CHEM 834: Computational Chemistry

1
CHEM 834 Computational Chemistry
Potential energy surfaces
March 3, 2008
2
Today

• overview of computational chemistry

• exploring potential energy surfaces

• gaussian/gaussview tutorial

Course website
http//www.chem.queensu.ca/people/faculty/Mosey/ch
em834.htm

• lecture notes
• announcements

3
Computational Chemistry
application of mathematical models and
simulations to chemical problems using computers
Includes (list is not exhaustive)

• molecular energies ? reaction energies, barriers

• molecular geometries

• electron distributions ? reactivities

• spectroscopic properties e.g. UV/VIS, IR/Raman,
  NMR, etc.

• molecular dynamics ? reaction rates and
  mechanisms, protein folding

• physical properties e.g. mechanical hardness

• quantitative structure-activity relationships

Doesnt Include (list is not exhaustive)

• solving rate equations

• deconvoluting spectra

4
Computational Chemistry
general details of a calculation
energy calculation methods

• calculate molecular energy

property calculation methods
guess of molecular structure

• molecular structure
• relative energies
• reaction pathways
• simulated spectra
• mechanical properties
• molecular dynamics
• etc.

5
Computational Chemistry
two main categories of methods
1. Energy Calculation Methods

• relate the energy of a molecule to its geometry

• fundamental to all calculations

• are used to construct the potential energy surface

• include

1. force-fields/molecular mechanics
- ball-and-spring approach
- neglects electronic structure
2. ab initio methods
- fully quantum mechanical treatment of the
electronic wavefunction
3. semi-emiprical molecular orbital methods
- ab initio methods with an approximate/parameteri
zed Hamiltonian
4. density functional theory methods
- fully quantum mechanical treatment of the
electron density
6
Computational Chemistry
two main categories of methods
2. Property Calculation Methods

• evaluate specific properties of a system

• use information from energy calculation methods

• include

1. geometry optimization

• determine structures and energies of reactants,
  products and transitions states

• gives reaction energetics

2. molecular dynamics

• follow spatiotemporal evolution of a chemical
  system

acceleration
positions
velocity
7
Computational Chemistry
two main categories of methods
2. Property Calculation Methods

• evaluate specific properties of a system

• use information from energy calculation methods

• include

3. statistical mechanical treatments

• predict thermodynamic properties

4. normal mode analysis in the harmonic
approximation

• gives vibrational frequencies, IR/Raman spectra

8
Potential Energy Surface

• structure-energy relationship is fundamental to
  computational chemistry

• the potential energy surface (PES) describes this
  relationship

• the PES is calculated at 0K

• energy calculation methods provide access to the
  PES

in molecular mechanics
in quantum mechanics
these methods give the energy for a single
nuclear configuration RI
we have to do a lot of calculations to construct
an entire PES
9
1-D Example N2

• calculate energy at several distinct values of
  dN-N

• called a potential energy surface scan

• interpolate between points to construct PES

• gives relevant structural and energetic
  information

10
2-D Example Ozone Isomerization

dO1-O2
O1
O1
O1
O2
O2
O2
O2
O2
O2
aO2-O1-O2
ozone
iso-ozone
build up the PES

• systematically vary aO2-O1-O2 and dO1-O2 on a
  grid and calculate the energy at each set of
  values

• requires a large number of calculations due to
  2-D nature of system

2-D potential energy surface
11
2-D Example Ozone Isomerization

dO1-O2
O1
O1
O1
O2
O2
O2
O2
O2
O2
aO2-O1-O2
ozone
iso-ozone
can identify

• reactant and product structures

iso-ozone
ozone

• local minima on PES

2-D potential energy surface
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2-D Example Ozone Isomerization

dO1-O2
O1
O1
O1
O2
O2
O2
O2
O2
O2
aO2-O1-O2
ozone
iso-ozone
can identify

• reactant and product structures

transition state

• local minima on PES

• transition state structure

• saddle point connecting local minima

2-D potential energy surface
13
2-D Example Ozone Isomerization

dO1-O2
O1
O1
O1
O2
O2
O2
O2
O2
O2
aO2-O1-O2
ozone
iso-ozone
can identify

• reactant and product structures

reaction path

• local minima on PES

• transition state structure

• saddle point connecting local minima

• reaction pathway

• lowest energy pathway connecting local minima

2-D potential energy surface
14
2-D Example Ozone Isomerization

dO1-O2
O1
O1
O1
O2
O2
O2
O2
O2
O2
aO2-O1-O2
ozone
iso-ozone
reaction coordinate

• lowest energy path on full PES connecting
  reactant, product, and transition state

• often plotted to aid in interpreting data

useful for identifying

• reaction energies

• reaction barriers

• reactant, product and transition state geometries

15
What About Higher-Dimensional Cases?
virtually all chemical systems have more than 2
degrees of freedom

• in general, a molecule has 3N-6 degrees of
  freedom (3N-5 for linear molecules) ? PES is a
  hypersurface

Problem we cannot plot a surface with more than
2 degrees of freedom
Solution treat hypersurface as mathematical
entity

• employ mathematical definitions to identify
  relevant points on the PES

• better than manually scanning the PES

• discreteness of points in a PES scan means we may
  miss minima

• mathematical search for minima and transition
  states usually involves fewer calculations than a
  scan (particularly for large systems)

So, what are we actually interested in doing?

• want to identify minima and saddle-points
  (transition states) on the PES

• mathematically, these are stationary points

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Stationary Points
reactants, products and transition states are
stationary points

• mathematically, a stationary point is defined by

along all directions, qi
There are different kinds of stationary points

• reactants and products are local minima

• transition states are saddle points

• minimum in all directions

• maximum in only one direction, minima in all
  others

• lowest energy minimum on the entire PES is called
  the global minimum

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2-D Example Ozone Isomerization

dO1-O2
O1
O1
O1
O2
O2
O2
O2
O2
O2
aO2-O1-O2
ozone
iso-ozone
can identify

• reactant and product structures

reaction path

• local minima on PES

• transition state structure

• saddle point connecting local minima

• reaction pathway

• lowest energy pathway connecting local minima

2-D potential energy surface
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Stationary Points
Local minimum (reactant or product)

• for all qi

• for all qi

• stationary point

• positive curvature in all directions

Saddle Point (transition state)

• for all qi

• for exactly one qi

• stationary point

• negative curvature in one direction

• maximum along that direction

• for all other qi

• positive curvature in all other directions

• minimum along all other directions

19
Geometry Optimization

• process for locating a minimum or transition
  state on the PES

• also called relaxation (at least for finding
  minima)

• most common type of calculation performed in
  computational chemistry

Why?

• need geometries/energies of minima and transition
  states to

• determine reaction mechanisms and associated
  energetics

• determine preferred molecular geometries for
  calculations of other properties

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

• lets us find minima/transition states without any
  exact information regarding those points on the
  PES

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Newton-Raphson Method

• basis for geometry optimization procedures used
  in most computational chemistry codes

consider a 1-D harmonic potential
energy, E
bond length, q
minimum (q0, E0)
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Newton-Raphson Method

• in general, PES is not harmonic ? cannot find q0
  in 1 step

• 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

• NOTE doesnt just follow reaction coordinate,
  but optimizes over whole PES

initial guess structure (qi)
energy, E
. . .
local minimum 1
local minimum 2
reaction coordinate, q
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Newton-Raphson Method
in general, optimization is done using 3N
cartesian coordinates
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Newton-Raphson Method
to do an optimization, we need
qi

• column vector of 3N cartesian coordinates
  defining the molecular geometry

• initial guess structure should look like
  anticipated minimum or transition state

• can build with molecular editors

gi

• column vector of 3N derivatives of the energy wrt
  cartesian coordinates

• represent the forces on the nuclei

• 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

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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
6. update Hessian
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 lt 4.5 x 10-4 Hartree/bohr
- RMS force lt 3.0 x 10-4 Hartree/bohr
- other geometry-based criteria
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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
(modify H for TS)
6. update Hessian
7. repeat 3 through 6 until convergence criteria
are met
Initial guess

• finding a saddle point is difficult ? need good
  guess structure, accurate Hessian

- preliminary scan along reaction coordinate

• guess structure by

- interpolate between reactant and product
structures
- chemical intuition
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Characterizing Stationary Points

• convergence criteria are based only on derivatives

• definitions of minima and transition states
  involve 2nd derivatives

minimum
for all qi
transition state
for exactly one qi
for all other qi
need 2nd derivatives to characterize
minima/transition states
? frequency calculation and normal mode analysis
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Normal Mode Calculation
O
normal modes
bend
H
H

• collective vibrational motions of atoms

• simplest vibrational motions in a molecule

symmetric stretch
O
H
H

• probed in IR/Raman experiments

• 3N-6 modes/molecule (3N-5 if linear)

asymmetric stretch
O
H
H
frequencies in harmonic approximation
PES is approximately harmonic at stationary points
(units of cm-1)
k force constant
? reduced mass of mode
c speed of light
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Normal Mode Calculation
normal modes are eigenvectors of the Hessian, H,
and force constants are the eigenvalues
diagonalization

• square matrix, A, can be broken down into the
  product of three square matrices

• P square matrix, columns are eigenvectors

• D diagonal matrix, diagonal elements are
  eigenvalues

for normal modes
P
H
K
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Normal Mode Calculation

• columns are nuclear displacements associated with
  the normal mode

• first index designates direction, second
  indicates mode

• modes 1 6 are translations and rotations, 7
  3N are vibrations

• diagonal elements are force constants, k

• k1 k6 correspond to translations/rotations, k7
  k3N are vibrations

• mass-weighting gives the vibrational frequencies

• P and K are useful for

1. characterizing stationary points
2. visualizing normal modes
3. calculating zero-point vibrational energies
4. simulating IR/Raman spectra (a future lecture)
5. finite temperature corrections (a future
lecture)
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Characterizing Stationary Points
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
transition state
for exactly one qi

• one negative element in H ? one diagonal element
  of K is negative

? if k is negative, the frequency is imaginary
transition states are characterized as having
exactly one imaginary frequency
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Characterizing Stationary Points
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 185.6i cm-1
v 2449.6i cm-1
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Characterizing Transition States
How do you know you have the correct transition
state?

• Look at the structure. Does it look right?

• pretty weak confirmation method, but helpful
  sometimes

• The energy of the transition state must be higher
  than the reactant and product.

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Characterizing Transition States
How do you know you have the correct transition
state?

• Look at the structure. Does it look right?

• pretty weak confirmation method, but helpful
  sometimes

• The energy of the transition state must be higher
  than the reactant and product.

• 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?

• 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.

34
Relative Energies
once you identify the reactant, product, and
transition state

• energies can be calculated relative to reactant

energy (kcal/mol)

• sets reactant energy to 0

• all energies must be calculated at the same level
  of theory!!!

reactants
TS
product
For reactions with multiple reactants or products
e.g. A B ?TS ? C D

• 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

35
Zero-Point Vibrational Energies
Quantum mechanics

• molecules vibrate even at 0K

• Heisenberg uncertainty principle says we cant
  know simultaneously the position and velocity of
  a particle

• if atoms stopped vibrating they would have no
  velocities and well-defined positions

36
Zero-Point Vibrational Energies
zero-point vibrational energy has small effect on
relative energies
without ZPVE
with ZPVE
88.0

• nearly cancels out for reaction energies

84.3

• effect is often more pronounced for barriers

energy (kcal/mol)

• transition states often have many weak vibrations
  with low frequencies

14.6
14.5

• ZPVE correction is smaller than for reactants and
  products

0.0

• often included in reaction energies and barriers
  if frequencies are available

37
Thermal Corrections
energy calculation methods provide the potential
energy of the system at 0K
Meanwhile, experiments

• are done at finite temperatures and pressure

• 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 an macroscopic
  ensemble

• statistical mechanics will let us determine how
  thermal energy is distributed in the molecule

• provides access to enthalpies, entropies, and
  free energies

38
Partition Functions and Thermodynamics
Partition function, Q

• describes distribution of energy in an ensemble
  of molecules

• in the canonical ensemble (constant N, V, T)

Ei energy of state i
kB Boltzmanns constant 1.38 x 10-23 J/K
Connection to Thermodynamics
added as corrections to the electronic potential
energy
39
Simplification of the Partition Function
assume ideal gas behaviour ? molecules do not
interact

• Ei sum of energies of molecules in state i

?1i energy of molecule 1 in state i

• Q can be broken down into a product of partition
  functions for each molecule

gk degeneracy of state k
40
Simplification of the Partition Function
For 1 molecule
decompose ?k into main contributions
substitute q(V,T) into expression for Q and then
calculate internal energy and entropy
41
Electronic Partition Function

• closed shell molecules ? gi 1

• excited states are inaccessible at normal
  temperatures ? only consider ground state

• set ground state energy as 0

multiplicity
Rideal gas constant assumes NNA
42
Translational Partition Function
for an ideal gas
(for a mole of particles)
putting qtrans into Q yields
(recall equipartition principle)
43
Rotational Partition Function
assume molecule is a rigid rotator ? no
rotational-vibrational coupling
IA rotational moment of inertia ? depends on
geometry and atomic masses
? factor accounting for rotational symmetry
putting qrot into Q yields
(recall equipartition principle)
44
Vibrational Partition Function
assume normal modes behave as quantum mechanical
harmonic oscillators
j(i) vibrational state j of normal mode i
For one mode

• assumes Evib0 when k0

For 3N-6 modes
45
Vibrational Partition Function
assume normal modes behave as quantum mechanical
harmonic oscillators
46
Partition Functions and Thermodynamics
within ideal gas approximation
frequencies depend on structure of the system
frequency calculations are needed to calculate
the internal energy at a given temperature
completely independent of the system
enthalpy
PV RT for ideal gas
47
Partition Functions and Thermodynamics
within ideal gas approximation

• Selec, Strans, Srot can be calculated directly
  from atomic structure

• Svib requires a frequency calculation

free energy

• can calculate G from definitions of U, H, and S

• must perform frequency calculation to employ
  these definitions

48
Relative Enthalpies/Free Energies
once you identify the reactant, product, and
transition state

• enthalpies/free energies can be calculated
  relative to reactant

energy (kcal/mol)

• sets reactant energy to 0

• all energies must be calculated at the same level
  of theory!!!

reactants
TS
product
For reactions with multiple reactants or products
e.g. A B ?TS ? C D

• 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

49
Summary of PES-related Concepts

• Potential energy surface (PES)

-describes relationship between molecular
structure and energy
-PES evaluated with energy calculation methods
(future lectures)

• Stationary Points

-points on the PES where
-reactants and products are local minima (all
positive 2nd derivatives)
-transition states are saddle-points (exactly one
negative 2nd derivative)

• Geometry Optimization

-procedure for identifying stationary points on
the PES

• 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.
50
How to Explore the PES with Gaussian/Gaussview
Gaussian (www.gaussian.com)

• computational chemistry software package

• performs molecular mechanics, ab initio, density
  functional theory, and semi-empirical molecular
  orbital calculations

• calculates a wide range of properties

• performs geometry optimizations and frequency
  calculations

Gaussview (www.gaussian.com)

• graphical user interface for Gaussian

• can build molecules, set-up input files, submit
  Gaussian calculations, and visualize results

Gaussian03 and Gaussview are available on
department computer cluster (Rm. 100)
Gaussian03 and Gaussview can be installed on
other department owned computers (ask me)
51
Gaussian Input File
the Gaussian input file has the following form
(http//www.gaussian.com/g_ur/m_input.htm)
1. Link 0 Commands -set up memory limits, etc.
Line starts with . Optional.
2. Route Section
-specifies the details of the calculation
-can be multiple lines with max. 80 characters
-each line in Route Section must start with
3. Blank Line
-tells program Route Section is done
4. Title
5. Blank Line
-tells program Title is done
6. Charge and Multiplicity
7. Molecular Geometry
-provide the atomic coordinates
-Cartesian or Z-matrix format
8. Blank Line
-tells program the input file is done
52
Gaussian Input File
example input for water
link 0 commands
route line
charge and multiplicity
geometry in cartesian coordinates
53
Geometry Specification
2 commons ways to specify the molecular geometry
1. Cartesian coordinates

• atomic symbol, x, y, z coordinates of each nucleus

• Gaussian expects values in Angstroms

• convenient because most molecular building
  programs will output Cartesian coordinates

2. Z-matrix coordinates

• also called internal coordinates

• specify positions of atoms relative to one
  another using bond lengths, angles and dihedral
  angles (3N-6 variables)

• one section specifies connectivity, second
  section specifies values of variables
  corresponding to bond lengths, etc.

• Gaussian expects values in Angstroms and degrees

• convenient for PES scans because bonds and angles
  are defined explicitly

54
Z-matrix Input
connectivity specification
C
C 1 B1
H 1 B2 2 A1
H 1 B3 2 A2 3 D1
H 2 B4 1 A3 3 D2
H 2 B5 1 A4 5 D3

• 1st column specifies atom type

• 2nd column defines a bond, e.g. the 1 in line 2
  indicates that atom 2 is bonded to atom 1

• 3rd column gives the label of a variable
  corresponding to the bond length

• 4th column defines an angle, e.g. the 2 in line
  3 indicates that the 3rd atom forms a 3-1-2
  (H3-C1-C2) angle

• 5th line gives the label of a variable containing
  the value of the dihedral angle

• 6th line defines a dihedral angle, e.g. the 3
  in line 4 indicates that the 4th atom forms a
  dihedral 4-1-2-3 (H4-C1-C2-H3) dihedral angle

• 7th line gives the label of a variable containing
  the value of the dihedral angle

55
Z-matrix Input
connectivity specification
C
C 1 B1
H 1 B2 2 A1
A3
H 1 B3 2 A2 3 D1
B4
H 2 B4 1 A3 3 D2
H 2 B5 1 A4 5 D3
D2
example

• line 5 means a hydrogen atom is bonded atom 2
  with a bond distance of B4, forms an angle with
  atoms 2 and 1 with a value of A3, and forms a
  dihedral angle with atoms 2, 1, and 3 with a
  value of D2

56
Z-matrix Input
connectivity specification
C
C 1 B1
H 1 B2 2 A1
A3
H 1 B3 2 A2 3 D1
B4
H 2 B4 1 A3 3 D2
H 2 B5 1 A4 5 D3
D2
variables
B11.5
B21.1

• can simplify by taking advantage of symmetry

B31.1

• expect C-H bonds to be same lengths

B41.1

• use variable B2 for all C-H bonds

B51.1
A1120.0

• expect H-C-C angles to be the same

A2120.0

• use variable A1 for all H-C-C angles

A3120.0
A4120.0
D10.0
D20.0
D3180.0
57
Z-matrix Input
connectivity specification
C
C 1 B1
H 1 B2 2 A1
A3
H 1 B2 2 A1 3 D1
B4
H 2 B2 1 A1 3 D2
H 2 B2 1 A1 5 D3
D2
variables
B11.5
B21.1

• can simplify by taking advantage of symmetry

A1120.0

• expect C-H bonds to be same lengths

D10.0

• use variable B2 for all C-H bonds

D20.0
D3180.0

• expect H-C-C angles to be the same

• use variable A1 for all H-C-C angles

• careful, though

• assigning the same label to two or more geometric
  variables means they have to remain equal
  throughout entire calculation

58
Route Line

• specifies type of calculation that is to be
  performed

• line starts with , can only be 80 characters
  in length

• can have multiple lines

• line contains method, basis set and keywords with
  options in parentheses

method/basis_set keyword1(options)
keyword2(options)
keyword3(options) keyword4(options)

• must be followed by a blank line

• full list of keywords and options available at

http//www.gaussian.com/g_ur/keywords.htm

• relevant keywords for exploring the PES

• scan ? perform a scan along predefined coordinates

• opt ? perform a geometry optimization to a
  minimum or transition state

• freq ? perform a frequency/normal mode calculation

59
Output

• gaussian output files will usually end with .log
  or .out

• contains a lot of information ? contents depend
  on type of calculation

• units are usually Hartree for energy and Angstrom
  for distance (but not always)

1 Hartree 627.51 kcal/mol
1 Angstrom 1.0 x 10-10 m
Things to look for in the output

• molecular structure ? look for a line saying
  Input orientation

• molecular energy ? look for a line saying SCF
  Done

• convergence in optimization ? look for a line
  saying Maximum Force

• summary of a rigid scan ? look for a line saying
  Summary of the potential surface scan

• summary of a relaxed scan ? look for a line
  saying Summary of Optimized Potential Surface
  Scan

• frequency information ? look for a line saying
  Harmonic frequencies

60
Example Calculations
1. Single point calculation of ethane
2. Rigid scan of the C-C bond in ethane
3. Geometry optimization of (CH3)2CO
4. Transition state search for (CH3)2CO ?
CH3C(OH)CH2
5. Relaxed scan of the O-H bond for (CH3)2CO ?
CH3C(OH)CH2
6. Frequency calculation of (CH3)2CO
61
Single Point Calculation
Single Point Calculation

• calculate the energy for a specific geometry

• provides 1 point on the potential energy surface

• the geometry is not updated or changed

• simplest, yet most fundamental, type of
  calculation

This example

• calculate the energy of ethane

• geometry provided in Z-matrix format

• calculation performed at hf/3-21G level of theory
  (more on this in future lectures)

62
Single Point Calculation - Input

• route line specifies method used in calculation

• coordinates of ethane in Z-matrix format

63
Single Point Calculation - Output
in a single point calculation we may want to
determine

• the energy of the given structure

• other properties of the system

• well look into those more in future lectures

64
Rigid Scan
Rigid Scan Calculation

• calculate the energies for a series of structures

• structures are based on predetermined changes to
  an initial structure, e.g. varying a bond length
  or changing an angle

• all non-scanned geometric variables are fixed at
  their original values

• provides a series of points on the potential
  energy surface

This example

• perform a rigid scan by increasing the C-C bond
  length in ethane

• geometry provided in Z-matrix format ? required
  by gaussian to do a scan

• calculation performed at hf/3-21G level of theory
  (more on this in future lectures)

65
Rigid Scan - Input

• keyword scan request a rigid scan of the
  selected variables

• hf/3-21G specifies level of theory for the
  calculation

• nosymm tells the program to set the initial
  symmetry to C1

• scans often change the symmetry of the system,
  causing the calculation to fail

• B4 1.000000 60 0.1 tells the program to
  increase the value of variable B4 from an initial
  value of 1.0 Ang in 60 steps of 0.1 Ang

• this will result in 61 single point calculations
  with B4 ranging from 1.0 to 7.0 Ang in increments
  of 0.1 Ang

• all other variables will remain fixed at their
  original values

• more than one variable can be selected for
  scanning in a given input

• if two or more variables are scanned, the energy
  will be calculated for all possible combinations
  of the scanned variables

66
Rigid Scan - Output
in a rigid scan calculation we may want to
determine

• a series of energies as a function of the changed
  geometric variable

• value of the coordinate that was scanned

• energy in Hartree at each step of the scan

Note if multiple coordinates were scanned the
summary will contain multiple columns for those
coordinates
. . .
. . .
. . .
67
Geometry Optimization
Geometry Optimization

• minimize the energy of a molecule by iteratively
  modifying its structure

• provides the energetically-preferred structure of
  a molecule

• the located structure will correspond to the
  local minimum nearest on the potential energy
  surface to the input structure

• suitable for determining the structures and
  energies of reactants and products

This example

• optimize the geometry of (CH3)2CO starting from a
  structure built with gaussview using standard
  bond lengths and angles

• geometry provided in Z-matrix format

• calculation performed at hf/3-21G level of theory
  (more on this in future lectures)

68
Geometry Optimization Input

• keyword opt requests a geometry optimization to
  a minimum energy structure

• hf/3-21G specifies level of theory for the
  calculation

• nosymm tells the program to set the initial
  symmetry to C1

• geometry optimizations sometimes change the
  symmetry of the system, causing the calculation
  to fail

• the minimum energy structure may not have the
  same symmetry as the initial structure

69
Geometry Optimization Output
in a geometry optimization we may want to
determine

• a stationary point corresponding to a minimum
  energy structure

? need to monitor whether the convergence
criteria are met
first step
intermediate step
final step
70
Geometry Optimization Output
in a geometry optimization we may want to
determine

• a stationary point corresponding to a minimum
  energy structure

? need to monitor whether the convergence
criteria are met

• energy of the optimized structure

? energy statement in step where convergence
criteria are met
? last energy statement in the output file
71
Geometry Optimization Output
in a geometry optimization we may want to
determine

• a stationary point corresponding to a minimum
  energy structure

? need to monitor whether the convergence
criteria are met

• energy of the optimized structure

? energy statement in step where convergence
criteria are met
? last energy statement in the output file

• geometry of the optimized structure

? follows statement where convergence criteria
are met
72
Transition State Optimization
Transition State Optimization

• iteratively modifying a molecular structure to
  arrive at a transition state

• the located structure should correspond to a
  saddle-point on the potential energy surface

• input structure must be reasonably close to the
  transition state

• require an accurate Hessian

• suitable for determining the structures and
  energies of transition states

This example

• find the transition state for the reaction
  (CH3)2CO ? CH3C(OH)CH2

• geometry provided in Z-matrix format

• calculation performed at hf/3-21G level of theory
  (more on this in future lectures)

73
Transition State Optimization Input

• keyword opt requests a geometry optimization to
  a minimum energy structure

• option ts requests a transition state
  optimization

• option calcfc requests that the Hessian is
  calculated analytically at the first optimization
  step

• option noeigen requests that the Hessian is not
  tested throughout the calculation. If testing is
  permitted, the calculation often fails because at
  some steps the Hessian may not have the correct
  number of imaginary frequencies.

• hf/3-21G specifies level of theory for the
  calculation

• nosymm tells the program to set the initial
  symmetry to C1

• geometry optimizations sometimes change the
  symmetry of the system, causing the calculation
  to fail

• the minimum energy structure may not have the
  same symmetry as the initial structure

74
Transition State Optimization Output

• the output is the same as for a geometry
  optimization

75
Relaxed Scan
Relaxed Scan Calculation

• calculate the energies for a series of structures

• structures are based on predetermined changes to
  an initial structure, e.g. varying a bond length
  or changing an angle

• all non-scanned geometric variables are
  optimized, while scanned variables are held fixed

• provides a series of points on the potential
  energy surface

• useful for generating guess structures of
  transition states

This example

• perform a relaxed scan of the O-H bond for
  (CH3)2CO ? CH3C(OH)CH2

• geometry provided in Z-matrix format ? required
  by gaussian to do a scan

• calculation performed at hf/3-21G level of theory
  (more on this in future lectures)

76
Relaxed Scan - Input

• keyword opt requests a geometry optimization to
  a minimum energy structure

• option z-matrix requests that the optimization
  is performed using z-matrix coordinates

• hf/3-21G specifies level of theory for the
  calculation

• nosymm tells the program to set the initial
  symmetry to C1

• geometry optimizations sometimes change the
  symmetry of the system, causing the calculation
  to fail

• the minimum energy structure may not have the
  same symmetry as the initial structure

• B7 2.63480418 S 8 -0.2 tells the program to
  decrease the value of variable B7 from its
  initial value in 8 steps of 0.2 Ang

• this will result in 9 calculations where the
  geometry is optimized except for bond length B7,
  which is held at the selected value

• all other variables will remain fixed at their
  original values

• more than one variable can be selected for
  scanning in a given input

• if two or more variables are scanned, the energy
  will be calculated for all possible combinations
  of the scanned variables

77
Relaxed Scan - Output
in a relaxed scan calculation we may want to
determine

• a series of energies and optimized structures as
  a function of the changed geometric variable

step number in scan
energy at each step in Hartree
optimized z-matrix coordinates at each step
78
Frequency Calculation
Frequency Calculation

• calculate the normal modes and associated
  vibrational frequencies for the input structure

• used to characterize stationary points as minima
  or transition states

• used to calculate zero-point vibrational energies

• used to calculate thermal corrections to the
  potential energy

• used to simulate IR/Raman spectra (future lecture)

This example

• perform a frequency calculation of (CH3)2CO

• geometry provided in Z-matrix format ? geometry
  obtained through a previous optimization

• calculation performed at hf/3-21G level of theory
  (more on this in future lectures)

79
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

80
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
81
Frequency Calculation - Output
in a frequency calculation we may want to
determine

• normal modes and vibrational frequencies

• zero-point vibrational energies

• thermal corrections to the potential energy

82
Next Time

• overview of gaussian/gaussview

• gaussian/gaussview tutorial