lecture # 5.pptxvvhjjjgguiyffuffyuffyufg

Introduction to
Computational
Chemistry
Dr. Naveen Kosar
Assistant Professor
Department of Chemistry
School of Science, UMT
CH-435 Spring 2022

Objectives
Optimization procedure using Newton-Raphson method
Find out Minima and transition state

3
Geometry Optimization
• process for locating a minimum or transition state on the PES
• most common type of calculation performed in computational chemistry
• also called relaxation (at least for finding minima)
Basic idea:
• iterative process
• start with a ‘guess’ structure and use information of the PES to change
the coordinates such that the final structure is a stationary point
Why?
• need geometries/energies of minima and transition states to:
1. determine reaction mechanisms and associated energetics
2. determine preferred molecular geometries for calculations of other
properties
• lets us find minima/transition states without any exact information
regarding those points on the PES

4
Newton-Raphson Method
• basis for geometry optimization procedures used in most computational
chemistry codes
consider a 1-D harmonic potential
 
2
0 0
i i
E E k q q
  
 
0
2 i
i
E
k q q
q
 

 
 

 
2
2
2
i
E
k
q
 


 

 
 
2
0
2 i
i i
E E
q q
q q
 
 
 
 
 
 
 
   
2
0 2
i
i i
E E
q q
q q
 
 
 
   
 
 
   
guess structure (qi, Ei)
bond length, q
energy,
E
minimum (q0, E0)
step
finds minimum in 1 step!!

5
• in general, PES is not harmonic  cannot find q0 in 1 step
initial guess structure (qi)
local minimum 2
reaction coordinate, q
energy,
E
• instead, we iterate until forces on nuclei (derivatives, gradients) are small
.
.
.
• with a reasonable guess structure, most programs will find the nearest local minimum
in 10 – 20 iterations
• transition states can be more difficult to locate
2
1 2
i i
i i
E E
q q
q q

 
 
 
   
 
 
   
2
2 1 2
1 1
i i
i i
E E
q q
q q
 
 
 
 
 
   
 
 
   
2
1 2
1 1
n n
n n
E E
q q
q q

 
 
 
 
   
 
 
   
local minimum 1
• NOTE: doesn’t just follow reaction coordinate, but optimizes over whole PES

6
in general, optimization is done using 3N cartesian coordinates
1
2
3
i
i
i i
Ni
q
q
q
q
 
 
 
 
 
 
 
q

 
 
 
1
2
3
i
i
i
i
N i
E q
E q
E
q
E q
 
 
 
 
 
  
 
   

 
 
 
 
 
g

     
     
     
2 2 2
1 1 1 2 1 3
2 2 2
2
2 1 2 2 2 3
2
2 2 2
3 1 3 2 3 3
N
i i i
N
i i i
i
i
N N N N
i i i
E q q E q q E q q
E q q E q q E q q
E
q
E q q E q q E q q
 
     
 
 
     
 

 
 
 
  
 
 
     
 
 
H


   

2
1
1 1
2
i i i i i i
i i
E E
q q
q q

 
 
 
 
    
 
 
 
   
q q H g

7
to do an optimization, we need:
qi
• initial guess structure should look like anticipated minimum or transition state
• can build with molecular editors
• column vector of 3N cartesian coordinates defining the molecular geometry
gi
• column vector of 3N derivatives of the energy wrt cartesian coordinates
• calculated by the program  ~20% of effort in geometry optimization
Hi
• “Hessian” matrix containing 3Nx3N 2nd
derivatives of the energy wrt
cartesian coordinates
• 2nd
derivatives are expensive to calculate analytically
• generally estimated during optimization:
 
2
b i
a b a
i
E q
E
q q q
  
 


 
 
 
• represent the forces on the nuclei: a a
F E q
  

8
Finding Minima
1. construct a “guess” structure  optimization will go to nearest
minimum
2. build up initial Hessian, Hi (approximate from low-level calculation)
3. calculate derivatives, gi
4. compare derivatives against convergence criteria
5. update geometry:
1
1
i i i i

  
q q H g
6. update Hessian:
 
2
b i
a b a
i
E q
E
q q q
  
 


 
 
 
7. repeat 3 through 6 until convergence criteria are met
Convergence Criteria
• impossible to find exact location of minimum
• instead geometry optimization stops when all derivatives are small
• in most codes:- maximum force < 4.5 x 10-4
Hartree/bohr
- RMS force < 3.0 x 10-4
Hartree/bohr
- other geometry-based criteria

9
Finding Transition States
1. construct a “guess” structure  need good guess for transition
states
2. build up initial Hessian, Hi (calculate analytically)
3. calculate derivatives, gi
4. compare derivatives against convergence criteria
5. update geometry:
1
1
i i i i

  
q q H g
6. update Hessian:  
2
b i
a b a
i
E q
E
q q q
  
 


 
 
 
7. repeat 3 through 6 until convergence criteria are met
Initial guess
• finding a saddle point is difficult  need good guess structure, accurate
Hessian
• guess structure by: - preliminary scan along reaction coordinate
- interpolate between reactant and product structures
- chemical intuition
(modify H for TS)

10
Characterizing Stationary Points
• convergence criteria are based only on derivatives
• definitions of minima and transition states involve 2nd
derivatives:
2
2
0
i
E
q



minimum: for all qi
transition state:
2
2
0
i
E
q



for exactly one qi
2
2
0
i
E
q



for all other qi
need 2nd
derivatives to characterize minima/transition states
 frequency calculation and normal mode analysis

11
Normal Mode Calculation
• collective vibrational motions of atoms
normal modes:
• simplest vibrational motions in a molecule
• probed in IR/Raman experiments
• 3N-6 modes/molecule (3N-5 if linear)
frequencies in harmonic approximation:
1
2
1
2
k
c

 
 
  
 
 (units of cm-1
)
k = force constant:
2
2
E
k
q
 

 

 
 = reduced mass of mode
c = speed of light
PES is approximately harmonic at stationary
points
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
800.0
900.0
1000.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
calculated PES
harmonic fit
dN-N [Å]
energy
[kcal/mol]
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
800.0
900.0
1000.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
calculated PES
harmonic fit
dN-N [Å]
energy
[kcal/mol]
O
H H
bend:
O
H H
symmetric
stretch:
O
H H
asymmetric
stretch:

12
1
2
1
2
k
c

 
 
  
 

2
2
0
i
E
q



minimum:
for all qi
• all positive elements in H  all diagonal elements of K are positive
 if k is positive, the frequency is positive
minima are characterized as having all positive frequencies
2
2
0
i
E
q



transition state:
for exactly one qi
• one negative element in H  one diagonal element of K is negative
1
2
1
2
k
c

 
 
  
 
  if k is negative, the frequency is imaginary
transition states are characterized as having exactly one imaginary
frequency

13
You get the right number of frequencies. But, does the
structure make sense?
minimum: • look at the structure and you can usually tell
transition state: • there are a lot of saddle points on the PES
• the transition state for your reaction should be
the saddle point that connects the reactants and
products
• example keto-enol tautomerization
correct TS incorrect TS
v = 2449.6i cm-1 v = 185.6i cm-1

14
Characterizing Transition States
How do you know you have the correct transition state?
1. Look at the structure. Does it look right?
• pretty weak confirmation method, but helpful sometimes
2. The energy of the transition state must be higher than the reactant
and product.
0
10
20
30
40
50
reaction coordinate, q [arb]
energy,
E
[kcal/mol]
reactant product
transition state
0
10
20
30
40
50
reaction coordinate, q [arb]
energy,
E
[kcal/mol]
reactant product
transition state

Tutorial: 01
TS is somewhat halfway between two minima

16
Characterizing Transition States
How do you know you have the correct transition state?
1. Look at the structure. Does it look right?
• pretty weak confirmation method, but helpful sometimes
2. The energy of the transition state must be higher than the reactant
and product.
3. Animate the normal mode with the imaginary frequency
• the normal mode with the imaginary frequency corresponds to the motion
of the molecule over the energy barrier. Does this motion connect the
reactants to the products?
4. Perform an intrinsic reaction coordinate (IRC) calculation
• IRC calculations start at the transition state and follow the lowest energy
pathway in each direction along the imaginary normal mode. Do these
pathways lead to the reactants and products?
• IRC calculations are expensive and rarely necessary. But, they are a
surefire way to characterize the transition state.

17
Relative Energies
once you identify the reactant, product, and transition state
rxn prod react
E E E
 
†
TS react
E E E
 
• energies can be calculated
relative to reactant
• sets reactant energy to 0
• all energies must be
calculated at the same
level of theory!!!
energy
(kcal/mol)
rxn
E
†
E
For reactions with multiple reactants or products:
e.g. A + B TS†
 C +
D
rxn prod react
E E E
 
 
†
TS react
E E E
  
• energies are calculated
separately for each molecule
• e.g. do a calculation on A and
a calculation on B, not a
calculation on A and B
together
reactants TS product

18
Zero-Point Vibrational Energies
• molecules vibrate even at 0K
Quantum mechanics:
® Heisenberg uncertainty principle
says we can’t know simultaneously
the position and velocity of a
particle
® if atoms stopped vibrating they would
have no velocities and well-defined
positions
Zero-Point Vibrational Energy:
• vibrational energy with n = 0
• added to potential energy of the molecule
 
   
  0.5
tot I I i
i
E R E R h 
  
potential energy from
an energy calculation
zero-point vibrational
energy from normal
mode calculation
h = Planck’s constant
vi = frequencies from normal
= 6.626x10-34
Js-1
mode calculation
n = vibrational state
2
1
2
V kx

internuclear separation, x
energy
n = 0
n = 1
n = 2
n = 3
n = 4
0 0.5
E h

 
0.5 1
n
E n h
 
2
1
2
V kx

internuclear separation, x
energy
n = 0
n = 1
n = 2
n = 3
n = 4
0 0.5
E h

 
0.5 1
n
E n h
 

19
Zero-Point Vibrational Energies
zero-point vibrational energy has small effect on relative energies
energy
(kcal/mol)
0.0
14.6
14.5
88.0
84.3
without ZPVE
with ZPVE
CH3
C
O
CH3
CH3
C
OH
CH2
CH3
C
O
CH2
H
• nearly cancels out for
reaction energies
• often included in reaction
energies and barriers if
frequencies are available
• effect is often more
pronounced for barriers
® transition states often
have many weak
vibrations with low
frequencies
® ZPVE correction is
smaller than for reactants
and products
 
   
  0.5
tot I I i
i
E R E R h 
  

20
Thermal Corrections
energy calculation methods provide the potential
energy of the system at 0K
• are done at finite temperatures and pressure
Meanwhile, experiments:
• are governed by enthalpies and free energies
how do we incorporate thermal and pressure
effects into our calculated energies?
• use statistical mechanics to explore the properties of the
system in a macroscopic ensemble
• statistical mechanics will let us determine how thermal
energy is distributed in the molecule
• provides access to enthalpies, entropies, and free energies

Enthalpies
The sum of the internal energy of the system plus the product of the pressure of the gas in the
system and its volume
H = E +PV
Enthalpy of the reaction (∆H)
The difference between the sum of the enthalpies of the product and the sum of the enthalpies of the
reactant:
∆H = ∑ nHproduct - ∑mHreactant
Exothermic reaction: Reaction in which a system release heat to its surrounding
∆H is negative (∆H < 0)
Endothermic reaction: Reaction in which a system absorb heat from its surrounding
∆H is positive (∆H > 0)

Entropies
A measure of the unavailable energy in a closed thermodynamic system that
is also usually considered to be a measure of the system's disorder, that is a
property of the system's state, and that varies directly with any reversible
change in heat in the system and inversely with the temperature of the
system.
Free Energies
The energy with a chemical reaction that can be used to do work. The
free energy of a system is the sum of its enthalpy plus the product of the
temperature and the entropy of the system.
G = H- TS
SPONTANEOUS: G is negative ( G < 0)
NON-SPONTANEOUS: G is positive ( G > 0)
EQUILIBRIUM: G = 0

Free energies
The sum of the internal energy of the system plus the product of the pressure of the gas in the
system and its volume
H = E +PV
Enthalpy of the reaction (∆H)
The difference between the sum of the enthalpies of the product and the sum of the enthalpies of the
reactant:
∆H = ∑ nHproduct - ∑mHreactant
Exothermic reaction: Reaction in which a system release heat to its surrounding
∆H is negative (∆H < 0)
Endothermic reaction: Reaction in which a system absorb heat from its surrounding
∆H is positive (∆H > 0)

24
Summary of PES-related Concepts
1. Potential energy surface (PES)
-describes relationship between molecular structure and energy
-PES evaluated with energy calculation methods (future lectures)
2. Stationary Points
-points on the PES where
1 2 3 6
0
N
E E E
q q q 
  
   
  

-reactants and products are local minima (all positive 2nd
derivatives)
-transition states are saddle-points (exactly one negative 2nd
derivative)
3. Geometry Optimization
-procedure for identifying stationary points on the PES
4. Normal Mode Analysis/Frequency Calculation
-relate vibrational frequencies to 2nd
of energy wrt coordinates
-can be used to characterize stationary points and calculate ZPVEs
-only valid at stationary points
-provide access to enthalpies, entropies and free energies in ideal gas approx.

Vibrational frequencies are computed by determining the
second derivatives of the energy with respect to the
Cartesian nuclear coordinates and then transforming to
mass-weighted coordinates. This transformation is only valid at
a stationary point. Thus, it is meaningless to compute
frequencies at any geometry other than a stationary point
for the method used for frequency determination.

Include Opt and Freq in the route section:
geometry optimization followed by frequency calculation
Prior Geometry Optimization
Job Type (keyword)
Opt +Freq Keyword
Same level of theory and basis set for opt and Freq

27
Frequency Calculation - Input
• keyword ‘freq’ requests a frequency calculation
• ‘hf/3-21G’ specifies level of theory for the calculation
• coordinates in Z-matrix format, but frequency calculations can
also be performed with cartesian coordinates
• structure must correspond to a stationary point
• recall, frequency calculations in harmonic approximation are
only valid at stationary points

28
Frequency Calculation - Output
in a frequency calculation we may want to determine:
• normal modes and vibrational frequencies
mode
frequencies in cm-1
normal mode
displacements

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