Essentials of Computational Chemistry

More documents

Recommendations

Info

∆G 12.2 COMPUTING FREE-ENERGY DIFFERENCES 435 0 l 1 Figure 12.2 A typical FEP diagram showing the free-energy change in the forward (above) and reverse (below) directions for a λ-coupled mutation because neither endpoint is evaluated using a Hamiltonian that is more than 0.5dλ from the Hamiltonian used to generate the ensemble. Further discussion of technical points and error analysis is deferred to Section 12.2.6. 12.2.3 Slow Growth and Thermodynamic Integration In Eq. (12.16), one may imagine taking λ intervals so small that E on any given interval is arbitrarily close to zero. In that case, we may represent the exponential as a truncated power series, deriving 〈A〉B −〈A〉A = lim dλ→0 λ=0 1 kBT ln 1 + (Eλ+dλ − Eλ) kBT λ (12.18) This expression may be further simplified by noting that ln(1 + x) is well approximated by x for sufficiently small values of x, so that we may write 〈A〉B −〈A〉A = lim dλ→0 λ=0 = lim dλ→0 λ=0 = lim dλ→0 λ=0 1 kBT (Eλ+dλ − Eλ) kBT 1 〈(Eλ+dλ − Eλ)〉λ λ 1 (Eλ+dλ − Eλ) (12.19)
436 12 EXPLICIT MODELS FOR CONDENSED PHASES The removal of the ensemble average over the λ ensemble in the final line on the r.h.s. reflects the protocol of this technique, the so-called slow-growth method. It is assumed that if the Hamiltonian is infinitesimally perturbed at every step in the simulation, then the system will constantly be at equilibrium (following some initial period of equilibration), so separate ensemble averages need not be acquired. In practice, then, the slow-growth technique is rather different from FEP when it comes to evaluating E. Since each change in λ is also a step in the simulation, all of the intrasolvent energy terms change in addition to the solvent–solute interaction terms. With respect to the latter terms, however, the evaluation is similar to FEP in that chimeric molecules are involved. A third simulation protocol for determining Helmholtz free-energy differences can be illustrated from further manipulation of Eq. (12.19). Thus we may write 〈A〉B −〈A〉A = lim 1 〈(Eλ+dλ − Eλ)〉 λ dλ→0 λ=0 1 (Eλ+λ − Eλ) = lim λ→0 λ=0 1 ∂E = dλ 0 ∂λ λ λ λ λ 1 ∂E ≈ ∂λ λ (12.20) λ=0 where we first recognize the calculus relationship between the sum appearing on the r.h.s. in the second line and the definite integral in the third line (and simultaneously the definition of the partial derivative), and we then approximate the definite integral as a sum over small intervals. While the transformation from line 3 to line 4 may appear to simply reverse the transformation from line 2 to line 3, this is not the case, because the partial derivative remains in its analytic form; this is possible because most simulations evaluate the energy using E(λ) functions that are trivially differentiated. Moreover, λ in the final line is no longer infinitesimally small, i.e., this is a standard estimation of an integral by division of the integration range into discrete intervals with the function approximated over each interval by a single value, in this case the value at the start of the interval. This process defines the thermodynamic integration (TI) method. [TI can be derived in a much more rigorous and general way, and indeed, FEP may be regarded as a special case of TI; interested readers are referred to the bibliography at the end of the chapter.] It is evident that TI and FEP are similar in that they involve multiple simulations over different windows λ, with accuracy expected to increase when more and smaller windows are employed. However, there are key differences as well. In TI, the ensemble average for one value of λ is not used to evaluate any energies involving a different value of λ; only the ensemble average of the energy derivative is accumulated. Moreover, different forms of E(λ) may be conveniently evaluated, corresponding to different mutation paths from A λ
Page 2 and 3:
Essentials of Computational Chemist
Page 4 and 5:
Essentials of Computational Chemist
Page 6 and 7:
For Katherine
Page 8 and 9:
viii CONTENTS 2.5 Menagerie of Mode
Page 10 and 11:
x CONTENTS 7 Including Electron Cor
Page 12 and 13:
xii CONTENTS 11.4 Strengths and Wea
Page 14 and 15:
Preface to the First Edition Comput
Page 16 and 17:
PREFACE TO THE FIRST EDITION xvii d
Page 18 and 19:
xx PREFACE TO THE SECOND EDITION to
Page 20 and 21:
Known Typographical and Other Error
Page 22 and 23:
1 What are Theory, Computation, and
Page 24 and 25:
Activation free energy (kcal mol
Page 26 and 27:
1.3 COMPUTABLE QUANTITIES 5 etc.) f
Page 28 and 29:
a a r 1 1.3 COMPUTABLE QUANTITIES 7
Page 30 and 31:
1.3 COMPUTABLE QUANTITIES 9 path is
Page 32 and 33:
1.4 COST AND EFFICIENCY 11 this is
Page 34 and 35:
1.4 COST AND EFFICIENCY 13 kinds of
Page 36 and 37:
Physical quantity (unit name) BIBLI
Page 38 and 39:
2 Molecular Mechanics 2.1 History a
Page 40 and 41:
2.2 POTENTIAL ENERGY FUNCTIONAL FOR
Page 42 and 43:
Page 44 and 45:
Page 46 and 47:
Page 48 and 49:
Page 50 and 51:
Page 52 and 53:
Page 54 and 55:
Page 56 and 57:
Page 58 and 59:
Page 60 and 61:
2.3 FORCE-FIELD ENERGIES AND THERMO
Page 62 and 63:
Heat of formation A E nb (A) 2.4 GE
Page 64 and 65:
gradient vector, g, which is define
Page 66 and 67:
Eq. (2.38) as 2.4 GEOMETRY OPTIMIZA
Page 68 and 69:
2.4 GEOMETRY OPTIMIZATION 47 that t
Page 70 and 71:
2.4 GEOMETRY OPTIMIZATION 49 is opp
Page 72 and 73:
Table 2.1 Force fields Name (if any
Page 74 and 75:
12 Davies, E. K. and Murrall, N. W.
Page 76 and 77:
5 (MM2(85), MM2(91), Chem-3D) Compr
Page 78 and 79:
2.5 MENAGERIE OF MODERN FORCE FIELD
Page 80 and 81:
Page 82 and 83:
Page 84 and 85:
2.6 FORCE FIELDS AND DOCKING 63 Fig
Page 86 and 87:
2.7 CASE STUDY: (2R ∗ ,4S ∗ )-1
Page 88 and 89:
REFERENCES 67 Cramer, C. J. 1994.
Page 90 and 91:
3 Simulations of Molecular Ensemble
Page 92 and 93:
3.2 PHASE SPACE AND TRAJECTORIES 71
Page 94 and 95:
m −b r eq −b 3.3 MOLECULAR DYNA
Page 96 and 97:
and 3.3 MOLECULAR DYNAMICS 75 p(t +
Page 98 and 99:
3.3 MOLECULAR DYNAMICS 77 step mean
Page 100 and 101:
3.3 MOLECULAR DYNAMICS 79 of course
Page 102 and 103:
3.4 MONTE CARLO 81 One way to reduc
Page 104 and 105:
3.5 ENSEMBLE AND DYNAMICAL PROPERTY
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
3.6 KEY DETAILS IN FORMALISM 89 Fig
Page 112 and 113:
3.6 KEY DETAILS IN FORMALISM 91 alc
Page 114 and 115:
3.6 KEY DETAILS IN FORMALISM 93 mor
Page 116 and 117:
3.6 KEY DETAILS IN FORMALISM 95 der
Page 118 and 119:
3.6 KEY DETAILS IN FORMALISM 97 to
Page 120 and 121:
3.8 CASE STUDY: SILICA SODALITE 99
Page 122 and 123:
BIBLIOGRAPHY AND SUGGESTED ADDITION
Page 124 and 125:
REFERENCES 103 Jaqaman, K. and Orto
Page 126 and 127:
106 4 FOUNDATIONS OF MOLECULAR ORBI
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
132 5 SEMIEMPIRICAL IMPLEMENTATIONS
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
Page 186 and 187:
166 6 AB INITIO HARTREE-FOCK MO THE
Page 188 and 189:
Page 190 and 191:
Page 192 and 193:
Page 194 and 195:
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
Page 204 and 205:
Page 206 and 207:
Page 208 and 209:
Page 210 and 211:
Page 212 and 213:
Page 214 and 215:
Page 216 and 217:
Page 218 and 219:
Page 220 and 221:
Page 222 and 223:
Page 224 and 225:
204 7 INCLUDING ELECTRON CORRELATIO
Page 226 and 227:
Page 228 and 229:
Page 230 and 231:
Page 232 and 233:
Page 234 and 235:
Page 236 and 237:
Page 238 and 239:
Page 240 and 241:
Page 242 and 243:
Page 244 and 245:
Page 246 and 247:
Page 248 and 249:
Page 250 and 251:
Page 252 and 253:
Page 254 and 255:
Page 256 and 257:
Page 258 and 259:
Page 260 and 261:
Page 262 and 263:
Page 264 and 265:
Page 266 and 267:
Page 268 and 269:
Page 270 and 271:
250 8 DENSITY FUNCTIONAL THEORY num
Page 272 and 273:
252 8 DENSITY FUNCTIONAL THEORY for
Page 274 and 275:
254 8 DENSITY FUNCTIONAL THEORY fun
Page 276 and 277:
256 8 DENSITY FUNCTIONAL THEORY Not
Page 278 and 279:
258 8 DENSITY FUNCTIONAL THEORY emp
Page 280 and 281:
260 8 DENSITY FUNCTIONAL THEORY whe
Page 282 and 283:
Page 284 and 285:
264 8 DENSITY FUNCTIONAL THEORY [As
Page 286 and 287:
266 8 DENSITY FUNCTIONAL THEORY som
Page 288 and 289:
268 8 DENSITY FUNCTIONAL THEORY for
Page 290 and 291:
Page 292 and 293:
272 8 DENSITY FUNCTIONAL THEORY Ano
Page 294 and 295:
274 8 DENSITY FUNCTIONAL THEORY Eve
Page 296 and 297:
276 8 DENSITY FUNCTIONAL THEORY val
Page 298 and 299:
278 8 DENSITY FUNCTIONAL THEORY acc
Page 300 and 301:
280 8 DENSITY FUNCTIONAL THEORY als
Page 302 and 303:
282 8 DENSITY FUNCTIONAL THEORY Tab
Page 304 and 305:
Page 306 and 307:
Page 308 and 309:
Page 310 and 311:
Page 312 and 313:
Page 314 and 315:
294 8 DENSITY FUNCTIONAL THEORY con
Page 316 and 317:
Page 318 and 319:
Page 320 and 321:
300 8 DENSITY FUNCTIONAL THEORY dou
Page 322 and 323:
302 8 DENSITY FUNCTIONAL THEORY Cho
Page 324 and 325:
9 Charge Distribution and Spectrosc
Page 326 and 327:
9.1 PROPERTIES RELATED TO CHARGE DI
Page 328 and 329:
Page 330 and 331:
Page 332 and 333:
Page 334 and 335:
Page 336 and 337:
Page 338 and 339:
Page 340 and 341:
Page 342 and 343:
Page 344 and 345:
Page 346 and 347:
Page 348 and 349:
H H P O H H H F Cl Cl P P F H F F P
Page 350 and 351:
9.3 SPECTROSCOPY OF NUCLEAR MOTION
Page 352 and 353:
Page 354 and 355:
Energy hn 0→1 9.3 SPECTROSCOPY OF
Page 356 and 357:
Page 358 and 359:
Page 360 and 361:
Page 362 and 363:
Page 364 and 365:
9.4 NMR SPECTRAL PROPERTIES 345 The
Page 366 and 367:
9.4 NMR SPECTRAL PROPERTIES 347 Tab
Page 368 and 369:
9.5 CASE STUDY: MATRIX ISOLATION OF
Page 370 and 371:
REFERENCES 351 The use of computed
Page 372 and 373:
REFERENCES 353 Rablen, P. R., Pearl
Page 374 and 375:
356 10 THERMODYNAMIC PROPERTIES of
Page 376 and 377:
358 10 THERMODYNAMIC PROPERTIES whe
Page 378 and 379:
360 10 THERMODYNAMIC PROPERTIES tha
Page 380 and 381:
362 10 THERMODYNAMIC PROPERTIES (Th
Page 382 and 383:
364 10 THERMODYNAMIC PROPERTIES the
Page 384 and 385:
Page 386 and 387:
368 10 THERMODYNAMIC PROPERTIES Ene
Page 388 and 389:
Page 390 and 391:
372 10 THERMODYNAMIC PROPERTIES 10.
Page 392 and 393:
374 10 THERMODYNAMIC PROPERTIES gen
Page 394 and 395:
Page 396 and 397:
Page 398 and 399:
Page 400 and 401:
382 10 THERMODYNAMIC PROPERTIES Tab
Page 402 and 403: 384 10 THERMODYNAMIC PROPERTIES Clo
Page 404 and 405: 386 11 IMPLICIT MODELS FOR CONDENSE
Page 446 and 447: 12 Explicit Models for Condensed Ph
Page 448 and 449: 12.2 COMPUTING FREE-ENERGY DIFFEREN
Page 462 and 463: 12.4 SOLVENT MODELS 445 In some cas
Page 464 and 465: 12.4 SOLVENT MODELS 447 increases t
Page 466 and 467: 12.5 RELATIVE MERITS OF EXPLICIT AN
Page 468 and 469: 12.5 RELATIVE MERITS OF EXPLICIT AN
Page 470 and 471: 12.6 CASE STUDY: BINDING OF BIOTIN
Page 472 and 473: REFERENCES 455 Levy, R. M. and Gall
Page 474 and 475: 13 Hybrid Quantal/Classical Models
Page 476 and 477: 13.2 BOUNDARIES THROUGH SPACE 459 a
Page 478 and 479: 13.2 BOUNDARIES THROUGH SPACE 461 i
Page 480 and 481: 13.2 BOUNDARIES THROUGH SPACE 463 T
Page 482 and 483: 13.2 BOUNDARIES THROUGH SPACE 465 W
Page 484 and 485: 13.3 BOUNDARIES THROUGH BONDS 467 t
Page 486 and 487: 13.3 BOUNDARIES THROUGH BONDS 469 o
Page 488 and 489: 180 180 4 10 15 15 20 150 6 8 150 8
Page 490 and 491: 13.3 BOUNDARIES THROUGH BONDS 473 O
Page 492 and 493: 13.3.3 Frozen Orbitals 13.3 BOUNDAR
Page 494 and 495: Link atom LSCF GHO 13.4 EMPIRICAL V
Page 496 and 497: 13.4 EMPIRICAL VALENCE BOND METHODS
Page 498 and 499: 13.4 EMPIRICAL VALENCE BOND METHODS
Page 500 and 501: 13.5 CASE STUDY: CATALYTIC MECHANIS
Page 502 and 503:
REFERENCES 485 Gao, J., Amara, P.,
Page 504 and 505:
14 Excited Electronic States 14.1 D
Page 506 and 507:
14.1 DETERMINANTAL/CONFIGURATIONAL
Page 508 and 509:
14.1 DETERMINANTAL/CONFIGURATIONAL
Page 510 and 511:
14.2 SINGLY EXCITED STATES 493 and
Page 512 and 513:
14.2 SINGLY EXCITED STATES 495 Tabl
Page 514 and 515:
14.2 SINGLY EXCITED STATES 497 wher
Page 516 and 517:
14.3 GENERAL EXCITED STATE METHODS
Page 518 and 519:
14.3 GENERAL EXCITED STATE METHODS
Page 520 and 521:
14.4 SUM AND PROJECTION METHODS 503
Page 522 and 523:
14.4 SUM AND PROJECTION METHODS 505
Page 524 and 525:
14.5 TRANSITION PROBABILITIES 507 o
Page 526 and 527:
14.5 TRANSITION PROBABILITIES 509 I
Page 528 and 529:
14.6 SOLVATOCHROMISM 511 overlap) l
Page 530 and 531:
14.7 CASE STUDY: ORGANIC LIGHT EMIT
Page 532 and 533:
BIBLIOGRAPHY AND SUGGESTED ADDITION
Page 534 and 535:
REFERENCES 517 Kim, K., Shavitt, I,
Page 536 and 537:
520 15 ADIABATIC REACTION DYNAMICS
Page 538 and 539:
Page 540 and 541:
Page 542 and 543:
Page 544 and 545:
Page 546 and 547:
Page 548 and 549:
Page 550 and 551:
Page 552 and 553:
Page 554 and 555:
Page 556 and 557:
Page 558 and 559:
Page 560 and 561:
Page 562 and 563:
Page 564 and 565:
Appendix A Acronym Glossary Note: B
Page 566 and 567:
ACRONYM GLOSSARY 551 ECP Effective
Page 568 and 569:
ACRONYM GLOSSARY 553 mPW1S mPW1PW91
Page 570 and 571:
ACRONYM GLOSSARY 555 SF-CISD Spin-f
Page 572 and 573:
558 APPENDIX B H 3 C CH 3 H H H a
Page 574 and 575:
560 APPENDIX B Linear molecule? no
Page 576 and 577:
562 APPENDIX B Table B.5 Product ru
Page 578 and 579:
Appendix C Spin Algebra C.1 Spin Op
Page 580 and 581:
function we have SPIN ALGEBRA 567
Page 582 and 583:
SPIN ALGEBRA 569 + = 1 1 ( 2 2 α(
Page 584 and 585:
C.3 UHF Wave Functions SPIN ALGEBRA
Page 586 and 587:
SPIN ALGEBRA 573 = 〈cs s |H |cs s
Page 588 and 589:
Appendix D Orbital Localization D.1
Page 590 and 591:
ORBITAL LOCALIZATION 577 〈|H |〉
Page 592 and 593:
E rel (kcal mol −1 ) 10 8 6 4 2 0
Page 594 and 595:
582 INDEX B3PW91, 266-267, 284, 288
Page 596 and 597:
584 INDEX Convergence (continued) f
Page 598 and 599:
Global minimum, 23, 46, 97, 146, 38
Page 600 and 601:
Matrix diagonalization, (see also S
Page 602 and 603:
primitive, 168-173 Slater-type, 134
Page 604 and 605:
Reductive dechlorination, 422-424 R
Page 606 and 607:
SYBYL, 58 Symmetry, 182-188, 209, 2
show all

Essentials of Computational Chemistry

Create successful ePaper yourself

Delete template?

Save as template?