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

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.

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

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

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

WHY? Evolution of Computational Chemistry Advantages 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

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

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.

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

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)

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

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

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)

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

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

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-1 g(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

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)

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)

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

HOW? How do the computer programs work? User Interface 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

HOW? Molecular Mechanics Quantum Mechanical Methods 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

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)

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)]

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)

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)

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

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

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

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.

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.

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

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

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

Bond Stretching 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

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

Molecular Mechanics

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

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.

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)

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

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

Protein G Angela Gronenborn, NIH

5-Enolpyruvylshuikimate-3-phosphate Synthase

Acknowledgements Dr. John K.Young, Washington State University