1、 8.3 The Huckel Moleculor Orbital method (HMO)Butadiene , e.g.： 61s+4(1s22s22px12py12pz0)=26 AOHMO approximation : 4 pz.In his approach,The orbitals are treated separately from the orbitals, and the latter form a rigid framework that determine the shape of the molecule. HMO deal with conjugated mole

2、cules. Huckel approximation IHMO is suggested by Eric Hckel in 1931.Butadiene4 pz of C atoms The energy and coefficients satisfy the following equations:let The best molecular orbitals are those which minimise the total energy. This is achieved by imposing the condition:： Huckel approximation II： no

3、n-trivial solutions ： These values, called the non-trivial solutions to these equations, occur when:letThis determinant can be easily multiplied out to give:x4-3 x2+1=01=0.37171+0.60152+0.60153+0.371742=0.60151+0.371720.371730.601543=0.601510.371720.37173+0.601544=0.371710.60152+0.601530.37174 0, so

4、 E1 E2 E3 E4We obtain four values of E, which is reasonable since we expect to find four molecular orbitals.Delocalization energyTotal energy E=2E1+2E2 =2(+1.62)+ 2(+0.62) =4 +4.48Energy levelsOccupied orbitalUnfilled orbitalC = C C = CE= 4 +4E- E=0.48Frontier orbitalsThe highest occupied molecular

5、orbital, HOMOThe lowest unfilled molecular orbital, LUMOThe frontier orbitals are important because they are largely responsible for many of the chemical and spectroscopic properties of the molecule. Give each of the following terms a brief explanation:1. LCAO2. Eigenvalue3. Born-Oppenheimer approxi

6、mation4. LUMO5. HOMO6. Fermion7. DegeneracyAnswer the following questions: 1. What are the assumptions of ab initio MO theory ?2. What are the assumptions of quantum mechanics?3. Atoms are composed of electrons and nuclear. Why an atom is only described by the states of the electrons of the atom?4.

7、What is the Hckel method (HMO)?5.How to convert physical observables into operators in quantum mechanics?Exercise20 Crystal Structure and SurfaceIntroduction20.1 Course content Crystal BasicsSymmetry20.2 Atomic and molecular arrangementMany chemical (including biochemical) and physical properties de

8、pend on crystal structure and knowledge of crystallography is essential if the properties of materials are to be understood and exploited.Cu and Si fcc C graphiteDiamondFullerrenesNanotube- 1D crystal Graphene(Novoselov et al, Science 306, 2004)A.K. Geim and K.S. Novoselov, University of ManchesterC

9、lassifications of solids by atomic arrangement ordered disorderedatomic arrangement regular random*order long-range short-range*name crystalline amorphous “crystal” “glass”Definition - CrystalA homogenous solid formed by a repeating, three-dimensional pattern of atoms, ions, or molecules and having

10、fixed distances between constituent parts.Basic Characteristic of CrystalsHomogeneity Under macroscopic observation,the physics effect and chemical composition of a crystal are the same.Anisotropy Physical properties of a crystal differ according to the direction of measurement.Self-limitation prope

11、rtySymmetryX-ray diffractionIdeal CrystalAn ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed by the infinite repetition of these identical structural units in space.Structure can be described in terms of a lattice, with a group of atoms attache

12、d to each lattice point. The group of atoms is the basis.0.3 Bravais LatticeCrystals are composed of a periodic array of atoms:The structure of all crystals can be described in terms of a lattice, with a group of atoms attached to each lattice point called basis:basis ( motif ) + lattice = crystal s

13、tructure0.3.1 1-D LatticeA basis of atoms is attached to a lattice point and each atom in the basis is specified by:Translational group (algebra)where m=0, 1, 2The basis consists of one or several atoms.Lattice translation vectors (geometry)Groups The elements of a set G together with a product rule

14、 form a group G if:G,H G, GHG (closure).F,G,H G, F (GH )=(FG )H (associativity).An element EG exists such that EG =GE = GGG (identity).For each GG there exists an element G -1G such that G -1 G =GG-1 = E (inverse).If in addition GH -HG G,H G, G is Abelian.To describe a crystal, it is necessary to sp

15、ecify three things:What is the latticeWhat is the basisWhat are the lattice translation vectors0.3.2 2-D LatticeIn two dimensions, if any lattice point is chosen as the origin, the position of any other lattice point is defined by the vectorTranslational groupwhere m, n =0, 1, 2graphiteNaCl (100)0.3

16、.3 3-D LatticeIn 3 dimensions, if any lattice point is chosen as the origin, the position of any other lattice point is defined by the vectorTranslational groupwhere m, n , p =0, 1, 2 A unit cell can be chosen in a variety of ways. It is conventional to choose the cell that represents the full symme

17、try of the lattice. In this rectangular lattice, the rectangular unit cell would normally be adopted.0.3.4 Unit cell2-D Unit cellprimitive unit cellA unit cell is a parallel-sided (but not necessarily rectangular) figure from which the entire crystal structure can be constructed by using only transl

18、ations (not reflections, rotations, or inversions).3-D Unit cellA unit cell is a parallel-sided (but not necessarily rectangular) figure from which the entire crystal structure can be constructed by using only translations (not reflections, rotations, or inversions).0.4 Packing of rigid spheresSimple cubic packing packing efficiency PE= (1/8x8)(4/3 pi r3) / (2r)3 = pi/6 = 52.35% atomic coor