1549380323-Statistical Mechanics Theory and Molecular Simulation

664 Energies and forces The second technical problem associated with the Ewald sum is the high com- putational overhead associat ...

Energies and forces 665 0 1 2 3 4 5 6 7 8 x 0.0 0.2 0.4 0.6 0.8 1.0 M (n x) n= 2 n= 4 n= 6 n= 8 Fig. B.2 Cardinal B-spline funct ...

666 Energies and forces ×Mn(uz,i−kz−j 3 Nl). (B.30) Because the computational overhead of a fast Fourier transform of lengthNlsc ...

Appendix C Proof of the Trotter theorem The Trotter theorem figures prominently throughout the book, inboth the develop- ment of ...

668 Trotter theorem ( PˆhQˆh− 1 ) ψ=h ( Aˆ+Bˆ ) ψ+O(h), (C.7) whereO(h) denotes any vectorφsuch that lim h→ 0 ||φ|| h = 0. (C.8) ...

Trotter theorem 669 ∣ ∣ ∣ ∣ ∣ ∣ [( PˆhQˆh )n −Rˆnh ] ψ ∣ ∣ ∣ ∣ ∣ ∣= ∣ ∣ ∣ ∣ ∣ ∣ ( PˆhQˆh−Rˆh ) Rˆ(n−1)hψ +PˆhQˆh ( PˆhQˆh−Rˆh ) ...

Appendix D Laplace transforms The Laplace transform of a functionf(t) is just one of a general class of integral transforms of t ...

Laplace transforms 671 ̃g(s) ̃h(s) = lim a→∞ ∫a 0 dxe−sxg(x) ∫a−x 0 dye−syh(y). (D.4) The use of the upper limita−xrather than s ...

672 Laplace transforms From a numerical standpoint, a forward Laplace transform can beperform straight- forwardly, while a numer ...

Laplace transforms 673 Im(s) Re(s) Poles of f (s) ~ Fig. D.1Bromwich contour, which contains all of the poles off ̃(s). Laplace ...

674 Laplace transforms ̃x(s) = 1 s^2 +a^2 ̃x(s) s^2 . (D.19) Solving this for ̃x(s), we obtain ̃x(s) = 1 s^2 −a^2 , (D.20) and, ...

References Abrams, J. B., Rosso, L., and Tuckerman, M. E. (2006). Efficient and precise solvation free energies via alchemical a ...

676 References Berne, B. J. (1971). Time-dependent properties in condensed media. InPhysical Chemistry(ed. H. Eyrine, D. Henders ...

References 677 for beta hairpin folding from combined parallel tempering and metadynamics. J. Am. Chem. Soc., 128 , 13435. Calde ...

678 References Chipot, C.and Pohorille, A. (eds) (2007).Free Energy Calculations. Springer–Verlag, Berlin, Heidelberg, New York. ...

References 679 Dellago, C., Bolhuis, P. G., and Geissler, P. L. (2002). Transition pathsampling. Adv. Chem. Phys., 123 , 1. Deng ...

680 References Fl ̈ugge, S. (1994).Practical Quantum Mechanics 2nd edn.Springer, Berlin. Frenkel, D. and Smit, B. (2002). Unders ...

References 681 McGraw-Hill, New York, NY. Hohenberg, P. and Kohn, W. (1964). Inhomogeneous electron gas.Phys. Rev., 136 , B864. ...

682 References a thermal environment.J. Stat. Mech.: Theor. and Expt., P09005. Kadanoff, L. P. (1966). Scaling laws for Ising mo ...

References 683 Langevin, P. (1908). Sur la th ́eorie du mouvement Brownien.Comptes rendus heb- domadaires des s ́eances de l’aca ...