Editing Magnetic circular dichroism (section)

=== 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.