once e is known. one obtain F(t)=exp(-i Et/h) and the full wave function can be written as p(qi, t)=H(qi) exp(-i Et/h) For the above example, the time dependence is expressed by F(t)=exp(-iti m2t2/2M)/t) In summary, whenever the Hamiltonian does not depend on time explicitly, one can solve the time-independent Schrodinger equation first and then obtain the time dependence as exp(-i Et/h)once the energy e is known. In the case of molecular structure theory, it is a quite daunting task even to approximately solve the full Schrodinger equation because it is a partial differential equation depending on all of the coordinates of the electrons and nuclei in the molecule For this reason, there are various approximations that one usually implements when attempting to study molecular structure using quantum mechanics 3. The born-Oppenheimer Approximation 31
31 once E is known, one obtains F(t) = exp( -i Et/ h), and the full wave function can be written as Y(qj ,t) = Y(qj ) exp (-i Et/ h). For the above example, the time dependence is expressed by F(t) = exp ( -i t { m2 h2 /2M }/ h). In summary, whenever the Hamiltonian does not depend on time explicitly, one can solve the time-independent Schrödinger equation first and then obtain the time dependence as exp(-i Et/ h) once the energy E is known. In the case of molecular structure theory, it is a quite daunting task even to approximately solve the full Schrödinger equation because it is a partial differential equation depending on all of the coordinates of the electrons and nuclei in the molecule. For this reason, there are various approximations that one usually implements when attempting to study molecular structure using quantum mechanics. 3. The Born-Oppenheimer Approximation
One of the most important approximations relating to applying quantum mechanics to molecules is known as the Born-Oppenheimer(BO) approximation The basic idea behind this approximation involves realizing that in the full electrons-plus- nuclei Hamiltonian operator introduced above H=2{-(2/2me)02/oqn2+1/22e2 za +2ai-(h2/2ma)a2/0qa2+ 1/2 2b ZaZbe 2/ra. b j the time scales with which the electrons and nuclei move are generally quite different In particular, the heavy nuclei (i.e, even a h nucleus weighs nearly 2000 times what an electron weighs)move (i.e, vibrate and rotate)more slowly than do the lighter electrons Thus, we expect the electrons to be able to"adjust "their motions to the much more slowly moving nuclei. This observation motivates us to solve the Schrodinger equation for the movement of the electrons in the presence of fixed nuclei as a way to represent the fully-adjusted state of the electrons at any fixed positions of the nuclei The electronic Hamiltonian that pertains to the motions of the electrons in the e of so-called clamped nuclei H=∑1{-(2/2me)a2/q2+1/2e2 e2/r produces as its eigenvalues through the equation
32 One of the most important approximations relating to applying quantum mechanics to molecules is known as the Born-Oppenheimer (BO) approximation. The basic idea behind this approximation involves realizing that in the full electrons-plusnuclei Hamiltonian operator introduced above H = Si { - (h2/2me) ¶ 2/¶qi 2 + 1/2 Sj e2/ri,j - Sa Zae 2/ri,a } + Sa { - (h2/2ma) ¶ 2/¶qa 2+ 1/2 Sb ZaZbe 2/ra,b } the time scales with which the electrons and nuclei move are generally quite different. In particular, the heavy nuclei (i.e., even a H nucleus weighs nearly 2000 times what an electron weighs) move (i.e., vibrate and rotate) more slowly than do the lighter electrons. Thus, we expect the electrons to be able to “adjust” their motions to the much more slowly moving nuclei. This observation motivates us to solve the Schrödinger equation for the movement of the electrons in the presence of fixed nuclei as a way to represent the fully-adjusted state of the electrons at any fixed positions of the nuclei. The electronic Hamiltonian that pertains to the motions of the electrons in the presence of so-called clamped nuclei H = Si { - (h2/2me) ¶ 2/¶qi 2 + 1/2 Sj e2/ri,j - Sa Zae 2/ri,a } produces as its eigenvalues through the equation
Hv(alaa=E(qa),(aila) energies E(qa) that depend on where the nuclei are located (i.e., the iga) coordinates). As its eigenfunctions, one obtains what are called electronic wave functions yx(qlq)i which also depend on where the nuclei are located The energies Ex(a ) are what we usually call potential energy surfaces. An example of such a surface is shown in Fig. 1.5 Transition Structure A Secord Order Saddle Point Transition Structure B Product A Inimum io for Product B Second orde Saddle pcint Valley Ridge nection Point Figure 1. 5. Two dimensional potential energy surface showing local minima, transition states and paths connecting them This surface depends on two geometrical coordinates( qa) and is a plot of one particular eigenvalue e(q)vs. these two coordinates 33
33 H yJ (qj |qa ) = EJ (qa ) yJ (qj |qa ) energies EK(qa ) that depend on where the nuclei are located (i.e., the {qa} coordinates). As its eigenfunctions, one obtains what are called electronic wave functions {yK(qi |qa )} which also depend on where the nuclei are located. The energies EK(qa ) are what we usually call potential energy surfaces. An example of such a surface is shown in Fig. 1.5. Figure 1. 5. Two dimensional potential energy surface showing local minima, transition states and paths connecting them. This surface depends on two geometrical coordinates {qa} and is a plot of one particular eigenvalue EJ (qa ) vs. these two coordinates
Although this plot has more information on it than we shall discuss now, a few features are worth noting. There appear to be three minima(i.e, points where the derivative of E, with respect to both coordinates vanish and where the surface has positive curvature). These points correspond, as we will see toward the end of this introductory material, to geometries of stable molecular structures. The surface also displays two first-order saddle points(labeled transition structures A and b)that connee the three minima. These points have zero first derivative of e, with respect to both coordinates but have one direction of negative curvature. As we will show later, these points describe transition states that play crucial roles in the kinetics of transitions among the three stable geometries Keep in mind that Fig. 1. 5 shows just one of the E surfaces, each molecule has a ground-state surface(.e, the one that is lowest in energy as well as an infinite number of excited-state surfaces. Let's now return to our discussion of the bo model and ask what one does once one has such an energy surface in hand The motion of the nuclei are subsequently within the BO model, assumed to obey a Schrodinger equation in which 2af-(h2/2ma)a2/10qa2+ 1/2 2b ZaZbe2/rab)+ek(ga) defines a rotation-vibration Hamiltonian for the particular energy state ex of interest The rotational and vibrational energies and wave functions belonging to each electronic state (i.e, for each value of the index K in e(aa)) are then found by solving a Schrodinger equation with such a Hamiltonian This bo model forms the basis of much of how chemists view molecular structure and molecular spectroscopy. For example as applied to formaldehyde H,c=o, we speak of the singlet ground electronic state(with all electrons spin paired and
34 Although this plot has more information on it than we shall discuss now, a few features are worth noting. There appear to be three minima (i.e., points where the derivative of EJ with respect to both coordinates vanish and where the surface has positive curvature). These points correspond, as we will see toward the end of this introductory material, to geometries of stable molecular structures. The surface also displays two first-order saddle points (labeled transition structures A and B) that connect the three minima. These points have zero first derivative of EJ with respect to both coordinates but have one direction of negative curvature. As we will show later, these points describe transition states that play crucial roles in the kinetics of transitions among the three stable geometries. Keep in mind that Fig. 1. 5 shows just one of the EJ surfaces; each molecule has a ground-state surface (i.e., the one that is lowest in energy) as well as an infinite number of excited-state surfaces. Let’s now return to our discussion of the BO model and ask what one does once one has such an energy surface in hand. The motion of the nuclei are subsequently, within the BO model, assumed to obey a Schrödinger equation in which Sa { - (h2/2ma) ¶ 2/¶qa 2+ 1/2 Sb ZaZbe 2/ra,b } + EK(qa ) defines a rotation-vibration Hamiltonian for the particular energy state EK of interest. The rotational and vibrational energies and wave functions belonging to each electronic state (i.e., for each value of the index K in EK(qa )) are then found by solving a Schrödinger equation with such a Hamiltonian. This BO model forms the basis of much of how chemists view molecular structure and molecular spectroscopy. For example as applied to formaldehyde H2C=O, we speak of the singlet ground electronic state (with all electrons spin paired and
ccupying the lowest energy orbitals)and its vibrational states as well as the n>I*and I>I* electronic states and their vibrational levels. Although much more will be said about these concepts later in this text, the student should be aware of the concepts of electronic energy surfaces (i.e, the EK(ga)))and the vibration- rotation states that belong cach such surface Having been introduced to the concepts of operators, wave functions, the Hamiltonian and its Schrodinger equation, it is important to now consider several examples of the applications of these concepts. The examples treated below were chosen to provide the reader with valuable experience in solving the Schrodinger equation; they were also chosen because they form the most elementary chemical models of electronic motions in conjugated molecules and in atoms, rotations of linear molecules, and vibrations of chemical bond Il. Your First Application of Quantum Mechanics- Motion of a Particle in One Dimension This is a very important problem whose solutions chemists use to model a wide variety of phenomena Let's begin by examining the motion of a single particle of mass m in one direction which we will call x while under the influence of a potential denoted v(x). The classical expression for the total energy of such a system is E=p/2m+ V(x), where p is the momentum of the particle along the x-axis. To focus on specific examples, consider how
35 occupying the lowest energy orbitals) and its vibrational states as well as the n® p* and p ® p* electronic states and their vibrational levels. Although much more will be said about these concepts later in this text, the student should be aware of the concepts of electronic energy surfaces (i.e., the {EK(qa )}) and the vibration-rotation states that belong to each such surface. Having been introduced to the concepts of operators, wave functions, the Hamiltonian and its Schrödinger equation, it is important to now consider several examples of the applications of these concepts. The examples treated below were chosen to provide the reader with valuable experience in solving the Schrödinger equation; they were also chosen because they form the most elementary chemical models of electronic motions in conjugated molecules and in atoms, rotations of linear molecules, and vibrations of chemical bonds. II. Your First Application of Quantum Mechanics- Motion of a Particle in One Dimension. This is a very important problem whose solutions chemists use to model a wide variety of phenomena. Let’s begin by examining the motion of a single particle of mass m in one direction which we will call x while under the influence of a potential denoted V(x). The classical expression for the total energy of such a system is E = p2 /2m + V(x), where p is the momentum of the particle along the x-axis. To focus on specific examples, consider how