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Solution to 10.675 Assignment #1
1.
Derive an expression for the total ground state energy of a twoelectron system using
HartreeFock theory. This should be done in terms of spin orbitals. Identify the coulomb
and exchange integrals, and explain their physical significance.
Solution:
Twoelectron system is fairly easy to deal with. Since we are doing the general derivation
for Nelectron system in the next problem, we will omit the repeated work here. Please
refer to chapter 2.3.1 in Szabo and Ostlund’s
Modern Quantum Chemistry
(pages 6466)
for the detailed derivation of twoelectron system.
2.
Make yourself comfortable with the general derivation of this expression for N
electrons.
Solution:
For Nelectron system, the Hamiltonian is:
N
N
N
1
(
H
=
∑
i
h
)
+
∑
∑
r
ij
(1)
i
=
1
i
=
1
j
>
i
Where,
i
h
)
−
=
1
∇
i
2
−
∑
Z
A
(
2
A
r
iA
In HartreeFock theory, we choose the form of the wave function to be that of single
determinant, formed from spin orbitals:
N
!

ψ
>=

χ
(1)
χ
(2)
⋅⋅⋅
χ
(
N
)
>=
1
∑
(
−
1)
P
n
P
{
χ
(1)
χ
(2)
⋅⋅⋅
χ
(
N
)}
(2)
m
n
k
N
!
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 Spring '03
 RogerD.Kamm
 Electron, pH, Trigraph, Pauli exclusion principle, spin orbitals

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