Atomic Quantum Mechanics - Hydrogen Atom (15.1-15.3)

1
Atomic Quantum Mechanics - Hydrogen Atom
(15.1-15.3)

• Assuming an atom doesnt move in space
  (translate), the SE is reduced to solving for the
  electrons only
• For single electron atoms and ions (e.g.,
  hydrogen), only the attraction of the electron to
  the nucleus is needed in the potential energy
  term
• For many electron atoms and ions, repulsion
  between the electrons is needed in the PE
• For the hydrogen atom, the solution to the SE is
  a set of atomic orbitals
• The wavefunction involves three quantum numbers
  (n, l, ml) since three sets of boundary
  conditions are needed (1 for r, ?, and f)
• The number of nodes in the wavefunction increases
  with increasing values of n and l
• The energy of the hydrogen atom only depends on
  the principle quantum number n

2
Atomic QM Many Electron Atoms (15.6-15.7)

• When more than one electron is present, Vee must
  be included in SE
• Electrons are in constant motion, so potential
  energy is a constantly changing variable
  (electron correlation)
• SE is no longer exactly solvable!
• Solutions to many electron atom SE are very
  similar to hydrogen orbitals
• The wavefunction for each electron is a
  hydrogen-like orbital (1s, 2s, 2p, etc.)
• The energy associated with each electron now
  depends on n and l
  (orbital filling diagram)
• Two electrons can have the same principle,
  angular, and magnetic quantum numbers
• Electron configurations are used to show what the
  atomic wavefunction looks like
• Electrons posses another quantum number
  associated with their spin
• Electrons have another quantum number called the
  spin quantum number ( 1/2)
• Pauli exclusion principle states no two electrons
  can have the same quantum numbers, so one
  electron in an orbital must be spin-up and the
  other spin-down

3
Atomic QM to Molecular QM (16.4-16.6)

• Solution of SE for molecules is more complicated
  due to much larger number of electrons and
  multiple nuclei
• SE is still not exactly solvable since more than
  one electron is involved
• Atomic orbitals are not appropriate since
  multiple nuclei are involved
• Just as atoms combine to form molecules, atomic
  orbitals (AO) should combine to form molecular
  orbitals (MO)
• Linear combination of atomic orbitals (LCAO) is
  an approximation used to solve the molecular SE
• When creating MOs from AOs, there is a one-to-one
  correspondence
• Atomic orbital overlap is the driving force in
  whether an appropriate MO is generated (this
  included orbital phases)
• MOs have similar properties to AOs (and other
  wavefunctions)
• Two electrons can reside in each MO
• MOs are orthogonal to one another
• Energy order is related to nodal character

4
Hydrogen Wavefunctions
5
Radial Nodes in Hydrogen Orbitals
6
Angular Nodes in Hydrogen Orbitals
7
Probability Distribution Functions for Hydrogen
Orbitals
8
Orbital Filling Diagram
9
Atomic Orbital Overlap