This chapter discusses the central concepts and equations of quantum theory which
demonstrates how particles spread through space like waves and are described mathematically
by a wavefunction. It examines how the wavefunction is interpreted and introduces the
uncertainty principle, which is one of the most profound departures of quantum mechanics
from classical mechanics. It also highlights the dynamical properties of a system that are
contained in the wavefunction and are obtained by solving the Schrödinger equation. The
chapter reconciles the facts that atoms and molecules can possess only certain energies,
that waves exhibit the properties of particles, and that particles exhibit the properties of
waves. It reviews the classical concept that have been accommodated by the development of
quantum mechanics, in which equations are set up to treat a particle as spread through space
like a wave.

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### Chapter

## The dynamics of microscopic systems

### Chapter

## The quantum mechanics of motion

This chapter pays particular attention to the quantum mechanics of motion. It is not possible to explain the structures and properties of atoms and molecules, where the role of electrons is central, without using quantum mechanics. Many of the processes of biochemistry involve the transfer of electrons and protons from a donor to an acceptor, and this migration can be understood only in terms of quantum theory. Here, the chapter introduces three fundamental types of motion — translation, rotation, and vibration — along with their characteristic quantum mechanical behaviours. It shows how various unexpected properties emerge, such as the ability of particles to penetrate into classically forbidden regions, the quantization of energy and angular momentum, and the existence of energy that cannot be removed from the system.

### Chapter

## Wavefunctions

This chapter examines the interpretation of the wavefunction, and specifically what
it reveals about the location of a particle. In quantum mechanics, all the
properties of a system are expressed in terms of a wavefunction, which is obtained
by solving the equation proposed by Erwin Schrödinger. Indeed, wavefunctions provide
the essential foundation for understanding the properties of electrons in atoms and
molecules, and are central to explanations in chemistry. The chapter then considers
how, according to the Born interpretation, the probability density at a point is
proportional to the square of the wavefunction at that point. A wavefunction is
normalized if the integral over all space of its square modulus is equal to 1.
Ultimately, a wavefunction must be single-valued, continuous, not infinite over a
finite region of space, and have a continuous slope. The quantization of energy
stems from the constraints that an acceptable wavefunction must satisfy.

### Chapter

## Translation

This chapter describes a particle confined to a finite region of space that can possess
only certain discrete energies. It also looks at the corresponding wavefunctions. It
analyzes the quantization of energy that emerges as a natural consequence of solving the
Schrödinger equation and the conditions imposed on it. It also introduces a striking
non-classical feature of small particles, which describes their ability to penetrate into
and through regions where classical physics would forbid them to be found. The chapter talks
about the translational energy levels of a particle which are confined to a finite region of
space and are quantized. It explores how quantum mechanics reveal the ability of a particle
to penetrate into and through regions where classical physics would forbid it to be
found.

### Book

### T. P. Softley

Atomic Spectra starts off by looking at quantum mechanics and the relationship of quantum mechanics with light. The next chapter considers the structure and spectrum of the hydrogen atoms. The text also covers the spectrum of the helium atom. Finally, the text examines the spectra of many-electron atoms.

### Book

### H. Grant Guy and Richards W. Graham

Computational Chemistry starts by arguing that the uses of computers in chemistry are many and varied. This ranges from the modelling of solid state systems to the design of complex molecules which can be used as drugs. This text introduces the many methods currently used by practising computational chemists and shows the value of computers in modern chemical research. The text describes the various computational techniques available and explains how they can be applied to single molecules, to assemblies of molecules, and to molecules undergoing reaction. An introductory chapter outlines the hardware and software available, and looks at some applications and developments. Subsequent chapters cover quantum mechanics, molecular mechanics, statistical mechanics, the modelling of biomolecules, and drug design. Whilst emphasizing the use of computers to model biological systems, the chapters explain how the methods can be applied to a whole range of chemical problems.

### Chapter

## Translational motion

This chapter evaluates translational motion, motion through space, which is one of
the fundamental types of motion treated by quantum mechanics. According to quantum
theory, a particle constrained to move in a finite region of space is described by
only certain wavefunctions and can possess only certain energies. That is,
quantization emerges as a natural consequence of solving the Schrödinger equation
and the conditions imposed on it. The solutions also expose a number of
non-classical features of particles, especially their ability to tunnel into and
through regions where classical physics would forbid them to be found. Light
particles are more able to tunnel through barriers than heavy ones. The chapter also
considers the occurrence of degeneracy, which is a consequence of the symmetry of
the system.

### Chapter

## Operators and observables

This chapter explains that a central feature of quantum theory is its
representation of observables by ‘operators’, which act on the wavefunction and
extract the information it contains. It shows how operators are constructed and
used. Observables are represented by hermitian operators, which have real
eigenvalues and orthogonal eigenfunctions. When the system is not described by a
single eigenfunction of an operator, it may be expressed as a superposition of such
eigenfunctions. One consequence of the use of operators is the ‘uncertainty
principle’, one of the most profound departures of quantum mechanics from classical
mechanics. The uncertainty principle restricts the precision with which
complementary observables may be specified and measured simultaneously.

### Chapter

## Quantum Mechanics and Spectroscopy

This chapter reviews the basic quantum mechanics of spectroscopy, analyzing forms of spectroscopy based on the measurement of energy as an atom or molecule undergoes a change in quantum states. It looks at different aspects of quantum mechanics in order to understand instrumental techniques and analysis involving the interaction of light and matter. It also looks at how spectroscopy explores particular energy level differences between quantum states in molecules and atoms. The chapter stresses the importance of quantum mechanics in the development of spectroscopic techniques used to investigate atomic composition and chemical structures. It cites an example that outlines the quantum mechanic events that led to the line spectrum of hydrogen in a gas in a discharge tube.

### Chapter

## Quantum Chemistry

This chapter discusses quantum chemistry. Pretty much all of computational chemistry relies on quantum mechanics in the sense that molecular systems follow the laws of quantum mechanics. But ‘quantum chemistry‘ has a more specific meaning: it is the study of chemistry through the use of approximate solutions to the electronic Schrödinger equation. The chapter describes the background and principles of quantum chemistry, with a focus on the conceptually most important approximate approach, which is Hartree–Fock theory. In principle, the time-dependent Schrödinger equation could be used to predict what happens in chemical reactions. The basic approximation in Hartree–Fock theory is an assumption that electrons move independently of one another throughout the molecular system. The chapter then looks at the calculation of a Hartree–Fock wavefunction.

### Book

### James Keeler and Peter Wothers

Why Chemical Reactions Happen provides all the tools and concepts needed to think like a chemist. The text takes a unified approach to the subject, aiming to help with the development of a real overview of chemical processes, by avoiding the traditional divisions of physical, inorganic, and organic chemistry. To understand how chemical reactions happen we need to know about the bonding in molecules, how molecules interact, what determines whether an interaction is favourable or not, and what the outcome will be. Answering these questions requires an understanding of topics from quantum mechanics, through thermodynamics, to ‘curly arrows’.

### Chapter

## Probability II

### Partition functions and wavefunctions

This chapter assesses partition functions and wavefunctions. Probability is best treated statistically when we deal with a vast number of particles. Statistical probabilities are usually considered in terms of a certainty or a proportion rather than simpler probabilities. The chapter then looks at the Boltzmann distribution, which describes the population of the energy levels which has the greatest weight and therefore the highest probability. We can further simplify the Boltzmann distribution by writing it in terms of the molecular partition function. The chapter also considers equilibrium constants, residual entropy, quantum mechanics, and tunnelling probability. Quantum tunnelling can occur when a particle tunnels through an energy barrier that classical mechanics would prevent.

### Chapter

## The origins of quantum mechanics

This chapter traces the origins of quantum mechanics. The classical mechanics
developed by Isaac Newton in the seventeenth century is an extraordinarily
successful theory for describing the motion of everyday objects and planets.
However, late in the nineteenth century, scientists started to make observations
that could not be explained by classical mechanics. They were forced to revise their
entire conception of the nature of matter and replace classical mechanics by a
theory that became known as quantum mechanics. The chapter begins by looking at
energy quantization, considering black-body radiation, heat capacity, and atomic and
molecular spectra. It then discusses wave–particle duality, which is the recognition
that the concepts of particle and wave blend together.

### Book

### Peter Atkins, Julio de Paula, and Ronald Friedman

Physical Chemistry starts off by looking at the foundations of the subject and provides the reader with some mathematical background. It then looks at quantum mechanics, taking into consideration the quantum mechanics of motion, molecular structure, and molecular symmetry. It then considers Fourier transforms, molecular spectroscopy, magnetic resonance, statistical thermodynamics, probability theory, the first law of thermodynamics, and multivariate calculus. The final part of the book examines physical equilibria, chemical equilibria, molecular motion, chemical kinetics, reaction dynamics, and processes in fluid systems and solid states.

### Chapter

## Quantum mechanics and spectroscopy

This chapter addresses quantum mechanics and spectroscopy. Two key ideas emerge from the theory of quantum mechanics: the first is that the electron is described by a wavefunction, and the second is that the energy of the electron is quantized. The wavefunction is important since its square gives the probability density of the electron. Quantization means that the energy can only have certain discrete values, rather than being allowed to vary continuously. The chapter looks at how these wavefunctions and energy levels arise in quantum mechanics, and how they can actually be computed in some simple cases. This will involve setting up and then solving the famous Schrödinger equation. Quantum mechanics is not restricted to the description of electrons in atoms and molecules, but can also be used to describe the motion of molecules in space (translation), as well as the rotation and vibration of molecules.

### Book

### Robert M. Granger, Hank M. Yochum, Jill N. Granger, Karl D. Sienerth, Robert M. Granger, Hank M. Yochum, Jill N. Granger, and Karl D. Sienerth

Instrumental Analysis XE
begins by presenting the analyst's toolbox. The next chapter covers quantum mechanics and spectroscopy. Other topics include optics, instrumental electronics, signals and noise and signal processing, molecular ultraviolet and visible spectroscopy, and atomic absorption spectroscopy. The text then goes on to cover luminescence spectroscopy, atomic emission spectroscopy, infrared spectroscopy, and raman spectroscopy. There are also chapters on mass spectroscopy, nuclear magnetic resonance spectroscopy, and liquid chromatography. Next, there are chapters on gas chromatography and electrophoresis. Finally, the text looks at potentiometry and probes and statistical data analysis.

### Chapter

## Determinants

This chapter takes a closer look at the theory of determinants, preparing the reader for the topic of matrix algebra. Determinants have certain symmetry properties that have made them an important tool in quantum mechanics. The chapter starts by explaining how the system of three linear equations in three unknowns can be solved. It also shows the solution of linear equations—introducing the Cramer's rule. The chapter also tackles the general properties of determinants such as transposition, multiplication by a constant, additional rule, interchange of rows or columns, equal two rows or columns, proportional rows or columns, among others. Furthermore, it explains the reduction of determinants to triangular form and describes alternating functions.

### Chapter

## Electrons in atoms

This chapter explores electrons in atoms. It mentions how a chemical bond involves the sharing of electrons. The chapter refers to the concept behind quantum mechanics as the behaviour of electrons in atoms and molecules. It notes the significance of atomic energy levels, atomic orbitals, and electronic configuration by explaining hydrogen orbitals. Photoelectron spectroscopy (PES) provides a particularly direct way of looking into the energy levels of electrons in atoms and molecules. The chapter explores how wavefunctions predict the spatial distribution of electrons. It also includes the usage of radial distribution functions to work out the total probability of finding the electron at a set distance from the nucleus. In addition, electronegativity is related to the atom which attracts more of the electron density.

### Chapter

## Molecular Mechanics Methods

This chapter evaluates molecular mechanics methods. In this approach, a known chemical bonding pattern is assumed and used to define preferred bond lengths and angles, and thereby an energy expression that takes into account distortions away from these ideal values. For a given bonding environment, the type of energy terms needed, and the numerical parameters within the energy expression, are transferable from one system to another. Hence, general forcefields can be constructed with quite general applicability. The chapter describes how the energy terms and parameters are chosen, based on input from experiment and quantum chemistry. Molecular mechanics can be applied to large systems due to its efficiency, allowing calculations on liquids, solutions, and solids. This frequently makes use of periodically repeating models and the chapter looks at special measures needed to treat such models. Finally, it discusses the type of software used for molecular mechanics.

### Chapter

## Hybrid and Multi-Scale Methods

This chapter addresses hybrid and multi-scale methods. Chemistry of a ‘core’ system is frequently perturbed by the ‘environment’. If the perturbation is weak, as is often the case, it is acceptable to perform calculations on the core part only. The effects of the environment can in many cases be described using continuum models, which can conveniently be coupled to quantum chemical calculations. It is also possible to devise hybrid methods, in which the atoms making up the core and the environment in the model are treated at different levels of theory. One very popular family of hybrid methods treats the core quantum mechanically (QM) and the environment with molecular mechanics (MM), and these methods are referred to as QM/MM methods.

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