3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fund of Mat Sci: Bonding – Lecture 10MOLECULES FROM ATOMSImage of a journal article removed for copyright reasons.See Slater, J. C., and G. F. Koster. "Simplified LCAO Method for the Periodic Potential Problem." Physical Review 94, no. 6 (1954).3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Homework for Mon Oct 17• Study: 25.2, 25.4, 24.4, 24.63.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Last time: 1. Variational principle 2. Application to the hydrogen atom 3. LCAO – optimize the coefficients4. Hydrogen molecular ion3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Energy of a MoleculeeeNeeNNETV V V−−−=+ + +3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Linear Combination of Atomic Orbitals• Trial wavefunction is a linear combination of atomic orbitals – the variationalparameters are the coefficients:()()11 21trial s A s BcrRcrRΨ=Ψ − +Ψ −rrrr3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Formation of a Bonding OrbitalSee animation at http://winter.group.shef.ac.uk/orbitron/MOs/H2/1s1s-sigma/index.html3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Formation of an Antibonding OrbitalSee animation at http://winter.group.shef.ac.uk/orbitron/MOs/H2/1s1s-sigma-star/index.html3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Bonding and Antibonding (II) 6420-2-4-681 2 3 4σ*1s energyuσg1s energyBorn-Oppenheimer energyof exact orbitalsEBO/eVR/10-10muOrbital Energies for the σg 1s and σ* 1s LCAO Molecular Orbitals.Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Many-Atoms Hamiltonian211ˆˆ ˆ2||||IeieN eeiiI ijiijiIZTV VrrrR−−>=− ∇ = − =−−∑∑∑ ∑∑rrrrˆˆˆ ˆeeNeeNNHTV V V−−−=+ + +3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Molecular Orbitals From Atomic Orbitals1 1, 2 1, 3 1, 4 1,52, 62, 72, 82,abcdxyztrial s H s H s H s HsCpCpCpCcccccc c cΨ=Ψ +Ψ +Ψ +Ψ+Ψ +Ψ +Ψ +Ψ3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)sp3hybridizationsp3Hypbridization. Source: Wikipedia.Diagram of s and p orbitals combining to create sp3 hybridization removed for copyright reasons. See p. 440, figure 12-5 in Petrucci, R. H., W. S. Harwood, and F. G. Herring. General Chemistry: Principles and Modern Applications. 8th ed. Upper Saddle River, NJ: Prentice Hall, 2002.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)sp3hybridizationImages of sp3 hybridization in CH4 and NH3 removed for copyright reasons. See pp. 441-442, figures 12-6 and 12-7 in Petrucci, R. H., W. S. Harwood, and F. G. Herring. General Chemistry: Principles and Modern Applications. 8th ed. Upper Saddle River, NJ: Prentice Hall, 2002.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Great gases and liquids……3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)sp2hybridizationsp2Hypbridization. Source: Wikipedia.Diagram of s and p orbitals combining to create sp2hybridization removed for copyright reasons. See p. 442, figure 12-8 in Petrucci, R. H., W. S. Harwood, and F. G. Herring. General Chemistry: Principles and Modern Applications. 8th ed. Upper Saddle River, NJ: Prentice Hall, 2002.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Carbon CompoundsSource: Wikipedia3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)sp hybridizationDiagram of s and p orbitals combining to create sp hybridization removed for copyright reasons.See p. 443, figure 12-9 in Petrucci, R. H., W. S. Harwood, and F. G. Herring. General Chemistry: Principles and Modern Applications. 8th ed. Upper Saddle River, NJ: Prentice Hall, 2002.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Ethyne (Acetylene)Source: WikipediaDiagram of the formation of σ and π bonds in ethyneremoved for copyright reasons.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Formation of a σ Bonding OrbitalSee animation at http://winter.group.shef.ac.uk/orbitron/MOs/N2/2pz2pz-sigma/index.html3.012 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)Ethene (Ethylene)Source: WikipediaDiagram of the formation of σ and π bonds in etheneremoved for copyright reasons.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Ethane (saturated)Source: WikipediaDiagram of ethane removed for copyright reasons. See Goodman et al., Nature (2001).3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Bond Lengths and Bond EnergiesTables of Average Bond Lengths and Average Bond Energies removed for copyright reasons.See p. 420, table 11.2, and p. 422, table 11.3, in Petrucci, R. H., W. S. Harwood, and F. G. Herring. General Chemistry: Principlesand Modern Applications. 8th ed. Upper Saddle River, NJ: Prentice Hall,
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