C7790 Introduction to Molecular Modelling -1Lesson
7
Quantum Mechanics I
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/2020 Distant Form of Teaching: Rev1
C7790 Introduction to Molecular Modelling -2-
Context
t
𝐸(𝑡)
Time average: Ensemble average:
𝐸𝑖
Thermodynamic properties:
H, S, G, …
Mechanical properties:
Ei
equilibrium (binding affinities)
kinetics
Molecular modelling
How to describe interactions?
Phenomenological thermodynamics
Statistical thermodynamics
Monte Carlo simulations
Molecular Dynamics
C7790 Introduction to Molecular Modelling -3Quantum
mechanics
C7790 Introduction to Molecular Modelling -4Chemical
system
molecule
atom
electrons
chemistry
physics
nuclei
protons,
neutronswave character
electron nucleus
macrosystem
In chemistry, a nucleus is
considered to be a mass object
with a positive charge equal to a
proton number. Thus, nuclear
structure is not taken into
account.
C7790 Introduction to Molecular Modelling -5Complication
in description
2
2
0
1
c
v
vm
h
p
h
−==
Elementary particles and objects composed of them (nuclei) do not obey classical
physics laws. Particles exhibits a dual character. A particle with a momentum p
also behaves like a wave with the wavelength .
de Broglie hypothesis
Confirmed by numerous experiments, such as the passage of electrons through
single/double slits.
diffraction on one slit passage of electrons through two slits.
1923
C7790 Introduction to Molecular Modelling -6Foundation
of Quantum Mechanics
Wave character of particles require special approaches to describe their
behaviors.
Schrödinger equation (SR) is a foundation of quantum mechanics (QM).
t
t
itH


=
),(
),(ˆ r
r

  time-dependent Schrödinger equation
Hamiltonian (operator)
(it defines the system, i.e., the number of particles and
how they interact with each other, or how they interact
with their surroundings)
wave function
(it defines a state of the
system)
Legend:
r - position vector of particles, t - time
i - imaginary unit, h - Planck constant,
ħ - reduced Planck constant 2
h
=
C7790 Introduction to Molecular Modelling -7-
Hamiltonian
VTH
i
i

+=  ˆˆ
potential energy
operator
kinetic energy operator for
particle i
Hamiltonian (operator of the total energy):
2
2
2
ˆ −=
m
T

Kinetic energy operator:
(particle movement)
2
2
2
2
2
2
2
zyx 

+


+


=
Laplacian in Cartesian coordinates
Potential energy operator:
(interaction between particles)
),( tVV r=

potential energy itself
It describes particle motions and
interactions.
C7790 Introduction to Molecular Modelling -8Wave
function
➢ it describe a state of the system
➢ it can be a complex function
➢ physical interpretation is difficult
➢ square value of the wave function is related to probability density
 dkk )()(*
rr
the probability to find the system in the
configuration r in a volume element d
1)()(*
=Ω
rr  dkk
The probability that we will find particles
in the entire space is 100 %.
probability density
probability
C7790 Introduction to Molecular Modelling -9Interpretation
of QM
4.1 Classification adopted by Einstein
4.2 The Copenhagen interpretation (Copenhagen Convention)
4.3 Many worlds
4.4 Consistent histories
4.5 Ensemble interpretation, or statistical interpretation
4.6 de Broglie–Bohm theory
4.7 Relational quantum mechanics
4.8 Transactional interpretation
4.9 Stochastic mechanics
4.10 Objective collapse theories
4.11 von Neumann / Wigner interpretation: consciousness causes the collapse
4.12 Many minds
4.13 Quantum logic
4.14 Quantum information theories
4.15 Modal interpretations of quantum theory
4.16 Time-symmetric theories
4.17 Branching space-time theories
4.18 Other interpretations
www.wikipedia.com
C7790 Introduction to Molecular Modelling -10Interpretation
of QM
4.1 Classification adopted by Einstein
4.2 The Copenhagen interpretation (Copenhagen Convention)
4.3 Many worlds
4.4 Consistent histories
4.5 Ensemble interpretation, or statistical interpretation
4.6 de Broglie–Bohm theory
4.7 Relational quantum mechanics
4.8 Transactional interpretation
4.9 Stochastic mechanics
4.10 Objective collapse theories
www.wikipedia.com
Copenhagen interpretation* is an interpretation of quantum mechanics that is most
prevalent among physicists. According to this interpretation, probabilistic nature of
quantum mechanical predictions cannot be explained in some other way, such as
unknown (hidden) deterministic theory.
Quantum mechanics provides probabilistic results because the universe itself is
probabilistic rather than deterministic.
*mainly due to theoretical physics Niels Bohr
C7790 Introduction to Molecular Modelling -11Interpretation
of QM
4.1 Classification adopted by Einstein
4.2 The Copenhagen interpretation (Copenhagen Convention)
4.3 Many worlds
4.4 Consistent histories
4.5 Ensemble interpretation, or statistical interpretation
4.6 de Broglie–Bohm theory
4.7 Relational quantum mechanics
4.8 Transactional interpretation
4.9 Stochastic mechanics
4.10 Objective collapse theories
Fundamental problems:
➢ Is it apparatus of quantum mechanics (particles) also applicable to macrosystems?
➢ Paradoxes:
➢ Schrödinger's cat
➢ Wigner's friend
Castelvecchi, D. Reimagining of Schrödinger's Cat Breaks Quantum Mechanics and
Stumps Physicists. Nature 2018, 561 (7724), 446–447.
C7790 Introduction to Molecular Modelling -12Uncertainty
principle
Heisenberg's uncertainty principle (also uncertainty relation) is a mathematical property
of two complementary quantities. Heisenberg's principle says that the more accurately we
determine one of the complementary properties, the less accurately we can determine the
other - no matter how accurate instruments are.
2

 px
The most common relations:
uncertainty in particle positioning
uncertainty in determining the momentum (velocity) of a particle
2

 tE
uncertainty in determining the energy of the system
uncertainty in determining time at which we measured the energy
position/momentum
energy/time
C7790 Introduction to Molecular Modelling -13Uncertainty
principle
Heisenberg's uncertainty principle (also uncertainty relation) is a mathematical property
of two complementary quantities. Heisenberg's principle says that the more accurately we
determine one of the complementary properties, the less accurately we can determine the
other - no matter how accurate instruments are.
2

 px
The most common relations:
uncertainty in particle positioning
uncertainty in determining the momentum (velocity) of a particle
2

 tE
uncertainty in determining the energy of the system
uncertainty in determining time at which we measured the energy
Heisenberg is stopped by the traffic police. The policeman asks him, "Do you
know how fast you drove?" Heisenberg replies, "No, but I know where I am."
position/momentum
energy/time
C7790 Introduction to Molecular Modelling -14System
energy
Ensemble average:
𝐸𝑖
Thermodynamic properties:
H, S, G, …
Mechanical properties:
Ei
Statistical thermodynamics
Monte Carlo simulations
C7790 Introduction to Molecular Modelling -15System
energy
2

 tE
t
t
itH


=
),(
),(ˆ r
r

 
time-dependent Schrödinger equation Heisenberg's uncertainty principle
the system state described by the wave function
is known at the exact moment in time
energy of the system cannot be determined
?
C7790 Introduction to Molecular Modelling -16Schrödinger
equation
t
t
itH


=
),(
),(ˆ r
r

  time-dependent Schrödinger equation
)()(ˆ rr kkk EH  =
time separation
time independent Schrödinger equation
)()(),( tft rr  =
time (t) and configuration
(r) are set independent of
each other
)(
)(
tEf
dt
tdf
i =
C7790 Introduction to Molecular Modelling -17Time
independence
)()(),( tft rr  =
Time (t) and particle configuration (r) are considered as
independent variables. Consequently, the state description is
also independent to time and configuration.
)()()( BPAPBAP =
The following applies to independent events:
the join probability of A
and B events probability of event A
probability of event B
A similar approximation is used for:
• Born-Oppenheimer approximation
• separation of translational, rotational, and vibrational movements
• one-electron approximations (Hartree-Fock method)
C7790 Introduction to Molecular Modelling -18Schrödinger
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.
+
C7790 Introduction to Molecular Modelling -19System
vs State (inaccurate example)
! Very rough comparison not taking into account the probabilistic behavior of quantum
systems !
geomagSystem definition:
The Hamiltonian indicates the number of spheres and
connectors (particles) and their mutual interaction.
System status:
Determined by the wave function, which indicates the actual arrangement of balls and
connectors in space.
state A state B
http://www.magnetickysvet.cz
C7790 Introduction to Molecular Modelling -20-
Summary
➢ Molecules are composed from atoms. Atoms are composed from electrons and nuclei.
➢ Electrons, nuclei, and atoms (and molecules) are small and exhibit dual character
(wave/particle).
➢ Behavior of particles and their assemblies can be described by time-independent
Schrodinger equation.
➢ Solution of Schrödinger equation provides all possible microstates and their energies.
)()(ˆ rr kkk EH  =
C7790 Introduction to Molecular Modelling -21-
Summary
➢ Molecules are composed from atoms. Atoms are composed from electrons and nuclei.
➢ Electrons, nuclei, and atoms (and molecules) are small and exhibit dual character
(wave/particle).
➢ Behavior of particles and their assemblies can be described by time-independent
Schrodinger equation.
➢ Solution of Schrödinger equation are all possible microstates and their energies.
)()(ˆ rr kkk EH  =
? ➢ Probabilistic description of the structure in the
given state
➢ Unsolvable for microstates of macrosystems (>
1023 atoms)
➢ Practically impossible to solve even for small
chemical systems (hydrogen molecule)