CHEM 834: Computational Chemistry

1
CHEM 834 Computational Chemistry
Parameterized Methods and Properties
April 7, 2009
2
Topics
last time

• properties

• more properties

today
Exam remined
Room 202 Chernoff Hall, 2-4 pm, April 8, 2009
3
UV/Vis Spectra
UV/Vis probes the electronic excitations of
molecules
Excited state

• weve already discussed how to get wavefunctions
  for excited electronic states

• standard methods include CIS, TD-DFT, and ZINDO

electronic energy
Ground state
generic coordinate
4
UV/Vis Spectra
UV/Vis probes the electronic excitations of
molecules
Excited state

• each state has its own potential energy surface

• if the system remains in an excited state long
  enough it will optimize its geometry

electronic energy
Ground state

• so, there are several relevant energy differences

generic coordinate
5
UV/Vis Spectra
UV/Vis probes the electronic excitations of
molecules
Excited state
Vertical absorption

• an excitation occurs while the geometry of the
  system fixed at the minimum energy ground state
  geometry

electronic energy
Ground state

• this is what Gaussian calculates by default

?Evert, absorption
generic coordinate
6
UV/Vis Spectra
UV/Vis probes the electronic excitations of
molecules
Excited state
Vertical emission

• an excitation occurs while the system remains
  fixed at the minimum energy geometry on the
  excited state PES

• in Gaussian

electronic energy
Ground state

• optimize the geometry on the excited state
  potential energy surface

?Evert, emission

• keep the geometry fixed and calculated the
  vertical excitation from the ground state to the
  excited state

• the results will be -?Evert,emission

generic coordinate
7
UV/Vis Spectra
UV/Vis probes the electronic excitations of
molecules
Excited state
?Eadiabatic

• the difference in energy between the optimized
  geometries on the ground state and excited state
  PESs

• in Gaussian, you have to do geometry
  optimizations on each surface

electronic energy
Ground state
?Eadiabatic
generic coordinate
8
UV/Vis Spectra
weve already looked at how to calculate excited
states with Gaussian/Gaussview
to do an excited state calculation
change Ground State to an excited state method
9
UV/Vis Spectra
weve already looked at how to calculate excited
states with Gaussian/Gaussview
to do an excited state calculation
pick number of excited states to solve
pick state of interest

• Gaussian will print out detailed analysis of this
  state

• geometry optimization will be performed on this
  state (you also have to specify opt on the route
  line)

10
UV/Vis Spectra
the output can be used to construct UV/Vis spectra
intensity (? f)
wavelength
11
UV/Vis Spectra
the output can be used to construct UV/Vis spectra
General comments

• you will not get peak widths

• excited state energies can be very inaccurate

• intensity are usually qualitatively correct

• trends and examining molecular orbitals involved
  in the transitions are most relevant when
  comparing with experiment

intensity (? f)
wavelength
12
NMR
NMR calculations involve determining the energy
difference between a system in the presence and
absence of an magnetic field
two magnetic fields matter
1. external field
2. internal field of nucleus
magnetic fields perturb the kinetic energy

• motion of electrons generates electronic magnetic
  moments

• evaluating these integrals gives chemical shifts
  and spin-spin couplings

• these integrals are very difficult to treat
  computationally

• simulation lags behind experiment a lot in terms
  of simulating NMR

13
NMR
to evaluate the NMR integrals, we have to define
an origin for the magnetic field called a gauge
origin

• defining an origin for the magnetic field means
  that different molecular orbitals will be
  different distances away from the origin

• there are two standard ways to overcome this issue

• Gauge Invariant Atomic Orbitals (GIAO)

• the NMR calculation is performed using a set of
  atomic orbitals that are dependent on the gauge
  origin

• a standard basis set is used for the wavefunction
  optimization and it is projected onto the GIAOs
  for the NMR calculation

• Individual Gauge for Localized Orbitals (IGLO)

• each molecular orbital gets its own gauge origin

14
NMR
general comments on NMR calculations

• GIAO method is generally more robust than IGLO

• huge basis sets are needed ? at least triple-?
  with lots of polarization and diffuse functions

• ECPs should not be used

• NMR chemical shifts are really affected by subtle
  interactions between the valence and core
  electrons

• replacing core electrons with an ECP, limits the
  ability to describe those interactions

• relativistic effects can be important for heavier
  elements ? you may have to use a relativistic
  Hamiltonian

• calculations give absolute shifts, but in
  experiments the shifts are given relative to a
  reference compound. You will have to do
  additional calculations on the reference compound
  to compare with experiment.

• simulating proton NMR is difficult to do with
  high accuracy, other elements with wider ranges
  of chemical shifts exhibit better accuracy

• spin-spin coupling calculations are not yet
  routine

15
NMR in Gaussian/Gaussview
Gaussian can perform NMR calculations with
several different methods

• specify nmr method on the route line

• methods include giao, cgst, igaim

• you still have to specify a quantum chemical
  method and basis set

• methods that can be used include

• you still have to specify a quantum chemical
  method and basis set

• you should do a geometry optimization first

Example input
16
NMR in Gaussian/Gaussview
Gaussian can perform NMR calculations with
several different methods
Example output
17
NMR in Gaussian/Gaussview
you can set up NMR calculations through Gaussview
select NMR from the Job Type menu
18
NMR in Gaussian/Gaussview
you can set up NMR calculations through Gaussview
you still have to select a method and basis set
select the NMR method
19
NMR in Gaussian/Gaussview
you can plot the NMR spectrum from Results ? NMR
benzaldehyde
13C NMR
reference molecule
20
NMR in Gaussian/Gaussview
you can plot the NMR spectrum from Results ? NMR
benzaldehyde
1H NMR
reference molecule
21
Thermodynamic Quantities
reactions are governed by changes in free
energies, enthalpies, etc.
from experiments, we evaluate changes in enthalpy
with heats of formation
heats of formation relative to a standard state
standard states are usually the most forms of the
constituent atoms of the molecules involved in
the reaction at 273K and 1 atm
22
Thermodynamic Quantities
reactions are governed by changes in free
energies, enthalpies, etc.
in quantum chemical calculations we use relative
electronic energies
what is the standard state?
the energy we get in a quantum chemical
calculation is relative to having all the
electrons and nuclei infinitely separated
23
Thermodynamic Quantities
reactions are governed by changes in free
energies, enthalpies, etc.
energy changes in experiments are usually
measured for large numbers of molecules
in the first lecture, we looked at a statistical
mechanical treatment of molecules
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
24
Thermodynamic Quantities
reactions are governed by changes in free
energies, enthalpies, etc.
energy changes in experiments are usually
measured for large numbers of molecules
in the first lecture, we looked at a statistical
mechanical treatment of molecules
Thermodynamic quantities

• frequency calculations give the partition
  function and associated thermodynamic quantities

• how do we connect them to chemical phenomena?

25
Equilibrium Constants
if we have an ensemble of molecules at
equilibrium, some will be found in each potential
energy minimum
from 1nd year thermodynamics
energy
A
B
reaction coordinate
for multiple minima
B
energy
A
C
reaction coordinate
26
Rate Constants
we can also use thermodynamic quantities to
estimate rate constants with transition state
theory
assumptions

• there exists an activated complex at the
  transition state

A
A
B

• the reactants are activated complex are in
  equilibrium

A

• progress from the activated complex to the
  products is irreversible

energy
A
B
reaction coordinate
27
Rate Constants
we can also use thermodynamic quantities to
estimate rate constants with transition state
theory
overall
kact

• the rate is determined by the conversion of the
  activated complex into B

k
A
A
B
kdeact
A
energy
A
B
reaction coordinate
equilibrium constant between A and A
28
Rate Constants
we can also use thermodynamic quantities to
estimate rate constants with transition state
theory
from thermodynamics, we know
29
Rate Constants
we can also use thermodynamic quantities to
estimate rate constants with transition state
theory
going back to our rate expression
recall, the partition function describes how
energy is distributed among the various degrees
of freedom
For the reactant
3 translational degrees of freedom
3 rotational degrees of freedom
3N-6 vibrational degrees of freedom
30
Rate Constants
we can also use thermodynamic quantities to
estimate rate constants with transition state
theory
going back to our rate expression
recall, the partition function describes how
energy is distributed among the various degrees
of freedom
For the transition state
3 translational degrees of freedom
3 rotational degrees of freedom
3N-7 real vibrations
1 imaginary vibration
31
Rate Constants
we can also use thermodynamic quantities to
estimate rate constants with transition state
theory
for the transition state
the partition function for the imaginary
vibration can be approximated as
imaginary frequency
32
Rate Constants
we can also use thermodynamic quantities to
estimate rate constants with transition state
theory
so, the rate constant becomes
but
so
33
Rate Constants
we can also use thermodynamic quantities to
estimate rate constants with transition state
theory
consider the meaning of the imaginary frequency

• the imaginary frequency corresponds to motion
  over the barrier along the reaction coordinate

?

• the frequency of the vibration determines how
  fast reactants are converted to products

energy

• so, we set ? equal to k

reaction coordinate
34
Rate Constants
we can also use thermodynamic quantities to
estimate rate constants with transition state
theory
yields
35
Rate Constants
we can also use thermodynamic quantities to
estimate rate constants with transition state
theory
putting it all together gives
36
Rate Constants
we can also use thermodynamic quantities to
estimate rate constants with transition state
theory

• if you know the free energy barrier for a
  reaction you can estimate the rate constant

• estimates will generally be too high (even with
  an exact barrier)

• other approaches exist, such as harmonic
  transition state theory and variational
  transition state theory

37
Beyond the Ideal Gas
frequency calculations determine thermodynamic
quantities within the ideal gas approximation

• the ideal gas approximation allows the easy
  calculation of the partition function

• however, real molecules under real experimental
  conditions rarely behave like ideal gases

• so, how do we measure thermodynamic properties of
  realistic systems

in experiments, we measure average properties of
a chemical system
example

• the temperature of a liquid is a measure of the
  average kinetic energy of the atoms in the liquid

38
Beyond the Ideal Gas
we can get more realistic thermodynamic
quantities by taking averages
2 kinds of averages

• ensemble average

• the average of a property in a large collection
  of molecules at a given instant in time

• time average

• the average of a property in a single molecule
  averaged over a long period of time

how do we calculate these averages?
Monte Carlo calculations ? give ensemble averages
molecular dynamics ? gives time averages
39
Monte Carlo Calculations
evaluate the properties of a system of molecules
in random configurations (states)
Consider a two state system
state 1 C-C bond length is 1.5 Å
state 2 C-C bond length is 1.3 Å
there is a 60 chance a molecule is in state 1
and a 40 chance a molecule is in state 2
If there are N molecules
fraction of molecules in states 1 and 2
the average C-C bond length is
40
Monte Carlo Calculations
evaluate the properties of a system of molecules
in random configurations (states)
From 1st year thermodynamics
so, we can get an ensemble average from
we have to sum over several different states in
which the system can exist

• different configurations of molecules

• different molecular geometries

41
Monte Carlo Calculations
evaluate the properties of a system of molecules
in random configurations (states)
Monte Carlo calculations
random change
state 2
state N
state 1
after sampling lots of states, you can
accurately evaluate
42
Monte Carlo Calculations
evaluate the properties of a system of molecules
in random configurations (states)
Monte Carlo calculations
after sampling lots of states, you can
accurately evaluate

• techniques exist to sample the most relevant
  states

• this gives an ensemble average

• there is no notion of time

• new states are selected randomly, not based on
  how the system should move in reality

• these methods are not used to study reactions,
  and are usually performed with force-fields

43
Molecular dynamics
evaluate the time average by simulating how a
molecule changes over time
nuclei treated as classical particles
potential energy of the system
force on atom I
position of atom I
mass of atom I
acceleration of atom I
44
Molecular dynamics
evaluate the time average by simulating how a
molecule changes over time
nuclei treated as classical particles
positions and velocities are determined with the
velocity Verlet algorithm
45
Molecular dynamics
evaluate the time average by simulating how a
molecule changes over time
in general

• ?t must be short enough to capture the fastest
  vibrations in a molecule

• fastest vibrations are X-H bonds 1014 Hz

• so, ?t 1.0 fs ? 10-15 seconds

• 1 million energy and force evaluations are needed
  to simulate a nanosecond (compare with the 20
  energy and force evaluations needed to do a
  geometry optimization)

46
Molecular dynamics
what a molecular dynamics simulation looks like
molecular dynamics simulation of 1,5 hexadiene
47
Molecular dynamics
evaluate the time average by simulating how a
molecule changes over time
but, molecular dynamics is used more often to
study a systems behaviour

• the kinetics from a molecular dynamics simulation
  are meaningful

• changes in the system are based on actual
  physical laws, not random changes as in Monte
  Carlo

• we have a real notion of time in these simulations

• we can study the system under particular
  conditions

e.g. controlling the velocities, lets us
consider specific temperatures
48
Molecular dynamics
how do we get the forces on the nuclei
force-fields
spring potential

• treat bonds like springs

energy

• U ? ? for large RAB

ab initio potential

• bonds can never break

ab initio methods

• energy is calculated with quantum chemical methods

RAB
RAB

• can describe changes in bonding, e.g. reactions

49
Molecular dynamics
how do we get the forces on the nuclei
1. Force-fields

• treat molecules as atoms connected by springs

• very low computational effort

• can simulate large systems (millions of atoms)

• can simulate long time scales (milliseconds)

• cannot describe chemical reactions

• used to study processes like protein folding, or
  to simulate liquids, etc.

2. Quantum chemistry

• treat electrons with quantum mechanics

• very high computational effort

• can simulate small systems (hundreds of atoms)

• can simulate short time scales (nanoseconds)

• can describe chemical reactions

• used to identify and study chemical reactions

50
Ab initio Molecular dynamics
t-Bu ZDDP
t-Bu ZDDP at 1500K
51
Molecular dynamics
A common simulation strategy
Study reactions quantitatively with DFT
Identify reactions with AIMD
52
Molecular dynamics
molecular dynamics is a way of simulating the
propagation of a molecular system in space and
time
thermodynamics

• molecular dynamics can be used to evaluate time
  averages

• real notion of time allows one to evaluate rate
  constants (if you run the simulation long enough)

• can perform simulations under specific
  thermodynamic conditions

qualitative insight

• these simulations are often called computational
  experiments

• molecular dynamics gives qualitative insight into
  how chemical systems change over time

• ab initio molecular dynamics can be used to
  identify new reactions

in my experience, the qualitative aspects of
molecular dynamics simulations are more useful
than the quantitative results because of the
small systems and short times scales explored in
these simulations