2.2 Molecular Hamiltonian (Bunker and Jensen 1998)
9
V NN =
1
4πε 0
N
α=1
N
β>α
Z α Z β e
2
R αβ
=
⎡
⎣
N
α=1
N
β>α
Z α Z β
R αβ
⎤
⎦
(2.2d)
V ee =
1
4πε 0
n
i=1
n
j>i
e
2
r i j
=
⎡
⎣
n
i=1
n
j>i
1
r i j
⎤
⎦
(2.2e)
is the Laplace operator, M α and Z α are the mass and atomic number of the
nucleus α, m is the mass of the electron, r iα is the distance between electron i and
nucleus α, and similar definitions hold for r ij and R αβ . The terms in brackets on the
right are in atomic units (au). It is common to use them in order to eliminate the
fundamental physical constants from the electronic Hamiltonian. They are chosen
such that: = m = e
2
4πε 0 = 1.
The atomic unit of length is the Bohr:
a 0 =
4πε 0
2
me 2 = 5.29177210903(80) × 10
−11 m
The atomic unit of energy is the Hartree, E H , which is the Coulomb repulsion
between two electrons separated by 1 Bohr:
E H =
2
ma
2
0
= 4.3597447222071(85) × 10
−18 J.
The atomic unit of mass is the mass of the electron: m e = 9.1093837015(28)×
10
−31 kg.
The CODATA (Committee on Data for Science and Technology) internationally
recommended values of the fundamental constants may be found at: https://physics.
nist.gov/cuu/Constants.
2.3 Born–Oppenheimer Approximation and Electronic
Hamiltonian (Bunker and Jensen 2000)
This Hamiltonian is too complicated to be solved exactly. As first proposed by Born
and Oppenheimer (BO), the nuclear kinetic energy T N is first neglected. The justification of the BO approximation is that the heavy nuclei move much more slowly
than the light electrons (it also assumes that the momentum of the electrons and the
nuclei is of the same order of magnitude). In the remaining electronic Hamiltonian
H e , the nuclear positions R enter as parameters
H e ψ
(e)
= E
BO
ψ
(e)
(2.3)
Varying the position of R in small steps, one obtains E
bo as a function of R. This is
the potential energy (hyper)surface (PES): E
bo (R). Its global minimum corresponds
to the equilibrium structure of the molecule. As the nuclear masses are absent in H e ,
9
V NN =
1
4πε 0
N
α=1
N
β>α
Z α Z β e
2
R αβ
=
⎡
⎣
N
α=1
N
β>α
Z α Z β
R αβ
⎤
⎦
(2.2d)
V ee =
1
4πε 0
n
i=1
n
j>i
e
2
r i j
=
⎡
⎣
n
i=1
n
j>i
1
r i j
⎤
⎦
(2.2e)
is the Laplace operator, M α and Z α are the mass and atomic number of the
nucleus α, m is the mass of the electron, r iα is the distance between electron i and
nucleus α, and similar definitions hold for r ij and R αβ . The terms in brackets on the
right are in atomic units (au). It is common to use them in order to eliminate the
fundamental physical constants from the electronic Hamiltonian. They are chosen
such that: = m = e
2
4πε 0 = 1.
The atomic unit of length is the Bohr:
a 0 =
4πε 0
2
me 2 = 5.29177210903(80) × 10
−11 m
The atomic unit of energy is the Hartree, E H , which is the Coulomb repulsion
between two electrons separated by 1 Bohr:
E H =
2
ma
2
0
= 4.3597447222071(85) × 10
−18 J.
The atomic unit of mass is the mass of the electron: m e = 9.1093837015(28)×
10
−31 kg.
The CODATA (Committee on Data for Science and Technology) internationally
recommended values of the fundamental constants may be found at: https://physics.
nist.gov/cuu/Constants.
2.3 Born–Oppenheimer Approximation and Electronic
Hamiltonian (Bunker and Jensen 2000)
This Hamiltonian is too complicated to be solved exactly. As first proposed by Born
and Oppenheimer (BO), the nuclear kinetic energy T N is first neglected. The justification of the BO approximation is that the heavy nuclei move much more slowly
than the light electrons (it also assumes that the momentum of the electrons and the
nuclei is of the same order of magnitude). In the remaining electronic Hamiltonian
H e , the nuclear positions R enter as parameters
H e ψ
(e)
= E
BO
ψ
(e)
(2.3)
Varying the position of R in small steps, one obtains E
bo as a function of R. This is
the potential energy (hyper)surface (PES): E
bo (R). Its global minimum corresponds
to the equilibrium structure of the molecule. As the nuclear masses are absent in H e ,