C7790 Introduction to Molecular Modelling -1Petr
Kulhánek
kulhanek@chemi.muni.cz
National Centre for Biomolecular Research, Faculty of Science
Masaryk University, Kamenice 5, CZ-62500 Brno
Lesson 13
Potential Energy Surface I
JS/2022 Present Form of Teaching: Rev3
C7790 Introduction to Molecular
Modelling
TSM Modelling Molecular Structures
C9087 Computational Chemistry for Structural Biology
C7790 Introduction to Molecular Modelling -2-
Context
microworldmacroworld
equilibrium (equilibrium constant)
kinetics (rate constant)
free energy
(Gibbs/Helmholtz)
partition function
phenomenological thermodynamics
statistical thermodynamics
microstates
(mechanical properties, E)
states
(thermodynamic properties, G, T,…)
microstate ≠ microworld
Description levels (model chemistry):
• quantum mechanics
• semiempirical methods
• ab initio methods
• post-HF methods
• DFT methods
• molecular mechanics
• coarse-grained mechanics
Structure
EnergyFunction
Simulations:
• molecular dynamics
• Monte Carlo simulations
• docking
• …
C7790 Introduction to Molecular Modelling -3-
Revision
t
t
itH


=
),(
),(ˆ x
x

 
time dependent Schrödinger equation
C7790 Introduction to Molecular Modelling -4-
Revision
t
t
itH


=
),(
),(ˆ x
x

 
time dependent Schrödinger equation
)()(ˆ xx kkk EH  =
time independent Schrödinger equation
)()(),( tft xx  =
system can exist in several quantum states, each state is described by
wavefunction k and has energy Ek
C7790 Introduction to Molecular Modelling -5-
Revision
t
t
itH


=
),(
),(ˆ x
x

 
)()(ˆ xx kkk EH  =
)()(),( tft xx  =
),()(),(ˆ RrRRr mmme EH = )()(ˆ
, RR llVRTlR EH  =
Born- Oppenheim approximation
time independent Schrödinger equation
time dependent Schrödinger equation
electron motion in the static field of nuclei
electronic properties
nuclei motion in effective field of electrons
vibration, rotation, translation
),()(),( RrRRr = 
C7790 Introduction to Molecular Modelling -6Hydrogen
molecule
HW: What is the dissociation energy of H2, D2, and T2?
H2 H· H·+
DGr = ?
DEr = ?
approximation
do not consider thermal effects
),()(),(ˆ RrRRr mmme EH =
Energy is a function of nuclei positions.
The function and its projections to lower dimensional configurational spaces are
called potential energy surface.
What is the potential energy surface for H2, D2, and T2?
Do they differ?
C7790 Introduction to Molecular Modelling -7H2
- Potential Energy Surface
2H·
DEr
????dissociation
energy????
H2
What is the energy of dissociated state?
How is it related to reference states?
Two suitable reference states:
➢ standard QM reference state (infinite
separation of electrons and nuclei, no
kinetic energy) - negative energy
➢ dissociated state is considered as a
reference with zero energy
bound state
dissociated state
REMEMBER:
➢ The reference state represents well defined state with
well defined energy, usually zero.
➢ Its choice is arbitrary, but it must be consistent for all
compounds and their states.
),()(),(ˆ RrRRr mmme EH = positions of electrons (r) and nuclei (R)
distance between two
hydrogen atoms
C7790 Introduction to Molecular Modelling -8Recall
Hamiltonian of chemical system
 = = == =
+−+−=
n
i
n
ij ij
N
i
n
j ij
i
N
i
N
ij ij
ji
n
i
i
rr
Z
r
ZZ
m
H
11 111
2 1
2
1ˆ
Hamiltonian of chemical system consisting of N nuclei of mass M and charge Z
and n electrons is given by:
kinetic energy operator (ELECTRONS ONLY) potential energy
electrons electron-electron
electron-nucleus
nucleus-nucleus
Nuclei motion (nuclei mass) is not considered in the BO approximation.
Schrödinger equation:
),()(),(ˆ RrRRr mmme EH =
C7790 Introduction to Molecular Modelling -9H2,
D2, T2 - Potential Energy Surface
DEr(r0)
H2
r0
DEr(r0)
D2
r0
DEr(r0)
T2
r0
Potential energy surfaces are the same!
r0 is the same as well.
Reason:
All three systems are chemically identical
(two electrons and two +1 charged nuclei).
C7790 Introduction to Molecular Modelling -10H2,
D2, T2 - Potential Energy Surface
DEr(r0)
H2
r0
DEr(r0)
D2
r0
DEr(r0)
T2
r0
What about vibrations? Do they contribute and how?
𝐸 𝑉 = 𝑣 +
1
2
ℎ𝜐
Consider harmonic oscillator (approximation):
v = 0, 1, 2, …
characteristic frequency
non-zero energy even in the ground vibration state!!!
C7790 Introduction to Molecular Modelling -11H2,
D2, T2 - Vibrations
𝐸 𝑉 = 𝑣 +
1
2
ℎ𝜐
Harmonic oscillator:
𝜐 =
1
2𝜋
𝐾
𝜇
v = 0, 1, 2, …
characteristic frequency
force constant; does it differ?
reduced mass (clearly this differs among
H2, D2, and T2)
C7790 Introduction to Molecular Modelling -12H2,
D2, T2 - Vibrations
𝐸 𝑉 = 𝑣 +
1
2
ℎ𝜐
Harmonic oscillator:
𝜐 =
1
2𝜋
𝐾
𝜇
v = 0, 1, 2, …
characteristic frequency
reduced mass (clearly this differs among
H2, D2, and T2)
𝑉(𝑟) =
1
2
𝐾 𝑟 − 𝑟0
2
𝜕𝑉(𝑟)
𝜕𝑟
= 𝐾 𝑟 − 𝑟0
𝜕2
𝑉(𝑟)
𝜕𝑟2
= 𝐾
first derivative
with respect to r
second derivative
with respect to r
What about the force constant?
Harmonic potential:
The force constant can be determined from the PES curvature
at equilibrium distance (r0) in harmonic approximation.
tangent curvature
All three systems
have the same PES
and thus the same K
as well .
force constant; does it differ?
C7790 Introduction to Molecular Modelling -13H2,
D2, T2 - PES + Vibrations
DEr
H2
r0
DEr
D2
r0
DEr
T2
r0
𝜐 =
1
2𝜋
𝐾
𝜇
bigger mass -> smaller frequency -> lower energy
< <|DEr||DEr| |DEr|
r0 r0 r0= =
𝐸 𝑉 = 𝑣 +
1
2
ℎ𝜐
Ev
re re re~ ~ observable equilibrium bond lengths
impact of anharmonicity and QM
character of vibrations
!! not in scale !!
C7790 Introduction to Molecular Modelling -14-
Revision
),()(),(ˆ RrRRr mmme EH = )()(ˆ
, RR llVRTlR EH  =
lVRTmoptmk EREE ,, )( +=
total energy of the state
electronic energy part
vibration, rotation,
translation energy part
optimal geometry, at
which Em is minimal
nuclei motion in effective field of electrons
vibration, rotation, translation
electron motion in the static field of nuclei
electronic properties
C7790 Introduction to Molecular Modelling -15Structure
vs system state
✓
✓
𝐸 = 𝐸(𝑟𝑜) + 𝐸 𝑉𝑅𝑇
only part of the quantum
state description
!!!!!!
EVRT is nonzero even at 0 K
because of vibrational (and
translational) energy.
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
approximations approximations
C7790 Introduction to Molecular Modelling -17Quantum
vs Classical description
➢ Fully QM
➢ QM, MM + QM harmonic
approximation, or similar
➢ QM, MM + path integral
molecular dynamics
➢ QM, MM, CG + classical
nuclei/atom motions,
molecular dynamics
(MD)
thermal energy not shown in graphs is 1/2kBT
(equipartition principle) in all cases (fully
quantum/classical)
NO DIFFERENCE
C7790 Introduction to Molecular Modelling -18-
Summary
𝐸 = 𝐸(𝑟𝑜) + 𝐸 𝑉𝑅𝑇
r0
a) we need to find
a potential energy minimum
b) we can further evaluate vibrations
from the PES curvature at the minimum
➢ PES cannot describe mass effect of nuclei; it only describes electronic effects.
➢ Isotope effects can be measured experimentally
➢ Primary Isotope Effect (kinetics)
➢ Secondary Isotope Effect (kinetics)
➢ It can be even tasted by your tongue, see:
Ben Abu, N.; Mason, P. E.; Klein, H.; Dubovski, N.; Ben Shoshan-Galeczki, Y.;
Malach, E.; Pražienková, V.; Maletínská, L.; Tempra, C.; Chamorro, V. C.; Cvačka, J.;
Behrens, M.; Niv, M. Y.; Jungwirth, P. Sweet Taste of Heavy Water. Communications
Biology 2021, 4 (1), 1–10. https://doi.org/10.1038/s42003-021-01964-y.
To characterize a quantum state:
(too difficult to calculate, thus it is usually neglected)