Atomic Quantum Mechanics - Hydrogen Atom (15.1-15.3)• Assuming an atom doesn’t 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, θ, and φ)– 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

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VNe (r) = −Ze2

4πε0r

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Vee (r12) =e2

4πε0r12

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En = −mee

4

8ε02h2n2

n =1, 2, 3, ...

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”

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

Hydrogen Wavefunctions

Radial Nodes in Hydrogen Orbitals

Angular Nodes in Hydrogen Orbitals

Probability Distribution Functions for Hydrogen Orbitals

Orbital Filling Diagram

Atomic Orbital Overlap