Statistical Mechanics and Multi-Scale Simulation Methods ChBE 591-009

1. Statistical Mechanics and Multi-Scale Simulation MethodsChBE 591-009 Prof. C. Heath Turner Lecture 07 • Some materials adapted from Prof. Keith E. Gubbins: http://gubbins.ncsu.edu • Some materials adapted from Prof. David Kofke: http://www.cbe.buffalo.edu/kofke.htm

2. Semiempirical Molecular Orbital Theory • Background • HF MO theory limited due to computational complexity • approximations (fitting to experimental data) to improve speed • approximations to improve accuracy • Implementations • Most demanding step of HF calculations: J and K integrals – numerical solutions required, N4 scaling • SOLUTION: estimate these terms • J (coulomb integral): repulsion between 2 e-. Estimate: If basis functions of e- #1 are far from e- #2, integral is zero. • Electron Correlation: dE of He with and without e- correlation = 26 kcal/mol • Analytic Derivatives: geometry optimizations need to calculate dE/dr. Early approximations of HF were used to develop analytic derivatives. • Semiempirical Methods • Hückel Theory/Extended Hückel Theory • CNDO, INDO, NNDO • MINDO/3, MNDO, AM1, PM3 • SAR • QM/MM

3. Theory and Simulation Scales Continuum Methods Based on SDSC Blue Horizon (SP3) 512-1024 processors 1.728 Tflops peak performance CPU time = 1 week / processor TIME/s 100 Atomistic SimulationMethods Mesoscale methods (ms) 10-3 Lattice Monte Carlo Brownian dynamics Dissipative particle dyn (ms) 10-6 Semi-empirical (ns) 10-9 methods Monte Carlo molecular dynamics (ps) 10-12 Ab initio methods tight-binding MNDO, INDO/S (fs) 10-15 10-10 10-9 10-8 10-7 10-6 10-5 10-4 (mm) (nm) LENGTH/meters NC State University 2002

4. Semiempirical Molecular Orbital Theory • Hückel Theory (Erich Hückel) • Illustration of LCAO approach • Used to describe unsaturated/aromatic hydrocarbons • Developed in the 30’s (not used much today) • Assumptions: • Basis set = parallel 2p orbitals, one per C atom (designed to treat planar hydrocarbon p systems) • Sij = dij(orthonormal basis set) • Hii = a (negative of the ionization potential of the methyl radical) • Hij= b (negative stabilization energy). 90º rotation removes all bonding, thus we can calculate DE:DE = 2Ep - Ep where Ep = a and Ep = 2a + 2b (as shown below) • 5. Hij= between carbon 2p orbitals more distant than nearest neighbors is set to zero.

5. Semiempirical Molecular Orbital Theory • Hückel Theory: Application – Allyl System (C3H5) • 3 carbon atoms = 3 carbon 2p orbitals • Construct the secular equation: Q: What are the possible energy values (eigenvalues)? Q: What is the lowest energy eigenvalue? Q: What is the molecular orbital associated with this energy? Solve: Answer:

6. Semiempirical Molecular Orbital Theory • Hückel Theory: Application – Allyl System (C3H5) • We need an additional constraint: normalization of the coefficients: ** Second subscript has been added to designate the 1st energy level (bonding). Lowest energy Molecular Orbital: Next energy level: Highest energy level: Allyl cation: 2e- =

7. Semiempirical Molecular Orbital Theory Picture taken from: CJ Cramer, Essentials of Computational Chemistry, Wiley (2004).

8. Semiempirical Molecular Orbital Theory • Extended Hückel Theory (EHT) • All core e- are ignored (all modern semiempirical methods use this approximation) • Only valence orbitals are considered, represented as STOs (hydrogenic orbitals). Correct radial dependence. Overlap integrals between two STOs as a functions of r are readily computed. (HT assumed that the overlap elements Sij = dij). The overlap between STOs on the same atom is zero. • Resonance Integrals: • For diagonal elements: Hmm= negative of ionization potential (in the appropriate orbital). Valence shell ionization potentials (VSIPs) have been tabulated, but can be treated as an adjustable parameter. • For off-diagonal elements: Hmn is approximated as (C is an empirical constant, usually 1.75): • RESULT: the secular equation can now be solved to determine MO energies and wave functions. • ** Matrix elements do NOT depend on the final MOs (unlike HF)  the process is NOT iterative. Therefore, the solution is very fast, even for large molecules.

9. Semiempirical Molecular Orbital Theory • Extended Hückel Theory (EHT) • Performance Issues: • PESs are poorly reproduced • Restricted to systems for which experimental geometries are available. • Primarily now used on large systems, such as extended solids (band structure calculations) • EHT fails to account for e- spin – cannot energetically distinguish between singlets/triplets/etc. • Complete Neglect of Differential Overlap (CNDO) • Strategy – replace matrix elements in the HF secular equation with approximations. • Basis set is formed from valence STOs • Overlap matrix, Smn = dmn • All 2-e- integrals are simplified. Only the integrals that have m and n as identical orbitals on the same atom and orbitals l and s on the same atom are non-zero (atoms may be different atoms). Mathematically:

10. Semiempirical Molecular Orbital Theory • Complete Neglect of Differential Overlap (CNDO) • The following two-electron integrals remain, and are abbreviated as gAB (A and B correspond to atoms A and B): • These integrals can be calculated explicitly from the STOs or they can be treated as a parameter. Popular parametric form comes from Pariser-Parr approximation (IP=ionization potential, EA=electron affinity): • The one-electron integrals for the diagonal matrix elements can be approximated as (m is centered on atom A):

11. Semiempirical Molecular Orbital Theory • Complete Neglect of Differential Overlap (CNDO) • The one-electron integrals for the off-diagonal matrix elements (m is centered on atom A and n is centered on atom B): • Smn shown above is the overlap matrix element computed using the STO basis set. This is different than defined previously for the secular equation. b is an adjustable parameter, seen before in Hückel theory. • PERFORMANCE • Reduced number of 2e- integrals from N4 to N2 • The 2e- integrals can be solved algebraically • CNDO not good for predicting molecular structures • The Pariser-Parr-Pople (PPP) is a CNDO model that is used some today for conjugated p systems.

12. Semiempirical Molecular Orbital Theory • Intermediate Neglect of Differential Overlap (INDO) • Modification of CNDO method to permit more flexibility in modeling e-/e- interactions on the same center. • Modification introduces different values for the unique one-center two-electron integrals. These values are empirical, adjustable parameters. • INDO method predicts valence bond angles with greater accuracy than CNDO. • INDO geometry still rather poor, but does a good job of predicting spectroscopic properties. • Modified Intermediate Neglect of Differential Overlap (MINDO) • Previously, semiempirical methods were problem specific • MINDO developed to become more robust • Every parameter treated as a free variable, within the limits of physical rules (by incorporating penalty functions)

13. Semiempirical Molecular Orbital Theory • Neglect of Diatomic Differential Overlap (NDDO) • Relaxes constraints on the two-center two-electron integrals. Thus all integrals of (mn|ls) are retained provided that m and n are on the same center and that l and s are on the same center. • Most modern semiempirical models are NDDO models: MNDO, AM1, PM3 • Structure-Activity Relationships (SAR) • Used in the pharmaceutical industry to understand the structure-activity relationship of biological molecules (use NDDO models) • Drug companies use SARs to screen through candidate molecules in order to identify potentially active species – design drugs and predict activity. • ALSO: • Quantitative Structure-Activity Relationship (QSAR) – for more information on the methodology see the guide developed by Network Science Corporation (http://www.netsci.org/Science/Compchem/feature19.html) • Quantitative Structure-Property Relationship (QSPR) – correlation developed between structure and physical properties. Typically used to design materials and predict properties.