Computational Chemistry for Dummies

1. Computational Chemistry for Dummies Svein Saebø Department of Chemistry Mississippi State University

2. Computational Chemists / Theoretical Chemists • Computational Chemists use existing computer software (often commercial) to study problems from chemistry • Theoretical Chemists develop new computational methods and algorithms.

3. Theoretical / Computational Chemistry • Tool: Modern Computer • Application of • Mathematics • Physics • Computer Science • to solve chemical problems

4. Chemistry • Molecular Science • Studies of molecules • Large Molecules, macromolecules: • Proteins, DNA • Biochemistry, Medicine, Molecular Biology • Other polymers • Material Science (physics)

5. Computational Chemistry • WHY • do theoretical calculations? • WHAT • do we calculate? • HOW are the calculations carried out?

6. WHY? • Evolution of Computational Chemistry • Confirmation of experimental results • Interpretation of experimental results • assignment • Prediction of new results • The truth is experimental! • Advantages • Avoid experimental difficulties • Safety • Cost • Widely used by chemical and pharmaceutical industry • Visualization

7. WHAT? • Molecular System • One or several molecules • Collection of atoms • Structure (geometry): • 3-dimensional arrangement of these atoms

8. WHAT? • Molecular Potential Surfaces • A molecular system with N atoms is described by 3N Cartesian (x,y,z) or 3N-6 internal coordinates (bond lengths, angles, dihedral angles) • R = {q1 ,q2 ,q3 q4 ,….. q3N-6} • Potential Energy Surface (PES) : E(R) • the energy as a function of the three-dimensional arrangement of the atoms.

9. Diatomic Molecule Only one coordinate:R= bond length Potential Surface: E(R)

10. E(R) • Morse Potential: E=D(1-exp(-F(R-R0))2 • Parabola: E=1/2 F (R-R0)2 • First derivative: dE/dR = F (R-R0) • Second derivative d2E/dR2 =F • (force-constant, Hooks Law) • Vibrational Frequency n=1/(2p) (F/m)

11. Intercept through a PES. • Stationary points • Minima • Saddle points (transition states)

12. Potential Surfaces • We are normally interested in stationary points • Global Minimum : Equilibrium Structure • Local Minima: Other (stable?) forms of the system • Saddle Points: Transition States

13. Stationary Points • Mathematical Concept dE / dqi = 0 for all i Slope of potential energy curve = 0 • Minimum: second derivatives positive • Maximum: second derivatives negative • Saddle Point: All second derivatives positive except one (negative)

14. WHAT do we calculate? • Energy: E(q1,q2,q3,….q3N-6) • Gradients: dE/dqi • Force Constants: Fi,j = d2E/dqidqj • + Other second derivatives with respect to • nuclear coordinates • electric , magnetic fields

15. Gradients (dE/dq) • Needed for automatic determination of structure • Force (f) in direction of coordinate q • f = -(dE/dq) = -F (R-R0) • Geometry relaxed until the forces vanish • Quadratic surface: R+f/F=R-F(R-R0)/F=R0

16. Force Relaxation: • (1) start with an initial guess of the geometry Rn and the force constants F (a matrix) (n=1) • (2) calculate the energy E(Rn), and the gradient g(Rn) at Rn • (3) get an improved geometry: • Rn+1= Rn–F-1g(Rn) • (4) Check the largest element of g(R) • If larger than THR (e.g. 10-6) n=n+1 go to (2) • If smaller than THR – finished • The final result does not depend on F

17. Optimization Methods • Calculation of the gradient at several geometries provide information about the force-constants F • Widely used optimizations methods: • Newton Raphson • Steepest Decent • Conjugate gradient • Variable Metric (quasi Newton)

18. WHY Second Derivatives? • provide many important molecular(spectroscopic) properties • Twice with respect to the nuclear coordinates • F=d2E/dqidqj Force Constants Vibrational Frequencies • Dipole moment derivatives  IR intensities • Polarizability Derivatives  Raman Intensities • Once with respect to external magnetic field, once with respect to magnetic moment. • Magnetic shielding- chemical shifts (NMR)

19. Summary (What?) • Common for all Computational chemistry Methods: • Potential Energy Surfaces • Normally seeking a local minimum (or a saddle point) • Get energy and structure • Spectroscopic properties are normally only calculated by quantum mechanical methods

20. HOW? • How do the computer programs work? • Many different computational chemistry programs • Wide range of accuracy • Low price - low accuracy • User Interface • Input/output • Many modern programs are very user friendly • menu-driven point and click

21. HOW? • Molecular Mechanics • Based on classical mechanics • No electrons or wave-function • Inexpensive; can be applied to very large systems (e. g. proteins) • Quantum Mechanical Methods • Seek approximate solutions of the Schrödinger equation for the system HY = E Y

22. Quantum Mechanical Methods • Exact solutions only for the hydrogen atom • s, p, d, f functions • Molecules: LCAO-approximation • fi = S CimCm • {C} H-like atomic functions: Basis set (AO) • {f} Molecular orbitals (MO)

23. Quantum Mechanical Methods • Semiempirical Methods • Use parameters from experiments • Inexpensive, can be applied to quite large systems • Ab Initio Methods • Latin: From the beginning • No empirical data used [except the charge of the electron (e) and the value of Planck’s constant (h)]

24. Semi empirical Methods • p-electrons only • Hückel, PPP (Pariser Parr Pople) • Semi empirical MO methods • Extended Hückel • CNDO, INDO, NDDO • MINDO, MNDO,AM1,PM3…(Dewar)

25. Ab Initio Methods • Hartree-Fock (SCF) Method • based on orbital approximation • Single Configuration • Wave-Function:Y a single determinant • Each electron is interacting with the average of the other electrons • Absolute Error (in the energy) : ~1% • Formal Scaling : N4 (N number of basis functions)

26. Ab Initio Methods • Electron Correlation • Ecorr = E(exact) - EHF • Many Configurations (or determinants) • MP2-scale as N5 • CI, QCISD, CCSD, MP3, MP4(SDQ) - scale as N6

27. Power-Law Scaling • Ab Initio Methods: N4 – N6 • N proportional to the size of the system • Double the size: Price increases by a factor of ~60! (from 1 minute to 1 hour) • Increase computational power with a factor of 1000 • 1000 ~ 3.56 • Could only do systems 3-4 time as big as today

28. The Future of Quantum Chemistry • Dirac (1929): • “The underlying physical laws necessary for the mathematical theory of 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” • 1950’s “it is wise to renounce at the outset any attempt at obtaining precise solutions of the Schrodinger equation for systems more complicated than the hydrogen molecule ion” Levine: “Quantum Chemistry” Fifth Edition

29. Future of Quantum Chemistry • The difficulties at least partly overcome by application of high speed computers • 1998 Nobel Price Committee: (Chemistry Price shared by Walter Kohn and John Pople): • “Quantum Chemistry is revolutionizing the field of chemistry” • We are able to study chemically interesting systems, but not yet biologically interesting systems using quantum mechanical methods.

30. Low-Scaling Methods for Electron Correlation • Low Scaling MP2 • Near linear scaling for large systems • formal scaling N5 • Applied to polypeptides (polyglycines up to 50 glycine units, C100H151N50O51) • Saebo, Pulay, J. Chem. Phys. 2001, 115, 3975. • Saebo in: “Computational Chemistry- Review of Current Trends” Vol. 7, 2002.

31. Molecular Mechanics • Based on classical mechanics • No electrons or wave-function • Inexpensive; can be applied to very large systems, macromolecules. • Polymers • Proteins • DNA

32. Molecular Mechanics • E=Estr + Ebend + Eoop + Etors + Ecross + EvdW + Ees • Estr bond stretching • Ebend bond-angle bending • Eoop out of plane • Etors internal rotation • Ecrosscombinations of these distortions • Non-bonded interactions: • EvdW van der Waals interactions • Ees electrostatic

33. Force Fields • The explicit form used for each of these contributions is called the force field. • Will consider bond stretching as an example

34. E(R)=D(1-exp(-F(R-R0))2 D - dissociation energy F - force-constant R0 - ‘natural’ bond length The parameters D, F, R0 are part of the so-called force-field. The values of these parameters are determined experimentally or by ab initio calculations Bond Stretching

35. Force-fields • Similar formulas and parameters can be defined for: • Bond angle bending • Out of plane bending • Twisting (torsion) • Hydrogen bonding , etc.

36. Molecular Mechanics

37. Force-fields • Each atom is assigned to an atom type based on: • atomic number and • molecular environment • Examples: • Saturated carbon (sp3) • Doubly bonded carbon (sp2) • Aromatic carbon • Carbonyl carbon…..

38. Force-fields • An energy function and parameters (D, F, R0) are assigned to each bond in the molecule. • In a similar fashion appropriate functions and parameters are assigned to each type of distortion • Hydrogen bonds and non-bonding interactions are also accounted for.

39. Commonly Used Force Fields • Organic Molecules: • MM2, MM3, MM4 (Allinger) • Peptides,proteins, nucleic acids • AMBER (Assisted Model Building with Energy Refinement) (Kollman) • CHARMM (Chemistry at HARvard Molecular Modeling (Karplus) • MMFF94 (Merck Molecular Force Field) (Halgren)

40. More Force Fields.. • CFF93, CFF95 (Consistent Force Field) • Hagler (Biosym, Molecular Simulations) • SYBYL or TRIPOS (Clark)

41. Computational Chemistry and NMR • Powerful technique for protein structure determination competitive with X-ray crystallography. • NOE: Nuclear Overhauser Effect • Proton-proton distances • Structure optimized under NOE constraints • Chemical shifts are also used

44. Protein G Angela Gronenborn, NIH

46. 5-Enolpyruvylshuikimate-3-phosphate Synthase

47. Acknowledgements • Dr. John K.Young, Washington State University