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

Computational Chemistry for Computational Chemistry for DummiesDummies

Svein Saebø

Department of Chemistry

Mississippi State University

Computational Chemists / Computational Chemists / Theoretical ChemistsTheoretical Chemists

Computational Chemists use existing computer software (often commercial) to study problems from chemistry

Theoretical Chemists develop new computational methods and algorithms.

Theoretical / Computational Theoretical / Computational ChemistryChemistry

Tool: Modern ComputerApplication of

– Mathematics– Physics– Computer Science– to solve chemical problemschemical problems

Chemistry Chemistry

Molecular Science– Studies of molecules

Large Molecules, macromolecules:– Proteins, DNA

Biochemistry, Medicine, Molecular Biology

– Other polymers Material Science (physics)

Computational ChemistryComputational Chemistry

WHY– do theoretical calculations?

WHAT– do we calculate?

HOWare the calculations carried out?

WHY?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

WHAT?WHAT?

Molecular System– One or several molecules

Collection of atomsStructure (geometry):

– 3-dimensional arrangement of these atoms

WHAT?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 MoleculeDiatomic Molecule

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

E(E(RR))

Morse Potential: E=D(1-exp(-F(R-R0))2

Parabola: E=1/2 F (R-R0)2

First derivative:E/R = F (R-R0)

Second derivativeE/R2 =F – (force-constant, Hooks Law)

Vibrational Frequency =1/(2) (F/)

Intercept through a PES.Intercept through a PES.

Stationary points– Minima– Saddle points (transition states)

Potential SurfacesPotential 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 PointsStationary Points

Mathematical ConceptE / qi = 0 for all i

Slope of potential energy curve = 0

Minimum: second derivatives positiveMaximum: second derivatives negativeSaddle Point: All second derivatives

positive except one (negative)

WHAT do we calculate?WHAT do we calculate?

Energy: E(q1,q2,q3,….q3N-6)

Gradients: E/qi

Force Constants: Fi,j = 2E/qiqj

+ Other second derivatives with respect to – nuclear coordinates

– electric , magnetic fields

Gradients (Gradients (E/E/qq))

Needed for automatic determination of structure

Force (f) in direction of coordinate q– f = -(E/q) = -F (R-R0)

Geometry relaxed until the forces vanish– Quadratic surface:

R+ f/F=R-F(R-R0)/F=R0

Force Relaxation: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 MethodsOptimization 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?WHY Second Derivatives?

provide many important molecular(spectroscopic) properties

– Twice with respect to the nuclear coordinates F=2E/qiqj 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?)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?

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

HOW?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 H = E

Quantum Mechanical MethodsQuantum Mechanical Methods

Exact solutions only for the hydrogen atom– s, p, d, f functions

Molecules: LCAO-approximation– i = Ci

– {} H-like atomic functions: Basis set (AO)– {} Molecular orbitals (MO)

Quantum Mechanical MethodsQuantum Mechanical Methods

Semiempirical Methods– Use parameters from experiments– Inexpensive, can be applied to quite large

systemsAb 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 MethodsSemi empirical Methods

-electrons only– Hückel, PPP (Pariser Parr Pople)

Semi empirical MO methods– Extended Hückel– CNDO, INDO, NDDO– MINDO, MNDO,AM1,PM3…(Dewar)

Ab Initio MethodsAb Initio Methods

Hartree-Fock (SCF) Method– based on orbital approximation– Single Configuration

Wave-Function: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 MethodsAb 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 ScalingPower-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 The Future of Quantum ChemistryChemistry

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 ChemistryFuture 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 Low-Scaling Methods for Electron Correlation 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 MechanicsMolecular Mechanics

Based on classical mechanicsNo electrons or wave-functionInexpensive; can be applied to very large

systems, macromolecules.– Polymers– Proteins– DNA

Molecular MechanicsMolecular 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 FieldsForce Fields

The explicit form used for each of these contributions is called the force field.

Will consider bond stretching as an example

Bond StretchingBond 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-fieldsForce-fields

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

Molecular MechanicsMolecular Mechanics

Force-fieldsForce-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…..

Force-fieldsForce-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 FieldsCommonly 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..More Force Fields..

CFF93, CFF95 (Consistent Force Field)– Hagler (Biosym, Molecular Simulations)

SYBYL or TRIPOS (Clark)

Computational Chemistry and Computational Chemistry and NMRNMR

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 Protein G Angela Gronenborn, NIHAngela Gronenborn, NIH

5-Enolpyruvylshuikimate-3-5-Enolpyruvylshuikimate-3-phosphate Synthasephosphate Synthase

AcknowledgementsAcknowledgements

Dr. John K.Young, Washington State University