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Molecular Mechanics&

Quantum Chemistry

- Science Honors Program -

Computer Modeling and Visualization in Chemistry

Eric KnollEric Knoll

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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

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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

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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

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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

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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)

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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

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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

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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

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Force Fields

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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

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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

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Force Fields: Typical Energy Functions2

0

20

12 6

1( )

2

1( )

2

[1 cos( )]2

( )

[ ]

rbonds

angles

n

torsions

improper

i j

elec ij

ij ij

LJ ij ij

U k r r

k

Vn

V improper torsion

q q

r

A B

r r

Bond stretches

Angle bending

Torsional rotation

Improper torsion (sp2)

Electrostatic interaction

Lennard-Jones interaction

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Bonding Terms: bond stretch

Most often Harmonic

Morse Potential for dissociation studies

20 )(

2

1rrkV r

bondsbond

bonds

rraMorse DeDV 2)( ]1[ 0

Harmonic Potential

bond length

Vbond

r0

Morse Potential

bond length

Vm

ors

e

Two new parameters: D: dissociation energy a: width of the potential well

r0

D

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Bonding Terms: angle bending

Most often Harmonic

20 )(

2

1 kVangles

angle

Harmonic Potential

angle

Vangle

0

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What do these FF parameters look like?

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Atom types (AMBER)

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Bond Parameters

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Angle Parameters

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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

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/

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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

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Quantum Mechanics

- Science Honors Program -

Computer Modeling and Visualization in Chemistry

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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

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Quantum Stuff

Photoelectric effect: particle-wave duality of light de Broglie equation: particle-wave duality of matter

Heisenberg Uncertainty principle: Δx Δp ≥ h

h h

p mv

h h

p mv

h h

p mv

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What is an Atom?

Protons and neutrons make up the heavy, positive core, the NUCLEUS, which occupies a small volume of the atom.

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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:

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Problem:

Acceleration of Electron in Classical Theory

Bohr Model: Circular Orbits, Angular Momentum Quantized

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Photoelectric EffectPhotoelectric 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.

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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

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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?

Interpretations of Quantum Mechanics

1. The Realist Position The particle really was at point C

2. The Orthodox Position The particle really was not anywhere

3. The Agnostic Position Refuse to answer

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Atomic Orbitals – Wave-particle duality.

Traveling waves vs. Standing Waves.

Atomic and Molecular Orbitals are 3-D STANDING WAVESthat have stationary states.

Schrodinger developed this theory in the 1920’s.

Example of 1-D guitar string standing wave.

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Weird Quantum Effect: Quantum Tunneling

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Schrödinger Equation

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:

EH

2 2

2( ) ( ) ( )

2

dV x x E x

m dx

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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.

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.

Wavefunctions: ψ

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s Orbitals

Boundary surface encloses surface with a > 90% probability of finding electron

Wavefunctions of s orbitals of higher energy have more complicated radial variation with nodes.

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Schrodinger Eq. is an Eigenvalue problem

Classical-mechanical quantities represented by linear operators: Indicates that operates on f(x) to give a new

function g(x). Example of operators

ˆ ( ) ( )Af x g x

A

A

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Schrodinger Eq. is an Eigenvalue problem

Classical-mechanical quantities represented by linear operators: Indicates that operates on f(x) to give a new

function g(x). Example of operators

ˆ ( ) ( )Af x g x

A

A

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What is a linear operator?

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Schrodinger Eq. is an Eigenvalue problem

Schrodinger Equation:

A

2 2

2( ) ( ) ( )

2

ˆ ( ) ( )

dV x x E x

m dx

H x E x

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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:

A

*

all space

( ) ( ) 1r r dr

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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:

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 n n nA a

ˆn n nA a

*

all space

ˆa A dr

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Hamiltonian for a Molecule

(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

nuclei

BA AB

BAelectrons

ji ij

nuclei

A iA

Aelectrons

iA

nuclei

A Ai

electrons

i e r

ZZe

r

e

r

Ze

mm

2222

22

2

22ˆ H

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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

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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:

Od

d

*

*O

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Variational Theorem the expectation value of the Hamiltonian is the variational energy

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

exactEEd

d

var*

* ˆ

H

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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

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Born-Oppenheimer Approximation

freeze the nuclear positions (nuclear kinetic energy is zero in the electronic Hamiltonian)

calculate the electronic wavefunction and energy

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

2 2 22ˆ

2

electrons electrons nuclei electronsA

el ii i A i je iA ij

e Z e

m r r

H

*

*

ˆˆ ,

el el el

el el el

el el

dE E

d

H

H

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Hartree Approximation

assume that a many electron wavefunction can be written as a product of one electron functions

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

)()()(),,,( 321321 rrrrrr

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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

n

nnn n

n

n

n

21222

111

)()1()1(

)()2()1(

)()2()1(

1

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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

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Basis Functions

’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

c

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Slater-type Functions

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

)2/exp(32/)(

)2/exp(96/)(

)exp(/)(

2

2/1522

2

2/1522

1

2/1311

rxr

rrr

rr

pppx

sss

sss

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Gaussian-type Functions

die off too quickly for large r no cusp at nucleus all two electron integrals can be done analytically

)exp(/2048)(

)exp(9/2048)(

)exp(/128)(

)exp(/2)(

24/137

224/137

24/135

24/13

rxyrg

rxrg

rxrg

rrg

xy

xx

x

s

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Roothaan-Hall Equations

choose a suitable set of basis functions

plug into the variational expression for the energy

find the coefficients for each orbital that minimizes the variational energy

c

d

dE

*

*

var

H

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Fock Equation take the Hartree-Fock wavefunction

put it into the variational energy expression

minimize the energy with respect to changes in the orbitals

yields the Fock equation

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d

dE

*

*

var

H

iii F

0/var iE

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Fock Equation

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

iii F

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Fock Operator

kinetic energy operator

nuclear-electron attraction operator

22

2ˆ

em

T

nuclei

A iA

Ane r

Ze2

V

KJVTF ˆˆˆˆˆ NE

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Fock Operator

Coulomb operator (electron-electron repulsion)

exchange operator (purely quantum mechanical -arises from the fact that the wavefunction must switch sign when you exchange to electrons)

ijij

j

electrons

ji d

r

e }{ˆ2

J

jiij

j

electrons

ji d

r

e }{ˆ2

K

KJVTF ˆˆˆˆˆ NE

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Solving the Fock Equations

1. obtain an initial guess for all the orbitals i

2. use the current I to construct a new Fock operator

3. solve the Fock equations for a new set of I

4. if the new I are different from the old I, go back to step 2.

iii F

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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)

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Hartree-Fock Orbitals

for homonuclear diatomic molecules, the Hartree-Fock 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

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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?

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LCAO Approximation

c

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 (LCAO)

σ – bond H2

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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

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Fock matrix and Overlap matrix

Fock matrix

overlap matrix

dF F

dS

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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

one electron integrals are fairly easy and few in number (only N2)

two electron integrals are much harder and much more numerous (N4)

dh ne )ˆˆ( VT

2112

)2()2(1

)1()1()|( ddr

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Solving the Roothaan-Hall Equations

1. choose a basis set

2. calculate all the one and two electron integrals

3. obtain an initial guess for all the molecular orbital coefficients Ci

4. use the current Ci to construct a new Fock matrix

5. solve F Ci = i S Ci for a new set of Ci

6. if the new Ci are different from the old Ci, go back to step 4.

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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

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