Module 8 1. Force field methods

Transcription

1 Module 8 1. Force field methods Updated May 30, 2016

2 For many problems, a full quantum chemical treatment is not possible. Ribozyme model: Courtesy of åkan ugosson Complementary/integrated methods are required. We need alternative ways to express V(x)! Reproduced from Bolhuis et al. Annual Review Phys. Chem. 53,291 (2002)

3 The electronic Schrödinger equation Ψ(X, x) = E el Ψ(X, x) is the electronic amiltonian: T e + V Ne + V ee + V NN T e = r i 2 kinetic energy of electrons V Ne = Z A r ia V ee = r ij electron nuclear attraction electron electron repulsion V NN = 1 2 ZAZ B R AB nuclear nuclear repulsion

4 Nuclear motion From the Born-Oppenheimer approximation the Schrödinger equation for nuclear motion: χ(x, t) (T N + V N(X))χ(X, t) = i t V N(X) is the potential energy at the nuclear geometry X. It is obtained by solving the electronic Schrödinger equation at that geometry. In this potential V N(X), the energy ɛ depends explicitly solely on the nuclear coordinates (X). V N X

5 Sampling the phase space in complex systems In systems with many degrees of freedom, it is impossible to scan the whole multi-dimensional (3N-6) potential energy landscape. Reproduced from Bolhuis et al. Ann. Rev. Phys. Chem. 53,291 (2002) Monte Carlo simulations: Algorithm to sample the distribution corresponding to V N(X). Molecular dynamics simulations: The classical motion of the atoms are calculated: F i = ma i F i = VN({x i}) x i Ab initio MD: Forces calculated on the fly. Classical MD: Pre-calculated analytical force field.

6 Molecular mechanics (MM) or Force field (FF) methods Approximations: 1. Classical nuclear motion F i = ma i, F i = VN({x i}) x i. 2. V N(X) is approximated by a parametrized function: E(X) What is the functional form of E? Reproduced from Bolhuis et al. Ann. Rev. Phys. Chem. 53,291 (2002) Should describe the variations in potential energy with respect to nuclear coordinates. Notice! Only description of accessible region required

7 The force field energy Intramolecular degrees of freedom Bond stretching C Angle bending Torsions C Intermolecular degrees of freedom Relative positions and orientations

8 The force field energy E FF = E str + E bend + E tors + E vdw + E el (+E cross) E str: bond stretching E bend : angle bending Pair interactions E tors: torsion (bond rotation) E vdw : van der Waals interaction δ E el : electrostatic interaction Non-bonded interactions: E vdw + E el δ+ δ+ Ribozyme model: Courtesy of åkan ugosson E cross: couplings between E str, E bend, E tors Functional form of each term? (Parametrization empirical/ab initio)

9 General purpose force fields Molecules are composed of structurally similar units. Note: Many different atom types, e.g. C sp 3 -carbon C sp 2 -carbon, alkene C sp 2 -carbon, carbonyl C sp 2 -carbon, aromatic C sp-carbon C cyclopropane-carbon C radical-carbon etc, etc Assumption: For a particular pair of atom types i) the optimal bond distance is the same, and ii) the energy change upon distortion is the same (force constant) Similar for bend, torsion etc.

10 The ethyl benzene molecule What type of carbon is this? sp 3 -carbon sp 2 -carbon, alkene sp 2 -carbon, carbonyl sp 2 -carbon, aromatic sp-carbon cyclopropane-carbon radical-carbon

11 The ethyl benzene molecule What type of carbon is this? sp 3 -carbon sp 2 -carbon, alkene sp 2 -carbon, carbonyl sp 2 -carbon, aromatic sp-carbon cyclopropane-carbon radical-carbon

12 What can force fields be used for? 1. Determine structures (minimize E FF with respect to the nuclear coordinates) 2. Compare energies of different structures (more difficult) 3. Simulate systems at finite temperature, free energy 4. Establish rigorous link to statistical mechanics 5. Dynamical information... diffusion/free energy 1. Relatively fast (many medium sized molecules) 2. Large systems (e.g. proteins) 3. Combine with quantum mechanics: QM/MM Limitations of force field methods? Accuracy Predictability Transferability Reactions very difficult Garbage in garbage out

13 Many different force field methods exist. They differ e.g. in: 1. Types of interactions included 2. Functional form of each term 3. Inclusion of cross terms 4. Type of information used for fitting the parameters: i) empirical different types of experimental data ii) ab initio data from electronic structure calculations Different purposes of different methods: Class I methods: proteins, DNA, etc (multi-purpose) Class II methods: smaller molecules ( high accuracy)

14 Typical system to have in mind: Ribozym model: Courtesy of åkan ugosson

15 The bond stretch energy Limitation: Quantization! Only v = 0 populated! Taylor expansion for the energy: E str(r AB R AB 0 ) = E(0) + de dr (RAB R AB 0 ) de E(0) 0, dr = 0 (equilibrium) Second order = harmonic approximation: E str(r AB R AB 0 ) = k AB (R AB R AB 0 ) 2 Parameters: k AB : force constant d 2 E dr 2 (RAB R AB 0 ) R AB 0 : natural bond length Improvements: Third and fourth order terms (more parameters) Morse potential Licensed under CC BY-SA 3.0 Alternative: Morse potential: E Morse( R) = D e(1 e α R ) 2 ( R = R R 0) Goes to dissociation energy at large R, parameters: D e, α, R 0.

16 The bending energy Taylor expansion and harmonic approximation: A C θ B E bend (θ ABC θ ABC 0 ) = k ABC (θ ABC θ ABC 0 ) 2 Parameters: k ABC : force constant θ ABC 0 : natural bond angle igher terms: mainly cubic Special problems: Di-or trivalent central atom: barrier for 180 Special atom types for small rings

17 The torsion energy 1. The energy is periodic in ω 2. Low cost for distortion large distortions Dihedral angle: Taylor expansion not useful, use Fourier series (periodic): C E tors(ω) = n V ncos(nω) O n = 1 periodic by 360, n = 2 periodic by 180, n = 3 periodic by 120 V n determines the barrier height Improper dihedral: The general expression used (several minima common). C E tors(ω ABCD ) = 1 2 V1ABCD (1 + cos(ω ABCD )) V2ABCD (1 cos(2ω ABCD ))+ 1 2 V3ABCD (1+cos(3ω ABCD )) O For each angle ω ABCD : three parameters: V ABCD 1, V ABCD 2, V ABCD 3

18 Electrostatic energy (non-bonded atoms) The distribution of electrons gives positive and negative parts of the molecules, which interact with each other. 1. Assign partial atomic charges Q x R AB E el (R AB ) = QA Q B ɛr AB ɛ can be used to describe the screening by solvent molecules (atoms). 2. Alternative: Assign dipoles µ x to the bonds in the molecule µa µ B E el (R AB ) = ɛ(r AB ) (cos χ 3 cos 3 αa cos α B ) R AB distance, χ, α A, α B angles between the dipoles. Q A A Q B B Charge method most often used Easier to parametrize Anisotropy and polarization effects: multipole distributions and induced dipoles

19 van der Waals energy non-bonded atoms 1. Repulsive part: overlap of electron clouds (short distance), exchange interaction (Pauli principle, QM) 2. Attractive part: dispersion forces, induced dipoles, quadrupoles, etc. Dipole dipole: (R AB ) 6 Dipole quadrupole: (R AB ) 8 etc CAB E vdw (R AB ) = E rep(r AB ) (R AB ) 6 E rep(r AB )? E vdw / cm σ ε R AB / Å

20 van der Waals energy non-bonded atoms 1. Repulsive part: overlap of electron clouds (short distance), exchange interaction (Pauli principle, QM) 2. Attractive part: dispersion forces, induced dipoles, quadrupoles, etc. Dipole dipole: (R AB ) 6 Dipole quadrupole: (R AB ) 8 etc CAB E vdw (R AB ) = E rep(r AB ) (R AB ) 6 E rep(r AB )? E vdw / cm σ ε R AB / Å

21 van der Waals energy non-bonded atoms 1. Repulsive part: overlap of electron clouds (short distance), exchange interaction (Pauli principle, QM) 2. Attractive part: dispersion forces, induced dipoles, quadrupoles, etc. Dipole dipole: (R AB ) 6 Dipole quadrupole: (R AB ) 8 etc CAB E vdw (R AB ) = E rep(r AB ) (R AB ) 6 E rep(r AB )? E vdw / cm σ ε R AB / Å

22 Potential forms 1. Lennard Jones potential E LJ(R) = C1 R R ( 6 ( ) 12 R0 E LJ(R) = ɛ 2 R 12 C2 ( R0 R ) 6 ) C 1 and C 2 are parameters ɛ and R 0 are parameters 2. Buckingham potential E vdw (Ra) = Ae BR C R 6 A, B, and C are parameters 3. Morse potential E Morse(R) = D ( 1 e αr) 2 D and α are parameters Short distance: Morse better than e BR better than Lennard Jones For practical purposes: no big difference in accuracy Lennard Jones potential computationally most efficient Some force fields have a special treatment of hydrogen bonds

23 Force field parameters The number of parameters needed is very large. Not all bond combinations are realistic. Example: 71 atom types (MM2 1991) Total number Actual number E vdw E str E bend E tors Effectively 30 atom types form bonds with each other. For each parameter: 3 4 independent exp. data needed Alternative: QM calculations Note: i. Parameters only partially transferable between programs ii. Additional parameters must be determined in the same way as the old ones

24 Intermolecular non-bonded energy terms Include E vdw /E el for atom pairs separated by three or more bonds. E vdw /E el contributions grow as the square of the system size Compare: E str/e bend /E tors contributions grow linearly with the system size. Use cutoff parameter for non-bonded terms, longer cutoff distance for E el than for E vdw. Or in periodic systems E el : Ewald sum method (N 3/2 ) Fast multipole method (linear)