Molecular Modeling : Beyond Empirical Equations Quantum Mechanics Realm

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

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

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

Schrodinger’s Equation • - Hamiltonian operator • Gravity?

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

Electronic Schrodinger Equation • Solutions: • , the basis set, are of a known form • Need to determine coefficients (cm) • 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)

Hydrogen Molecule VBT • HOMO • LUMO

Hydrogen Molecule • Bond Density

Ab Initio/DFT • Complete Description! • Generic! • Major Drawbacks: • Mathematics can be cumbersome • Exact solution only for hydrogen • Informatics • 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) • 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)

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

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

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

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

Gaussian Type Orbitals • Primitives • Shapes typical of H-atom orbitals (s,p,d etc) • Contracted • 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 • H(# of basis functions=1)-1s • Li-Ne(# of basis functions=5) 1s,2s,2px, 2y, 2pz

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

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

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

Polarization • Addition of higher angular momentum functions • HCN • 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) • Anions • Radicals • Excited States • Van der Waals complexes (Gilbert) • Ex. Benzene-Dimers (Gilbert) • w/o Diffuse functions T-shaped optimum • w/Diffuse functions parallel-displaced optimum

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

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

Computational Reminders • HF typically scales N4 • 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)

Basis STO-3G(minimal 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) Quick Guide

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

Modeling Nuclear Motion (Vibrations)Harmonic Oscillator Hamiltonian

Molecular Modeling : Beyond Empirical Equations Quantum Mechanics Realm