34
2 Experimental and Computational Methods
4. Repulsive electron-electron electrostatic interaction
5. Repulsive nuclear-nuclear electrostatic interaction
Terms 3–5 depend on the spatial separation between electron and nuclear spatial
coordinates − → r i and
− →
R A , respectively; hence r i A =
− → r i −
− →
R A
.
The Schrödinger equation therefore ‘simply’ states that using the mathematical
formulation of the Hamiltonian, the energy of a system can be extracted from its wavefunction. Unfortunately, due to the infamous many-body problem, the Schrödinger
equation cannot be solved exactly for multi-electron systems, and a variety of
approximations must be made.
Arguably the most important approximation in quantum chemistry is the BornOppenheimer approximation (BOA) [3]. Within the BOA, the significant difference
in mass between a nucleus and an electron is considered (ca. 1800 times greater
in
1 H, and > 20000 times greater in
14 C). The electrons can therefore be taken as
moving in a field of fixed nuclear geometry, hence term 2 of Eq. 2.2 disappears
and term 5 becomes a constant described by Coulomb’s law. The Hamiltonian (and
wavefunction) can therefore be split into electronic and nuclear components, to be
solved independently. The electronic Hamiltonian, ˆ
H elec becomes,
H
elec = −
1
2
N
i=1
∇
2
i −
N
i=1
M
A=1
Z A
r i A
+
N
i=1
N
j>i
1
r i j
= ˆ
T + ˆ
V Ne + ˆ
V ee
(2.3)
The total energy therefore comes from the sum of electronic energy, E elec and the
constant nuclear repulsion term E tot = E elec + E nucl , where
E nucl =
M
A=1
M
B>1
Z A Z B
R AB
(2.4)
This includes all of the energy of the system under fixed nuclear geometry.
Despite the great simplifications obtained via the BOA, no solution to the manybody problem is achieved. The third term of Eq. 2.3, ˆ
V ee , cannot be solved explicitly
for multi-electron systems, and elements of terms one and two are beyond reach.
Methods designed to deal with this issue using a variety of approximations have
been developed and are widely employed in the field of Computational Chemistry.
2.1.2 Hartree-Fock Theory
The simplest ab initio method is the Hartree-Fock (HF) scheme. The HF method
forms the base for nearly all wavefunction based theories. Noting that the solution of the complete N-electron wavefunction cannot be solved exactly, HF theory
2 Experimental and Computational Methods
4. Repulsive electron-electron electrostatic interaction
5. Repulsive nuclear-nuclear electrostatic interaction
Terms 3–5 depend on the spatial separation between electron and nuclear spatial
coordinates − → r i and
− →
R A , respectively; hence r i A =
− → r i −
− →
R A
.
The Schrödinger equation therefore ‘simply’ states that using the mathematical
formulation of the Hamiltonian, the energy of a system can be extracted from its wavefunction. Unfortunately, due to the infamous many-body problem, the Schrödinger
equation cannot be solved exactly for multi-electron systems, and a variety of
approximations must be made.
Arguably the most important approximation in quantum chemistry is the BornOppenheimer approximation (BOA) [3]. Within the BOA, the significant difference
in mass between a nucleus and an electron is considered (ca. 1800 times greater
in
1 H, and > 20000 times greater in
14 C). The electrons can therefore be taken as
moving in a field of fixed nuclear geometry, hence term 2 of Eq. 2.2 disappears
and term 5 becomes a constant described by Coulomb’s law. The Hamiltonian (and
wavefunction) can therefore be split into electronic and nuclear components, to be
solved independently. The electronic Hamiltonian, ˆ
H elec becomes,
H
elec = −
1
2
N
i=1
∇
2
i −
N
i=1
M
A=1
Z A
r i A
+
N
i=1
N
j>i
1
r i j
= ˆ
T + ˆ
V Ne + ˆ
V ee
(2.3)
The total energy therefore comes from the sum of electronic energy, E elec and the
constant nuclear repulsion term E tot = E elec + E nucl , where
E nucl =
M
A=1
M
B>1
Z A Z B
R AB
(2.4)
This includes all of the energy of the system under fixed nuclear geometry.
Despite the great simplifications obtained via the BOA, no solution to the manybody problem is achieved. The third term of Eq. 2.3, ˆ
V ee , cannot be solved explicitly
for multi-electron systems, and elements of terms one and two are beyond reach.
Methods designed to deal with this issue using a variety of approximations have
been developed and are widely employed in the field of Computational Chemistry.
2.1.2 Hartree-Fock Theory
The simplest ab initio method is the Hartree-Fock (HF) scheme. The HF method
forms the base for nearly all wavefunction based theories. Noting that the solution of the complete N-electron wavefunction cannot be solved exactly, HF theory