Atomic Orbitals, Electron Configurations, and Atomic Spectra 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. 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.

The Structure of the H-atom The Coulombic energy The Hamiltonian

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

Some Radial Wavefunctions Orbital n l Rn,l 1s 1 2(Z/ao)3/2 e-/2 2s 2 1/(2 21/2) (Z/ao)3/2(2-1/2) e-/4 2p + 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 For all other orbitals

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.

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. An s orbital has a finite, nonzero value at the nucleus.

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) Hydrogen atom – all subshells have the same energy!

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

The Orbital Approximation For many electron atoms Think of the individual orbitals as resembling the hydrogenic orbitals

The Hamiltonian in the Orbital Approximation For many electron atoms 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 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 Hartree - 1928

SCF Method #2 The SE is separated into n equations of the type 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 State J

Calculating the L value Add the individual l values according to a Clebsch-Gordan series 2 L+1 orientations

What do the L values mean? Term S 1 P 2 D 3 F 4 G

The Multiplicity (S) Add the individual s values according to a Clebsch-Gordan series

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-values in j-j Coupling Add the individual j values according to a Clebsch-Gordan series

Note – upper term precedes lower term by convention Selection Rules Any state of the atom and any spectral transition can be specified using term symbols! Note – upper term precedes lower term by convention

Selection Rules #2 These selection rules arise from the conservation of angular momentum Note – J=0  J=0 is not allowed

The Effects of Magnetic Fields The electron generates an orbital magnetic moment The energy

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.