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

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

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

4. Estimating Energy Recall E classical quantal Recall E classical quantal

5. Approximate Methods SCF (Self Consistent Field) Method (a.ka. Mean Field or Hartree Fock) 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 (V external) Pick single electron and average influence of remaining electrons as a single force field (V 0 external) Then solve Schrodinger equation for single electron in presence of field (e.g. H-atom problem with extra force field) 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 Perform for all electrons in system Combine to give system wavefunction and energy (E Combine to give system wavefunction and energy (E) Repeat to error tolerance (E Repeat to error tolerance (E i+1 -E i )

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

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

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

9. Matrix Algebra

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

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

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

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

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

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

16. Slater Determinants Ex. Hydrogen molecule Ex. Hydrogen molecule

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

18. Neglect of Differential Overlap (NDO) CNDO (1965, Pople et al) CNDO (1965, Pople et al) MINDO (1975, Dewar ) MINDO (1975, Dewar ) MNDO (1977, Thiel) MNDO (1977, Thiel) INDO (1967, Pople et al) INDO (1967, Pople et al) ZINDO ZINDO SINDO1 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

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

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

21. General Reccommendations More accurate than empirical methods More accurate than empirical methods Less accurate than ab initio methods Less accurate than ab initio methods Inorganics and transition metals Inorganics and transition metals Pretty good geometry OR energies Pretty good geometry OR energies Poor results for systems with diffusive interactions (van der Waals, H-bonded, radicals etc.) 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 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) One-electron overlap integral for different electrons is zero (as in Huckel Theory) Two-electron integrals are zero if basis functions not identical Two-electron integrals are zero if basis functions not identical

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

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