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

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28 CHAPTER 3. MOLECULES Toy Model to which atom an electron belongs is by the AOs. Furthermore, as Löwdin’s method was used for the orthogonalization, it is straightforward to use Löwdin partitioning for population analysis. This is equivalent to performing the partitioning in the orthogonalized basis χ ′ = S 1/2 χ, where S ′ jk = δ jk . Summing over all MO, as in eq. 3.15, the total number of electrons N is written as: N = ∑ i = ∑ i = ∑ i n i n i ∑ j,k∈AO n i ∑ j∈AO C ′ ji C′ ki δ jk C ′2 ji . Using the sum over all j ∈ AO, N can be partitioned such that for an atom A and AOs i on A (i@a) ρ A = ∑ i n i ∑ j@A C ′2 ji . The charge on A is then the sum between the nuclear charge and the electronic population: Q A = Z A − ρ A . This population analysis combines well with the choice of a Slater-type orbital basis set. The orbitals describe the wave functions close to the atom they are centered on, thus their LCAO coefficient can describe the electron population on that atom. As an example, the overlap matrix constructed for propenamide is shown (tab. 3.6). Only the SMILES C = CC(= O)N was needed for its generation and the subsequent calculation of energy levels (fig. 3.8) and charge distribution (fig. 3.7). −0.03 −0.07 0.60 O −1.02 −0.10 NH 2 Figure 3.7: Charge distribution in propenamide (in electron charges).
Table 3.6: Overlap matrix for propenamide (fig. 3.5). First and second rows are the atom and the the type of the AO. The indices next to the atom names are given for better orientation and correspond to the indices in the propenamide SMILES C 1 (H 2 )(H 3 ) = C 4 (H 5 )C 6 (= O 7 )N 8 (H 9 )H 10 . C 1 H 2 C 1 H 3 C 1 C 4 C 4 H 5 C 4 C 6 C 6 O 7 C 6 N 8 N 8 H 9 N 8 H 10 O 7 O 7 C 1 C 4 C 6 O 7 N 8 sp 2 s sp 2 s sp 2 sp 2 sp 2 s sp 2 sp 2 sp 2 sp 2 sp 2 sp 2 sp 2 s sp 2 s sp 2 sp 2 p p p p p 1 .65 0 0 0 .077 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .65 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 .65 0 .077 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .65 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 .77.077 0 .077 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .077 0 .077 0 .77 1 0 0 0 .077 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .077 0 1 .65 0 .077 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .65 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .077 0 0 0 1 .77.077 0 .077 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .077.077 0 .77 1 0 .068 0 .073 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .077 0 1 .68 0 .073 0 0 0 0 .068.068 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .068.68 1 .068 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .077 0 0 .068 1 .73.073 0 .073 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .073.073 0 .73 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .073 0 1 .63 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .63 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .073 0 0 0 1 .63 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .63 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .068 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .068 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 .38 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .38 1 .38 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .38 1 .26.31 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .26 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .31 0 1 3.4. WAVE FUNCTION ANALYSIS 29
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 25 and 26: 3.1. EXTENDED HÜCKEL THEORY 15 is
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: 3.4. WAVE FUNCTION ANALYSIS 27 Derw
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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