實驗九. 預測化學反應途徑與反應速率 組員名單: 段振斌 49712048 目的、原理(一) 李厚寬 49712025 原理(二) 、(三) 簡 薇 49712016 原理(四) 、步驟 實驗目的 1‧以電腦為工具探討化學反應。 2‧解分子的 electronic Schrödinger equation 藉以預測:分子結構、反應途徑、過渡態、中 間產物、產物 3‧熟悉 分子記算程式:Gaussian 03、Gauss view 分子繪圖程式:Chem Draw 4‧計算反應速率常數,並得出產量。 實驗原理 一. The Born-Oppenheimer appoximation 二. The Hartree-Fock equation 三. 6-31G basis set 四.計算反應速率常數 (Eyring equation) B ORN -O PPENHEIMER A PPROXIMATION Molecular Hamiltonian: First term: The kinetic energy of the nuclei Second term: The kinetic energy of the electrons Third term: The potential energy of the repulsions between the nuclei Fourth term: The potential energy of the attractions between the electrons and nuclei Fifth term: The potential energy of the repulsions between the electrons 資料來源:Levine Quantum Theory 第五版 13.1節 A S AN EXAMPLE , CONSIDER H2 ˆ 2 - 2 2 2m p - 2 2 2m p 2me 2 1 2 2me 2 2 e e e e e e - - - r r1 r1 r2 r2 r12 2 2 2 2 2 2 α、 β為兩氫原子之原子核,下標1、2 為電子1和電子2,mp為質子之質量 資料來源:Levine Quantum Theory 第五版 13.1節 利用S CHRÖDINGER 方程式找能量 :電子 :原子核 Purely electronic Hamiltonian: 資料來源:Levine Quantum Theory 第五版 13.1節 : The nuclear repulsion Purely electronic energy: ˆ E el el el el 資料來源:Levine Quantum Theory 第五版 13.1節 Re:位能達最低點時之平衡距離 De:達平衡時之解離能 D0:分子的振動能階之最低能量(零點能量) ν=0:zero point energy=1/2hν Purely nuclear Hamiltonian: ˆ E N N N 將電子的運動與原子核的運動用數學方法來進行分離 的過程及稱為Born-Oppenheimer approximation 因此我們將wave function寫成: qi ,q el qi ;q N q 但B.O approxmation在計算 時,並不考慮核之間的作用力 資料來源:Levine Quantum Theory 第五版 13.1節 H ARTREE - FOCK APPROXIMATION ◎ Consider a simpler N-el system: (neglect the el-el repulsion) N H h(i ) i 1 Recall: → where h(i) is the oper- ator describing the K.E and P.E of electron i. Now we can write: ◎ Hartree product: because H is the sum of one-el Hamiltonians , a wave function which is a simple product of the wave functions for each electron . ψHP(x1,x2,…,xN) = χi(x1) χj(x2)˙˙˙χk(xN) An eigenfunction of H : Such a many-electron wave function is termed a “Hartree product”. NOTE: The Hartree product does not satisfy the “antisymmetry principle” . ◎If we put electron-one in χi and electron-two in χj ,we have: ψHP12(x1,x2)= χi(x1) χj(x2)---------------(1) ψHP21(x1,x2)= χi(x2) χj(x1)---------------(2) We can obtain a wave function to satisfy the antisymmetry principle by taking the appropriate linear combination of these two HP. It can be rewritten as a determinant and is called a“slater determinant”. i( x1) j ( x1) i( x2) j ( x2) 1/ 2 ( x1, x2) 2 -1/2 ψ(x1, x2,...,xN) = (N!) i ( x1) j ( x1) k ( x1) i ( x 2) j ( x 2 ) k ( x 2 ) i ( xN ) j ( xN ) k ( xN ) Clearly, ψ(x1,x2) = -ψ(x2,x1) (fermion) ◎Hartree-Fock equation where f(i) is an effec- tive one-el operator called the Fock operator 1 2 M ZA f (i) i HF (i) 2 A1 riA Where νHF is the average potential experienced by the i-th electron due to the presence of the other electrons. The potential energy of interaction between Q1Q 2 Point charges Q1 and Q2 is V 12 4 0 r12 Q1 2 V 12 dv 2 4 0 r 12 2 e s 2 Q1 e 2 V 12 e'2 s2 2 r 12 2 e e' 2 4 0 dv2 Adding in the interactions w/ the other el’ we have: n V 12 V 13 ... V 1n e' j 2 2 sj r 2 1j dvj The Born-Oppenheimer approximation is inherently assumed. Relativistic effects are completely neglected. The variational solution is assumed to be a linear combination of a finite number of basis functions. Each energy eigenfunction is assumed to be describable by a single Slater determinant. The mean field approximation is implied. 6-31G ◎basis BASIS SET set: a mathematical description of orb- itals of a system, which is used for approximate theoretical calculation or modeling. It is a set of basic functional building blocks can be stacked or added to have the features we need. a11 a2 2 ... ann curgu u χ稱為收斂高斯函數(contracted Gaussian) g為初始高斯函數( primitive Gaussian) → 例如STO- 3G: 就是以3個初始高斯函數( primitive Gaussian)來組 合成一個基底函數組(basis set) ◎6-31G →內殼層(inner shell):每個原子軌域(AO)以一個 基底函數來表示,此基底函數是由6個初始函數線性組 合而成。 →價殼層(valence shell):由兩個基底函數組合而 成每一的基底函數則分別是由3及1個初始函數所構成。 “*” 號代表極化函數(polarized functions),第一個*號表 示重原子中加入更高階的角動量函數,以苯為例,重原子為碳 原子,因此加入六個d形式的基底函數。第二個*號代表再每一 個氫原子中加入三個p形式的基底函數。“+” 號表加入擴散 函數(diffuse function),因此每一個碳原子中需加入三個p 及一個s形式的基底函數;另外,第二個+號表示每一個H原子 中加入一個s形式函數。 Eyring equation k P A B C d [ P] V k [C ] dt C :activated com plex or transition state 1. C is in pre equilibrium with A、B if gases, P C / P P C P RT[C ] P [C ] P ( PA / P )(PB / P ) PAPB RT[ A]RT[ B] RT [ A][B ] nA (ideal gas: P V nRT PA RT [A]RT ) V RT [C ] [ A][B ] P d[P ] RT k [C ] φ k Κ [A][B] k2[A][B] dt P RT k2 φ k Κ t o find k2: get (1) k (2) Κ P 2. V (1) t o get k : k κ κ : t ransmission coefficient (2) t o get Κ : (equilibrium const ant for A B C ) φ q J, m νJ ΔrE0 /RT recall, K ( ) e J NA NA q φc ΔrE /RT 0 Κ φ φ e qA qB where ΔrE0 E 0 (C ) - E 0 (A) - E 0 (B) φ q J st andardmolarpart it ionfunct ion vibrat ional mode q 1 1 e h / kT h 1 kT 利用泰勒展開式 x2 h e 1 x , nowthat x 2! kT 1 1 kT q h h 1 (1 ) 1 (1 ) h kT kT x qC qC ,T qC , qC ,R qC , E kT ( qC ,T qC , R qC , E qC , ) h kT 代入 qC h kT qC N q kT A C rE 0 / RT kT rE 0 / RT h e e h q A q B h q A qB NA RT RT kT k h P P kT RT h P kT C 為Eyring equation h 已知k 2 以上資料來源 : Atkins CH24.4 實驗步驟 點選gview.exe