Post on 29-Dec-2015
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
re
Vo
4
2
The Hamiltonian
re
mm
VEEH
oN
Ne
e
nucleus,Kelectron,K
422
22
22
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
nl,n
l
l,nl,n e Ln
NrR 2
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
224 rrP
For all other orbitals
22 rRrrP
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 orbitalsEffective 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 orbitalsThe first shelln = 1 The shell nearest the nucleusl = 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 electronss = + 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 signExchange 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 upElectrons are added to hydrogenic orbitals as Z increases.
Many Electron Species
The Schrödinger equation cannot be solved exactly for the He atom
12
2
2
2
1
2
22
21
2
4
1
2
re
rZe
rZe
mH
o
e
The Orbital Approximation
For many electron atoms
2121 rrr,r
Think of the individual orbitals as resembling the hydrogenic orbitals
The Hamiltonian in the Orbital Approximation
For many electron atoms
21
2211
2211
2121
r,rE
rErE
rHrH
rrHr,rH
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. Non-penetrating Orbitals
s orbitals – penetrating orbitalsp 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
...,,r,,r 22221111
Hartree - 1928
SCF Method #2
The SE is separated into n equations of the type
iiiio
ie
Er
Zem
2
22
4
1
2
Note – Ei is the energy of theorbital 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 electronsCalculation 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?
P23
3
StateJ
Multiplicity
Calculating the L value
Add the individual l values according to a Clebsch-Gordan series
21
212121 21
ll,...,
ll,ll,llL
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
21
212121 21
ss,...,
ss,ss,ssS
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
SL,...,
2SL,1SL,SLJ
J-values in j-j Coupling
Add the individual j values according to a Clebsch-Gordan series
21
212121 21
jj,...,
jj,jj,jjJ
Selection Rules
Any state of the atom and any spectral transition can be specified using term symbols!
S P21
2
23
3
Note – upper term precedes lower term by convention
Selection Rules #2
These selection rules arise from the conservation of angular momentum
1 0,J
1 0,L 0,S
Note – J=0 J=0 is not allowed
The Effects of Magnetic Fields
The electron generates an orbital magnetic moment
lB
le
Z
m
mme
2
The energy BmE lBml
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.