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IntroductionIntroduction

What is Computational Chemistry? Use of computer to help solving chemical

problemsChemical Problems

Computer Programs

Physical Models

Math formulas

Physical & Chemical Properties

Chemical Systems Geometrical Arrangements of the nuclei

(atoms/molecules) Relative Energies Physical & Chemical Properties Time dependence of molecular structures

and properties Molecular interactions

System Description Fundamental Units

– elementary units (quarks/electrons/nuetron …)– atoms/Molecules– Macromolecules/Surfaces– Bulk materials

Starting Condition Interaction Dynamical Equation

Molecular Structure Arrangement of nuclei/groups of nuclei Coordination Systems

– Cartesian coordinate (x,y,z)– Spherical coordinate (r,,)– Internal coordinate (r,a,d)

x

y

z

1

x1

y1

z1

r

z

r1

r2a

Fundamental Forces The interaction between particles can be

described in terms of either forces (F) or potentials (V)

r

VrF

)( VdrrF )(

r

r

V

Force Particle Relative strength

Range

Gravitational Mass particles 10-40

Electromagnetic Charged particle 1

Week Interaction Quarks & Leptons 0.001 <10-15

Strong Interaction Quarks 100 <10-15

r

VrF

)(

r

qqCrV jielecijelec )(

r

mmCrV jigravijgrav )(

Potential Energy Surface (PES) The concept of potential energy surfaces is

central to computational chemistry The challenge for computational chemistry

is to explore potential energy surfaces with methods that are efficient and accurate enough to describe the chemistry of interest

Potential Energy Curve Potential Energy between two atoms

+

-

+-

V = Vw/s + Vpn + Vee + Vpp

E r

Potential Energy Surfaces

Product

Reactant

Potential energy depends on many structural variables

r1r2

degree

E

0 60 120 180 240 300 360

Cl

Cl

Cl

Cl

Cl

Cl

Important Features of PES Equilibrium molecular structures correspond to the

positions of the minima in the valleys on a PES Energetics of reactions can be calculated from the

energies or altitudes of the minima for reactants and products

A reaction path connects reactants and products through a mountain pass

A transition structure is the highest point on the lowest energy path

Reaction rates can be obtained from the height and profile of the potential energy surface around the transition structure

The shape of the valley around a minimum determines the vibrational spectrum

Each electronic state of a molecule has a separate potential energy surface, and the separation between these surfaces yields the electronic spectrum

Properties of molecules such as dipole moment, polarizability, NMR shielding, etc. depend on the response of the energy to applied electric and magnetic fields

Classical & Quantum Mechanics Newtonian Mechanic Quantum Mechanic

maF H

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 to months of

supercomputer time Some compromises necessary between

cost and accuracy of modeling methods

Plastic molecular models Assemble from standard parts Fixed bond lengths and coordination geometries Good enough from qualitative modeling of the

structure of some molecules Easy and cheap to use Provide a good feeling for the 3 dimensional

structure of molecules No information on properties, energetics or

reactivity

Molecular mechanics Ball and spring description of molecules Better representation of equilibrium geometries than

plastic models Able to compute relative strain energies Cheap to compute Lots of empirical parameters that have to be carefully

tested and calibrated 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

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

Ab Initio Molecular Orbital Methods 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 Expensive

Molecular Modeling Software Many packages available on numerous

platforms Most have graphical interfaces, so that

molecules can be sketched and results viewed pictorially

Will use a few selected packages to simplify the learning curve

Experience readily transferred to other packages

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

Modeling Software, cont’d Gaussian 03

– semi-empirical and ab initio molecular orbital calculations

– available on Mac (OS 10), Windows and Unix (we will probably use all three versions, depending on which classroom we are in)

GaussView– graphical user interface for Gaussian

Modeling Software, cont’d Software for marcomolecular modeling and

molecular dynamics will be determined later (depends on what is freely available and is capable of meeting our needs)

Force Field Methods Stretching Energy Bending Energy Torsion Energy Van der Waals Energy Electrostatic Energy

– Charges/dipoles– multipoles/polarizabilities

Cross terms

Molecular Mechanics PES calculated using empirical potentials

fitted to experimental and calculated data composed of stretch, bend, torsion and

non-bonded components

E = Estr + Ebend + Etorsion + Enon-bond

e.g. the stretch component has a term for each bond in the molecule

Bond Stretch Term many force fields use just a quadratic term, but the energy

is too large for very elongated bonds

Estr = ki (r – r0)2

Morse potential is more accurate, but is usually not used because of expense

Estr = De [1-exp(-(r – r0)]2

a cubic polynomial has wrong asymptotic form, but a quartic polynomial is a good fit for bond length of interest

Estr = { ki (r – r0)2 + k’i (r – r0)3 + k”i (r – r0)4 } The reference bond length, r0, not the same as the

equilibrium bond length, because of non-bonded contributions

Angle Bend Term usually a quadratic polynomial is sufficient

Ebend = ki ( – 0)2

for very strained systems (e.g. cyclopropane) a higher polynomial is better

Ebend = ki ( – 0)2 + k’i ( – 0)3 + k”i ( – 0)4 + . . .

alternatively, special atom types may be used for very strained atoms

Torsional Term most force fields use a single cosine with

appropriate barrier multiplicity, n

Etors = Vi cos[n( – 0)]

some use a sum of cosines for 1-fold (dipole), 2-fold (conjugation) and 3-fold (steric) contributions

Etors = { Vi cos[( – 0)] + V’i cos[2( – 0)]

+ V”i cos[3( – 0)] }

Non-Bonded Terms Lennard-Jones potential

– EvdW = 4 ij ( (ij / rij)12 - (ij / rij)6 )– easy to compute, but r -12 rises too rapidly

Buckingham potential– EvdW = A exp(-B rij) - C rij

-6 – QM suggests exponential repulsion better, but is

harder to compute tabulate and for each atom

– obtain mixed terms as arithmetic and geometric means

AB = (AA + BB)/2; AB = (AA BB)1/2

Applications