Introduction to Computational Quantum Chemistry
Lesson 3: the Potential Energy Surfaces (PES)
(Prepared by Radek Marek Research Group)
Lesson 03 - Potential Energy Surface 1
Potential Energy Surface
The Potential Energy Surface (PES) is a mathematical function that gives
the internal energy of a molecule as a function of geometrical modulations
as it stretches, bends, torsions, breaks, etc.
The PES of a molecule is obtained by the total interactions of:
nuclear-nuclear repulsion
electron-electron interaction
electron-nuclear attraction
The topology (surface) of the PES is dependent on what methods it is
derived from
Molecular Mechanics is based in some classical parameters and
can provide approximate results.
Quantum Mechanics is more accurate and exact and in principle
and works for any molecule but computationally expensive.
(Prepared by Radek Marek Research Group)
Lesson 03 - Potential Energy Surface 2
Geometry Description
Coordinates Notation Degrees of freedom
Cartesian x, y, z 3N
Spherical r, θ, φ 3N
Internal R, A, D 3N-6 (3N-5)
x
y
z
(x, y, z)
x
y
z
x
y
z
(r, , )
r
O
O
H
H RHO
R'HO
ROO
AHOO
A'HOO
DHOOH
(Prepared by Radek Marek Research Group)
Lesson 03 - Potential Energy Surface 3
The Born-Oppenheimer Approximation
the Born-Oppenheimer Approximations allows separation of electronic
and nuclear degrees of freedom "simplifies things" even though they are
coupled by the electron-nuclear potential energy VeN(r,R).
Electrons are much lighter than the nuclei, thus with respect to electrons,
the nuclei are almost stationary.
fix the nuclei at some chosen configuration Ra
solve for the motion of the electrons for this nuclear configuration,
giving an electronic energy Ee(Ra)
repeat for other nuclear configurations Rb of interest, building up a
Potential Energy Surface Ee(Rb).
(Prepared by Radek Marek Research Group)
Lesson 03 - Potential Energy Surface 4
PES of a Diatomic Molecules
can be visualized as function of energy versus internuclear
distance (single internal coordinate)
Morse potential:
V (r) = De(1 − e−a(r−re)
)2
(1)
where
De, is the depth of potential well
a, controls the width of the potential
r, is the internuclear separation
re, is the equilibrium distance
-5
0
5
10
15
1 2 3 4 5 6
Energy
Distance
Morse potential
(Prepared by Radek Marek Research Group)
Lesson 03 - Potential Energy Surface 5
PES of Multiatomic Systems
impossible to visualize more than 2 variables and 1 energy dimension
cuts from multidimensional space (hyperspace) where all other degrees
of freedom are kept fixed
Energy
Reaction coordinate
Educts
Transition State
Products
(Prepared by Radek Marek Research Group)
Lesson 03 - Potential Energy Surface 6
Important Points on the PES
stationary points:
∂E
∂qi
= 0 (2)
local minimum:
∂2E
∂q2
i
> 0 for all degrees of freedom (3)
nth order saddle point:
∂2E
∂q2
i
< 0 for n degrees of freedom (4)
(Prepared by Radek Marek Research Group)
Lesson 03 - Potential Energy Surface 7
Zero-Point Energy (ZPE) Corrections
Vibrational corrections for 0K (ground vibrational state)
“Cancel out” for energy differences
Energy
Reaction coordinate
First vibrational state
Energy of state
(Prepared by Radek Marek Research Group)
Lesson 03 - Potential Energy Surface 8
Zero-Point Energy (ZPE) Corrections
Vibrational corrections for 0K (ground vibrational state)
“Cancel out” for energy differences
Energy
Reaction coordinate
First vibrational state
Energy of state
Ea
(Prepared by Radek Marek Research Group)
Lesson 03 - Potential Energy Surface 8
Zero-Point Energy (ZPE) Corrections
Vibrational corrections for 0K (ground vibrational state)
“Cancel out” for energy differences
Energy
Reaction coordinate
First vibrational state
Energy of state
Ea
ΔE
(Prepared by Radek Marek Research Group)
Lesson 03 - Potential Energy Surface 8
Hessian Index
a calculated optimized geometry needs a vibrational analysis
to verify its location in the PES via the Hessian index
the Hessian index is the number of negative eigenvalues of
the force constant matrix (i.e. imaginary frequencies). For
a stationary point, this corresponds to the number of internal
degrees of freedom along which that point is a potential
energy maximum. The Hessian index is:
0 for minima
1 for transition states
> 1 for higher-order saddle points
(Prepared by Radek Marek Research Group)
Lesson 03 - Potential Energy Surface 9
Relaxed vs Rigid PES
there are two types of mapping the PES namely Rigid and
Relaxed scanning
RIGID, means scanning the energetics of the molecule by
only changing specific angle while all the bond lengths at
their fixed position
RELAXED, means scanning the energetics while holding
the dihedral angle constant at a series of values, and
relaxing the remainder of the coordinates.
the second method (RELAXED Scan) will give us the true
minimum energy path while the first method is merely an
approximation to the minimum energy path.
(Prepared by Radek Marek Research Group)
Lesson 03 - Potential Energy Surface 10
ACTIVITY
Activities related to Potential Energy Surface scans will be
included in
Lesson 7 : PES Scan, Reaction Coordinates, and
Transition State Search
(Prepared by Radek Marek Research Group)
Lesson 03 - Potential Energy Surface 11
Introduction to Computational Quantum Chemistry
END
(Prepared by Radek Marek Research Group)
Lesson 03 - Potential Energy Surface 12