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Molecular Modeling: Beyond Empirical Equations Quantum Mechanics Realm C372 Introduction to Cheminformatics II Kelsey Forsythe Atomistic Model History Atomic Spectra Plum-Pudding Model Neils Bohr (circa 1913) Wave-Particle Duality Planck (circa 1905) Planetary Model J. J. Thomson (circa 1900) UV Catastrophe-Quantization Balmer (1885) DeBroglie (circa 1924) Uncertainty Principle (Heisenberg) Schrodinger Wave Equation Erwin Schrodinger and Werner Heisenberg(1926) Classical vs. Quantum 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 Schrodinger’s Equation Hˆ E Hˆ - Hamiltonian operator Hˆ Tˆ Vˆ N Gravity? i 2 2mi N 2 C i j e ie j ri r j Hydrogen Molecule Hamiltonian Hˆ Tˆ Vˆ 2 Hˆ 2 2p1 2p 2 e21 e22 m m m m p p e e 1 1 1 1 1 1 C re1e 2 rp1 p 2 rp1e1 rp1e 2 rp 2e1 rp 2e 2 Born-Oppenheimer Approximation (Fix nuclei) Hˆ el Tˆel Vˆel nuclei Vnuclei 2 2 2 1 1 1 1 1 1 e 1 e 2 ˆ H el C C 2 me me rp1 p 2 re1e 2 rp1e1 rp1e 2 rp 2e1 rp 2e 2 Now Solve Electronic Problem Electronic Schrodinger Equation Solutions: F (r ) c m m (r ) m m (r ), 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 HOMO 1 ( A B ) 2 LUMO LUMO 1 ( A B ) 2 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 F LCAO (r ) c m m (r ) m N m Cmje mj r 2 j Expansion Coefficient Contraction coefficient (Static for calculation) STO(Slater Type Orbital): ~Hydrogen Atom Solutions GTO(Gaussian Type Orbital): m More Amenable to computation mj r 2 STO vs. GTO GTO Improper behavior for small r (slope equals zero at nucleus) Decays too quickly Basis Sets Basis Sets Molecular Orbital F (r ) cm m (r ) What “we” do!! m Atomic Orbital N m Cmj j STO GTO/CGTO j Optimized using atomic ab initio calculations PGTO j e j r 2 Gaussian Type Orbitals Primitives ,n,l,m (r,, ) NYl,m (, )r (2n2l ) r 2 e 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 (2primitives 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 Basis set size BO not withstanding 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) Quick Guide Basis Meaning STO-3G(minimal basis) 3-21G-6-311G(split-valence basis) */** 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 NMR – Magnetic Spin Microwave – Rotations Modeling Nuclear Motion (Vibrations) Harmonic Oscillator Hamiltonian 2 1 ˆ H (r ) (r ) 2 2 r 2 8.35E-28 8.35E-28 8.35E-28 8.35E-28 1.4E-18 8.35E-28 8.35E-28 8.35E-28 1.2E-18 8.35E-28 8.35E-28 1E-18 8.35E-28 8.35E-28 8E-19 8.35E-28 8.35E-28 8.35E-28 6E-19 8.35E-28 8.35E-28 4E-19 8.35E-28 8.35E-28 2E-19 8.35E-28 8.35E-28 8.35E-28 0 8.35E-28 0 8.35E-28 8.77567E+14 20568787140 2.03098E-18 1.05374E-18 8.77567E+14 8.77567E+14 8.77567E+14 8.77567E+14 8.77567E+14 8.77567E+14 8.77567E+14 8.77567E+14 8.77567E+14 8.77567E+14 8.77567E+14 8.77567E+14 8.77567E+14 8.77567E+14 8.77567E+14 8.77567E+14 8.77567E+14 8.77567E+14 8.77567E+14 8.77567E+14 0.5 1 8.77567E+14 20568787140 20568787140 20568787140 20568787140 20568787140 20568787140 20568787140 20568787140 20568787140 20568787140 20568787140 20568787140 20568787140 20568787140 20568787140 20568787140 20568787140 20568787140 20568787140 20568787140 1.5 20568787140 1.54682E-18 1.34201E-18 1.15913E-18 9.96207E-19 8.51451E-19 7.23209E-19 6.09973E-19 5.10362E-19 4.2311E-19 3.47061E-19 2.81155E-19 2.24426E-19 1.75987E-19 1.35031E-19 1.0082E-19 7.26787E-20 4.99924E-20 3.22001E-20 1.87901E-20 2 9.29638E-21 2.5 3.29443E-21 8.82365E-19 8.02375E-19 7.26185E-19 6.53795E-19 5.85205E-19 5.20415E-19 4.59425E-19 4.02235E-19 3.48845E-19 2.99255E-19 2.53465E-19 2.11475E-19 1.73285E-19 1.38895E-19 1.08305E-19 8.15147E-20 5.85247E-20 3.93347E-20 2.39447E-20 31.23547E-20 3.5 4.56475E-21 Empirical for Hydrogen Molecule9.66155E-19 8.77567E+14Potential 20568787140 1.77569E-18 4