Presentation on theme: "Atomic Orbitals, Electron Configurations, and Atomic Spectra"— Presentation transcript:
1 Atomic Orbitals, Electron Configurations, and Atomic Spectra Chemistry 330Atomic Orbitals, Electron Configurations, and Atomic Spectra
2 The Hydrogen SpectrumThe 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.
3 Photon EmissionEnergy 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.
4 The Hydrogen AtomThe 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.When the electron has zero orbital angular momentum, the effective potential energy is the Coulombic potential energy. When the electron has nonzero orbital angular momentum, the centrifugal effect gives rise to a positive contribution which is very large close to the nucleus. We can expect the l = 0 and l 0 wavefunctions to be very different near the nucleus.
5 The Structure of the H-atom The Coulombic energyThe Hamiltonian
6 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.
7 The SolutionsThe solution to the SE for the H-atom separates into two functionsRadial functions (real)Spherical Harmonics (complex functions)
8 Radial Wavefunctions The radial wavefunctions products of the Laguerre polynomialsExponentially decaying function of distance
14 Some Pretty PicturesThe radial distribution functions for the 1s, 2s, and 3s, orbitals.
15 Boundary SurfacesThe boundary surface of an s orbital, within which there is a 90 per cent probability of finding the electron.
16 Radial Distribution Function For spherically symmetric orbitalsFor all other orbitals
17 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.For a 1s electron in hydrogen, P is a maximum when r is equal to the Bohr radius a0.
18 The Dependence of on rClose 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.An s orbital has a finite, nonzero value at the nucleus.
19 Hydrogen Energy Levels The energy levels of a hydrogen atom. The values are relative to an infinitely separated, stationary electron and a proton.
20 Energy Level Designations The energy levels of the hydrogen atomsubshellsthe numbers of orbitals in each subshell (square brackets)Hydrogen atom – all subshells have the same energy!
21 Many-Electron AtomsScreening or shielding alters the energies of orbitalsEffective nuclear charge – ZeffCharge felt by electron in may electron atoms
22 Quantum NumbersThree quantum numbers are obtained from the radial and the spherical harmonicsPrincipal quantum number n. Has integer values 1, 2, 3Azimuthal quantum number, l. Its range of values depends upon n: it can have values of 0, up to n – 1Magnetic quantum number, ml . It can have values -l … 0 … +lStern-Gerlach experiment - spin quantum number, ms. It can have a value of -½ or +½
23 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 subshells = -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
24 The Pauli PrincipleExchange the labels of any two fermions, the total wavefunction changes its signExchange the labels of any two bosons, the total wavefunction retains its sign
25 The Spin Pairings of Electrons Pair electron spins - zero resultant spin angular momentum.Represent by two vectors on conesWherever one vector lies on its cone, the other points in the opposite direction
26 Aufbau Principle Building up Electrons are added to hydrogenic orbitals as Z increases.
27 Many Electron SpeciesThe Schrödinger equation cannot be solved exactly for the He atom
28 The Orbital Approximation For many electron atomsThink of the individual orbitals asresembling the hydrogenic orbitals
29 The Hamiltonian in the Orbital Approximation For many electron atomsNote – if the electrons interact, the theory fails
30 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 energyLess shielded, lower energy
31 Penetrating Vs. Non-penetrating Orbitals s orbitals – penetrating orbitalsp orbitals – less penetrating.d, f – orbitals – negligible penetration of electrons
32 Shielding #2Electrons 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!
33 The Self Consistent Field (SCF) Method A variation function is used to obtain the form of the orbitals for a many electron speciesHartree
34 SCF Method #2 The SE is separated into n equations of the type Note – Ei is the energy of theorbital for the ith electron
35 SCF Method #3The 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!
36 SCF CalculationsThe 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.
37 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.
38 Spin-Orbit CouplingSpin-orbit coupling is a magnetic interaction between spin and orbital magnetic moments.When the angular momenta areParallel – the magnetic moments are aligned unfavourablyOpposed – the interaction is favourable.
39 Term Symbols Origin of the symbols in the Grotian diagram for He? MultiplicityStateJ
40 Calculating the L value Add the individual l values according to a Clebsch-Gordan series2 L+1 orientations
42 The Multiplicity (S)Add the individual s values according to a Clebsch-Gordan series
43 Coupling of Momenta Two regimes Russell-Saunders coupling (light atoms)Heavy atoms – j-j couplingTerm symbols are derived in the case of Russell-Saunders coupling may be used as labels in j-j coupling schemesNote – some forbidden transitions inlight atoms are allowed in heavy atoms
44 J values in Russell-Saunders Coupling Add the individual L and S values according to a Clebsch-Gordan series
45 J-values in j-j Coupling Add the individual j values according to a Clebsch-Gordan series
46 Note – upper term precedes lower term by convention Selection RulesAny state of the atom and any spectral transition can be specified using term symbols!Note – upper term precedeslower term by convention
47 Selection Rules #2These selection rules arise from the conservation of angular momentumNote – J=0 J=0is not allowed
48 The Effects of Magnetic Fields The electron generates an orbital magnetic momentThe energy
49 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.