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Fundamentals of Electron Absorption Spectroscopy

Electron absorption spectroscopy uses the property that certain substances absorb frequencies in the region of the electromagnetic spectrum of visible light (400-700 nm) and emit a corresponding wavelength which results in the color of the substance. Metal coordination complexes usually have intense colors due to $d\longrightarrow d$ transitions that occur due to the absorption of light.

The wavelength of the color of light absorbed is reflected by the complementary color. The complementary color can be determined by a color wheel. The color reflected is opposite to the color absorbed. Therefore, if a compound absorbs light of the wavelength of red, the compound will reflect and appear to be green.

An emission of light can excite electrons of a molecule into a higher energy state.The energy due to the absorption is related to the wavelength by

{\textstyle E=h\nu }

where E is energy, h is Planck's constant and

\nu

is the frequency which is related to the wavelength by

c=\lambda \upsilon

where c is the speed of light (m/s),

\lambda

is the wavelength (nm), and

\nu

is the frequency (Hz).

The absorption bands that result due to excitation of electrons reveal the wavelengths of different transitions that occur. Organic compounds such as polyethylene have longer wavelengths which absorb in the visible region of the electromagnetic spectrum . Such transitions involve $\pi \longrightarrow \pi ^{*}$ transitions and $n\longrightarrow \pi$ transitions. Beta-carrotene, absorbs in the wavelength of blue light and emits in the orange range of the light absorption spectrum. Beta-carrotene is known for giving the orange color to carrots.^[1] However, metal coordination complexes such as [Cu(H₂O)₆]²⁺ obtain its blue color due to $d\longrightarrow d$ transitions.

Beer-Lambert Law

The Beer-Lambert law relates absorption and concentration of a solution to produce spectra that helps determine what corresponding species are in solution.^[2]

$A=\log(I/Io)=\epsilon lc$

A-absorption (dimensionless) I_o- Original intensity emitted

$\epsilon$ - Molar absorption coefficient (1/M*cm) I-Final intensity recorded

l-length of path (cm)

c-concentration of solution (Mol/L)

The law states that the absorbance of a solution is equal to the concentration of the solution multiplied by the path length the light traveled and the molar absorption coefficient. The molar absorption coefficient is a property that is unique for different substances. It represents the amount of light per mole that is absorbed by the substance. A spectrophotometer is used to measure the absorbance of a solution as light is emitted to it. The intensity of light passing through a solution can either be reflected, absorbed by the molecules, scattered, and/or transmitted through. I_o is reported as the intensity of light initially emitted through and I is the intensity after the light has passed through the solution. If the intensity of the light after passing through the solution is lower than the intensity initially emitted then it can be concluded that the solution absorbed part of the light emitted. The spectrophotometer usually reports in percent transmittance. A plot of absorbance vs. wavelength gives a characteristic plot of the components in solution which can be compared to determine what components are in the solution.

Selection Rules

The intensity of color of certain coordination complexes has to do with whether certain transitions are allowed or forbidden. The molar absorption coefficient is relatively small for coordination complexes compared to organic molecules due to selection rules that make most $d\longrightarrow d$ transitions forbidden.

Spin Selection Rule

Transitions between states of different spin multiplicities are not allowed.^[3]

This implies that the spin multiplicity between two different states must remain the same. This is given by $S=2\Sigma s+1$ where S is the spin multiplicity and s is the overall spin state where spin up is +1/2 and spin down is -1/2.

Laporte's Selection Rule

Transitions between states of same symmetry with respect to the center of inversion are forbidden.^[3]

Thus gerade to gerade $(g\longrightarrow g)$ transitions are forbidden and ungerade to gerade $(g\longrightarrow u)$ are allowed. According to these rules $d\longrightarrow d$ transitions are not allowed since they are still the same symmetry. This rule affects only complexes with a center of symmetry. Therefore it only applies to octahedral complexes and not to tetrahedral complexes.

Despite the selection rules, $d\longrightarrow d$ transitions are still observed due to different mechanisms that helps complexes obey the selection rules.

Mechanisms that Bypass Selection Rules

Vibronic Coupling

In quantum mechanics, the rigid rotor model, assumes that the the bond length between two molecules is fixed^[4]. Vibrations are neglected as the amplitude caused by the vibrations is minor compared to the bond length. However, some bonds in transition metal complexes can cause a temporary change to the symmetry of the complex. Octahedral complexes can lose their center of symmetry after they undergo vibrations which would no longer violate Laporte's rule and allow $d\longrightarrow d$ transitions.^[3]

Orbital Mixing

Tetrahedral complexes usually absorb better than octahedral complexes due to their ability to mix orbitals. Since the p and d orbitals have the same symmetry, the orbitals are allowed to mix unlike octahedral complexes in which the two types of orbitals have different symmetries. Tetrahedral complexes are mostly sp³ and sd³ hybridized. d orbitals are considered to be gerade while p orbitals are considered to be ungerade. Therefore, when the two orbitals mix, the transition results in a mixture of d to p transitions which no longer violates Laporte’s selection rule.^[5]

Oh	E	8C₃	6C₂	6C₄	3C₂	i	6S₄	8S₆	3σ_h	6σ_d
A_1g	1	1	1	1	1	1	1	1	1	1		(z² + x²+ y²)
A_2g	1	1	-1	-1	1	1	-1	1	1	-1
E_g	2	-1	0	0	2	2	0	-1	2	0		(2z² - x²-y₂, x² - y²)
T_1g	3	0	-1	1	-1	3	1	0	-1	-1	(R_x ,R_y ,R_z)
T_2g	3	0	1	-1	-1	3	-1	0	-1	1		(xz,yz,xy)
A_1u	1	1	1	1	1	-1	-1	-1	-1	-1
A_2u	1	1	-1	-1	1	-1	1	-1	-1	1
E_u	2	-1	0	0	2	-2	0	1	-2	0
T_1u	3	0	-1	1	-1	-3	-1	0	1	1	(x,y,z)
T_2u	3	0	1	-1	-1	-3	1	0	1	-1

From the character table, it is notable that complexes with octahedral structures have different symmetry for p and d orbitals while tetrahedral complexes have the same symmetry (as noted in red).

Td	E	8C₃	3C₂	6S₄	6σ_d
A₁	1	1	1	1	1		(z² + x²+ y²)
A₂	1	1	1	-1	-1	(R^x + R^y+ R^z)
E	2	-1	2	0	0		(2z² - x²-y₂, x² - y²)
T₁	3	0	-1	1	-1
T₂	3	0	-1	-1	1	(x,y,z)	(xy,yz,xz)

Spin Orbit Coupling

Spin forbidden transitions can occur if the complex undergoes spin-orbit coupling. This occurs when the angular momenta of the electrons and the spin angular momenta of the electrons interact with each other producing microstates with different value of quantum numbers. This splits the states into different energy values which would now make it allowable to excite to another state with a different spin multiplicity. However, the molar absorption coefficient is so small for these transitions that they produce weak bands for first row transition metals complexes. The effect is more notable in second and third row transition metal complexes.

Charge Transfer Spectra

Coordination complexes even without d electrons can also have vibrant colors due to charge transfer between ligands and a metal center. Charge Transfer absorption bands do not arise from $d\longrightarrow d$ transitions but instead from electrons that are excited from the metal to the ligand and vice versa.^[6]

Metal to Ligand Charge Transfer (MLCT)

A metal has its electrons excited to the ligand’s orbitals. The ligand is usually a $\pi$ acceptor in which a vacant $\pi$ * orbital receives an electron after it has been excited from the metal orbitals. The metal usually has a low oxidation state and common ligands are CO, CN^-, NO.

Ligand to Metal Charge Transfer (LMCT)

A ligand has its electrons excited to the metal orbitals. The ligand is usually a $\pi$ donor. This usually results in the reduction of the metal. Common Ligands are Cl, Br, I.

However, complexes such as Cr(CO)₆ which has a carbonyl as the ligand can undergo both MLCT and LMCT due to the fact that it has $\sigma$ donor and $\pi$ acceptor orbitals.

^ Oxtoby, D. , Gillis, H., Campion, A. (2012). Principles of Modern Chemistry. United States: Brooks/Cole, Cengage Learning. ISBN 9780840049315.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Russell, A., Henary, M. (1999). Chemistry Experiments for Physical Science and Engineering Majors. United States: Burgess International Group. pp. 48–50. ISBN 0808725165.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ ^a ^b ^c Miessler, G. , Tarr, D. (1999). Inorganic Chemistry. New Jersey: Prentice Hall Inc. ISBN 0138418918.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Mcquarrie, D. (2008). Quantum Chemistry. United States: University Science Books. ISBN 9781891389504.
^ Carter, Robert (1998). Molecular Symmetry and Group Theory. New York: John Wiley & Sons, Inc. pp. 248–256. ISBN 0471149551.
^ Pfenning, Brian (2015). Principles of Inorganic Chemistry. New Jersey: John Wiley & Sons Inc. ISBN 978-1-118-85910-0.

[1] Oxtoby, D. , Gillis, H., Campion, A. (2012). Principles of Modern Chemistry. United States: Brooks/Cole, Cengage Learning. ISBN 9780840049315.{{cite book}}: CS1 maint: multiple names: authors list (link)

[2] Russell, A., Henary, M. (1999). Chemistry Experiments for Physical Science and Engineering Majors. United States: Burgess International Group. pp. 48–50. ISBN 0808725165.{{cite book}}: CS1 maint: multiple names: authors list (link)

[:0-3] Miessler, G. , Tarr, D. (1999). Inorganic Chemistry. New Jersey: Prentice Hall Inc. ISBN 0138418918.{{cite book}}: CS1 maint: multiple names: authors list (link)

[4] Mcquarrie, D. (2008). Quantum Chemistry. United States: University Science Books. ISBN 9781891389504.

[5] Carter, Robert (1998). Molecular Symmetry and Group Theory. New York: John Wiley & Sons, Inc. pp. 248–256. ISBN 0471149551.

[6] Pfenning, Brian (2015). Principles of Inorganic Chemistry. New Jersey: John Wiley & Sons Inc. ISBN 978-1-118-85910-0.

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