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Explore the principles of quantum mechanics applied to chemistry and its applications in understanding molecular properties, reactions, and spectroscopy.
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بسم الله الرحمن الرحيم INTRODUCTION IN QUANTUM CHEMISTRY • In the late seventeenth century, Isaac Newton discovered classical mechanics, the laws of motion of macroscopic objects. • In the early twentieth century, physicists found that classical mechanics does not correctly describe the behavior of very small particles such as the electrons and nuclei of atoms and molecules. • The behavior of such particles is described by a set of appropriate concepts and equations, which is called quantum mechanics. • Quantum chemistry applies quantum mechanics to problems in chemistry.
Physical chemists use quantum mechanics 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 properties (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 reactions, thereby allowing estimation of rate constants • to understand intermolecular forces; and to deal with bonding in solids • Inorganic chemists use ligand field theory, an approximate quantum-mechanical method, to predict and explain the properties of transition-metal complex ions.
Organic chemists use quantum mechanics to: • estimate the relative stabilities of molecules • to calculate properties of reaction intermediates • to investigate the mechanisms of chemical reactions • to analyze NMR spectra • Biochemists are beginning to benefit from quantum-mechanical studies of conformations of biological molecules, enzyme-substrate binding, and solvation of biological molecules. • Several companies sell quantum-chemistry software for doing molecular quantum-chemistry calculations. These programs are designed to be used by all kinds of chemists, not just quantum chemists.
Essential concepts of physics Energy The central concept of all explanations in physical chemistry, as in so many other branches of physical science, is that of energy, the capacity to do work. • energy is conserved Kinetic and potential energy The kinetic energy, EK , of a body is the energy the body possesses as a result of its motion. For a body of mass m travelling at a speed v
The potential energy, Epor V, of a body is the energy it possesses as a result of its position • No universal expression for the potential energy can be given because it depends on the type of interaction the body experiences. • The zero of potential energy is arbitrary. • For example, the gravitational potential energy of a body is often set to zero at the surface of the Earth; the electrical potential energy of two charged particles is set to zero when their separation is infinite. • The total energy is the sum of the kinetic and potential energies of a particle: E = EK + EP The SI unit of energy is the joule , which is defined as 1 J = 1 kg m2 s-2
The rate of change of energy is called the power, P, expressed as joules per second, or watt, W: 1W = 1 J s-1 Classical mechanics Classical mechanics describes the behavior of objects in terms of two equations. One expresses the fact that the total energy is constant in the absence of external forces The other expresses the response of particles to the forces acting on them. The trajectory in terms of the energy The velocity, v, of a particle is the rate of change of its position:
The velocity is a vector, with both direction and magnitude. The magnitude of the velocity is the speed, v. The linear momentum, p, of a particle of mass m is related to its velocity, v, by P = mv Like the velocity vector, the linear momentum vector points in the direction of travel of the particle (Fig. 1). In terms of the linear momentum, the total energy of a particle is This equation can be used to show that a particle will have a definite trajectory, or definite position and momentum at each instant
For example, consider a particle free to move in one direction (along the x-axis) in a region where V = 0 (so the energy is independent of position). Because v = dx/dr, it follows from eqns. that A solution of this differential equation is The linear momentum is a constant: Hence, if we know the initial position and momentum, we can predict all later positions and momenta exactly.
Newton's second law • The force, F, experienced by a particle free to move in one dimension is related to its potential energy, V, by This relation implies that the direction of the force is towards decreasing potential energy • Newton's second law of motion states that the rate of change of momentum is equal to the force acting on the particle. In one dimension: Because p = m(dx/dt) in one dimension, it is sometimes more convenient to write this equation as
The second derivative, d2x/dt2, is the acceleration of the particle, its rate of change of velocity (in this instance, along the x-axis). • It follows that, if we know the force acting everywhere and at all times, then solving eqn. will also give the trajectory. • For example, it can be used to show that, if a particle of mass m is initially stationary and is subjected to a constant force F for a time t, then its kinetic energy increases from zero to • Because the applied force, F, and the time, t, for which it acts may be varied at will, the solution implies that the energy of the particle may be increased to any value.
In brief, classical physics 1) predicts a precise trajectory for particles, with precisely specified locations and momenta at each instant. 2) allows the translational, rotational, and vibrational modes of motion to be excited to any energy simply by controlling the forces that are applied. • Careful experiments show that classical mechanics fails when applied to the transfers of very small energies and to objects of very small mass.
Waves • Waves are disturbances that travel through space with a finite velocity • Waves can be characterized by a wave equation, a differential equation that describes the motion of the wave in space and time. • Harmonic waves are waves with displacements that can be expressed as sine or cosine functions. A harmonic wave at a particular instant in time. A is the amplitude and λ is the wavelength.
Electromagnetic Radiation There are many kinds of waves, such as water waves, sound waves, and light waves. In 1873 James Clerk Maxwell proposed that visible light consists of electromagnetic waves. The significance of Maxwell’s theory is that it provides a mathematical description of the general behavior of light. In particular, his model accurately describes how energy in the form of radiation can be propagated through space as vibrating electric and magnetic fields. The electromagnetic field • In classical physics, electromagnetic radiation is understood in terms of the electromagnetic field • Electromagnetic field, an oscillating electric and magnetic disturbance that spreads as a harmonic wave through empty space, the vacuum. • The wave travels at a constant speed called the speed of light, c, which is about 3×108 m s−1.
An electromagnetic field has two components 1. an electric field that acts on charged particles (whether stationary of moving) 2. a magnetic field that acts only on moving charged particles. These two components have the same wavelength and frequency, and hence the same speed, but they travel in mutually perpendicular planes • The waves associated with electromagnetic radiation are called electromagnetic waves Electromagnetic radiation is the emission and transmission of energy in the form of electromagnetic waves.
The electromagnetic field is characterized by 1. wavelength, (lambda), the distance between the neighboring peaks of the wave 2. Frequency, v (nu), the number of times per second at which its displacement at a fixed point returns to its original value, the frequency is measured in hertz, where 1Hz = 1s−1 The wavelength and frequency of an electromagnetic wave are related by =c Therefore, the shorter the wavelength, the higher the frequency.
EXAMPLE The wavelength of the green light from a traffic signal is centered at 522 nm. What is the frequency of this radiation?
Light As An Electromagnetic Wave • Electromagnetic waves would be reflected by metal mirrors • would be refracted by dielectrics like glass • would exhibit polarization and interference • and would travel outward from the wire through a vacuum with a speed of 3.0 × 108 m/s.
The failures of classical physics 1. Black-body radiation • When burner of a stove is heated, it first turns dull red and becomes more and more red as the temperature increases. • We know that as body is heated even further, the radiation becomes white and even becomes blue as the temperature becomes higher and higher. • A heated iron bar glowing red hot becomes white hot when heated further. When solids are heated, they emit electromagnetic radiation over a wide range of wavelengths. Examples of radiation from heated solids. • The dull red glow of an electric heater • The bright white light of a tungsten light bulb
At high temperatures, an appreciable proportion of the radiation is in the visible region of the spectrum. • and a higher proportion of short-wavelength blue light is generated as the temperature is raised. • Measurements showed that the amount of radiation energy emitted by an object at a certain temperature depends on its wavelength. Fig. Emission from a glowing solid. Note that the amount of radiation emitted (the area under the curve) increases rapidly with increasing temperature
Black body is an object that emits and absorbs all frequencies of radiation uniformly. • The radiation emitted by black body is called black body radiation. From a classical viewpoint, Rayleigh and Jeans derived an expression depending on the assumption: The radiation emitted by the body was due to the oscillation of the electrons in the constituent particles of material body. • Electrons oscillate in an antenna to give off radio waves. • The oscillations occur at a much higher frequency and hence we find frequencies in the visible, infrared, and ultraviolet regions • A good approximation to a black body is a pinhole in an empty container maintained at a constant temperature, because any radiation leaking out of the hole has been absorbed and re-emitted inside so many times that it has come to thermal equilibrium with the walls.
The electromagnetic field is a collection of oscillators of all possible frequencies. They regarded the presence of radiation of frequency v (and therefore of wavelength = c/v) as signifying that the electromagnetic oscillator of that frequency had been excited.
dE is the energy density, in the range of wavelengths between and + d, with unit (Jm−3) (rho), the density of states, with unit (Jm−4) is the proportionality constant between d and the energy density dE k is Boltzmann's constant (k = 1.381 x 10-23 J K-1). • The Rayleigh-Jeans law is quite successful at long wavelengths (low frequencies) • It fails badly at short wavelengths (high frequencies). Thus, as decreases, r increases without going through a maximum.
Ultraviolet catastrophe : a large amount of energy is radiated in the high-frequency region of the electromagnetic spectrum. Implicit in the derivation of Rayleigh and Jeans is the assumption that the energy of the electronic oscillators responsible for the emission of the radiation could have any value whatsoever. (One of the basic assumptions of classical physics) All possible energies are allowed.
The Planck distribution • The German physicist Max Planck studied black-body radiation from the viewpoint of thermodynamics. Assumption: the energy of the oscillators had to be proportional to an integer multiple of the frequency, in an equation that E=nhv n = 0, 1, 2, 3, 4,….. where h is called Planck’s constant v is the frequency of radiation Planck said that atoms and molecules could emit (or absorb) energy only in discrete quantities • Planck gave the name quantum to the smallest quantity of energy that can be emitted (or absorbed) in the form of electromagnetic radiation. • The energy E of a single quantum of energy is given by E=hv • The limitation of energies to discrete values is called the quantization of energy.
On the basis of this assumption, Planck was able to derive the Planck distribution: Plank distribution law for black body radiation • Plank was able to show that this equation gives excellent agreement with the experimental data for all frequencies and temperatures if h has the value 6.626 × 10-34 Js
The Planck distribution accounts very well for the experimentally determined distribution of black-body radiation.
Photoelectric effect A phenomenon in which electrons are ejected from the surface of certain metals exposed to light of at least a certain minimum frequency, called the threshold frequency (v0). • In 1902 Lennard obtained the following results after careful measurements: • The kinetic energy (m/2)v2 of the photoelectrons is dependent solely on the wavelength λ of the incident light, not on its intensity! • The number of ejected photoelectrons is proportional to the light intensity. Fig. Photoelectric effect apparatus. • There is no measurable time delay between irradiation and electron ejection. • Below the threshold frequency no electrons were ejected no matter how intense the light.
Kmax is easily measured by applying a retarding voltage and gradually increasing it until the most energetic electrons are stopped and the photocurrent becomes zero. At this point, • Where me is the mass of the electron, max is the maximum electron speed, e is the electronic charge, and Vs is the stopping voltage. Kmax v v0 Fig. A graph showing the dependence of Kmax on light frequency.
The photoelectric effect could not be explained by the wave theory of light. • Using Planck’s quantum theory of radiation as a starting point, Einstein, however, made an extraordinary assumption. The energy of light is not distributed evenly over the classical wavefront, but is concentrated in discrete regions (or in “bundles”), called quanta, each containing energy, hv. These particles of light are now called photons. each photon must possess energy E, given by the equation Photon with energy hv Fig. (a) A classical view of a traveling light wave. (b) Einstein’s photon picture of “a traveling light wave.”
These particles of light are now called photons. each photon must possess energy E, given by the equation This situation is summarized by the equation Where is the work function The work function is the minimum energy required to detach an electron from a given substance
v v02 v01 Fig. Universal characteristics of all metals undergoing the photoelectric effect. • With any theory, one looks not only for explanations of previously observed results but also for new predictions.
Example The work function of nickel equals 5.0 eV. Find (a) the threshold wavelength for nickel and (b) the maximum electron speed for a wavelength of 195 nm.
Atomic and molecular spectra • Spectroscopy, the detection and analysis of the electromagnetic radiation absorbed, emitted, or scattered by a substance. • The record of the intensity of light transmitted or scattered by a molecule as a function of frequency (v), wavelength (), or wavenumber (ṽ= v/c) is called its spectrum Fig. A region of the spectrum of radiation emitted by excited iron atoms consists of radiation at a series of discrete wavelengths (or frequencies).
This observation can be understood if the energy of the atoms or molecules is also confined to discrete values, for then energy can be discarded or absorbed only in discrete amounts • Then, if the energy of an atom decreases by DE, the energy is carried away as radiation of frequency v, and an emission 'line', a sharply defined peak, appears in the spectrum. DE=hv Wave-particle duality • The energies of the electromagnetic field and of oscillating atoms are quantized. • That electromagnetic radiation-which classical physics treats as wave-like-actually also displays the characteristics of particles.
The observation that electromagnetic radiation of frequency v can possess only the energies 0, hv, 2hv, ... suggests that it can be thought of as consisting of 0, 1, 2, ... particles, each particle having an energy hv. • Then, if one of these particles is present, the energy is hv, if two are present the energy is 2hv, and so on. • These particles of electromagnetic radiation are now called photons.
Bohr’s Theory Of The Hydrogen Atom • Emission spectra, that is, either continuous or line spectra of radiation emitted by substances. • The emission spectrum of a substance can be seen by energizing a sample of material either with thermal energy or with some other form of energy • The emission spectra of atoms in the gas phase produce bright lines in different parts of the visible spectrum. • These line spectra are the light emission only at specific wavelengths.
Rydberg was able to represent the wavelengths of all of the spectral lines of hydrogen atoms with a single empirical formula: where n1 and n2 are two positive integers and RH is a constant known as Rydberg’s constant, equal to 1.09677581×107 m−1 if the wavelengths are measured in a vacuum. • Classical physics was unable to explain this relationship.
A simplified version of Bohr’s assumptions is: 1. The hydrogen atom consists of a positive nucleus of charge e and an electron of charge e−moving around it in a circular orbit. 2. The angular momentum of the electron is quantized: Its magnitude can take on one of the values h/2π, 2h/2π, 3h/2π, 4h/2π, . . . , nh/2π, where h is Planck’s constant and where n is an integer (a quantum number). 3. Maxwell’s equations do not apply. Radiation is emitted or absorbed only when a transition is made from one quantized value of the angular momentum to another. (stationary electron orbits) 4. The wavelength of emitted or absorbed light is given by the Planck–Einstein relation, with the energy of the photon equal to the difference in energy of the initial and final states of the atom. 5. In all other regards, classical mechanics is valid.
Consider a particle rotating in a plane about a fixed center as in the following Fig. Let vrot be the frequency of rotation (cycle/second). The velocity of the particle is =2rvrot=rrot Where rot=2vrot and is called angular velocity • The kinetic energy of the revolving particle is Where I = mr2 is the moment of inertia
We can make the correspondences and Im • Where and I are angular quantities and and m are linear quantities. • According to this correspondence, there should be a quantity I corresponding to the linear momentum • Linear momentum p = m, In fact the quantity l, defined by • Kinetic energy can be written in terms of momentum. For linear system And for rotating system,
From general physics that a particle revolving around a fixed point experiences an outward acceleration, and requires an inward force (f) to keep it moving in a circular orbit where is the speed of the electron, m is its mass, and r is the radius of the orbit • The nucleus is much more massive than the electron and the electron moves about it almost as if the nucleus were stationary. • Coulomb’s law for the force between a charge Q1 and another charge Q2 separated by a vacuum is where 0 is the permittivity of the vacuum, equal to 8.8545×10−12 C2 N−1 m−2, and where r12 is the distance between the charges. • If the two charges are of opposite sign, the force is negative, corresponding to attraction.
The force holding the electron on a circular orbit is supplied by the Coulombic force of attraction between the proton and the electron (Q1 = Q2= e) If we equate Coulomb’s force with f , then we have Eq. 1 The angular momentum of the electron must be quantized according to assumption 2, and is given by where the quantum number n is a positive integer Using where ( called h-bar) and occurs often in quantum mechanics.
Solving for the speed and the result is substituted into Eq. (1). The resulting equation is solved for the radius of the orbit (r) to give
Thus we see that the radii of the allowed orbits, or Bohr orbits, are quantized The orbit with the smallest radius is the orbit with n=1: r= 0.529 Å The radius of the first Bohr orbit is often denoted by a0 • The potential energy for an electron of charge e−at a distance r from a nucleus of charge e is The negative sign here indicates the proton and electron attract each other
From The kinetic energy is given by The total energy of the hydrogen atom is given by the formula: The only allowed values for of r are those given by And so if we substitute into the previous equation, then we find that the allowed energies are
n=1 corresponds to the lowest energy (ground-state energy) the states of higher energy are called excited states and are generally unstable with respect to the ground state. The energy is quantized and is determined by the value of the quantum number, n. With the accepted values of the physical constants, we can write where the electron-volt (eV) is the energy required to move an electron through a potential difference of 1 volt:
Bohr postulated that a photon is emitted or absorbed only when the electron makes a transition from one energy level to another. The energy of an emitted or absorbed photon is equal to the difference between two quantized energies of the atom: where n2 is the quantum number of the final state and n1 is the quantum number of the initial state of the atom. Setting E=hv : is called the Bohr frequency condition
