Editing Magnetic circular dichroism (section)

== Theory ==
Consider a system of localized, non-interacting absorbing centers. Based on the semi-classical radiation absorption theory within the electric dipole approximation, the electric vector of the circularly polarized waves propagates along the +z direction. In this system, <math>\omega=2\pi\nu</math> is the [[angular frequency]], and <math>\tilde{n}</math> = n – ik is the [[complex refractive index]]. As the light travels, the attenuation of the beam is expressed as<ref name="steph"/>

:<math> I(z) = I(0) \exp(-2\omega kz/c) </math> 

where <math>I(z)</math> is the intensity of light at position <math>z</math>, <math>k</math> is the absorption coefficient of the medium in the <math>z</math> direction, and <math>c</math> is the speed of light. Circular dichroism (CD) is then defined by the difference between left (<math>-</math>) and right (<math>+</math>) circularly polarized light, <math>\Delta k = k_- - k_+</math>, following the sign convention of natural optical activity. In the presence of a static, uniform external magnetic field applied parallel to the direction of propagation of light,<ref name=buck/> the Hamiltonian for the absorbing center takes the form <math>\mathcal{H}(t) = \mathcal{H}_0 + \mathcal{H}_1(t)</math> for <math>\mathcal{H}_0</math> describing the system in the external magnetic field and <math>\mathcal{H}_1(t)</math> describing the applied electromagnetic radiation. The absorption coefficient for a transition between two eigenstates of <math>\mathcal{H}_0</math>, <math>a</math> and <math>j</math>, can be described using the electric dipole transition operator <math>m</math> as 

:<math>
[k_\pm (a \to j)] = \int_{0}^{\infty} k_\pm (a \to j) d \omega = \frac{\pi^2}{\hbar} (N_a - N_j) \left(\frac{\alpha^2}{n}\right) \left| \langle a | m_\pm | j \rangle \right|^2 
</math>
:<math>
[\Delta k (a \to j)] = \int_{0}^{\infty} \Delta k (a \to j) d \omega = \frac{\pi^2}{\hbar} (N_a - N_j) \left(\frac{\alpha^2}{n}\right) \left( \left| \langle a | m_- | j \rangle \right|^2  - \left| \langle a | m_+| j \rangle \right|^2 \right) 
</math>

The <math>(\alpha^2/n)</math> term is a frequency-independent correction factor allowing for the effect of the medium on the light wave electric field, composed of the permittivity <math>\alpha</math> and the real refractive index <math>n</math>.

=== Discrete line spectrum ===
In cases of a discrete spectrum, the observed <math>\Delta k</math> at a particular frequency <math>\omega</math> can be treated as a sum of contributions from each transition, 

:<math>\Delta k_\mathrm{obs}(\omega) = \sum_{a,j} \Delta k_{a\to j}(\omega) = \sum_{a,j} [\Delta k_{a\to j}]f_{ja}(\omega)</math> 

where <math>\Delta k_{a\to j}(\omega)</math> is the contribution at <math>\omega</math> from the <math>a\to j</math> transition, <math>[\Delta k_{a\to j}]</math> is the absorption coefficient for the <math>a\to j</math> transition, and <math>f_{ja}(\omega)</math> is a bandshape function (<math>\textstyle{\int_{0}^{\infty} f_{ja}(\omega) d \omega = 1}</math>). Because eigenstates <math>a</math> and <math>j</math> depend on the applied external field, the value of <math>\Delta k_\mathrm{obs}(\omega)</math> varies with field. It is frequently useful to compare this value to the absorption coefficient in the absence of an applied field, often denoted 

:<math>k^0(\omega) = \sum_{a,j} k^0_{a\to j}(\omega) = \sum_{a,j} [k^0_{a\to j}]f^0_{ja}(\omega)</math> 

When the [[Zeeman effect]] is small compared to zero-field state separations, line width, and <math>kT</math> and when the line shape is independent of the applied external field <math>H</math>, first-order perturbation theory can be applied to separate <math>\Delta k</math> into three contributing [[Faraday effect|Faraday]] terms, called <math>\mathcal{A}_1</math>, <math>\mathcal{B}_0</math>, and <math>\mathcal{C}_0</math>. The subscript indicates the moment such that <math>\mathcal{A}_1</math> contributes a derivative-shaped signal and <math>\mathcal{B}_0</math> and <math>\mathcal{C}_0</math> contribute regular absorptions. Additionally, a zero-field absorption term <math>\mathcal{D}_0</math> is defined. The relationships between <math>\Delta k</math>, <math>k^0</math>, and these Faraday terms are 
:<math>\Delta k_{A\to J}(\omega) = -\frac{4}{3} \gamma N^0_A \left\{\frac{\mathcal{A}_1(A\to J)}{\hbar} \frac{\partial f^0_{ja}(\omega)}{\partial \omega} + \left[\mathcal{B}_0(A\to J) + \frac{\mathcal{C}_0(A\to J)}{k_BT}\right]f^0_{ja}(\omega)\right\}H </math>
:<math>k^0_{A\to J}(\omega) = \frac{2}{3} \gamma N_A^0 \mathcal{D}_0(A\to J) f^0_{ja}(\omega)</math> 
for external field strength <math>H</math>, Boltzmann constant <math>k_B</math>, temperature <math>T</math>, and a proportionality constant <math>\gamma</math>. This expression requires assumptions that <math>j</math> is sufficiently high in energy that <math>N_j \approx 0</math>, and that the temperature of the sample is high enough that magnetic saturation does not produce nonlinear <math>\mathcal{C}</math> term behavior. 
Though one must pay attention to proportionality constants, there is a proportionality between <math>\Delta k</math> and [[Molar attenuation coefficient|molar extinction coefficient]] <math>\epsilon</math> and absorbance <math>A/Cl</math> for concentration <math>C</math> and path length <math>l</math>. 

These Faraday terms are the usual language in which MCD spectra are discussed. Their definitions from perturbation theory are<ref name=steph2>{{cite book|author=Stephens, P. J.|title=Advances in Chemical Physics |journal=Adv. Chem. Phys.|date=1976|volume=35|pages=197–264|doi=10.1002/9780470142547.ch4|chapter=Magnetic Circular Dichroism|isbn=9780470142547}}</ref> 

:<math>\begin{align}
  \mathcal{A}_1 &= -\frac{1}{d_A} \sum_{\alpha,\lambda} \left( \langle J_\lambda |L_z+2S_z| J_\lambda\rangle - \langle A_\alpha |L_z+2S_z| A_\alpha\rangle\right) \times \left(|\langle A_\alpha|m_-|J_\lambda\rangle|^2 - \langle A_\alpha|m_+|J_\lambda\rangle|^2\right) \\
  \mathcal{B}_0 &= \frac{2}{d_A} \Re \sum_{\alpha,\lambda}\left[ \sum_{K\neq J,\kappa} \frac{1}{E_K-E_J} \langle J_\lambda |L_z+2S_z| K_\kappa\rangle \times \left(\langle A_\alpha|m_-|J_\lambda\rangle\langle K_\kappa|m_+|A_\alpha\rangle - \langle A_\alpha|m_+|J_\lambda\rangle\langle K_\kappa|m_-|A_\alpha\rangle\right) \right. \\
 &\qquad \left. + \sum_{K\neq A,\kappa} \frac{1}{E_K-E_A} \langle K_\kappa |L_z+2S_z| A_\alpha\rangle \times \left( \langle A_\alpha|m_-|J_\lambda\rangle\langle J_\lambda|m_+|K_\kappa\rangle - \langle A_\alpha|m_+|J_\lambda\rangle\langle J_\lambda|m_-|K_\kappa\rangle \right) \right] \\
  \mathcal{C}_0 &= \frac{1}{d_A} \sum_{\alpha,\lambda} \langle A_\alpha|L_z+2S_z|A_\alpha\rangle \times \left(|\langle A_\alpha|m_-|J_\lambda\rangle|^2 - \langle A_\alpha|m_+|J_\lambda\rangle|^2\right) \\
  \mathcal{D}_0 &= \frac{1}{2d_A} \sum_{\alpha,\lambda} \left(|\langle A_\alpha|m_-|J_\lambda\rangle|^2 + \langle A_\alpha|m_+|J_\lambda\rangle|^2\right) 
\end{align}</math> 
where <math>d_A</math> is the degeneracy of ground state <math>A</math>, <math>K</math> labels states other than <math>A</math> or <math>J</math>, <math>\alpha</math> and <math>\lambda</math> and <math>\kappa</math> label the levels within states <math>A</math> and <math>J</math> and <math>K</math> (respectively), <math>E_X</math> is the energy of unperturbed state <math>X</math>, <math>L_z</math> is the <math>z</math> angular momentum operator, <math>S_z</math> is the <math>z</math> spin operator, and <math>\Re</math> indicates the real part of the expression. 

===Origins of A, B, and C Faraday Terms===

[[File:MCD-ABC-terms.svg|thumb|upright=2|<math>\mathcal{A}_1</math>, <math>\mathcal{B}_0</math>, and <math>\mathcal{C}_0</math> term intensity mechanisms for magnetic circular dichroism (MCD) signal]]

The equations in the previous subsection reveal that the <math>\mathcal{A}_1</math>, <math>\mathcal{B}_0</math>, and <math>\mathcal{C}_0</math> terms originate through three distinct mechanisms. 

The <math>\mathcal{A}_1</math> term arises from Zeeman splitting of the ground or excited degenerate states. These field-dependent changes in energies of the magnetic sublevels causes small shifts in the bands to higher/lower energy. The slight offsets result in incomplete cancellation of the positive and negative features, giving a net derivative shape in the spectrum. This intensity mechanism is generally independent of sample temperature.

The <math>\mathcal{B}_0</math> term is due to the field-induced mixing of states. Energetic proximity of a third state <math>|K\rangle</math> to either the ground state <math>|A\rangle</math> or excited state <math>|J\rangle</math> gives appreciable [[Zeeman effect|Zeeman coupling]] in the presence of an applied external field. As the strength of the magnetic field increases, the amount of mixing increases to give growth of an absorption band shape. Like the <math>\mathcal{A}_1</math> term, the <math>\mathcal{B}_0</math> term is generally temperature independent. Temperature dependence of <math>\mathcal{B}_0</math> term intensity can sometimes be observed when <math>|K\rangle</math> is particularly low-lying in energy. 

The <math>\mathcal{C}_0</math> term requires the degeneracy of the ground state, often encountered for paramagnetic samples. This happens due to a change in the [[Boltzmann distribution|Boltzmann population]] of the magnetic sublevels, which is dependent on the degree of field-induced splitting of the sublevel energies and on the sample temperature.<ref>{{cite journal|author1=Lehnert, N. |author2=DeBeer George, S. |author3=Solomon, E. I. |author-link3=Edward I. Solomon |journal=Current Opinion in Chemical Biology|date=2001|volume=5|pages=176–187|doi=10.1016/S1367-5931(00)00188-5|pmid=11282345|title=Recent advances in bioinorganic spectroscopy|issue=2}}</ref> Decrease of the temperature and increase of the magnetic field increases the <math>\mathcal{C}_0</math> term intensity until it reaches the maximum (saturation limit). Experimentally, the <math>\mathcal{C}_0</math> term spectrum can be obtained from MCD raw data by subtraction of MCD spectra measured in the same applied magnetic field at different temperatures, while <math>\mathcal{A}_1</math> and <math>\mathcal{B}_0</math> terms can be distinguished via their different band shapes.<ref name=solomon/>

The relative contributions of A, B and C terms to the MCD spectrum are proportional to the inverse line width, energy splitting, and temperature:
:<math>A:B:C = \frac{1} {\Delta \Gamma} : \frac {1} {\Delta E} : \frac{1}{kT}</math>
where <math>\Delta \Gamma</math> is line width and <math>\Delta E</math> is the zero-field state separation. For typical values of <math>\Delta \Gamma</math> = 1000&nbsp;cm<sup>−1</sup>, <math>\Delta E</math> = 10,000&nbsp;cm<sup>−1</sup> and <math>kT</math> = 6&nbsp;cm<sup>−1</sup> (at 10&nbsp;K), the three terms make relative contributions 1:0.1:150. So, at low temperature the <math>\mathcal{C}_0</math> term dominates over <math>\mathcal{A}_1</math> and <math>\mathcal{B}_0</math> for paramagnetic samples.<ref>{{cite journal|author1=Neese, F. |author2=Solomon, E. I. |author-link2=Edward I. Solomon |journal=Inorg. Chem.|date=1999|volume=38|pages=1847–1865|doi=10.1021/ic981264d|pmid=11670957|title=MCD C-Term Signs, Saturation Behavior, and Determination of Band Polarizations in Randomly Oriented Systems with Spin S >/= (1)/(2). Applications to S = (1)/(2) and S = (5)/(2)|issue=8}}</ref>