Lecture 6 MO Theory of Extended Molecules and Solids Suggested reading: Chapter 2.12 & 3.19-3.2 Reminder: Quiz Wednesday

Why is MO Theory Important? 1)) Predicting d the h shapes h off molecules l l 2) Understanding the electronic properties of molecules and solids 3) Applications to nanoscience

3) MO Theory of Nanostructures

Au shell SiO2

Halas Group, Rice University

3) MO Theory of Nanostructures

Au shell SiO2

Halas Group, Rice University

Nanoparticle Application: Optical Cancer Therapy

Atwater, “The Power of Plasmonics,” Scientific American

1. Shapes of Molecules W ter (H2O) Water Beryllium Hydride (BeH2)

Correlation diagram: shows how one set of orbitals evolves into another as a parameter (such as a bond angle) is changed

Walsh Diagram for XH2

SALCs for XH2: Linear Molecule

   c1  2 s  c2 g

   2 p u

y ,z

   c3  2 p  c4 u

y

SALCs for XH2: Angular Molecule

 a1  c1  2 s  c2  2 p  c3 z

 b1   2 p

x

 b 2  c4  2 p  c5 y

Walsh Diagram for XH2 XH2 molecules with four or less valence l electrons l t will ill be b linear li XH2 with five to eight valence electrons is predicted to be angular

BeH2

BH2

CH2

NH2

OH2

180o

131o

136o

103o

105o

Hybridization Molecular shapes can also be intuited by hybridization - a mixing of atomic orbitals (predates MO theory, but still is conceptually useful)

Si Hybridization

Isolated Si atom

Si just before bonding

An isolated Si atom has two electrons in the 3s and two electrons in the 3p orbitals. When Si is about to bond, the one 3s orbital and the three 3p orbitals interfere to form four hybridized y orbitals, hyb, called spp3 orbitals, which are directed toward the corners of a tetrahedron. The hyb orbital has a large major lobe and a small back lobe. Each hyb orbital takes one of the four valence electrons. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw‐Hill, 2005)

Types of Hybridization sp3 hybridization: Hybridized orbitals formed from an “s” orbital and 3 “p” orbitals. sp2 hybridization: Hybridized orbitals formed from an “s” orbital and 2 “p” orbitals. Bond angles of 120o sp hybridization: Hybridized orbitals formed from an “s” orbital and 1 “p” p orbitals. Bond angles of 180o

Geometries of Hybrid Orbitals

Note: Hybridization incorrectly predicts the photoelectron spectra of many molecules molecules. Also, hybridization over-emphasizes the localization of bonding electrons. It should only be used for a qualitative understanding of molecular structure.

2. Electronic Properties of Molecules & Solids

a=0.154 0 154 nm

a=0.142 nm

Graphene: Top 10 list 10) The largest aromatic molecule: One atom thick planar sheet of sp2 bonded carbon 9) First isolated in 2004 using scotch tape 8) Linear dispersion: completely free electrons 7) Resistivity of 10-6Ohm-cm, less than that of Ag: the lowest resistivityy at room temperature p 6) Absorbs πα~2.3% of white light 5) Thermal conductivity exceeding diamond 4) Breaking strength 200x steel  strongest material 3) Large surface area to volume ratio  ideal sensor 2) St Strong response tto applied li d fi fields ld ((modulators, d l t FET FETs)) 1) Zero effective mass electrons and holes

Three-atom system: three energy levels

The energies of the three molecular orbitals, labeled a, b, and c, in a system with three H atoms.

N-atom system: N energy levels

The formation of 2s energy band from the 2s orbitals when N Li atoms (1s2 2s1) come together to form the Li solid. There are N 2s electrons, but 2N states in the band. The 2s band is therefore only half full. The atomic 1s orbital is close to the Li nucleus and remains undisturbed in the solid.

Band theory of solids

As Li atoms are brought together from infinity, the atomic orbitals overlap and give rise to bands. Outer orbitals overlap first. The 3s orbitals give rise to the 3s band, 2p orbitals to the 2p p to p produce a single g band in which the energy gy is nearlyy band, etc. The various bands overlap continuous. Note: As with MO theory of molecules, we can no longer consider the electrons as belonging to specific atoms – they are shared among the entire solid.

Band theory of solids

In a metal, metal the various energy bands overlap to give a single energy band that is only partially full of electrons. There are states with energies up to the vacuum level, where the electron is free.

Band theory of solids: The Fermi Level

Typical T i l electron l t energy band b d diagram di for f a metal. t l All the th valence l electrons l t are in an energy band, which they only partially fill. The top of the band is the vacuum level, where the electron is free from the solid (PE = 0).

How are the energies of electrons distributed in a band? 1 state with the highest energy

Many states with comparable intermediate energies (given by the # of nodes)

1 state with h the h lowest energy

Density of States, g(E)

g(E) is the number of states (molecular orbitals) in the energy interval E to (E+dE) per unit volume of the sample. sample The total number of states per unit volume up to some energy E’ is:

S V

(E ) 

E'

 g ( E ) dE 0

Deriving the density of states in a metal Let’ss consider the electrons in the solid to be in a 3D potential well of Let volume V, with sides L. The energy of the electrons is:

En1n2n3



h2 2 2 2  n  n  n 1 2 3 2 8mL



In 2D, we only have n1 and n2. Each state, state or electron wavefunctions in the crystal, crystal can be represented by a box at n1, n2.

How many combinations of n1 and n2 have h e an n energy energ less than E’?

Deriving the density of states in a metal Let’ss consider the electrons in the solid to be in a 3D potential well of Let volume V, with sides L. The energy of the electrons is:

En1n2n3



h2 2 2 2  n  n  n 1 2 3 2 8mL



In 2D, we only have n1 and n2. Each state state, or electron wavefunctions in the crystal crystal, can be represented by a box at n1, n2.

The area contained by n1 and n2 approximates a circle with area ¼(πn’2)

Deriving the density of states in a metal In three dimensions, the volume defined by a sphere of radius n' and the positive axes n1, n2, and n3, contains all the possible combinations of positive n1, n2, and n3 values that satisfy n12  n22  n32  n2

V sphere

1   n '3 6

Each orbital can hold two electrons, so the total number of states up to some quantum number n’ is:

S ( n )  2V sphere

1   n '3 3

Deriving the density of states in a metal S ( n )  2V sphere

En1n2n3



1   n '3 3



2 2 h2 h n 2 2 2 n1  n2  n3   2 8mL 8mL2

S (E ') 

2 ' 8 mL E 2 n'  h2

 L3 (8 mE ' ) 3 / 2 33hh 3

Since L3 is the volume of the solid, the total number of states per unit volume is:

SV ( E ' ) 

g

(E ) 

 (8 mE E ' )3/ 2 3h 3





dS V m   8 21 / 2  2e  dE h 

3/ 2

E 1/ 2

Applying the Density of States: STM Image of an elliptical atomic corral (Co atoms on a Cu surface)

From Manorahan et al. Nature, Feb. 3, 2000

Co Cu (111)

dI/dV     (d density of sstates)

Scanningg tunnelingg microscope tip

Topograaphy

The Quantum Mirage: An application of the DOS

Mirage

Manoharan, H. C.; Lutz, C. P.; Eigler, D. M. Nature 403 (2000)

Co Cu (111)

dI/dV     (d density of sstates)

Scanningg tunnelingg microscope tip

Topograaphy

The Quantum Mirage: An application of the DOS

Mirage

Manoharan, H. C.; Lutz, C. P.; Eigler, D. M. Nature 403 (2000)

Photonic Crystals: An application of the optical y of states density

http://photonics.tfp.uni‐karlsruhe.de/research.html

From Wikipedia: “Some structures can completely inhibit the propagation of light of certain colors (energies), creating a photonic bandgap: the DOS is zero for those photon energies.”

Application the DOS: Fermi Energy At T=0,, the Fermi energy gy is the chemical potential p of the system. y For a metal, we can readily determine the chemical potential by integrating the density of states: S V

(E ) 

EF

 g ( E ) dE  0

S  2V sphere

 (8 mE F ) 3 / 2 3h 3

3 1 1 3   n '  Nq ( ) V 2 3

N: Number b off atoms in solid ld q: Number of electrons contributed per atom V: volume of solid ½: Each state can hold two electrons

h 2 3Nq 3 Nq 2 / 3 EF  ( ) 8me V

Energy bands arise from bonding & antibonding wavefunctions

Left: potential energy versis atomic separation. At the equilibrium distance for Si, a bandgap is formed. Right: Simple energy band diagram of a semiconductor. CB is the conduction band and VB is the valence band. AT 0 K, the VB is full with all the valence electrons.

Extra Slides: 1 Structure 1. S and dB Bond dP Properties i 2. Determining Oxidation Numbers 3 Symmetry 3. S

Structure and Bond Properties Covalent radius: equilibrium bond lengths can be empirically and approximately partitioned into contributions from each atom of the bonded pair Bond dissociation enthalpy: the standard reaction enthalpy for the reaction: AB(g) A(g)+B(g) Pauling electronegativity can be determined from mean A-B bond enthaplies. Binary compounds with electronegativity differences greater than 1.7 tend to be ionic. Oxidation states are determined by exaggerating the ionic character of a bond. (Charge an atom would have if it acquired the two electrons of a bond.) bond )

Determining Oxidation Number

Symmetry operations