700 Index
Scalar potential, 531
Scale invariance, 635
Scaling factor, length, 642
Scaling hypothesis, 634
Scaling relations, 634
Scaling variable, 644
Schr ̈odinger equation, 373, 377, 395, 399, 534
single-particle, 410
Second fluctuation dissipation theorem, 579
Second law of thermodynamics, 59–61
Second quantization, 413
Separatrix, 9
SHAKE algorithm, 105–107, 256–261
Shear rate, 505, 519, 521, 523, 526, 527
Shooting move, 309–312, 596–598
Slater determinant, 393, 412, 415
SLLOD algorithm, 517
Smooth particle-mesh Ewald summation,
663–666
Spectral density, 570, 575
State function, 56, 315, 320
State vector, 368
Stationary phase method, 283
Steady state, 497
Step function, Heaviside, 82, 131, 605
Stirling’s approximation, 90
Strain-rate tensor, 524
Strang splitting formula, 110
Structure factor, 161, 662, 664, 665
partial, 161
Sublimation curve, 610
Supercritical fluid region, 610
Susceptibility, complex
definition, 554
relation to spectrum, 554–557
Switching function, 127, 318
Symplectic condition
definition, 24
harmonic oscillator, 24
integrator, 101
quaternions, 119–121
shadow Hamiltonian, 121–122
symplectic map, 121
velocity Verlet, 113
Tensor product, 370
Thermal equilibrium, 84–87
Thermal reservoir, 137
Thermodynamic equilibrium, 56
Thermodynamic integration, 318–321
Thermodynamic limit, 81, 83, 90, 95, 143,
271, 278, 417, 424, 437, 628
energy-level spacing, 403, 414
equivalence of ensembles, 96, 265, 415
Thermodynamic state, 56
Thermodynamic system, 56
Thermostat
isokinetic, 202–207
Nos ́e, 180–183
Nos ́e–Hoover, 185, 190–191
Nos ́e–Hoover chain, 192–197
Nos ́e–Poincar ́e, 183–184
Third law of thermodynamics, 62
Time correlation function
classical, 502
calculation of, 512–517
energy, 589
Green–Kubo theory, 508, 510
harmonic oscillator, 512
Mori–Zwanzig theory, 602
velocity, 583
vibrational dephasing, 587–591
correlation time, 503
harmonic approximation, 550
infrared spectrum, 550–551
Onsager regression hypothesis, 504
properties of, 503–504
quantum, 545, 548
centroid molecular dynamics
approximation, 562–564
coordinate-space representation, 558
eigenstate representation, 558
harmonic oscillator, 548–550
Kubo-transformed, 561–562
linear response theory, 553–556
ring-polymer molecular dynamics
approximation, 564–565
self-consistent test, 565
symmetrized, 559–561
random force, 579
Time-ordered exponential, 541
Time-ordering operator, 539
Torque, 42
Trajectory, 3
action integral, 29
central potential, 15
chaotic, 128
classical correlation function calculation,
513–517
conservation of phase space distribution,
69
constrained system, 35
correlation time, 503–504
damped-driven oscillator, 497
driven oscillator, 496
ergodic hypothesis, 98
harmonic oscillator, 8, 496
hill potential, 9
histogram test, 359–360
instanton, 492
Jarzynski’s equality, 326–328
kink, 492
Langevin harmonic oscillator, 595
metadynamics, 355–357
numerical
global error, 111
harmonic oscillator, 125
shadow Hamiltonian, 122
Verlet algorithm, 100
phase space, 6, 21