Download Electronic structure and spectroscopy

Electronic structure and spectroscopy
Péter G. Szalay
Institute of Chemistry, Eötvös Loránd University, Budapest
Course material at:
http://www.chem.elte.hu/departments/elmkem/szalay/szalay_files/Pavlodar/
2013. október 21.
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1.
Today’s concept of the atom
Shortly after Dalton’s atomic theory became widely accepted (matter consists of undividable particles called atoms) experiments by physicists indicated that the atoms indeed
have some structure.
1.1.
Discovery of electron
Joseph John Thomson (1856-1940)
His famous experiment with the cathode ray tube (1897):
In a cathode ray tube, independently of the material of the cathode, the same event
can be observed: light spots (flashes) appear on the screen, i.e. a particles leave the
cathode and fly against the plate. In an electric field, this particle deviates towards the
positive pole, i.e. it must have a negative charge. Its mass (measured by magnitude of
the deviation) is much smaller than the mass of the atom (in fact Thomson meassured
the ratio of mass and charge). He called it a „Corpuscle”, the name „electron” was later
introduced by G. Johnstone Stoney (1853-1928), who worked with electricity and called
the unit of electricity as electron.
Thomson was also a bit skeptical: "Could anything at first sight seem more impractical
than a body which is so small that its mass is an insignificant fraction of the mass of an
atom of hydrogen?" (Thomson)
He got the Nobel prize in 1906.
But how the atom should look like? There is a positive charge with large mass and
a tiny electron bearing the negative charge. It can be like a plum pudding; electrons are
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the plums, and the pudding is the distributed positive charge.
1.2.
Discovery of the nucleus
Ernest Rutherford (1871-1937)
Experiment: alpha particles originating from nuclear decay were used to bomb gold foil
(Geiger és Mardsen):
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The atom must have a nucleus which is much smaller than the atom and it is bearing
the positive charge.
„It is like to fire with a cannon on a sheet of paper and it would bounce back”
Rutherford’s atomic model (1911): an atom consists of a positively charged small nucleus
bearing almost all mass of it; the electrons are orbiting around it like the moon around
the Earth.
There are lots of unanswered question, though. For example, why the electron does not
fall into the nucleus? (This system is different from the Earth-Moon system, since it has
charges!!!)
Today we know that even the nucleus has a structure: it consists of protons and neutrons,
and even these can be divided into smaller elementary particles. For chemistry this is,
however, not relevant, chemistry in most part can be described by a nucleus „surrounded”
by the electrons. This is, on the other hand, already very relevant for chemical analysis!
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2.
Development of quantum mechanical view
The atomic theory allowed the development of modern chemistry, but lots of questions
remained unanswered, and in particular the WHY is not being explained:
• What is the binding force between atoms. It is not the charge since atoms are
neutral. Why can even two atoms of the same kind (like H-H) form a bond?
• Why atoms can form molecules only with certain rates?
• What is the reason of the periodic table of Mendeleev?
At the turning of the 19th and 20st century new experiments appeared which could
not be explained by the tools of the classical (Newtonian) mechanics. For the new theory
new concepts were needed:
• quantization: the energy can not have arbitrary value
• particle-wave dualism
⇒ development of QUANTUM MECHANICS
The new theory was developed along a long root (which in time was not that long at
all!). We will follow this route now and stop at the most important steps.
2.1.
Introduction: same basic terms related to light
In the strict sense, „light” is a narrow range of electromagnetic radiation, what we can
sense with our eyes. In physics very often the term „light” is used for the entire spectrum.
The electromagnetic radiation consists of oscillating magnetic and electric fileds wich are
perpedicular to each other and the direction of its propagation.
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Basic terms:
• ν: frequency of the oscillation [1/s]
• ν ∗ : wavenumber [1/m]
• λ: wave length [m]
• c: speed of light
• polarization: there is oscillation only in a plane
Important relations:
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Magspingerjesztés
Ranges of the electromagnetic radiation:
Ionizáció
1
λ
Molekularezgések
gerjesztése
ν∗ =
Molekulákforgásának
gerjesztése
c
ν
Elektrongerjesztés
Maggerjesztések
λ=
What is spectroscopy?
The matter can absorbe or emit light. The absorbed/emitted light can be divided into
its components, and these will be characteristic for the matter.
The light, thus, can be divided into its components, for example by a prism.
2.2.
2.2.1.
Observations leading to quantum mechanics
Black Body Radiation
Planck in 1900 came up with a new, unusual explanation: according to his theory,
the energy of the radiation is quantized, it can only be hν, 2hν, 3hν ..., thus it does not
change continuously. Here h is the so called Planck constant: h = 6.626 · 10−34 Js
(Planck himself did not like his own theory, since it required an assumption (postulate),
i.e. the existence of constant h; he wanted to derive this from the existing theory. He
was not successful with this; now we know it is not possible to derive since it follows from
a new theory. Thus, despite of his genius discovery, he could not participate in further
development of quantum mechanics. )
2.2.2.
Heat capacity
According to the Dulong-Petit rule, the heat capacity is given by cv,m ≈ 3R, i.e. it is
independent of temperature. This is valid at temperatures people could investigate until
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the 19th century, but then it turned out that at low temperatures it goes to zero:
Einstein explained this using Planck’s idea: the matter is also quantized, the oscillators
(vibrations) can not have any energy, like the oscillators causing the black body radiation.
(The final form of the theory with several oscillators was derived by Debye.)
2.2.3.
Photoelectric effect
Shining light on the metal plate can result in electric current in the circuit. However,
there is a threshold frequency, below this there is no current, irrespective of the intensity
of the light, i.e.
• below the threshold frequency, no electron leaves the metal plate
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• increasing the intensity of the light, the energy of the emitted electron does not
change, only their number grows.
According to the measurements, the following relation exists between the kinetic
energy of the electron (Te ) and the frequency of the light (ν):
Tel = hν − A
where A depends on the quality of the metal plate (called work function).
Explanation was given again by Einstein using the quantization introduced by Planck:
the light consist of tiny particles which can have energy of hν only. (Note that Planck
opposed the use of his „uncompleted” theory!!)
2.2.4.
The Compton effect
A photon collides with a resting electron, it looses energy. Therefore its frequency also
changes. The photon acted as a particle in this experiment!! Note a wave scattering on
an object would not change its wave length or frequency!!!
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2.2.5.
Scattering of electron beam
Experiment by Davisson and Germer (1927), as well as G.P. Thomson (1928). There are
interference circles on the photographic plate, just like in case of X-ray radiation → electron acted as a wave.
2.2.6.
The hydrogen atom
Hydrogen atom has four lines in his emission spectra in the visible range (experiment by
Ångsröm):
The position of the lines have been described by Balmer (so called Balmer formula):
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1
1
= R 2− 2
λ
2
n
10
n = 3, 4, 5, 6
(1)
where R is the so called Rydberg constant, λ is the wave length.
The energy of the hydrogen atom must be quantized, too!!
Explanation by Bohr: in his atomic model, the electron must fullfil some „quantum”
relations:
• in case of orbits having certain radius, the electron do not dissipate energy; these
are the so called stationary states;
• if the electron jumps from one orbit to the other, it emits (or absorbes) energy in
form of electromagnetic field („light”).
• the possible values for the energy are:
E = −
1 e2
2n2 a0
n is real number
(2)
(e is the charge of the electron, a0 unit length (1 bohr)).
Gives the Balmer formula back. HOMEWORK: SHOW THAT THIS IS TRUE.
However, can not be applied for the He or any other atom!!!
2.2.7.
Summary
Event
New term
black body radiation
energy quantized (hν)
photoelectric effect
energy of the light is quantized
heat capacity at low temperature
matter is quantized
goes to zero
Compton effect
electromagnetic radiation
acts like a particle
scattering of electron
electron acts like a wave
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Discoverer
Planck (1900)
Einstein (1905)
Einstein (1905),
Debye
Compton (1923)
Davisson (1927),
G.P. Thomson (1928)
Remember:
• ν is the frequency of light
• λ is the wave length of light (λ = νc )
• c speed of light
• h = 6.626 10−34 Js a Planck constant
• h̄ =
h
2π
Important consequence of all these: particle-wave dualism (dual nature of the
matter)
The existing theories need to be revised completely! Although Bohr could „fix” this old
theory with quantum condition to describe the hydrogen atom, but the theory does not
work for other atoms!
New theory:
• Heisenberg (1925): Matrix mechanics
• Schrödinger (1926): Wave mechanics
It turned out later that the two theories are equivalent, they use only a slightly
different mathematics. Now we call this theory as (non-relativistic) quantum
mechanics.
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2.3.
2.3.1.
Basic concepts of quantum mechanics
Operators
What is an operator? It acts on a function and produces an other function:
Âf (x) = g(x)
There are special functions called eigenfunctions of an operator: if the operator acts
on its eigenfunction, it results in a constant times the same function:
Âf (x) = af (x)
where a is a constant, the eigenvalue of the operator.
Demonstration: Assume that
d2
 =
dx2
Then cos(x) is an eigenfunction of this operator since:
 cos(x) =
d2 cos(x)
= − cos(x)
dx2
its eigenvalue being −1.
2.3.2.
Schrödinger-equation
The stationary states of a system (e.g. atom, molecule) can be obtained by solving the
so called (time independent) Schrödinger-equation:
ĤΨ = EΨ
(3)
with:
• Ĥ being the Hamilton operator of the system;
• Ψ is the state function of the system;
• E is the energy of the system.
This is an eigenvalue equation, Ψ being the eigenfunction of Ĥ, E is the eigenvalue. This
has to be solved in order to obtain the states of, e.g. molecules.
According to Dirac (1929) the whole chemistry is included in this equation:
P. A. M. Dirac, "Quantum Mechanics of Many-Electron Systems", Proceedings of the Royal
Society of London, Series A, Vol. CXXIII (123), April 1929, pp 714.:
„The general theory of quantum mechanics is now almost complete ... The underlying physical
laws necessary for the mathematical theory of a large part of physics and the whole of chemistry
are thus completely known, and the difficulty is only that the exact application of these equations
leads to equations much too complicated to be soluble. It therefore becomes desireable that
approximate practical methods of applying quantum mechanics should be developed, which
can lead to an explanation of the main features of complex atomic systems without too much
computation."
My own interpretation (2013) :
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• to describe molecules one need quantum mechanics;
• we need to develop methods which can give more and more accurate solutions to
the Schrödinger equation
• we also need approximate methods which support chemical intuition without expensive calculations.
In practice: one has to approximate Ψ. Chemists are very good at this!
2.3.3.
The Hamilton operator
The Hamilton operator of the system (Ĥ) consists of the sum of the kinetic (T̂ ) and the
potential energy (V̂ ) operators.
Ĥ = T̂ + V̂
(4)
the form of T̂ is the same for all systems, while the potential energy represents the molecule
by including the interactions between the electrons and nuclei.
2.3.4.
State function
In quantum mechanics the state of the system is represented by the wave function (or
state function) which depends on the coordinates of the particles:
Ψ = Ψ(x, y, z) = Ψ(r)
(5)
Ψ = Ψ(x1 , y1 , z1 , x2 , y2 , z2 , ..., xn , yn , zn ) = Ψ(r1 , r2 , ..., rn )
(6)
or in case of n particles:
The wave function has no physical meaning, but its square, the so called probability
density can be given a probability interpretation:
Ψ∗ (x0 , y 0 , z 0 ) · Ψ(x0 , y 0 , z 0 )dx dy dz
(7)
is the probability of finding a particle at point (x0 , y 0 , z 0 ) (more precisely in the infinitesimal proximity).
Shorter notation: Ψ∗ Ψdv or |Ψ|2 dv
We have to chose the wave function normalized, otherwise the valószínűség of finding
the particle in the whole space would not be one:
Z Z Z
Ψ∗ · Ψ dx dy dz = 1
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(8)
2.3.5.
Other physical quantities
Other physical properties are also described by operators. Most important operators are:
• position operator: x̂ (x̂f (x) = xf (x))
∂
• momentum operator: p̂x = −ih̄ ∂x
• kinetic energy operator: T̂ =
1 2
p̂
2m
2
2
h̄ d
1
= − 2m
≡ − 2m
∆
dx2
• angular momentum operator: ˆl = (ˆlx , ˆly , ˆlz )
• ...
Operators of position (x̂) and momentum (p̂x ) do not commute, meaning that
[x̂, p̂x ] = x̂p̂x − p̂x x̂ = −ih̄
(9)
which means that the two operators can not be interchanged (the results depends on,
which operator acts first).
Consequence: the two quantities (coordinate and momentum) can be measured in the
same time, the product of their uncertainty (∆x and ∆px must be larger the a given value:
1
h̄
2
This is the famous Heisenberg uncertainty principle.
∆x · ∆px ≥
2.3.6.
(10)
„The particle in the box” model
This is a very instructive model system which shows nicely the new properties of quantum
objects:
Hamiltonian:
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V (x) = 0, 0 < x < L
V (x) = ∞, otherwise
Within the box of length L the Hamiltonian is equal to the kinetic energy:
Ĥ = T̂ +V (x),
| {z }
0
The particle can not leave the box, the probability finding it outside the box is zero,
therefore the wave function must also vanish there. The keep the wave function continuos,
it has to vanish at the walls, as well (boundary condition):
Ψ(0) = Ψ(L) = 0
(11)
Therefore the Schrödinger equation to solve reads:
T̂ Ψ(x) = EΨ(x)
(12)
After a short (and instructive) calculation one gets the following result:
h2
; n = 1, 2, ...
2
s 8mL
2
π
Ψ(x) =
sin n x
L
L
E = n2 ·
The form of the wave function:
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Notes:
• The energy is quantized, it grows quadratically with the quantum number n, it is
invers proportional to L2 and m.
If L → ∞, E2 − E1 ∼
L = ∞.
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L2
→ 0. This means, the quantization disappears with
The same is true for growing mass m → ∞.
• There is a zero point energy (ZPE)!
The energy is not 0 for the lowest level (ground state).
If, however, L → ∞, E0 → 0.
Why is ZPE there? This is an unknown term for classical mechanics!
It can be explained by the uncertainty principle: ∆x · ∆p ≥ 21 h̄.
Since here we have V̂ = 0, E ∼ p2 , i.e. the energy of the particle originates in his
momentum only. Assume that E = 0, than p = 0, therefore ∆x = ∞, which is a
contradiction since ∆x ≤ L, the particle must be in the box. We conclude that the
energy can never get zero, since in this case its uncertainty would also be zero which
is possible only for very large box where the uncertainty of the coordinate is large.
Or alternatively, one can also say: if L → 0 =⇒ ∆x → 0 =⇒ ∆p → ∞ =⇒ ∆E →
∞. This means the energy of all levels MUST BE larger and larger if the size of the
box gets smaller.
• Wave function: the larger n is the more nodes are on the wave function: ground
state has none, first excited state has one, etc. (Node: where the wave function
changes sign).
• How does the solution looks like in 3D?
π 2 h̄2
E =
2m
n2a n2b n2c
+ 2 + 2 ,
a2
b
c
!
where a, b, c are the three measures of the box and na , nb , nc = 1, 2, ... are the the
quantum numbers.
If a = b = L, then
na
1
2
1
nb
1
1
2
E
h2
8mL2
2
5
5
We have found degeneracy which is caused by symmetry of the system (two measures
are the same).
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