Essentials of Computational Chemistry

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9.3 SPECTROSCOPY OF NUCLEAR MOTION 339 Table 9.2 Mean absolute errors in harmonic vibrational frequencies over a 32-molecule G2 subset (cm −1 ) a Level of theory Error MO theoretical methods HF/6-311G(3df,2p) 144 MP2/6-31G(d,p) 99 CCSD(T)/6-311G(3df,2p) LSDA functionals 31 SVWN/6-31G(d,p) GGA functionals 75 BLYP/6-311G(d,p) 59 BPW91/6-311G(d,p) 69 PWPW91/6-311G(d,p) 66 mPWPW91/6-311G(d,p) Hybrid functionals 66 BH&HLYP/6-311G(d,p) 100 B1LYP/6-311G(d,p) 33 B1PW91/6-311G(d,p) 48 mPW1PW91/6-311G(d,p) 39 B3LYP/6-311G(d,p) 31 B3PW91/6-311G(d,p) 45 mPW3PW91/6-311G(d,p) 37 a Test set includes 32 molecules containing only first-row atoms, see Johnson, Gill, and Pople (1993). Data from Adamo and Barone (1998). error remains. CCSD(T) and some of the hybrid levels of density functional theory show the highest accuracies. In general, the BLYP combination seems to be more accurate than BPW91, whether pure or hybrid in formulation, but PWPW91 is nearly as accurate as BLYP, again whether pure or hybrid in formulation. Of some interest in the error analysis is the degree to which the error is systematic. Although HF errors are large, they are very systematic. HF overemphasizes bonding, so all force constants are too large, and thus so are all frequencies. However, application of a constant scaling factor to the HF frequencies improves their accuracy enormously (Pople et al. 1993). Scott and Radom studied this issue in detail for eight different levels of theory using a database of 122 molecules and 1066 fundamentals (i.e., measured, anharmonic vibrational frequencies) and a summary of their results, together with a few other recommended scaling factors, is provided in Table 9.3 (Scott and Radom 1996; see also, Wong 1996). Note that even though the scale factor required for the HF/6-31G(d) level of theory is substantial, reducing every frequency by more than 10%, the final accuracy is quite high – better than the considerably more expensive MP2. Note also that the pure DFT functional BLYP requires essentially no scaling, i.e., its errors are random about the experimental values,
340 9 CHARGE DISTRIBUTION AND SPECTROSCOPIC PROPERTIES Table 9.3 Scale factors and post-scaling errors in vibrational frequencies from different levels of theory a Level of theory Scale factor RMS error (cm −1 ) Outliers (%) b AM1 0.9532 126 15 PM3 0.9761 159 17 HF/3-21G 0.9085 87 9 HF/6-31G(d) 0.8953 50 2 HF/6-31G(d,p) 0.8992 53 3 HF/6-311G(d) c 0.9361 32 HF/6-311G(d,p) 0.9051 54 3 HF/LANL2DZ c 0.9393 49 MP2/6-31G(d) 0.9434 63 4 MP2/6-31G(d,p) d 0.9646 MP2/pVTZ e 0.9649 70 QCISD/6-31G(d) 0.9537 37 2 BLYP/6-31G(d) 0.9945 45 2 BLYP/6-311G(d) c 1.0160 38 BLYP/LANL2DZ c 1.0371 47 BP86/6-31G(d) 0.9914 41 2 B3LYP/6-31G(d) 0.9614 34 1 0.9664 f 46 0.9800 g B3LYP/6-311G(d) c 0.9739 38 B3LYP/pVTZ d 0.9726 42 B3LYP/6-311+G(3df,2p) g 0.9890 B3LYP/6-311++G(3df,3pd) f 0.9542 31 B3LYP/LANL2DZ c 0.9978 45 B3PW91/6-31G(d) 0.9573 34 2 B3PW91/pVTZ d 0.9674 43 VSXC/6-31G(d) 0.9659 48 VSXC/6-311++G(3df,3pd) f 0.9652 37 a Data from Scott and Radom (1996) unless otherwise indicated. b Number of frequencies still in error by more than 20% of the experimental value after scaling. c From analysis of 511 frequencies in 50 inorganic molecules (Bytheway and Wong 1998). d Pople et al. (1993). e From analysis of 900 frequencies for 111 molecules comprised of first- and second-row atoms and hydrogen (Halls, Velkovski, and Schlegel 2001). f From analysis of 110 frequencies for 31 small molecules having only first-row atoms and hydrogen (Jaramillo and Scuseria 1999). g Bauschlicher and Partridge (1995). while the hybrid functionals require scale factors consistent with their inclusion of some HF character. Thus, including HF character results in proportionately too high predictions in vibrational frequencies, although the scaling procedure is very effective here as well. Finally, the errors in the semiempirical levels are quite high, and scaling is only modestly helpful. For those looking for the highest accuracy, the U.S. National Institute of Standards and Technology (NIST) maintains a web facility that permits users to select a focused set of molecules from NIST’s computational chemistry database (presumably based upon the user’s interest in a structurally related unknown) and then to compute least-squares best scaling
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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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Page 324 and 325: 9 Charge Distribution and Spectrosc
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390 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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