Essentials of Computational Chemistry

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280 8 DENSITY FUNCTIONAL THEORY also lead to very poor predictions along coordinates for bond dissociation (Bally and Sastry 1997; Zhang and Wang 1998; Gräfenstein, Kraka, and Cremer 2004), nucleophilic substitution (Adamo and Barone 1998; Gritsenko et al. 2000), competing cycloaddition pathways (Jones et al. 2002), and rotation about single bonds in conjugated systems, like the benzylic bond in styrene (Choi, Kertesz, and Karpfen 1997). Note that, because electron correlation often stabilizes delocalized electronic structures over localized ones, HF theory tends to be inaccurate for such systems in the opposite direction from DFT, and thus, again, hybrid ACM functionals tend to show improved performance by an offsetting of errors. A number of different methods have been proposed to introduce a self-interaction correction into the Kohn–Sham formalism (Perdew and Zunger 1981; Kümmel and Perdew 2003; Gräfenstein, Kraka, and Cremer 2004). This correction is particularly useful in situations with odd numbers of electrons distributed over more than one atom, e.g., in transition-state structures (Patchkovskii and Ziegler 2002). Unfortunately, the correction introduces an additional level of self-consistency into the KS SCF process because it depends on the KS orbitals, and it tends to be difficult and time-consuming to converge the relevant equations. However, future developments in non-local correlation functionals may be able to correct for self-interaction error in a more efficient manner. 8.6 General Performance Overview of DFT While the cases noted in the immediately preceding section illustrate certain pathological failures of current DFT functionals, the general picture for DFT is really quite bright. For the ‘average’ problem, DFT is the method of choice to achieve a particular level of accuracy at lowest cost. With the appearance of each new functional, there has tended to be at least one paper benchmarking the performance of that functional on a variety of standard test sets (for energies, structures, etc.) and there is now a rather large body of data that is somewhat scattered and disjoint with respect to individual functional performance. The comparisons made below are designed to provide as broad a coverage as possible without becoming unwieldy, and as such are not necessarily exhaustive. 8.6.1 Energetics Exact DFT is an ab initio theory (even if most modern implementations may be regarded as having a semiempirical flavor) and like other such theories its quality with respect to energetic predictions is usually judged based on its performance for atomization energies. Table 8.1 collects average unsigned and maximum absolute errors in atomization energies as computed for various functionals, and for some other computational methodologies, over several different test sets of increasing complexity. The G2/97 and G3/99 test sets (columns D and E) include substituted hydrocarbons, radicals, inorganic hydrides, unsaturated ring hydrocarbons, and polyhalogenated organics and inorganics. While effort is made to describe all levels of theory accurately, it should be noted that in many cases geometries are optimized (and zero-point vibrational energies computed) using a basis set smaller than that used to compute the atomization energies. In addition, some results for the larger test sets include
8.6 GENERAL PERFORMANCE OVERVIEW OF DFT 281 Table 8.1 Mean absolute errors and maximum errors, where available, in atomization energies for different methods over several different test sets (kcal mol −1 ) Level of theory Test sets a A B C D E MO theoretical methods MNDO 9.3 (116.7) AM1 7.8 (58.2) PM3 7.0 (32.2) MNDO/d 7.3 (33.9) HF/6-31G(d) 85.9 80.5 150.6 b (184.3) HF/6-31G(d,p) 119.2 HF/6-311G(2df,p) 82.0 146.2 b HF/6-311+G(3df,2p) 74.5 144.4 b 148.3 c 211.5 c (170.0) (344.1) (582.2) MP2/6-31G(d) 22.4 16.0 38.1 b (40.3) MP2/6-31G(d,p) 23.7 22.0 MP2/6-311+G(3df,2p) 7.3 9.7 b (25.4) MP4/6-31G(2df,p) 13.5 b QCISD/6-31G(d) 28.8 51.7 b MC-QCISD 1.7 b CCSD(T)/6-311G(2df,p) 11.5 G2(MP2) 1.8 (8.8) CBS-4 2.7 (12.9) G2 1.2 1.2 1.4 (4.9) (10.6) CBS-q 2.3 (11.4) CBS-Q 1.0 1.4 b 1.2 (3.3) (8.1) G3(MP2) 1.2 1.2 (9.3) G3 1.1 0.9 b 0.9 1.1 (4.0) (4.9) (7.1) G3S(MP2) 1.2 1.3 G3S 0.9 b 1.0 1.1 MCG3 1.0 b (continued overleaf )
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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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Page 324 and 325: 9 Charge Distribution and Spectrosc
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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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13.4 EMPIRICAL VALENCE BOND METHODS
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13.5 CASE STUDY: CATALYTIC MECHANIS
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REFERENCES 485 Gao, J., Amara, P.,
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14 Excited Electronic States 14.1 D
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14.1 DETERMINANTAL/CONFIGURATIONAL
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14.1 DETERMINANTAL/CONFIGURATIONAL
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14.2 SINGLY EXCITED STATES 493 and
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14.2 SINGLY EXCITED STATES 495 Tabl
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14.2 SINGLY EXCITED STATES 497 wher
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14.3 GENERAL EXCITED STATE METHODS
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14.3 GENERAL EXCITED STATE METHODS
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14.4 SUM AND PROJECTION METHODS 503
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14.4 SUM AND PROJECTION METHODS 505
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14.5 TRANSITION PROBABILITIES 507 o
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14.5 TRANSITION PROBABILITIES 509 I
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14.6 SOLVATOCHROMISM 511 overlap) l
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14.7 CASE STUDY: ORGANIC LIGHT EMIT
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BIBLIOGRAPHY AND SUGGESTED ADDITION
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REFERENCES 517 Kim, K., Shavitt, I,
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520 15 ADIABATIC REACTION DYNAMICS
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Appendix A Acronym Glossary Note: B
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ACRONYM GLOSSARY 551 ECP Effective
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ACRONYM GLOSSARY 553 mPW1S mPW1PW91
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ACRONYM GLOSSARY 555 SF-CISD Spin-f
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558 APPENDIX B H 3 C CH 3 H H H a
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560 APPENDIX B Linear molecule? no
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562 APPENDIX B Table B.5 Product ru
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Appendix C Spin Algebra C.1 Spin Op
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function we have SPIN ALGEBRA 567
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SPIN ALGEBRA 569 + = 1 1 ( 2 2 α(
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C.3 UHF Wave Functions SPIN ALGEBRA
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SPIN ALGEBRA 573 = 〈cs s |H |cs s
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Appendix D Orbital Localization D.1
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ORBITAL LOCALIZATION 577 〈|H |〉
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E rel (kcal mol −1 ) 10 8 6 4 2 0
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582 INDEX B3PW91, 266-267, 284, 288
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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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