1. Computational Chemistry for Dummies
Svein Saebø
Department of Chemistry
Mississippi State University

2. Computational Chemists / Theoretical Chemists
Computational Chemists use existing computer software (often commercial) to study problems from chemistry
Theoretical Chemists develop new computational methods and algorithms.

3. Theoretical / Computational Chemistry
Tool: Modern Computer
Application of
Mathematics
Physics
Computer Science
to solve chemical problems

4. Chemistry Molecular Science Large Molecules, macromolecules:
Studies of molecules
Large Molecules, macromolecules:
Proteins, DNA
Biochemistry, Medicine, Molecular Biology
Other polymers
Material Science (physics)

5. Computational Chemistry
WHY
do theoretical calculations?
WHAT
do we calculate?
HOW
are the calculations carried out?

6. WHY? Evolution of Computational Chemistry Advantages
Confirmation of experimental results
Interpretation of experimental results
assignment
Prediction of new results
The truth is experimental!
Advantages
Avoid experimental difficulties
Safety
Cost
Widely used by chemical and pharmaceutical industry
Visualization

7. WHAT? Molecular System Collection of atoms Structure (geometry):
One or several molecules
Collection of atoms
Structure (geometry):
3-dimensional arrangement of these atoms

8. WHAT? Molecular Potential Surfaces
A molecular system with N atoms is described by 3N Cartesian (x,y,z) or 3N-6 internal coordinates (bond lengths, angles, dihedral angles)
R = {q1 ,q2 ,q3 q4 ,….. q3N-6}
Potential Energy Surface (PES) : E(R)
the energy as a function of the three-dimensional arrangement of the atoms.

9. Diatomic Molecule Only one coordinate:R= bond length
Potential Surface: E(R)

10. E(R) Morse Potential: E=D(1-exp(-F(R-R0))2 Parabola: E=1/2 F (R-R0)2
First derivative: dE/dR = F (R-R0)
Second derivative d2E/dR2 =F
(force-constant, Hooks Law)
Vibrational Frequency n=1/(2p) (F/m)

11. Intercept through a PES.
Stationary points
Minima
Saddle points (transition states)

12. Potential Surfaces We are normally interested in stationary points
Global Minimum : Equilibrium Structure
Local Minima: Other (stable?) forms of the system
Saddle Points: Transition States

13. Stationary Points Mathematical Concept
dE / dqi = 0 for all i
Slope of potential energy curve = 0
Minimum: second derivatives positive
Maximum: second derivatives negative
Saddle Point: All second derivatives positive except one (negative)

14. WHAT do we calculate? Energy: E(q1,q2,q3,….q3N-6) Gradients: dE/dqi
Force Constants: Fi,j = d2E/dqidqj
+ Other second derivatives with respect to
nuclear coordinates
electric , magnetic fields

15. Gradients (dE/dq) Needed for automatic determination of structure
Force (f) in direction of coordinate q
f = -(dE/dq) = -F (R-R0)
Geometry relaxed until the forces vanish
Quadratic surface:
R+ f/F=R-F(R-R0)/F=R0

16. Force Relaxation:
(1) start with an initial guess of the geometry Rn and the force constants F (a matrix) (n=1)
(2) calculate the energy E(Rn), and the gradient g(Rn) at Rn
(3) get an improved geometry:
Rn+1 = Rn –F-1 g(Rn)
(4) Check the largest element of g(R)
If larger than THR (e.g. 10-6) n=n+1 go to (2)
If smaller than THR – finished
The final result does not depend on F

17. Optimization Methods
Calculation of the gradient at several geometries provide information about the force-constants F
Widely used optimizations methods:
Newton Raphson
Steepest Decent
Conjugate gradient
Variable Metric (quasi Newton)

18. WHY Second Derivatives?
provide many important molecular(spectroscopic) properties
Twice with respect to the nuclear coordinates
F=d2E/dqidqj Force Constants Vibrational Frequencies
Dipole moment derivatives  IR intensities
Polarizability Derivatives  Raman Intensities
Once with respect to external magnetic field, once with respect to magnetic moment.
Magnetic shielding- chemical shifts (NMR)

19. Summary (What?) Common for all Computational chemistry Methods:
Potential Energy Surfaces
Normally seeking a local minimum (or a saddle point)
Get energy and structure
Spectroscopic properties are normally only calculated by quantum mechanical methods

20. HOW? How do the computer programs work? User Interface
Many different computational chemistry programs
Wide range of accuracy
Low price - low accuracy
User Interface
Input/output
Many modern programs are very user friendly
menu-driven point and click

21. HOW? Molecular Mechanics Quantum Mechanical Methods
Based on classical mechanics
No electrons or wave-function
Inexpensive; can be applied to very large systems (e. g. proteins)
Quantum Mechanical Methods
Seek approximate solutions of the Schrödinger equation for the system HY = E Y

22. Quantum Mechanical Methods
Exact solutions only for the hydrogen atom
s, p, d, f functions
Molecules: LCAO-approximation
fi = S CimCm
{C} H-like atomic functions: Basis set (AO)
{f} Molecular orbitals (MO)

23. Quantum Mechanical Methods
Semiempirical Methods
Use parameters from experiments
Inexpensive, can be applied to quite large systems
Ab Initio Methods
Latin: From the beginning
No empirical data used [except the charge of the electron (e) and the value of Planck’s constant (h)]

24. Semi empirical Methods
p-electrons only
Hückel, PPP (Pariser Parr Pople)
Semi empirical MO methods
Extended Hückel
CNDO, INDO, NDDO
MINDO, MNDO,AM1,PM3…(Dewar)

25. Ab Initio Methods Hartree-Fock (SCF) Method
based on orbital approximation
Single Configuration
Wave-Function:Y a single determinant
Each electron is interacting with the average of the other electrons
Absolute Error (in the energy) : ~1%
Formal Scaling : N4 (N number of basis functions)

26. Ab Initio Methods Electron Correlation Ecorr = E(exact) - EHF
Many Configurations (or determinants)
MP2-scale as N5
CI, QCISD, CCSD, MP3, MP4(SDQ) - scale as N6

27. Power-Law Scaling Ab Initio Methods: N4 – N6
N proportional to the size of the system
Double the size: Price increases by a factor of ~60! (from 1 minute to 1 hour)
Increase computational power with a factor of 1000
1000 ~ 3.56
Could only do systems 3-4 time as big as today

28. The Future of Quantum Chemistry
Dirac (1929):
“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”
1950’s
“it is wise to renounce at the outset any attempt at obtaining precise solutions of the Schrodinger equation for systems more complicated than the hydrogen molecule ion”
Levine: “Quantum Chemistry”
Fifth Edition

29. Future of Quantum Chemistry
The difficulties at least partly overcome by application of high speed computers
1998 Nobel Price Committee: (Chemistry Price shared by Walter Kohn and John Pople):
“Quantum Chemistry is revolutionizing the field of chemistry”
We are able to study chemically interesting systems, but not yet biologically interesting systems using quantum mechanical methods.

30. Low-Scaling Methods for Electron Correlation
Low Scaling MP2
Near linear scaling for large systems
formal scaling N5
Applied to polypeptides (polyglycines up to 50 glycine units, C100H151N50O51)
Saebo, Pulay, J. Chem. Phys. 2001, 115, 3975.
Saebo in: “Computational Chemistry- Review of Current Trends” Vol. 7, 2002.

31. Molecular Mechanics Based on classical mechanics
No electrons or wave-function
Inexpensive; can be applied to very large systems, macromolecules.
Polymers
Proteins
DNA

32. Molecular Mechanics
E=Estr + Ebend + Eoop + Etors + Ecross + EvdW + Ees
Estr bond stretching
Ebend bond-angle bending
Eoop out of plane
Etors internal rotation
Ecross combinations of these distortions
Non-bonded interactions:
EvdW van der Waals interactions
Ees electrostatic

33. Force Fields
The explicit form used for each of these contributions is called the force field.
Will consider bond stretching as an example

34. Bond Stretching E(R)=D (1-exp(-F(R-R0))2 D - dissociation energy
F force-constant
R ‘natural’ bond length
The parameters D, F, R0 are part of the so-called force-field.
The values of these parameters are determined experimentally or by ab initio calculations

35. Force-fields Similar formulas and parameters can be defined for:
Bond angle bending
Out of plane bending
Twisting (torsion)
Hydrogen bonding , etc.

36. Molecular Mechanics

37. Force-fields Each atom is assigned to an atom type based on: Examples:
atomic number and
molecular environment
Examples:
Saturated carbon (sp3)
Doubly bonded carbon (sp2)
Aromatic carbon
Carbonyl carbon…..

38. Force-fields
An energy function and parameters (D, F, R0) are assigned to each bond in the molecule.
In a similar fashion appropriate functions and parameters are assigned to each type of distortion
Hydrogen bonds and non-bonding interactions are also accounted for.

39. Commonly Used Force Fields
Organic Molecules:
MM2, MM3, MM4 (Allinger)
Peptides,proteins, nucleic acids
AMBER (Assisted Model Building with Energy Refinement) (Kollman)
CHARMM (Chemistry at HARvard Molecular Modeling (Karplus)
MMFF94 (Merck Molecular Force Field) (Halgren)

40. More Force Fields.. CFF93, CFF95 (Consistent Force Field)
Hagler (Biosym, Molecular Simulations)
SYBYL or TRIPOS (Clark)

41. Computational Chemistry and NMR
Powerful technique for protein structure determination competitive with X-ray crystallography.
NOE: Nuclear Overhauser Effect
Proton-proton distances
Structure optimized under NOE constraints
Chemical shifts are also used

44. Protein G Angela Gronenborn, NIH

46. 5-Enolpyruvylshuikimate-3-phosphate Synthase

47. Acknowledgements
Dr. John K.Young, Washington State University