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.