Theoretische Chemie (TC II) Computational Chemistry

Transkript

1 Theoretische Chemie (TC II) Computational Chemistry Lecture 12 27/01/2012 Irene Burghardt Vorlesung: 27/01/2012, 01/02/2012, Mi 11h45-13h15, Fr 8h-9h30 Praktikum (gemäß Ankündigung, statt Vorlesung): Fr Computing Center (BCC) Web site: 1

2 Starting point: the molecular Hamiltonian all electrons and nuclei Ĥ = ˆT e + ˆT N + ˆV e + ˆV N + ˆV en = N e i=1 h 2 2m e 2 i N N a=1 + e2 { Ne 4πɛ 0 i=1 h 2 2M a 2 a N e j>i 1 r ij + N N N N a=1 b>a Z a Z b r ab N e i=1 N N a=1 Z } a r ia i h t ψ = Ĥψ Ĥψ = Eψ 2

3 But we might eventually do classical molecular dynamics (MD) simulations, e.g., for proteins Quantum classical transition due to decoherence Do any quantum effects survive? 3

4 Of course the electrons always need a quantum mechanical treatment but they are usually integrated out so as to yield effective potentials for the nuclear motion (Born-Oppenheimer approach) 4

5 Two Steps 1 Born-Oppenheimer approximation: separate the electronic and nuclear motions and generate effective potentials for the nuclear motion 2 follow the dynamics of the nuclei, either using quantum dynamics (time-dependent Schrödinger equation) or else using classical dynamics (Newton s equations)( ) ( )... or else using a variety of semiclassical and mixed quantum-classical approaches 5

6 Step 1: the Born-Oppenheimer approximation Max Born Robert Oppenheimer perturbative expansion in orders of the mass ratio m/m 1/1836 6

7 Born-Oppenheimer basics Using the idea of an adiabatic separability of the time scales for electronic vs. nuclear motion, separate the total Hamiltonian Ĥ T = ˆT e + ˆV e + ˆT N + ˆV N + ˆV en = ˆT N + Ĥ el and solve the electronic Schrödinger equation first disregarding ˆT N : Ĥ el ψ n (r el R) = ɛ n (R)ψ n (r el R) The eigenvalues ɛ n (R) depend parametrically on the nuclear coordinate(s) and constitute the Born-Oppenheimer surfaces 7

8 Born-Oppenheimer, cont d If we assume that the overall wavefunction (electronic + nuclear) can be written as Ψ T (r el, R) = ψ n (r el R)χ n (R) we obtain nuclear motion in terms of the TDSE for the nuclear wavefunction χ n (R, t) on the nth Born-Oppenheimer surface: i h χ ( n t = h2 ) 2M 2 R + ɛ n(r) χ n Thus we have separated the electronic-nuclear problem into two parts: Ĥ el ψ n (r el R) = ɛ n (R)ψ n (r el R) ; i h χ n = ( ˆT N + ɛ n (R)) χ n 8

9 BO surfaces for the hydrogen molecule molecular orbitals = bonding and non-bonding combinations of atomic orbitals parametric dependence on the internuclear distance potentials 9

10 Born-Oppenheimer surfaces for the I 2 molecule note that in the BO picture, the nuclei move only on a given BO surface at a time thus, the multiple crossings of the I 2 poentials are indicative of a breakdown of the BO approximation (see Lecture 2) 10

11 Step 2: calculate the dynamics of the nuclei i h Ψ t = ( ˆT + ˆV )Ψ or q = p m ṗ = V In many applications, Newton s equations are used! 11

12 Trajectory dynamics: the Heisenberg picture of classical mechanics trajectory = chronological sequence of (coordinate-space or phase-space) configurations 12

13 Trajectories in phase space Hamilton s equations: (yielding Newton s equations, m q = ( V / q)): q = p/m ṗ = V / q or: q ṗ = H p H q = H q H p where we used the classical Hamiltonian: H = T + V = p2 2m + V (q) the antisymmetric matrix ((0, 1), ( 1, 0)) reflects the symplectic structure of classical mechanics 13

14 Time-evolving phase-space densities: the Schrödinger picture of classical mechanics f(q, p, t) = dq dp δ(q q(t))δ(p p(t))f(q, p, t 0 ) Liouville equation: f t = {H, f} = H p = p m f q + H q f q + V q f p f p Lf picture: Martens & co where we introduced the Poisson bracket {, } and the Liouvillian L 14

15 Liouville s theorem df dt = 0 where df dt = f t + f q q + f pṗ phase space continuity equation the density keeps its size: it is incompressible each trajectory keeps its weight as a function of time 15

16 Special case: equilibrium distribution f t = 0 The distribution function does not evolve in time but the trajectories do! Typical equilibrium ensembles: microcanonical ensemble (NVE): all microstates with fixed total energy: Q NV E = Γ δ(h(γ) E) canonical ensemble (NVT): assembly of all microstates with fixed N, V, but energy can fluctuate according to T : Q NV T = Γ exp( H(Γ)/kT ) 16

17 MD: Solve Newton s Equations in a Simulation Box Each atom feels forces on it from other atoms: F 1 = dv(x 1,...,x N ) dx 1 where V(x 1,...,x N ) is the interaction potential Newton s Equations: the acceleration of the particle is proportional to the force: F 1 = mẍ 1 ma 1 where a 1 is the acceleration experienced by particle 1 17

18 Newton s Equations at Work calculate force at given particle position x 1 update position (x) and velocity (v = ẋ) from Newton s equation: x 2 = x 1 + v 1 t v 2 = v 1 + a 1 t = v 1 1 m dv dx t x1 the sequence of positions (x 1,x 2,...) define the particle s trajectory 18

19 Which Potentials (Force Fields)? dihedral angle typical force fields: V bond (r) = 1 2 k(r r 0) 2 V angular (φ) = 1 (1 + cos(mφ δ))2 2 V vdw (r ij ) = C 12 r 12 ij C 6 r 6 ij nonbonded interaction 19

20 Harmonic oscillator potential (V ) vs. kinetic energy (K) change in the course of the trajectory while the total energy (E) remains constant 20

21 Phase space picture y(t) = b cosωt p(t) = b mω sinωt ẏ = p m ṗ = dv dy V (y) = mω 2 y 2 21

22 Potential landscapes characteristic points: minima, maxima, saddle points 22

23 Potentials in 2D V (x, y) = 1 2 k xx k yy 2 V (x, y) = 1 2 k xx k yy 2 minimum saddle point 23

24 Equilibrium ensembles Typical equilibrium ensembles: microcanonical ensemble (NVE): all microstates with fixed total energy: Q NV E = q,p δ(h(q, p) E) canonical ensemble (NVT): assembly of all microstates with fixed N, V, but energy can fluctuate according to T : Q NV T = q,p exp( H(q, p))/kt ) 24

25 Properties = ensemble averages A = dq dp A(q, p)p (q, p) where P (q, p) = probability of being at a given phase-space point: P NV T (q, p) = 1 ( Q exp E(q, p) ) kt phase space averages in many dimensions may be hard to calculate! 25

26 Ergodic theorem: ensemble averages are equivalent to time averages A ensemble = 1 τobs τ obs τ =1 A(q(τ ), p(τ )) if trajectories explore all of phase space with time 26

27 Conformational analysis Ramachandran map = conformational energy surface usually represented as contour map showing accessible conformations example above: dipeptide as model for protein conformation 27

28 Correlation functions e.g., 2-time velocity autocorrelation function 28