This Physical Chemistry lecture uses graphs from the following textbooks:

This Physical Chemistry lecture uses graphs from the following textbooks:

P.W. Atkins, Physical Chemistry, 7. ed., Oxford University Press, Oxford 2003

G. Wedler, Lehrbuch der Physikalischen Chemie, 4. ed., Wiley-VCH, Weinheim 1997

Some Spectroscopic Techniques in Chemical Industry

Infrared (IR) Spectroscopy: molecular rotation and vibration (chemical identity, structure, and concentration)

Near Infrared (NIR) Spectroscopy: molecular vibration (chemical identity and concentration)

Raman Spectroscopy (RS): molecular rotation and vibration (chemical identity and structure)

Ultraviolet and Visible (UV/VIS) Spectroscopy: electronic transitions in atoms and molecules(Atomic Absorption Spectroscopy (AAS)) (chemical identity and concentration)

Nuclear Magnetic Resonance (NMR) Spectroscopy: nuclear orientation (chemical identity and structure)

X-Ray Photoelectron Spectroscopy (XPS): determination of surface composition (surface stoichiometry and concentration)

Mass Spectrometry (MS): determination of atomic and molecular masses (chemical identity and structure)

Spectroscopy: The analysis of the electromagnetic radiation emitted, absorbed, or scattered by atoms or molecules

The electromagnetic spectrum and the classification of the spectral regions. The band at the bottom indicates the types of transitions that absorb or emit in the various regions (‘nuclear magnetism’ refers to the types of transition used in NMR spectroscopy, ‘nuclear’ to transitions within the nucleus).

The origin of quantum mechanics

Spectroscopy is not understandable on the basis of classical mechanics, which

(1) predicts a precise trajectory for particles, with precisely specified locations and momenta at each instant, and

(2) allows the translational, rotational, and vibrational modes of motion to be excited to any energy simply by controlling the forces that are applied.

These conclusions agree with everyday experience. Everyday experience, however, does not extend to individual atoms. In particular, experiments have shown that systems can take up energy only in discrete amounts.

This observation is known as the

failure of classical physicswhich means that classical mechanics fails when applied to the transfers of very small quantities of energy and to objects of very small mass.

Most prominent examples are e.g. the black-body radiation, atomic and molecular spectra, the photoelectric effect, and the diffraction of particles.

Black-body radiation

An experimental representation of a black body is a pinhole in an otherwise closed container. The radiation is reflected many times within the container and comes to thermal equilibrium with the walls at a temperature T. Radiation leaking out through the pinhole is characteristic of the radiation within the container, and only a fundtion of temperature. The energy distribution in a black-body cavity at several temperatures.

Note how the energy density increases in the visible region as the temperature is raised, and how the peak shifts to shorter wavelength. The total energy density (the area under the curve) increases as the temperature is increased (as T4).

Empirical laws:

(1) Position of the maximum:

Tmax = const. = 0.288 Kcm(Wien displacement law)

e.g. the sun: T6000 K max480 nm

(2) Exitance, i.e. the power emitted by a region of surface divided by the area of the surface:

M = T4 = 5,6710-8 Wm-2K-4

(Stefan-Boltzmann law), „T4 law“

Problem: Energy density distribution

according to Maxwell‘s electro-dynamic theory:

and : frequencyE(): energy density distributionU: oscillator strength of the excited

electronsc: speed of light

classical treatment : U = kT

(reason: 2 quadratic degrees of freedom for each vibration)

this implies that all frequencies should be uniformly excited!

dUc

dE 2

2

dTkc

dE 2

2

with:

follows

Rayleigh-Jeans law

agrees withexperiment onlyfor large

theory predictsE() for 0

„classical ultravioletcatastrophe“

dc

dc

2,

dTkc

dE 4

The problem was solved by Max Planck. He could account for the observed distribution of energy if he supposed that the permitted energies of an electromagnetic oscillator of frequency are integer multiples of h:

E = n·h· n = 0, 1, 2, …

where h is a fundamental constant known as Planck’s constant (h=6.62608·10-34 J·s).

After introduction of a „mean oscillator energy“

the Planck distribution could be derived:

b

a

e

hU

kTh 1/ a: energy of oscillation with frequency

1/b: probability for excitation of this particular oscillation

de

hcdE

dec

hdE

kThc

kTh

1

1

/5

2

/2

3

a) fits experimental curve very well

b) for h 0 the Planck distribution would approach the Rayleigh-Jeans law

The Planck distribution

The most compelling evidence for the quantization of energy comes from the observation of the frequencies of radiation absorbed and emitted by atoms and molecules.

Atomic and molecular spectra

A region of the spectrum of radiation emitted by excited iron atoms consists of radiation at a series of discrete wavelength (or frequencies).

When a molecule changes its state, it does so by absorbing or emitting radiation at definite frequencies. This spectrum is part of that due to the electronic, vibrational and rotational excitation of sulphur dioxide (SO2) molecules. This observation suggests that molecules can possess only discrete energies, not arbitrary energy.

Spectral lines can be accounted for if we assume that a molecule emits a photon as it changes between discrete energy levels. Note that high-frequency radiation is emitted when the energy change is large.

Wave-particle dualityThe particle character of electromagnetic radiation

The observation that electromagnetic radiation of frequency can possess only the energies 0, h, 2h, … suggests that it can be thought of as consisting of 0, 1, 2, … particles, each particle having the energy h. These particles of electromagnetic radiation are now called photons. The observation of discrete spectra from atoms and molecules can be pictured as the atom or molecule generating a photon of energy h when it discards an energy of magnitude E, with E = h.

Further evidence for the particle-like character of radiation comes from the measurement of the energies of electrons produced in the photoelectric effect. This effect is the ejection of electrons from metals when they are exposed to ultraviolet radiation. The experimental characteristics of the photoelectric effect are summarized on the next transparency.

In the photoelectric effect, it is found that no electrons are ejected when the incident radiation has a frequency below a value characteristic of the metal, and above that value, the kinetic energy of the photoelectrons varies linearly with the frequency of the incident radiation.

Wave-particle dualityThe particle character of electromagnetic radiation (cont’d)

1. No electrons are ejected, regardless of the intensity of the radiation, unless the frequency exceeds a threshold value characteristic of the metal.

2. The kinetic energy of the ejected electrons increases linearly with the frequency of the incident radiation but is independent of the intensity of the incident radiation.

3. Even at low light intensities, electrons are ejected immediately if the frequency is above threshold.

The photoelectric effect can be explained if it is supposed that the incident radiation is composed of photons that have energy proportional to the frequency of the radiation. (a) The energy of the photon is insufficient to drive an electron out of the metal. (b) The energy of the photon is more than enough to eject an electron, and the excess energy is carried away as the kinetic energy of the photoelectron (the ejected electron

These observations suggest an ejection of the electron after collision with a particle-like projectile that carries enough energy to eject the electron from the metal. If we suppose that the projectile is a photon of energy h, then the conservation of energy requires that the kinetic energy of the ejected electron should obey

½ mev2 = h -

In this expression is a characteristic of the metal called its work function, the energy required to remove the electron from the metal to infinity (Einstein, 1905).

Wave-particle dualityThe wave character of particles

In 1927, Davisson and Germer observed diffraction of electrons by a crystal of nickel, which acted as a diffraction grating. Diffraction is a characteristic property of waves because it occurs when there is interference between their peaks and troughs. Depending on whether the interference is constructive or destructive, the result is a region of enhanced or diminished intensity.

Left: The Davisson-Germer experiment. The scattering of an electron beam from a nickel crystal shows a variation of intensity characteristic of a diffraction experiment in which waves interfere constructively and destructively in different directions.

310 eV

3 grids

fringe field electrode

2 MCPs

sreen

electron gun

x 100 ... 10000

single crystal

Top: Exploded viwe of a modern low-energy electron diffraction (LEED) apparatus and diffraction pattern from CaF2(111).

Wave-particle dualityThe wave character of particles (cont’d)

Already in 1924 the French physicist Louis de Broglie had suggested that any particle, not only photons, travelling with a linear momentum p should have a wavelength given by the de Broglie relation:

That is, a particle with a high linear momentum has a short wavelength (see figure). Macroscopic bodies have such high momenta (even if they are moving slowly) that their wavelength are undetectably small, and the wave-like properties cannot be observed.

Examples:Electron, kinetic energy 100 eV: = 1.22·10-10 mNeutron, kinetic energy 300 K: = 1.78·10-10 mMan, m=75 kg, v=1 m·s-1: = 8.83·10-36 m

An illustration of the de Broglie relation between momentum and wavelength. A wave is associated with a particle (later this will be seen to be the wavefunction of the particle). A particle with high momentum has a short wavelength, and vice versa.

h

p

Atomic structure and atomic spectra: First attemptsThe spectrum of atomic hydrogen

Already in 1855 the Swiss schoolteacher Johann Balmer pointed out that (in modern terms) the wavenumbers of the emission lines which were observed in the visible region when an electric discharge is passed through gaseous hydrogen fit the expression

The lines this formula describes are now called the Balmer series. When further lines were discovered in the ultraviolet (Lyman series) and in the infrared (Paschen series), the Swedish spectroscopist Johannes Rydberg noted (in 1890) that all of them were described by the expression

with n1=1 (the Lyman series), 2 (the Balmer series), and 3 (the Paschen series), and that in each case n2=n1+1, n1+2, … . The constant RH is now called the Rydberg constant for the hydrogen atom.

The spectrum of atomic hydrogen. Both the observed spectrum and its resolution into overlapping series are shown. Note that the Balmer series lies in the visible region.

2 2

1 1

2 n

1H H2 2

1 2

1 1R R 109677 cm

n n

Atomic structure and atomic spectra: First attemptsThe spectrum of atomic hydrogen

The Rydberg formula strongly suggests that the wavenumber of each spectral line can be written as the difference of two terms, each of the form

The Ritz combination principle states that the wavenumber of any spectral line is the difference between two terms:

It is readily explained in terms of photons and the conservation of energy. Thus, a spectroscopic line arises from the transition of an atom from one energy level (a term) to another (another term) with the emission of the difference in energy as a photon (see figure). This interpretation leads to the Bohr frequency condition, which states that, when an atom changes its energy by E, the difference is carried away as a photon of frequency , where

Energy is conserved when a photon is emitted, so the difference in energy of the atom before and after the emission must be equal to the energy of the photon emitted.

Hn 2

RT

n

1 2T T

E h

Atomic structure and atomic spectra: First attemptsThe Bohr model of atomic hydrogen

• One of the most famous of the obsolete theories of the hydrogen atom, proposed by the Swedish scientist Niels Bohr in 1913.

• Idea: The electron surrounds the nucleus like planets the sun.

• The Coulombic force of attraction (centripetal force) is balanced by the centrifugal effect of the orbital motion.

• In addition to the frequency condition, Bohr introduced two postulates:a) The electron does not radiate.b) The angular momentum is limited to integral values of ħ (h/2): L=nħ.

• Under these conditions the energy and the radius of the electron’s orbit around the nucleus can be calculated, and are found to be a function of the quantum number n:

2 2 40 e

n n2 2 2e 0

h n m er E n 1,2,3,...

m e 8 h n

Atomic structure and atomic spectra: First attemptsThe Bohr model of atomic hydrogen (cont’d)

• Result: discrete orbits and energy levels!!!

• Radius for n=1: r1=5.292·10-11 m atomic length unit, Bohr radius a0.

• Comparison to Rydberg’s formula:

excellent agreement between model and experiment!

• Advantage of Bohr’s model:- very simple and intuitive description- quantitative agreement

• Disadvantages of Bohr’s model:- only good for one-electron systems (H, He+, Li2+, …)- wrong assumptions (no well-defined orbits, … )- agreement to experiment due to coincidence!

quantum mechanical description required for exact treatment!

4eA A

n A H2 20

m eE EE E R

h cn 8 h

Quantum Theory: An IntroductionThe Schrödinger equation

• Quantum mechanics acknowledges the wave-particle duality of matter by supposing that, rather than travelling a definite path, a particle is distributed through space like a wave. The mathematical representation of the wave that in quantum mechanics replaces the classical concept of trajectory is called a wavefunction, (psi).

• In 1926, the Austrian physicist Erwin Schrödinger proposed an equation for finding the wavefunction of any system. The time-independent Schrödinger equation for a particle of mass m moving in one dimension with energy E is

The factor V(x) is the potential energy of the particle at the point x, and ħ (which is read h-cross or h-bar) a modification of Planck’s constant: ħ = h/2.

• The Schrödinger equation should be regarded as a postulate, like Newton’s equations of motion. However, it is (at least partially) possible to justify it.

• It can be shown that the term with the second derivative of the wavefunction with respect to the coordinate in space corresponds to the classical kinetic energy, the product V(x)· to the potential energy, and E· therefore to the total energy.

2 2

2

dV(x) E

2m dx

Quantum Theory: An IntroductionThe Schrödinger equation (cont’d)

• For a three-dimensional system

where 2 (‘del squared’ or ‘nabla squared’) is

• In systems with spherical symmetry (in polar coordinates)

• In the general case the Schrödinger equation is written

where H is the hamiltonian operator for the system (a Hamiltonian is a function which expresses the energy in terms of its momentum and positional coordinates).

22 V(x,y,z) E

2m

2 2 22

2 2 2x y z

2 22 2 2

2 2 2 2

2 1 1 1and sin

r r sinr r sin

22H E with H V

2m

Quantum Theory: An IntroductionBorn’s interpretation of the wavefunction

• In principle, the wavefunction contains all the dynamical information about the system it describes. We will concentrate on the location of the particle.

• The Born interpretation focuses on the square of the wavefunction (or the square modulus, ||2=*, if is complex). For a one-dimensional system:

If the wavefunction of a particle has the value at some point x, then the probability of finding the particle between x and x+dx is proportional to ||2dx.

Thus, ||2 is the probability density, and to obtain the probability it must be multiplied by the length of the infinitesimal region dx. The wavefunction itself is often called the probability amplitude.

• For a particle free to move in three dimensions (for example, an electron near a nucleus in an atom), the wavefunction depends on the point dr with coordinates x, y and z, and the interpretation of (r) is as follows:

If the wavefunction of a particle has the value at some point r, then the probability of finding the particle in an infinitesimal volume d = dx dy dz at that point is proportional to ||2d.

Quantum Theory: An IntroductionBorn’s interpretation of the wavefunction (cont’d)

• Thus, there is no direct significance in the negative (or complex) value of a wavefunction: only the square modulus, a positive quantity, is directly physically significant, and both negative and positive regions of a wavefunction may correspond to a high probability of finding a particle in a region *.

The wavefunction is a probability amplitude in the sense that its square modulus (* or ||2) is a probability density. The probability of finding a particle in the region dx located at x is proportional to ||2dx.

The Born interpretation of the wavefunction in three-dimensional space implies that the probability of finding the particle in the volume element d = dx dy dz at some location r is proportional to the product of d and the value of ||2 at that location.

The sign of a wavefunction has no direct physical significance: the positive and negative regions of this wavefunction both correspond to the same probability distribution (as given by the square modulus of and dpicted by the density of shafing).

* Later we shall see that the presence of positive and negative regions of a wavefunction is of great indirect interest, because it gives rise to the possibility of constructive and destructive interference between different wavefunctions.

Quantum Theory: An IntroductionHeisenberg’s uncertainty principle

• It is impossible to specify simultaneously, with arbitrary precision, both the momentum and the position of a particles (Werner Heisenberg, 1927).

Left: The wavefunction for a particle at a well-defined location is a sharply spiked function that has zero amplitude everywhere except at the particles position.

Right: The wavefunction for a particle with an ill-defined location can be regarded as the superposition of several wavefunctions that interfere constructively in one place but destructively elsewhere. As more waves are used in the superposition (as given by the numbers attached to the curves), the location becomes more precise at the expense of uncertainty in the particles momentum. An infinite number of wavefunctions is needed to construct the wavefunction of a perfectly localized particle.

• A quantitative version of this result is

In this expression p is the ‘uncertainty’ in the linear momentum parallel to the axis q, and q is the uncertainty in position along that axis.

1p q

2

Quantum Theory: An IntroductionHeisenberg’s uncertainty principle (cont’d)

• The uncertainties are precisely defined, for they are the root mean square deviations of the properties from their mean values:

If there is complete certainty about the position of the particle (q=0), the only way to satisfy the above equation is for p=, which implies complete uncertainty about the momentum.

• The Heisenberg uncertainty principle is more general than the above equation suggests. It implies to any pair of observables called complementary observables, which are defined in terms of the properties of their operators.

1/ 2 1/ 22 22 2p p p q q q

Using Quantum Theory: Techniques and ApplicationsOne-Dimensional Translational Motion

A particle in a boxAssume a box in which a particle is confined between two walls at x=0 and x=L: The potential energy is zero inside the box but rises to infinity at the walls. This model is an idealization of the potential energy of a gas-phase molecule that is free to move in a one-dimensional container. However, it is also the basis of the treatment of the electronic structure of metals and a primitive treatment of conjugated molecules (like e.g. CH2=CH-CH=CH2).

The Schrödinger equation between the walls is the same as for a free particle:

The solution of this 2nd order differential equation is simply:

A particle in a one-dimensional region with impenetrable walls. Its potential energy is zero between x=0 and x=L, and rises abruptly to infinity as soon as it touches the walls.

2 2

2

dE

2m dx

2 2

k kk

(x) Asin(kx) Bcos(kx) E2m

Since eikx=cos(x)±i·sin(x) this is

equivalent to k(x) = C·eikx + D·e-ikx

Using Quantum Theory: Techniques and ApplicationsOne-Dimensional Translational Motion (cont’d)

A particle in a box: Boundary conditions

For a free particle, any value of Ek is acceptable.

When the particle is confined within a region, the acceptable wavefunctions must satisfy certain boundary conditions, or constraints on the function at certain locations:

It is physically impossible to find the particle with an infinite potential energy, so the wavefunction must be zero where V is infinite, at x<0 and x>L, and

the continuity of the wavefunction requires it to vanish just inside the well at x=0 and x=L, so k(0)=0 and k(L)=0.

For the wavefunction k(x) = A·sin(k·x) + B·cos(k·x) this implies that B=0, and kL=n, with n=1,2,3,…: k(x) = A·sin(n··x/L)

The integral of 2 over all the space available to the particle, i.e. the probability to find it somewhere, is equal to 1. Therefore normalization yields the complete solution to the problem:

L L !2 2 2

0 0

1/ 22n

Ax

dx A sin dx 1L L

Using Quantum Theory: Techniques and ApplicationsOne-Dimensional Translational Motion (cont’d)

(a)The first two wavefunctions, (b) the corresponding probability distributions, and (c) representation of the probability distributions in terms of the darkness of the shading.

The allowed energy levels for a particle in a box. Note that the energy levels increase as n2, and that their spearation increases as the quantum number increases. Note that E=0 is impossible.

The first five normalized wavefunctions of a particle in a box. Each wavefunction is a standing wave, and succsessive functions possess one more half wave and a correspondingly shorter wavelength.

1/ 2

n

2 2

n

2 n x(x) sin

L L

for 0

n hE

8md

L

x L

an

• Consider a two-dimensional version of the particle in a box. The particle is now confined to a rectangular surface of length L1 in x-direction and length L2 in y-direction. The energy is zero everywhere except at the walls where it rises to infinity. The Schrödinger equation becomes now a function of both x and y:

• This problem can be solved by the separation of variables technique, which divides the equation into (in this case) two ordinary differentials, one for each variable:

Using Quantum Theory: Techniques and ApplicationsTwo-Dimensional Translational Motion

A two-dimensional square well. The particle is confined to the plane bounded by impenetrable walls. As soon as it touches the walls its potential energy rises to infinity.

2 2 2

2 2

d dE

2m dx dy

2 2 2 2

x y2 2

x y

d X d YE X E Y

2m 2mdx dy

E E

(x,y) X(x) )

E

Y(y

Each of the two previous differential equations is the same as the on-dimensional square-well Schrödinger equation. Thus, the result for the latter can be directly adapted:

Using Quantum Theory: Techniques and ApplicationsTwo-Dimensional Translational Motion (cont’d)

The wavefunctions of a particle confined to a rectangular surface depicted as contours of equal amplitude. (a) n1=1, n2=1, (b) n1=1, n2=2, (c) n1=2, n2=2, and (d) n1=2, n2=2.

1 21 2

2 2 21 2

n ,n 2 21

1 2

2n ,n 1 21/ 2

1 21 2

n n hE

8mL L

n x n x2(x,y) sin sin 0 x L 0 y L

L LL L

An interesting feature of the solutions for a particle in a two-dimensional box is obtained when the plane wave is square, with L1=L2=L.

In this case, e.g. the wavefunctions with n1=1, n2=2 and n1=2, n2=1 have the same energy:

Apparently different wavefunctions are degenerate, which means they correspond to the same energy although their quantum numbers are different.

Using Quantum Theory: Techniques and ApplicationsDegeneracy

The wavefunctions of a particle confined to a square surface. Note that one wavefunction can be converted into the other by a rotation of the box by 90°. The two functions correspond to the same energy. Degeneracy and symmetry are closely related.

2 2

1,2 2,121,2 2, 215h 5h

E E8mL

2 x 2 x 2 2 x xsin sin sin sin

L L L L L 8mLL

If the potential energy of a particle does not rise to infinity when it is in the walls of the container, and E<V, the wavefunction does not decay abruptly to zero.

If the walls are thin (so that V falls to zero after a finite distance), the wavefunction oscillates inside the box, varies smoothly within the wall, and oscillates again outside the box.

The conditions of continuity inside the box, within the wall, and outside of the box enable us to obtain the solution of the Schrödinger equation.

Using Quantum Theory: Techniques and ApplicationsTunneling

A particle incident on a barrier from the left has an oscillating wavefunction, but inside the barrier there are no oscillations (for E<V). If the barrier is not too thick, the wavefunction is nonzero at its opposite face, and so oscillations begin there (only the real component of the wavefunction is shown).

The wavefunctions of a particle confined to a square surface. Note that one wavefunction can be converted into the other by a rotation of the box by 90°. The two functions correspond to the same energy. Degeneracy and symmetry are closely related.

The transmission probability, T, of a particle to travel through the wall is given by:

where ħ=(2m(V-E)1/2) and =E/V. For high, wide barriers, in the sense that L»1, this simplifies to:

Using Quantum Theory: Techniques and ApplicationsTunneling (cont’d)

The transmission probabilities for passage through a barrier. The horizontal axis is the energy of the particle expressed as a multiple of the barrier height. The curves are labelled with the values of L(2mV)1/2/ħ. The graph on the left is for E<V and that on the right for E>V. Note that T>0 for E<V, whereas classically T would be zero. However, T<1 for E>V whereas classically T would be 1.

12L Le eT 1

16 1

2 LT 16 1 e

Using Quantum Theory: Techniques and ApplicationsTunneling (cont’d) and Microscopy

The wavefunction of a heavy particle decays more rapidly inside a barrier than that of a light particle. Consequently, a light particle has a greater probability of tunneling through the barrier.

The central component in a scanning tunneling microscope (STM) is an atomically sharp needle (Pt or W) which is scanned across the surface of a conducting solid. When the tip is brought very close to the surface, electrons tunnel across the intervening space. In the constant-current mode of operation, the stylus moves up and down corresponding to the topography of the surface, which, including and adsorbates, can be mapped on an atomic scale.

For the case of separation of internal from external motion and using the Coulomb potential energy of the electron in a hydrogen atom it is now straightforward to write down its Schrödinger equation, i.e. for an electron orbiting a nucleus (in this case a single proton with Z=1):

Due to the huge difference in mass between the nucleus and the electron, the reduced mass µ can - in excellent approximation be replaced by the mass of the electron: µme.

Because the potential energy is centrosymmetric (independent of angle), one can suspect that the equation is separable into radial and angular components.

Using Quantum Theory: Atomic Spectra and StructureThe Hydrogen Atom

Spherical polar coordinates. A particle on the surface of a sphere of radius r, can be described by its colatitude, , and the azimuth, .

2 2 2 22 2

0 0

Ze ZeH E

2 4 r 2 4nd

ra

(r, , ) R(r) Y( , )

An atomic orbital is a one-electron wavefunction for an electron in an atom.

Acceptable solutions for the hydrogen atom’s wavefunction can be found only for integral values of a quantum number n, and the allowed energies are:

n is the principal quantum number which determines the energy of the electron, and can take the values n = 1, 2, 3, … .

The radial contribution R(r) depends on one more quantum number, l. An electron in an orbital with quantum number l has an angular momentum of {l(l+1)}1/2ħ, with l = 0, 1, 2, 3, … n-1.

The angular contribution depends on l and ml. An electron in an orbital with quantum number ml has a z-component of angular momentum mlħ with ml = 0, ±1, ±2, …, ±l.

Orbitals with quantum numbers l = 0, 1, 2, 3, … are usually called s, p, d, f … .

Using Quantum Theory: Atomic Spectra and StructureAtomic Orbitals, Energy and Quantum Numbers

The energy levels of a hydrogen atom. The values are relative to an infinitely separated, stationary electron and a proton

4

n 2 2 2 20

eE

32 n

Using Quantum Theory: Atomic Spectra and StructureRadial Wavefunctions

The radial wavefunctions for the first few states of hydrogenic atoms (i.e. atoms with one electron only) of atomic number Z. Note that the s orbitals have a finite and nonzero value at the nucleus. The horizontal scales are different in each case: orbitals with high principal quantum numbers are relatively distant from the nucleus. Remember that s l=0, p l=1, d l=2 …

Using Quantum Theory: Atomic Spectra and StructureAngular Wavefunctions

A representation of the angular wavefunctions for l = 0, 1, 2, and 3. The distance of a point on the surface from the origin is proportional to the square modulus of the amplitude of the wavefunction at that point.

Top: The permitted orientations of angular momentum when l=2. This representation is too specific because the azimuthal direction of the vector (its angle around z) is undeterminable.

Right: (a) A summary of above figure. However, because the azimuthal angle around z is undeterminable, a better representation is (b) where each vector lies on its cone.

Using Quantum Theory: Atomic Spectra and StructureThe Hydrogen Atom: Shells and Subshells

Left: The energy levels of the hydrogen atom showing the subshells and (in square brackets) the numbers of orbitals in each subshell. In hydrogenic atoms, all orbitals of a given shell have the same energy (this is not the case in systems with more than one electron !).

Right: The organization of orbitals (white squares) into subshells (characterized by l) and shells

(characterized by n).

All the orbitals of a given value of n are said to form a single shell. In a hydrogenic atom all orbitals belonging to the same shell have the same energy. It is common to revere to successive shells by letters:

n = 1 2 3 4…K L M N…

The orbitals with the same value of n but different values of l are said to form a subshell of a given shell. These subshells are generally also referred to by letters:

l = 0 1 2 3 4 5 6…s p d f g h i…

Using Quantum Theory: Atomic Spectra and StructureThe Hydrogen Atom’s Orbitals: s Orbitals

Left: Representation of the (a) 1s and (b) 2s hydrogenic orbitals in terms of their electron densities (as represented by the density of shading).

Top: The variation of the mean radius of a hydrogenic atom with the principal and orbital momentum quantum numbers. Note that the mean radius lies in the order d < p < s.

The orbital occupied in the ground state is the one with n=1 (and therefore with l=0 and ml=0). Its wavefunction is:

This wavefunction is independent of angle and has the same value at all points of constant radius; that is, the 1s orbital is spherically symmetrical. It has the maximum value at r=0. It follows that the most probable point where the electron will be found is the nucleus itself!

All s orbitals are spherically symmetrical, but differ in the number of radial nodes (0 for 1s, 1 for 2s, 2 for 3s, …).

0r / a

1/

20

023 2

0 e

4(a 52.9177 pm; Bohr radius)

me

e

1

a

Using Quantum Theory: Atomic Spectra and StructureThe Hydrogen Atom’s Orbitals: Radial Distribution Functions

A constant-volume electron sensitive detector (the small cube) gives its greatest reading at the nucleus, and a smaller reading elsewhere. The same reading is obtained anywhere on a circle of given radius: the s orbital is spherically symmetrical.

The wavefunction tells us, through the value of 2, the probability of finding an electron in any region. Imagine a probe with a volume d and sensitive to electrons, which can be moved around the hydrogen atom. The reading of this detector is shown in the figure to the right.

Now consider the probability of finding the electron anywhere on a spherical shell of thickness dr at a radius r. The sensitivity volume is now the volume of the shell, which is 4r2dr. Thus the probability to find the electron in a distance r is P(r)dr= 4r22dr, the result of which is shown in the lower figure.

For orbitals that are not spherically symmetrical, the more general expression r2R(r)2dr has to be used, where R(r) is the radial wavefunction of the orbital in question.

The radial distribution function P gives the probability that the

electron will be found anywhere in a shell of radius r. For a 1s

electron in hydrogen, P is a maximum when r is eaual to the

Bohr radius a0 (!). The value of P is equal to the reading that a

detector shaped like a spherical shell would give as its radius is

varied.

Using Quantum Theory: Atomic Spectra and StructureThe Hydrogen Atom’s Orbitals: p Orbitals

A p electron has nonzero angular momentum, which has a profound effect on the shape of the wavefunction close to the nucleus, for p orbitals have zero amplitude at r=0. This effect can be classically understood in terms of the centrifugal effect of the orbital angular momentum, which tends to fling the electrons away from the nucleus.

Since the solutions of Schrödinger’s equation usually contain imaginary contributions for p, d, f… orbitals, they are usually represented as purely real linear combinations of the latter, since each of these is a solution of the Schrödinger equation, too.

These linear combinations are standing waves with no net angular orbital momentum around the z-axis, as they are superpositions of states with equal and opposite values. The px orbital has the same shape as a pz orbital, but is directed along the x-axis; the py-orbital is similarly directed along the y-axis.

Left: Close to the nucleus, p orbitals are proportional to r, d orbitals are proportional to r2, and f orbitals are proportional to r3. Electrons are progressively excluded from the neighbourhood as l increases. An s orbital has a finite, nonzero value at the nucleus.

Top: The boundary surface of p orbitals. A nodal plane passes through the nucleus and separates the two lobes of each orbital.

Using Quantum Theory: Atomic Spectra and StructureThe Hydrogen Atom’s Orbitals: d Orbitals

When n=3, l can be 0, 1, or 2. As a result, this shell consists of one 3s orbital, three 3p orbitals, and 53d orbitals.

The five d orbitals have ml= +2, +1, 0, -1, -2, and correspond to five different angular momenta around the z-axis (but the same magnitude of angular momentum around the z-axis, because l=2 in each case).

As for the case of p orbitals, d orbitals with opposite sign of ml (and hence opposite sign of motion around the z-axis) may be combined in pairs to give standing waves, whose boundary surfaces are shown below.

These real combinations are shown here as an example:

The boundary surfaces of d orbitals. Two nodal planes in each orbital at intersect at the nucleus and separate the lobes of each orbital. The dark and light areas denote regions of opposite sign of the wavefunction.

2 2 2

xy yz zx

2 2 2 2x y z

d xyf(r) d yzf(r) d zxf(r)

1 1d (x y )f(r) d 3 3z r f(r)

2 2

SpectroscopyWhat’s behind it?

All changes on atomic and molecular scale correspond to transitions between discrete energy levels.

These energy levels, and the transition between them, can be described via the SCHRÖDINGER equation.

SPECTROSCOPY: Determination of the transition frequencies with the goal to obtain information with respect to chemical identity and/or atomic or molecular structure.









Photoelectron Spectroscopies: XPS and UPS

Ionisation of atoms and molecules with monochromatic X-ray or ultraviolett radiation:

Photoelectrons with well-defined, substance-specific energy kin b SpE E e

Extreme surface sensitivity:

2-3 layers only

An example: XPS* spectrum of stainless steel

*ESCA (Electron Spectroscopy for Chemical Analysis) is another acronym for XPS

- Chemical Identity

- Quantitative Chemical Analysis

- Oxidation States / Chemical Surroundings

- Adsorbate Species

IR(ambient pressure)

VT-AFM/STM

XPS/UPS etc.

Preparation

Load Lock









General Features of Spectroscopy

Relation between frequency, wavelength, and wavenumber:

c

c

1

: wavelength

: frequency

: wavenumber (usually measured in [cm ])

c : speed of light

• In emission spectroscopy, a molecule undergoes a transition from a state of high energy E1 to a state of lower energy E2 and emits the excess energy as a photon.

• In absorption spectroscopy, the net absorption of nearly monochromatic (single-frequency) incident radiation is monitored as the radiation is swept over a range of frequencies.

• The energy, h, of the photon emitted or absorbed, and therefore the frequency of the radiation emitted or absorbed, is given by the Bohr condition

h = E1 – E2

• Emission and absorption spectroscopy give the same information about energy level separation, but absorption spectroscopy is more common in the lab.

Spectrometers: The Sources

Symbolic representation of the principal components used in obtaining an absorption spectrum:

Sources: either polychromatic, i.e. spanning a wide range of frequencies, or monochromatic, i.e. spanning a very narrow range of frequencies around a central value

Many commercial spectrometers take advantage of black-body radiation from hot materials, which can provide radiation spanning a wide range of frequencies

Common lab sources: Mercury lamp for far-infrared radiation (FIR; 35 – 200 cm-1) Globar (SiC, heated to 1500 K) for mid-infrared radiation

(MIR; 200 – 4000 cm-1) Quartz-tungsten-halogen lamp for near-infrared, visible

and near-ultraviolet radiation (NIR – VIS – UV; 320 – 2500 nm, i.e. 4000 - 30000 cm-1; T3000 K)

Xenon discharge lamp for ultraviolet radiation (T6000 K)

A Special Source: Synchrotron Storage Rings

For certain applications, synchrotron radiation from a synchrotron storage ring is appropriate.

A synchrotron storage ring consists of an electron beam travelling in a circular path of several meters in diameter. As electrons travelling in a circle are constantly accelerated by the forces that constrain them to their path, they generate radiation. Synchrotron radiation spans a wide range of frequencies, including the infrared and X-rays.

Synchrotron radiation is much more intense than can be obtained by most conventional sources. The disadvantage of the source is that it is so large and costly that it is essentially a national facility, not a laboratory commonplace (e.g. BESSY II in Berlin-Adlershof).

A synchrotron storage ring. The electrons injected into the ring from the linear accelerator and booster synchrotron are accelerated to high speed (almost the speed of light) in the main ring. An electron in a curved path is subject to constant acceleration, and an accelerated charge radiates electromagnetic energy.

Spectrometers: Dispersing Elements

The dispersing element in most absorption spectrometers operating in the ultraviolet to near-infrared region of the spectrum is a diffraction grating, which consists of a glass or ceramic plate into which fine grooves have been cut and covered with a reflective aluminium coating. The grating causes interference between waves reflected from its surface, and constructive interference occurs when

n = d(sin - sin )

where n=1, 2, … is the diffraction order, is the wavelength of the diffracted radiation, d is the distance between grooves, is the angle of incidence of the beam, and is the angle of emergence of the beam.

In a monochromator, a narrow exit slit allows only a narrow range of wavelength to reach the detector. Turning the grating around an axis perpendicular to the incident and diffracted beams allows different wavelength to be analysed.

In a polychromator, there is no slit and a broad range of wavelengths can be analysed simultaneously by array detectors.

A polychromatic beam is dispersed by a diffraction grating into three component wavelength 1, 2 and 3. In the configuration shown only radiation with 2 passes through a narrow slit and reaches the detector. Rotating the diffraction grating in the direction shown by the double arrows allows 1 and 3 to reach the detector.

Spectrometers: Fourier Transform Techniques

Modern spectrometers, particularly those operating in the infrared and near-infrared, now almost always use Fourier transform (FT) techniques of spectral detection and analysis.

The heart of a FT spectrometer is a Michelson interferometer. A Michelson interferometer works by splitting the beam from the sample into two and introducing a varying path difference, p, into one of them (moveable mirror). When the two components recombine, there is a phase difference between them, and they interfere either constructively or destructively depending on the difference in the path lengths. The detected signal oscillates as the two components alternately come in and out of phase as the path difference is changed.

Left: A Michelson interferometer. The beam-splitting element divides the incident beam into two beams with a path difference p that depends on the location of the mirror M1. The compensator ensures that both beams pass through the same thickness of material

Top: Interferograms produced as the path length p is changed in the interferometer. In the left example only a single-frequency component is present in the radiation, while in the other one several (in this case, three) frequencies are present.

Spectrometers: Fourier Transform Techniques (cont’d)

The problem is to find I( ), the variation of intensity with wavenumber, which is the spectrum one requires, from the record of values of intensity as a function of path difference, I(p). This step is a standard technique of mathematics, and is the ‘Fourier transformation’ step from which this form of spectroscopy takes its name. Specifically:

where I(0) is the intensity for p=0. This integration is carried out in a computer connected to the spectrometer.

A major advantage of the Fourier transform procedure is that all the radiation emitted by the source is monitored continuously. This is in contrast to a spectrometer in which a monochromator discards most of the generated radiation. As a result, Fourier transform spectrometers have a higher sensitivity than conventional spectrometers.

Another advantage of Fourier transform spectrometers is their superior resolution. It is determined by the maximum path length difference, pmax, of the interferometer:

To achieve a resolution of 0.1 cm-1 requires a maximum path length difference of about 10 cm, corresponding to only 5 cm travel of the moving mirror. This resolution would be very difficult to achieve with a monochromator.

The three frequency components and their intensities that account for the appearance of the interferogram on the previous transparency. This spectrum is the Fourier transfrom of the interferogram, and is a depiction of the contributing frequencies.

0

1I( ) 4 I(p) I(0) cos2 pdp

2

max

1

p

The intensities of spectral lines: The Beer-Lambert Law

The ratio of the transmitted intensity, I, to the incident intensity, I0, at a given frequency is called the transmittance, T, of the sample at that frequency:

It is found empirically that the transmitted intensity varies with the length, l, of the sample and the molar concentration, [J], of the absorbing species J in accord with the Beer-Lambert law:

The quantity is called the molar absorption coefficient (formerly, and still widely in use, the ‘extinction coefficient’). The molar absorption coefficient depends on the frequency of the incident radiation and is greatest where the absorption is most intense.

It is sensible to introduce the absorbance, A, of a sample at a given wavenumber as

Then the Beer-Lambert law becomes

The product [J]l was formerly known as the optical density of the sample. For known the measurement of A allows determination of the concentration of species J.

0

IT

I

[J]l0I I 10

0 orI

A log A logTI

A [J]l

Molecular Vibrations: IR and NIR Spectroscopy

The wavenumbers of molecular vibrations are typically in the range from 400 to 4000 cm -1, and can be excited by (mid-) infrared light. Vibrational frequencies above ~1000 cm -1 can in general be attributed to specific functionalities (e.g. C-H, C-O, C-C or other bonds), those below ~1000 cm -1 are usually due to more complex vibrations of the whole molecule. From the infrared spectrum unknown compounds can be identified. The vibrational spectrum is a ‘fingerprint’ of the molecule. From comparison to libraries even complex mixtures can be analysed.

e,g, hexane C6H14 and xylene C8H10

Additional information is available in the near-infrared (NIR) region (~4000 - ~10000 cm -1), where overtones and combination bands are found.

5001000150020002500300035004000

Wavenumber cm-1

020

40

60

80

100

Tra

nsm

ittanc

e [%

]

Hook's law:

F D x

Potential V:

dVF

dx 21

V D x2

2

2

d xD x m

dt

0

1 D

2 m

or 0

1 D

2

Combination with Newton‘s law:

Quantum Mechanics:2

22 2

d 2 1E Dx 0

dx 2

0

1E h v

2

Ideal:

Real:

Molecular Vibrations: The Background

The vibrational frequency is absolutely sensitive on

- the force constants- the masses

Identification of different chemical species

Quantitation using Beer-Lambert’s law

Analysis of complex mixture by automatic comparison to spectra libraries

How to do it: Fourier Transform Infrared Spectroscopy (FTIRS)

- measure the light intensity modulation as function of mirror movement in an interferometer simultaneous measurement of all frequencies!

- calculation of the frequency spectrum from the interferogram by Fourier transformation (after apodisation, zerofilling, and phase correction).

B( ) I(x)cos(2 x)dx

I(x) B( )cos(2 x)d

Raman Spectroscopy

In Raman spectroscopy, molecular energy levels are explored by examining the frequencies present in the radiation scattered by molecules. In a typical experiment, a monochromatic incident laser beam is passed through the sample and the radiation scattered from the front face of the sample is monitored.

About 1 in 107 of the incident photons collide with the molecules, give up some of their energy which serves to excite vibrations or rotations of the molecules, and emerge with a lower energy. These scattered photons constitute the lower-frequency Stokes radiation from the sample. Other incident photons may collect energy from the molecules (if they are already excited) and emerge as higher-frequency anti-Stokes radiation. The component of radiation scattered into the forward direction without change of frequency is called Rayleigh radiation.

Raman spectroscopy and regular absorption spectroscopy often give complementary information. In case of molecules with a centre of inversion a molecular vibration is either Raman or infrared active.

A common arrangement adopted in Raman spectroscopy. A laser beam first passes through a lens and than through a small hole in a mirror with a curved reflecting surface. The focused beam strikes the sample and scattered light is both deflected and focused by the mirror. The spectrum is analysed by a monochromator or an interferometer.

In conventional Raman spectroscopy, the incident radiation does not match an absorption frequency of the molecule´, and there is only a ‘virtual’ transition to an excited state.

Electronic Transitions: UV-VIS Spectroscopy

The energies needed to change the electron distributions of molecules are of the order of several electronvolts (1 eV is equivalent to about 8000 cm-1 or 100 kJ mol-1). Consequently, the photons emitted or absorbed when such changes occur lie in the visible and ultraviolet regions of the spectrum. In some cases the relocation of electrons may be so extensive that it results in ionization or dissociation of the molecule.

The nuclei in a molecule are subjected to different forces after an electronic transisition has occurred, and the molecule may respond by starting to vibrate. The resulting vibrational structure of electronic transitions can be resolved for gaseous samples, but in a liquid or solid the lines usually merge together and result in a broad, almost featureless band. Superimposed on the vibrational transition of a molecule in the gas phase is an additional structure that arises from rotational transitions.

Due to the broad absorptions, UV-VIS spectroscopy is usually not the method of choice for the analysis of complex mixtures. However, in case of compounds with rather specific absorption frequencies it can be used for e.g. determination of concentrations, control of purity et cetera.

The absorption spectrum of chlorophyll in the visible region. Note that it absorbs in the blue and red regions, and that green light is not absorbed

appreciably.









Nuclear Magnetic Resonance (NMR) Spectroscopy

The Stern-Gerlach experiment provided evidence for electron spin. It turned out that many nuclei also possess spin angular momentum. Orbital and spin angular momenta give rise to magnetic moments, and to say that electrons and nuclei have magnetic moments means, to some extent, that they behave like little bar magnets. Therefore, one can expect that the application of a magnetic field should affect atoms and molecules.

The spin quantum number, I, of a nucleus is a fixed characteristic property and may be an integer or a half-integer but is never negative. A nucleus with spin quantum number I has the following properties:

1. An angular momentum of magnitude {I(I+1)}1/2ħ.2. A component of angular momentum mIħ on an arbitrary axis, where mI = I, I-1, …, -I.3. If I>0, a magnetic moment with a constant magnitude and an orientation that is determined by the value of mI.

Several important nuclei, like 1H. 13C, 19F, and 31P have I = ½.

Number of protons Number of neutrons I

even even 0

odd odd Integer (1, 2, 3, …)

even odd half-integer (1/2, 3/2, 5/2, …)

odd even half-integer (1/2, 3/2, 5/2, …)

Nuclear constitution and the nuclear spin quantum number

Nuclear Magnetic Resonance (NMR) Spectroscopy (cont’d)

If a nucleus with a magnetic moment is placed in a magnetic field of strength B0 in the z-direction, the (2I+1) orientations of the nucleus have different energies, which are given by

Lis the so-called Larmor frequency, the magnetogyric ratio for the nucleus under consideration.

The energy separation between the mI=+½ ( or ) and the mI=-½ ( or ) states of spin-½ nuclei, which are nuclei with I=½, is

and resonant absorption occurs when the resonance condition h=ħB0 is fulfilled. That is, an transition occurs at =L.

I

0m I L L

BE mh

2

0 LE E E B h

The interactions between the ms states of an electron (or mI states of a I=½ nucleus) and an external magnetic field may be visualized as the precession of the vectors representing the angular momentum on the cones drawn here.

The nuclear spin energy levels of a spin-½ nucleus with positive magnetogyric ratio (e.g. 1H or 13C) in a magnetic field. Resonance occurs when the energy separation of the levels matches the energy of the photons in the electromagnetic field.

NMR: The Spectrometer

In its simplest form, nuclear magnetic resonance (NMR) is the study of the properties of molecules containing magnetic nuclei by applying a magnetic field and observing the frequency of the resonant electromagnetic field absorption. Larmor frequencies of nuclei at the fields normally applied lie in the radiofrequency region of the electromagnetic spectrum, so NMR is a radiofrequency technique. For example, at 12 T, protons come into resonance at about 500 MHz (the Larmor frequency of the electromagnetic field). For such high magnetic fields, usually superconducting magnets are used.

NMR spectra have proved invaluable in chemistry (and in particular organic chemistry), for they reveal so much structural information. A magnetic nucleus is a very sensitive, noninvasive probe of the surrounding electronic structure.

Nowadays the NMR technique is not only important in chemistry, where most of the modern instruments are based on pulse techniques. One of the most striking applications is in medicine, where magnetic resonance imaging (MRI) is a portrayal of the concentrations of protons in a solid object. A great advantage of MRI is that it can display soft tissue, such as a cross-section through a patient’s head.

The layout of a typical NMR spectrometer. The link from the transmitter to the detector indicates that the high frequency of the transmitter is subtracted from the high frequency received signal to give a low frequency signal for processing.

NMR: The Chemical Shift

Nuclear magnetic moments interact with the local magnetic field, which is influenced by the electron density (electrons also have a magnetic moment) as well as the chemical surroundings with respect to other magnetic nuclei. These contributions are proportional to the applied field, and it is conventional to write

where the dimensionless quantity is called the shielding constant of the nucleus.

Because the total local field is Bloc = B0 + B = (1-)B0 the Larmor frequency is

It is convenient to express the resonance frequencies in terms of an empirical quantity called the chemical shift, which is related to the difference between the resonance frequency, , and that of a resonance standard, 0:

The standard for protons is the proton resonance 0 in tetramethylsilane (Si(CH3)4), commonly referred to as TMS, which bristles with protons and dissolves without reaction in many liquids.

loc 0L

B B(1 )

2 2

0B B

06

010

NMR: The Chemical Shift (cont’d)

The range of typical chemical shifts for (a) 1H resonances and (b) 13C resonances.

The 1H NMR spectrum of ethanol. The bold letters denote the protons giving rise to the resonance peak, and the step-like curve is the integrated signal. The CH3 protons form one group of nuclei with 1. The two CH2 protons are in a different part of the molecule, experience a different local magnetic field, and resonate at 3. Finally, the OH proton is in another environment, and has a chemical shift of 4. The increasing value of (that is the decrease in shielding) is consistent with the electron-withdrawing power of the O atom, which reduces the electron density at the O atom most. The integration of peak intensities yields 1:2:3, in agreement with the (in this case known) chemical composition.

NMR: The Fine Structure

The splitting of resonances into individual lines is called the fine structure. It arises because each magnetic nucleus may contribute to the local field experienced by the other nuclei and so modify their resonance frequencies, The strength of the interactions is expressed in terms of the scalar spin coupling constant, J, and reported in Hertz (Hz).

For the example ethanol (CH3CH2OH) which is shown below again, this means:

- The three protons of the CH3 group split the resonance of the CH2 protons into a 1:3:3:1 quartet.- Likewise, the two protons of the CH2 group split the resonance of the CH3 protons into a 1:2:1 triplet

with the same splitting J. - All the lines mentioned so far are split into doublets by the OH proton, but the splitting cannot be detected because the OH protons migrate rapidly from molecule to molecule and their effect averages to zero.

Documents

This Physical Chemistry lecture uses graphs from the following textbooks: