Introduction

What is Computational Chemistry?  Use of computer to help solving chemical problems Chemical 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 x1x1 y1y1 z1z1 r  z r1r1 r2r2 a

Fundamental Forces  The interaction between particles can be described in terms of either forces (F) or potentials (V) r r V

ForceParticle Relative strength Range Gravitational Mass particles  Electromagnetic Charged particle1  Week Interaction Quarks & Leptons0.001< Strong Interaction Quarks100<10 -15

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 = V w/s + V pn + V ee + V pp E r

Potential Energy Surfaces Product Reactant  Potential energy depends on many structural variables r1r1 r2r2

degree E 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

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 = E str + E bend + E torsion + E non-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 E str =  k i (r – r 0 ) 2  Morse potential is more accurate, but is usually not used because of expense E str =  D e [1-exp(-  (r – r 0 )] 2  a cubic polynomial has wrong asymptotic form, but a quartic polynomial is a good fit for bond length of interest E str =  { k i (r – r 0 ) 2 + k’ i (r – r 0 ) 3 + k” i (r – r 0 ) 4 }  The reference bond length, r 0, not the same as the equilibrium bond length, because of non-bonded contributions

Angle Bend Term  usually a quadratic polynomial is sufficient E bend =  k i (  –  0 ) 2  for very strained systems (e.g. cyclopropane) a higher polynomial is better E bend =  k i (  –  0 ) 2 + k’ i (  –  0 ) 3 + k” i (  –  0 )  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 E tors =  V i cos[n(  –  0 )]  some use a sum of cosines for 1-fold (dipole), 2- fold (conjugation) and 3-fold (steric) contributions E tors =  { V i cos[(  –  0 )] + V’ i cos[2(  –  0 )] + V” i cos[3(  –  0 )] }

Non-Bonded Terms  Lennard-Jones potential –E vdW =  4  ij ( (  ij / r ij ) 12 - (  ij / r ij ) 6 ) –easy to compute, but r -12 rises too rapidly  Buckingham potential –E vdW =  A exp(-B r ij ) - C r ij -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