Download Electrophilic Additions to Double Bonds

- Science Honors Program Computer Modeling and Visualization in Chemistry
Molecular Mechanics
&
Quantum Chemistry
Eric Knoll
1
Jiggling and Wiggling

Feynman Lectures on Physics
Certainly no subject or field is making more progress on
so many fronts at the present moment than biology, and
if we were to name the most powerful assumption of all,
which leads one on and on in an attempt to understand
life, it is that all things are made of atoms, and that
everything that living things do can be understood in
terms of the jigglings and wigglings of atoms.
-Feynman, 1963
2
Types of Molecular Models




Wish to model molecular structure, properties
and reactivity
Range from simple qualitative descriptions to
accurate, quantitative results
Costs range from trivial (seconds) to months of
supercomputer time
Some compromises necessary between cost
and accuracy of modeling methods
3
Molecular mechanics
Pros
 Ball and spring description of molecules
 Better representation of equilibrium geometries than plastic models
 Able to compute relative strain energies
 Cheap to compute
 Can be used on very large systems containing 1000’s of atoms
 Lots of empirical parameters that have to be carefully tested and
calibrated
Cons
 Limited to equilibrium geometries
 Does not take electronic interactions into account
 No information on properties or reactivity
 Cannot readily handle reactions involving the making and breaking
of bonds
4
Semi-empirical
molecular orbital methods





Approximate description of valence electrons
Obtained by solving a simplified form of the Schrödinger
equation
Many integrals approximated using empirical
expressions with various parameters
Semi-quantitative description of electronic distribution,
molecular structure, properties and relative energies
Cheaper than ab initio electronic structure methods, but
not as accurate
5
Ab Initio Molecular Orbital Methods
Pros
 More accurate treatment of the electronic distribution using the full
Schrödinger equation
 Can be systematically improved to obtain chemical accuracy
 Does not need to be parameterized or calibrated with respect to
experiment
 Can describe structure, properties, energetics and reactivity
Cons
 Expensive
 Cannot be used with large molecules or systems (> ~300 atoms)
6
Molecular Modeling Software



Many packages available on numerous
platforms
Most have graphical interfaces, so that
molecules can be sketched and results viewed
pictorially
We use Spartan by Wavefunction
 Spartan has
 Molecular Mechanics
 Semi-emperical
 Ab initio
7
Modeling Software, cont’d

Chem3D
 molecular
mechanics and simple semi-empirical
methods
 available on Mac and Windows
 easy, intuitive to use
 most labs already have copies of this, along with
ChemDraw

Maestro suite from Schrödinger
 Molecular
Mechanics: Impact
 Ab initio (quantum mechanics): Jaguar
8
Modeling Software, cont’d

Gaussian 2003
 semi-empirical
and ab initio molecular orbital
calculations
 available on Mac (OS 10), Windows and Unix

GaussView
 graphical
user interface for Gaussian
9
Force Fields
10
Origin of Force Fields
Quantum Mechanics
The underlying physical laws necessary for the
mathematical theory of a large part of physics and the
whole of chemistry are thus completely known, and the
difficulty is only that the exact application of these laws
leads to equations much too complicated to be soluble. It
therefore becomes desirable that approximate practical
methods of applying quantum mechanics should be
developed, which can lead to an explanation of the main
features of complex atomic systems without too much
computation.
-- Dirac, 1929
11
What is a Force Field?

Force field is a collection of parameters for a
potential energy function
U ( x)  mx
F  ma

Parameters might come from fitting against
experimental data or quantum mechanics
calculations
12
Force Fields: Typical Energy Functions
1
U   kr (r  r0 ) 2
bonds 2
1
  k (   0 ) 2
angles 2
Vn
 
[1  cos(n   )]
torsions 2


V (improper torsion)
improper

Angle bending
Torsional rotation
Improper torsion (sp2)
qi q j
elec
 [
LJ
Bond stretches
rij
Aij
rij
12
Electrostatic interaction

Bij
rij
6
]
Lennard-Jones interaction
13
Bonding Terms: bond stretch
Harmonic Potential
Most often Harmonic
Vbond
1
  kr (r  r0 ) 2
bonds2
Vbond

r0
bond length
Morse Potential for
dissociation studies
VMorse 
 a ( r  r0 )
2
D
[
e

1
]
D

bonds
Two new parameters:
D: dissociation energy
a: width of the potential well
Vmorse

Morse Potential
D
r0
bond length
14
Bonding Terms: angle bending
Most often Harmonic
1
Vangle   k (   0 ) 2
angles 2
Vangle

Harmonic Potential
0
angle
15
What do these FF
parameters look like?
16
Atom types (AMBER)
17
Bond Parameters
18
Angle Parameters
19
Applications







Protein structure prediction
Protein folding kinetics and
mechanics
Conformational dynamics
Global optimization
DNA/RNA simulations
Membrane proteins/lipid
layers simulations
NMR or X-ray structure
refinements
20
Molecular Dynamics Simulation Movies

An example of how force fields andm olecular
mechanics are used. Molecular mechanics are
used as the basis for the molecular dynamics
simulations in the below movies.

http://www.ks.uiuc.edu/Gallery/Movies/

http://chem.acad.wabash.edu/~trippm/Lipids/
21
Limitations of MM




MM cannot be used for reactions that break or
make bonds
Limited to equilibrium geometries
Does not take electronic interactions into
account
No information on properties or reactivity
22
- Science Honors Program Computer Modeling and Visualization in Chemistry
Quantum Mechanics
23
MM vs QM





molecular mechanics uses empirical functions for the
interaction of atoms in molecules to calculate energies and
potential energy surfaces
these interactions are due to the behavior of the electrons
and nuclei
electrons are too small and too light to be described by
classical mechanics
electrons need to be described by quantum mechanics
accurate energy and potential energy surfaces for
molecules can be calculated using modern electronic
structure methods
24

h
h

p mv
Quantum Stuff


Photoelectric effect: particle-wave duality of light
de Broglie equation: particle-wave duality of matter
h
h
 
p mv

Heisenberg Uncertainty principle:
Δx Δp ≥ h
25
What is an Atom?
Protons and neutrons make up the heavy, positive core, the
NUCLEUS, which occupies a small volume of the atom.
26
J J Thompson in his plum pudding
model. This consisted of a matrix of
protons in which were embedded
electrons.
Ernest Rutherford (1871 – 1937) used
alpha particles to study the nature of
atomic structure with the following
apparatus:
27

Problem:
Acceleration of
Electron in Classical
Theory

Bohr Model: Circular
Orbits, Angular
Momentum Quantized
28
Photoelectric Effect
Photoelectric Effect: the ejection of electrons from the
surface of a substance by light; the energy of the electrons
depends upon the wavelength of light, not the intensity.
29
DeBroglie: Wave-like properties of matter.




If light is particle (photon)
with wavelength, why not
matter, too?
E=hv  mc2=hv=hc/λ
 λ=h/mc  λ=h/p
DeBroglie Wavelength
30
Wavelengths:




DeBroglie Wavelength λ = h/p = h/(mv)
h = 6.626 x 10-34 kg m2 s-1
What is wavelength of electron moving at
1,000,000 m/s. Mass electron = 9.11 x 10-31 kg.
What is wavelength of baseball (0.17kg) thrown
at 30 m/s?
31
Interpretations of Quantum Mechanics

1. The Realist Position
 The

2. The Orthodox Position
 The

particle really was at point C
particle really was not anywhere
3. The Agnostic Position
 Refuse
to answer
32
Atomic Orbitals – Wave-particle duality.
Traveling waves vs. Standing Waves.
Atomic and Molecular Orbitals are 3-D STANDING WAVES
that have stationary states.
Schrodinger developed this theory in the 1920’s.
Example of 1-D guitar string standing wave.
33
Weird Quantum Effect: Quantum Tunneling

34
Schrödinger Equation
Ĥ  E

H is the quantum mechanical Hamiltonian for the system
(an operator containing derivatives)
E is the energy of the system
 is the wavefunction (contains everything we are
allowed to know about the system)
||2 is the probability distribution of the particles

Schrodinger Equation in 1-D:



d 2

 V ( x) ( x)  E  ( x)
2
2m dx
2
35
Atomic Orbitals:
How do electrons move around the nucleus?
Density of shading represents
the probability of finding an
electron at any point.
The graph shows how probability
varies with distance.
Wavefunctions: ψ
Since electrons are particles that have wavelike properties, we cannot
expect them to behave like point-like objects moving along precise
trajectories.
Erwin Schrödinger: Replace the precise trajectory of particles by a
wavefunction (ψ), a mathematical function that varies with position
Max Born: physical interpretation of wavefunctions. Probability of finding a
particle in a region is proportional to ψ2.
36
s Orbitals
Wavefunctions of s orbitals of
higher energy have more
complicated radial variation
with nodes.
Boundary surface encloses surface with
a > 90% probability of finding electron
37
Â
Schrodinger Eq. is an Eigenvalue problem


Classical-mechanical quantities represented by linear
ˆ ( x)  g ( x)
operators: Af
 Indicates that  operates on f(x) to give a new
function g(x).
Example of operators
38
Â
Schrodinger Eq. is an Eigenvalue problem


Classical-mechanical quantities represented by linear
ˆ ( x)  g ( x)
operators: Af
 Indicates that  operates on f(x) to give a new
function g(x).
Example of operators
39
What is a linear operator?
40
Â
Schrodinger Eq. is an Eigenvalue problem

Schrodinger Equation:
2


d 2
 V ( x)   ( x)  E  ( x)

2
 2m dx

Hˆ  ( x)  E  ( x)
41
Â
Postulates of Quantum Mechanics

The state of a quantum-mechanical system is completely
specified by the wave function ψ that depends upon the
coordinates of the particles in the system. All possible
information about the system can be derived from ψ. ψ
has the important property that
ψ(r)* ψ(r) dr
is the probability that the particle lies in the interval dr,
located at position r.
Because the square of the wave function has a
probabilistic interpretation, it must satisfy the following
condition:

* (r ) (r )dr  1
all space
42
ˆ  n  an  n
A
Postulates of Quantum Mechanics


To every observable in classical mechanics there
corresponds a linear operator in quantum mechanics.
In any measurement of the observable associated with
the operator , the only values that will ever be observed
are the eigenvalues an, which satisfy the eigenvalue
equation:
Aˆ  n  an  n

If a system is in a state described by a normalized wave
function Ψ, then the average value of the observable
corresponding to is given by:
a 

 Aˆ dr
*
all space
43
44
Hamiltonian for a Molecule
ˆ 
H
electrons

i
  2 2 nuclei   2 2 electronsnuclei  e 2 Z A electrons e 2 nuclei e 2 Z A Z B
i  
A   
 

2me
riA
rij
rAB
A 2m A
i
A
i j
A B
(Terms from left to right)
 kinetic energy of the electrons
 kinetic energy of the nuclei
 electrostatic interaction between the electrons and the
nuclei
 electrostatic interaction between the electrons
 electrostatic interaction between the nuclei
45
Solving the Schrödinger Equation




analytic solutions can be obtained only for very
simple systems, like atoms with one electron.
particle in a box, harmonic oscillator, hydrogen
atom can be solved exactly
need to make approximations so that molecules
can be treated
approximations are a trade off between ease of
computation and accuracy of the result
46
47
Expectation Values



for every measurable property, we can construct
an operator
repeated measurements will give an average
value of the operator
the average value or expectation value of an
operator can be calculated by:
  Ôd 
  d
*
*
O
48
Variational Theorem

the expectation value of the Hamiltonian is the variational energy
* ˆ

 Hd
  d
*




 Evar  Eexact
the variational energy is an upper bound to the lowest energy of the
system
any approximate wavefunction will yield an energy higher than the
ground state energy
parameters in an approximate wavefunction can be varied to
minimize the Evar
this yields a better estimate of the ground state energy and a better
approximation to the wavefunction
49
Born-Oppenheimer Approximation

the nuclei are much heavier than the electrons and move
more slowly than the electrons

in the Born-Oppenheimer approximation, we freeze the
nuclear positions, Rnuc, and calculate the electronic
wavefunction, el(rel;Rnuc) and energy E(Rnuc)

E(Rnuc) is the potential energy surface of the molecule
(i.e. the energy as a function of the geometry)

on this potential energy surface, we can treat the motion
of the nuclei classically or quantum mechanically
50
Born-Oppenheimer Approximation

freeze the nuclear positions (nuclear kinetic energy is zero in the
electronic Hamiltonian)
ˆ 
H
el
electrons

i

calculate the electronic wavefunction and energy
ˆ   E ,
H
el
el
el


 2 2 electrons nuclei e2 Z A electrons e 2
i   
 
2me
riA
rij
i
A
i j
E
* ˆ

 el Hel el d
*

 el el d
E depends on the nuclear positions through the nuclear-electron
attraction and nuclear-nuclear repulsion terms
E = 0 corresponds to all particles at infinite separation
51
Hartree Approximation

assume that a many electron wavefunction can be
written as a product of one electron functions
(r1 , r2 , r3 ,)   (r1 ) (r2 ) (r3 ) 


if we use the variational energy, solving the many
electron Schrödinger equation is reduced to solving a
series of one electron Schrödinger equations
each electron interacts with the average distribution of
the other electrons
52
Hartree-Fock Approximation

the Pauli principle requires that a wavefunction for electrons
must change sign when any two electrons are permuted

the Hartree-product wavefunction must be antisymmetrized

can be done by writing the wavefunction as a determinant

1 (1) 1 (2)  1 (n)
1 2 (1) 2 (2)  2 (n)
n




n (1) n (1)  n (n)
 1 2  n
53
Spin Orbitals

each spin orbital I describes the distribution of one electron

in a Hartree-Fock wavefunction, each electron must be in a different
spin orbital (or else the determinant is zero)

an electron has both space and spin coordinates

an electron can be alpha spin (, , spin up) or beta spin (, , spin
up)

each spatial orbital can be combined with an alpha or beta spin
component to form a spin orbital

thus, at most two electrons can be in each spatial orbital
54
Basis Functions
   c  





’s are called basis functions
usually centered on atoms
can be more general and more flexible than atomic
orbitals
larger number of well chosen basis functions yields
more accurate approximations to the molecular orbitals
55
Slater-type Functions



1/ 2

3
1s (r )   1s /  exp(  1s r )
1/ 2

5
 2 s (r )   2 s / 96 r exp(  2 s r / 2)
1/ 2

5
 2 px (r )   2 p / 32 x exp(  2 p r / 2)








exact for hydrogen atom
used for atomic calculations
right asymptotic form
correct nuclear cusp condition
3 and 4 center two electron integrals cannot be done
analytically
56
Gaussian-type Functions





1/ 4

3
g s (r )  2 /  exp(  r 2 )

5
3 1/ 4
g x (r )  128 / 
x exp(  r 2 )

7
3 1/ 4 2
g xx (r )  2048 / 9
x exp(  r 2 )

7
3 1/ 4
g xy (r )  2048 / 
xy exp(  r 2 )






die off too quickly for large r
no cusp at nucleus
all two electron integrals can be done analytically
57
Roothaan-Hall Equations

choose a suitable set of basis functions
   c  


plug into the variational expression for the energy
 Ĥd


   d
*
Evar

*
find the coefficients for each orbital that minimizes the
variational energy
58
Fock Equation

take the Hartree-Fock wavefunction
  1 2  n

put it into the variational energy expression
Evar 

*

 Ĥd
*

  d
minimize the energy with respect to changes in the orbitals
Evar / i  0

yields the Fock equation
F̂i   ii
59
Fock Equation
F̂i   ii




the Fock operator is an effective one electron
Hamiltonian for an orbital 
 is the orbital energy
each orbital  sees the average distribution of all the
other electrons
finding a many electron wavefunction is reduced to
finding a series of one electron orbitals
60
Fock Operator
ˆ V
ˆ  Jˆ  K
ˆ
Fˆ  T
NE

kinetic energy operator

nuclear-electron attraction operator
2


2
ˆT 

2me
V̂ne 
nuclei

A
 e2 Z A
riA
61
Fock Operator
ˆ V
ˆ  Jˆ  K
ˆ
Fˆ  T
NE

Coulomb operator (electron-electron repulsion)
e2
  j rij  j d }i
electrons
Jˆ i  {

j

exchange operator (purely quantum mechanical -arises
from the fact that the wavefunction must switch sign
when you exchange to electrons)
e2
  j rij i d } j
electrons
ˆ  {
K
i

j
62
Solving the Fock Equations
F̂i   ii
1.
2.
3.
4.
obtain an initial guess for all the orbitals i
use the current I to construct a new Fock operator
solve the Fock equations for a new set of I
if the new I are different from the old I, go back to
step 2.
63
Hartree-Fock Orbitals




for atoms, the Hartree-Fock orbitals can be computed
numerically
the  ‘s resemble the shapes of the hydrogen orbitals
s, p, d orbitals
radial part somewhat different, because of interaction
with the other electrons (e.g. electrostatic repulsion
and exchange interaction with other electrons)
64
Hartree-Fock Orbitals

for homonuclear diatomic molecules, the HartreeFock orbitals can also be computed numerically (but
with much more difficulty)

the  ‘s resemble the shapes of the H2+ orbitals

, , bonding and anti-bonding orbitals
65
Recall:
Valence Bond Theory vs.
Molecular Orbital Theory
For Polyatomic Molecules:
Valence Bond Theory: Similar to drawing Lewis structures. Orbitals for
bonds are localized between the two bonded atoms, or as a lone pair of
electrons on one atom. The electrons in the lone pair or bond do NOT
spread out over the entire molecule.
Molecular Orbital Theory: orbitals are delocalized over the entire
molecule.
Which is more correct?
66
LCAO Approximation




numerical solutions for the Hartree-Fock orbitals only practical for
atoms and diatomics
diatomic orbitals resemble linear combinations of atomic orbitals
e.g. sigma bond in H2
  1sA + 1sB
for polyatomics,
approximate the
molecular orbital
by a linear
combination of
atomic orbitals
σ – bond H2
(LCAO)
   c  

67
Roothaan-Hall Equations

basis set expansion leads to a matrix form of the Fock
equations
F Ci = i S Ci

F – Fock matrix

Ci – column vector of the molecular orbital coefficients

I – orbital energy

S – overlap matrix
68
Fock matrix and Overlap matrix

Fock matrix
F     F̂ d

overlap matrix
S       d
69
Intergrals for the Fock matrix

Fock matrix involves one electron integrals of kinetic
and nuclear-electron attraction operators and two
electron integrals of 1/r
ˆ V
ˆ )  d
h     (T
ne



one electron integrals are fairly easy and few in
number (only N2)
1
(  |  )     (1)  (1)   (2)  (2)d 1d 2
r12
two electron integrals are much harder and much
more numerous (N4)
70
Solving the Roothaan-Hall Equations
1.
2.
3.
4.
5.
6.
choose a basis set
calculate all the one and two electron integrals
obtain an initial guess for all the molecular orbital
coefficients Ci
use the current Ci to construct a new Fock matrix
solve F Ci = i S Ci for a new set of Ci
if the new Ci are different from the old Ci, go back to
step 4.
71
Solving the Roothaan-Hall Equations




also known as the self consistent field (SCF) equations, since each
orbital depends on all the other orbitals, and they are adjusted until
they are all converged
calculating all two electron integrals is a major bottleneck, because
they are difficult (6 dimensional integrals) and very numerous
(formally N4)
iterative solution may be difficult to converge
formation of the Fock matrix in each cycle is costly, since it involves
all N4 two electron integrals
  1 2  n
72