C372_L4_qm - Beyond Empirical Equations Quantum Mechanics...

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Unformatted text preview: Beyond Empirical Equations Quantum Mechanics Realm C372 Introduction to Cheminformatics II Kelsey Forsythe Molecular Modeling: Atomistic Model History Atomic Spectra PlumPudding Model Balmer (1885) UV CatastropheQuantization J. J. Thomson (circa 1900) Planck (circa 1905) Planetary Model WaveParticle Duality Neils Bohr (circa 1913) DeBroglie (circa 1924) Uncertainty Principle (Heisenberg) Erwin Schrodinger and Werner Heisenberg(1926) Schrodinger Wave Equation 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 Zeropoint energy Schrodinger's Equation = EY HY Hamiltonian operator H H =T +V - N i 2 2mi N 2 i< j C e ie j ri - rj Gravity? Hydrogen Molecule Hamiltonian 2 2 2 H = T +V h2 2 H =- 2 2 2 2 h 21 h e2 h h p p2 e1 + + + + h m p m p me me h h h 1 1 1 1 1 1 h h Ch + - - - - hre1e 2 rp1 p 2 rp1e1 rp1e 2 rp 2 e1 rp 2 e 2 h BornOppenheimer Approximation (Fix nuclei) 2 2 2 H el = Tel + Vel - nuclei + Vnuclei 2 2 h2 h e1 h e 2 h h 1 1 1 1 1 h 1 h 2 =- h H el + + Ch - - - - +C 2 h me me rp1 p 2 hre1e 2 rp1e1 rp1e 2 rp 2 e1 rp 2 e 2 h Now Solve Electronic Problem Electronic Schrodinger Equation Solutions: v ( r ) = F v c m * F m (r ) m v, the basis set, are of a known form m ( r ) 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 (f A + f B ) 2 LUMO YLUMO = 1 (f A - f B ) 2 Hydrogen Molecule Bond Density Ab Initio/DFT Complete Description! Generic! Major Drawbacks: Mathematics can be cumbersome Exact solution only for hydrogen Approximate solution time and storage intensive Informatics 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. Hatom problem with extra force field) 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 (Ei+1Ei) Recall Schrodinger Equation Quantum vs. Classical Born Oppenheimer HartreeFock (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 v ( r ) = v c m * F m (r ) - z mj r 2 m N Expansion Coefficient Contraction coefficient (Static for calculation) - z mj r Fm = j Cmj e STO(Slater Type Orbital): ~Hydrogen Atom Solutions m GTO(Gaussian Type Orbital): m More Amenable to computation - z 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 v v (r ) = o cm m (r ) What "we" do!! m Atomic Orbital m = h Cmj c j GTO/CGTO STO N j Optimized using atomic ab initio calculations PGTO j e - z j r2 Gaussian Type Orbitals Primitives ,n,l,m (r,q,j ) = NYl,m (q,j )r (2n- 2- l ) - zr 2 e Shapes typical of Hatom 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 LiNe(# of basis functions=5) 1s,2s,2p , 2 , 2p x y z Types: Basis Sets STOnG(n=integer)Minimal Basis Set 321G (Split Valence Basis Sets) Approximates shape of STO using single contraction of n PGTOs (typically, n=3) Intuitive The universe is NOT spherical!! Core AOs 3PGTOs Valence AOs with 2 contractions, one with 2 primitives and other with 1 primitive Types: Basis Sets 321G(*)Use of d orbital functions (2nd row atoms only)ad hoc 631G*Use of d orbital functions for nonH atoms 631G**Use of d orbital functions for H as well Examples C STO3GMinimal Basis Set 321G basisValence Double Zeta 3 primitive gaussians used to model each STO # basis functions = 5 (1s,2s,32p's) 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,62p's) Polarization Addition of higher angular momentum functions HCN Addition of pfunction 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. BenzeneDimers (Gilbert) w/o Diffuse functions Tshaped optimum w/Diffuse functions paralleldisplaced optimum Computational Limits HartreeFock 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) MollerPlesset (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 STO3G(minimal basis) 321G6311G(split valence basis) */** +/++ Meaning 3 PGTO used for each STO/atomic orbital Additional basis functions for valence electrons Addition of dtype 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 h2 2 H ( r ) = - 2 8.35E28 8.35E28 8.35E28 8.35E28 1.4E-18 8.35E28 8.35E28 8.35E28 1.2E-18 8.35E28 8.35E28 1E-18 8.35E28 8.35E28 8E-19 8.35E28 8.35E28 8.35E28 6E-19 8.35E28 8.35E28 4E-19 8.35E28 8.35E28 2E-19 8.35E28 8.35E28 8.35E28 0 8.35E28 0 8.35E28 8.77567E+14 1 + ( r ) 2 r 2 2.03098E18 1.05374E18 1.54682E18 1.34201E18 1.15913E18 9.96207E19 8.51451E19 7.23209E19 6.09973E19 5.10362E19 4.2311E19 3.47061E19 2.81155E19 2.24426E19 1.75987E19 1.35031E19 1.0082E19 7.26787E20 4.99924E20 3.22001E20 1.87901E20 2 9.29638E21 2.5 3.29443E21 8.82365E19 8.02375E19 7.26185E19 6.53795E19 5.85205E19 5.20415E19 4.59425E19 4.02235E19 3.48845E19 2.99255E19 2.53465E19 2.11475E19 1.73285E19 1.38895E19 1.08305E19 8.15147E20 5.85247E20 3.93347E20 2.39447E20 1.23547E20 3 3.5 4.56475E21 Empirical Potential for Hydrogen Molecule 8.77567E+14 20568787140 1.77569E18 9.66155E19 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 1 20568787140 0.5 1.5 8.77567E+14 20568787140 20568787140 4 ...
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This note was uploaded on 06/28/2011 for the course C 372 taught by Professor Yoonsuplee during the Spring '11 term at Korea Advanced Institute of Science and Technology.

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