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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

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

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

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

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)

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

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

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

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

What is a Force Field?

- Force field is a collection of parameters for a potential energy function

- Parameters might come from fitting against experimental data or quantum mechanics calculations

Force Fields: Typical Energy Functions

Bond stretches

Angle bending

Torsional rotation

Improper torsion (sp2)

Electrostatic interaction

Lennard-Jones interaction

Bonding Terms: bond stretch

- Most often Harmonic
- Morse Potential for dissociation studies

r0

D

r0

Two new parameters:

D: dissociation energy

a: width of the potential well

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

ApplicationsMolecular 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/

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

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

Quantum Stuff

- Photoelectric effect: particle-wave duality of light
- de Broglie equation: particle-wave duality of matter
- Heisenberg Uncertainty principle:Δx Δp ≥ h

What is an Atom?

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

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:

Bohr Model: Circular Orbits, Angular Momentum Quantized

- Problem: Acceleration of Electron in Classical Theory

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.

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

DeBroglie: Wave-like properties of matter.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

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.

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:

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.

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

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

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

Schrodinger Eq. is an Eigenvalue problem

- Schrodinger Equation:

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) dris 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:

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:

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

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

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:

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

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

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

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

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

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

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

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

Gaussian-type Functions

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

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

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

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

Fock Operator

- kinetic energy operator
- nuclear-electron attraction operator

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)

Solving the Fock Equations

- 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.

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)

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

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?

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 (LCAO)

σ – bond H2

Roothaan-Hall Equations

- basis set expansion leads to a matrix form of the Fock equations

FCi = iSCi

- F – Fock matrix
- Ci – column vector of the molecular orbital coefficients
- I – orbital energy
- S – overlap matrix

Fock matrix and Overlap matrix

- Fock matrix
- overlap matrix

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)

Solving the Roothaan-Hall Equations

- 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 FCi = iSCi for a new set of Ci
- if the new Ci are different from the old Ci, go back to step 4.

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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