Advanced Quantum Chemistry III: Part 6

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1 Advanced Quantum Chemistry III: Part 6 Norio Yoshida Kyushu University Last updated Winter Term 1

2 Quantum Chemistry for Condensed Phase Liquid phase Solid phase Biological systems 2

3 Divide the system Intermolecular Intramolecular Quantum Classical Biological systems Solute- solvent systems Mul - scale Mul - physics 3

4 Modeling inter- and intra- molecular interac on 4

+ ˆ V AB Ψ AB, E B 0 = Ψ B 0 Ĥ B Ψ B 0 Molecule A Intermolecular

5 Intermolecular interac ons Total energy of the system E Total = Ψ AB Energy of isolated system E A 0 = Ψ A 0 Ĥ A Ψ A 0 Ĥ A + Ĥ B + ˆ V AB Ψ AB, E B 0 = Ψ B 0 Ĥ B Ψ B 0 Molecule A Intermolecular interac on energy Molecule B V A B = E Total (E A 0 + E B 0 ) 5

6 Components of intermolecular interac on energy V A B = E ES + E PL + E EX + E CT + E MIX E ES : Electrosta c energy E PL : Polariza on energy E EX : Exchange energy E CT : Charge transfer energy E MIX : Coupling term ( E disp : Dispersion) Kitaura, K. and Morokuma, K., Int. J. Quantum Chem., X, (1976) 6

7 MM force field Nbond MM 1 r V ( R) = k R R 2 bond( ij ) N angle( ijk ) 2 ( 0) ij ij ij angle 1 θ + k 2 dihedral ( ijkl ) n( dihedral ) 2 ( 0 θ θ ) ijk ijk ijk Ndihedral N φn d Vijkl 1 cos( n + ' nφ ) ijkl γ ( ijkl 2 ) * Natom ' Aij B ( LJ ij + δij , i < j + ) Rij Rij,* N atom ' q q ( cl i j + δij +, i < j + ) εrij,* Intramolecular Intermolecular 7

8 Intramolecular force field 1 V k R R Nbond MM r intra ( R) = ij ij ij bond( ij ) 2 N angle( ijk ) 2 ( 0) angle 1 θ + k 2 dihedral ( ijkl ) n( dihedral ) 2 ( 0 θ θ ) ijk ijk ijk i q ijk R ij j f ijkl Ndihedral N φn d Vijkl 1 cos( n + % nφ ) ijkl γ & ijkl 2 ' ( k l 8

9 Bond and Angle N bond bond(ij ) 1 2 k ij r ( 0 R ij R ) 2 ij k r ij : spring constant i j R ij R ij 0 N angle angle(ijk ) 1 2 θ k ijk ( 0 θ ijk θ ) 2 ijk k θ ijk : spring constant i k θ ijk j θ ijk 0 9

10 N dihedral N φn V d ijkl dihedral(ijkl) n(dihedral) 2 Dihedral % n & 1 cos nφ ijkl γ ijkl ( ) ' ( n=1 j k l φ ijkl φ ijkl 0 n=2 i φ ijkl 0 10

11 11

12 Intermolecular force field V MM inter ( R) Natom # Aij B $ LJ ij = δij & 12 6 ' i < j &( Rij Rij ') # q q $ Natom cl i j + δij & ' i < j εrij &( ') 12

13 Determine the parameter sets 13

14 Different chemical bonds have different parameters Parameters are assigned typical amino acids, nucleic acids and organic compounds. Famous parameter set OPLS AMBER CHARMM MM3 MM parameter set AMBER force field; JACS 117, (1995)

Ndihedral N φn d Vijkl 1 cos( n + ' nφ ) ijkl γ ( ijkl 2 ) * Natom ' Aij B ( LJ

15 Nbond MM 1 r V ( R) = k R R 2 bond( ij ) N angle( ijk ) MM parameter set 2 ( 0) ij ij ij angle 1 θ + k 2 dihedral ( ijkl ) n( dihedral ) 2 ( 0 θ θ ) ijk ijk ijk Ndihedral N φn d Vijkl 1 cos( n + ' nφ ) ijkl γ ( ijkl 2 ) * Natom ' Aij B ( LJ ij + δij , i < j + ) Rij Rij,* N atom ' q q ( cl i j + δij +, i < j + ) εrij,*

16 Parameter fi ng Calculate ab ini o value of energy Determine the parameters to fit the ab ini o values i j R ij 16

17 Effective charge on atom: Electrostatic potential (ESP) method Population analysis Mulliken population analysis Löwdin population analysis Natural population analysis Electrostatic potential (ESP) method The electrostatic potential generated by electron density distribution is very different from that by population analysis The effective charges must be determined to reproduce the electrostatic potential around the molecule

18 ESP method Set of the effective charges (q a ) are determined to minimize I. N grid I = ω ( α u(r α )!u(r α )) 2 N + 2λ q a N e α a u(r α ) = N a Z a r α R a tr(pa(r α )) Lagrange multiplier u(r α ) = N a q n a r α R a + constraint of total charge conservation N a q e a r α R a q = q n ca 1 + N + e ca 1 1 t 1a 1 1 t a ab = N grid 1 c α r aα r bα N grid α { B} a = ω α µν 1a 1 { } a = tr(pb a ) χ * µ (r ')χ ν (r ') r αa r α r ' dr ' { A(r α )} µν = χ * (r')χ µ ν (r') r α r' dr'

19 Restrained ESP method Set of the effective charges (q a ) are determined to minimize I. N grid & I RESP = ω ( α u(r α ) u(r α )) 2 N QM + 2λ ( q a Q tot α ' a ) N + + QM g q 2 a a * a Harmonic penalty function 0

20 VDW parameters Adjustable parameter σ (Diameter of Atom of Atom group) ε (Energe c parameter) Fit σ and ε to reproduce the density and enthalpy of liquids. ex. sp 2 C and aroma c H Monte Carlo simula on on benzene liquid and adjus ng the σ and ε to reproduce the density and enthalpy of liquid benzene. 20

21 QM/MM method for liquid phase 21

22 Quantum- Classical Quantum Mechanics/Molecular Mechanics (QM/MM) method a) Solute molecules are described by ab ini o QM methods. Solvent molecules are treated by MM. Modeling intermolecular interac ons Lennard- Jones poten al Coulomb interac on between point charge etc. a) M.J. Field, P.A. Bash, M. Karplus, J. Comput. Chem., 11 (1990)

23 Strategy Solute ( 溶質 ) Quantum mechanics(qm) Solute- solvent interac on ( 溶質溶媒相互作用 ) Solvent ( 溶媒 ) Molecular Mechanics(MM) 23

24 QM/MM Hamiltonian Ĥ QM /MM = N elec 1 p 2 2 i i N elec N QM A Z A + i r ia N elec 1 + i> j r ij N QM A>B Z A Z B r AB QM N elec N MM q a + i a r ia N QM N MM A a Z A q a r Aa MM +V QM MM QM- MM MM +V MM MM 24

25 QM/MM energy E = Ψ Hˆ Ψ QM / MM QM / MM total E QM / MM el ( D, d) Z q NQM NQM NMM ZAZB A a MM VQM MM + A> B rab A a raa 1 E h D g d NBF NBF NBF NBF NBF NBF QM / MM QM / MM el ( D, d) = µν µν + µνσλ µνσλ µ ν 2 µ ν σ λ h = q ( ) ( ) NMM QM / MM * a µν = hµν + dx1φ µ x1 φν x1 a r1 a V MM MM 25

26 Interaction between QM-MM 1. Electrosta c interac on - Interac on between QM electron and MM charge 2. Polariza on interac on - QM molecule: Polarized by the electrosta c fields from MM molecule - MM molecule: Fixed 3. Exchange repulsion and dispersion - LJ poten al (MM Force field) 4. Charge transfer - Not available

Calculate QM energy Calculate forces ac ng on solute molecules

27 QM/MM- MD or MC To sample the distribu on of solvent molecule, QM/MM is coupled with MD or MC Ini al structure/distribu on Calculate QM energy Calculate forces ac ng on solute molecules Calculate forces ac ng on solvent molecules Es mate new structure/ coordinate 27

28 1st deriva ve of energy over atoms coordinates For QM atom QM / MM N dh MM µν dhµν dhµν d * qa = + 1φ µ 1 φν 1 dra dra dra dr dx x A a r x 1a ( ) ( ) For MM atom QM / MM NQM NMM detotal d $ ZAq % a MM MM = & + VQM MM + VMM ' dra dra ( A a raa ) NBF NBF $ * d q + % + & φ ( ) φ ( )' D µ ν & dx x x (. / )' * a 1 µ 1, - ν 1 µν dra r1 a 28

29 Example Intramolecular proton transfer reac on of glycine N. Okuyama-Yoshida, K. et al. J. Chem. Phys., 113, 3519 (2000). N. Takenaka, et al. Theor. Chem. Acc., 130, 215 (2011). Y. Kitamura, et al. Chem. Phys. Lett., 514, 261 (2011). N. Takenaka, et al. J. Chem. Phys., 137, (2012). 29

30 References コンピュータシミュレーションの基礎 ( 第 2 版 ) 岡崎進吉井範行化学同人分子シミュレーション上田顕裳華房 Essen als of Computa onal Chemistry Cramer, WILLEY Understanding Molecular Simula on Frenkel and Smit, ACADEMIC PRESS 30

Advanced Quantum Chemistry III: Part 6

Advanced Quantum Chemistry III: Part 6 Norio Yoshida Kyushu University Last updated 14-1-15 2013 Winter Term 1 Quantum Chemistry for Condensed Phase Liquid phase Solid phase Biological systems 2 Divide