ch676 physical chemistry: principles and applications · ch676 physical chemistry: principles and...

CH676 Physical Chemistry: Principles and Applications

CH676 Physical Chemistry:Principles and Applications


Crystal Structure: XRD


Centering Allowed Peaks

I-centered → h+k+l is an even numberF-centered → hkl either all even or all oddC-centered → h+k is an even numberB-centered → h+l is an even numberA-centered → k+l is an even numberR-centered → -h+k+l is a multiple of 3

XRD: Diffraction Peak Positions


XRD: Diffraction Peak Intensities

I(hkl) = |S(hkl)|2 × Mhkl × LP(θ) × TF(θ)

S(hkl) = Structure FactorMhkl = MultiplicityLP(θ) = Lorentz & Polarization FactorsTF(θ) = Temperature factor (more correctly referred to as the displacement parameter)


XRD: Diffraction Peak Widths

Scherrer Equation

cos

2L

KB

• Peak width (B) is inversely proportional to crystallite size (L)

• P. Scherrer, “Bestimmung der Grösse und der inneren Struktur von Kolloidteilchen mittels Röntgenstrahlen,” Nachr. Ges. Wiss. Göttingen 26 (1918) pp 98-100.

• J.I. Langford and A.J.C. Wilson, “Scherrer after Sixty Years: A Survey and Some New Results in the Determination of Crystallite Size,” J. Appl. Cryst. 11 (1978) pp 102-113.

46.746.846.9 47.047.147.247.347.447.547.6 47.747.847.92 (deg.)

Inte

nsity

(a.u

.)

FWHM


Molecular Orbital Theory to Band Theory

Schrödinger equation

Four quantum numbers define the properties of each atomic orbital. Principle quantum number, n = 1,2,3,.. Azimuthal quantum number, l = 0(s), 1(p), 2(d), 3(f), n-1. Magnetic quantum number, ml = - l, ., l (i.e px, py, pz). Spin quantum number, ms = +1/2 or -1/2

Pauli Exclusion PrincipleNo two electrons can have the same set of quantum numbers (each orbital canhold 2 e-)

Hund’s RuleFor degenerate orbitals the lowest energy configuration maximizes the electronspin (no pairing of electrons if avoidable)


Molecular Orbital Theory to Band TheorySchrödinger equation

Four quantum numbers define the properties of each atomic orbital

Pauli Exclusion Principle

Hund’s Rule



Orbital Overlap:Molecular Orbital (MO) Theory



Orbital Overlap: Molecular Orbital (MO) TheoryThe overlap of two atomic orbitals is dependent upon:• Symmetry of the orbitals• Distance between the orbitals• Spatial extent of the orbitals• The energy difference between orbitalsIncreasing the overlap (spatial and energetic) leads to thefollowing:• Stabilization of the bonding MO• Destabilization of the antibonding MO• The antibonding MO is destabilized to a greater extent than the

bonding MO is stabilizedThe spatial overlap in a bond depends upon symmetry• It decreases as the number of nodal planes increases, σ > π• π bonds are more sensitive to changes in bond angle



MO Diagram for H2


Band TheoryReview – WavefunctionThe wave function does not represent any physical quantities. It should beunderstood merely as a mathematical description of a electron which enablesus to calculate its actual behavior in a convenient way. This thought probablysounds unfamiliar to a beginner in quantum physics. However, by repeatedexposure, one can become accustomed to this kind of thought.

Ψ = sin(kx - wt)


Band TheoryReview – Wave functionThe simplest waveforms is mathematically expressed by a sine (or a cosine)function. This simple disturbance is "harmonic wave." (Fourier transformationcan substitute any odd type of waveform by a series of harmonic waves, eachhaving a different frequency.) The properties of electrons will be described by aharmonic wave.

Ψ = sin(kx - wt)


The 2s orbitals have a lower energy than the 2porbitals.

The σ-bonds have a greater spatial overlapthan the π-bonds. This leads to a larger splittingof the bonding and antibonding orbitals.

The 2px and 2py π-interaction produces to twosets of degenerate orbitals.

In O2 there are 12 valence electrons and eachof the 2pπ* orbitals (πg) are singly occupied.Thus the bond order = 2, and O2 isparamagnetic.

1st Order MO Diagram for O2



2nd Order MO Diagram for O2 & N2

A more accurate depiction of thebonding takes into account mixingof of MOs with the same symmetry(σg

+ & σu+). The consequences of

this 2nd order effect are:

The lower energy orbital isstabilized while the higher energyorbital is destablized.

The s and p character of the σ MOsbecomes mixed.



Heteronuclear Case & ElectronegativityThe atomic orbitals of the more electronegative atom are lowered.The splitting between bonding and antibonding MO’s now has an ionic (Ei)and a covalent (Ec) component.The ionic component of the splitting (Ei) increases as the electronegativitydifference increases.The covalency and the covalent stabilization / destabilization decrease asthe electronegativity difference increases.The orbital character of the more electronegative atom is enhanced in thebonding MO and diminished in the antibonding MO.


AX2 Linear & BentIn bent H2O the O 2s σ* orbital and the O 2pxorbital are allowed to mix by symmetry,lowering the energy of the O 2px orbital. Nowthere is only one non-bonding orbital (O 2py)


Walsh’s RuleA molecule adopts the structure that best stabilizes the HOMO. If the HOMO isunperturbed the occupied MO lying closest to it governs the geometricalpreference.

2nd Order Jahn-Teller DistortionA molecule with a small energy gap between the occupied and unoccupiedMOs is susceptible to a structural distortion that allows intermixing betweenthem.




Structure & Properties of the Group 14 Elements

As you go proceed down the group thetendency for the s-orbitals to becomeinvolved in bonding diminishes. Thisdestabilizes tetrahedral coordinationand semiconducting/insulating behavior.

C

Pb


MO Diagram ReO66- Octahedron


The diagram to the left shows aMO diagram for a transition metaloctahedrally coordinated by σ-bonding ligands. (π-bonding hasbeen neglected)

Note that in an octahedron there isno mixing between s, p and d-orbitals.

For a main group metal the samediagram applies, but we neglectthe d-orbitals.


Band TheoryBand Structure ReO3 Band Structure

(aka Spaghetti diagram): MOdiagram with translational symmetrytaken into account.

Density of States (DOS)Integration of the band structure.Shows the # of available levelsbetween E and E+dE as dE → 0.


Band TheoryImportant Points of Band Structures

• What is being plotted?Energy vs. k, where k is the wavevector that gives the phase of the AOs as well as the wavelength of the electron wavefunction (crystal momentum).

• How many lines are there in a band structure diagram?As many as there are orbitals in the unit cell.

• How do we determine whether a band runs uphill or downhill?By comparing the orbital overlap at k=0 and k=π/a.

• How do we distinguish metals from semiconductors and insulators?The Fermi level cuts a band in a metal, whereas there is a gap between the filled and empty states in a semiconductor.

• Why are some bands flat and others steep?This depends on the degree of orbital overlap between building units.Wide bands → Large intermolecular overlap → delocalized e-

Narrow bands → Weak intermolecular overlap → localized e-


Band Theory

It is possible to construct a reasonable approximation of the DOS diagramfrom the MO diagram of the building block.

The energy levels of each block of bands (BOB) comes from the MO diagram (based on electronegativity and bonding interactions)

The area of each BOB is proportional to the number of MOs at that approximate energy.

Constructing a DOS Diagram: TiO2


Band TheoryChain of Atoms

2 H atoms Chain of 5 H atoms

Antibonding

Nonbonding

Bonding


Band TheoryInfinite 1D Chain of atomsThe wavefunction for each electronicState (MOs) is:

Ψk = Σ eiknaχn

(Euler's formula: mathematical formula for convertingrelationship between the trigonometric functions andthe complex exponential function.)

a is the lattice constant (spacing between Hatoms).

n identifies the individual atoms within thechain.

χn represents the atomic orbitals

k is a quantum # that identifies thewavefunction and tells us the phase of theorbitals.


Band TheoryInfinite 1D Chain of atoms

k = π/a

Ψπ/a = χ0 + (exp{iπ})χ1 + (exp{i2π})χ2 + (exp{i3π})χ3 + (exp{i4π})χ4 + …

k = π/2a

Ψπ/2a = χ0+(exp{iπ/2})χ1 + (exp{iπ})χ2 + (exp{i3π/2})χ3 + (exp{i2π})χ4 +…

k = 0

Ψ0 = χ0 + χ1 + χ2 + χ3 + χ4 +…


Band TheoryBand Structure of 1D Chain of atoms

• The band runs "uphill" (from 0 to π/a) because the in phase (at k=0)combination of orbitals is bonding and the out of phase (at k=π/a) isantibonding.

• The Fermi energy separates the filled states from the empty states (T = 0 K).

k = π/a

Ψπ/a = χ0 + (exp{iπ})χ1 + (exp{i2π})χ2 + (exp{i3π})χ3 + (exp{i4π})χ4 + …

k = π/2a

Ψπ/2a = χ0+(exp{iπ/2})χ1 + (exp{iπ})χ2 + (exp{i3π/2})χ3 + (exp{i2π})χ4 +…

k = 0

Ψ0 = χ0 + χ1 + χ2 + χ3 + χ4 +…


Band TheoryBand Structure of 1D Chain of atomsBand Structure: Linear Chain of F

Which is the correct band structure for a linear chain of F atoms?

A B


Band TheoryBand Structure of 1D Chain of atomsBand Structure: Linear Chain of F


Band TheoryBand Structure of 1D Chain of atomsEffect of Orbital Overlap

If we reduce the lattice parameter a it has the following effects:

• The band becomes more bonding at k=0

• The band becomes moreantibonding k=π/a. The increased antibonding is larger than theincreased bonding.

• The bandwidth increases.

• The electron mobility increases.

Wide bands → Good orbitaloverlap → High carrier mobility

Wide bands → Large intermolecular overlap → delocalized e-Narrow bands → Weak intermolecular overlap → localized e-


Band TheoryBand Structure of 3D Chain of atoms


Band TheoryThree Dimensions - Band Structure of Ba2SnO4

ch676 physical chemistry: principles and applications · ch676 physical chemistry: principles and...

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