3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fund of Mat Sci: Bonding – Lecture 11BONDING IN MOLECULESThe future of electronics ? A pentacene molecule (left) deposited on SiO2as a thin film (right)Image removed for copyright reasons.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Homework for Wed Oct 19• Study: 24.2, 24.4-6• Read math supplement of Engel-Reid (A.7 and A.8, working with determinants and working with matrices)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Last time: 1. Stability determined by the interplay of n-n, e-e-, e-n interactions and the quantum kinetic energy2. Many-electron wavefunction as product of single-particle orbitals (each one LCAO)3. Many-atom Hamiltonian4. sp, sp2 and sp3 hybridizations – bond lengths and bond energies3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Complexity of the many-body Ψ1( ,..., )nrrψψ=rr3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Mean-field approach• Independent particle model (Hartree): each electron moves in an effective potential, representing the attraction of the nuclei and the average effect of the repulsive interactions of the other electrons• This average repulsion is the electrostatic repulsion of the average charge density of all other electrons3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Hartree Equations)()()(),...,(22111 nnnrrrrrrLrrrrϕϕϕψ=•The Hartree equations can be obtained directly from the variational principle, once the search is restricted to the many-body wavefunctions that are written – as above – as the product of single spin-orbitals (i.e. we are working with independent electrons)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Hartree Equations2211( ) ( ) () ()2||iIijj jiiiiIjijiVR r r dr r rrrϕϕεϕ≠⎡⎤−∇+ − + =⎢⎥−⎢⎥⎣⎦∑∑∫rrrrrrrr3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)The self-consistent field• The single-particle Hartree operator is self-consistent ! I.e., it depends in itself on the orbitals that are the solution of all other Hartree equations• We have n simultaneous integro-differential equations for the n orbitals• Solution is achieved iteratively3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Differential AnalyzerVannevar Bush and the Differential Analyzer.Courtesy of the MIT Museum. Used with permission.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Spin-Statistics• All elementary particles are either fermions(half-integer spins) or bosons (integer)• A set of identical (indistinguishable) fermions has a wavefunction that is antisymmetric by exchange• For bosons it is symmetric),...,,...,,...,,(),...,,...,,...,,(2121 njknkjrrrrrrrrrrrrrrrrrrrψψ−=3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Slater determinant• An antisymmetric wavefunction is constructed via a Slater determinant of the individual orbitals (instead of just a product, as in the Hartree approach))()()()()()()()()(!1),...,,(22211121nnnnrrrrrrrrrnrrrrLrrMOMMrLrrrLrrrrrνβανβανβαϕϕϕϕϕϕϕϕϕψ=3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Pauli principle• If two states are identical, the determinant vanishes (i.e. we can’t have two electrons in the same quantum state)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Hartree-Fock Equations22*1()()21() ()||1() () () ()||iIiiIjjijijijj ijiVR r rrdrrrrrrrdrrrrλµλµµµλλµϕϕϕϕϕϕεϕ⎡⎤−∇+ − +⎢⎥⎣⎦⎡⎤−⎢⎥−⎢⎥⎣⎦⎡⎤=⎢⎥−⎢⎥⎣⎦∑∑∫∑∫rrrrrrrrrrrr rrr•The Hartree-Fock equations are, again, obtained from the variational principle: we look for the minimum of the many-electron Schrödinger equation in the class of all wavefunctions that are written as a single Slater determinantSlaterrrn=),...,(1rrψ3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Example: two electrons in H2()()1112 1 2 2 122,11 21() ()11(,) ()() ()()() ()22( ) full solution of (integro-differentail) Hartree-Fock equations, or() ( ) ( )()sA sBrrrr r r r rrrrrc rRc rR spinuprαβαβ α βαβαβαβϕϕψϕϕ ϕϕϕϕϕϕϕ⎡⎤==−⎣⎦==Ψ − +Ψ − × −rrrr r r r rrrrrrrr rr()()11 21() ()sA sBc r R c r R spin down=Ψ − +Ψ − × −rrrr3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)H2 and He2H21s 1s1σ*u1σgHe21s1s1σ*u1σgFigure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Symmetries• Rotation along molecular axis →σ• Nodal plane ┴ molecular axis →π• Parity upon inversion:()()() ( )rr r rrr→− ⇒Ψ =Ψ −Ψ=−Ψ −rr r rrr3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Formation of a π Bonding OrbitalSee animation at http://winter.group.shef.ac.uk/orbitron/MOs/N2/2px2px-pi/index.html3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)SymmetriesContour plots of several bonding and antibonding orbitals of H2+. Images removed for copyright reasons. See p. 528, figure 24.4 in Engel, T., and P. Reid. Physical Chemistry. Single volume ed. San Francisco, CA: Benjamin Cummings, 2005.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Homonuclear Diatomic Levels (I)Diagram of Orbital Regions for 2p atomic orbitals and LCAO molecular orbitals made from them removed for copyright reasons. See p. 667, figure 18.11 in Mortimer, R. G. Physical Chemistry. 2nd ed. San Diego, CA: Elsevier, 2000.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Homonuclear Diatomic Levels (II)2pxA2sA2pyA 2pzA πu2pxπ∗σ∗σ∗σg2sσg1s2sB1sA1sAπ∗σg2pzπu2py2pzB 2pxB 2pyBAtomic OrbitalsMolecular OrbitalsAtomic OrbitalsOrbital Energyg2pxu2pzg2pyu2sσ∗u1sFigure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Fluorine dimer F23σ*u2σ*u2σ*u3σg3σg2σg2σg2p 2p2s 2s1π*g1π*g1πu1πu+++++++++++++___________Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall
View Full Document
Unlocking...