Essentials of Computational Chemistry

More documents

Recommendations

Info

120 4 FOUNDATIONS OF MOLECULAR ORBITAL THEORY rarely sufficiently accurate for quantitative assessments. To improve our models, we need to take a more sophisticated accounting of many-electron effects. 4.5.1 Hartree-product Wave Functions Let us examine the Schrödinger equation in the context of a one-electron Hamiltonian a little more carefully. When the only terms in the Hamiltonian are the one-electron kinetic energy and nuclear attraction terms, the operator is ‘separable’ and may be expressed as H = N hi i=1 (4.32) where N is the total number of electrons and hi is the one-electron Hamiltonian defined by hi =− 1 2 ∇2 i − M Zk rik k=1 (4.33) where M is the total number of nuclei (note that Eq. (4.33) is written in atomic units). Eigenfunctions of the one-electron Hamiltonian defined by Eq. (4.33) must satisfy the corresponding one-electron Schrödinger equation hiψi = εiψi (4.34) Because the Hamiltonian operator defined by Eq. (4.32) is separable, its many-electron eigenfunctions can be constructed as products of one-electron eigenfunctions. That is HP = ψ1ψ2 ···ψN A wave function of the form of Eq. (4.35) is called a ‘Hartree-product’ wave function. The eigenvalue of is readily found from proving the validity of Eq. (4.35), viz., HHP = Hψ1ψ2 ···ψN N = hiψ1ψ2 ···ψN i=1 = (h1ψ1)ψ2 ···ψN + ψ1(h2ψ2) ···ψN + ...+ ψ1ψ2 ···(hNψN) = (ε1ψ1)ψ2 ···ψN + ψ1(ε2ψ2) ···ψN + ...+ ψ1ψ2 ···(εNψN) N = εiψ1ψ2 ···ψN i=1 N = εi i=1 HP (4.35) (4.36)
4.5 MANY-ELECTRON WAVE FUNCTIONS 121 where repeated application of Eq. (4.34) is used in proving that the energy eigenvalue of the many-electron wave function is simply the sum of the one-electron energy eigenvalues. Note that Eqs. (4.32)–(4.36) provide the mathematical rigor behind the Hückel theory example presented more informally above. Note that if every ψ is normalized then HP is also normalized, since |HP| 2 = |ψ1| 2 |ψ2| 2 ···|ψN| 2 . 4.5.2 The Hartree Hamiltonian As noted above, however, the Hamiltonian defined by Eqs. (4.32) and (4.33) does not include interelectronic repulsion, computation of which is vexing because it depends not on one electron, but instead on all possible (simultaneous) pairwise interactions. We may ask, however, how useful is the Hartree-product wave function in computing energies from the correct Hamiltonian? That is, we wish to find orbitals ψ that minimize 〈HP |H | HP〉. By applying variational calculus, one can show that each such orbital ψi is an eigenfunction of its own operator hi defined by hi =− 1 2 ∇2 i − M Zk + Vi{j} (4.37) rik k=1 where the final term represents an interaction potential with all of the other electrons occupying orbitals {j} and may be computed as Vi{j} = j=i ρj rij dr (4.38) where ρj is the charge (probability) density associated with electron j. The repulsive third term on the r.h.s. of Eq. (4.37) is thus exactly analogous to the attractive second term, except that nuclei are treated as point charges, while electrons, being treated as wave functions, have their charge spread out, so an integration over all space is necessary. Recall, however, that ρj =|ψj| 2 . Since the point of undertaking the calculation is to determine the individual ψ, how can they be used in the one-electron Hamiltonians before they are known? To finesse this problem, Hartree (1928) proposed an iterative ‘self-consistent field’ (SCF) method. In the first step of the SCF process, one guesses the wave functions ψ for all of the occupied MOs (AOs in Hartree’s case, since he was working exclusively with atoms) and uses these to construct the necessary one-electron operators h. Solution of each differential Eq. (4.34) (in an atom, with its spherical symmetry, this is relatively straightforward, and Hartree was helped by his retired father who enjoyed the mathematical challenge afforded by such calculations) provides a new set of ψ, presumably different from the initial guess. So, the one-electron Hamiltonians are formed anew using these presumably more accurate ψ to determine each necessary ρ, and the process is repeated to obtain a still better set of ψ. At some point, the difference between a newly determined set and the immediately preceding set falls below some threshold criterion, and we refer to the final set of ψ as the ‘converged’ SCF orbitals. (An example of a threshold criterion might be that the total electronic energy change by no more than 10 −6 a.u., and/or that the energy eigenvalue for each MO change by
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 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 152 and 153: 132 5 SEMIEMPIRICAL IMPLEMENTATIONS
Page 186 and 187: 166 6 AB INITIO HARTREE-FOCK MO THE
Page 188 and 189: 168 6 AB INITIO HARTREE-FOCK MO THE
Page 190 and 191:
170 6 AB INITIO HARTREE-FOCK MO THE
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 406 and 407:
Page 408 and 409:
Page 410 and 411:
Page 412 and 413:
Page 414 and 415:
Page 416 and 417:
Page 418 and 419:
Page 420 and 421:
Page 422 and 423:
Page 424 and 425:
Page 426 and 427:
Page 428 and 429:
Page 430 and 431:
Page 432 and 433:
Page 434 and 435:
Page 436 and 437:
Page 438 and 439:
Page 440 and 441:
Page 442 and 443:
Page 444 and 445:
Page 446 and 447:
12 Explicit Models for Condensed Ph
Page 448 and 449:
12.2 COMPUTING FREE-ENERGY DIFFEREN
Page 450 and 451:
Page 452 and 453:
∆G 12.2 COMPUTING FREE-ENERGY DIF
Page 454 and 455:
Page 456 and 457:
Page 458 and 459:
Page 460 and 461:
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?