C7790 Úvod do molekulového modelování -1C7790
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
Lesson 6
Thermodynamics & Modelling
Statistical Thermodynamics
C7790 Úvod do molekulového modelování -2Thermodynamics
& Modelling
Δ𝐺𝑟
0
= −𝑅𝑇 ln 𝐾
𝐾 =
𝐶 𝑟
𝑐
𝐷 𝑟
𝑑
𝐴 𝑟
𝑎
𝐵 𝑟
𝑏
Fundamental relation
Δ𝐺𝑟
0
= 𝑐Δ𝐺𝑓,𝐶
0
+ 𝑑Δ𝐺𝑓,𝐷
0
− 𝑎Δ𝐺𝑓,𝐴
0
+ 𝑏Δ𝐺𝑓,𝐵
0
A
B
C
D
We only need to know
the properties of individual
components involved in the
reaction at standard conditions (or
at different conditions, which are
well defined).
What do we need to know?
solution at
equilibrium
We need to know the composition of
solution at equilibrium.
C7790 Úvod do molekulového modelování -3Thermodynamics
& Modelling
Δ𝐺𝑟
0
= −𝑅𝑇 ln 𝐾
𝐾 =
𝐶 𝑟
𝑐
𝐷 𝑟
𝑑
𝐴 𝑟
𝑎
𝐵 𝑟
𝑏
Fundamental relation
Δ𝐺𝑟
0
= 𝑐Δ𝐺𝑓,𝐶
0
+ 𝑑Δ𝐺𝑓,𝐷
0
− 𝑎Δ𝐺𝑓,𝐴
0
+ 𝑏Δ𝐺𝑓,𝐵
0
What do we need to know?
solution at
equilibrium
A
B
C
D
We only need to know
the properties of individual
components involved in the
reaction at standard conditions (or
at different conditions, which are
well defined).
We need to know the composition of
solution at equilibrium.
easier for modelling It is hard or impossible to model.
A B
C
D
D
CD
D B
C
C7790 Úvod do molekulového modelování -4-
Overview
C7790 Úvod do molekulového modelování -5-
Overview
bridge
C7790 Úvod do molekulového modelování -6-
Statistical
Thermodynamics
Or what you should already know….
C7790 Úvod do molekulového modelování -7Two
approaches - I
1. Phenomenological approach:
Thermodynamics examines the interrelationships between quantities that characterize
the macroscopic state of the system and changes in these quantities in physical
processes. Many of the features of the system can be clarified without a thorough
knowledge of its internal structure. It is based on several axiomatically pronounced
(and experimentally confirmed) laws, which, in connection with the known properties
of the system, served to derive other properties and relationships. The state of the
system is described using state functions and equations, which determine the
relationships between individual state functions.
Level of description:
▪ state functions
▪ state equations
▪ thermodynamic theorems
wikipedia.cz, simplified
C7790 Úvod do molekulového modelování -8Two
approaches - II
2. Statistical approach:
Statistical physics (statistical mechanics) relates two levels of description of physical
reality, namely the macroscopic and microscopic levels. In a more traditional sense, it
deals with the study of the properties of macroscopic systems or systems, considering
the microscopic structure of these systems (statistical thermodynamics). The founders
were Ludwig Boltzmann and Josiah Willard Gibbs.
Level of description:
▪ particles and interactions between them
▪ equations of motions
wikipedia.cz, simplified
C7790 Úvod do molekulového modelování -9System
properties
t
)(tM
Time average:
=
tott
otot
dttM
t
M )(
1
The observable value ( ഥ𝑀) of the property M can be determined by two approaches:
snapshot of the system at time t is called
a microstate
C7790 Úvod do molekulového modelování -10System
properties
t
)(tM
Time average: Ensemble average:
=
=
K
i
iiMpM
1
2/6 2/6 1/6 1/6
See later: We can run Monte Carlo
simulations to get value of property by
molecular modelling.
The observable value ( ഥ𝑀) of the property M can be determined by two approaches:
=
tott
otot
dttM
t
M )(
1
See later: We can run molecular
dynamics simulations to get value of
property by molecular modelling.
snapshot of the system at time t is called
a microstate
𝑀𝑖
C7790 Úvod do molekulového modelování -11Statistical
ensemble
prototype
Statistical ensemble (Gibbs ensemble) is thought construction, in which the ensemble is
formed by large number of copies of the system (prototype), whose thermodynamic
properties we want to determine.
Each replica of the prototype is located in exactly one microstate.
Interactions between prototype replicas are very weak (it is practically possible to neglect
them), however, sufficient for the ansemble to be located in thermodynamic equilibrium.
=
=
K
i
ii MpM
1
iM
Statistical view:
𝑝𝑖 =
𝑛 𝑖
∗
𝐿
=?
L number of copies of the prototype
K number of microstates that the prototype can
acquire
ni number of prototypes in microstate i
pi the probability of prototype occurrence in the given
microstate i
C7790 Úvod do molekulového modelování -12Types
of statistical ensembles
The most common types of statistical ensembles include:
▪ microcanonical ensemble (NVE) - the prototype contains a constant number of
particles, has a constant volume, and energy
▪ grand canonical ensemble (mVT) - the prototype has a constant chemical potential,
volume, and temperature
▪ canonical ensemble (NVT) - the prototype contains a constant number of particles,
has a constant volume, and temperature
C7790 Úvod do molekulového modelování -13Canonical
ensemble
Consider the system (prototype), which has a constant
number of molecules, a constant volume, and temperature.
NVT NVT
NVT NVT
NVT NVT
NVT NVT
NVT NVT
NVT NVT
NVT NVT
NVT NVT
prototype
1. The ensemble total energy is equal to the
sum of the energies of the subsystems (the
copies do not interact with each other) energy
conservation:
𝐸 = ෍
𝑖=1
𝐿
𝐸𝑖 = ෍
𝑖=1
𝐾
𝑛𝑖 𝐸𝑖
LUUe =
(internal energy) (Helmholtz energy)
LSSe =
(entropy)
LFFe =
Two constrains apply:
2. The sum of number of prototypes in
given microstate must be constant and
equal to L (total number of prototype
copies):
𝐿 = ෍
𝑖=1
𝐾
𝑛𝑖
C7790 Úvod do molekulového modelování -14Entropy
of canonical ensemble
WkS Be ln=
kB - Boltzmann constant kB - is not 1.0 because the
definition of absolute
temperature
It can be shown that the entropy of the statistical ensemble is related to the statistical
weight W.
The statistical weight W determines number of possible ensemble implementations.
=
= K
i
i
K
n
L
nnW
1
1
!
!
),...,( L - number of copies of the prototype
K - number of microstates
ni - number of prototypes of the given microstate inumber of microstates
correction for indistinguishability of individual microstates
number of all combinationsStatistical weight
C7790 Úvod do molekulového modelování -15Entropy
of canonical ensemble, II
Because L is a large number, there is an implementation for which the statistical weight of
the distribution W* dominates over others.
𝑊∗(𝑛1
∗
, . . . , 𝑛 𝐾
∗
) >> 𝑊𝑜𝑡ℎ𝑒𝑟𝑠
Then, we search for the ensemble composition, in which its entropy reaches maximum
value and all imposed constraints are fulfilled.
𝑆 𝑒 = 𝑘 𝑏 ln 𝑊 (𝑛1
∗
, . . . , 𝑛 𝐾
∗
) = 𝑘 𝑏 ln
𝐿!
ς𝑖=1
𝐾
𝑛𝑖!
→ max!
𝐸 = ෍
𝑖=1
𝐿
𝐸𝑖 = ෍
𝑖=1
𝐾
𝑛𝑖 𝐸𝑖
Two constrains apply:
𝐿 = ෍
𝑖=1
𝐾
𝑛𝑖
C7790 Úvod do molekulového modelování -16Canonical
ensemble - solution
kB - Boltzmann constant
T - absolute temperature
The final result:
Q
e
e
e
p
i
j
i E
K
j
E
E
i


 −
=
−
−
==
1
*
=
−
=
K
j
E j
eQ
1

Canonical partition function:
it is a normalization value
TkB
1
=
Value of observable property:
𝑀 = ෍
𝑖=1
𝐾
𝑝𝑖
∗
𝑀𝑖
Partition function determines various
thermodynamic properties.
C7790 Úvod do molekulového modelování -17Thermodynamic
properties
Entropy:
𝑆 =
𝑈
𝑇
+ 𝑘 𝐵 ln 𝑄
Helmholtz energy F:
QTkF B ln−=
TSUF −=
Q
eE
e
eE
pEU
i
j
i E
K
i
i
K
j
E
E
K
i
i
i
K
i
i


 −
=
=
−
−
=
=



 === 1
1
1*
1
Internal energy:
VN
B
T
Q
TkU
,ln
ln








=
=
−
=
K
j
E j
eQ
1

Canonical partition function:
Canonical ensemble
C7790 Úvod do molekulového modelování -18Partition
function and modelling
Helmholtz energy F:
QTkF B ln−==
−
=
K
j
E j
eQ
1

Canonical partition function:
Evaluation of all (or important) microstates Molecular dynamics
(classical continuous system)
𝑄 =
1
ℎ3
ඵ
Ω
𝑒−𝛽𝐻(𝒙,𝒑)
𝑑𝒙𝑑𝒑
h - Planck constant
H - Hamiltonian (energy of the system)
For example:
• ideal gas model with contributions from
• electronic microstates
• vibration microstates
• rotation microstates
• translation microstates
• Monte Carlo simulations
E1, E2, E3, …
We need to find discrete energies of
microstates.
We need to solve describe energy
evolution in time.
C7790 Úvod do molekulového modelování -19Partition
function and modelling
Helmholtz energy F:
QTkF B ln−==
−
=
K
j
E j
eQ
1

Canonical partition function:
Consider only the most important microstate
approximation
E1, E2, E3, …
The most important microstate is the
microstate with the lowest energy.
𝐹 = 𝐸1
Very often used for qualitative consideration
or when computationally demanding
methods are employed (typically quantum
chemical calculations).
C7790 Úvod do molekulového modelování -20-
Summary
bridge