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Quantum Chemistry Fifth Edition IRA N. LEVINE Chemistry Department Brooklyn College City University of New York Brooklyn, New York q bp 4 So 3 > PRENTICE HALL, Upper Saddle River, New Jersey 07458 Library of Congress Cataloging-in-Publication Data Levine, Ira N. Quantum chemistry / Yra N. Levine —S5th ed. p em Includes bibliographical references and index. ISBN 0-13-685512-1 1. Quantum chemistry. LTisle. QD462148 2000 S4L2B—dez so-8558 cIP Senior Editor: John Challice Executive Managing Editor: Kathleen Schiapareili Assistant Managing Editor: Lisa Kinne Manufacturing Manager: Trudy Piscioti Cover Designer: Bruce Kenselaar Ast Director: Jayne Conte Production Supesvision/Composition: Accu-color/NK Graphics O 2000, 1991 by Prentice-Hall, Inc. Upper Saddle River, New Jersey 07458 AMI rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. Previous editions copyright O 1970, 1974, 1983 by Allyn and Bacon, Inc, Printed in the United States of America 10 987654321 ISBN 0-13-L85512-1 PRENTICE-HALL [INTERNATIONAL (UK) LIMITED, London PRENTICS-HALL OF AUSTRALIA PTY. LIMITED, Sydney PrentiCE-HALL CANADA INC., Toronto PRENTICE-HALL HISPANOAMERICANA, S.A., Mexico PrEnTICE-HALL OF INDIA PRIVATE LIMITED, Net Delhi PRENTICE-HALL OF JAPAN, INC., Tokyo Prenticr-HALL (Singapore) PrE. LTD. Epirora PRENTICE-HALL DO BRAsiL, LTDA., Rio de Janeiro Contents PREFACE ix 1 THE SCHRÔDINGER EQUATION 1 1.1 Quantum Chemistry, 1 1.2 Historical Background of Quantum Mechanics, 2 1.3 The Uncertainty Principle, 5 1,4 The Time-Dependent Schrôdinger Equation, 7 1.5 The Time-Independent Schródinger Equation, 12 1.6 Probability, 14 1.7 Complex Numbers, 16 1.8 Uniis, 18 1.9 Summary, 18 THE PARTICLE IN A BOX 21 2.1 Differential Equations, 21 2.2 Particle in a One-Dimensional Box, 22 2.3 The Free Particle in One Dimension, 28 24 Particle in a Rectangular Well, 29 2.5 Tunneling,31 2.6 Summary, 32 OPERATORS 35 3.1 Operators, 35 3.2 Eigenfunctions and Eigenvalues, 39 3.3 Operators and Quantum Mechanics, 40 3.4 The Three-Dimensional Many-Particle Schrôdinger Equation, 46 3.5 The Particle in a Three-Dimensional Box, 49 3.6 Degeneracy, 52 3.7 Average Values, 53 3.8 Requirements for an Acceptable Wave Function, 57 3.9 Summary, 58 THE HARMONIC OSCILLATOR 62 4,1 Power-Series Solution of Differential Equations, 62 4.2 The One-Dimensional Harmonic Oscillator, 65 4.3 Vibration of Molecules, 74 4.4 Numerical Solution of the One-Dimensional Time-Independent Schródinger Equation, 78 4.5 Summary, 89 Contents ANGULAR MOMENTUM 94 5.1 Simultaneous Specification of Several Properties, 94 5.2 Vectors, 97 5.3 Angular Momentum of a One-Particle System, 102 5.4 The Ladder-Operator Method for Angular Momentum, 115 5.5 Summary, 120 THE HYDROGEN ATOM 123 6.1 The One-Particle Central-Force Problem, 123 6.2 Noninteracting Particles and Separation of Variables, 125 6.3 Reduction of the Two-Particle Problem to Two One-Particle Problems, 127 6.4 The Two-Particle Rigid Rotor, 130 6.5 The Hydrogen Atom, 134 6.6 The Bound-State Hydrogen-Atom Wave Functions, 142 6.7 Hydrogenlike Orbitals, 150 6.8 The Zeeman Effect, 154 6.9 Numerical Solution of the Radial Schródinger Equation, 157 6.10 Summary, 158 THEOREMS OF QUANTUM MECHANICS 163 7.1 Introduction, 163 7.2 Hermitian Operators, 164 7.3 Expansion in Terms of Figenfunctions, 170 7.4 Eigenfunctions of Commuting Operators, 175 7.5 Parity, 178 7.6 Measurement and the Superposition of States, 182 7.7 Position Eigenfunctions, 187 7.8 The Postulates of Quantum Mechanics, 190 7.9 Measurement and the Interpretation of Quantum Mechanics, 194 7.10 Matrices, 198 7.11 Summary, 201 THE VARIATION METHOD 208 8.1 The Variation Theorem, 208 8.2 Extension of the Variation Method, 212 8.3 Determinants, 213 8.4 Simultaneous Linear Equations, 217 8.5 Linear Variation Functions, 220 8.6 Matrices, Eigenvalues, and Eigenvectors, 228 8.7 Summary, 235 Contents vii 13 ELECTRONIC STRUCTURE OF DIATFOMIC MOLECULES 366 13.1 The Born-Oppenheimer Approximation, 366 13.2 Nuclear Motion in Diatomic Molecules, 370 13.3 Atomic Units, 375 13.4 The Hydrogen Molecule Ion, 376 13.5 Approximate Treatments of the H; Ground Electronic State, 381 13.6 Molecular Orbitals for H; Excited States, 390 13.7 MO Configurations of Homonuclear Diatomic Molecules, 396 13.8 Electronic Terms of Diatomic Molecules, 402 13.9 The Hydrogen Molecule, 407 13.10 The Valence-Hond Treatment of H,, 410 13,11 Comparison of the MO and VB Theories, 414 13.12 MO and VB Wave Functions for Homonuclear Diatomic Molecules, 416 13.13 Excited States of H;, 419 13.14 Electron Probability Density, 421 13.15 Dipole Moments, 423 13.16 The Hartree-Fock Method for Molecules, 426 13.17 SCF Wave Functions for Diatomic Molecules, 436 13.18 MO Treatment of Heteronuclear Diatomic Molecules, 439 13.19 VB Treatment of Heteronuclear Diatomic Molecules, 442 13.20 The Valence-Electron Approximation, 443 13.21 CIWave Functions, 444 13.222 Summary, 451 14 THE VIRIAL THEOREM AND THE HELLMANN-FEYNMAN THEOREM 459 14.1 The Virial Theorem, 459 14.2 The Virial Theorem and Chemical Bonding, 466 14.3 The Hellmann-Feynman Theorem, 469 14.4 The Electrostatic Theorem, 472 14.5 Summary, 478 15 AB INITIO AND DENSITY-FUNCTIONAL TREATMENTS OF MOLECULÊS 480 15.1 Ab Tnitio, Density-Functional, Semiempirical, and Molecular-Mechanics Methods, 480 15.2 Electronic Terms of Polyatomic Molecules, 481 15.3 The SCF MO Treatment of Polyatomic Molecules, 485 15.4 Basis Functions, 486 15.5 Speeding Up Hartree-Fock Calculations, 494 15.6 The SCF MO Treatment of H,0, 498 15.7 Population Analysis, 505 15.8 The Molecular Electrostatic Potential and Atomic Charges, 508 15.9 Localized MOs,511 viii Contents 16 17 15,10 The SCF MO Treatment of Methane, Ethane, and Ethylene, 517 15.11 Molecular Geometry, 528 15.12 Conformational Searching, 539 15.13 Molecular Vibrational Frequencies, 545 15.14 Thermodynamic Properties, 548 15.15 Ab Initio Quantum Chemistry Programs, 550 15.16 Performing Ab Initio Calculations, 551 15.17 Configuration Interaction, 557 15.18 Mgller-Plesset (MP) Perturbation Theory, 563 15.19 The Coupled-Cluster Method, 568 15.20 Density-Functional Theory, 573 15.21 Composite Methods for Energy Calculations, 592 15.22 Solvent Effects, 593 15.23 Relativistic Effects, 602 15.24 Valence-Bond Treatment of Polyatomic Molecules, 604 15.25 The Generalized Valence-Bond Method, 612 15.26 Chemical Reactions, 613 SEMIEMPIRICAL AND MOLECULAR-MECHANICS TREATMENTS OF MOLECULES 626 16.1 Semiempirical MO Treatments of Planar Conjugated Molecules, 626 16.2 The Free-Electrov MO Method, 627 16.3 The Hiickel MO Method, 629 16.4 The Pariser-Parr-Pople Method, 650 16.5 General Semiempirical MO Methods, 652 16.6 The Molecular-Mechanies Method, 664 16.7 Empirical and Semiempirical Treatments of Solvent Effects, 680 16.8 Chemical Reactions, 684 COMPARISONS OF METHODS 693 17.1 Molecular Geometry, 693 17.2 Energy Changes, 696 17.3 Other Properties, 703 174 Hydrogen Bondinp, 705 17.5 Concelusion, 707 17.6 The Future of Quantum Chemistry, 708 APPENDIX 710 BIBLIOGRAPHY 712 ANSWERS TO SELECTED PROBLEMS 715 INDEX 721 x Preface and other needed topics. Rather than putting all the mathematics in an introductory chapter or a series of appendices, I have integrated the mathematics with the physics and chemistry. Immediate application of the mathematics to solving a quantum- mechanical problem will make the mathematics more meaningful to students than would separate study of the mathematics. I have also kept in mind the limited physics background of many chemistry students by reviewing topics in physics. “This book has benefited from the reviews and suggestions of Leland Allen, N. Colin Baird, James Bolton, Donald Chesnut, Melvyn Feinberg, Gordon A. Gallup, David Goldberg, Warren Hehre, Hans Jaffé, Neil Kestner, Harry King, Peter Kollman, Errol Lewars, Joel Liebman, Frank Meeks, Robert Metzger, William Palke, Gary Pfeiffer, Russell Pitzer, Kenneth Sando, Harrison Shull, James J. P. Stewart, Richard Stratt, Arieh Warshel, and Michael Zerner. Portions of the fifth edition were reviewed by Steven Bernasek, W. David Chandler, R. James Cross, David Farrelly, Tracy Ham- ilton, John Head, Miklos Kertesz, Mel Levy, Pedro Muifio, Sharon Palmer, and John S. Winn. Robert Gotwals” online computational chemistry course at the North Carolina Supercomputing Center allowed me to get experience using a supercomputer. I wish to thank these people and several anonymous reviewers. I would appreciate receiving any suggestions that readers may have for improv- ing the book, Tra N. Levine INLevineEbrooklyn.cuny.edu Quantum Chemistry «Ming, 5 É CHAPTER Tí The Schródinger Equation 1.1 QUANTUM CHEMISTRY In the late seventeenth century, Isaac Newton discovered classical mechanies, the laws of motion of macroscopic objects. In the early twentieth century, physicists found that classical mechanies does not correctly describe the behavior of very small particles such as the ejectrons and nuclei of atoms and molecules. The behavior of such particles is described by a set of laws called quantum mechanics. Quantum chemistry applies quantum mechanics to problems in chemistry. The influence of quantum chemistry is evident in all branches of chemistry. Physical chemists use quantum mechanies to calculate (with the aid of statistical mechanics) thermodynamic properties (for example, entropy, heat capacity) of gases; to interpret molecular spectra, thereby allowing experimental determination of molecular proper- ties (for example, bond lengths and bond angles, dipole moments, barriers to internal rotation, energy differences between conformational isomers); to calculate molecular properties theoretically: to calculate properties of transition states in chemical reac- tions, thereby allowing estimation of rate constants; to understand intermolecular forces; and to deal with bonding in solids. Organic chemists use quantum mechanics to estimate the relative stabilities of molecules, to calculate properties of reaction intermediates, to investigate the mecha- nisms of chemical reactions, and to analyze NMR spectra. Analytical chemists use spectroscopic methods extensively. The frequencies and intensities of lines in a spectrum can be properly understood and interpreted only through the use of quantum mechanics. Inorganic chemists use ligand field theory, an approximate quantum-mechanical method, to predict and explain the properties of transition-metal complex ions. Although the large size of biologically important molecules makes quantum- mechanical calculations on them extremely hard, biochemists are beginning to benefit from quantum-mechanical studies of conformations of biological molecules, enzyme- substrate binding, and solvation of biological molecules. Nowadays, several companies sell quantum-chemistry software for doing molec- ular quantum-chemistry calculations. These programs are designed to be used by all Xinds of chemists, not just quantum chemists. 2 Chapter 1 The Schrôdinger Equation 1.2 HISTORICAL BACKGROUND OF QUANTUM MECHANICS The development of quantum mechanics began in 1900 with Planck's study of the light emitted by heated solids, so we start by discussing the nature of light. In 1801, Thomas Young gave convincing experimental evidence for the wave nature of light by observing diffraction and interference when light went through two adjacent pinholes. (Diffraction is the bending of a wave around an obstacle, Interference is the combining of two waves of the same frequency to give a wave whose disturbance at each point in space is the algebraic or vector sum of the disturbances at that point resulting from each interfering wave. See any first-year physics text.) About 1860, James Clerk Maxwell developed four equations, known as Maxwell's equations, which unified the laws of electricity and magnetism. Maxwell's equations predicted that an accelerated electric charge would radiate energy in the form of electromagnetic waves consisting of osciliating electric and magnetic fields. The speed predicted by Maxwell's equations for these waves turned out to be lhe same as the experimentally measured speed of light. Maxwell concluded that light is an elec- tromagnetic wave. In 1888, Heinrich Hertz detected radio waves produced by accelerated electric charges in a spark, as predicted by Maxwell's equations. This convinced physicists that light is indeed an electromagnetic wave. AJ electromagnetic waves travel at speed c = 2.998 x 10º'm/s in vacuum. The frequency » and wavelength À of an electromagnetic wave are related by Av=€ (Lis (An equation with an asterisk after its number should be memorized.) Various conven- tional labels are applied to electromagnetic waves depending on their frequency. In order of increasing frequency are radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays. We shall use the term light to denote any kind of electromagnetic radiation. Wavelengths of visible and ultraviolet radiation were formerly given in angsíroms (À) and are now given in nanometers (nm): inm=10m, 1Ã=10-m=o0Olnm (LI Im the late 1800s, physicists measured the intensity of light at various frequencies emitted by a heated blackbody at a fixed temperature. A blackbody is an object that absorbs all light falling on it. A good approximation to a blackbody is a cavity with a tiny hole. When physicists used statistical mechanics and the electromagnetic-wave model of light to predict the intensity-versus-frequency curve for emitted blackbody radiation, they found a result in complete disagreement with the high-frequency por- tion of the experimental curves. In 1900, Max Planck developed a theory that gave excellent agreement with lhe observed blackbody-radiation curves. Planck assumed that the atoms of the blackbody could emil light energy oniy in amounts given by Av, where v is the radiation's fre- quency and A is a proportionality constant, called Planck's constant. The value h= 66x 104].s gave curves that agreed with the experimental blackbody curves. Planck's work marks the beginning of quantum mechanics. Planck's hypothesis that only certain quantities of light energy can be emitted 4 Chapter? The Schrôdinger Equation the positive charge were spread throughout the atom (as J. J. Thomson had proposed in 1904), once the high-energy alpha particle penetrated the atom, the repulsive force would fall off, becoming zero at the center of the atom, according to classical electro- statics. Hence Rutherford concluded that such large deflections could occur only if the positive charge were concentrated in a tiny, heavy nucleus. An atom contains a tiny (107! to 10"! em radius), heavy nucleus consisting of neutrons and Z protons, where Z is the atomic number. Outside thg nucleus there are Z electrons. The charged particles interact according to Coulomb's law. (The nucleons are held together in the nucleus by strong, short-range nuclear forces, which will not concern us.) The radius of an atom is about one angstrom, as shown, for example, by results from the kinetic theory of gases. Molecules have more than one nucleus. The chemical properties of atoms and molecules are determined by their elec- tronic structure, and so the question arises as to the nature of the motions and energies of the electrons. Since the nucleus is much more massive than the electron, we expect the motion of the nucleus to be slight compared with the electrons' motions. In 1911, Rutherford proposed his planetary model of the atom in which the elec- trons revolved about the nucleus in various orbits, just as the planets revolve about the sun. However, there is a fundamental difficulty with this model. According to classical electromagnetic theory, an accelerated charged particle radiates energy in the form of electromagnetic (light) waves. An electron circling the nucleus at constant speed is being accelerated, since the direction of its velocity vector is continually changing. Hence the electrons in the Rutherford model should continually lose energy by rad- iation and therefore would spiral toward the nucleus. Thus, according to classical (nineteenth-century) physics, the Rutherford atom is unstable and would collapse. A possible way out ef this difficulty was proposed by Niels Bohr in 1913, when he applied the concept of quantization of energy to the hydrogen atom. Bohr assumed that the energy of the electron in a hydrogen atom was quantized, with the electron constrained to move only on one of a number of allowed circles. When an electron makes a transition from one Bohr orbit to another, a photon cf light whose frequency v satisfies Eupper” Flower = hv (14)* is absorbed or emitted, where Eupper and Elower Are the energies of the upper and lower states (conservation of energy). With the assumption that an electron making a transi- tion from a free (ionized) state to one of the bound orbits emits a photon whose fre- quency is an integral multiple of one-half the classical frequency of revolution of the electron in the bound orbit, Bohr used classical mechanics to derive a formula for the hydrogen-atom energy levels. Using (1.4), he obtained agreement with the observed hydrogen spectrum. However, attempts to fit the helium spectrum using the Bohr the- ory failed. Moreover, the theory could not account for chemical bonds in molecules. The basic difficulty in the Bohr model arises from the use of classical mechanics to describe the electronic motions in atoms. The evidence of atomic spectra, which show discrete frequencies, indicates that only certain energies of motion are allowed; the electronic energy is quantized. However, classical mechanics allows a continuous range of energies. Quantization does occur in wave motion; for example, the funda- mental and overtone frequencies of a violin string. Hence Louis de Broglie suggested Section 1.3 The Uncertainty Principle 5 in 1923 that the motion of electrons might have a wave aspect; that an electron of mass m and speed v would have a wavelength h h 1 (1.5) associated with it, where p is the linear momentum. De Broglie arrived at Eq. (1.5) by reasoning in analogy with photons. The energy of any particle (including a photon) can be expressed, according to Einstein's special theory of relativity,as E = me? where cis the speed of light and 2 is the particle's relativistic mass (not its rest mass). Using Eptoton = hv, we get mc? = hy = hc/A and À = h/mc = h/p for a photon traveling at speed e. Equation (1.5) is the corresponding equation for an electron. mn 1927, Davisson and Germer experimentally confirmed de Broglie's hypothesis by reflecting electrong from metals and observing diffraction effects. In 1932, Stem observed the same effects with helium atoms and hydrogen molecules, thus verifying that the wave effects are not peculiar to electrons, but result from some general law of motion for microscopic particles. Thus electrons behave in some respects like particles and in other respects like waves. We are faced with the apparently contradictory “wave-particle duality” of mat- ter (and of light). How can an electron be both a particle, which is a localized entity, and a wave, which is nonlocalized? The answer is that an electron is neither a wave nor a particle, but something else. An accurate pictorial description of an electron's behav- ior is impossible using the wave or particle concept of classical physics. The concepts of classical physics have been developed from experience in the macroscopic world and do not properly describe the microscopic world. Evolution has shaped the human brain to allow it to understand and deal effectively with macroscopic phenomena. The human nervous system was not developed to deal with phenomena at the atomic and molecular level, so it is not surprising if we cannot fully understand such phenomena. Although both photons and electrons show an apparent duality, they are not the same kinds of entities. Photons always travel at speed c and have zero rest mass; electrons always have v < c and a nonzero rest mass. Photons must always be treated relativistically, but electrons whose speed is much Jess than c can be treated nonrela- tivistically. 1.3 THE UNCERTAINTY PRINCIPLE Let us consider what effect the wave-particle duality has on attempts to measure simultaneously the x coordinate and the x component of linear momentum of a micro- scopic particle. We start with a beam of particles with momentum p, traveling in the y direction, and we let the beam fall on a narrow slit. Behind this slit is a photographic plate. See Fig. 1.1. Particles that pass through the slit of width w have an uncertainty w in their x coordinate at the time of going through the slit. Calling this spread in x values Ax, we have Ax = w. Since microscopic particles have wave properties, they are diffracted by the slit producing (as would a light beam) a diffraction pattem on the plate. The height of the graphin Fig. 1.1 is a measure of the number of particles reaching a given point. The dif- Section 1.4 | The Time-Dependent Schródinger Equation 7 FIGURE 1.2 Calculation of first diffraction minimum, indicating that the product of the uncertainties in x and p, is of the order of magnitude of Planck's constant. In Section 5.1 we will give a precise statistical definition of the uncertainties and a precise inequality to replace (1.7). Although we have demonstrated (1.7) for only one experimental setup, its valid- ity is general, No matter what attempts are made, the wave-particle duality of micro- scopic “particles” imposes a limit on our ability to measure simultaneously the position and momentum of such particies. The more precisely we determine the position, the less accurate is our determination of momentum. (In Fig. 1.1, sin = A/w, so narrow- ing the slit increases the spread of the diffraction pattern.) This limitation is the uncer- tainty principle, discovered in 1927 by Werner Heisenberg. Because of the wave-particle duality, the act of measurement introduces an uncontrollable disturbance in the system being measured. We started with particles having a precise value of p, (zero). By imposing the slit, we measured the x coordinate of the particles to an accuracy w, but this measurement introduced an uncertainty into the p, values of the particles. The measurement changed the state of the system. 1.4 THE TIME-DEPENDENT SCHRÓDINGER EQUATION Classical mechanics applies only to macroscopic particles. For microscopic “particles” we require a new form of mechanics, called quantum mechanies. We now consider some of the contrasts between classical and quantum mechanics. For simplicity a one- particle, one-dimensional system will be discussed, In classical mechanics the motion of a particle is governed by Newton's second law: dx ar (1.8)*+ F-=ma-m where Fis the force acting on the particle, m is its mass, and t is the time; a is the accel- eration, given by a = dv/dt = (d/dt(dx/at) = dºx/d?, where v is the velocity. Equation (1.8) contains the second derivative of the coordinate x with respect to time. To solve it, we must carry out two integrations. This introduces two arbitrary constants ey and c; into the solution, and x=g(t, ce) (1.9) 8 Chapter1 The Schrôdinger Equation where g is some function of time. We now ask: What information must we possess at a given time & to be able to predict the future motion of the particle? If we know that at to the particle is at point xy, we have *o = Elo €C9) (1.10) Since we have two constants to determine, more information is needed. Differentiating (1.9), we bave dx d ( ) C-v=0 tac dt de Sb Cu em If we also know that at time & the particle has velocity v, then we have the additional relation d = 800163) Qua) =tg We may then use (1.40) and (1.11) to solve for c, and c, in terms of xp and vw. Knowing c, and c,, we can use Eg. (1.9) to predict the exact future motion of the particle. As an example of Egs. (1.8) to (1.1t), consider the vertical motion of a particle in the earth's gravitational field. Let the x axis point upward. The force on the particle is downward andis 7 = —mg, where g is the gravitational acceleration constant. Newton's second law (1.8) is —mg = m d?x/df?, so dºx/df? = —g. A single integration gives dx/dt = —gt + c. The arbitrary constant c, can be found if we know that at time £ the particle had velocity w. Since v = dx/dt, we have mw = —gf tc and c; = vw + gh Therefore dx/dt = —gt + go + % Integration gives x = —igt? + (gty + mt + c TÊ we know that at time £ the particle had position xo, then xo = —1giê + (gto + voy + & and c; = xo Sgt — voto. The expression for x as a function of time becomes x= Ig? + (go + ol + xo bg — auto ora = xo belt — to) + ut — to). Know ing x, and v at time to, we can predict the future position of the particle. The classical-mechanical potential energy V of a particle moving in one dimen- sion is defined to satisfy aV(x, t)/9x = —F(x,t) (LIZ) For example, for a particle moving in the earth's gravitational field, 9V/0x = —F = mg and integration gives V = mgx + c, where cis an arbitrary constant. We are free to set the zero level of potential energy wherever we please. Choosing c = 0, we have V = mgx as the potential-energy function, The word state in classical mechanics means a specification of the position and velocity of each particle of the system at some instant of time, plus specification of the forces acting on the particles. According to Newton's second law, given the state of a system at any time, its future state and future motions are exactly determined, as shown by Egs. (1.9)-(1.11). The impressive success of Newton's laws in explaining planetary motions led many philosophers to use Newton's laws as an argument for philosophical determinism. The mathematician and astronomer Laplace (1749-1827) assumed that the universe consisted of nothing but particles that obeyed Newton's laws. Therefore, given the state of the universe at some instant, the future motion of 
