Molecular Modeling: Semi-Empirical Methods

1. Molecular Modeling: Semi-Empirical Methods C372 Introduction to Cheminformatics II Kelsey Forsythe

2. Semi-Empirical Methods • Advantage • Faster than ab initio • Less sensitive to parameterization than MM methods • Disadvantage • Accuracy depends upon parameterization

3. Semi-Empirical Methods • Ignore Core Electrons • Approximate part of HF integration

4. Estimating Energy • Recall Eclassical <E>quantal

5. Approximate Methods • SCF (Self Consistent Field) Method (a.ka. Mean Field or Hartree Fock) • Pick single electron and average influence of remaining electrons as a single force field (V0 external) • Then solve Schrodinger equation for single electron in presence of field (e.g. H-atom problem with extra force field) • Perform for all electrons in system • Combine to give system wavefunction and energy (E) • Repeat to error tolerance (Ei+1-Ei)

6. Estimating Energy • Want to find c’s so that

7. Estimating Energy • F simulataneous equations gives a matrix equation

8. Matrix Algebra • Finding determinant akin to rotating matrix until diagonal ( )

9. Matrix Algebra

10. Huckel Theory • Assumptions • Atomic basis set - parallel 2p orbitals • No overlap between orbitals, • 2p Orbital energy equal to ionization potential of methyl radical (singly occupied 2p orbital) • The  stabilization energy is the difference between the 2p-parallel configuration and the 2p perpendicular configuration • Non-nearest interactions are zero

11. Ex. Allyl (C3H5) • One p-orbital per carbon atom - basis size = 3 • Huckel matrix is • Resonance stabilization same for allyl cation, radical and anion (NOT found experimentally)

12. Ex. Allyl (C3H5) • Huckel matrix (determinant form)-resonance (beta represents overlap/interaction between orbitals) In matrix (determinant form) • Energy of resonance system. Note the lowest energy is less than the isolated orbital/AO due (this is resonance stabilization) • Huckel matrix (determinant form)-no resonance • Energy of three isolated methylene sp2 orbitals Overlap between orbital 1 and orbital 2 (hence matrix element H12)

13. Extended Huckel Theory (aka Tight Binding Approximation) • Includes non-nearest neighbor orbital interactions • Experimental Valence Shell Ionization Potentials used to model matrix elements • Generally applicable to any element • Useful for calculating band structures in solid-state physics

14. Beyond One-Electron Formalism • HF method • Ignores electron correlation • Effective interaction potential • Hatree Product- Fock introduced exchange – (relativistic quantum mechanics)

15. HF-Exchange • For a two electron system • Fock modified wavefunction

16. Slater Determinants • Ex. Hydrogen molecule

17. Beyond One-Electron Formalism • HF method • Ignores electron correlation • Effective interaction potential • Hatree Product- Fock introduced exchange – (relativistic quantum mechanics)

18. CNDO (1965, Pople et al) MINDO (1975, Dewar ) MNDO (1977, Thiel) INDO (1967, Pople et al) ZINDO SINDO1 STO-basis (/S-spectra,/2 d-orbitals) /1/2/3, organics /d, organics, transition metals Organics Electronic spectra, transition metals 1-3 row binding energies, photochemistry and transition metals Neglect of Differential Overlap (NDO)

19. Semi-Empirical Methods • SAM1 • Closer to # of ab initio basis functions (e.g. d orbitals) • Increased CPU time • G1,G2 and G3 • Extrapolated ab initio results for organics • “slightly empirical theory”(Gilbert-more ab initio than semi-empirical in nature)

20. Semi-Empirical Methods • AM1 • Modified nuclear repulsion terms model to account for H-bonding (1985, Dewar et al) • Widely used today (transition metals, inorganics) • PM3 (1989, Stewart) • Larger data set for parameterization compared to AM1 • Widely used today (transition metals, inorganics)

21. General Reccommendations • More accurate than empirical methods • Less accurate than ab initio methods • Inorganics and transition metals • Pretty good geometry OR energies • Poor results for systems with diffusive interactions (van der Waals, H-bonded, radicals etc.)

22. Complete Neglect of Differential Overlap (CNDO) • Overlap integrals, S, is assumed zero

23. Neglect of Differential Overlap (NDO) Gives rise to overlap between electronic basis functions of different types and on different atoms

24. Complete Neglect of Differential Overlap (CNDO) • One-electron overlap integral for different electrons is zero (as in Huckel Theory) • Two-electron integrals are zero if basis functions not identical

25. Intermediate Neglect of Differential Overlap (CNDO) • Overlap integrals, S, is assumed zero

26. Eigenvalue Equation • Matrix * Vector = Matrix (diagonal) * Vector • Schrodinger’s equation! The solutions to this differential equation are equal to the solutions to the matrix eigenvalue equation