Essentials of Computational Chemistry

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290 8 DENSITY FUNCTIONAL THEORY Table 8.4 (continued) Level of theory IP test sets EA test sets G2 a G2/97 b G2 G2/97 mPWPW91/6-311+G(3df,2p) 0.16 0.12 TPSS/6-311++G(3df,3pd) 0.24 c 0.14 c (1.22) (0.82) VSXC/6-311+G(3df,2p) 0.1 0.23 c 0.13 c (0.4) (1.20) (0.78) XLYP/6-311+G(3df,2p) 0.19 0.12 Hybrid functionals BH&HLYP/6-311+G(3df,2p) 0.21 0.25 B1B95/numeric 0.1 (0.4) B97/numeric 0.1 0.1 (0.6) (0.4) B3LYP/cc-pVDZ 0.2 B3LYP/aug-cc-pVDZ 0.2 B3LYP/6-31+G(d) 0.2 0.2 B3LYP/6-311+G(3df,2p) 0.17 0.18 c 0.10 0.12 c (0.8) (1.57) (0.5) (1.10) B3LYP/aug-cc-pVTZ 0.2 0.1 B3LYP*/TZVPP 0.4 0.1 (0.7) (0.4) B3P86/6-311+G(3df,2p) 0.64 0.55 c 0.59 0.59 c (2.13) (1.63) B3PW91/6-311+G(3df,2p) 0.16 0.19 c 0.10 0.14 c (1.58) (1.08) O3LYP/6-311+G(3df,2p) 0.14 0.11 PBE1PBE/6-311+G(3df,2p) 0.16 0.20 c 0.13 0.16 c (0.7) (1.61) (0.3) (1.09) PW1PW91/6-311+G(3df,2p) 0.16 0.11 mPW1PW91/6-311+G(3df,2p) 0.16 0.11 TPSSh/6-311++G(3df,3pd) 0.23 c 0.16 c (1.41) (0.95) X3LYP/6-311+G(3df,2p) 0.15 0.09 a 38 IPs and 25 EAs, see Curtiss et al. (1991). b 83 IPs and 58 EAs, see Curtiss et al. (1998) and also http://chemistry.anl.gov/compmat/g3theory.htm. c IP set includes three additional data for toluene, aniline, and phenol; geometries and ZPVE obtained at the B3LYP/6-31G(2df,p) level with scaling of vibrational frequencies by 0.9854; electronic energies use 6-311++ G(3df,3pd) basis set; see Staroverov et al. (2003). draw from the table are for the most part similar to those noted above based on atomization energies, except that there is now much less, if any, preference for hybrid functionals over pure functionals, so long as P86 is avoided. In analyses of other systems, including metal
8.6 GENERAL PERFORMANCE OVERVIEW OF DFT 291 atoms, Rienstra-Kiracofe et al. (2002) and Bauschlicher and Gutsev (2002) have separately noted that B3LYP with large basis sets seems to be particularly robust. Finally, atomic and molecular proton affinities (PAs) have also been evaluated for various functionals for ammonia, water, acetylene, silane, phosphine, silylene, hydrochloric acid, and molecular hydrogen. For G2 and G3 theories, the mean unsigned error in PAs is 1.1 and 1.3 kcal mol −1 , respectively. At the SVWN, BLYP, BP86, BPW91, B3LYP, B3P86, and B3PW91 levels (using the 6-311+G(3df,2p) basis set), the corresponding errors are 5.8, 1.8, 1.5, 1.5, 1.3, 1.1, and 1.2 kcal mol −1 , respectively (quantitatively similar results have also been obtained with more modern functionals). The much cheaper hybrid DFT methods are thus entirely competitive with G2 and G3, although the data set is perhaps too small to come to any firm conclusions on this topic (cf. Pokon et al. 2001). 8.6.2 Geometries Analytic first derivatives are available for almost all density functionals, and as a result geometry optimization can be carried out with facility. The performance of the various functionals is usually quite good when it comes to predicting minimum energy structures. As summarized in Table 8.5, bond lengths at the LDA level for molecules composed of firstand second-row atoms are typically as good as those predicted from MP2 optimizations, with both these levels being somewhat improved over HF theory. The use of GGA functionals does not usually result in much improvement over the LDA level. However, the GGA functionals tend to systematically overestimate bond lengths. As noted in Section 6.4.2, the HF level tends to systematically underestimate bond lengths. Thus, it should come as no surprise that the hybrid ACM functionals, which mix the two, give noticeable improvement in predicted bond lengths (of course, the improvements are on the order of 0.005 ˚A, and it should be noted that most molecular properties are very little affected by such small variations in bond lengths). Very small improvements in geometrical accuracy are usually noted with increasing basis-set size beyond those listed in Table 8.5. Accuracies in bond angles for all flavors of DFT average about 1 ◦ , the same as is found for HF and MP2. Similarly, the limited amount of data available for dihedral angles suggests that HF, MP2, and DFT all perform equivalently in this area. Table 8.5 also indicates that the most popular functionals fail to be as accurate for molecules containing third-row main-group elements as they are for molecules made up of elements from the first two rows. The LYP correlation functional seems to perform particularly badly, while the PW91 functional is more robust. It is, however, for the transition metals themselves that DFT has proven to be a tremendous improvement over HF and post-HF methods, particularly for cases where the metal atom is coordinatively unsaturated. The narrow separation between filled and empty d-block orbitals typically leads to enormous non-dynamical correlation problems with an HF treatment, and DFT is much less prone to analogous problems. Even in cases of a saturated coordination sphere, DFT methods typically significantly outperform HF or MP2. Jonas and Thiel (1995) used the BP86 functional to compute geometries for the neutral hexacarbonyl complexes of Cr, Mo, and W, the pentacarbonyl complexes of Fe, Ru, and Os, and the tetracarbonyl
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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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9.3 SPECTROSCOPY OF NUCLEAR MOTION
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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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