1. Molecular Mechanics & Quantum Chemistry
- Science Honors Program -
Computer Modeling and Visualization in Chemistry
Molecular Mechanics & Quantum Chemistry
Eric Knoll

2. 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

3. 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

4. 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

5. 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

6. 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)

7. 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

8. 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

9. 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

10. Force Fields

11. 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

12. 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

13. Force Fields: Typical Energy Functions
Bond stretches
Angle bending
Torsional rotation
Improper torsion (sp2)
Electrostatic interaction
Lennard-Jones interaction

14. 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

15. Bonding Terms: angle bending
Most often Harmonic q0

16. What do these FF parameters look like?

17. Atom types (AMBER)

18. Bond Parameters

19. Angle Parameters

20. 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

21. 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.

22. 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

23. Quantum Mechanics - Science Honors Program -
Computer Modeling and Visualization in Chemistry
Quantum Mechanics

24. 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

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

26. What is an Atom?
Protons and neutrons make up the heavy, positive core, the NUCLEUS, which occupies a small volume of the atom.

27. J J Thompson in his plum pudding model
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:

28. Problem: Acceleration of Electron in Classical Theory
Bohr Model: Circular Orbits, Angular Momentum Quantized

29. 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.
Very intense red light had no effect; vs. very low intensity blue light ejects electrons - goes against classical theory, which thought that just intensity counted.
Intensity does have some effect: #of photons  #of ejected electrons. Total energy of light is function of wavelength & intensity.
Photoelectric effect is for slabs of metal, not individual atoms or molecules. But similar concept. In Spartan we calculate energy for an individual atom in the gas phase – no other atoms near by affecting it.
Einstein – commonly thought to have won Nobel prize for general theory of relativity. No.
1905: photoelectric quation n (1/2)mev2 = hv – hv0.
With individual atoms and molecules, have discrete energy levels.

30. 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

31. Wavelengths: DeBroglie Wavelength λ = h/p = h/(mv)
h = x kg m2 s-1
What is wavelength of electron moving at 1,000,000 m/s. Mass electron = 9.11 x kg.
What is wavelength of baseball (0.17kg) thrown at 30 m/s?
7.3 x 10e-10 = 7.3 A
1.3 x 10e-34 = 1.3 x 10e-24 A

32. 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

33. 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.
In his theory of the photoelectric effect, Einstein treated light as having particle-like as well aas wave-like properties. In 1924, the young French physicist Louis de Broglie posed this question: if light, which we think of as a wave, can have particle-like properties, then why cannot particles of matter have wave-like properties? (recall, photons have no mass, but electrons do, a very small amount much smaller than a proton or neutron, but electrons still have mass).
In fact all matter has wave like properties. The wavelength is related to the to the momentum of the particle through the formula: lamda = h/p. Where lamda is the de Broglie wavelength.
EXAMPLE: What is the wavelength of an electron moving at 1.0 x 10^6 m/s?
ANSWER: 7.3 Angstroms, which is a dozen times larger than the radius of a ground state H-atom – thus clearly, the wave-like properties of electrons are critical to understanding atomic structure.
EXAMPLE: What is the wavelength of a baseball (mass 0.17 kg) that is thrown at 30 m/s?
ANSWER: 1.3 x 10^-34 m. This wavelength is FAR to small to be observed and is never considered when studying the motion of heavy objects.
So electrons have wave-like properties.
Constructive & destructive interference – like water. In 1927 C. Davisson and L.H. Germer showed that a crystal in fact diffracts electrons and that de Broglie’s relationship correctly predicts their wavelengths.
You’re use to thinking about traveling waves – light moving at the speed of light, or water ripples moving across a pond.
There is another type of wave – a standing wave. Atomic orbitals are standing waves (“matter-waves”) that have stationary states - or states that are stable indefinitely.

34. Weird Quantum Effect: Quantum Tunneling

35. 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:

36. 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.
Schrodinger Equation
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.

37. s Orbitals
Talk about:
+/- phase nodes probability density surface
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

38. 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

39. 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

40. What is a linear operator?

41. Schrodinger Eq. is an Eigenvalue problem
Schrodinger Equation:

42. 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:

43. 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:

45. 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

46. 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

48. 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:

49. 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

50. 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

51. 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

52. 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

53. 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

54. 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

55. 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

56. 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

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

58. 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

59. 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

60. 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

61. Fock Operator kinetic energy operator
nuclear-electron attraction operator

62. 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)

63. 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.

64. 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)

65. 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

66. 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?
Valence Shell (for an atom): the outermost unfilled shell of electrons surrounding the nucleus of an atom. These are the electrons that participate in bonding!
Core electrons: the electrons in the inner part of atom

67. LCAO Approximation σ – bond H2
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

68. 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

69. Fock matrix and Overlap matrix

70. 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)

71. 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 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.

72. 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