Chemistry Week 8

Chemistry Week 8 - Chemical Bonding in Molecules and Solids...

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Chemical Bonding in Molecules and Solids TEXTBOOK READING : BLB-10 , Chapters 8 and 9, pp. 300-397. The basis for chemical bonding models in molecules and solids also relies on solving the Schrödinger wave equation. For such structures, this is an extremely complicated problem, and cannot be solved exactly because of the many particles making up a molecule, i.e., protons, neutrons (in the nuclei) and electrons. To see how complicated the problem can be, even for a very simple molecule, let’s look at the complete Schrödinger wave equation for the H 2 molecule. In this equation, we have H atom #1 (proton 1 + electron 1) and H atom #2 (proton 2 + electron 2); the masses of the protons are M , the coordinates are R i = ( X i , Y i , Z i ); the masses of the electrons are m , the coordinates are r i = ( x i , y i , z i ). 22 2 2 2 2 2 2 2 2 2 2 11 1 2 222 2 2 2 2 2 2 2 111222 2 21 1122 2 12 21 1 2 8 8 hdddddd M d Xd Yd Zd Z m d xd yd zd z ee e e π ⎡⎤ ⎛⎞ −+ + + + + ⎢⎥ ⎜⎟ ⎝⎠ + + + + +−+ −− −++ ⎣⎦ RR r R rR rR rR rr () ( ) 1 212 12 1 ,; , , QQ Q E Ψ= Ψ rrRR The expression in [ ]’s is the kinetic energy + potential energy operator and includes terms for the nuclei and the electrons. In order, they are: (i) kinetic energy of the nuclei; (ii) kinetic energy of the electrons; (iii) repulsion between nuclei p 1 and p 2 ; (iv) attractions between electron and proton within each atom ( e 1 and p 1 ; e 2 and p 2 ); (v) attractions between electron and proton between atoms ( e 1 and p 2 ; e 2 and p 1 ); and (vi) repulsion between electrons e 1 and e 2 . The solutions to this equation are the wavefunctions, which are given with respect to the coordinates of the electrons and protons, and the energies, which depend solely on the positions of the nuclei (protons). There are quantum numbers { Q }, but these are no longer { n , l , m }. To simplify this equation, we use the fact that the protons are 2000 × more massive than the electrons, so that the kinetic energies of the protons (speeds) are much smaller than those of the electrons, and are considered to be zero, i.e., stationary nuclei at R 1 and R 2 . This is called the Born-Oppenheimer approximation . The new equation to solve now involves just the coordinates of the electrons, while the coordinates of the nuclei are fixed. ( ) 2222 2 2 2 2 2 2 2 1 2 11 2 2 1 2 8 ,, Q m d z E e + + + + Ψ + + R R
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This equation is typically solved for several different settings of the nuclear coordinates, R 1 and R 2 . In this way, the relationship between the energy and the structure (distance between the nuclei) can be assessed. As with the Schrödinger wave equation for atoms, we will be interested only in the solutions to this wave equation for molecules. Methods to solve this equation involve computation. (1) Molecular Orbitals TEXTBOOK READING : BLB-10 , Chapters 8.2-4; 9.4, 6-8, pp. 303-317; 360-361, 367-386. Practice PROBLEMS: (Ch 9) 59, 60, 62, 66, 68, 70 Just like atoms, the electronic structures of molecules and solids are described by orbitals , but we can no longer use the same quantum numbers, because the geometrical structure of molecules and solids is not the same as atoms.
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This note was uploaded on 03/27/2008 for the course CHEM 201 taught by Professor Miller during the Fall '07 term at Iowa State.

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Chemistry Week 8 - Chemical Bonding in Molecules and Solids...

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