Semiempirical, empirical and hybrid methods

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1 Semiempirical, empirical and hybrid methods Gérald MONARD Théorie - Modélisation - Simulation UMR 7565 CNRS - Université de Lorraine Faculté des Sciences - B.P Vandœuvre-les-Nancy Cedex - FRANCE

2 Outline. Semiempirical QM methods NDDO methods DFTB methods e.g., Organic reaction energies and barriers 2. Empirical Methods Molecular Mechanics (MM) e.g., Protein folding 3. Hybrid methods Combined QM/MM e.g., Reaction free energies in proteins

3 Standard QM algorithm (Hartree-Fock) Roothan Equations (closed shells) Total energy E = 2 µ ν occ Density matrix element P µν = 2 j P µν [H c µν + F µν ] + A c µj c νj Fock matrix element F µν = H c µν + λ bielectronic integrals (µν λ η) = B>A Z A Z B R AB (c µj : M.O. coefficients) [ P λη (µν λη) ] η 2 (µη λν) χ µ ()χ ν () r 2 χ λ (2)χ η (2) dr dr 2 The Roothan equations FC = SCε (ε : M.O. eigenvalues) (S : overlap matrix C : M.O. coefficient matrix)

4 Hartree-Fock SCF algorithm. Compute mono- and bielectronic integrals O(N 4 ) 2. Build core hamiltonian (invariant) H c 3. Guess an initial density matrix 4. Build the Fock matrix F 5. Orthogonal transformation using S /2 O(N 3 ) F C = εc 6. Diagonalization of the Fock matrix F O(N 3 ) The C coefficients are obtained 7. Inverse transformation C C 8. Build the new density matrix O(N 3 ) Back to 4. unless convergence

5 The QM scaling problem energy of a water cluster (3-2G basis set) energy of a water cluster (6-3G* basis set) 35 3 B3LYP/3-2G BLYP/3-2G CCSD(T)/3-2G MP2/3-2G HF/3-2G 35 3 B3LYP/6-3G* BLYP/6-3G* CCSD(T)/6-3G* MP2/6-3G* HF/6-3G* wall clock CPU time (seconds) wall clock CPU time (seconds) number of water molecules number of water molecules (H 2 O) n water cluster (n from to 26) energy calculations Gaussian G9.B (NProcShared=4, Mem=8Gb, MaxDisk=36Gb) Wall clock time limit: hour Intel(R) Xeon(R) CPU E GHz (8 cores) 32Gb RAM wall clock CPU time (seconds) energy of a water cluster (6-3+G** basis set) number of water molecules B3LYP/6-3+G** BLYP/6-3+G** CCSD(T)/6-3+G** MP2/6-3+G** HF/6-3+G**

6 Quantum Chemistry is CPU intensive Theoretical CPU scaling order for different QM methods QM method Scaling semiempirical O(N 3 ) DFT O(N 3 ) ab initio O(N 4 ) MP2 O(N 5 ) Full CI O(exp N ) The (H 2 O) n example: n max in /2 hour (4 cores) HF BLYP B3LYP MP2 CCSD(T) 3-2G G* G**

7 How to solve the QM scaling problem? Moore s Law: CPU power doubles every 8 months doubling a molecular system is possible: O(N 3 ) scaling: every 8x3 months = 4.5 years O(N 4 ) scaling: every 6 years O(N 5 ) scaling: every 7.5 years, etc. Parallelism is not a valid option in the long run Good speeds-up are difficult to obtain (Amdahl s Law) non linear scaling of the standard algorithms change the methods: use approximate quantum methods semiempirical QM methods molecular mechanics (MM) force fields combined QM/MM methods change the algorithms Linear scaling algorithms

8 They are as old as ab initio methods Semiempirical methods PPP (Pariser-Parr-Pople) method 95s Extended Huckel method 96s CNDO 96s INDO 96s etc. A shared assumption ab initio (HF) calculations are too time consuming the equations are simplified to yield accessible timings for real molecules some parameters are introduced to correct the loss of information these parameters are obtained from experimental data empirical parameters (hence the term semiempirical methods)

9 NDDO based semiempirical methods NDDO: Neglect of Diatomic Differential Overlap Most modern semiempirical methods are NDDO based: MNDO (977) AM (985) PM3 (989) PDDG/PM3 & PDDG/MNDO (22) PM6 (27) PM7 (23) and going... They are based on a simplification of the Hartree-Fock equations

10 NDDO approximations Only valence shell electrons are considered core electrons are taken into account by reducing the nuclei charges (effective nuclei charge) and by introducing empirical functions to model the interactions between (nuclei+core electrons) and the other particles A minimal basis set is used. Usually: minimal Slater Type Orbital basis set ZDO approximation (Zero Differential Overlap): All products between basis functions corresponding to a single electron but centered on different atoms are neglected: ϕ A µ (i).ϕ B ν (i) = if A B

11 Consequences of the ZDO approximation The overlap matrix S is equal to unity: S = I There is no orthogonalization step in the SCF procedure one-electron three-center integrals (two centers for the basis functions and one center for the operator) are considered to be equal to zero All three-center and four-center bielectronic integrals are neglected (these are the most numerous integrals) The number of integrals scales as O(N 2 ) (where N is the number of basis functions) Z A Z B in HF equations is replaced by A A>B R AB A A>B f AB (R AB ) a parameterized core-core repulsion function

12 Roothan Equations (closed shells) Standard NDDO algorithm Total energy E = 2 µ ν occ Density matrix element P µν = 2 j P µν [H c µν + F µν ] + A c µj c νj Fock matrix element F µν = H c µν + λ B>A f AB R AB (c µj : M.O. coefficients) [ P λη (µν λη) ] η 2 (µη λν) bielectronic integrals (µν λ η) = fast analytical functions The Roothan equations FC = Cε (ε : M.O. eigenvalues)

13 PM3 vs. ab initio energy of a water cluster (3-2G basis set vs. PM3) energy of a water cluster (3-2G basis set vs. PM3) 35 B3LYP/3-2G BLYP/3-2G B3LYP/3-2G BLYP/3-2G CCSD(T)/3-2G MP2/3-2G CCSD(T)/3-2G MP2/3-2G 3 HF/3-2G PM3 8 HF/3-2G PM3 wall clock CPU time (seconds) wall clock CPU time (seconds) number of water molecules number of water molecules (H 2 O) n water cluster (n from to 26) energy calculations Gaussian G9.B Wall clock time limit: hour Intel(R) Xeon(R) CPU E GHz (8 cores) 32Gb RAM 3-2G: NProcShared=4 Mem=8Gb MaxDisk=36Gb PM3: NProcShared=

14 Determination of the semiempirical parameters The semiempirical parameters are optimized (=fitted) to reproduce a given set of experimental data from small molecules in gas phase: geometrical structures heat of formation ( H f ) dipolar moments ionization potentials MNDO, AM, and PM3, etc. are different because they make use of different semiempirical equations, different number of parameters, different number of optimized parameters (experimental parameters vs. optimized parameters), and different sets of experimental data.

15 Advantages and disadvantages of the semiempirical methods A lot faster than Hartree-Fock and post-hartree-fock methods. Electronic correlation is implicitly taken into account through the use parameters fitted from experimental data. Give, when properly used, better results than Hartree-Fock method. The quality of a semiempirical computation is dependant on the way the semiempirical parameters have been fitted: experimental = small gas phase domain of validity for data molecule semiempirical methods!

16 Semiempirical: what is it good for? enthalpies, heats of formation gas phase geometries (stable structures) of small molecules transition state geometries frequency calculations intermolecular interactions ( currently being improved)

17 Density Functional Tight Binding () Another approximate QM method Similar, in some sense, to NDDO/MNDO based methods: 2-3 orders of magnitude faster than ab initio methods minimal AO basis set (Slater type-orbitals) valence electrons only diatomic (no 3- or 4- center terms) O(N 3 ) bottleneck (diagonalization) But DFT based approach rather than a Hartree-Fock approximation

18 Density Functional Tight Binding (2) Another approximate QM method Similar, in some sense, to NDDO/MNDO based methods: 2-3 orders of magnitude faster than ab initio methods minimal AO basis set (Slater type-orbitals) valence electrons only diatomic (no 3- or 4- center terms) O(N 3 ) bottleneck (diagonalization) But DFT based approach rather than a Hartree-Fock approximation Expansion of the Energy using atomic densities with E[ρ] = E [ρ ] + E [ρ,δρ] + E 2 [ρ,(δρ) 2 ] + E 3 [ρ,(δρ) 3 ] +... ρ = atoms ρ A (ρ A : atomic densities, pre-computed) A δ ρ = deviation from the reference = the unknown

19 Density Functional Tight Binding (3) Different schemes based on the order of the expansion DFTB DFTB2, also referred to as SCC-DFTB DFTB3 E DFTB3 = 2 V rep AB AB + i MOs µ A ν B n i C µi C νi H µν 2 q A q B γab h + AB 3 ( q A ) 2 q B Γ AB AB

20 Density Functional Tight Binding (4) Different schemes based on the order of the expansion DFTB DFTB2, also referred to as SCC-DFTB DFTB3 E DFTB3 = 2 V rep AB AB + i MOs µ A ν B n i C µi C νi H µν 2 q A q B γab h + AB 3 ( q A ) 2 q B Γ AB AB E [ρ ] = 2 V rep AB AB short-ranged function with exp. decay

21 Density Functional Tight Binding (5) Different schemes based on the order of the expansion DFTB DFTB2, also referred to as SCC-DFTB DFTB3 E DFTB3 = 2 V rep AB AB + i MOs µ A ν B n i C µi C νi H µν 2 q A q B γab h + AB 3 ( q A ) 2 q B Γ AB AB H µν =< µ H ν >=< µ H[ρ A + ρ B ] ν >

22 Density Functional Tight Binding (6) Different schemes based on the order of the expansion DFTB DFTB2, also referred to as SCC-DFTB DFTB3 E DFTB3 = 2 V rep AB AB + i MOs µ A ν B n i C µi C νi H µν 2 q A q B γab h + AB 3 ( q A ) 2 q B Γ AB AB E 2 [ρ,(δρ) 2 ] = 2 AB q A q B γ h AB δ q q monopole approximation (Mulliken point charges) γ h AB : analytical function which converges to /R AB

23 Density Functional Tight Binding (7) Different schemes based on the order of the expansion DFTB DFTB2, also referred to as SCC-DFTB DFTB3 E DFTB3 = 2 V rep AB AB + i MOs µ A ν B n i C µi C νi H µν 2 q A q B γab h + AB 3 ( q A ) 2 q B Γ AB AB E 3 [ρ,(δρ) 3 ] = 3 AB( q A ) 2 q B Γ AB Γ derivative of γ h with respect to charges

24 Semiempirical methods Selected reviews Thiel, W., WIRE Comput. Mol. Sci., 24, 4(2), Christensen, A. S.; Kubar, T.; Cui, Q. and Elstner, M., Chem. Rev., 26, 6(9),

25 Selected SE Example () Gruden, M.; Andjeklovic, L.; Jissy, A. K.; Stepanovic, S.; Zlatar, M.; Cui, Q. and Elstner, M., J. Comput. Chem., 27, 38(25),

26 Selected SE Example () Gruden, M.; Andjeklovic, L.; Jissy, A. K.; Stepanovic, S.; Zlatar, M.; Cui, Q. and Elstner, M., J. Comput. Chem., 27, 38(25),

27 Selected SE Example () Gruden, M.; Andjeklovic, L.; Jissy, A. K.; Stepanovic, S.; Zlatar, M.; Cui, Q. and Elstner, M., J. Comput. Chem., 27, 38(25),

28 Molecular mechanics: chemistry without electrons How can we further speed up the calculations? In many problems, an accurate description of the electronic wavefunctions is not necessary This is true when no chemical change is performed along a simulation Molecular Mechanics is a simplification of the description of a molecular system at the atomic level where no explicit electrons are considered the energy of a system is then defined solely by the positions of the nuclei (Born-Oppenheimer approximation)

29 Quantum Mechanics around the equilibrium structure water molecule O H H Symetric stretch (3657 cm - ) O H H Asymetric stretch (3776 cm - ) O H H Bend (595 cm - ) Deformation around the equilibrium geometry can be modelled using harmonic potentials.

30 Quantum Mechanics around the equilibrium structure Many water molecules Water molecules in interactions: van der Waals contacts: [ ( ) 2 ( ) ] 6 E ij vdw = ε σij σij ij 2 R ij R ij electrostatic dipole-dipole interactions replaced by charge-charge interactions: E elec = i i>j 4πε q i q j r ij

31 Molecular Mechanics Molecular Mechanics (MM) is the application of the Newtonian mechanics (classical mechanics) to molecular systems. In a molecule, each atom is considered as a point charge The point charges interact using a parametrized force field A force field is an equation describing all possible interactions in a molecular system associated with pre-defined parameters: force field = equation + parameters In most cases, the connectivity of the system remains constant ( no chemical reaction)

32 Molecular Interactions described by a force field Bond stretching Angle bending Bond rotation (torsion) Out of plane (improper torsion) δ+ δ δ+ Non bonded interactions (electrostatic) Non bonded interactions (van der Waals)

33 Transferability / Additivity Molecular Mechanics is based on two main assumptions: Transferability: properties of chemical subgroups are similar either in small molecules or large compounds (e.g.: a carbonyl C=O group has very similar stretching properties in H 2 CO or in a, atom structure) Additivity: effective molecular energy can be expressed as a sum of potentials describing all interactions in the molecular system: van der Waals and electrostatic interactions (non-bonded interactions) bond length and angle deviations, internal torsion flexibility, etc. (bonded interactions)

34 Example of a force field: AMBER AMBER: general force field for the description of proteins and nucleic acids (DNA, RNA). E pot = bonds angles b 2 k b(r r b ) 2 + a 2 k a(θ θ a ) 2 dihedrals + + d atoms i V n n 2 { atoms j>i ( + cos(nω γ)) q i q j + ε ij 4πε ε r r ij [ (σij r ij ) 2 2 ( σij r ij ) 6 ]}

35 An example using the AMBER force field (ff3) N-methylacetamide Atom Residue Number Name Name Number HH3 ACE 2 CH3 ACE 3 2HH3 ACE 4 3HH3 ACE 5 C ACE 6 O ACE 7 N NME 2 8 H NME 2 9 CH3 NME 2 HH3 NME 2 2HH3 NME 2 2 3HH3 NME 2

36 AMBER atom types and atom charges (ff3) N-methylacetamide Atom Number Name Type Charge HH3 HC.76 2 CH3 CT HH3 HC HH3 HC.76 5 C C O O N N H H.29 9 CH3 CT HH3 H.627 2HH3 H HH3 H.627

37 N-methylacetamide AMBER bond types (ff3) Bond Number k b r b CT HC CT C C O C N N H N CT CT H

38 N-methylacetamide AMBER angle types (ff3) Angle Number k a r a HC CT HC HC CT C CT C O CT C N O C N C N H C N CT H N CT N CT H H CT H

39 AMBER dihedral and improper types (ff3) N-methylacetamide Dihedral Number n V n γ HC CT C O HC CT C N 3.. CT C N H CT C N CT O C N H O C N CT C N CT H 3.. H N CT H 3.. Improper Number n V n γ H N C CT O C N CT 2. 8.

40 N-methylacetamide AMBER van der Waals types (ff3) Atom type σ i ε i C CT H.6.57 HC H N O [ ( E ij vdw = ε σij ij R ij ) 2 ( ) ] 6 σij 2 R ij σ ij = σ i + σ j and ε ij = ε i ε j

41 Some usual force fields AMBER Assisted Model Building and Energy Refinement (UCSF) specialized in the modelization of proteins and nucleic acids (DNA, RNA) CHARMm Chemistry at HARvard Macromolecular Mechanics (Harvard, Strasbourg) specialized in the modelization of proteins MM2, MM3, MM4 Allinger Molecular Mechanics (UGA) specialized in organic compounds MMFF94 Merck Molecular Force Field (Merck Res. Lab.) specialized in organic compounds OPLS Optimized Potentials for Liquid Simulations (Yale) AMOEBA Polarizable force field for water, ions and proteins (WUSTL) etc.

42 Simulating infinite systems

43 Simulating infinite systems Periodic Boundary Conditions (PBC): a molecular system is enclosed in a box (the unit cell) and is replicated infinitely in the three space dimensions (the images). Minimum Image Convention: Only the coordinates of the unit cell is recorded. As an atom leaves the unit cell by crossing the boundary, an image enters to replace it. the total number of particles is conserved.

44 Long-range electrostatic interactions The coulomb energy in periodic domains (neutral system): E elec = 2 n i j q i q j r i r j + n The sum is conditionnally convergent (= slow convergence, if any) cut-off: if r ij > r cut-off r ij = non-physical but speeds up computations

45 The Ewald Summation The coulomb sum can be converted in a sum of two absolutely and rapidly convergent series in direct and reciprocal space. This conversion is accomplished by adding to each point charge a Gaussian charge density of opposite value and same magnitude as the point charge: ρ i ( r) = q i α 3 exp( α 2 r 2 )/ π 3 () where α is a positive parameter which determine the width of the gaussians This charge distribution screens the interaction between neighbouring point charges. fast convergence in the direct space. The distribution of opposite gaussian charges converges quickly in the reciprocal space using a Fourier transform.

46 It is demonstrated (by Ewald, 92): The Ewald Summation E elec = U r + U m + U with U r the direct sum, U m the reciprocal sum, and U the self-interacting term (which corrects the interactions between the counter charges introduced in the system). U r = 2 i,j q i q j 4πε n erfc(α r i r j + n ) r i r j + n U m q i q j exp( (π m/α) = 2πL 3 i,j 4πε 2 + 2π i m.( r i r j )) m 2 (3) m U = α π qi 2 (4) i with m = 2π n L a reciprocal space vector, and erfc(x) = erf(x) = 2 x π e u2 du (2)

47 Particle Mesh Ewald (Darden et al., 993) The computation time of the Ewald summations grows as O(N 2 ) where N is the number of particles in the periodic systems. To speed up computations, the Particle Mesh Ewald (PME) method has been designed. Its computation grows as O(N log N). It is based on the use of a cut-off in the direct space and the use of Fast Fourier Transform (FFT) in the reciprocal space.

48 Selected MM Examples () Freddolino, P. L.; Liu, F.; Gruebele, M. and Schulten, K., Biophys. J., 28, 94(), L75 7 Freddolino, P. L. and Schulten, K., Biophys. J., 29, 97(8),

49 Selected MM Examples (2) Nguyen, H.; Maier, J.; Huang, H.; Perrone, V. and Simmerling, C., J. Am. Chem. Soc., 24, 36(4), Brute force folding of protein structures using a classical force field and molecular dynamics Reference: 7 proteins with known 3D structures Start: elongated structures Methods & Tools: Amber force field (Molecular Mechanics) Implicit solvent (Generalized Born model) Replica-Exchange Molecular Dynamics Graphical Processing Units (Gamers GPUs from NVIDIA)

50 Selected MM Examples (2) Nguyen, H.; Maier, J.; Huang, H.; Perrone, V. and Simmerling, C., J. Am. Chem. Soc., 24, 36(4),

51 QM/MM Methods: Foundations How to simulate a reactive molecular system with explicit solvent molecules? Quantum Mechanics Description of the electrons and nuclei behavior Allows the breaking and forming of covalent bonds CPU time intensive limited to small systems Molecular Mechanics Atoms = interacting point charges Bad description of chemical reaction Fast computations suitable for large systems

52 QM/MM Methods: Foundations General Idea â Partionning of the total system â Active part = small number of atoms Description by Quantum Mechanics (QM) + the quantum part â Rest of the system Description by Molecular Mechanics (MM) + the classical part â The MM part acts as a perbutation to the QM part â The coupling is called a QM/MM method

53 QM/MM Methods QM/MM Hamilonians H = H QM + H MM + H QM/MM H QM/MM describes the interactions between the quantum part and the classical part The QM hamiltonian H QM = 2 e- i i e- i nuclei Z K + K r ik e- i e- i>j r ij + nuclei K nuclei K>L Z K Z L R KL

54 QM/MM Methods The MM hamiltonian H MM = bonds b + 2 k b(r r b ) 2 + { atoms j>i atoms i angles a q i q j + ε 4πε ε r r ij ij 2 k a(θ θ a ) 2 dihedrals + d [ ( ) 2 σij 2 r ij V n ( + cos(nω γ)) n 2 ( ) ]} 6 σij r ij The QM/MM hamiltonian H QM/MM = e- classical Q C r ic i C }{{} e charge interactions + nuclei classical Z K Q C R KC K C }{{} nuclei - charge interactions +V van der Waals QM/MM

55 QM/MM Methods re-writing of the equations into electrostatic and non-electrostatic interactions H = H elec + H non-elec e- e- nuclei Z K H elec = 2 i + i i K r ik + i i>j r ij }{{} standard equations e- e- e- classical Q C r ic i C }{{} wavefunction polarization by external charges van der Waals H non-elec = H MM + VQM/MM + = H MM + V van der Waals QM/MM nuclei K classical Z K Q C + C R KC + V nuclei QM+QM/MM nuclei K nuclei K>L Z K Z L R KL

56 QM/MM Implementations QM/MM Methods H elec, and V nuclei QM+QM/MM can be computed using a standard quantum mechanics code. The term describing the electrons-classical charge interaction is incorporated into the core Hamiltonian of the quantum subsystem. H MM, and V van der Waals QM/MM are computed using standard molecular mechanics code and are relatively easy to implement.

57 Calibrating QM/MM interactions QM/MM Methods The calibration of the QM/MM interactions is the main problem facing QM/MM methods The QM/MM interaction should reproduce quantitatively the interaction between the classical and the quantum parts as if the system was computed fully quantum mechanically The quantitative reproduction of the QM/MM interactions depends on three points. The choice of Q C or more in general the choice of the MM force field van der Waals 2. The choice of the van der Waals parameters to describe VQM/MM 3. The way the classic charges polarize the quantum subsystem

58 QM/MM Methods The choice of Q C Q C must be chosen to reproduce the electrostatic field due to the MM part onto the QM part It is a good approximation to take the charge definition from an empirical force field and incorporate those charges into H elec Because MM charges are designed to properly reproduce electrostatic potentials However MM charges can differ greatly between force fields No systematic studies so far

59 QM/MM Methods The choice of the van der Waals components Specific sets of van der Waals parameters and potential energy should be redefined to properly reproduce non-electrostatic QM/MM interactions This has been accomplished for solute/water interactions (Laaksonen, 999; Ruiz-López, 2; etc) and for some proteins systems (Freindorf, 25) all these parameters are QM and basis sets dependent

60 Classical charge polarization QM/MM Methods ab initio: similar to electron-nuclei interaction H core = H core electrons i classical Q C C r ic H core µν = < µ H core ν > = H core µν i < µ Q C ν > C r ic semiempirical: not similar to electron-nuclei interaction some conditions must be fulfilled (Luque, 2)

61 Cutting Covalent Bonds Classical Part C C Incomplete valency Quantum Part Link Atoms Connection Atoms Local Self Consistent Field Generalized Hybrid Orbitals

62 Link atom method (Field, 99) Cutting Covalent Bonds A monovalent atom is added along the X Y bond = the link atom Usually the link atom is an hydrogen, but some implementations use a halogen-like fluorine or chlorine Interaction with the MM part? It should interact with the MM part, except for the few closest atoms (Reuter, 2) The link atom can be free or constrained along the X Y bond Easiest implementation Give accurate answers as long as it is placed sufficiently far away from the reactive atoms (3-4 covalent bonds)

63 Cutting Covalent Bonds Connection atoms (Antes, 999; Zhang, 999) A monovalent pseudo-atom is added at the Y position = the connection atom Its behavior mimics the behavior of a methyl group semiempirical: Antes and Thiel, 999 DFT (pseudo-potential): Zhang, Lee and Yang, 999 Pro: no supplementary atom (MM: Y atom; QM: connection atom) Con: Need to reparametrize each covalent bond type (C-C, C-N, etc)

64 Cutting Covalent Bonds Local Self Consistent Field (Rivail, 994) the two electrons of the frontier bond are described by a strictly localized bond orbital (SLBO) its electronic properties are considered as constant during the chemical reaction Using model systems and the MM transferability assumption of bond properties, it is possible to determine the representation of the SLBO in the atomic orbital basis set of the quantum part By freezing this representation, the other QM molecular orbitals, orthogonal to the SLBOs, are generated using a local self consistent procedure

65 Cutting Covalent Bonds Local Self Consistent Field (Rivail, 994) To simplify:. The MOs describing the frontier bonds are known (transferable SLBO extracted from a model system) 2. The other MOs describing the rest of the quantum fragment are built orthogonally to the frozen orbitals with a local SCF procedure. LSCF is available at the semiempirical and ab initio levels Pro: no supplementary atom, proper chemical description of the X Y bond Con: difficult to implement, especially in ab initio

66 Cutting Covalent Bonds Generalized Hybrid Orbitals (Gao, 998) Extension of the LSCF method the classical frontier atom is described by a set of orbitals divided into two sets of auxiliary and active orbitals The latter set is included in the SCF calculation, while the former generates an effective core potential for the frontier atom Available at the semiempirical and ab initio levels Pros and Cons similar to LSCF

67 ONIOM Methods Some peculiar QM/MM methods: ONIOM-like methods Size of the system What we would like to model Large (+2) 2 Small () Low Level High Level Level of computations E total = E+2 Low + E High E Low

68 ONIOM Methods Different Approaches IMOMM: QM/MM with no MM charge inclusion into the QM core hamiltonian (no QM polarization in the original version) IMOMO: QM/QM (low level QM polarization) ONIOM: N-layered scheme E total = E+2+3 Low + E+2 Medium E+2 Low + E High E Medium Note to Gaussian Users: please use the EmbedCharge keyword Cutting covalent bonds Link atom scheme

69 Availability of QM/MM methods Commercial and academic software (non exhaustive list) On the MM side: On the QM side: AMBER BOSS GROMACS + Gaussian/GAMESS/CPMD Other software (non exhaustive list) CP2K CPMD (with GROMOS) Gaussian9 ONIOM implementation NWCHEM Qsite ChemShell: a layer on top of other QM and MM software (Daresbury, UK P. Sherwood) Tinker-Gaussian (Nancy, France X. Assfeld & M. F. Ruiz-López) Tinker-Molcas (Marseille, France N. Ferré)

70 Seminal papers QM/MM Methods: Foundations Warshel, A. and Karplus, M., J. Am. Chem. Soc., 972, 94(6), Warshel, A. and Levitt, M., J. Mol. Biol., 976, 3, Singh, U. C. and Kollman, P. A., J. Comput. Chem., 986, 7, Field, M.; Bash, P. and Karplus, M., J. Comput. Chem., 99,, Selected reviews Åqvist, J. and Warshel, A., Chem. Rev., 993, 93, Monard, G. and Jr., K. M., Acc. Chem. Res., 999, 32(), 94 9 Monard, G.; Prat-Resina, X.; González-Lafont, A. and Lluch, J., Int. J. Quant. Chem., 23, 93(3), Lin, H. and Truhlar, D. G., Theor. Chem. Acc., 27, 7, 85 99

71 Selected QM/MM Example Ugur, I.; Marion, A.; Aviyente, V. and Monard, G., Biochemistry, 25, 54(6), Deamidation in Triosephosphate Isomerase

72 Selected QM/MM Example Ugur, I.; Marion, A.; Aviyente, V. and Monard, G., Biochemistry, 25, 54(6), Deamidation in Triosephosphate Isomerase c r i t i c a l di s t a nc e As n@cg Ψasn Gl

73 Selected QM/MM Example Ugur, I.; Marion, A.; Aviyente, V. and Monard, G., Biochemistry, 25, 54(6), Deamidation in Triosephosphate Isomerase

74 Selected QM/MM Example Ugur, I.; Marion, A.; Aviyente, V. and Monard, G., Biochemistry, 25, 54(6), Selection of the semiempirical QM methods ab initio reference asn r eact i ve conf or mer 8. 6 TS t et

75 Selected QM/MM Example Ugur, I.; Marion, A.; Aviyente, V. and Monard, G., Biochemistry, 25, 54(6), Selection of the semiempirical QM methods ab initio reference asn r eact i ve conf or mer 8. 6 TS t et

76 Selected QM/MM Example Ugur, I.; Marion, A.; Aviyente, V. and Monard, G., Biochemistry, 25, 54(6), Selection of the semiempirical QM methods ab initio reference asn r eact i ve conf or mer 8. 6 TS t et

77 Selected QM/MM Example Ugur, I.; Marion, A.; Aviyente, V. and Monard, G., Biochemistry, 25, 54(6), Selection of the semiempirical QM methods ab initio reference asn r eact i ve conf or mer 8. 6 TS t et

78 Selected QM/MM Example Ugur, I.; Marion, A.; Aviyente, V. and Monard, G., Biochemistry, 25, 54(6), Selection of the semiempirical QM methods ab initio reference asn r eact i ve conf or mer 8. 6 TS t et

79 Selected QM/MM Example Ugur, I.; Marion, A.; Aviyente, V. and Monard, G., Biochemistry, 25, 54(6), Selection of the semiempirical QM methods ab initio reference asn r eact i ve conf or mer 8. 6 TS t et

80 Selected QM/MM Example Ugur, I.; Marion, A.; Aviyente, V. and Monard, G., Biochemistry, 25, 54(6), QM/MM reaction free energies (SCC-DFTB/Amber) asn TS t et asn TS t et 4 26 asn TS t et 2 34 asn TS t et 28 49