Fundamentals of Quantum Mechanics for Chemistry G´eraldMONARD Equipe de Chimie et Biochimie Th´eoriques UMR 7565 CNRS - Universit´eHenri Poincar´e Facult´edes Sciences - B.P. 239 54506 Vandœuvre-les-Nancy Cedex - FRANCE http://www.monard.info/ Outline . 1. Fundamentals of Quantum Mechanics for Chemistry 3 Hartree-Fock methods 3 Density Functional Theory 3 The QM scaling problem 3 Semiempirical methods 3 Molecular Mechanics 2. Fundamentals of QM/MM methods 3 Partionning 3 QM/MM interactions 3 Cutting covalent bonds 3 ONIOM 3 Some available software 3. Selected QM/MM applications 3 Solvent effects 3 Spectroscopy 3 Biochemistry . Outline 4. Fundamentals of Linear Scaling methods 3 QM Bottlenecks 3 General ideas and solutions 3 Some available software 5. Focus on some Linear Scaling methods 3 CG-DMS 3 Mozyme 3 Divide & Conquer 6. Selected Linear Scaling applications 3 Energy Decomposition; Charge Transfer & Polarization 3 Born-Oppenheimer Molecular Dynamics 7. Parallelization of QM/MM and Linear Scaling methods Fundamentals of Quantum Mechanics for Chemistry (1) Some approximations And there was the Schr¨odinger equation. H0Ψ0 = E0Ψ0 where: H0 is an Hamiltonian operator that describes a molecular system Ψ0 is a wavefunction (solution of the Schr¨odingerequation) that describe a state of the system E0 the energy associated to Ψ0 + 1 equation + 2 unknowns (given H0) = +¥ solutions! ­ From now on: 3 ground state 3 closed shell 3 non-relativistic Fundamentals of Quantum Mechanics for Chemistry (2) Born-Oppenheimer approximation ­ nuclei are fixed point charges ­ only electrons are represented by a wavefunction Ψ HΨ = Eelec Ψ (1) H = Te + VeN + Vee 1 −Z 1 K (2) = − ∑∆i + ∑∑ + ∑∑ 2 i i K riK i i>j rij | {z } | {z } | {z } kinetic energy e−-nuclei inter. e−{e− inter. E = Eelec + Enuclei Z Z = hΨjHjΨi + ∑ ∑ K L K L>K RKL Fundamentals of Quantum Mechanics for Chemistry (3) Orbital approximation ­ each electron is described by a mono-electronic wavefunction: the Molecular Orbital (MO) Ψ(1;2;:::;n) = y1(1)y2(2):::yn(n) ­ all MOs are combined in a Slater determinant (Pauli principle) Linear Combination of Atomic Orbitals (LCAO) ­ Each MO yi is developed on a basis set of functions fm : the Atomic Orbitals (AO) AOs yi = ∑ cmi fm m ­ the real coefficients cim are the unknown of the problem Fundamentals of Quantum Mechanics for Chemistry (4) Variational Principle ­ The eletronic energy Eelec corresponds to a minimum with respect to each MO yi ¶E 8i elec = 0 ¶yi + Hartree-Fock equations 8i Fyi = ei yi where F is the mono-electronic Fock operator The Hartree-Fock method (1) The Fock operator c F(1) = H (1) + ∑[Jj (1) − Kj (1)] j ­ Hc(1) is the one-electron core Hamiltonian c 1 ZK H (1) = − ∆1 − ∑ 2 K R1K ­ Jj (1) is the Coulomb operator, Z ∗ 1 Jj (1) = yj (2) yj (2)dt2 (2) r12 ­ Kj (1) is the exchange operator Z ∗ 1 Kj (1)yi (1) = yj (1) yj (2) yi (2)dt2 (2) r12 The Hartree-Fock method (2) The Roothan-Hall equations ­ N electrons, fm AO basis set, closed shell, ground state + occupied MO = 2 electrons; virtual MO = 0 electron AO yi = ∑ cmi fm m ­ the Hartree-Fock equations can be re-written: FC = SCe with 3 C the matrix of cmi coefficients 3 e the diagonal energy matrix 3 S the overlap matrix: Smn =< fm jfn > The Hartree-Fock method (3) The Density Matrix From the MO coefficient cmi , it is possible to build a density matrix whose elements are: MO Pmn = ∑ nj cmj cnj with nj = 0 or 2 (occupation number) j The Fock matrix AO AO 1 F = Hc + P (mnjlh) − (mhjln) mn mn ∑∑ lh 2 l h with: Z Z 1 (mnjlh) = fm (1)fn (1) fl (2)fh (2)dr1dr2 (1) (2) r12 The Hartree-Fock energy 1 Eelec = ∑∑Pmn [Hmn + Fmn ] 2 m n The Hartree-Fock method (4) The Hartree-Fock algorithm 1. Compute mono- and bielectronic integrals 2. Build core hamiltonian (invariant) Hc 3. Guess an initial density matrix 4. Build the Fock matrix F 5. Orthogonal transformation using S1=2 F0C0 = eC0 6. Diagonalization of the Fock matrix F0 The C0 coefficients are obtained 7. Inverse transformation C0 ! C 8. Build the new density matrix Back to 4. unless convergence The Hartree-Fock method (5) The Hartree-Fock method is an ab initio method ­ No (empirical) parameters ¡ ab initio method ­ Orbital approximation ¡ No electronic correlation Other ab initio methods (post-Hartree-Fock methods) ­ Møller-Plesset Perturbation Theory (MP2, MP4, etc) ­ Configuration Interaction (CI) ­ Coupled Cluster (e.g. CCSD(T)) ­ MultiConfigurational Self-Consistent-Field (MCSCF) Density Functional Theory (1) Background ­ DFT (Density Functional Theory) methods are (almost) ab initio methods which include electronic correlation at a cost similar to a Hartree-Fock calculation. In most cases, a DFT calculation is even less costly than a HF calculation. ­ DFT methods relie on the Hohenberg-Kohn theorem (1964) which states that the ground state energy E of a system is a functional of the electronic density of this system, r(~r). Any electronic density r0(~r) other than the real electronic density will necessary lead to a higher energy. (variational principle) Density Functional Theory (2) A different approach To the opposite of ab initio methods, DFT methods try to find a simple 3-dimensional r(~r) function and not a complex 3N-dimensional wave function. r : R3 −! R ~r 7−! r(~r) From the Hohenberg-Kohn theorem, the energy E depends on the electronic density. It is said that E is a functional of the electronic density: 3 E : R ! R −! R r 7−! E [r(~r)] Density Functional Theory (3) The Kohn-Sham approach Let's write: E [r(~r)] = U [r(~r)] + T [r(~r)] + Exc [r(~r)] with: ­ U [r(~r)] the classical electrostatic energy nuclei Z −Z r(~r) 1 ZZ r(~r)r(~r 0) U [r(~r)] = A d~r + d~rd~r 0 ∑ ~ ~0 A j~r − RAj 2 j~r − r j | {z } | {z } electron-nuclei attraction electron-electron repulsion ­ T [r(~r)] is defined as the kinetic energy of a system with the same electronic density r(~r) but in which the electrons don't interact ­ Exc [r(~r)] the rest of the energy: exchange and electronic correlation contributions to the total energy + the difference between T [r(~r)] and the real kinetic energy Density Functional Theory (4) Kohn-Sham orbitals Kohn and Sham suggest to decompose the total electronic density into a sum of individual contributions for each electron: N Na b a b r(~r) = ∑ri (~r) + ∑ri (~r) i i N Na b 2 j a ~ j2 b ~ = ∑ yi (r) + ∑ yi (r) i i (a : high spin; b : low spin) a b yi , yi : Kohn-Sham molecular orbitals, or \auxiliary" orbitals Density Functional Theory (5) The Kinetic Energy operator is defined using Kohn-Sham orbitals One can then define T [r(~r)]: Ns Z ∆ T [r(~r)] = ys (~r) − ys (~r)d~r ∑ ∑ i 2 i s=a;b i Be careful: T [r(~r)] is not a real functional of the density since it is only defined using Kohn-Sham molecular orbitals (and not using r(~r)). Density Functional Theory (6) The Kohn-Sham Equations By applying the variational principle to the ground state energy: ¶E [r(~r)] ¶E [r(~r)] = = 0 ¶ra (~r) b i ¶ri (~r) One can find the one-electron Kohn-Sham equations: hKSyi = ei yi with hKS: the one-electron Kohn-Sham operator ∆ Z Z (~r 0) A r ~0 hKS = − − + dr + Vxc ∑ ~ ~0 2 A j~r − RAj j~r − r j ¶E [r(~r)] and V = xc : the exchange-correlation potential xc ¶r(~r) Density Functional Theory (7) Exchange-correlation functionals If the \true" exchange-correlation functional was known, the Kohn-Sham equations would give the exact electronic density of a ground state system. This is not the case ! + various approximations are to be made + it exists various exchange-correlation models Z First case: Exc = exc (r):r(~r)d~r + local methods Z ~ Second case: Exc = exc r;∇r :r(~r)d~r non-local methods + gradient corrected GGA (Generalized Gradient Approximation) Density Functional Theory (8) Exchange-correlation potential models: Local density methods Usually, the exchange and the correlation contributions are separated: Exc = Ex + Ec Ex Ec ­ LDA: Local Density Approximation ­ VWN (Vosko-Wilk-Nusair) Z 2 ­ PZ (Perdew-Zunger) −3e 2 1=3 Ex = r(~r) 3p r(~r) d~r 4p ­ PW92 (Perdew-Wang, 1992) (exact exchange energy in a ­ etc. homogeneous electron gas) Density Functional Theory (9) Exchange-correlation potential models: Non-local density methods Ex Ec ­ PW86 (Perdew-Wang, 1986) ­ LYP (Lee-Yang-Parr) PW 86 LDA 2 4 61=15 ­ PW91 ex = ex 1 + ax + bx + cx ­ PBE (Perdew-Burke-Ernzerhof) j~∇r with x = ­ P86 r4=3 and a, b, and c real parameters Exc ­ B88 (Becke 1988) 2 B88 LDA 1=3 x ­ BP86 = B88 + P86 ex = ex − br 1 + 6bx sinh−1 x ­ BLYP = B88 + LYP with b an atomic parameter ­ PBE ­ PW91 ­ BPW91 = B88 + PW91 ­ PBE ­ etc. Density Functional Theory (10) Exchange-correlation potential models: Hybrid methods In the hybrid methods, the exchange energy contains a part of \exact" exchange energy calculated in a similar manner as Hartree-Fock exchange energy (but using Kohn-Sham orbitals) Ex.: B3LYP B3LYP LDA HF LDA B88 LDA LYP LDA Exc = Exc + a0(Ex − Ex ) + ax (Ex − Ex ) + ac (Ec − Ec ) with a0 = 0:20, ax = 0:72, ac = 0:81 Common hybrid methods: B3LYP, PBE0, PBE1PBE The QM scaling problem (1) energy of a water cluster (3-21G basis set) energy of a water cluster (6-31G* basis set) 3500 B3LYP/3-21G 3500 B3LYP/6-31G* BLYP/3-21G BLYP/6-31G* CCSD(T)/3-21G CCSD(T)/6-31G* MP2/3-21G MP2/6-31G* 3000 HF/3-21G 3000 HF/6-31G* 2500 2500 2000 2000 1500 1500 wall clock CPU time (seconds) 1000 wall clock CPU time (seconds) 1000 500 500 0 0 0 50 100 150 200 0 50 100 150 200 number of water molecules number of water molecules ­ (H O) water cluster energy of a water cluster (6-311+G** basis set) 2 n 3500 B3LYP/6-311+G** BLYP/6-311+G** CCSD(T)/6-311+G** (n from 1 to 216) MP2/6-311+G**