Fundamentals Of Quantum Chemistry
Fundamentals ofQuantum Chemistry
Fundamentals ofQuantum ChemistryMolecular Spectroscopyand Modern ElectronicStructure ComputationsMichael MuellerRose-Hullman Institute of TechnologyTerre Haute, IndianaKLUWER ACADEMIC PUBLISHERSNEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW
eBook ISBN:Print ISBN:0-306-47566-90-306-46596-5 2002 Kluwer Academic PublishersNew York, Boston, Dordrecht, London, MoscowPrint 2001 Kluwer Academic/Plenum PublishersNew YorkAll rights reservedNo part of this eBook may be reproduced or transmitted in any form or by any means, electronic,mechanical, recording, or otherwise, without written consent from the PublisherCreated in the United States of AmericaVisit Kluwer Online at:and Kluwer's eBookstore ne.com
ForewordAs quantum theory enters its second century, it is fitting to examine justhow far it has come as a tool for the chemist. Beginning with Max Planck’sagonizing conclusion in 1900 that linked energy emission in discreet bundlesto the resultant black-body radiation curve, a body of knowledge hasdeveloped with profound consequences in our ability to understand nature.In the early years, quantum theory was the providence of physicists andcertain breeds of physical chemists. While physicists honed and refined thetheory and studied atoms and their component systems, physical chemistsbegan the foray into the study of larger, molecular systems. Quantum theorypredictions of these systems were first verified through experimentalspectroscopic studies in the electromagnetic spectrum (microwave, infraredand ultraviolet/visible), and, later, by nuclear magnetic resonance (NMR)spectroscopy.Over two generations these studies were hampered by two majordrawbacks: lack of resolution of spectroscopic data, and the complexity ofcalculations.This powerful theory that promised understanding of thefundamental nature of molecules faced formidable challenges.Thefollowing example may put things in perspective for today’s chemistryfaculty, college seniors or graduate students: As little as 40 years ago, forcefield calculations on a molecule as simple as ketene was a four to five yeardissertation project. The calculations were carried out utilizing the bestmainframe computers in attempts to match fundamental frequencies toexperimental values measured with a resolution of five to ten wavenumbersv
viForewordin the low infrared region! Post World War II advances in instrumentation,particularly the spin-offs of the National Aeronautics and SpaceAdministration (NASA) efforts, quickly changed the landscape of highresolution spectroscopic data.Laser sources and Fourier transformspectroscopy are two notable advances, and these began to appear inundergraduate laboratories in the mid-1980s. At that time, only chemistswith access to supercomputers were to realize the full fruits of quantumtheory. This past decade’s advent of commercially available quantummechanical calculation packages, which run on surprisingly sophisticatedlaptop computers, provide approximation technology for all chemists.Approximation techniques developed by the early pioneers can now becarried out to as many iterations as necessary to produce meaningful resultsfor sophomore organic chemistry students, graduate students, endowed chairprofessors, and pharmaceutical researchers.The impact of quantummechanical calculations is also being felt in certain areas of the biologicalsciences, as illustrated in the results of conformational studies of biologicallyactive molecules. Today’s growth of quantum chemistry literature is as fastas that of NMR studies in the 1960s.An excellent example of the introduction of quantum chemistrycalculations in the undergraduate curriculum is found at the author’sinstitution. Sophomore organic chemistry students are introduced to the PCSpartan program to calculate the lowest energy of possible structures.The same program is utilized in physical chemistry to compute the potentialenergy surface of the reaction coordinate in simple reactions. Biochemistrystudents take advantage of calculations to elucidate the pathways to creationof designer drugs.This hands-on approach to quantum chemistrycalculations is not unique to that institution. However, the flavor of thedepartment’s philosophy ties in quite nicely with the tone of this textbookthat is pitched at just the proper level, advanced undergraduates and firstyear graduate students.Farrell BrownProfessor Emeritus of ChemistryClemson University
PrefaceThis text is designed as a practical introduction to quantum chemistry forundergraduate and graduate students. The text requires a student to havecompleted a year of calculus, a physics course in mechanics, and a minimumof a year of chemistry. Since the text does not require an extensivebackground in chemistry, it is applicable to a wide variety of students withthe aforementioned background; however, the primary target of this text isfor undergraduate chemistry majors.The text provides students with a strong foundation in the principles,formulations, and applications of quantum mechanics in chemistry. Forsome students, this is a terminal course in quantum chemistry providingthem with a basic introduction to quantum theory and problem solvingtechniques along with the skills to do electronic structure calculations - anapplication that is becoming increasingly more prevalent in all disciplines ofchemistry. For students who will take more advanced courses in quantumchemistry in either their undergraduate or graduate program, this text willprovide a solid foundation that they can build further knowledge from.Early in the text, the fundamentals of quantum mechanics are established.This is done in a way so that students see the relevance of quantummechanics to chemistry throughout the development of quantum theorythrough special boxes entitled Chemical Connection. The questions in theseboxes provide an excellent basis for discussion in or out of the classroomwhile providing the student with insight as to how these concepts will beused later in the text when chemical models are actually developed.vii
viiiPrefaceStudents are also guided into thinking “quantum mechanically” early inthe text through conceptual questions in boxes entitled Points of FurtherUnderstanding. Like the questions in the Chemical Connection boxes, thesequestions provide an excellent basis for discussion in or out of theclassroom. These questions move students from just focusing on therigorous mathematical derivations and help them begin to visualize theimplications of quantum mechanics.Rotational and vibrational spectroscopy of molecules is discussed in thetext as early as possible to provide an application of quantum mechanics tochemistry using model problems developed previously.Spectroscopyprovides for a means of demonstrating how quantum mechanics can be usedto explain and predict experimental observation.The last chapter of the text focuses on the understanding and theapproach to doing modern day electronic structure computations ofmolecules. These types of computations have become invaluable tools inmodern theoretical and experimental chemical research. The computationalmethods are discussed along with the results compared to experiment whenpossible to aide in making sound decisions as to what type of Hamiltonianand basis set that should be used, and it provides a basis for usingcomputational strategies based on desired reliability to make computationsas efficient as possible.There are many people to thank in the development of this text, far toomany to list individually here. A special thanks goes out to the students overthe years who have helped shape the approach used in this text based onwhat has helped them learn and develop interest in the subject.Terre Haute, INMichael R. Mueller
AcknowledgmentsClemson UniversityFarrell B. BrownUniversity of ClevelandCollege of Applied ScienceRita K. HessleyDaniel L. Morris, Jr.Rose-Hulman Institute of TechnologyGerome F. WagnerRose-Hulman Institute of TechnologyThe permission of the copyright holder, Prentice-Hall, to reproduce Figure7-1 is gratefully acknowledged.The permission of the copyright holder, Wavefunction, Inc., to reproduce thedata on molecular electronic structure computations in Chapter 9 isgratefully acknowledged.ix
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ContentsChapter 1. Classical Mechanics1.11.21.3Newtonian Mechanics, 1Hamiltonian Mechanics, 3The Harmonic Oscillator, 5Chapter 2. Fundamentals of Quantum Mechanics2.12.22.32.42.52.62.72.81The de Broglie Relationship, 14Accounting for Wave Character in MechanicalSystems, 16The Born Interpretation, 18Particle-in-a-Box, 20Hermitian Operators, 27Operators and Expectation Values, 27The Heisenberg Uncertainty Principle, 29Particle in a Three-Dimensional Box andDegeneracy, 33xi14
ContentsxiiChapter 3. Rotational Motion3.13.2Particle-on-a-Ring, 37Particle-on-a-Sphere, 42Chapter 4. Techniques of amentals of Spectroscopy, 113Rigid Rotor Harmonic Oscillator Approximation(RRHO), 115Vibrational Anharmonicity, 128Centrifugal Distortion, 132Vibration-Rotation Coupling, 135Spectroscopic Constants fromVibrational Spectra, 136Time Dependence and Selection Rules, 140Chapter 7. Vibrational and RotationalSpectroscopy of Polyatomic Molecules7.185Harmonic Oscillator, 85Tunneling, Transmission, and Reflection, 96Chapter 6. Vibrational/Rotational Spectroscopy ofDiatomic Molecules6.16.254Variation Theory, 54Time-Independent Non -Degenerate PerturbationTheory, 60Time-Independent Degenerate PerturbationTheory, 76Chapter 5. Particles Encountering a FinitePotential Energy5.15.237Rotational Spectroscopy of LinearPolyatomic Molecules, 150Rotational Spectroscopy of Non-LinearPolyatomic Molecules, 156Infrared Spectroscopy ofPolyatomic Molecules, 168150
ContentsxiiiChapter 8. Atomic Structure and Spectra8.18.28.38.48.58.6One-Electron Systems, 177The Helium Atom, 191Electron Spin, 199Complex Atoms, 200Spin-Orbit Interaction, 207Selection Rules and Atomic Spectra, 217Chapter 9. Methods of Molecular ElectronicStructure Computations9.19.29.39.49.59.69.7177222The Born-Oppenheimer Approximation, 222TheMolecule, 224Molecular Mechanics Methods, 232Ab Initio Methods, 235Semi-Empirical Methods, 249Density Functional Methods, 251Computational Strategies, 255Appendix I. Table of Physical Constants259Appendix II. Table of Energy Conversion Factors260Appendix III. Table of Common Operators261Index262
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Chapter 1Classical MechanicsClassical mechanics arises from our observation of matter in themacroscopic world. From these everyday observations, the definition ofparticles is formulated. In classical mechanics, a particle has a specificlocation in space that can be defined precisely limited only by theuncertainty of the measurement instruments used. If all of the forces actingon the particle are accounted for, an exact energy and trajectory for theparticle can be determined. Classical mechanics yields results consistentwith experiment on macroscopic particles; hence, any theory such asquantum mechanics must yield classical results at these limits.There are a number of different techniques used to solve classicalmechanical systems that include Newtonian and Hamiltonian mechanics.Hamiltonian mechanics, though originally developed for classical systems,has a framework that is particularly useful in quantum mechanics.1.1 NEWTONIAN MECHANICSIn the mechanics of Sir Isaac Newton, the equations of motion areobtained from one of Newton’s Laws of Motion: Change of motion isproportional to the applied force and takes place in the direction of the force.Force,is a vector that is equal to the mass of the particle, m, multipliedby the acceleration vector1
2Chapter 1If the resultant force acting on the particle is known, then the equation ofmotion (i.e. trajectory) for the particle can be obtained. The acceleration isthe second time derivative of position, q, which is represented asThe symbol q is used as a general symbol for position expressed in anyinertial coordinate system such as Cartesian, polar, or spherical. A doubledot on top of a symbol, such asrepresents the second derivative withrespect to time, and a single dot over a symbol represents the first derivativewith respect to time.The systems considered, until later in the text, will be conservativesystems, and masses will be considered to be point masses. If a force is afunction of position only (i.e. no time dependence), then the force is said tobe conservative. In conservative systems, the sum of the kinetic andpotential energy remains constant throughout the motion. Non-conservativesystems, that is, those for which the force has time dependence, are usuallyof a dissipation type, such as friction or air resistance. Masses will beassumed to have no volume but exist at a given point in space.Example 1-1Problem: Determine the trajectory of a projectile fired from a cannonwhereby the muzzle is at an angle from the horizontal x-axis and leavesthe muzzle with a velocity ofAssume that there is no air resistance.Solution: This problem is an example of a separable problem: the equationsof motion can be solved independently in the horizontal and verticalcoordinates. First the forces acting on the particle must be obtained in thetwo independent coordinates.
3Classical MechanicsThe forces generate two differential equations to be solved. Uponintegration, this results in the following trajectories for the particle along thex and y-axes:The constantandrepresent the projectile at the origin (i.e. initial time).1.2 HAMILTONIAN MECHANICSAn alternative approach to solving mechanical problems that makes someproblems more tractable was first introduced in 1834 by the Scottishmathematician Sir William R. Hamilton. In this approach, the Hamiltonian,H, is obtained from the kinetic energy, T, and the potential energy, V, of theparticles in a conservative system.The kinetic energy is expressed as the dot product of the momentum vector,divided by two times the mass of each particle in the system.The potential energy of the particles will depend on the positions of theparticles. Hamilton determined that for a generalized coordinate system, theequations of motion could be obtained from the Hamiltonian and from thefollowing identities:
4Chapter 1andSimultaneous solution of these differential equations through all of thecoordinates in the system will result in the trajectories for the particles.Example 1-2Problem: Solve the same problem as shown in Example 1-1 usingHamiltonian mechanics.Solution: The first step is to determine the Hamiltonian for the problem.The problem is still separable and the projectile will have kinetic energy inboth the x and y-axes. The potential energy of the particle is due togravitational potential energy given asNow the Hamilton identities in Equations 1-5 and 1-6 must be determinedfor this system.
Classical Mechanics5The above formulations result in two non-trivial differential equations thatare the same as obtained in Example 1-1 using Newtonian mechanics.This will result in the same trajectory as obtained in Example 1-1.Notice that in Hamiltonian mechanics, initially the momentum of theparticles is treated separately from the position of the particles. This methodof treating the momentum separately from position will prove useful inquantum mechanics.1.3 THE HARMONIC OSCILLATORThe harmonic oscillator is an important model problem in chemicalsystems to describe the oscillatory (vibrational) motion along the bondsbetween the atoms in a molecule. In this model, the bond is viewed as aspring with a force constant of k.Consider a spring with a force constant k such that one end of the springis attached to an immovable object such as a wall and the other is attached toa mass, m (see Figure 1-1). Hamiltonian mechanics will be used; hence, thefirst step is to determine the Hamiltonian for the problem. The mass isconfined to the x-axis and will have both kinetic and potential energy. Thepotential energy is the square of the distance the spring is displaced from itsequilibrium position,times one-half of the spring force constant, k(Hooke’s Law).
6Chapter 1Taking the derivative of the Hamiltonian (Equation 1-7) with respect toposition and applying Equation 1-5 yields:Taking the derivative of the Hamiltonian (Equation 1-7) with respect tomomentum and applying Equation 1-6 yields:The second differential equation yields a trivial result:however, the first differential equation can be used to determine thetrajectory of the mass m. The time derivative of momentum is equivalent tothe force, or mass times acceleration.
7Classical MechanicsorThe solution to this differential equation is well known. One solution isgiven below.Another mathematically equivalent solution can be found by utilizing thefollowing Euler identitiesandThis results in the following mathematically equivalent trajectory as inEquation 1-9:The value of is the equilibrium length of the spring. Since the productofmust be dimensionless, the constant must have units of inverse timeand must be the frequency of oscillation. By taking the second timederivative of either Equation 1-9 or 1-11 results in the following expression:
8Chapter 1By comparing Equation 1-12 with Equation l-8b, an expression forreadily obtained.isSince the sine and cosine functions will oscillate from 1 to –1, the constantsa and b in Equation 1-9 and likewise the constants A and B in Equation 1-11are related to the amplitude and phase of motion of the mass. There are noconstraints on the values of these constants, and the system is not quantized.A model can now be developed that more accurately describes a diatomicmolecule. Consider two masses,andseparated by a spring with aforce constant k and an equilibrium length of as shown in Figure 1-2. TheHamiltonian is shown below.
Classical Mechanics9Note that the Hamiltonian appears to be inseparable. Making a coordinatetransformation to a center-of-mass coordinate system can make this problemseparable. Define r as the displacement of the spring from its equilibriumposition and s as the position of the center of mass.As a result of the coordinate transformation, the potential energy for thesystem becomes:Now the momentum and must be transformed to the momentum in thes and r coordinates. The time derivatives of r and s must be taken andrelated to the time derivatives of and
10Chapter 1From Equations 1-14 and 1-15, expressions forcan be obtained.The momentum terms andmass coordinates s and r.The reduced mass of the system,This reduces the expressions forandin terms ofandare now expressed in terms of the center ofis defined asandandto the following:
Classical Mechanics11The Hamiltonian can now be written in terms of the center-of-masscoordinate system.A further simplification can be made to the Hamiltonian by recognizing thatthe total mass of the system, M, is the sum ofandRecall that the coordinate s corresponds to the center of mass of thesystem whereas the coordinate r corresponds to the displacement of thespring. This ensures that r and s are separable. It can be concluded that thekinetic energy termmust correspond to the translation of the entire system in space. Since it isthe vibrational motion that is of interest, the kinetic term for the translationof the system can be neglected in the Hamiltonian. The resultingHamiltonian that corresponds to the vibrational motion is as follows:Notice that the Hamiltonian in Equation 1-19 is identical in form to theHamiltonian in Equation 1-7 solved previously. The solution can be inferredfrom the previous result recognizing that when the spring is in itsequilibrium positionthen(refer to Equation 1-14).
12Chapter 1This example demonstrates a number of important techniques in solvingmechanical problems. A mechanical problem c
quantum mechanics must yield classical results at these limits. There are a number of different techniques used to solve classical mechanical systems that include Newtonian and Hamiltonian mechanics. Hamiltonian mechanics, though originally developed for classical systems, has a framework that is p
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