A Toy Model of Chemical Reaction Networks - TBI - UniversitÃ¤t Wien

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14 CHAPTER 3. MOLECULES Schrödinger equation ĤΨ = EΨ ⇑ ⇓ Born-Oppenheimer LCAO and Extended Hückel VSEPR and Tight Binding Generalized Eigenvalue Problem ⎛ ⎞ ⎛ ⎞ . .. . .. ⎜ ⎟ ⎜ ⎟ ⎝ H AOi −AO j ⎠C = ⎝ S AOi −AO j ⎠CE . .. . .. Figure 3.1: Schematic derivation of the used electronic calculation method. H AOi −AO i , S AOi −AO j and K are given parameters (see equations 3.13). the Schrödinger equation is separated into a part describing the electronic wave function Ψ el for fixed nuclei, Ĥ el Ψ el = E el Ψ el , (3.3) and a part describing the nuclear wave function. The electronic Hamilton operator is: ( ) ∑e − Ĥ el = − 1 2 ˆ∇ ∑nuc 2 Z I ∑e − ∑e − 1 ∑nuc ∑nuc Z I Z J i − + , (3.4) r i I iI r i j>i ij r } {{ } }{{} I J>I IJ } {{ } ĝ ij V n ĥ i + where ˆ∇ is the nabla operator, Z i is the nuclear charge of the atom i and r ij
3.1. EXTENDED HÜCKEL THEORY 15 is the distance between the nuclei or electrons i and j. e − and nuc indicate sums running over all electrons and nuclei, respectively. V n , the electrostatic nuclear repulsion, does not depend on electron coordinates and is thus an additive constant to the final energy. For more clarity, it is omitted from the following equations. Hartree proposed the orbital approximation, in which the wave function is built from Slater determinants, ∣ ∣∣∣∣∣∣∣∣ φ 1 (1) φ 2 (1) · · · φ N (1) Ψ SD = √ 1 φ 1 (2) φ 2 (2) · · · φ N (2) . N! . . .. . (3.5) . φ 1 (N) φ 2 (N) · · · φ N (N) ∣ The φ are one-electron wave functions, called molecular orbitals (MO) or spin orbitals. The energy of a single Slater determinant will be used lateron for the variational principle. It may be written as ∫ E el = Ψ SD Ĥ el Ψ SD d τ (3.6) and decomposed into one-electron or core integrals ∫ φ i (i)ĥiφ i (i) dτ, twoelectron Coulomb integrals ∫ φ i (i)φ j (j)ĝ ij φ i (i)φ j (j) dτ, and two-electron exchange integrals ∫ φ i (i)φ j (j)ĝ ij φ j (i)φ i (j) dτ. Semi-empirical methods, Extended Hückel Theory [62, 65] for instance, now further approximate the wave function by building it from one single Slater determinant, thus neglecting electron correlation. First, using the basis set or LCAO approximation, each φ is expanded as a linear combination of atomic orbitals χ (LCAO) : φ i = ∑ j C ij χ j . (3.7) The definition of the electron density ρ in LCAO will be needed in sect. 3.4 and will be shortly presented here. The electron density or the probability of finding an electron in a MO φ i , occupied by n i electrons, at the position defined by the position vector r is ρ i (r) = n i φ 2 i (r) . (3.8)
Page 1 and 2: A Toy Model of Chemical Reaction Ne
Page 3: i Dank an alle, die zum Gelingen di
Page 7: v Zusammenfassung Chemische Reaktio
Page 10 and 11: viii CONTENTS B GRW 61 C Organic re
Page 12 and 13: 2 CHAPTER 1. INTRODUCTION ▽ △
Page 14 and 15: 4 CHAPTER 1. INTRODUCTION ¡ ¡ ¡
Page 16 and 17: 6 CHAPTER 1. INTRODUCTION and symme
Page 18 and 19: 8 CHAPTER 2. ARTIFICIAL CHEMISTRIES
Page 20 and 21: 10 CHAPTER 2. ARTIFICIAL CHEMISTRIE
Page 22: 12 CHAPTER 2. ARTIFICIAL CHEMISTRIE
Page 27 and 28: 3.2. FURTHER APPROXIMATIONS 17 K. F
Page 29 and 30: 3.2. FURTHER APPROXIMATIONS 19 Figu
Page 31 and 32: 3.2. FURTHER APPROXIMATIONS 21 are
Page 33 and 34: 3.3. CHEMICAL STRUCTURE REPRESENTAT
Page 35 and 36: 3.3. CHEMICAL STRUCTURE REPRESENTAT
Page 37 and 38: 3.4. WAVE FUNCTION ANALYSIS 27 Derw
Page 39 and 40: Table 3.6: Overlap matrix for prope
Page 41 and 42: 3.5. PERFORMANCE 31 3.5 Performance
Page 43 and 44: 3.6. FRONTIER MOLECULAR ORBITAL THE
Page 45 and 46: Chapter 4 Chemical reactions A mole
Page 47 and 48: 4.1. GRAPH REWRITING 37 Graphical t
Page 49 and 50: 4.1. GRAPH REWRITING 39 C C C C C C
Page 51 and 52: 4.2. GRAPH REWRITE ENGINE 41 O O O
Page 53 and 54: Chapter 5 Reaction networks 5.1 Net
Page 55 and 56: 5.3. NETWORK PROPERTIES 45 O 3 NO 3
Page 57 and 58: 5.3. NETWORK PROPERTIES 47 (see ch.
Page 59 and 60: Chapter 6 Computational results Whe
Page 61 and 62: O O O O O O O O O O O O O O O O O O
Page 63 and 64: 6.3. GRAPH-THEORETIC PROPERTIES 53
Page 65 and 66: Chapter 7 Conclusion and Outlook Th
Page 67 and 68: Using the rules of mass action kine
Page 69 and 70: Appendix A Parameters The energy ca
Page 71 and 72: Appendix B GRW GRW is implemented i
Page 73 and 74:
Appendix C Organic reactions Fromht
Page 75 and 76:
Bibliography [1] R. Albert and A.-L
Page 77 and 78:
BIBLIOGRAPHY 67 [25] O. Diels and K
Page 79 and 80:
BIBLIOGRAPHY 69 [50] J. Gasteiger a
Page 81 and 82:
BIBLIOGRAPHY 71 [76] W. Langenbeck.
Page 83 and 84:
BIBLIOGRAPHY 73 [102] S. Schuster,
Page 85:
BIBLIOGRAPHY 75 [126] R. Woodward a
Page 89 and 90:
The Wooing of Archibald Modern tren
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A Toy Model of Chemical Reaction Networks - TBI - UniversitÃ¤t Wien

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