C7790 Introduction to Molecular Modelling -1Lesson
11
Quantum Mechanics III
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 -3-
Context
Quantum Mechanics
➢ It can properly describe systems composed of atoms, which are further composed from
electrons and atom nuclei (dual character - particle/wave).
➢ Microstate energies are solution of time-independent Schrödinger equation.
)()(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)
➢ Analytically solvable for simple systems
C7790 Introduction to Molecular Modelling -4QM
Description of
Simple Systems
➢ hydrogen atom
➢ harmonic oscillator
➢ rigid rotator
➢ particle in potential well
➢ hydrogen molecule
approximate description for
➢ vibrational
➢ rotational
➢ translational
motions
C7790 Introduction to Molecular Modelling -5Hydrogen
Atom
C7790 Introduction to Molecular Modelling -6Hydrogen
atom
[xp, yp, zp]
[xe, ye, ze]
r
Hamiltonian
r
e
mM
H ep
2
0
2
2
2
2
4
1
22
ˆ

−−−=

operator describing
proton motion electrostatic interaction
between proton and electron
operator describing
electron motion
mM
Mm
+
=
Motion of two bodies can be described by motion of one body with a reduced weight:
What is the reduced mass of hydrogen atom (proton/electron)?
M = 1836 au
m = 1 au
 = 0.99945 au practically the same weight
C7790 Introduction to Molecular Modelling -7Hydrogen
atom
[xe, ye, ze]
r
r
e
m
H e
2
0
2
2
4
1
2
ˆ

−−

m
mM
Mm

+
=
x
y
z
Cartesian vs spherical coordinates
[xe, ye, ze]
r
x
y
z
[r,q,f]
r
x
y
z
222
eee zyxr ++=
q
F
2
2
222
2
2
2
sin
1
sin
sin
11
qq
q
qq 

+









+









=
rrr
r
rr
C7790 Introduction to Molecular Modelling -8Hydrogen
atom - solution
),,(),,(ˆ qq rErH kkk =
),()(),,( ,, fqq mllnk YrRr =
Solution:
2
2
1
n
Ek −=
radial part of the wave function
angular (angular) part of the wave function (WF)
quantum numbers:
n - principal quantum number (1,2,3 ...)
l - angular quantum number (0, ..., n-1 = s, p, d, f, g,...)
m - magnetic quantum number (-l, ..., 0, ..., l)
2
00
22
8 na
eZ
Ek

−=
Z - proton number
e - electron charge
0 - vacuum permittivity
a0 - Bohr radius
in atomic units:
C7790 Introduction to Molecular Modelling -9Hydrogen
atom - solution
radial component of the wave function
angular component of the wave function
C7790 Introduction to Molecular Modelling -10-
Summary
➢ Hydrogen atom and hydrogen like atoms (atom cations with one electron) are only
chemical systems, whose SE is solvable analytically.
➢ Allowed energy is discretized (quantized) and dependent only on the principal quantum
number.
➢ Hydrogen atom WF is a foundation for atomic orbitals employed by quantum chemistry
methods.
a) The hydrogen atom has degenerate states, i.e., states with the same n have the same
energy.
b) Atoms with more electrons.
C7790 Introduction to Molecular Modelling -11SR
solution for simple systems
➢ hydrogen atom
➢ harmonic oscillator
➢ rigid rotator
➢ particles in potential well
approximate description for
➢ vibratory
➢ rotational
➢ translational
motions
C7790 Introduction to Molecular Modelling -12Harmonic
Oscilator
C7790 Introduction to Molecular Modelling -13Harmonic
oscillator
Hamiltonian
)(
22
ˆ 2
2
2
2
2
1
1
2
rV
mm
H +−−=

( )2
0
2
1
)( rrKrV −=
m1 m2
spring with stiffness K
( )0)( rrKrF −=
the force is proportional to the deviation
from the equilibrium position
)(
2
ˆ 2
2
rVH +−=

 ( )2
0
2
1
)( rrKrV −=
22
21
mm
mm
+
=
Simplification:
r0
C7790 Introduction to Molecular Modelling -14Harmonic
oscillator - solution
)()(ˆ rErH kkk  =
Solution:
)()( rr vk =






+=
2
1
vEk
quantum numbers:
v - vibrational quantum number (0,1,2,3 ...)


K
=angular frequency
C7790 Introduction to Molecular Modelling -15-
Summary
➢ Quantum harmonic oscillator cannot have zero energy in the ground state.
➢ This intrinsic behaviour can be explained by uncertainty principle.
➢ For low vibrational numbers, the highest probability for particle finding is at
equilibrium distance (this is opposite to the classical harmonic oscillator
behavior).
➢ Energies are equidistant.
2

 px
position/momentum
- no motion (exact momentum)
- position at potential bottom (exact position)
C7790 Introduction to Molecular Modelling -16Harmonic
vs anharmonic oscillator
Morse potential
( )
( )2
0
1)( rra
e eDrV −−
−=
Harmonic potential
( )2
0
2
1
)( rrKrV −=
eD
a
K
2
=
Simplified description of vibrational motion. A more accurate empirical description is given
by Morse's potential.
C7790 Introduction to Molecular Modelling -17Rigid
Rotor
C7790 Introduction to Molecular Modelling -18Rigid
rotor
Hamiltonian
2
2
2
2
2
1
1
2
22
ˆ −−=
mm
H

m1 m2
r0
with constraint r= r0
[x,y,z]
r0
x
y
z
2
2
2
ˆ −=


H
22
21
mm
mm
+
=
Simplification:
with constraint r= r0
C7790 Introduction to Molecular Modelling -19Rigid
Rotor - solution
),(),(ˆ qq kkk EH =
),(),( , fqq mlk Y=
Solution:
)1(
2
2
+= ll
I
El
 angular part of the wave function
quantum numbers:
l - angular quantum number (0,1,2, ...)
m - magnetic quantum number (-l,...,0,...,l)
2
0
rI =moment of inertia
C7790 Introduction to Molecular Modelling -20Particle
in a Box
C7790 Introduction to Molecular Modelling -21Particle
in a box
m
=V
0=V
L
2
2
2
ˆ −=
m
H

Hamiltonian
Solution:
2
2
22
2
n
mL
En

=
quantum numbers:
n - quantum number (1,2, ...)
1D potential box (the infinite potential well) is infinitely
deep, so the probability of particle finding outside the box
is zero.
with constraint
0)( =r
for r > L and r < 0






= x
L
n
An

 sin
For a multi-dimensional potential box (3D), the
dimensions can be replaced by the box volume.
standing waves
C7790 Introduction to Molecular Modelling -22Hydrogen
Molecule
➢ Many electron atoms (He, Li, ...)
➢ Born-Oppenheimer approximation
➢ One-electron approximation
➢ …
➢ Many atom (=many electron) molecules
➢ Born-Oppenheimer approximation
➢ One-electron approximation
➢ …
C7790 Introduction to Molecular Modelling -23-
Revision
t
t
itH


=
),(
),(ˆ x
x
f
f 
časově závislá Schrödingerova rovnice
C7790 Introduction to Molecular Modelling -24-
Revision
t
t
itH


=
),(
),(ˆ x
x
f
f 
time-dependent Schrödinger equation
)()(ˆ xx kkk EH  =
time-independent Schrödinger equation
)()(),( tft xx f =
system can exist in several quantum states described by
wavefunction Yk and energy Ek
C7790 Introduction to Molecular Modelling -25-
Revision
t
t
itH


=
),(
),(ˆ x
x
f
f 
time-dependent Schrödinger equation
)()(ˆ xx kkk EH  =
time-independent Schrödinger equation
)()(),( tft xx f =
),()(),(ˆ RrRRr mmme EH Y=Y )()(ˆ
, RR llVRTlR EH  =
)(),()( RRrx  Y=
electron motion in the static field of nuclei
electronic properties
nuclei motion in effective field of electrons
vibration, rotation, translation
Born- Oppenheimer approximation
C7790 Introduction to Molecular Modelling -26-
Revision
),()(),(ˆ RrRRr mmme EH Y=Y )()(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 -27-
Revision
),()(),(ˆ RrRRr mmme EH Y=Y )()(ˆ
, RR llVRTlR EH  =
je možné obdobným
způsobem dále rozdělit na
samostatné příspěvky
vibrační, rotační a translační
kTjRiVlVRT EEEE ,,,, ++=
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 -28Structure
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 -29-
Homework
1. What is the order of the dissociation energies of H2 (hydrogen molecule), D2
(deuterium molecule), and T2 (tritium molecule)?
Help:
▪ vibrations are quantized
▪ neglect rotation and translation (why?)
hvEV 





+=
2
1
Focus on the ground state (1s+1s) only:
vibrational quantum
number 0,1,2,...
Total energy of the ground state:
)0()( =+= vErEE Vo