Molecular Mechanics

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Molecular MechanicsQuantum Chemistry
- Science Honors Program - Computer Modeling and
Visualization in Chemistry
Eric 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
  1000s 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 (gt
  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, contd

• 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, contd

• 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

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

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Force Fields Typical Energy Functions

• 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

r0
D
r0
Two new parameters D dissociation energy a
width of the potential well
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Bonding Terms angle bending

• Most often Harmonic

q0
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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

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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 ?x ?p h
  

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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 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.
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DeBroglie Wave-like properties of matter.

• If light is particle (photon) with wavelength,
  why not matter, too?
• Ehv ? mc2hvhc/?
• ? ?h/mc ? ?h/p
• DeBroglie Wavelength

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Wavelengths

• DeBroglie Wavelength ? 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?

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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 WAVES that
have stationary states. Schrodinger developed
this theory in the 1920s.
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

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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.
Wavefunctions ?
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
(?), a mathematical function that varies with
position Max Born physical interpretation of
wavefunctions. Probability of finding a particle
in a region is proportional to ?2.
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s Orbitals
Wavefunctions of s orbitals of higher energy have
more complicated radial variation with nodes.
Boundary surface encloses surface with a gt 90
probability of finding electron
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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

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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

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What is a linear operator?
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Schrodinger Eq. is an Eigenvalue problem

• Schrodinger Equation

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Postulates of Quantum Mechanics

• The state of a quantum-mechanical system is
  completely specified by the wave function ? that
  depends upon the coordinates of the particles in
  the system. All possible information about the
  system can be derived from ?. ? has the important
  property that ?(r) ?(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

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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 ?, then the average
  value of the observable corresponding to is
  given by

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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

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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

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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

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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(relRnuc) 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

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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

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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

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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

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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

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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

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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

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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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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

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Fock Operator

• kinetic energy operator
• nuclear-electron attraction operator

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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)

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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.

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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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RecallValence 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. Molecula
r Orbital Theory orbitals are delocalized over
the entire molecule. Which is more correct?
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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)

s 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

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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)

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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