electron-electron Coulomb repulsions in its derivation. Nevertheless, the success of this model made it clear that discrete emission spectra could only be explained by introducing the concept that not all orbits were “allowed”. Only special orbits that obeyed a constructive-interference condition were really accessible to the electron’s motions. This idea that not all energies were allowed, but only certain “quantized” energies could occur was essential to achieving even a qualitative sense of agreement with the experimental fact that emission spectra were discrete. In summary, two experimental observations on the behavior of electrons that were crucial to the abandonment of Newtonian dynamics were the observations of electron diffraction and of discrete emission spectra. Both of these findings seem to suggest that electrons have some wave characteristics and that these waves have only certain allowed (i.e., quantized) wavelengths. So, now we have some idea about why Newton’s equations fail to account for the dynamical motions of light and small particles such as electrons and nuclei. We see that extra conditions (e.g., the Bragg condition or constraints on the de Broglie wavelength) could be imposed to achieve some degree of agreement with experimental observation
