Essentials of Computational Chemistry

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532 15 ADIABATIC REACTION DYNAMICS we simply project out motion along the MEP at point s and evaluate the partition functions (and the ZPVE) for the remaining degrees of freedom, all of which are bound and at their local minima for that choice of s, in the usual fashion. In practice, minimization of the rate constant with respect to s is accomplished by standard search techniques for situations where analytic gradients of the function to be minimized are not available. Since the partition functions depend on both s and T , changes in T may lead to changes in the value of s minimizing Eq. (15.35). In other words, the variational transition state can move with changes in temperature. Such detailed surveys of the potential energy surface are computationally fairly demanding since not only are energies sought at every point but also gradients (to assess the degree to which one is still on the MEP) and force constants for all bound degrees of freedom (to compute the vibrational partition functions). If the potential energy surface for the reacting system can be described by an analytic function, then the computational complexity is not much worse than for any typical propagation of a molecular mechanics MD trajectory. However, while molecular mechanics can accurately describe regions of the PES in the close vicinity of minima, it is much more difficult to develop analytic functions for regions within which bond breaking (or making) are taking place and, moreover, to connect any such functions to the ‘usual’ ones about minima so that the entire PES can be described in a smooth, differentiable way. One alternative to having a global analytic function is simply to compute, on the fly, energies, derivatives, and force constants whenever they are required. Such an approach, usually called ‘direct dynamics’ (Truhlar and Gordon 1990), in principle permits the use of sophisticated quantum mechanical methods at every point. While this will typically improve on the quality of the computed quantities compared to a global analytic function, this improvement comes at the expense of potentially having to do many, many such QM calculations (this latter point is especially telling when one is actually propagating a trajectory, as opposed to simply looking for the variational TS). These constraints have provided much of the motivation for the development of rapid MM and QM/MM models, like the MCMM and VB methods discussed in Chapters 2 and 13, that can accurately reproduce features of the PES in the vicinity of TS structures, where VTST optimization takes place. Note that the conventional TST expression is simply the special case of VTST where evaluation is done exclusively for s = 0. As such, the VTST rate constant will always be less than or equal to the conventional TST rate constant (equal in the event that s = 0 minimizes Eq. (15.35)). Put differently, when very accurate potential energy surfaces are available, the conventional TST rate constant is typically an overestimate of the exact classical rate constant. (Note that it is possible, however, for a compensating or even offsetting error to arise from overestimation of the barrier height if the potential energy surface is not very accurate.) Allison and Truhlar have compared TST and VTST to accurate solution of the timedependent Schrödinger equation for a number of three-atom chemical reactions (it is only for such small systems that the accurate solution of the time-dependent Schrödinger equation is practical) and those results are listed in Table 15.1. On the high-quality surfaces available for this comparison, VTST is typically accurate to within 50% at temperatures above 600 K.
15.3 TRANSITION-STATE THEORY 533 Table 15.1 Logarithmically averaged percent errors in TST and VTST compared to accurate quantal rate constants for a series of 3-atom reactionsa T (K) Number of reactions LAPE (%) TST VTST 200 37 1480 1952 250 40 452 569 300 48 283 296 400 49 131 148 600 37 65 51 1000 34 53 24 1500 26 63 18 2400 8 139 21 a Allison and Truhlar 1998. Logarithmically averaged percent errors treat each factor of 2, whether an overestimate or an underestimate, as a 100% error. At the highest temperatures, VTST is about five times more accurate than canonical TST, but even canonical TST is still accurate to within about a factor of 2. Note that improved performance of VTST as temperature increases is to be expected since entropic effects increasingly dominate under those conditions, and it is primarily entropy that moves the optimal dividing surface away from the potential energy saddle point. At low temperatures, TST appears to outperform VTST, but that is an artifact of not considering tunneling contributions to the rate constant (see Section 15.3.3). Tunneling effects are included in the accurate quantal rate constants; since TST usually overestimates the classical rate, it is accidentally in better agreement with the quantal rate, which is always increased over the exact classical rate by tunneling. Note that, with the minimized rate constant in hand, a generalized activation free energy can be defined as the difference between the free energy of the reactants and that for the point smin. Note also that for the computation of isotope effects, VTST proceeds exactly like conventional TST, except that there is no requirement at a given temperature that the value of s that minimizes the rate constant for the light-atom-substituted system will be the same value of s that minimizes the rate constant for the heavy-atom-substituted system. Each must be determined separately, at which point the ratio of rate constants for that temperature may be expressed. 15.3.3 Quantum Effects on the Rate Constant The metaphor invoked in Section 15.2 of a reacting system as a cloud wandering through a mountain pass is, by virtue of being macroscopic, necessarily a classical metaphor. In visualizing that situation, we accept as a given that those portions of the cloud below the level of the pass (i.e., at too low an energy) fail to go through and portions above the pass always do. Like the cloud in the mountain pass, the probability of transmission from
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Essentials of Computational Chemist
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Essentials of Computational Chemist
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For Katherine
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viii CONTENTS 2.5 Menagerie of Mode
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x CONTENTS 7 Including Electron Cor
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xii CONTENTS 11.4 Strengths and Wea
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Preface to the First Edition Comput
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PREFACE TO THE FIRST EDITION xvii d
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xx PREFACE TO THE SECOND EDITION to
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Known Typographical and Other Error
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1 What are Theory, Computation, and
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Activation free energy (kcal mol
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1.3 COMPUTABLE QUANTITIES 5 etc.) f
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a a r 1 1.3 COMPUTABLE QUANTITIES 7
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1.3 COMPUTABLE QUANTITIES 9 path is
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1.4 COST AND EFFICIENCY 11 this is
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1.4 COST AND EFFICIENCY 13 kinds of
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Physical quantity (unit name) BIBLI
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2 Molecular Mechanics 2.1 History a
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2.2 POTENTIAL ENERGY FUNCTIONAL FOR
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2.3 FORCE-FIELD ENERGIES AND THERMO
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Heat of formation A E nb (A) 2.4 GE
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gradient vector, g, which is define
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Eq. (2.38) as 2.4 GEOMETRY OPTIMIZA
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2.4 GEOMETRY OPTIMIZATION 47 that t
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2.4 GEOMETRY OPTIMIZATION 49 is opp
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Table 2.1 Force fields Name (if any
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12 Davies, E. K. and Murrall, N. W.
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5 (MM2(85), MM2(91), Chem-3D) Compr
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2.5 MENAGERIE OF MODERN FORCE FIELD
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2.6 FORCE FIELDS AND DOCKING 63 Fig
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2.7 CASE STUDY: (2R ∗ ,4S ∗ )-1
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REFERENCES 67 Cramer, C. J. 1994.
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3 Simulations of Molecular Ensemble
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3.2 PHASE SPACE AND TRAJECTORIES 71
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m −b r eq −b 3.3 MOLECULAR DYNA
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and 3.3 MOLECULAR DYNAMICS 75 p(t +
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3.3 MOLECULAR DYNAMICS 77 step mean
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3.3 MOLECULAR DYNAMICS 79 of course
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3.4 MONTE CARLO 81 One way to reduc
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3.5 ENSEMBLE AND DYNAMICAL PROPERTY
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3.6 KEY DETAILS IN FORMALISM 89 Fig
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3.6 KEY DETAILS IN FORMALISM 91 alc
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3.6 KEY DETAILS IN FORMALISM 93 mor
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3.6 KEY DETAILS IN FORMALISM 95 der
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3.6 KEY DETAILS IN FORMALISM 97 to
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3.8 CASE STUDY: SILICA SODALITE 99
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BIBLIOGRAPHY AND SUGGESTED ADDITION
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REFERENCES 103 Jaqaman, K. and Orto
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106 4 FOUNDATIONS OF MOLECULAR ORBI
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132 5 SEMIEMPIRICAL IMPLEMENTATIONS
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166 6 AB INITIO HARTREE-FOCK MO THE
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204 7 INCLUDING ELECTRON CORRELATIO
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250 8 DENSITY FUNCTIONAL THEORY num
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252 8 DENSITY FUNCTIONAL THEORY for
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254 8 DENSITY FUNCTIONAL THEORY fun
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256 8 DENSITY FUNCTIONAL THEORY Not
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258 8 DENSITY FUNCTIONAL THEORY emp
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260 8 DENSITY FUNCTIONAL THEORY whe
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264 8 DENSITY FUNCTIONAL THEORY [As
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266 8 DENSITY FUNCTIONAL THEORY som
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268 8 DENSITY FUNCTIONAL THEORY for
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272 8 DENSITY FUNCTIONAL THEORY Ano
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274 8 DENSITY FUNCTIONAL THEORY Eve
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276 8 DENSITY FUNCTIONAL THEORY val
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278 8 DENSITY FUNCTIONAL THEORY acc
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280 8 DENSITY FUNCTIONAL THEORY als
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282 8 DENSITY FUNCTIONAL THEORY Tab
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294 8 DENSITY FUNCTIONAL THEORY con
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300 8 DENSITY FUNCTIONAL THEORY dou
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302 8 DENSITY FUNCTIONAL THEORY Cho
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9 Charge Distribution and Spectrosc
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9.1 PROPERTIES RELATED TO CHARGE DI
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H H P O H H H F Cl Cl P P F H F F P
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9.3 SPECTROSCOPY OF NUCLEAR MOTION
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Energy hn 0→1 9.3 SPECTROSCOPY OF
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9.4 NMR SPECTRAL PROPERTIES 345 The
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9.4 NMR SPECTRAL PROPERTIES 347 Tab
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9.5 CASE STUDY: MATRIX ISOLATION OF
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REFERENCES 351 The use of computed
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REFERENCES 353 Rablen, P. R., Pearl
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356 10 THERMODYNAMIC PROPERTIES of
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358 10 THERMODYNAMIC PROPERTIES whe
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360 10 THERMODYNAMIC PROPERTIES tha
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362 10 THERMODYNAMIC PROPERTIES (Th
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364 10 THERMODYNAMIC PROPERTIES the
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368 10 THERMODYNAMIC PROPERTIES Ene
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372 10 THERMODYNAMIC PROPERTIES 10.
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374 10 THERMODYNAMIC PROPERTIES gen
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382 10 THERMODYNAMIC PROPERTIES Tab
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384 10 THERMODYNAMIC PROPERTIES Clo
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386 11 IMPLICIT MODELS FOR CONDENSE
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12 Explicit Models for Condensed Ph
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12.2 COMPUTING FREE-ENERGY DIFFEREN
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∆G 12.2 COMPUTING FREE-ENERGY DIF
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12.4 SOLVENT MODELS 445 In some cas
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12.4 SOLVENT MODELS 447 increases t
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12.5 RELATIVE MERITS OF EXPLICIT AN
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12.5 RELATIVE MERITS OF EXPLICIT AN
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12.6 CASE STUDY: BINDING OF BIOTIN
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REFERENCES 455 Levy, R. M. and Gall
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13 Hybrid Quantal/Classical Models
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13.2 BOUNDARIES THROUGH SPACE 459 a
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13.2 BOUNDARIES THROUGH SPACE 461 i
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13.2 BOUNDARIES THROUGH SPACE 463 T
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13.2 BOUNDARIES THROUGH SPACE 465 W
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13.3 BOUNDARIES THROUGH BONDS 467 t
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13.3 BOUNDARIES THROUGH BONDS 469 o
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180 180 4 10 15 15 20 150 6 8 150 8
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13.3 BOUNDARIES THROUGH BONDS 473 O
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13.3.3 Frozen Orbitals 13.3 BOUNDAR
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Link atom LSCF GHO 13.4 EMPIRICAL V
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13.4 EMPIRICAL VALENCE BOND METHODS
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Page 536 and 537: 520 15 ADIABATIC REACTION DYNAMICS
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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
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Page 578 and 579: Appendix C Spin Algebra C.1 Spin Op
Page 580 and 581: function we have SPIN ALGEBRA 567
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Page 584 and 585: C.3 UHF Wave Functions SPIN ALGEBRA
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Page 590 and 591: ORBITAL LOCALIZATION 577 〈|H |〉
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Page 594 and 595: 582 INDEX B3PW91, 266-267, 284, 288
Page 596 and 597: 584 INDEX Convergence (continued) f
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Global minimum, 23, 46, 97, 146, 38
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Matrix diagonalization, (see also S
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primitive, 168-173 Slater-type, 134
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Reductive dechlorination, 422-424 R
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SYBYL, 58 Symmetry, 182-188, 209, 2
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