Electronic Structure Notes

Investigating Electronic Structure

Course Information

• This course consists of 7 lectures and 2 tutorials.
• Instructor: iain.wright@ed.ac.uk, Office 283.

Course Aim

The aim of this course is to make students aware of the experimental tools available to the inorganic chemist to probe the electronic structure of transition metal complexes. This encompasses:

1. Electronic Absorption Spectroscopy
   • Russel-Saunders Coupling Schemes
   • Correlation Diagrams
   • Tanabe-Sugano Diagrams
2. Electron Paramagnetic Resonance
3. Voltammetry

• Reference: Inorganic Chemistry by Weller et al, 7th ed., Chapter 20.

Molecular Orbitals

• Different types of molecular orbitals result depending on the nature of the atomic orbitals involved.
• These can be classified in familiar ways.
• Example: MO diagram for carbon monoxide (CO) showing 2s and 2p atomic orbitals of C and O combining to form sigma ($\sigma$), pi ($\pi$), sigma star ($\sigma$), and pi star ($\pi$) molecular orbitals.

Frontier Orbitals

• When electrons from each atom are introduced into the molecular orbital diagram, the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are identified.

Electronic Structure

The vast majority of the chemical and physical properties of a transition metal complex are dictated by:

1. The energy of the frontier orbitals.
2. The energy gap between these orbitals.
3. The symmetry of the orbitals.
4. The number of electrons (e.g., a pair of electrons or a radical).

• Methods are needed to probe the frontier orbitals of molecular materials to understand them and devise ways to use them for any given application.

Electronic Structure - Contents

The following methods will be studied for electronic structure elucidation:

1. Electronic Spectroscopy
   • Characterization of electronic transitions between MOs.
2. Voltammetry
   • Characterization of the energy of available MOs.
3. Electron Paramagnetic Resonance Spectroscopy
   • Characterization of SOMOs (semi-occupied molecular orbitals).

Electronic Structure – Learning Outcomes

By the end of the lecture series, students should be able to:

1. Electronic Spectroscopy
   • Predict the UV/visible absorbance spectrum expected for a particular transition metal complex.
   • Construct and interpret Tanabe-Sugano diagrams.
2. Voltammetry
   • Interpret and critically evaluate experimentally derived voltammograms.
   • Plan a voltammetric experiment.
3. Electron Paramagnetic Resonance Spectroscopy
   • Interpret and critically evaluate experimentally derived EPR spectra.

Electronic Spectroscopy

• Electromagnetic radiation of sufficient energy can induce transitions of an electron from one orbital to another.
• This is the underlying principle of Electronic Spectroscopy, also known as UV/vis Absorbance Spectroscopy.
• In Electronic Spectroscopy, a solution of the analyte is illuminated with light.
• Photons are either absorbed by the analyte, causing electronic transitions, or pass straight through the sample where they are detected.

Electronic Spectroscopy cont.

• The color of any molecule is dictated by the wavelengths of light it absorbs.
• The wavelengths of light absorbed are dictated by the electronic structure of the molecule.
• Governed by selection rules which dictate how “allowed” or “forbidden” any transition is.

UV/vis Absorbance Spectroscopy

• Usually presented as absorbance (A) versus wavelength ($\lambda$) in nanometers (nm).
• $h \nu = \Delta E$, where h is Planck's constant, $<br /> u$ is frequency, and $\Delta E$ is the energy difference between the ground state and excited state.

UV/vis Absorbance Spectroscopy Note

• Sometimes presented as absorbance versus wavenumber ($\bar{\nu}$) in centimetres ($cm^{-1}$).
• Wavenumber is the number of wavelengths per centimetre, so $\bar{\nu} = 1/\lambda$ with units $cm^{-1}$.
• Be comfortable with converting between nm and $cm^{-1}$.

UV/vis Absorbance Spectroscopy - Key Information

Two key pieces of information emerge:

• $\lambda_{max}$ = energy of transition
• $\epsilon_{max}$ = probability of transition

Energy Gap Calculation

• Electromagnetic radiation of sufficient energy can promote an electronic transition between the HOMO and the LUMO.
• $h\nu = \Delta E$
• This gives $\Delta E$ in Joules (J) and will be a very small number ($10^{-19}$ J).
• $\nu = \frac{c}{\lambda}$, where c is the speed of light and $\lambda$ is the wavelength.
• $\Delta E = h\nu$
• $\therefore E = \frac{hc}{\lambda}$
• $h = 6.626 \times 10^{-34} Js$
• $\nu$ = frequency (Hz or $s^{-1}$)
• $c = 3 \times 10^8 ms^{-1}$
• Much more convenient to discuss $\Delta E$ in terms of electron volts (eV) which we can do by dividing by the charge of an electron ($e = 1.602 \times 10^{-19} C$).
• $\Delta E = \frac{hc}{\lambda e} = \frac{1.240 \times 10^{-6}}{\lambda}$
• If the UV/vis spectrum is in nanometers then just multiply by $10^9$ to convert from m to nm:
• $\Delta E = \frac{1240.68}{\lambda}$

Molar Absorption Coefficients

• Also called molar extinction coefficients and obtained from the absorbance A.
• Absorbance is related to the transmittance, T as shown:
• $T = \frac{I<em>t}{I</em>0}$, where $I<em>t$ is the transmitted light intensity and $I</em>0$ is the incident light intensity.
• $A = log \frac{I<em>0}{I</em>t}$
• $A = -log T$
• The value of A gives us an idea of how many electrons have undergone a transition from one orbital to another.
• The higher the value of A the more electrons have undergone a transition.

Molar Absorption Coefficients Cont.

The absorbance is therefore related to the probability of an electronic transition occurring.

• $A = \epsilon l c$
• A = absorbance (arbitrary units)
• $\epsilon$ = molar absorption coefficient ($M^{-1}cm^{-1}$)
• L = path length of cuvette (cm)
• c = concentration (M)
• This is expressed in the Beer-Lambert law.
• Calculation of $\epsilon$ in this fashion lets us establish how probable a transition is.
• This is dictated by the selection rules which govern how allowed or forbidden any electronic transition is.

Molar Absorption Coefficients - Selection Rules

• Spin selection rule: Upon excitation $\Delta S = 0$. Spin multiplicity must not change.
• Laporte selection rule: The change in total angular momentum can be $\Delta L = 0, \pm 1$ but L=0↔L=0 transitions are forbidden, or, transitions between states of the same symmetry (with respect to inversion) are forbidden!

Selection Rules - Example

Consider the $d^1$ complex $Ti(H<em>2O)</em>6^{3+}$:

• Spin selection rule: Upon excitation $\Delta S = 0$. Spin multiplicity must not change.
• Laporte rule: Transitions between states of the same symmetry (with respect to inversion) are forbidden!
• Symmetry forbidden $\epsilon \sim 10 M^{-1} cm^{-1}$

Selection Rules - Magnitude of Extinction Coefficient

• The magnitude of $\epsilon$ gives an indication of how allowed the transition is and thereby what rules are being broken.
• \epsilon < 1 M^{-1}cm^{-1}: Spin forbidden
• $\epsilon \approx 10-100 M^{-1}cm^{-1}$: d-d Laporte forbidden
• $\epsilon \approx 250-450 M^{-1}cm^{-1}$: d-d Laporte allowed
• \epsilon > 1000 M^{-1}cm^{-1}: Fully allowed, Charge transfer bands
• Insights into the electronic and molecular structure of a complex can be gained by considering the magnitude of $\epsilon$.

Charge transfer transitions - Octahedral Complex

• Metal: d-orbitals
• Ligand: ligand $\pi$*, ligand $\pi$, ligand $\sigma$
• MLCT (Metal to Ligand Charge Transfer)
• LMCT (Ligand to Metal Charge Transfer)

MO Diagram for [W(CO)6]

• Low oxidation state metal
• Relatively low lying ligand acceptor orbitals.

Ligand to Metal Charge Transfer

• Metal-based orbital gains an electron during the transition – metal is reduced.
• High oxidation state metal – low-lying empty orbitals
• Ligands will have lone pairs at relatively high energy
• Often absorb in the visible region.
• The permanganate ion $MnO_4^-$ ($\lambda$ = 525nm, $\epsilon = 2.5 \times 10^3 M^{-1}cm^{-1}$) is a good example.

Metal to Ligand Charge Transfer

• Ligand-based orbital gains an electron during the transition – metal is oxidized.
• Low oxidation state metal
• Ligands will have low-lying empty orbitals (often $\pi$* orbitals)
• Often absorb in visible region.
• $[Ru(2,2-bpy)<em>3]^{2+}$ ($\lambda</em>{max}$ = 452 nm, $\epsilon = 13 \times 10^3 M^{-1} cm^{-1}$) is a good example.

Metal to Ligand Charge Transfer Cont.

• Charge Transfer Bands
• Transition involves electron transfer between MOs based on different parts of the complex – high transition dipole moments.
• High intensity with typical $\epsilon$ ranging from 1000 to 100000 $M^{-1}cm^{-1}$ and beyond.
• High sensitivity to solvent polarity – solvatochromic.

Other Transitions

How many d-d transitions would one see for $[Cr(NH<em>3)</em>6]^{3+}$?

UV/Vis Spectrum for [Cr(NH3)6]3+

The spectrum shows multiple absorption bands at different wavelengths.

UV/Vis Spectrum for [Cr(NH3)6]3+ - Explanation

• Why is this spectrum so complicated?
• There are obviously more energy states allowed than we have considered to date.
• These arise through electron-electron repulsions.

Russell-Saunders Coupling

Russell-Saunders Coupling treats the system not as being composed of individual electrons but as the sum of all the electrons in the system.

Recap of quantum numbers:

• n = principal quantum number. n = 1, 2, 3, 4 etc
• l = orbital angular momentum quantum number. l = 0, 1, 2, 3 etc. (s, p, d, f etc.)
• $m<em>l$ = magnetic quantum number. $m</em>l$ = -l ≤ 0 ≤ +l
• $m<em>s$ = spin quantum number. $m</em>s$ = -½ or +½

Russell-Saunders Coupling - d2 Configuration

Consider a $d^2$ configuration in five degenerate d-orbitals…

Are the energies of these two configurations the same?

Clearly not – these two arrangements are in fact non-degenerate so we will need a means to define their relative energies.

This is what consideration of Russell-Saunders Coupling will provide.

Example – p2

• 3 p-orbitals
• 2 electrons
• p-orbital
  • $m_l$ = +1, 0, -1
• electron
  • $m_s$ = +½ , -½

Introduce the electrons into the orbitals (ignoring electronic repulsion).

Note the $m<em>l$ of the orbital chosen for each electron and use their direction of $m</em>s$ as a superscript for each electron like so: $(m<em>{l,1}^{m</em>{s,1}}, m<em>{l,2}^{m</em>{s,2}})$

Example – p2

$(m<em>{l,1}^{m</em>{s,1}}, m<em>{l,2}^{m</em>{s,2}})$

Example – p2 Cont.

Do these have the same energy?

No

These are examples of microstates and for our $p^2$ configuration we can generate many more….

Example – p2 Cont.

Two new quantum numbers need to be defined:

• L = total orbital angular momentum QN
• S = total spin QN

These relate to two more QNs:

• $M<em>L$ and $M</em>S$

These describe L and S for microstates.

Allowed values for $M_L$ depend upon L.

$M_L$ = +L, +(L-1)…. -L

So for a value of L = 2 $M_L$ can be +2, +1, 0, -1, -2

Allowed values for $M_S$ depend upon S.

$M_S$ = +S, +(S-1)…. -S

So for a value of S = 3/2 $M_S$ can be +3/2, +1/2, 0, -1/2, -3/2

Example – p2 Cont.

For a multi-electron system each electronic configuration can be described by $M<em>L$ and $M</em>S$.

Each combination of allowed values of $M<em>L$ and $M</em>S$ is a microstate.

$p^2$ has 15 microstates.

Russell-Saunders Coupling Scheme for $p^2$

Example – p2 Cont.

These microstates can be organised according to $M<em>L$ and $M</em>S$.
Now we can group the microstates which do have the same energy into states.

A state is a collection of degenerate microstates.

We will classify them using a Term Symbol. $^{2S+1}L$

Example – p2 Cont.

$^{2S+1}L$

• L Symbol
  • 0 = S
  • 1 = P
  • 2 = D
  • 3 = F
  • 4 = G
  • 5 = H
• 2S+1 Class
  • 1 = Singlet
  • 2 = Doublet
  • 3 = Triplet
  • 4 = Quartet
  • 5 = Quintet

Example – p2 Cont.

Start with the highest value of $M_L$.

$M_L$ = +2 so L = D

$M_S$ = 0 so 2S+1 = 1

So, Term = $^1D$

This Term includes all:

$M<em>L$ = +2 to -2, $M</em>S$ = 0

Example – p2 Cont.

Start with the highest value of ML.

The number of microstates in each state is defined by $(2S+1)(2L+1)$.

For $^1D$: $(2S+1)(2L+1) = (2(0)+1)(2(2)+1) = (1)(5) = 5$ microstates

There are 10 left – on to the next…

Example – p2 Cont.

Take the next highest value of $M_L$.

$M_L$ = +1 so L = P

$M_S$ = +1 so 2S+1 = 3

So, Term = $^3P$

This Term includes all:

$M_L$ = +1 to -1

$M_S$ = +1 to -1

Example – p2 Cont.

Start with the highest value of ML.

The number of microstates in each state is defined by $(2S+1)(2L+1)$.

For $^3P$: $(2S+1)(2L+1) = (2(1)+1)(2(1)+1) = (3)(3) = 9$ microstates

There is 1 left – on to the next…

Example – p2 Cont.

Take the next highest value of $M_L$.

$M_L$ = 0 so L = S

$M_S$ = 0 so 2S+1 = 1

So, Term = $^1S$

This Term includes all:

$M_L$ = 0

$M_S$ = 0

Example – p2 Cont.

Start with the highest value of $M_L$.

The number of microstates in each state is defined by $(2S+1)(2L+1)$.

For $^1S$: $(2S+1)(2L+1) = (2(0)+1)(2(0)+1) = (1)(1) = 1$ microstates

Example – p2 Cont.

We now have three term symbols which describe the different states of a $p^2$ configuration AND take account of electronic repulsion.

Which of these is our ground state and which are excited states?

Ground states lie the lowest in energy.

Example – p2 Cont.

Hund’s rules:

1. Term with the greatest spin multiplicity lies at the lowest energy.
2. For a given multiplicity, the greater the value of L for a term, the lower the energy.

Ground state = $^3P$

This does NOT hold for excited states!

Example – d2

• 5 d-orbitals
• 2 electrons
• d-orbital
  • $m_l$ = -2, -1, 0, +1, +2
• electron
  • $m_s$ = -½ , +½

There are 45 ways to introduce two electrons – 45 microstates.

The same treatment can be applied to obtain term symbols.

$(m<em>{l,1}^{m</em>{s,1}}, m<em>{l,2}^{m</em>{s,2}})$

Example – d2 Cont.

Example – d2 Cont.

Example – d2 Cont.

Example – d2 Cont.

Example – d2 Cont.

Ground state = $^3F$

Example – d2 Cont.

Ground state = $^3F$

Allowed excited state = $^3P$

Energy differences are what we will measure in electronic spectroscopy – NOT energy levels.

R-S Coupling – A recap

Russell-Saunders Coupling Schemes:

• Identify microstates.
• Group degenerate microstates into electronic states.
• States arise from electron-electron repulsion.
• Allocate Term Symbols to states.
• Identify Ground States.

These are Free Ion Terms describing:

• Naked metal atoms/ions
• Spherical geometry (symmetrical)
• Exist in different energies depending on where the electrons are.

How do we work out the energies of these states?

Racah Parameters

The Racah Parameters are A, B, and C and summarise how the energy of a Term is affected by interelectronic repulsions.

• A = Average of the total interelectron repulsion
• B = Relate to the repulsion energies between individual d-electrons
• C = Effectively provide a measure of how much different electrons see of each other.

This depends upon two things:

• Coulomb integral J(ab) – charge
• Exchange integral K(ab) – volume/space

Racah Parameters – d2

The energies of Free Ion Terms for $d^2$ configuration in terms of Racah Parameters are:

• E($^1$S) = A + 14B + 7C
• E($^1$G) = A + 4B + 2C
• E($^1$D) = A – 3B + 2C
• E($^3$P) = A + 7B
• E($^3$F) = A – 8B (GROUND STATE)

B is the important parameter to quantify.

A always cancels out and C ≈ 4B

C also only occurs in excited states which have different multiplicity than the ground state. Such excitations would be formally forbidden.

Racah Parameters – d2

Excitation from $^3F$ to $^3P$.

What will the energy difference be:

• E($^1$S) = A + 14B + 7C
• E($^1$G) = A + 4B + 2C
• E($^1$D) = A – 3B + 2C
• E($^3$P) = A + 7B
• E($^3$F) = A – 8B (GROUND STATE)
• $\Delta E = E(^3P) - E(^3F) = (A+7B) – (A-8B) = 15B$

So, if we use spectroscopy to measure we will get an energy gap = 15B.

Racah Parameters – d2 Cont.

• B increases as the oxidation state increases.
• B increases as the ion gets smaller!
• The size of the ion has a greater impact on electronic repulsion than the number of electrons.
• NB. Higher $cm^{-1}$ = Higher energy
• Correlation Diagrams and Tanabe-Sugano Diagrams are used to obtain both B and LFSE.

Russell-Saunders – Shorthand for GS

There is a quick way to obtain the ground state for any electronic configuration:

1. Add electrons maximising the value of ML.
2. Sum the ML for the electrons: $\Sigma M_L = L = 2+1+0 = 3 = F$
3. Sum the MS for the electrons: $\Sigma M_S = S = 1/2+1/2+1/2 = 3/2$
4. 2(S)+1 = 4

Ground state prediction = $^4F$

This works for any number of p, d, or f electrons.

Russell-Saunders – Shorthand for GS

There is a quick way to obtain the ground state for any electronic configuration:

1. Add electrons maximising the value of ML.
2. Sum the ML for the electrons: $\Sigma M_L = L = 3+2+1+0-1-2 = 3 = F$
3. Sum the MS for the electrons: $\Sigma M_S = S = 6(1/2) = 3$
4. 2(S)+1 = 7

Ground state prediction = $^7F$

These are free atoms – spherical – not complexes….

Complexes – Influence of Geometry

We will focus on the impact of ligands in an octahedral geometry.

The five 3d-orbitals

• Two point along the principle axes:
  • $d<em>{xy}, d</em>{xz}, d<em>{yz}, d</em>{x^2-y^2}, d_{z^2}$
• Three point between the principle axes:
• $D<em>t = \frac{4}{9}D</em>O$

$O_h$ Character Table

• A = Singly degenerate
• E = Doubly degenerate
• T = Triply degenerate
• We can use these to describe our d-electron configuration.

d2 Ground State

With help of the $O_h$ character table, the ground state is described as follows:

$^3T_{1g}$

• 2S+1 = 3
• g = gerade (centrosymmetric)
• 1 = arises from rotation/reflection axes (not important here).

d2 First Excited State

6 ways in total

Degenerate? No

d2 First Excited State Cont.

• 2 electrons in same plane
  • Coulomb Integral Up
  • Exchange Integral Up
  • Higher Energy
• 1 electrons in xy plane
  • 1 electron on z-axis
  • Coulomb Integral Reduced
  • Exchange Integral Reduced
  • Lower Energy

End up with two states of differing energies.

6 states will split into two T states

$^3T<em>{1g}$ & $^3T</em>{2g}$

d2 Second Excited State

d2 Second Excited State

Considering the free atom terms we obtained the weak field terms: $^3F, ^3P$

By introducing $O<em>h$ symmetry we obtained the strong field terms: $^3T</em>{1g}, ^3T<em>{1g} + ^3T</em>{2g}, ^3A_{2g}$

d2 Second Excited State Cont.

Considering the free atom terms we obtained the weak field terms: $^3F, ^3P$

By introducing $O<em>h$ symmetry we obtained the strong field terms: $^3T</em>{1g}, ^3T<em>{1g} + ^3T</em>{2g}, ^3A_{2g}$

$^3T_{1g} = -\frac{4}{5}\Delta O$ (ground state)

d2 Second Excited State Cont.

Considering the free atom terms we obtained the weak field terms: $^3F, ^3P$

By introducing $O<em>h$ symmetry we obtained the strong field terms: $^3T</em>{1g}, ^3T<em>{1g} + ^3T</em>{2g}, ^3A_{2g}$

$^3T_{1g} = -\frac{4}{5}\Delta O$ (ground state)

^3T{1g} & {}^3T{2g} = \frac{1}{5}\Delta O

d2 Second Excited State Cont.

Considering the free atom terms we obtained the weak field terms: $^3F, ^3P$

By introducing $O<em>h$ symmetry we obtained the strong field terms: $^3T</em>{1g}, ^3T<em>{1g} + ^3T</em>{2g}, ^3A_{2g}$

$^3T_{1g} = -\frac{4}{5}\Delta O$ (ground state)

^3T{1g} & {}^3T{2g} = \frac{1}{5}\Delta O

$^3A_{2g} = \frac{6}{5}\Delta O$

d2 Free Atom vs. Complex

d2 Free Atom vs. Complex

Correlation Diagram for a d2 Oh System

UV/Vis Spectrum for [Cr(NH3)6]3+ - Revisited

A need to understand this spectrum is where we started our journey.

Our model has advanced from free ions in a weak field to complexes ions in a strong field.

Now we need to be able to obtain quantitative information.

Correlation Diagram for a d2 Oh System

Correlation Diagram for a О d² System

Tanabe-Sugano Diagram for a О d² System

Tanabe-Sugano Diagrams for O̟ d² and d³

Tanabe-Sugano Diagrams for О d³

Tanabe-Sugano Diagrams for Oh d5

Using Tanabe-Sugano Diagrams

Estimating $\Delta O$ and the Racah parameter $B_{complex}$\

1. Determine the correct d-configuration for the metal ion
2. Choose the correct Tanabe-Sugano diagram
3. Measure $\lambda_{max}$ values from the spectrum of the two lowest energy d-d transitions
4. Convert $\lambda_{max}$ into wavenumbers ($1/\lambda$)
5. Calculate the energy ratio $E<em>2/E</em>1$
6. Use a ruler to find the point on the x-axis where this ratio holds for the relevant transitions
7. Measure E/B for the first transition
8. Solve for B and $\Delta O$

Nephelauxetic Parameter – b

The Racah Parameter B is a measure of the d-d electron repulsion on the metal.

Therefore if electron density is delocalised onto the ligands, through, for instance, $\pi$ back- bonding B will… decrease.

This is known as the nephelauxetic effect:

• Nephos = Cloud
• Auxesis = Growth
• $\beta = \frac{B<em>{complex}}{B</em>{free ion}}$

Nephelauxetic Parameter – b

The nephelauxetic effect occurs for two reasons.

1. The effective positive charge on the metal has decreased. Positive charge of the metal is reduced by negative charge on the ligands.
2. Overlapping with ligand orbitals and forming covalent bonds increases orbital size.
   A low $\beta$ means that there is a large amount of d-electron delocalisation onto the ligands = more covalent character.

• $\beta = \frac{B<em>{complex}}{B</em>{free ion}}$

Using Tanabe-Sugano Diagrams - Example

The electronic absorption spectrum of a V(III) complex exhibited two peaks at 600 nm and 400 nm.

The molar absorption coefficients were consistent with these transitions being spin allowed d-d transitions for an octahedral complex.

Calculate the values of the Racah parameter $B<em>{complex}$ and the $\Delta</em>o$ for the complex.

Using the values for $B_{ion}$ work out the nephelauxetic parameter for the complex.

Electronic Spectroscopy - Summary

• Reinforced basic concepts of UV/vis spectroscopy.
• Demonstrated that wavelengths/wavenumbers provide an energy scale.
• Demonstrated that extinction coefficients show allowed/forbidden transitions.
• Considered different types of electronic transitions.
• Identified that electron-electron repulsion – Russell-Saunders Coupling – results in non-degeneracy in MOs.
• Defined microstates as components of states.
• Derived Term Symbols for states in both free ions and in an $O_h$ ligand field.
• Generated Correlation diagrams and Tanabe-Sugano diagrams.
• Used Tanabe-Sugano diagrams to interpret spectra.