Chapter 2
Experimental and Computational
Methods
2.1 Computational Methods
2.1.1 The Schrödinger Equation
The ultimate goal of common quantum chemical approaches is to reach a (approximate) solution to the time independent, non-relativistic Schrödinger equation [1,
2],
H
− → x 1 , − → x 2 , . . . − →
x N ,
− →
R 1 ,
− →
R 2 , . . .
− →
R M
= E
− → x 1 , − → x 2 , . . . − →
x N ,
− →
R 1 ,
− →
R 2 , . . .
− →
R M
(2.1)
for a system of M nuclei and N electrons, where ˆ
H is the Hamilton operator (or
Hamiltonian) and is the wavefunction that describes the system. Note that Eq. 2.1
combines the electron spatial coordinate ( r , xyz) and spin coordinate (s i , either α
or β) into a single term ( x), and the nuclear spatial coordinates are denoted
R. The
Hamiltonian (here defined in atomic units) is a differential operator that describes
the total energy,
H
= −
1
2
N
i=1
∇
2
i −
1
2
M
A=1
1
M A
∇
2
i −
N
i=1
M
A=1
Z A
r i A
+
N
i=1
N
j>i
1
r i j
+
M
A=1
M
B>1
Z A Z B
R AB
(2.2)
In Eq. 2.2, the terms i and j are electron indices, and run over all N electrons, A
and B are the nuclear indices, running over all M nuclei with charge Z and mass M A .
The five terms in Eq. 2.2 thus describe:
1. Electron kinetic energy
2. Nuclear kinetic energy
3. Attractive electron-nucleus electrostatic interaction
© Springer Nature Switzerland AG 2020
A. A. L. Michalchuk, Mechanochemical Processes in Energetic Materials,
Springer Theses, https://doi.org/10.1007/978-3-030-56966-2_2
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