C7790 Introduction to Molecular Modelling -1Lesson
8
Quantum Mechanics II
C7790 Introduction to Molecular
Modelling
TSM Modelling Molecular Structures
Petr Kulhánek
kulhanek@chemi.muni.cz
National Centre for Biomolecular Research, Faculty of Science
Masaryk University, Kamenice 5, CZ-62500 Brno
PS/2021 Present Form of Teaching: Rev1
C7790 Introduction to Molecular Modelling -2Schrödinger
equation
)()(ˆ rr kkk EH  =
time independent Schrödinger equation
Hamiltonian (operator)
(it defines a system, i.e., number of
particles and how they interact with
each other)
wave function
(it defines a state k)
energy of state k
Solutions to the SR equation are pairs: k and Ek.
Each pair represent possible realization of the system (a microstate) and its
energy.
+
(!!!scalar value!!!)
C7790 Introduction to Molecular Modelling -3Hamiltonian
of Chemical System








+−+−−=  = = == ==
n
i
n
ij ij
N
i
n
j ij
i
N
i
N
ij ij
ji
n
i
i
N
i
i
i r
e
r
eZ
r
ZZ
mM
H
1
2
1 1101
2
2
1
2
2
4
1
2
1
2
ˆ


Hamiltonian of a chemical system, consisting of N nuclei of mass M and charge Z
and n electrons of mass m, is given by:
kinetic energy operator potential energy
nuclei electrons electron-electron
electron-core
core-core
Potential energy it is given by electrostatic interaction between charged particles:
ij
ji
r
qq
V
04
1

=Coulomb's law
motion interactions
C7790 Introduction to Molecular Modelling -4Structure
vs System State
✓
Hypothetical exact solution of time-independent Schrödinger equation (ground state):
It describes too many properties such as:
➢ electron density distribution
➢ distribution of nuclei due to translational,
rotational and vibrational movements of
the molecule
➢ and all their combinations
This is too complicated for subsequent analyzes.
H2O
?
C7790 Introduction to Molecular Modelling -5-
Born-Oppenheimer
Approximation
C7790 Introduction to Molecular Modelling -6Born-Oppenheimer
Approximation
 = = == ==
+−+−−=
n
i
n
ij ij
N
i
n
j ij
i
N
i
N
ij ij
ji
n
i
i
N
i
i
i rr
Z
r
ZZ
mM
H
11 111
2
1
2 1
2
11
2
1ˆ
),(),(ˆ RrRr  EH = WF provides a complicated description of the
system state.
Real positions of nuclei and electrons are known
only within the probabilistic description.
position of
electrons
The Born-Oppenheimer approximation separates motion of nuclei from electrons.
),()(),( RrRRr = 
motion of nuclei
motion of electrons in the static field of
nuclei
position of
nuclei
C7790 Introduction to Molecular Modelling -7Born-Oppenheimer
Approximation
 = = == ==
+−+−−=
n
i
n
ij ij
N
i
n
j ij
i
N
i
N
ij ij
ji
n
i
i
N
i
i
i rr
Z
r
ZZ
mM
H
11 111
2
1
2 1
2
11
2
1ˆ
),(),(ˆ RrRr  EH =
),()(),(ˆ RrRRr = ee EH )()(ˆ RR  VRTR EH =
electronic properties of molecule vibrational, rotational, translational
motions of molecule
)(),(),( RRrRr  =
Born-Oppenheimer approximation
C7790 Introduction to Molecular Modelling -8Electronic
Properties of System
),()(),(ˆ RrRRr = ee EH
 = = == =
+−+−=
n
i
n
ij ij
N
i
n
j ij
i
N
i
N
ij ij
ji
n
i
ie
rr
Z
r
ZZ
m
H
11 111
2 1
2
1ˆ
The energy is a function of the position of nuclei (atoms).
)(RE
concept of potential energy surfaces
R - determines the configuration of nuclei (atoms)
in space => structure for which we can determine
the energy
(function)
C7790 Introduction to Molecular Modelling -9Structure
vs System State
http://hypot.wordpress.com/2012/11/15/electron-density/
),()(),( RrRRr = 
Ground state of the water molecule (schematic):
distribution of electrons in the static field
of nuclei
it describes the overall state of the
system partially
O
H
H
C7790 Introduction to Molecular Modelling -10Structure
vs System State
http://hypot.wordpress.com/2012/11/15/electron-density/
),()(),( RrRRr = 
Ground state of the water molecule (schematic):
distribution of electrons in the static field
of nuclei
it describes the overall state of the
system partially
O
H
H
O
H
H
C7790 Introduction to Molecular Modelling -11Structure
vs System State
http://hypot.wordpress.com/2012/11/15/electron-density/
),()(),( RrRRr = 
ground state of the water molecule (schematic):
O
H
H
schematic representation of the molecular
structure - based on the distribution of
electron density
distribution of electrons in the static field
of nuclei
it describes the overall state of the
system partially
C7790 Introduction to Molecular Modelling -12Nuclear
Motions
)()(ˆ RR  VRTR EH =
?ˆ =RH
the nuclei are affected by the potential
a) electrostatic interaction of nuclei with each other
b) effective potential of electrons in the field of nuclei
scalar value (not a function)
Nuclei motions:
➢ vibrational
➢ rotational
➢ translational
it can be further approximated into individual motions and their contributions
using approximations based on a similar principle as used in the BO approximation
C7790 Introduction to Molecular Modelling -13Nuclear
Motions
)()(ˆ RR  VRTR EH =
)(
1
2
ˆ
1
2
2
RE
M
H e
N
i
i
i
R +−= =

value (not function)
Core movements:
➢ vibratory
➢ rotational
➢ translational
can be further approximated into individual movements and their contributions
using approximations based on a similar principle as used in the BO approximation
the nuclei are affected by the potential
a) electrostatic interaction of nuclei with each other
b) effective potential of electrons in the field of nuclei
C7790 Introduction to Molecular Modelling -14How
accurate is BO approximation?
The BO approximation recognizes the large difference between the electron mass and the
masses of atomic nuclei, and correspondingly the time scales of their motion.
https://en.wikipedia.org/wiki/Hartree_atomic_units
Atomic Units
[xp, yp, zp]
[xe, ye, ze]
r
hydrogen atom
Mp = 1836 au
me = 1 au
me
Mp
difference is bigger for
heavier elements
C7790 Introduction to Molecular Modelling -15-
Summary
➢ Born-Oppenheimer (BO) approximation is the most important approximation in
molecular modelling
➢ It is rather accurate because of significant difference between electron and nuclei
masses
➢ Electrons moves faster than nuclei (different time scales) and electrons can instantly
update their distributions once the nuclei position changes.
➢ BO approximation is foundations for all calculation methods (model chemistry) used in
molecular modelling
)(RE (function)
concept of potential energy surfaces
C7790 Introduction to Molecular Modelling -16Method
overview (model chemistry]
QM (Quantum mechanics) MM (Molecular mechanics) CGM (Coarse-grained mechanics)
)(RE )(RE )(RE
R - position of atom nuclei R - position of atoms R - position of beads