Quantum Mechanics - Prof. Eric R. Bittner - University of Houston

• Method development: The development and implementation of new theories and computational
strategies to take advantage of the increaseing power of computational hardware.
(Bigger, stronger, faster calculations)
• Application: The use of established methods for developing theoretical models of chemical
processes
Here we will go in to a brief bit of detail into various levels of theory and their implementation
in standard quantum chemical packages. For more in depth coverage, refer to
1. Quantum Chemistry, Ira Levine. The updated version of this text has a nice overview of
methods, basis sets, theories, and approaches for quantum chemistry.
2. Modern Quanutm Chemistry, A. Szabo and N. S. Ostlund.
3. Ab Initio Molecular Orbital Theory, W. J. Hehre, L. Radom, P. v. R. Schleyer, and J. A.
Pople.
4. Introduction to Quantum Mechanics in Chemistry, M. Ratner and G. Schatz.
8.4.1 The Born-Oppenheimer Approximation
The fundimental approximation in quantum chemistry is the Born Oppenheimer approximation
we discussed earlier. The idea is that because the mass of an electron is at least 10 −4 that of
a typical nuclei, the motion of the nuclei can be effectively ignored and we can write an electronic
Schrödinger equation in the field of fixed nuclei. If we write r for electronic coordinates and R
for the nuclear coordinates, the complete electronic/nuclear wavefunction becomes
Ψ(r, R) = ψ(r; R)χ(R) (8.80)
where ψ(r; R) is the electronic part and χ(R) the nuclear part. The full Hamiltonian is
H = Tn + Te + Ven + Vnn + Vee
(8.81)
Tn is the nuclear kinetic energy, Te is the electronic kinetic energy, and the V ’s are the electronnuclear,
nuclear-nuclear, and electron-electron Coulomb potential interactions. We want Ψ to be
a solution of the Schrödinger equation,
HΨ = (Tn + Te + Ven + Vnn + Vee)ψχ = Eψχ. (8.82)
So, we divide through by ψχ and take advantage of the fact that Te does not depend upon the
nuclear component of the total wavefunction
1
ψχ Tnψχ + 1
ψ Teψ + Ven + Vnn + Vee = E.
On the other hand, Tn operates on both components, and involves terms which look like
Tnψχ = �
n
− 1
(ψ∇
2Mn
2 nχ + χ∇ 2 nψ + 2∇χ · ∇nψ)
232