1549380323-Statistical Mechanics Theory and Molecular Simulation

564 Quantum time-dependent statistical mechanics definition holds for operatorsAˆandBˆthat are functions of momentum only. As di ...

Approximations 565 x ̇k= pk m p ̇k=− m β^2 P ̄h^2 [2xk−xk− 1 −xk+1]− ∂U ∂xk . (14.6.25) In this dynamics, no thermostats are use ...

566 Quantum time-dependent statistical mechanics 0 5 10 15 20 -1 -0.5 0 0.5 1 1.5 K xx (t ) RPMD CMD Exact 0 5 10 15 20 t -0.1 0 ...

Approximations 567 0 5 10 15 20 -0.4 -0.2 0 0.2 0.4 0.6 0.8 K xx (t ) RPMD CMD Exact 0 5 10 15 20 t -0.1 -0.05 0 0.05 0.1 0.15 K ...

568 Quantum time-dependent statistical mechanics 0 0.25 0.5 0.75 1 t (ps) 0 5 10 15 20 K vv (t ) (Å 2 /ps 2 ) CMD RPMD 0 0.035 0 ...

Problems 569 ∗b. Derive eqns. (14.6.8) through (14.6.10). 14.3. Derive eqn. (14.6.16). 14.4. A quantum harmonic oscillator of ma ...

570 Quantum time-dependent statistical mechanics d. Calculate the energy spectrumQ(ω) forω >0. Interpret your results, and in ...

Problems 571 ∗14.10. For the discrete correlation functionGAB,P(t) defined in eqn. (14.6.8), we could analyze the importance of ...

15 The Langevin and generalized Langevin equations 15.1 The general model of a system plus a bath Many problems in chemistry, bi ...

System coupled to a bath 573 the system coordinateqis a simple coordinate, such as a distance between two atoms or a Cartesian s ...

574 Langevin and generalized Langevin equations equivalent toV(q), the term in brackets represents the interaction between the s ...

Derivation of the GLE 575 whereRαβis an orthogonal matrix that diagonalizes the symmetric matrixH ̃αβvia H ̃diag=RTHR ̃ , whereR ...

576 Langevin and generalized Langevin equations q ̇= ∂H ∂p = p μ p ̇=− ∂H ∂q =− dV dq − ∑ α gαxα x ̇α= ∂H ∂pα = pα mα p ̇α=− ∂H ...

Derivation of the GLE 577 Taking the Laplace transform of both sides of the second line in eqn.(15.2.2) yields s^2 ̃xα(s)−x ̇α(0 ...

578 Langevin and generalized Langevin equations is large. This is just what we might expect for a real bath. Thus, in order to m ...

Derivation of the GLE 579 deterministic quantity. To understand whyR(t) can be treated as a random process, we note that a real ...

580 Langevin and generalized Langevin equations relatively high-density bath, which affects the system via only soft collisions ...

Derivation of the GLE 581 time of the memory kernel indicates that the bath, in reality, retains memory of the system motion for ...

582 Langevin and generalized Langevin equations The second limiting case we will consider is a sluggish bath that responds very ...

Examples 583 15.3 Analytically solvable examples based on the GLE In the next few subsections, a number of simple yet illustrati ...