Molecular Modeling : Beyond Empirical Equations Quantum Mechanics Realm C372 Introduction to Cheminformatics II Kelsey Forsythe

Atomistic Model History Atomic Spectra Atomic Spectra Balmer (1885) Balmer (1885) Plum-Pudding Model Plum-Pudding Model J. J. Thomson (circa 1900) J. J. Thomson (circa 1900) UV Catastrophe-Quantization UV Catastrophe-Quantization Planck (circa 1905) Planck (circa 1905) Planetary Model Planetary Model Neils Bohr (circa 1913) Neils Bohr (circa 1913) Wave-Particle Duality Wave-Particle Duality DeBroglie (circa 1924) DeBroglie (circa 1924) Uncertainty Principle (Heisenberg) Uncertainty Principle (Heisenberg) Schrodinger Wave Equation Schrodinger Wave Equation Erwin Schrodinger and Werner Heisenberg(1926) Erwin Schrodinger and Werner Heisenberg(1926)

Classical vs. Quantum Trajectory Trajectory Real numbers Real numbers Deterministic (“The value is ___”) Deterministic (“The value is ___”) Variables Variables Continuous energy spectrum Continuous energy spectrum Wavefunction Complex (Real and Imaginary components) Probabilistic (“The average value is __ ” Operators Discrete/Quantized energy Tunneling Zero-point energy

Schrodinger’s Equation - Hamiltonian operator - Hamiltonian operator Gravity? Gravity?

Hydrogen Molecule Hamiltonian Born-Oppenheimer Approximation (Fix nuclei) Born-Oppenheimer Approximation (Fix nuclei) Now Solve Electronic Problem Now Solve Electronic Problem

Electronic Schrodinger Equation Solutions: Solutions:, the basis set, are of a known form, the basis set, are of a known form Need to determine coefficients (c Need to determine coefficients (c m ) Wavefunctions gives probability of finding electrons in space (e. g. s,p,d and f orbitals) Wavefunctions gives probability of finding electrons in space (e. g. s,p,d and f orbitals) Molecular orbitals are formed by linear combinations of atomic orbitals (LCAO) Molecular orbitals are formed by linear combinations of atomic orbitals (LCAO)

Hydrogen Molecule HOMO HOMO LUMO LUMO VBT

Hydrogen Molecule Bond Density Bond Density

Ab Initio/DFT Complete Description! Complete Description! Generic! Generic! Major Drawbacks: Major Drawbacks: Mathematics can be cumbersome Mathematics can be cumbersome Exact solution only for hydrogen Exact solution only for hydrogen Informatics Informatics Approximate solution time and storage intensive Approximate solution time and storage intensive –Acquisition, manipulation and dissemination problems

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 )

Recall Schrodinger Equation Schrodinger Equation Schrodinger Equation Schrodinger Equation Quantum vs. Classical Quantum vs. Classical Quantum vs. Classical Quantum vs. Classical Born Oppenheimer Born Oppenheimer Born Oppenheimer Born Oppenheimer Hartree-Fock (aka SCF/central field) method Hartree-Fock (aka SCF/central field) method Hartree-Fock (aka SCF/central field) method Hartree-Fock (aka SCF/central field) method

Basis Sets Each atomic orbital/basis function is itself comprised of a set of standard functions Each atomic orbital/basis function is itself comprised of a set of standard functions STO(Slater Type Orbital): ~Hydrogen Atom Solutions GTO(Gaussian Type Orbital): More Amenable to computation Contraction coefficient (Static for calculation) Expansion Coefficient Atomic Orbital LCAO

STO vs. GTO GTO Improper behavior for small r (slope equals zero at nucleus) Decays too quickly

Basis Sets Optimized using atomic ab initio calculations What “we” do!! Basis Sets

Gaussian Type Orbitals Primitives Primitives Shapes typical of H-atom orbitals (s,p,d etc) Shapes typical of H-atom orbitals (s,p,d etc) Contracted Contracted Vary only coefficients of valence (chemically interesting parts) in calculation Vary only coefficients of valence (chemically interesting parts) in calculation

Minimum Basis Set (STO-3G) The number of basis functions is equal to the minimum required to accommodate the # of electrons in the system The number of basis functions is equal to the minimum required to accommodate the # of electrons in the system H(# of basis functions=1)-1s H(# of basis functions=1)-1s Li-Ne(# of basis functions=5) 1s,2s,2p x, 2 y, 2p z Li-Ne(# of basis functions=5) 1s,2s,2p x, 2 y, 2p z

Basis Sets Types: STO-nG(n=integer)-Minimal Basis Set STO-nG(n=integer)-Minimal Basis Set Approximates shape of STO using single contraction of n- PGTOs (typically, n=3) Approximates shape of STO using single contraction of n- PGTOs (typically, n=3) Intuitive Intuitive The universe is NOT spherical!! The universe is NOT spherical!! 3-21G (Split Valence Basis Sets) 3-21G (Split Valence Basis Sets) Core AOs 3-PGTOs Core AOs 3-PGTOs Valence AOs with 2 contractions, one with 2 primitives and other with 1 primitive Valence AOs with 2 contractions, one with 2 primitives and other with 1 primitive

Basis Sets Types: 3-21G(*)-Use of d orbital functions (2 nd row atoms only)-ad hoc 3-21G(*)-Use of d orbital functions (2 nd row atoms only)-ad hoc 6-31G*-Use of d orbital functions for non-H atoms 6-31G*-Use of d orbital functions for non-H atoms 6-31G**-Use of d orbital functions for H as well 6-31G**-Use of d orbital functions for H as well

Examples C STO-3G-Minimal Basis Set STO-3G-Minimal Basis Set 3 primitive gaussians used to model each STO 3 primitive gaussians used to model each STO # basis functions = 5 (1s,2s,3-2p’s) # basis functions = 5 (1s,2s,3-2p’s) 3-21G basis-Valence Double Zeta 3-21G basis-Valence Double Zeta 1s (core) electrons modeled with 3 primitive gaussians 1s (core) electrons modeled with 3 primitive gaussians 2s/2p electrons modeled with 2 contraction sets (2- primitives and 1 primitive) 2s/2p electrons modeled with 2 contraction sets (2- primitives and 1 primitive) # basis functions = 8 (1s,2s,6-2p’s) # basis functions = 8 (1s,2s,6-2p’s)

Polarization Addition of higher angular momentum functions Addition of higher angular momentum functions HCN HCN Addition of p-function to H (1s) basis better represents electron density (ie sp character) of HC bond Addition of p-function to H (1s) basis better represents electron density (ie sp character) of HC bond

Diffuse functions Addition of basis functions with small exponents (I.e. spatial spread is greater) Addition of basis functions with small exponents (I.e. spatial spread is greater) Anions Anions Radicals Radicals Excited States Excited States Van der Waals complexes (Gilbert) Van der Waals complexes (Gilbert) Ex. Benzene-Dimers (Gilbert) Ex. Benzene-Dimers (Gilbert) w/o Diffuse functions T-shaped optimum w/o Diffuse functions T-shaped optimum w/Diffuse functions parallel-displaced optimum w/Diffuse functions parallel-displaced optimum

Computational Limits Hartree-Fock limit Hartree-Fock limit NOT exact solution NOT exact solution Does not include correlation Does not include correlation Does not include exchange Does not include exchange Exact Energy* Basis set size Correlation/Exchange BO not withstanding

Correcting Approximations Accounting for Electron Correlations Accounting for Electron Correlations DFT(Density Functional Theory) DFT(Density Functional Theory) Moller-Plesset (Perturbation Theory) Moller-Plesset (Perturbation Theory) Configuration Interaction (Coupling single electron problems) Configuration Interaction (Coupling single electron problems)

Computational Reminders HF typically scales N 4 HF typically scales N 4 As increase basis set size accuracy/calculation time increases As increase basis set size accuracy/calculation time increases ALL of these ideas apply to any program utilizing ab initio techniques NOT just Spartan (Gilbert) ALL of these ideas apply to any program utilizing ab initio techniques NOT just Spartan (Gilbert)

Quick Guide Basis Basis STO-3G(minimal basis) STO-3G(minimal basis) 3-21G-6-311G(split-valence basis) 3-21G-6-311G(split-valence basis) */** */** +/++ +/++ Meaning 3 PGTO used for each STO/atomic orbital Additional basis functions for valence electrons Addition of d-type orbitals to calculation (polarization) ** (for H as well) Diffuse functions (s and p type) added ++ (for H as well)

Modeling Nuclear Motion IR - Vibrations IR - Vibrations NMR – Magnetic Spin NMR – Magnetic Spin Microwave – Rotations Microwave – Rotations

Modeling Nuclear Motion (Vibrations) Harmonic Oscillator Hamiltonian