Transcript Chemistry 330

Chemistry 330
Atomic Orbitals, Electron
Configurations, and Atomic
Spectra
The Hydrogen Spectrum
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.
Photon Emission
Energy is conserved
when a photon is
emitted, so the
difference in energy
of the atom before
and after the
emission event must
be equal to the
energy of the photon
emitted.
The Hydrogen Atom
The effective potential energy of an
electron in the hydrogen atom.
Electron with zero orbital angular
momentum the effective potential
energy is the Coulombic potential
energy.
The Structure of the H-atom
The Coulombic energy
e2
V
4or
The Hamiltonian
Ĥ  ÊK ,electron  ÊK ,nucleus  V̂
 2

e
2

e 
N 
2me
2mN
4or
2
2
2
The Separation of the Internal
Motion
The coordinates
used for discussing
the separation of the
relative motion of
two particles from
the motion of the
centre of mass.
The Solutions
The solution to the SE for the H-atom
separates into two functions


Radial functions (real)
Spherical Harmonics (complex functions)
Radial Wavefunctions
The radial wavefunctions products of
the


Laguerre polynomials
Exponentially decaying function of distance
l


R n,l r   Nn,l   L n,l e 2n
n
Some Radial Wavefunctions
Orbital
n
l
Rn,l
1s
1
0
2(Z/ao)3/2 e-/2
2s
2
0
1/(2 21/2) (Z/ao)3/2(2-1/2) e-/4
2p
2
+
1
1/(4 61/2) (Z/ao)3/2  e-/4
Electron has
nonzero orbital
angular momentum,
the centrifugal effect
gives rise to a
positive contribution
which is very large
close to the nucleus.
The Radial Wavefunctions
The radial wavefunctions of the first few
states of hydrogenic atoms of atomic
number Z.
Radial Wavefunctions
Radial Wavefunctions
Some Pretty Pictures
The radial distribution functions for the
1s, 2s, and 3s, orbitals.
Boundary Surfaces
The boundary
surface of an s
orbital, within which
there is a 90 per
cent probability of
finding the electron.
Radial Distribution Function
For spherically symmetric orbitals
Pr   4r 
2
2
For all other orbitals
Pr   r R r 
2
2
The P Function for a 1s Orbital
The radial
distribution function
P gives the
probability that the
electron will be
found anywhere in a
shell of radius r.
The Dependence of  on r
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
of the nucleus as l
increases.
Hydrogen Energy Levels
The energy levels of
a hydrogen atom.
The values are
relative to an
infinitely separated,
stationary electron
and a proton.
Energy Level Designations
The energy levels of
the hydrogen atom


subshells
the numbers of
orbitals in each
subshell (square
brackets)
Many-Electron Atoms
Screening or
shielding alters the
energies of orbitals
Effective nuclear
charge – Zeff

Charge felt by
electron in may
electron atoms
Quantum Numbers
Three quantum numbers are obtained from
the radial and the spherical harmonics




Principal quantum number n. Has integer values 1,
2, 3
Azimuthal quantum number, l. Its range of values
depends upon n: it can have values of 0, 1... up
to n – 1
Magnetic quantum number, ml . It can have
values -l … 0 … +l
Stern-Gerlach experiment - spin quantum number,
ms. It can have a value of -½ or +½
Atomic orbitals
The first shell
n=1
The shell nearest the nucleus
l =0
We call this the s subshell (l = 0)
ml = 0
There is one orbital in the subshell
s = -½
The orbital can hold two electrons
s = + ½ one with spin “up”, one “down”
No two electrons in an atom can have the same
value for the four quantum numbers: Pauli’s
Exclusion Principle
The Pauli Principle
Exchange the labels of any two
fermions, the total wavefunction
changes its sign
Exchange the labels of any two bosons,
the total wavefunction retains its sign
The Spin Pairings of Electrons
Pair electron spins zero resultant spin
angular momentum.
Represent by two
vectors on cones
Wherever one
vector lies on its
cone, the other
points in the
opposite direction
Aufbau
Principle
Building up
Electrons are added
to hydrogenic
orbitals as Z
increases.
Many Electron Species
The Schrödinger equation cannot be
solved exactly for the He atom



2
2
Ĥ 
1   2 
2me
2
2
2
2

1
Ze
Ze
e 

 

4o  r1
r2
r12 
The Orbital Approximation
For many electron atoms
r1 , r2   r1 r2 
Think of the individual orbitals as
resembling the hydrogenic orbitals
The Hamiltonian in the Orbital
Approximation
For many electron atoms
Ĥ r1 , r2   Ĥr1 r2 
Ĥ1r1   Ĥ2 r2 
 E1r1   E 2 r2 
 E r1 , r2 
Note – if the electrons interact, the theory fails
Effective Nuclear Charge.
Define Zeff = effective nuclear charge =
Z -  (screening constant)
Screening Effects (Shielding)
Electron energy is directly proportional
to the electron nuclear attraction
attractive forces,


More shielded, higher energy
Less shielded, lower energy
Penetrating Vs. Nonpenetrating Orbitals
s orbitals – penetrating orbitals
p orbitals – less penetrating.
d, f – orbitals – negligible penetration of
electrons
Shielding #2
Electrons in a given shell are shielded
by electrons in an inner shell but not by
an outer shell!
Inner filled shells shield electrons more
effectively then electrons in the same
subshell shield one another!
The Self Consistent Field
(SCF) Method
A variation function is used to obtain
the form of the orbitals for a many
electron species
  1 r1 , 1 , 1 2 r2 , 2 , 2 ...
Hartree - 1928
SCF Method #2
The SE is separated into n equations of
the type
 
1 Ze 
2
i 

  i  Ei i
4o ri 
 2me
2
 
2
Note – Ei is the energy of the
orbital for the ith electron
SCF Method #3
The orbital obtained (i) is used to
improve the potential energy function of
the next electron (V(r2)).
The process is repeated for all n
electrons
Calculation ceases when no further
changes in the orbitals occur!
SCF Calculations
The radial
distribution functions
for the orbitals of Na
based on SCF
calculations. Note
the shell-like
structure, with the
3s orbital outside
the inner K and L
shells.
The Grotian Diagram for the
Helium Atom
Part of the Grotrian
diagram for a helium
atom.
There are no
transitions between
the singlet and triplet
levels.
Wavelengths are
given in nanometres.
Spin-Orbit Coupling
Spin-orbit coupling is a
magnetic interaction
between spin and
orbital magnetic
moments.
When the angular
momenta are


Parallel – the magnetic
moments are aligned
unfavourably
Opposed – the
interaction is favourable.
Term Symbols
Origin of the symbols in the Grotian
diagram for He?
Multiplicity
3
State
P3
2
J
Calculating the L value
Add the individual l values according to
a Clebsch-Gordan series
L  l1  l2 , l1  l2  1, l1  l2  2
,..., l1  l2
2 L+1 orientations
What do the L values mean?
L
Term
0
S
1
P
2
D
3
F
4
G
The Multiplicity (S)
Add the individual s values according to
a Clebsch-Gordan series
S  s1  s 2 , s1  s 2  1, s1  s 2  2
,..., s1  s 2
Coupling of Momenta
Two regimes


Russell-Saunders coupling (light atoms)
Heavy atoms – j-j coupling
Term symbols are derived in the case of
Russell-Saunders coupling may be used
as labels in j-j coupling schemes
Note – some forbidden transitions in
light atoms are allowed in heavy atoms
J values in Russell-Saunders
Coupling
Add the individual L and S values
according to a Clebsch-Gordan series
J  L  S, L  S  1, L  S  2
,..., L  S
J-values in j-j Coupling
Add the individual j values according to
a Clebsch-Gordan series
J  j1  j2 , j1  j2  1, j1  j2  2
,..., j1  j2
Selection Rules
Any state of the atom and any spectral
transition can be specified using term
symbols!
3
P3 2 S 1
2
2
Note – upper term precedes
lower term by convention
Selection Rules #2
These selection rules arise from the
conservation of angular momentum
S  0, L  0,  1
J  0,  1
Note – J=0  J=0
is not allowed
The Effects of Magnetic Fields
The electron generates an orbital
magnetic moment
e
Z  
ml
2me
 Bml
The energy
E ml  BmlB
The Zeeman Effect
The normal Zeeman effect.



Field off, a single spectral line
is observed.
Field on, the line splits into
three, with different
polarizations.
The circularly polarized lines
are called the -lines; the
plane-polarized lines are
called -lines.