3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fund of Mat Sci: Bonding – Lecture 9VARIATIONS3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Homework for Fri Oct 14• Study: 21.4, 23.3 • Read: 23.4, 24.1, 24.23.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Last time: 1. Screening, and coupled, self-consistent Hartree equations for many-electron atoms2. 4thquantum number: spin3. Filling (auf-bau) of the periodic table4. Physical trends on sizes, IP, EA. (e.g., why He is smaller than H)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Variational Principle[]ˆHEΨΨΨ=ΨΨ3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Variational Principle[]ˆHEΨΨΨ=ΨΨIf , then Φ is the ground state wavefunction, and viceversa…[]0EEΨ≥[]0EEΨ=3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Good Quantum NumbersIf A commutes with the Hamiltonian, its expectation value does not change with time (it’s a constant of motion – if we are in an eigenstate, that quantum number will remain constant)ˆˆ1ˆˆ,dAdAAHdt dt ihΨΨ⎡⎤==⎣⎦3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Atomic Units•me=1, e=1, a0(Bohr radius)=1,1=h022141Energy of 1s electron=2(1 atomic unit of energy=1 Hartree=2 Rydberg=27.21 eVZnεπ=−3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Energy of an Hydrogen AtomˆHEΨΨ=ΨΨ3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Energy of an Hydrogen AtomˆHEαααααΨΨ=ΨΨ()expCrααΨ= −22223 211,22CCCrαα α α α απππαααΨΨ= Ψ−∇Ψ= Ψ−Ψ=−3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Hydrogen Molecular Ion H2+21111() ()2AB A BrErR R rR rRψψ⎡⎤⎛⎞⎢⎥⎜⎟−∇+ − − =⎜⎟⎢⎥−−−⎝⎠⎣⎦rrrrrr3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Linear Combination of Atomic Orbitals• Most common approach to find out the ground-state solution – it allows a meaningful definition of “hybridization”, “bonding” and “anti-bonding”orbitals.• Also knows as LCAO, LCAO-MO (for molecular orbitals), or tight-binding (for solids)• Trial wavefunction is a linear combination of atomic orbitals – the variational parameters are the coefficients:()()11 21trial s A s BcrRcrRΨ=Ψ − +Ψ −rrrr3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Linear Combination of Atomic Orbitals()()11 21trial s A s BcrRcrRΨ=Ψ − +Ψ −rrrr3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Bonding and Antibonding (I) Image of the orbital region for LCAO molecular orbitals removed for copyright reasons. See Mortimer, R. G. Physical Chemistry. 2nd ed. San Diego, CA: Elsevier, 2000, p. 657, figure 18.7.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Hydrogen Molecular Ion H2+• Born-Oppenheimer approximation: the electron is always in the ground state corresponding to the instantaneous ionic
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