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CH676 Physical Chemistry: Principles and Applications

CH676 Physical Chemistry:Principles and Applications

CH676 Physical Chemistry: Principles and Applications

Crystal Structure: XRD

CH676 Physical Chemistry: Principles and Applications

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

CH676 Physical Chemistry: Principles and Applications

XRD: Diffraction Peak Intensities

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

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

CH676 Physical Chemistry: Principles and Applications

XRD: Diffraction Peak Widths

Scherrer Equation

cos

2L

KB

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

P. Scherrer, Bestimmung der Grsse und der inneren Struktur von Kolloidteilchen mittels Rntgenstrahlen, Nachr. Ges. Wiss. Gttingen 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

CH676 Physical Chemistry: Principles and Applications

Molecular Orbital Theory to Band Theory

Schrdinger 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-)

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

CH676 Physical Chemistry: Principles and Applications

Molecular Orbital Theory to Band TheorySchrdinger equation

Four quantum numbers define the properties of each atomic orbital

Pauli Exclusion Principle

Hunds Rule

CH676 Physical Chemistry: Principles and Applications

Molecular Orbital Theory to Band Theory

Orbital Overlap:Molecular Orbital (MO) Theory

CH676 Physical Chemistry: Principles and Applications

Molecular Orbital Theory to Band Theory

Orbital Overlap:Molecular Orbital (MO) Theory

CH676 Physical Chemistry: Principles and Applications

Molecular Orbital Theory to Band 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, > bonds are more sensitive to changes in bond angle

CH676 Physical Chemistry: Principles and Applications

Molecular Orbital Theory to Band Theory

MO Diagram for H2

CH676 Physical Chemistry: Principles and Applications

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.

= sin(kx - wt)

CH676 Physical Chemistry: Principles and Applications

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.

= sin(kx - wt)

CH676 Physical Chemistry: Principles and Applications

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

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

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

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

1st Order MO Diagram for O2

Molecular Orbital Theory to Band Theory

CH676 Physical Chemistry: Principles and Applications

2nd Order MO Diagram for O2 & N2

A more accurate depiction of thebonding takes into account mixingof of MOs with the same symmetry(g+ & u+). The consequences ofthis 2nd order effect are:

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

The s and p character of the MOsbecomes mixed.

Molecular Orbital Theory to Band Theory

CH676 Physical Chemistry: Principles and Applications

Heteronuclear Case & ElectronegativityThe atomic orbitals of the more electronegative atom are lowered.The splitting between bonding and antibonding MOs 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.

Molecular Orbital Theory to Band Theory

AX2 Linear & Bent In bent H2O the O 2s * 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)

CH676 Physical Chemistry: Principles and Applications

Walshs 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.

Molecular Orbital Theory to Band Theory

CH676 Physical Chemistry: Principles and Applications

Molecular Orbital Theory to Band Theory

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

CH676 Physical Chemistry: Principles and Applications

MO Diagram ReO6 6- Octahedron

Molecular Orbital Theory to Band Theory

The diagram to the left shows aMO diagram for a transition metaloctahedrally coordinated by -bonding ligands. (-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.

CH676 Physical Chemistry: Principles and Applications

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.

CH676 Physical Chemistry: Principles and Applications

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=/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-

CH676 Physical Chemistry: Principles and Applications

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

CH676 Physical Chemistry: Principles and Applications

Band TheoryChain of Atoms

2 H atoms Chain of 5 H atoms

Antibonding

Nonbonding

Bonding

CH676 Physical Chemistry: Principles and Applications

Band TheoryInfinite 1D Chain of atomsThe wavefunction for each elect

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