(Computational Chemistry) Education Guiding Q

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? 2. 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?

1 Why Computational Chemistry?? • In 1929, 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!

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!!

Chemistry Today: A Different View Old Way New Way

Desired Properties e S t i a m l t M e e u r r l r p e a o r a t e s e C t u n r I e Synthesize Structural Compounds Design Build

2 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)

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.

Molecular Mechanics: Components • Bond stretching (l) • Bond Angle bending (θ) • Dihedral Angle rotation (Ф)

• Van der Waals forces H OO • Hydrogen bonding • Electrostatic interactions •Others 9

3 Equations: Bond Stretching • Use the Harmonic Oscillator Approximation: k 2 E =−s ()ll HOA 2 0

ks = force constant

l0 = equilibrium bond length

– Could include higher order terms:

k 2 2 3 E =−s ()ll1 −−−−−−− kll' ()() kll '' k ''' () ll ... HOA 2 0 []00 0 – MM2 (cubic); MM3 (quartic)

Equations: Bond Angle Bending • Mathematically similar to stretching:

1 2 Ek=−()θθ θθ2 0

where θ0 is the equilibrium angle

– Again, cubic and quartic terms give better fit with experimental values, but may not be included because of computational cost

Equations: Dihedral Angle Rotation • Use a sum of periodic functions:

EVtorsion =++++05. 12()( 1 cosφφ 05 . V 1 cos 2 )

0.5 V3()13++cosφ .....?

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. 12

4 Equations: Van der Waals & H-bonding • Lennard-Jones, or “6-12” potential: A B A = repulsive term E =− vdW r12r 6 B = attractive term • Hydrogen Bonding A B E =− HB r12r 10 – Often handled in the van der Waals and electrostatic terms – Sometimes, explicitly placed in a separate term • Called a “10-12” potential: Attractive region decays more rapidly with distance

Equations: Electrostatic Energy • Based on Coulomb’s Law: qqab Eelectro = εabr ab qn = (partial) charge on atom ε = dielectric constant r = interatomic distance Atom Charges: 1. Could be fixed values (easiest) 2. Values could also be calculated • Related to electronegativity of atom, as well as those atoms connected to it

Molecular Mechanics: Overall Energy • Also called the “steric” energy in MM – Summation of all the terms:

EEsteric=++ stretch EEE bend torsion+ vdW

+++EEEH− bonding electro other – 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

5 Parameters ~100 elements: N(N+1)/2 = 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 l0, just to cover bond stretching!

• Also need kθ, θ0, Vn, and φ values for all possible angles, A & B values, etc., etc. • To be thorough, would need ~108 parameters gathered from experimental data!!

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.

2. 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 2 2 eZ2 H =−hh ∇2 − ∇2 − k ∑ 22mmi ∑ k ∑∑ r i e k k i k ik e22eZZ ++ kl ∑ r ∑ r ij< ij kl< kl

6 Approximations Used 1. Born-Oppenheimer: Compared to e-’s, nuclei are stationary • Electrons move in a field of fixed nuclei 2. 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

MO Construction • Individual MO is defined as: N φϕ= ∑ aii i=1 Basis set = set of N functions φi, each associated with a molecular orbital expansion coefficient ai

• Variational Principle:

– We know that the set of all Ei will have a lowest energy value (ground state)

– Try different ai’s until energy minimum located • At this point, we have found the ground state

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 (φi) and a molecular geometry

2. Guess a set of ai’s defining the Fock matrix

3. Solve Roothaan-Hall equations to get new ai’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 21

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 over- estimated; Computationally expensive; Correlation energy is significant; Difficult to model reactions involving bond dissociation

Difficulties of HF Method •N4 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

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 – MP2, MP3, etc. • Computational cost is again very high

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 2. Don’t worry about Ψ at all – Instead, focus on the electron density → Density Functional Theory (DFT)

3. Semiempirical Methods • Simplified Hartree-Fock: – Want to avoid the computational time needed to solve N4 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

Simplifications 1. Only look at valence electrons – Core electrons subsumed into nucleus 2. 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

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 (∆fH) are the desired quantities • ZINDO/S parameterized to reproduce UV-Vis spectra • TNDO parameterized to reproduce NMR chemical shifts

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 29

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

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 Nd= ∫ ρ ()rr 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

DFT Process • The energy is minimized with respect to variations in ρ, subject to the constraint of charge conservation: Nd= ∫ ρ ()rr – 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

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:

ET(ρρρρρ) =++++ni( ) V ne( ) V ee( ) ∆∆ T( ) V ee( ρ)

where Tni = kinetic energy of non - interacting electrons,

VVne= nuclear - electron attraction, ee = 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

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 Exc • 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

Approaches to Exc • Terminology of density functionals:

1. Local: Simple dependence of Exc on ρ(r)

2. Nonlocal, or gradient corrected: Exc 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 Nd= ∫ ρ ()rr

12 Differences Between HF-SCF and DFT • DFT contains no approximations: It is exact

– All we need to know is Exc as a function of ρ

• As we will see, we must approximate Exc • 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

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

Density Functional Theory Advantages: Scales as N3 (HF was N4) • With few exceptions, DFT is the most cost- effective 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

13 Brief Comparison of Methods Things to consider: Semi- MM HF DFT Emp.

Results Good qual. Better Fast, large Advantages can be ~ and ~quant. accuracy molecules quantitative results than HF Parameters e- correlation Parameters Limited to Dis- may not be limits may not be smaller advantages available accuracy available systems Least Expense is Inexpensive Expense is Expense expensive quite high technique high technique

Units in Computational Chemistry

• Bohr: Atomic unit of Length (a0) – Equal to the radius of the first Bohr orbit for a hydrogen atom • 5.29 x 10-11m (0.0529 nm, 52.9pm, 0.529Å) • Hartree: Atomic unit of Energy – Equal to twice the energy of a ground state hydrogen atom • 627.51 kcal/mole • 2625.5 kJ/mole • 27.211 eV • 219474.6 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