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Quantum theory

and Atomic structure

CM1502

Chapter 1

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3

The Wave

nature

of light

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Amplitude is the height

of the crest or the depth

of the trough of each

wave.Amplitude is related to

the intensity of the

radiation which we

perceive as brightness inthe case of visible light.

The Wave nature of light

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5

Classical Distinction between

Wave and Particle

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Electromagnetic Spectrum

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Continuous Spectra

A warm solid, liquid and plasma will radiate atallwavelengths thus producing a continuousEM spectrum.

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Discrete Spectra 1

A warm gas emits EMR, but at certain specific

wavelengths thus producing a discrete EM

spectrum.

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Discrete Spectra 2

A gas can also absorb EMR and does so at

discrete wavelengths.

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Examples: Spectra of Stars

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Examples: Gas clouds in space 1

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Line spectra of Atoms

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Hydrogen Atom Spectrum 1

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Hydrogen Atom Spectrum 2

A high school maths teacher, Balmer, in 1885

noticed that the wavelengths of the visible lines of

Hs spectrum could be represented by the formula:

...5,4,3,1

4

112H

=

= nn

R

where RHis a constant.

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Hydrogen Atom Spectrum 3

Later Rydberg showed that all the H atoms absorption

and emission lines (not just those seen in the visible,

i.e., the Balmer series) could be represented by the

formula

where RHis, as before, a constant now known as theRydberg constant, 1/RH= 91.1763 nm.

...3,2,1...,3,2,1,111

111212

2

2

1

H +++==

= nnnnnnn

R

(1.2)

The Rydberg equation and the value of the constant are

based on data rather than theory

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The Unexplained

Why do the spectral lines of Hydrogen appear in apattern?

Is there any importance about this experimental

observation? Over a wide range of wavelengths, light is only

observed at certain discrete wavelengths

Light is quantized (?) It was taught before that light behaves as wave

only

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Explaining Rydbergs Formula

Quantization of Light

Max Planck first proposed that light could bequantized into little packets of energy.

The packets of energy, calledphotons, have an

energy value of,

hE=(1.3)

where his Plancks constant.

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Bohrs H atom 1 Bohr postulated that an atom could only exist in certain

allowed states of specific total energy Ewhich he called

stationary states. Atoms do not leak energy while in oneof its stationary states.

If an atom was not in its lowest energy state (groundstate), then it could make a downward transition, to astate of lower energy and in the process, emit a photon.

High energy atomic state = Eu

Low energy atomic state = El

.

Eu El= E=a photon energy = h.

Bohrs postulated fixed atomic states and energy levelsand this leads naturally to discrete spectra.

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The energy levels of hydrogenic

atoms/ions

En= -2.18 x 10-18 J Z2/n2

The negative sign for the energy appears

because we define the zero point of the atoms

energy when the electron is completely

removed from the attraction of the nucleus

A table top analogy for def in ing the

energy of a system.

If you define the zero point of your

textbooks potential energy when the

book is on the table, the energy is

negative when it is on the floor.

222

0

421

8 nh

eZE

n

= (1.4)

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The lowest energy state of hydrogen atom n=1 has

the energy of -13.6eV or -1312 KJ/mole.

The amount of energy needed to promote an atomfrom the ground state to a given excited state is

called excitation energy.

The amount of energy needed to remove an electronfrom an atom in ground state is called the ionization

energy.

The separation energy is energy needed to removean electron from an atom in any excited state.

H Atom Energy States

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n=1

n=

n=5

n=4

n=3

n=2 -13.6/22= -3.4 eV

-13.6/32= -1.51eV

-13.6/42=0.85eV

-13.6/52=-0.54eV

0

-13.6eV

1stexcited state

2nd excited state

Ground state

a

b

a: Ionization energy = +13.6eV

b: Excitation energy

c,d:Separation energies

Energy level diagram for hydrogen atom

c

d

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Applications of Bohrs equation for

energy levels of an atom

We can find the

-difference in energy between two levels.

-energy needed to ionize the H atom

-wavelength of the spectral line.

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Figure 7.11

The Bohr explanation of the three series of spectral lines.

E of emitted photon: UV series > VIS series > IR series

nin Rydberg equation23

A spectral line results because a photon of specific energy is emitted.

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Limitations of Bohrs Model

Works only for hydrogenic species(one electronspecies) such as H, He+, Li2+etc

Fails for atoms with more than one electron

because the e-e repulsions and additional nucleuselectron attractions create more complexinteractions.

He assumed that an atom has only certain energylevels in order to explain line spectra. However hehas no theoretical basefor the assumption.

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Louis de Broglieconsidered other

systems that displayonly certain allowedmotions such as thevibrations of a pluckedguitar string.

He proposed that ifenergy is particle-like,perhaps matter is wavelike.

If electrons havewavelike motion inorbits of fixed radii, theywould have only certainallowable frequenciesand energies.

de Broglie and Standing Waves

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Fundamental or 1

st

Harmonic

2ndHarmonic or 1stOvertone 3rdHarmonic or 2ndOvertone

Animations of standing waves

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Perhaps the electrons, fixed to occupy specific orbits,behaved like standing waves?

Took the formulae E= mc2and set it equal to E= hc/to obtain for a photon = h/mc, mis equivalentmass (not actual) of a photon.

For a particle, he substituted v(velocity of the particle)for c.

de Broglies Matter Waves

mv

h=particle (1.5)

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S

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Light and matter can interact.

Light can be absorbed and emitted by matter. The absorption and emission can be continuous ordiscrete.

Light is quantized, E= h.

The energy states of atoms (and molecules) arequantized.

Discrete spectra can be explained by atomsundergoing a transition and emitting or absorbing aphoton of energy.

Matter has wave properties.

The energy stationary-states of hydrogenic atoms(only) can be explained by electrons existing instanding-waves that surround the nucleus.

Summary

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The Schrdinger Wave Equation

Schrdinger eventually came up with the famous equation:

is the total energy operatoror Hamiltonianoperator. Itrepresents a set of mathematical operations that when

carried out with a particular wave function, yields one of the

allowed energy states of the atom. Thus each solution of

the equation gives an energy state associated with a givenatomic orbital.

is the wave function also a mathematical function.

Eis the total energy of the systemunder consideration it

is simply a number with units of J.

nnn EH =

(1.6)

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The Wave Function,

is just a function, like x2, or sinx, or ex. This function depends on the positions of all the

particles in the system under consideration.

For the hydrogen atom, it is a mathematical function of

the position of the electron and proton. If we were looking at H2O instead, then is a function

of the positions of the two H nuclei,and the O nucleusand the positions of each and every one of the 10electrons.

For each value of nwe have, we will have a different n. That is, 1, 2, 3etc.

Later we will see that these are the 1s, 2s, 3s,2pand 3detc. atomicorbitals.

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Solving the Schrdinger Equation

Solving the Schrdinger, therefore, involves finding the

correct mathematical function such that when we operate onthat function with we get a constant times the original

function back again.

If we can do this then the constant is the energy of thesystem.

Upon solving the Schrdinger equation it was found

thatthree integers, denoted as n, l, mlfully characterized

the functions that solved the Schrdinger equation.(According to Bohr, only n was needed)

( ) ( ) ( ) ,,,,,,,

ll mllnmln

YrRr =

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The Total Energy, En

Interestingly, the formula for the energies was the same as derivedfrom the Bohr and Bohr-de Broglie models, and did not depend on l,nor ml.

Recall that Enrepresented the energy of the H atom when its electronwas in orbit number n.

Thus the nhere reminds us that our system, here the H atom, couldexist in any number of energy states.

Each energy state being labeled by n, with the lowest being n= 1.

222

0

421

8 nh

eZE

n

=

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one electron

system

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Quantum Numbers n, l, and ml n the principal quantum number (QN). The energy of

hydrogenic atoms depends only on this quantumnumber. Can take the values 1, 2, 3,

l azimuthal QN, or orbital angular momentum QN. Is

associated with the allowed angular momentum of anelectron in an orbital. Can take the values, 0, 1, 2, , (n - 1)

ml magnetic QN, or orbital angular momentumprojection QN. Is associated with the orientation of theorbital angular momentum Can take the values, -l, -(l-1),-(l-2), , 0, 1, 2, , l, i.e., mlgoes

fromlto l in steps of one.

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Allowed Values of the Quantum

Numbers

Will denote here as (n, l, ml)

Some allowed values:

(1,0,0), (5,4,-1), (2,1,0), (2,0,0),

(4,3,3) Some impossible values:

(1,1,0), (5,4,-5), (2,1,-2), (2,0,1),

(0,3,3), (0,0,0)

By convention lower-caseletters have been used to

designate the lQN.

l Letter

0 s

1 p2 d

3 f

4 g5 h

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Another Way

At around the same time Schrdinger developed thematter wave equation, another scientist wasformulating quantum mechanics in an entirely differentway.

Heisenbergused linear algebra, or matrix algebra, anddeveloped quantum matrix mechanics.

A very important, finding from this approach is

Heisenberg Uncertainty Principle.

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The Heisenberg Uncertainty Principle

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The Heisenberg Uncertainty Principle

Because energy is directly related to p, what theuncertainty principle means for us is that becausethe energy of an atom is known with considerableaccuracy, the location of the electron within the atomis not known at all, accurately.

This means that nice circular orbits of electronsaround nuclei can not be correct.

Worse is that electron positions can only ever beknown in terms of probabilities rather than assigningto them a special spot around the atom.

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Th P b bl L ti f th l t

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The Probable Location of the electron While we cannot know the exact

position of the electron, we can know

where it probably is. i.e. where itspends most of its time.

2is called the probability density, ameasure of probability of finding the

electron in some tiny volume of theatom.

Electron probability density in theground state H-atom is shown in the

figure.

Theprobability density decreaseswith r but does not reach zero.

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Radial probability distribution

The total probability of finding the electron atsome distance rfrom the nucleus is called

radial probability distribution.

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Analyzing the Dart BoardZone N holes Area holes Prob %

50 169 3.14 53.79 11

40 358 9.42 37.98 24

30 401 15.71 25.53 27

20 268 21.99 12.19 18

10 154 28.27 5.45 10

Probability per unit area is

highest in the 50 zone 2

Most likely place to find

a dart, however, is in

the 30 zone RDF

Not as many holes per unit area as,but a lot more places for the holes to go.

Even more places for the holes to go, but far

too few holes per unit area

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Shape of s orbitals

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Shape of s orbitals

An orbital with l=0

has

a spherical shape

with the nucleus at

its centre is called s

orbital. Because asphere has only one

orientation, an s

orbital has only one

mlvalue.

Node is the regionwhere the probability

of finding the

electron drops to

zero.

( ) 0/2/300,0,11

,, ar

ear =

( ) 02/

0

2/3

00,0,2 2

24

1,,

are

a

rar

=

3,0,0(r,,) =

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Shape of 2p orbital

An orbital with l=1 is called p orbital and

has two regions of high probability, oneon the either side of the nucleus.

there are three possible ml values

Ml=-1,0,+1. Hence three possible

orientations in mutually perpendicular

directions.

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Shape of 3d orbital

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Shape of 3d orbital

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Shape of orbitals with higher l values

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Shape of orbitals with higher l values

Orbitals with l=3 are f orbitals.

They have 7 orientations.Given figure shows one of the seven

orientations.

What does an horbital look like?

Check out these site: http://www.orbitals.com/orb/orbtable.htm

http://winter.group.shef.ac.uk/orbitron/

Just for interest

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Radial Distribution

http://www.orbitals.com/orb/orbtable.htmhttp://winter.group.shef.ac.uk/orbitron/http://winter.group.shef.ac.uk/orbitron/http://www.orbitals.com/orb/orbtable.htm

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Radial Distribution

Functions/Bohrs radius

Note the probability maxima

occurs at the same orbit

radius fixed by Bohr

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Radial Distribution Functions

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Summary

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Summary

Atomic orbitals are solutions to the Schrdinger equationfor hydrogenic atoms.

Atomic orbitals are characterized by three quantumnumbers (QN). nthe principal QN and ranges from 1 up to infinity. The larger

the n, the more extended the orbital.

lthe orbital angular momentum QN, and ranges from 0 up to n-1.

lgives the shape of the orbital. l= 0 is an sorbital, l= 1 is a porbital, l= 2 is a dorbital, etc. mlthe orbital angular momentum projection QN, and ranges from

lup to lin steps of 1. mlgives the orientation of the orbital.

2is the probability density of finding the electron atposition (r,,) or (x,y,z).

The radial distribution function gives the probabilitydensity of finding the electron at a distance rfrom thenucleus, regardless of direction.

We finally arrive at the shapes of different atomicorbitals. CM1502 Sem2-2013-14