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A Short Summary of Quantum Chemistry
Quantum Chemistry is (typically) based on the nonrelativistic Schroedinger equation,
making the BornOppenheimer approximation. The Schroedinger equation is
H
total
Ψ
total
= E
Ψ
total
Where
E = an allowed energy of the system (the system is usually a molecule).
Ψ
total
= a function of the positions of all the electrons and nuclei and their spins.
H
total
= a differential operator constructed from the classical Hamiltonian H(p,q) = E by
replacing all the momenta p
i
with (h/2
π
i)
∂
/
∂
q
i
as long as all the p and q are Cartesian. If
you would prefer to use a nonCartesian coordinate system it is tricky to construct the
correct quantum mechanical H. The standard way to make Hamiltonians for arbitrary
coordinate systems is to use the “Podolsky trick”. For a system of nuclei and electrons in
a vacuum with no external fields, neglecting magnetic interactions, using atomic units:
H
total
=  ½
Σ
∇
i
2
/M
i
 ½
Σ
∇
n
2
+ Σ
Z
i
Z
j
/R
i
R
j
 
Σ
Z
i
/R
i
r
n
 +
Σ
1/r
m
r
n

The BornOppenheimer approximation is to neglect some of the terms coupling
the electrons and nuclei, so one can write:
Ψ
total
(R,r) =
Ψ
nuclei
(R)
Ψ
electrons
(r; R)
H
total
≅
T
nuclei
(P,R) + H
electrons
(p,r;R)
(ignores the dependence of H
electrons
on the momenta of the nuclei P)
then solve the Schroedinger equation for the electrons (with the nuclei fixed). The energy
we compute will depend on the positions R of those fixed nuclei, call it V(R):
H
electons
(p,r;R)
Ψ
electrons
(r; R) = V(R)
Ψ
electrons
(r; R)
Now we go back to the total Hamiltonian, and integrate over all the electron positions r,
ignoring any inconvenient terms, to obtain an approximate Schroedinger equation for the
nuclei:
<
Ψ
electrons
(r; R) H
total

Ψ
electrons
(r; R)>
≅
H
nuclei
=
T
nuclei
(P,R) + V(R)
Both approximate Schroedinger equations are still much too hard to solve exactly
(they are partial differential equations in 3N
particles
coordinates), so we have to
make more approximations. V
nuclei
is usually expanded to second order R about a
stationary point R
o
:
V
nuclei
≅
V
nuclei
(R
o
) + ½
Σ
(
∂
2
V/
∂
R
i
∂
R
j
) (R
i
R
oi
)(R
j
R
oj
)
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View Full Documentand then the translations, rotations, and vibrations are each treated separately,
neglecting any inconvenient terms that couple the different coordinates. In this
famous “rigidrotorharmonicoscillator (RRHO)” approximation, analytical
formulas are known for the energy eigenvalues, and for the corresponding
partition functions Q, look in any P.Chem. text.
This approximate approach has the important advantage that we do not need to
solve the Schroedinger equation for the electrons at very many R’s: we just need
to find a stationary point R
o
, and compute the energy and the second derivatives
at that R
o
. Many computer programs have been written that allow one to compute
the first and second derivatives of V almost as quickly as you can compute V. For
example, for the biggest calculation called for in problem 3 with 10 atoms and
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 Spring '03
 RogerD.Kamm
 Quantum Chemistry

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