3.012 Fund of Mat Sci: Bonding Lecture 11. The future of electronics? A pentacene molecule deposited on SiO 2. Image removed for copyright reasons.

Transcription

1 3.012 Fund of Mat Sci: Bonding Lectue 11 BONDING IN MOLECULES The futue of electonics? A pentacene molecule deposited on SiO 2 as a thin film Image emoved fo copyight easons Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

2 Homewok fo Wed Oct 19 Study: 24.2, Read math supplement of Engel-Reid A.7 and A.8, woking with deteminants and woking with matices Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

3 Last time: 1. Stability detemined by the inteplay of n-n, e- e-, e-n inteactions and the quantum kinetic enegy 2. Many-electon wavefunction as poduct of single-paticle obitals each one LCAO 3. Many-atom Hamiltonian 4. sp, sp 2 and sp 3 hybidizations bond lengths and bond enegies Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

4 Complexity of the many-body Ψ ψ = ψ,..., 1 n Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

5 Mean-field appoach Independent paticle model Hatee: each electon moves in an effective potential, epesenting the attaction of the nuclei and the aveage effect of the epulsive inteactions of the othe electons This aveage epulsion is the electostatic epulsion of the aveage chage density of all othe electons Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

6 Hatee Equations The Hatee equations can be obtained diectly fom the vaiational pinciple, once the seach is esticted to the many-body wavefunctions that ae witten as above as the poduct of single spin-obitals i.e. we ae woking with independent electons ψ,..., n = ϕ1 1 ϕ 2 2 Lϕ 1 n n Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

7 Hatee Equations i + V RI i + ϕj j dj ϕi i εϕi i I = 2 j i j i Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

8 The self-consistent field The single-paticle Hatee opeato is selfconsistent! I.e., it depends in itself on the obitals that ae the solution of all othe Hatee equations We have n simultaneous intego-diffeential equations fo the n obitals Solution is achieved iteatively Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

9 Diffeential Analyze Vanneva Bush and the Diffeential Analyze. Coutesy of the MIT Museum. Used with pemission Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

10 Spin-Statistics All elementay paticles ae eithe femions half-intege spins o bosons intege A set of identical indistinguishable femions has a wavefunction that is antisymmetic by exchange,,...,,...,,..., =,,...,,...,,..., ψ ψ 1 2 j k n 1 2 k j n Fo bosons it is symmetic Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

11 3.012 Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005 Slate deteminant An antisymmetic wavefunction is constucted via a Slate deteminant of the individual obitals instead of just a poduct, as in the Hatee appoach! 1,...,, n n n n n L M O M M L L ν β α ν β α ν β α ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ψ =

12 Pauli pinciple If two states ae identical, the deteminant vanishes i.e. we can t have two electons in the same quantum state Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

13 Hatee-Fock Equations The Hatee-Fock equations ae, again, obtained fom the vaiational pinciple: we look fo the minimum of the many-electon Schödinge equation in the class of all wavefunctions that ae witten as a single Slate deteminant 1 2 i + V RI i ϕλ i + 2 I µ µ 2 1 ϕµ j dj ϕλ i j i * 1 ϕµ j ϕµ i ϕλ j dj = εϕλ i j i ψ 1,..., n = Slate Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

14 Example: two electons in H 2 ϕ ϕ ψ, ϕ ϕ ϕ ϕ 1 α 1 β = = α 1 β 2 α 2 β 1 2 ϕα 2 ϕβ 2 2 ϕαβ, = full solution of intego-diffeential Hatee-Fock equations, o ϕ α = c1ψ1s RA + c2ψ1s RB spin up ϕ = c Ψ R + c Ψ R spin down β 1 1s A 2 1s B Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

15 H 2 and He 2 1σ* u 1σ* u 1s 1s 1s 1s 1σ g 1σ g H 2 He 2 Figue by MIT OCW Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

16 Symmeties Rotation along molecula axis σ Nodal plane containing molecula axis π Paity upon invesion: Ψ =Ψ Ψ = Ψ Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

17 Fomation of a π Bonding Obital See animation at Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

18 Symmeties Contou plots of seveal bonding and antibonding obitals of H 2 +. Images emoved fo copyight easons. See p. 528, figue 24.4 in Engel, T., and P. Reid. Physical Chemisty. Single volume ed. San Fancisco, CA: Benjamin Cummings, Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

19 Homonuclea Diatomic Levels I Diagam of Obital Regions fo 2p atomic obitals and LCAO molecula obitals made fom them emoved fo copyight easons. See p. 667, figue in Motime, R. G. Physical Chemisty. 2nd ed. San Diego, CA: Elsevie, Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

20 Homonuclea Diatomic Levels II π g 2p x σ u 2p z π g 2p y σ g 2pz 2p x A 2p y A 2p z A π u 2p x π u 2p y 2p z B 2p x B 2p y B σ u 2s Obital Enegy 2sA σ g 2s 2sB σ u 1s 1sA σ g 1s 1sA Atomic Obitals Molecula Obitals Atomic Obitals Figue by MIT OCW Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

21 + _ Fluoine dime F 2 3σ * u π * g 2p 1π * g 2p + + 1πu + + 1πu 3σg _ + 3σg 2σ * u 2s 2σg 2s + _ 2σ * u + + 2σg Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005 Figue by MIT OCW.

22 Homonuclea Diatomic Levels III Li 2 Be 2 B 2 C 2 N 2 O 2 F 2 σ σ Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005 Figue by MIT OCW.

23 Bond Ode Gaph of bond ode, bond enegy, bond length, and foce constant against numbe of electons. Removed fo copyight easons. See p. 535, figue in Engel, T., and P. Reid. Physical Chemisty. Single volume ed. San Fancisco, CA: Benjamin Cummings, Bond ode = ½ of [bonding electons - antibonding electons] Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005

24 Advanced Reading Mateial Futhe eadings fom less to moe advanced: Atkins Physical Chemisty Feeman & Co 2001 Thalle Visual Quantum Mechanics Telos 2000 Bansden & Joachain Quantum Mechanics 2nd ed Pentice Hall 2000 Bansden & Joachain Physics of Atoms and Molecules 2nd ed Pentice Hall Fundamentals of Mateials Science: Bonding - Nicola Mazai MIT, Fall 2005