Introduction to Computational Chemistry Computational (chemistry education) and/or. (Computational chemistry) education

Introduction to Computational Chemistry Computational (chemistry education) and/or (Computational chemistry) education First one: Use computational tools to help increase student understanding of material already covered in various courses Second one: Teach students about computational chemistry (molecular modeling) itself, in both courses and research projects 1 Guiding Questions 1. What is the role and purpose of molecular modeling?. What is the fundamental mathematical expression that needs to be solved? What are the terms, what is their significance, and what variations can be used? 3. What are the pros/cons of approximations that can be used in the calculations? How does the choice of approximation affect the results, computing time, etc.? Guiding Questions 4. There are four different methods commonly employed: molecular mechanics, semiempirical, ab initio, and density functional theory (DFT). What are these methods? How do they differ? 5. What are the fundamental units of measure commonly used by computational chemists? 6. What are some of the computer codes and platforms used to do computational chemistry? What are their pros/cons? 3 1

Why Computational Chemistry?? In 199, P.A.M. Dirac wrote: 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. Dirac didn t have access to digital computers, but we do! 4 Molecular Modeling Defined Provides information that is complementary to experimental data on the structures, properties, and reactions of substances Mainly based on one algorithm: Schrödinger s Equation Until recently, required the use of high performance computers (architecture) Modern PC s are now of sufficient power to run many molecular modeling codes Everyone now has access to this tool!! 5 Chemistry Today: A Different View Old Way New Way Correlate Structural Design Interpret Desired Properties Synthesize Build Measure Simulate Compounds 6

Some Molecular Modeling Codes CAChe(Computer Aided Chemistry) Spartan HyperChem PC Model Chem3D Gaussian Sybyl MOPAC (Molecular Orbital Package) GAMESS (General Atomic and Molecular Electronic Structure System) 7 1. Molecular Mechanics Apply classical mechanics to molecules No electrons, no orbital interactions!! Atoms are spheres with element dependent mass Bonds are springs that obey Hooke s Law: F = -kx where k is the force constant (for a specified bond type between certain atoms) Other types of springs represent bond angles, dihedral angles, etc. 8 Molecular Mechanics: Components Bond stretching (l) Bond Angle bending (θ) Dihedral Angle rotation (Ф) Van der Waals forces Hydrogen bonding O H O Electrostatic interactions Others 9 3

Equations: Bond Stretching Use the Harmonic Oscillator Approximation: ks EHOA = ( l l0) k s = force constant l 0 = equilibrium bond length E Could include higher order terms: HOA 3 [ ' 0 '' 0 ''' 0...] ks = ( l l0) 1 k ( l l ) k ( l l ) k ( l l ) MM (cubic); MM3 (quartic) 10 Equations: Bond Angle Bending Mathematically similar to stretching: 1 Eθ = kθ( θ θ0 ) where θ is the equilibrium angle 0 Again, cubic and quartic terms give better fit with experimental values, but may not be included because of computational cost 11 Equations: Dihedral Angle Rotation Use a sum of periodic functions: ( φ) ( φ) V ( 1+ cos 3φ ) +...? Etorsion = 05. V1 1+ cos + 05. V 1+ cos + 0.5 3 where Vn is the dihedral force constant, n is the periodicity, and φ is the dihedral angle Various contributions to the energy may have different periodicities Contributions may include dipole-dipole interactions, hyperconjugation, van der Waals, H- bonding, etc. 1 4

Equations: Van der Waals & H-bonding Lennard-Jones, or 6-1 potential: A B A = repulsive term EvdW = 1 6 r r B = attractive term Hydrogen Bonding Often handled in the van der Waals and electrostatic terms Sometimes, explicitly placed in a separate term Called a 10-1 potential: Attractive region decays more rapidly with distance E HB = A r B r 1 10 13 Equations: Electrostatic Energy Based on Coulomb s Law: qq a Eelectro r q n = (partial) charge on atom ε = dielectric constant r = interatomic distance b = εab ab Atom Charges: 1. Could be fixed values (easiest). Values could also be calculated Related to electronegativity of atom, as well as those atoms connected to it 14 Molecular Mechanics: Overall Energy Also called the steric energy in MM Summation of all the terms: Esteric = Estretch + Ebend + Etorsion + EvdW + EH bonding + Eelectro + Eother The collection of all the functional forms and the associated constants is called a force field BEWARE: Energies reported by MM are often meaningless are not externally referenced May be useful for conformers of same molecule 15 5

Parameters ~100 elements: N(N+1)/ = 5050 single bonds Multiple bonds: Define atom hybridizations ~300 atoms types: 45,150 force constants! Need 300 partial atomic charges; 45,150 values of ε (electrostatics); 45,150 values for A & B (van der Waals); 45,150 values of l 0, just to cover bond stretching! Also need k θ, θ 0, V n, and φ values for all possible angles, A & B values, etc., etc. To be thorough, would need ~10 8 parameters gathered from experimental data!! 16 Molecular Mechanics Advantages: Very fast, excellent structural results (for compounds with good parameters available), computationally inexpensive (can be applied to large molecules) Geometry optimization: Move all atoms until sum of all forces on each = 0 Disadvantages: ~80% of known compounds do not have parameters available No orbital information, can t look at reactions or transition states, can t predict reactivity, etc. 17. Ab Initio (Hartree-Fock) Method Based on Schrödinger s Equation: ĤΨ = EΨ Ĥ is the Hamiltonian operator E is the energy of the atom or molecule Ψ is the wavefunction, which is what we want h h H = i k m m i e k e + + r i< j ij k k< l ezz k r kl l i k ez r ik k 18 6

Approximations Used 1. Born-Oppenheimer: Compared to e - s, nuclei are stationary Electrons move in a field of fixed nuclei. Hartree-Fock: Separate Ψ (many-electron wavefunction) into a series of one-electron spin orbitals 3. LCAO (Linear Combination of Atomic Orbitals): MO s expressed as linear combinations of single electron atomic orbitals, represented by basis functions 19 MO Construction N Individual MO is defined as: φ = a iϕi i= 1 Basis set = set of N functions φ i, each associated with a molecular orbital expansion coefficient a i Variational Principle: We know that the set of all E i will have a lowest energy value (ground state) Try different a i s until energy minimum located At this point, we have found the ground state 0 Roothaan-Hall Equations Used to find the set of molecular orbital expansion coefficients that minimize the energy The solution process is iterative Process overview: 1. Select a basis set (φ i ) and a molecular geometry. Guess a set of a i s defining the Fock matrix 3. Solve Roothaan-Hall equations to get new a i s and a new Fock matrix 4. Repeat until a self-consistent solution is found (HF-SCF Method) This energy minimum is the ground state 1 7

Ab Initio Method Advantages: Ψ is often close enough so properties can be calculated via application of the appropriate operator Useful results can be obtained Can serve as a stepping stone to more advanced treatments Disadvantages: e - /e - repulsion is overestimated; Computationally expensive; Correlation energy is significant; Difficult to model reactions involving bond dissociation Difficulties of HF Method N 4 total integrals need to be evaluated N = number of basis functions used Computationally expensive Limited to study of smaller molecules Theoretical limitations HF treatment: Each electron experiences the others as an average distribution; no instantaneous e - /e - interaction is included Energies are often far from experimental values One electron nature of the operators used 3 Higher Level Methods Called electron correlation, or post-scf methods Deal with the electron correlation problem CI (Configuration Interaction) Extreme computational cost, but provides the most complete treatment of a molecular system possible Møller-Plesset Perturbation Theory MP, MP3, etc. Computational cost is again very high 4 8

Overcoming Limitations of HF-SCF Method 1. Since several approximations are made, introduce further simplifications to increase speed and accuracy Replace some integrals with parameterized values that will reproduce experimental results Constitutes Semiempirical techniques. Don t worry about Ψ at all Instead, focus on the electron density Density Functional Theory (DFT) 5 3. Semiempirical Methods Simplified Hartree-Fock: Want to avoid the computational time needed to solve N 4 integrals Introduce approximations that make HF theory more tractable AND somehow better account for correlation energy Improved chemical accuracy Different ways of doing this define the various semi-empirical methods 6 Simplifications 1. Only look at valence electrons Core electrons subsumed into nucleus. Neglect certain integrals 3. Parameterize other integrals using experimental data Different methods are parameterized to reproduce different properties 4. Use a minimal basis set 5. Employ a noniterative solution process 7 9

Semiempirical Methods So, the Hamiltonian is changed Calculated energy can be above OR below the exact energy (no Variational Principle anymore) Parameterization is used to reproduce experimental data Usually, molecular geometry and energy ( f H) are the desired quantities ZINDO/S parameterized to reproduce UV-Vis spectra TNDO parameterized to reproduce NMR chemical shifts 8 Some Semiempirical Methods Pariser-Parr-Pople MO Theory (PPP) Extended Hückel MO Theory (EHMO) Complete Neglect of Differential Overlap (CNDO) Intermediate Neglect of Diff. Overlap (INDO) Modified INDO (MINDO) Modified Neglect of Diatomic Overlap (MNDO) Austin Model 1 (AM1) Most popular Parametric Method 3 (PM3) More recent: SAM1, PM5 Different Hamiltonians, mainly differing in how the e - /e - repulsion is handled, are used in each 9 Semiempirical Advantages: Very fast, handles large molecules, get good qualitative and sometimes ~quantitative results Often works well for solution phase (water) studies Disadvantages: Not all atoms have parameters available Similarity between molecule and parameterization set Limited to ground state equilibrium geometries Can t calculate properties not addressed in parameterization process 30 10

4. Density Functional Theory Instead of Ψ, look at electron density HF-SCF Theory is 4N dimensional Electron density is 3N dimensional (N = # e - s) Get rid of one-dimension (spin) Hohenberg & Kohn (1964) The ground state energy E of an N-electron system is a functional of the electronic density ρ, and E is a minimum when evaluated with the exact ground state density Also N = ρ ( r) dr 31 What is a Functional? The energy depends on a complicated function, which is the e - density in 3-D space A function whose argument is also a function is called a functional The energy is a unique functional of ρ(r) A functional enables a function to be mapped to a number 3 DFT Process The energy is minimized with respect to variations in ρ, subject to the constraint of charge conservation: N = ρ ( r) d r Electron density is easier than dealing with Ψ Instead of attempting to calculate the full N- electron wavefunction, DFT only attempts to calculate the total electronic energy and the overall electron density distribution 33 11

Problem We have no idea of what the function describing the electron density looks like! Kohn and Sham (1965): Look at specific components of the energy functional: E( ρ) = Tni( ρ) + Vne( ρ) + Vee( ρ) + T( ρ) + Vee( ρ) where Tni = kinetic energy of non - interacting electrons, Vne = nuclear - electron attraction, Vee = classical electron - electron repulsion, T = correction to the kinetic energy from electron interaction, and Vee = sum of all non - classical corrections to the electron - electron replusion energy 34 Solution The first three terms from the previous slide have been seen before The last two terms are the troublesome ones They are usually combined together to form what is called the exchange-correlation energy E xc This term includes effects of quantum mechanical exchange and correlation The correlation energy is what was missing from HF theory, and was a major limitation 35 Approaches to E xc Terminology of density functionals: 1. Local: Simple dependence of E xc on ρ(r). Nonlocal, or gradient corrected: E xc depends on both ρ(r) and the gradient of ρ(r), ρ(r) Variational approach: Overall energy minimum corresponds to the exact ground state electron density (all other densities give higher energy) Further constraint: Number of electrons is fixed N = ρ ( r) d r 36 1

Differences Between HF-SCF and DFT DFT contains no approximations: It is exact All we need to know is E xc as a function of ρ As we will see, we must approximate E xc HF is a deliberately approximate theory so that we can solve the equations exactly So, with DFT our theory is exact and the equations are solved approximately, while with HF our theory is approximate but we can solve the equations exactly 37 Some Common Functionals B3LYP: Becke, Lee, Parr, and Yang; uses a hybrid exchange correlation energy functional B3PW91: Becke, Perdew, and Wang; The PW91 correlation functional replaces LYP PBE-96: Perdew, Burke, and Ernzerhof; an updated correlation functional VWN#5: Vosko, Wilk, and Nusair; another correlation functional BP86: Becke and Perdew; uses the Becke 1986 exchange energy and the Perdew 1996 GC correlation functional 38 Density Functional Theory Advantages: Scales as N 3 (HF was N 4 ) With few exceptions, DFT is the most costeffective method to achieve a given level of accuracy Electron correlation included with less expense Disadvantages: All e - are included Costs still large for molecules of interest With Ψ s apply the correct operator to determine the molecular property of interest With DFT, must know how the molecular property depends on the electron density 39 13

Brief Comparison of Methods Things to consider: MM HF Semi- Emp. DFT Advantages Fast, large molecules Results can be ~ quantitative Good qual. and ~quant. results Better accuracy than HF Disadvantages Parameters may not be available e - correlation limits accuracy Parameters may not be available Limited to smaller systems Expense Least expensive technique Expense is quite high Inexpensive technique Expense is high 40 Units in Computational Chemistry Bohr: Atomic unit of Length (a 0 ) Equal to the radius of the first Bohr orbit for a hydrogen atom 5.9 x 10-11 m (0.059 nm, 5.9pm, 0.59Å) Hartree: Atomic unit of Energy Equal to twice the energy of a ground state hydrogen atom 67.51 kcal/mole 65.5 kj/mole 7.11 ev 19474.6 cm -1 41 Lab Exercises Goal: Develop familiarity with the CAChe software Drawing and viewing molecules Running various calculations Determining bond angles and distances Remember: We have a wide variety of software available We will explore all of them this evening You may want to try some of the lab exercises using one of the other available programs 4 14