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Wavefunctions and Energy Levels
Since particles have wavelike properties cannot expect them
to behave like point-like objects moving along precise
trajectories.
Erwin Schrödinger: Replace the precise trajectory of particles
by a wavefunction (y), a mathematical function that varies
with position
Max Born: physical interpretation of wavefunctions.
Probability of finding a particle in a region is proportional
to y2.
y2 is the probability density. To calculate the probability that
a particle is in a small region in space multiply y2 by the
volume of the region.
Probability = y2 (x,y,z) dx dy dz
Schrödinger Equation
The Schrödinger equation describes the motion of a particle
of mass m moving in a region where the potential energy is
described by V(x).
2 y
d
-h
2 m dx2 + V(x) y = E y
(1-dimension)
Only certain wave functions are allowed for the electron in
an atom
The solutions to the equation defines the wavefunctions
and energies of the allowed states
An outcome of Schrödinger’s equation is that the particle
can only possess certain values of energy, i.e. energy of a
particle is quantized.
In the H atom the potential that the electron feels is the
electrostatic interaction between it and the positive
nucleus
V(r ) = - e2 / (4 p eo r)
r: distance between the electron and the nucleus.
Solve the Schrödinger equation to determune the allowed
energy levels of an electron in the H atom
Solution for allowed energy levels is:
En = - h R
n2
n = 1, 2,...
R = (me e4) / (8 h3 eo2) = 3.29 x 1015 Hz
n: principle quantum number.
Labels the energy levels
When n = 1 => ground state of
the H atom. Electron in its
lowest energy
n > 1 : excited states; energy
increases as n increases
E = 0 when n = ∞ , electron has
left the atom - ionization
Atomic Orbitals
Wavefunctions of electrons in atoms are called atomic
orbitals, have a dependence on position
Square of the wavefunction - probability density of electron
The wavefunction of an electron in a hydrogen atom is
specified by three quantum numbers, specifying energy
and probability of finding an electron.
1) Principle quantum number, n: specifies energy of the
orbitals. In a hydrogen atom, all atomic orbitals with the
same value of n have the same energy and are said to
belong to the same SHELL of the atom.
2) Orbital angular momentum quantum number, l
l = 0, 1, 2, …., n-1
Each value of l corresponds to a different type of orbital with
a different shape
The orbitals of a shell with principal quantum number n fall
into n groups, called SUBSHELLS; each subshell is
identified by a different l value.
l=0
l=1
l=2
l=3
s-orbitals
p-orbitals
d-orbitals
f-orbitals
Magnetic quantum number, ml : distinguishes the orbitals
within a subshell. Determines how the atom behaves in a
magnetic field.
ml = l, l -1, … - l
2 l + 1 ml values for each l
l = 1; ml = +1, 0, -1
n is related to the size of the orbital, l is related to its shape,
and ml is related to its orientation in space.
s orbitals: correspond to l = 0 and ml = 0
For Hydrogen atom the ground state is n = 0, l = 0 and ml = 0;
a s orbital
Density of shading
represents the
probability of finding an
electron at any point.
The graph shows how
probability varies with
distance
Wavefunctions of s orbitals of higher energy have more
complicated radial variation with nodes (points of zero
probability)
Boundary surface encloses
surface with a > 90% probability of
finding electron
electron density
wave function
radial probability
distribution
p orbitals: Three p orbitals l = 1, ml = +1, 0 - 1
d orbitals: Five p orbitals l = 2, ml = +2, +1, 0 - 1, -2
f orbitals: Seven f orbitals l = 3, ml = +3, +2, +1, 0 - 1, -2. -3
The three quantum numbers for an electron in a H atom in a
certain state are n = 4, l = 2, ml = -1. In what type of orbital
is the electron located?
Electron Spin
Spectral lines observed did not have exactly the same
frequencies as those calculated by Schrödinger.
S. Goudsmit and G. Uhlenbeck proposed electrons have spin.
Electrons behave like a spinning sphere, like a planet
rotating on its axis.
An electron has two spin states, represented by and  or a
and b.
Can think of these states as a counterclockwise () spin or a
clockwise (), both at the same rate.
Spin quantum number, ms, distinguishes the two spin states
ms = 1/2  electron
ms = - 1/2  electron
O. Stern and W. Gerlach
The state of an electron in a hydrogen atom is defined by the
four quantum numbers, n, l, ml, ms. As the values of n
increases, the size of the atom increases.
Many-Electron Atoms
Electronic Structure of H atom (Z = 1)
Electron in the lowest energy level - ground state of the atom,
n = 1 => 1s orbital
Quantum numbers of this 1s electron
n = 1, l = 0, ml = 0, ms = +1/2 or -1/2
If the electron acquires energy, the electron can undergo a
transition to the n = 2 shell and can occupy the 2s or one of
the three 2p orbitals (for H-atom all have the same energy)
The state of an electron in a H atom is defined by the four
quantum numbers n, l, ml, ms.
As the value of n increases, the size of the atom increases.
2 y
d
-h
2 m dx2 + V(x) y = E y
For H atom: V(r ) = - Z e2 / (4 p eo r)
(Z = 1 for H atom)
Many-electron atoms (Z > 1)
Electrons occupy orbitals like those of a H atom.
Energies of orbitals of many electron atoms are not the same
as those for the H atom.
Nuclear attraction for electrons is greater as Z increases
lowering the electrons’ energy; also have to account for
electron-electron repulsion.
In the Schrödinger equation, V(r ) has to account for both the
nuclear-electron attraction and the electron-electron
replusion
For example for He (Z = 2), V(r ) contains three terms
V(r ) = - [(2 e2) / (4 p eo r1)] - [(2 e2) / (4 p eo r2)] + e2 / (4 p eo r12)
attraction
attraction
repulsion
For many-electron atoms
The electron density of an isolated many-electron atom is ~
sum of the electron densities of each electron taken
individually
Every electron in an atom has a set of four quantum
numbers, n, l, ml and ms
The electron-electron repulsion opposes electron-nuclear
attraction.
The repulsion “shields” the electron from the full attraction of
the nucleus.
Electrons feel an “effective” nuclear charge which is less
than the full nuclear charge.
s orbitals have a non-zero probability density at the nucleus,
penetrate through inner shells
s electrons feel stronger nuclear attraction; are tightly bound
and hence lower in energy
p orbitals have zero probability density at the nucleus; less
penetrating than s and hence p electrons are higher in
energy.
d orbitals less penetrating than p and hence d electrons are
higher in energy than p
In many electron atoms, because of shielding and
penetration effects, order of the energy of orbitals in a given
shell is typically s < p < d < f.
Energies of orbitals depend on both n and l (not just n as in
the H atom)
Exclusion Principle
The electronic structure of an atom determines its chemical
properties.
Electron configuration - a list of all occupied orbitals of an
atom, with the number of electrons that each contains
Pauli Exclusion Principle: No more than two electrons may
occupy any given orbital. When two electrons occupy an
orbital their spins must be paired.
No two electrons in an atom can have the same set of
quantum numbers.
(a) Spins are paired if one is and the other .
Paired spins denoted as ; ms of each is different
(b) Two electrons have parallel spin if both spins are in the
same direction
Building Up: fill orbitals starting with the lowest energy
(aufbau principle), pairing electrons as determined by the
Pauli principle.
Order of energies of orbitals
5f
6d
7p
4f
5d
6p
3f
4d
5p
3d
4p
3p
2p
8s
7s
6s
5s
4s
3s
2s
1s