Molecular Mechanics & Quantum Chemistry

1. - 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. 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 Applications

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. http://www.ks.uiuc.edu/Gallery/Movies/ http://chem.acad.wabash.edu/~trippm/Lipids/

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

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.

30. 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.

31. 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?

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.

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. 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 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) 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:

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