A brief introduction to computational chemistry

Ria Armunanto

Austrian-Indonesian Center for Computational ChemistryChemistry Department, Mathematics and Natural Sciences

Universitas Gadjah Mada

Computational Chemistry

• Cheminformatics• Chemical calculations

– Interested in:• Structure• Properties• Reactivity

● Calculation methods:• Ab Initio Quantum

Mechanics• Density Functional

Theory (DFT)• Semi-Empirical

Methods• Molecular Mechanics

Chemical Calculation Types

• Single Point– Given geometry– Calculate total energy and other properties

• Geometry Optimization– Given starting structure– Calculate lowest energy conformation– Can also optimize to transition states

• Molecular Dynamics– Given starting structure– Follow system as it evolves through time– Forces from previous step used to calculate next step

• Vibrational Frequencies– Given geometry– Use matrix of second derivative of energy to generate vibrational

frequencies– Characterize the shape of potential energy surface

Quantum Mechanics

• All QM chemistry methods depend on solving the Schrödinger equation (SE.)

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• Can only solve directly for very small systems (1e-.)• Introduce approximations:

– Born-Oppenheimer Approx. – Electrons move in fixed nuclear frame.– Adiabatic Approx. – Restrict calculation to one electronic state.

• Now only need to solve electronic SE.

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Ab Initio Calculations: Introduction

• Purely theoretical methods. Only inputs to calculations are physical constants and system structure.

• Underlying mathematics are complex but not important for using the methods effectively.

• Advantages:– Most reliable computational methods available.– Systematically improvable.

• Disadvantages:– Limited size of system.– Problems for simpler methods with chemical reactions.

Ab Initio Calculations: Hartree-Fock (HF) Methods

• Replace n-electron SE by n 1-electron SEs (the HF equations.)

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• Each electron moves in average electric field due to other electrons (mean field approximation.)

• Require orbitals to solve equations to find orbitals?!?• Use the self-consistent field (SCF) method:

1. Begin with guess set of orbitals.2. Solve HF equations to yield new, improved set.3. Use new set in HF equations to yield further improved set.4. Iterate until system meets convergence criterion.

• HF method does not properly treat interelectronic term from SE.

Ab Initio Calculations: Electron Correlation

• Due to electron-electron repulsion term in SE.• Required for proper treatment of:– Dissociation processes.– Structures removed from equilibrium geometry.– Electronic excited states.

• Methods include:– Møller-Plesset Perturbation Theory (MP2, MP3, …)– Configuration Interaction (CIS, CISD, …)– Multi-configuration SCF (MCSCF, CASSCF)– Coupled Cluster (CCSD, CCSD(T))– (DFT)

Ab Initio Calculations: Summary

Method System Size (atoms)

Good for… Bad for…

HF <100 + Equilibrium geometries

+ Equilibrium frequencies

+ Relative energies of conformations

– Dissociative processes

– Structures far from equilibrium

MP2 <50 + Transition state calculations

+ Free radicals

+ Vibrational frequencies

– (Dissociative processes)

– Structures with excited state character

CISD <10 + High accuracy

+ Electronic excitations

– Medium to Large molecules

MCSCF <20 + Structures with excited state character

+ Dissociative processes

– Large molecules

CCSD(T) <20 + High accuracy

+ Dissociative processes

– Large molecules

– Electronic excitations

Ab Initio Calculations: Applications

• Calculation and stability of structures: equilibrium, transition state and reaction intermediates.

• Electronic properties – charge distribution, dipoles, polarizability, unpaired spin densities.

• Characterization of orbitals – predictions of reactivity.• Vibrational analysis – calculating IR and Raman spectra.• Electronic transitions, CD response, characterization of

excited states.

DFT: Introduction

• One-to-one mapping between electron density and wavefunction.

• Another strategy for solving electronic SE using functionals.• Includes some part of electron correlation.• Advantages:

– Treat larger molecules than possible using post-HF ab initio.– Particulary good for ground state, equilibrium structures.– Study condensed phase systems.

• Disadvantages:– Not systematically improvable. Poorly defined accuracy.– Bewildering choice of functionals– Difficulties describing some intermolecular interactions, particularly

those which involve dispersion forces.

DFT: Functionals

• Form of functional that maps electron density to electronic wavefunction is not known.

• Basic approximations (LDA) work well in solid state physics but not in molecular calculations.

• Next level (GGA) give good results for equilibrium molecular ground state geometries and energies.

• Many different advanced functionals have been developed.• Generally described by two parts:

– Exchange functional– Correlation functional

• For example, BLYP functional is made up of B (Becke exchange) and LYP (Lee, Yang and Parr correlation).

DFT: Hybrid Functionals

• Try to overcome difficulties of pure exchange functionals by mixing in component of exact exchange from HF theory.

• Parameters controlling amount of HF exchange mixed in are generally fitted to training set of molecules.– Not an ab initio method.

• Care needed when using hybrid functionals. Your system should resemble training set.

• Most common hybrid functional is B3LYP.– Becke exchange functional + 3 parameters controlling

inclusion of HF exchange.

DFT: Applications

• Calculation and stability of structures: equilibrium, transition state and reaction intermediates.

• Electronic properties – charge distribution, dipoles, polarizability, unpaired spin densities.

• Characterization of orbitals – predictions of reactivity.• Vibrational analysis – calculating IR and Raman spectra.

Basis Sets: Introduction

• Mathematical functions used in ab initio and DFT calculations to describe the electron distribution.

• Slater Type Orbitals (STO) correctly model the variation of electron density with distance from nucleus

• Gaussian Type Orbitals actually used as they are more efficient.

• Do not show correct behaviour though.

• Combinations of GTOs used to produce better approximation.

Basis Sets: In Real Calculations

• When one basis function represents one AO we have a minimal basis set.

• Multiple- (zeta) basis sets use more than one basis function per AO.

• Split-valence basis sets give more importance to the valence electrons than the core electrons.

• Polarization functions allow more flexibility in the shape of the electron density.

• Diffuse functions model electron density with a larger spatial extent.– Particularly useful for ions and excited states.

Basis Sets: Other Types

• Effective Core Potentials are used for elements in 4th period onwards.– Majority of electrons do not contribute to chemical properties

and reactivity.– Core electrons replaced by simple function.– Does not effect accuracy but increases speed.– Includes relativistic effects.

• Plane wave basis sets are often used in condensed phase calculations.– Periodically repeating trigonometric functions.

Ab Initio and DFT: EaStCHEM RCF

• Most popular methods on RCF.• Tasker – Ion extraction ligands.• McNab – Study of reaction pathways.• Robertson – Reactivity via frontier orbitals.• Mareque – Structure of zinc complexes.• Mount – Redox properties of indoles.• Rankin – Structures for GED experiments.• Morrison – Hydrogen bonding in crystals.• Madden – Ionic liquids.• Yellowlees – EPR properties.• And others…

Semi-Empirical Methods: Introduction

• Form of HF method, some parts simplified by using empirical data.• Advantages:

– Treat larger molecules than possible using ab initio.– Particulary good for organic molecules and reactions– Orbitals allow for prediction of reactions and properties

• Disadvantages:– Accuracy not as good as higher ab initio methods.– Does not work well for:

• Molecules involving H-bonding• Some transition states• Molecule types outside parameterization set.• Atoms that are poorly parameterized (e.g. nitrogen.)

– Unavailable for some atoms.

Semi-Empirical Methods: Theory

• All semi-empirical methods are based around 2 schemes for reducing the amount of computation:– Elimination of core electrons.– Reduction of number of 2e- integrals.

• First makes chemical sense.• Second is practical.• Fit parameters for different atomic types.• Latest versions of semi-empirical methods are called AM1 and

PM3.• All designed for organic systems.

Semi-Empirical Methods: Applications

• Calculation and stability of organic structures: equilibrium, transition state and reaction intermediates.

• Electronic properties – charge distribution and dipoles.• Characterization of orbitals – predictions of reactivity.• Vibrational analysis – calculating IR and Raman spectra.

• On the EaStCHEM RCF:– Hulme – Structures of organic molecules.– Tasker – Ion extraction ligands.

Classical Molecular Mechanics: Introduction

• Treat molecules as sets of spheres bound together by springs.– Spheres often correspond to atoms but there is no restriction.– ‘Springs’ (interaction potentials) define interactions between the spheres.

• Advantages:– Investigate larger systems than is possible using a QM methods.– Can look at biopolymers, liquids and extract information about bulk

properties.– Conceptually easy to understand.– Total flexibility in defining calculations.

• Disadvantages:– Can be problematic defining interaction potentials.– Difficult to set up calculations.– No information on electronic properties.

Classical Molecular Mechanics: Force Fields

• Collection of atom types and interaction potentials• Determined by fitting parameters to results from experiment or ab

initio calculations.• Usually based on pairwise sum over all spheres:

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• Many different force fields have been determined:

– Amber – for biomolecules.– MM3 – For single organic molecules.– OPLS-AA – For organic molecules in condensed phase.

Classical Molecular Mechanics: Force Field Issues

• Number of parameters quickly becomes unwieldy.• Consider all atoms up to Kr (ignoring noble gases):

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• May also be different hybridizations for atoms (3 for each):

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• Including all other parameters we need (bond angles, dihedral angles, VDW, …) leads to around 107 parameters!

• We have also left out 70 elements…• Defining all the force field parameters for a complex system is a problem.

Classical Molecular Mechanics: Applications

• Condensed phase systems and biomolecules.• Calculation and relative stability of structures.• Vibrational analysis.• Time evolution of system – dynamics.• Drug docking.• Modelling of configuration space – bulk

properties (via statistical mechanics).

• On the EaStCHEM RCF:– Hulme – Solution structures of cyclic polypeptides.

Chemical Calculation Types (Again)

• Single Point– Given geometry– Calculate total energy and other properties

• Geometry Optimization– Given starting structure– Calculate lowest energy conformation– Can also optimize to transition states

• Molecular Dynamics– Given starting structure– Follow system as it evolves through time– Forces from previous step used to calculate next step

• Vibrational Frequencies– Given geometry (and force field)– Use matrix of second derivative of energy to generate vibrational frequencies– Characterize the shape of potential energy surface

Further Information?

• Website– Information on software– Links to manuals and websites

• Wikipedia– Good starting place for theory– Links to more detailed information

• Have a lot fun