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Magnetically induced circular dichroism (MCD) is the differential absorption of left and right circularly polarized light, induced in a sample by a strong magnetic field oriented parallel to the direction of light propagation. In general, this means wrapping a circular dichrometer in a large electromagnet. If the magnet is water cooled, field strength of a couple of teslas are common. With a superconducting magnet, field strengths can reach 7 Teslas.

It was first shown by Faraday that optical activity could be induced in matter by a longitude magnetic field^[1]. The development of MCD really began in the 1930s when a quantum mechanical theory of MOR (magnetic optical rotatory dispersion) in regions outside absorption bands was formulated. The expansion of the theory to include MCD and MOR effects in the region of absorptions, which were referred to as “anomalous dispersions” was developed soon thereafter. There was, however, little effort made to refine MCD as a modern spectroscopic technique until the early 1960s. Since that time there have been numerous studies of MCD spectra for a very large variety of samples, including stable molecules in solutions, in isotropic solids, and in the gas phase, as well as unstable molecules entrapped in noble gas matrices. More recently, MCD has found useful application in the study of biologically important systems including metalloenzymes and proteins containing metal centers.^[2] ^[3]

The main focus of the MCD is to analyze the A, B, C terms of the spectra. Principles and examples will be explained later on this page.

Applicable Cases

MCD can be used as an optical technique for the detection of electronic structure of both the ground states and excited states. It is also a strong addition to the more commonly used absorption spectroscopy, and there are two reasons that explain this. First, a transition buried under a stronger transition can appear in MCD if the first derivative of the absorption is much larger for the weaker transition or it is of the opposite sign. Second, MCD will be found where no absorption is detected at all if ΔA > (ΔA_min) but A<A_min, where (ΔA)_min and A_min are the minimum of ΔA and A that are detectable. Typically, (ΔA_min) and A_min are of the magnitudes around 10^-5 and 10^-3 respectively. So, a transition can only be detected in MCD, not in the absorption spectroscopy, if ΔA/A > 10^-2. This happens in paramagnetic systems that are at lower temperature or that have sharp lines in the spectroscopy.^[4]

Differences between CD and MCD

In natural optical activity, the difference between the lcp light and the rcp light is caused by the dissymmetry of the molecules. Due to the handedness of the molecule, the absorption of the lcp light would be different from the rcp light. However, in MCD in the presence of a magnetic field, lcp and rcp no longer interact equivalently with the absorbing medium. Thus there is not the same direct relation between magnetic optical activity and molecular stereochemistry which would be expected, because it is found in natural optical activity. So, natural CD is a relatively rare phenomenon, while MCD is a property of all matter.^[5]

Spectrometer

Modern day spectrometers can achieve simultaneous measurement of absorbance and the MCD ΔA along the same light path.^[6] This eliminates error introduced through multiple measurements or different instruments that previously occurred before this advent. A MCD spectrometer begins with a light source that emits a monochromatic wave of light. This wave is passed through a Rochon prism linear polarizer, which separates the incident wave into two beams that are linearly polarized by 90 degrees. The two beams split into different paths- one beam (the extraordinary beam) traveling directly to a photomultiplier (PMT), and the other beam (the ordinary beam) passing through a photoelastic modulator (PEM). The PMT for the extraordinary beam detects the light intensity of the input beam, while the PEM causes a 1/4 wavelength shift that converts linearly polarized light into circularly polarized light. Linearly polarized light has two circular components with intensity represented by: File:Formula for intensity.png

The PEM will delay one component of linearly polarized light with a time dependence that advances the other component by 1/4 λ (hence, quarter-wave shift). The departing circularly polarized light oscillates between rcp and lcp in a sinusoidal time-dependence as depicted below:

The light finally travels through a magnet containing the sample, and the transmittance is recorded by another PMT. The schematic is given below:

The intensity of light from the ordinary wave that reaches the PMT is governed by the equation: File:Formula for the difference of the intensity.png

A_- and A₊ are the absorbances of lcp or rcp, respectively; ω is the modulator frequency- usually a high acoustic frequency such as 50 kHz; t is time; and δ₀ is the time dependent wavelength shift.

This intensity of light passing through the sample is converted into a two component voltage via a current/voltage amplifier. A DC voltage will emerge corresponding to the intensity of light passed through the sample. If there is a ΔA, then a small AC voltage with be present that corresponds to the modulation frequency, ω. From such voltage, ΔA and A can be derived using the following relations: File:The measument of det A.png

File:The measument of A.png

where V_ex is the voltage measured by the PMT from the extraordinary wave.

File:MCD actual instrument.jpg

Photograph given by permission of a low-temperature MCD Spectrometer Built at the University of Michigan.

MCD signals tend to be small, so the measurements often take time. To get a reasonable signal to noise ratio, the experiments can often take a few hours. Multiple spectra are often taken, digitized, accumulated and stored via computer. Although there is much overlap in the requirements and use of instruments, ordinary CD instruments are usually optimized for operation in the ultraviolet, approximately 170–300 nm, while MCD instruments are typically required to operate in the visible to near infrared, approximately 300–2000 nm.

The physical processes that lead to MCD are substantively different from those of CD. However, like CD, it is dependent on the differential absorption of left and right hand circularly polarized light. MCD will only exist at a given wavelength if the molecule has an optical absorption at that wavelength. This is distinctly different from the related phenomenon of Optical Rotatory Dispersion (ORD), which can be observed at wavelengths far from any absorption band.

Some superconducting magnets have a small sample chamber, far too small to contain the entire optical system. Instead, the magnet sample chamber has windows on two opposite sides. Light from the source enters one side, interacts with the sample (usually also temperature controlled) in the magnetic field, and exits through the opposite window to the detector. Optical relay systems that allow the source and detector each to be about a meter from the sample are typically employed. This arrangement avoids many of the difficulties that would be encountered if the optical apparatus had to operate in the high magnetic field, and also allows for a much less expensive magnet.

How it works

Basically, we consider the systems are of localized, non-interacting absorbing centers. Based on the semi-classical radiation absorption theory within the electric dipole approximation, the electric vector of the Circular Polarized waves propagates in the + z direction, where + and -denote to rcp light and lcp light respectively.

In this system, ω = 2nν is the circular frequency, and I_i = n-ik is the complex refractive index. As the light travels, the attenuation of the beam is expressed by

File:Intensity as the light travels.png

in which k means the absorption coefficient of the medium in the z direction Circular Dichroism (CD) is then defined through

File:Different absorpsion in terms of k.png

And it follows the sign convention of natural optical activity. In the presence of the static, uniform external magnetic field, The Hamiltonian for the absorbing center is of the form

File:Hamiltonian of the state in MCD.png

which is parallel to the direction of the propagation of the electric field k.^[7] Due to the transition between the two eigenstates of H₀, a and j, the absorption and the CD intensity would be determined as follows:

File:Intensity in terms of k.png

Discrete line spectrum

In a certain frequency range, if the transitions of the absorbing centers are relatively sparse, the absorption spectrum in this region would be of a series of lines. Each line would correspond to a single transition and be either completely or partially resolved from neighboring lines. Since it is of the discrete form, then the absorption coefficient and CD of the line are given by

When the Zeeman Effect is small compared to zero-field state separations, line width, and kT and when the line shape is independent of H, it could be obtain to first order in H for the expression of Δk, assuming J to be sufficiently high in energy that N_j = 0. And it is given by:

Δk is the field-dependent difference between lcp and rcp absorption, α is the electric permeability, n is the index of refraction, and H is the applied magnetic field, k is the Boltzmann constant and T is the temperature. Further, f is a band shape function and f’ is the correspondent first derivative. D is the dipole strength parameter, and it is defined as^[8] File:Definition of D term.png

There are various form of the formulas above. A more common form is in terms of A (absorption), which would be:

The constants A, B, and C are characteristic parameters specific to a given molecule and to a particular transition, which are referred to as A, B, and C terms, respectively, a nomenclature introduced by Stephens and dating back to Serber's early theory of MOR.

A term arises due to Zeeman splitting of the ground or excited degenerate states, like what is shown in the (2) in Figure 1 The small Zeeman splitting would make the oppositely signed transitions rcp light and lcp light almost cancel out with each other, which leads to the derivative shape of the band. B term is due to the field-induced mixing of states. As shown in (3) of the Figure 1, in order for there to be an appearance of B term, there must be another state K, which is closely related in energy to either the ground state or the excited states. Also, the energy between the ground state and the excited state should be high enough so that the excited state is not highly populated. Ususally, B term would have the absorption band shape. Otherwise, it will lead to the C term (4) of Figure 1, which requires the degeneracy of the ground state. This happens due to a change in the population of molecules over the Zeeman sublevels of a paramagnetic ground state. In addition, the C term is observed only for molecules with ground state paramagnetism. Importantly, A and B terms are independent of temperature while the C term is dependent on temperature. Decrease of the temperature and increase of the magnetic field would further increase the C term intensity until it reaches the maximum(saturation limit). Experimentally, the C term spectrum can be obtained from MCD raw data by substraction of MCD spectra measured in the same applied magnetic field at different temperatures, while A and B terms can be distinguished via their different band shapes. ^[9] The relative contributions of A, B and C terms to the MCD spectrum are proportional to the inverse line width, energy splitting, and temperature:

in which, ΔΓ is line width and ΔE is zero-field state separation magnitudes. Typically, the values (ΔГ= 1000 cm^-1, ΔE = 10,000 cm^-1 , and kT = 6 cm^-1 (at 10 K), the three terms will make relative contributions 1:0.1:150. So, at low temperature the C term will dominate over A and B term.^[10]

Examples about C terms

In the visible and near-ultraviolet regions, the hexacyanoferrate(III) ion exhibits three strong transitions, which have been ascribed to Ligand to Metal Charge Transfer (LMCT) transitions. The three intense bands are located at 24500, 32700, and 40500 cm^-1 in the spectra of the Fe (III) complex Fe(CN)₆^3-, which are all found lower in energy than the lowest energy intense band for the Fe(II) complex Fe(CN)₆^2- found at 46000 cm^-1.^[11] The red shift as the oxidation state of the metal increases is characteristic of LMCT bands.

File:Example C term figure 3.png

The dependence of the C term of MCD is explored here. The ground state of the anion is ²T_2g, which derives from the electronic configuration (t_2g)⁵. So, there would be an unpaired electron in the d orbital of Fe³⁺ From that, we can assign the three bands correspondent to the transitions ²t_2g→²t_1u¹, ²t_2g →²t_1u², ²t_2g →²t_2u. Noticed that two of the excited states are of the same symmetry, and based on the group theory, we can assert that they would actually mix with each other so that there are no pure σ, π character of the two t_1u states, but for t_2u, there would be no inter-mixing. It should be noted that A terms are also possible from the degenerate excited states, but the temperature-dependence studies showed that they are not as dependent as the C term.^[12]

File:Example C term figure 4(2).png

A more recent study was done on the MCD of Fe(CN)₆^3- embedded in a thin poly(vinyl alcohol) (PVA) film. We can clearly see that there is a dependence on temperature by the C term. With the equations that were given in the theory part, we can get the appropriate expression for C₀ and D₀. The observed values of C₀/D₀ at room temperature for the three bands in the Fe(CN)₆^3- spectrum are 1.2, -0.6, and 0.6 (with some experimental uncertainty in the last two values) respectively. Thus the signs we got from the experiment for the three C terms in the MCD spectra (positive, negative, and positive) establish the energy ordering as: ²t_2g→²t_1u²<²t_2g→²t_2u<²t_2g→²t_1u¹

Examples about A and B terms

Molecules possessing degenerate excited states will contain an A term if an improper or proper rotation axis greater than three exists. The B term will be evident for all molecules containing excited states that can mix. Diamagnetic samples with nondegenerate ground states with not have a C term. The A term consists of the transitions from the ground state to the orbitally degenerate states, which are split in the presence of the magnetic field according to the Zeeman effect. The corresponding B term for these transitions is proportional to the mixing/coupling between excited states. One case exemplifying MCD A and B terms is a square planar, d⁸ complex such as [(n-C₄H₉)₄N]₂Pt(CN)₄.^[13] The effects of spin-orbit coupling, negative terms, and pseudo terms are also demonstrated. File:Example for A and B term.png File:Example for A and B term MO diagram.png File:Example for A and B term spectormeter.png

From the molecular orbital diagram, the ground state is diamagnetic and the LUMO is the b_1g. The dipole allowed MLCT transitions are the a_1g, a_2u and the e_g, a_2u, resulting in both singlets and triplets of both A_2u and E_u states, respectively. Another transition, b_2u, a_2u yields weak and orbitally forbidden singlet and triplet B_1u states. Mixing of all these singlet and triplet states will occur and is attributed to the spin orbit coupling of platinum’s 5d orbitals (ϛ ~ 3500cm^-1). ¹A_2u will mix with ³E_u and give two A_2u states. ¹E_u, ³E_u, ³A_2u, and ³B_1u will also mix to give four E_u states. Diagonalization of spin-orbit secular determinants of these states resulted in close agreement with the experimental spectrum. The A term at 35400 cm^-1 is an E_u state. A positive A term and pseudo positive A term arise at 38400 cm^-1. The positive A term is the spin orbit E_u state, while the mixture of oppositely signed B terms of E_u and A_2u spin orbit states make a pseudo A term. The same situation happens again at 45500 cm^-1. The two B_1u states are assigned to the shoulder bands at ~ 42000cm^-1 and ~47000cm^-1. From these results the relative order of the d-orbitals are extrapolated as a_1g (d_z²) > e_g(d_xy,d_yz) > b_2g (d_xy).

Other fields

In biology, metalloproteins are the most likely candidates for MCD measurements, as the presence of metals with degenerate energy levels leads to strong MCD signals. In the case of ferric hemoproteins, MCD is capable of determining both oxidation and spin state to a remarkably exquisite degree. In regular proteins, MCD is capable of stoichiometrically measuring the tryptophan content of proteins, assuming there are no other competing absorbers in the spectroscopic system.

References

^ A. D. Buckingham and P. J. Stephens, Ann. Rev. Phys. Chem. 17,399 (1966)
^ W. Roy Mason, Practical Guide to Magnetic Circular Dichroism Spectroscopy, A, 2007 John Wiley & Sons.
^ P. N. Schatz, A. J. McCaffery
^ P. J. Stephens, Annu. Rev. Phys. Chem. 1974.25:201-232
^ Dennis Caldwell, Annu. Rev. Phys. Chem. 1971.22:259-278
^ G. A. Osborne, The Review of Scientific Instruments, Vol.44, 1973
^ Buckingham,Ann. Rev. Phys. Cher. 17: 399
^ P. J. Stephens, Annu. Rev. Phys. Chem. 1974.25:201-232
^ Solomon, E. I.; Pavel, E. G.; Loeb, K. E.; Campochiaro, C. Coord. Chem. Rev. 1995, 144, 369-460
^ Neese, F.; Solomon, E. I. Inorg. Chem. 1999, 38, 1847-1865.
^ Stephens, P. J. Inorg. Chem. 1965,4, 1690-1692
^ Upton,A. H. P.; Williamson, B. E., J. Phys. Chem., 1994, 98, 71
^ Isci, H.; Mason, W. R. Inorg. Chem. 1975, 14, 905

@@ Line 1: / Line 1: @@
 '''Magnetically induced circular dichroism (MCD)''' is the differential absorption of left and right [[circular polarization|circularly polarized]] light, induced in a sample by a strong [[magnetic field]] oriented parallel to the direction of light propagation. In general, this means wrapping a circular dichrometer in a large [[electromagnet]]. If the magnet is water cooled, field strength of a couple of teslas are common. With a [[superconducting magnet]], field strengths can reach 7 Teslas.
-It was first shown by [[Faraday#Diamagnetism|Faraday]] that optical activity could be induced in matter by a longitude magnetic field<ref>A.  D.  Buckingham  and  P.  J.  Stephens,  Ann.  Rev.  Phys. Chem.  17,399  (1966)</ref>. The development of MCD really began in the 1930s when a [[quantum mechanical]] theory of MOR (magnetic optical rotatory dispersion) in regions outside absorption bands was formulated. The expansion of the theory to include MCD and MOR effects in the region of absorptions, which were referred to as “anomalous dispersions” was developed soon thereafter. There was, however, little effort made to refine MCD as a modern spectroscopic technique until the early 1960s. Since that time there have been numerous studies of MCD spectra for a very large variety of samples, including stable molecules in solutions, in isotropic solids, and in the gas phase, as well as unstable molecules entrapped in noble gas matrices. More recently, MCD has found useful application in the study of biologically important systems including metalloenzymes and proteins containing metal centers.<ref>W. Roy Mason, Practical Guide to Magnetic Circular Dichroism Spectroscopy, A,  Copyright © 2007 John Wiley & Sons, Inc.</ref> <ref>P. N. Schatz, A. J. McCaffery, University of Virginia</ref>
+It was first shown by [[Faraday#Diamagnetism|Faraday]] that optical activity could be induced in matter by a longitude magnetic field<ref>A.  D.  Buckingham  and  P.  J.  Stephens,  Ann.  Rev.  Phys. Chem.  17,399  (1966)</ref>. The development of MCD really began in the 1930s when a [[quantum mechanical]] theory of MOR (magnetic optical rotatory dispersion) in regions outside absorption bands was formulated. The expansion of the theory to include MCD and MOR effects in the region of absorptions, which were referred to as “anomalous dispersions” was developed soon thereafter. There was, however, little effort made to refine MCD as a modern spectroscopic technique until the early 1960s. Since that time there have been numerous studies of MCD spectra for a very large variety of samples, including stable molecules in solutions, in isotropic solids, and in the gas phase, as well as unstable molecules entrapped in noble gas matrices. More recently, MCD has found useful application in the study of biologically important systems including metalloenzymes and proteins containing metal centers.<ref>W. Roy Mason, Practical Guide to Magnetic Circular Dichroism Spectroscopy, A, 2007 John Wiley & Sons.</ref> <ref>P. N. Schatz, A. J. McCaffery </ref>
 The main focus of the MCD is to analyze the A, B, C terms of the spectra. Principles and examples will be explained later on this page.
@@ Line 92: / Line 92: @@
 in which, ΔΓ is line width and ΔE is zero-field state separation magnitudes.
 Typically, the values (ΔГ= 1000 cm<sup>-1</sup>, ΔE = 10,000 cm<sup>-1</sup> , and kT = 6 cm<sup>-1</sup> (at 10 K), the three terms will make relative contributions 1:0.1:150. So, at low temperature the C term will dominate over A and B term.<ref>Neese, F.; Solomon, E. I. Inorg. Chem. 1999, 38, 1847-1865.</ref>
-==Examples about A and B terms==
-Molecules possessing degenerate excited states will contain an A term if an improper or proper rotation axis greater than three exists. The B term will be evident for all molecules containing excited states that can mix. Diamagnetic samples with nondegenerate ground states with not have a C term. The A term consists of the transitions from the ground state to the orbitally degenerate states, which are split in the presence of the magnetic field according to the Zeeman effect. The corresponding B term for these transitions is proportional to the mixing/coupling between excited states.
-	One case exemplifying MCD A and B terms is a square planar, d<sup>8</sup> complex such as  [(n-C<sub>4</sub>H<sub>9</sub>)<sub>4</sub>N]<sub>2</sub>Pt(CN)<sub>4</sub>.<ref>Isci, H.; Mason, W. R. Inorg. Chem. 1975, 14, 905</ref> The effects of spin-orbit coupling, negative terms, and pseudo terms are also demonstrated.
-[[File:Example_for_A_and_B_term.png]]
-[[File:Example_for_A_and_B_term_MO_diagram.png]]
-[[File:Example_for_A_and_B_term_spectormeter.png]]
-From the molecular orbital diagram, the ground state is diamagnetic and the LUMO is the b<sub>1g</sub>. The dipole allowed MLCT transitions are the a<sub>1g</sub>, a<sub>2u</sub> and the e<sub>g</sub>, a<sub>2u</sub>, resulting in both singlets and triplets of both A<sub>2u</sub>  and E<sub>u</sub> states, respectively. Another transition, b<sub>2u</sub>, a<sub>2u</sub> yields weak and orbitally forbidden singlet and triplet B<sub>1u</sub> states. Mixing of all these singlet and triplet states will occur and is attributed to the spin orbit coupling of platinum’s 5d orbitals (ϛ ~ 3500cm<sup>-1</sup>). <sup>1</sup>A<sub>2u</sub> will mix with <sup>3</sup>E<sub>u</sub> and give two A<sub>2u</sub> states. <sup>1</sup>E<sub>u</sub>, <sup>3</sup>E<sub>u</sub>, <sup>3</sup>A<sub>2u</sub>, and <sup>3</sup>B<sub>1u</sub> will also mix to give four E<sub>u</sub> states. Diagonalization of spin-orbit secular determinants of these states resulted in close agreement with the experimental spectrum.  The A term at 35400 cm<sup>-1</sup> is an E<sub>u</sub> state. A positive A term and pseudo positive A term arise at 38400 cm<sup>-1</sup>. The positive A term is the spin orbit E<sub>u</sub> state, while the mixture of oppositely signed B terms of E<sub>u</sub> and A<sub>2u</sub> spin orbit states make a pseudo A term. The same situation happens again at 45500 cm<sup>-1</sup>. The two B<sub>1u</sub> states are assigned to the shoulder bands at ~ 42000cm<sup>-1</sup> and ~47000cm<sup>-1</sup>.
-	From these results the relative order of the d-orbitals are extrapolated as a<sub>1g</sub> (d<sub>z<sup>2</sup></sub>) > e<sub>g</sub>(d<sub>xy</sub>,d<sub>yz</sub>) > b<sub>2g</sub> (d<sub>xy</sub>).
 ==Examples about C terms==
@@ Line 116: / Line 105: @@
 A more recent study was done on the MCD of Fe(CN)<sub>6</sub><sup>3-</sup> embedded in a thin poly(vinyl alcohol) (PVA) film. We can clearly see that there is a dependence on temperature by the C term. With the equations that were given in the theory part, we can get the appropriate expression for C<sub>0</sub> and D<sub>0</sub>. The observed values of C<sub>0</sub>/D<sub>0</sub> at room temperature for the three bands in the Fe(CN)<sub>6</sub><sup>3-</sup> spectrum are 1.2, -0.6, and 0.6 (with some experimental uncertainty in the last two values) respectively. Thus the signs we got from the experiment for the three C terms in the MCD spectra (positive, negative, and positive) establish the energy ordering as: <sup>2</sup>t<sub>2g</sub>→<sup>2</sup>t<sub>1u</sub><sup>2</sup><<sup>2</sup>t<sub>2g</sub>→<sup>2</sup>t<sub>2u</sub><<sup>2</sup>t<sub>2g</sub>→<sup>2</sup>t<sub>1u</sub><sup>1</sup>
+==Examples about A and B terms==
+Molecules possessing degenerate excited states will contain an A term if an improper or proper rotation axis greater than three exists. The B term will be evident for all molecules containing excited states that can mix. Diamagnetic samples with nondegenerate ground states with not have a C term. The A term consists of the transitions from the ground state to the orbitally degenerate states, which are split in the presence of the magnetic field according to the Zeeman effect. The corresponding B term for these transitions is proportional to the mixing/coupling between excited states.
+	One case exemplifying MCD A and B terms is a square planar, d<sup>8</sup> complex such as  [(n-C<sub>4</sub>H<sub>9</sub>)<sub>4</sub>N]<sub>2</sub>Pt(CN)<sub>4</sub>.<ref>Isci, H.; Mason, W. R. Inorg. Chem. 1975, 14, 905</ref> The effects of spin-orbit coupling, negative terms, and pseudo terms are also demonstrated.
+[[File:Example_for_A_and_B_term.png]]
+[[File:Example_for_A_and_B_term_MO_diagram.png]]
+[[File:Example_for_A_and_B_term_spectormeter.png]]
+From the molecular orbital diagram, the ground state is diamagnetic and the LUMO is the b<sub>1g</sub>. The dipole allowed MLCT transitions are the a<sub>1g</sub>, a<sub>2u</sub> and the e<sub>g</sub>, a<sub>2u</sub>, resulting in both singlets and triplets of both A<sub>2u</sub>  and E<sub>u</sub> states, respectively. Another transition, b<sub>2u</sub>, a<sub>2u</sub> yields weak and orbitally forbidden singlet and triplet B<sub>1u</sub> states. Mixing of all these singlet and triplet states will occur and is attributed to the spin orbit coupling of platinum’s 5d orbitals (ϛ ~ 3500cm<sup>-1</sup>). <sup>1</sup>A<sub>2u</sub> will mix with <sup>3</sup>E<sub>u</sub> and give two A<sub>2u</sub> states. <sup>1</sup>E<sub>u</sub>, <sup>3</sup>E<sub>u</sub>, <sup>3</sup>A<sub>2u</sub>, and <sup>3</sup>B<sub>1u</sub> will also mix to give four E<sub>u</sub> states. Diagonalization of spin-orbit secular determinants of these states resulted in close agreement with the experimental spectrum.  The A term at 35400 cm<sup>-1</sup> is an E<sub>u</sub> state. A positive A term and pseudo positive A term arise at 38400 cm<sup>-1</sup>. The positive A term is the spin orbit E<sub>u</sub> state, while the mixture of oppositely signed B terms of E<sub>u</sub> and A<sub>2u</sub> spin orbit states make a pseudo A term. The same situation happens again at 45500 cm<sup>-1</sup>. The two B<sub>1u</sub> states are assigned to the shoulder bands at ~ 42000cm<sup>-1</sup> and ~47000cm<sup>-1</sup>.
+	From these results the relative order of the d-orbitals are extrapolated as a<sub>1g</sub> (d<sub>z<sup>2</sup></sub>) > e<sub>g</sub>(d<sub>xy</sub>,d<sub>yz</sub>) > b<sub>2g</sub> (d<sub>xy</sub>).
 ==Other fields==