Statistical Mechanics and Multi- Scale Simulation Methods ChBE Prof. C. Heath Turner Lecture 07 Some materials adapted from Prof. Keith E. Gubbins: Some materials adapted from Prof. David Kofke:

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, N 4 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 H ü ckel Theory CNDO, INDO, NNDO MINDO/3, MNDO, AM1, PM3 SAR QM/MM

(nm) (  m) (fs) (ps) (ns) (  s) (ms) LENGTH /meters TIME/s Mesoscale methods Atomistic Simulation Methods Semi-empirical methods Ab initio methods Monte Carlo molecular dynamics tight-binding MNDO, INDO/S Continuum Lattice Monte Carlo Brownian dynamics Dissipative particle dyn Methods NC State University 2002 Theory and Simulation Scales Based on SDSC Blue Horizon (SP3) processors Tflops peak performance CPU time = 1 week / processor

Hückel Theory (Erich Hückel ) Illustration of LCAO approach Used to describe unsaturated/aromatic hydrocarbons Developed in the 30’s (not used much today) Assumptions: 1.Basis set = parallel 2p orbitals, one per C atom (designed to treat planar hydrocarbon  systems) 2.S ij =  ij (orthonormal basis set) 3.H ii =  (negative of the ionization potential of the methyl radical) 4.H ij =  (negative stabilization energy). 90º rotation removes all bonding, thus we can calculate  E:  E = 2E p - E  where E p =  and E  = 2  + 2  (as shown below) 5.H ij = between carbon 2p orbitals more distant than nearest neighbors is set to zero. Semiempirical Molecular Orbital Theory

Hückel Theory: Application – Allyl System (C 3 H 5 ) 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: Semiempirical Molecular Orbital Theory

Hückel Theory: Application – Allyl System (C 3 H 5 ) We need an additional constraint: normalization of the coefficients: ** Second subscript has been added to designate the 1 st energy level (bonding). Lowest energy Molecular Orbital: Next energy level: Highest energy level: Allyl cation: 2e - = Semiempirical Molecular Orbital Theory

Picture taken from: CJ Cramer, Essentials of Computational Chemistry, Wiley (2004). 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 S ij =  ij ). The overlap between STOs on the same atom is zero. Resonance Integrals: For diagonal elements: H   = 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: H  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. 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. Semiempirical Molecular Orbital Theory 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, S  =   All 2-e - integrals are simplified. Only the integrals that have  and as identical orbitals on the same atom and orbitals and  on the same atom are non-zero (atoms may be different atoms). Mathematically:

Semiempirical Molecular Orbital Theory Complete Neglect of Differential Overlap (CNDO) The following two-electron integrals remain, and are abbreviated as  AB (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 (  is centered on atom A):

Semiempirical Molecular Orbital Theory Complete Neglect of Differential Overlap (CNDO) The one-electron integrals for the off-diagonal matrix elements (  is centered on atom A and is centered on atom B): S  shown above is the overlap matrix element computed using the STO basis set. This is different than defined previously for the secular equation.  is an adjustable parameter, seen before in Hückel theory. PERFORMANCE Reduced number of 2e - integrals from N 4 to N 2 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  systems.

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)

Semiempirical Molecular Orbital Theory Neglect of Diatomic Differential Overlap (NDDO) Relaxes constraints on the two-center two-electron integrals. Thus all integrals of (  |  ) are retained provided that  and are on the same center and that and  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 ( ) Quantitative Structure-Property Relationship (QSPR) – correlation developed between structure and physical properties. Typically used to design materials and predict properties.