Download • Eψ = H ψ

Wave Functions for Hydrogen
Quantum Numbers
• Eψ = H ψ
– H (x,y,z) or H (r,θ,φ)

H 
2 2
e2
 
2
4 o r
• When solving the Schrödinger equation, three
quantum numbers are used.
– Principal quantum number, n (n = 1, 2, 3, 4, 5, …)
ψ(r,θ,φ) = R(r) Y(θ,φ)
– Secondary quantum number, l
R(r) is the radial wave function.
Y(θ,φ) is the angular wave function.
– Magnetic quantum number, ml
Quantum Numbers
Principle Shells and Subshells
• Principle electronic shell, n = 1, 2, 3…
– Orbital size
• Angular momentum quantum number,
l = 0, 1, 2…(n-1)
– Orbital shape
l = 0, s
l = 1, p
l = 2, d
l = 3, f
• Magnetic quantum number,
– Orbital orientation
– ml= -l …-2, -1, 0, 1, 2…+ l
En  
2
2me a02 n 2
same as for the Bohr atom
Interpreting and Representing
R
the
Orbitals of the Hydrogen
H
Atom.
Orbital Energies for 1 Electron Atom
• Note the relationship between number of orbitals within
s, p, d, and f and ml. Each principle shell contains n² orbitals.
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Visualizing Orbitals
• Wave functions for the first
five orbitals of hydrogen.
• The wave function is written
in spherical polar coordinates
with the nucleus at the origin
origin.
– Point in space defined by radius
r, and angles  and .
– r determines the orbital size.
–  and  determine the orbital
shape.
Visualizing Orbitals
Visualizing Orbitals
• A point in space
defined by radius r, and
angles  and .
• Chemists usually think
of orbitals in terms of
pictures.
– The space occupied by
an orbital is a 90%
probability of finding
an electron.
– A plot of the angular
part of the wave
function gives the
shape of the
corresponding orbital.
Visualizing Orbitals
• An orbital depicts the probability of finding an electron at
a given location.
– Node is a location where the electron is never found (ψ = 0)
• The radial part of the wave function describes how the
probability of finding an electron varies with distance from
the nucleus.
– Spherical nodes are generated by the radial portion of the wave
function, R(r).
• The angular part of the wave function describes how the
probability of finding an electron varies with orientation
from the nucleus.
– Nodal nodes are generated by the angular portion of the wave
function, Y(θ,φ) .
• s orbitals are spherical
• p orbitals have two lobes separated by a nodal plane.
– A nodal plane is a plane where the probability of finding an electron is zero
(here the yz plane).
• d orbitals have more complicated shapes due to the presence of two
nodal planes.
Probability Distribution for the s Orbitals
Radial Probability Distribution for 1s Orbital
No spherical nodes
1 spherical node
2 spherical nodes
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The Boundary Surface Representations
of All Three 2p Orbitals
p Orbitals
Nodal Plane
The Boundary Surfaces of All of the 3d Orbitals
Representation of the 4f Orbitals in Terms
of Their Boundary Surfaces
Orbital Nodes
1 Electron Atomic Orbital Example
An orbital is description by the quantum numbers n, l and ml
• Total number nodes (excluding r = 0 and ) is n-1
• The number of angular nodes is l whose location depends
on the value of ml
• The number of spherical nodes is n-1- l
So for a 4d orbital (n=4, l =2)
Would have 3 (n-1=3) total nodes; of which 2 (l =2) are
angular nodes and 1 (n-1- l =1) is spherical node
Which is larger?
• the H 2p orbital or the H 3p orbital
• the H 1s orbital or the Li2+ 1s orbital
Which is lower in energy?
• the H 2p orbital or the H 3p orbital
• a H 4s orbital or a H 4f orbital
How many nodes and where are they located in a
• 4s orbital
• 3py orbital
• 4dxz orbital
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Electron Spin: A Fourth Quantum Number
• The spin quantum number, ms, determines the number of
electrons that can occupy an orbital.
– ms = ±1/2
– Electrons described as “spin up” or “spin down”.
– An electron is specified by a set of four quantum numbers.
The Pauli Exclusion Principle
The Pauli Exclusion Principle states:
• No two electrons in an atom can have the same four
quantum numbers (n, l, ml, ms).
– Two electrons can have the same values of n, l, and ml,
but different values of ms.
– Two electrons maximum per orbital.
– Two electrons occupying the same orbital are spin
paired.
Multi-electron Atoms
• Hydrogen wavefunctions are for only one especies.
• Electron-electron repulsion in multi-electron
atoms.
– Use Hydrogen-like orbitals (by approximation).
Orbital Energies and Electron Configurations
• For electrons in larger orbitals, the charge “felt” is
a combination of the actual nuclear charge and
the offsetting charge of electrons in lower orbitals.
– The masking of the nuclear charge is called shielding.
– Shielding results in a reduced, effective nuclear charge.
A Comparison of the Radial Probability
Distributions of the 2s and 2p Orbitals
Penetration and Shielding
• Effective nuclear charge
allows for understanding of
the energy differences
between orbitals.
– 2s orbital: the small local
maximum close to the
nucleus
l
results
l in
i an
electron with a higher
effective nuclear charge.
– 2p orbital: lacks the local
minimum close to the
nucleus of the 2s orbital.
Zeff is the effective nuclear charge.
• Lower effective nuclear
charge for 2p electrons.
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The Radial Probability Distribution of the 3s Orbital
The Energies of Orbitals in Polyelectronic Atoms
A Comparison of the Radial Probability
Distributions of the 3s, 3p, and 3d Orbitals
Orbital Energies
Aufbau Process
As we add protons element by element across the periodic table,
electrons are similarly added into hydrogen-like orbitals,
starting with the lower-energy orbitals and building up.
We can determine the arrangement of electrons (how many
electrons in each orbital), called the electron configuration,
for all elements on the Periodic Table.
We follow two “rules” to do this
1. We saw that the Pauli Exclusion Principle states that no two
electrons can have the same four quantum numbers…so only
two electrons can be added to each orbital.
Orbital Diagram
• A notation that shows how many electrons an atom
has in each of its occupied electron orbitals.
Oxygen: 11s22s
O
2 22p
2 4
Oxygen:
1s
2s
2p
2. Hund’s Rule states that the lowest energy arrangement of
electrons is the one with the maximum number of unpaired
electrons (but still following the Pauli Exclusion Principle!)
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Orbital Filling
Ground State Electron Configurations
Z
1
Atom
Configuration
1s
2s
2p
2
3
4
5
6
7
8
9
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Filling the d Orbitals
Hund’s Rule and the Aufbau Principle
• The inner electrons, which lie closer to the nucleus, are referred to
as core electrons.
– Core electrons can be represented by the noble gas with the same electronic
configuration.
• The outer electrons are usually referred to as valence electrons.
– Valence electrons are shown explicitly when a noble gas shorthand is used to
write electronic configurations.
– Valence electrons determine reactivity.
Electron Configurations of Some Groups of Elements
The Periodic Table and Electron Configurations
• The periodic table and the electronic
configurations predicted by quantum mechanics
are related.
– The periodic table is broken into s, p, d, and f blocks.
– Elements in each block have the same subshell for the
highest electron.
– Structure of periodic table can be used to predict
electronic configurations.
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Electron Configurations Example
The Periodic Table and Electron Configurations
• For the following chemical species, what are the
expected electron configurations.
– Ca
– Fe
– Eu
• The shape of the periodic table can be broken down into blocks
according to the type of orbital occupied by the highest energy
electron in the ground state.
• We find the element of interest in the periodic table and write its core
electrons using the shorthand notation with the previous rare gas
element. Then we determine the valence electrons by noting where
the element sits within its own period in the table.
• What atom has the following electron
configurations?
– 1s2 2s2 2p6 3s2 3p5
– 1s2 2s2 2p6 3s2 3p6 3d7 4s2
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