Associated Difference SpectrumΒΆ

Suppose that there are \(k\) excited species, which contrbute difference absorption spectrum. Then transient difference absorption spectrum \(\Delta A(E, t)\) is represented by

\[\begin{equation*} \Delta A(E, t) = \sum_{i=1}^k c_i(E) y_i(t) \end{equation*}\]

, where \(c_i(E)\) and \(y_i(t)\) is difference absorption coefficient and population of \(i\)th excited species, respectively.

Suppose that one measure difference absorption spectrum with energy point \(E_1,\dotsc, E_n\) and time point \(t_1, \dotsc, t_m\).

Denote

\[\begin{align*} \Delta A &= \left[A(E_i, t_j)\right]_{i,j} \\ C &= \left[c_i(E_j)\right]_{i,j} \\ Y &= \left[y_i(t_j)\right]_{i,j} \end{align*}\]

Then \(\Delta A\), \(C\) and \(Y\) is \(n \times m\), \(k \times n\) and \(k \times m\) matrix, respectively.

Moreover

\[\begin{equation*} \Delta A = C^T Y \end{equation*}\]

Thus if one knows population matrix of excited species \(Y\) and want to deduced associated difference spectrum matrix \(C\), one should solve above equation. If \(m<k\) such equation is under determined, so to determine associated difference spectrum matrix \(C\) one need to measure \(m \geq k\) time points. However every experimental spectrum has unsystematic error, to correct such kind of error, one should measure at least \(k+1\) time points.

Suppose that one have measured \(m \geq k+1\) time points with error \(\mathrm{Err}\). Then such system is overdetermined, so our problem is changed to find best associated difference spectrum matrix \(C\) which minimizes

\[\begin{equation*} \chi^2 = \left \| \frac{\Delta A-C^T Y}{\mathrm{Err}} \right\|^2 \end{equation*}\]

To find such best \(C\), fix energy point \(E_i\) and denote scaled population matrix \(Y' = Y/\mathrm{Err}(E_i)\) and scaled transient absorption spectrum matrix \(\Delta A'(E_i) = \Delta A(E_i)/\mathrm{Err}(E_i)\). Then by the orthogonal projection theorem, the best coefficient \(C(E_i)\) at energy point \(E_i\) given as

\[\begin{equation*} C(E_i) = (Y'Y'^T)^{-1}Y'\left(\Delta A'(E_i)\right)^T \end{equation*}\]

The inverse of \(Y'Y'^T\) is called covarient matrix \(\mathrm{Cov} = (Y'Y'^T)^{-1}\). The error \(\sigma(E_i)\) of estimated associated difference spectrum matrix \(C(E_i)\) at energy point \(E_i\) is estimated to

\[\begin{equation*} \sigma(E_i) = \sqrt{\frac{\chi^2}{m-k}}\sqrt{\mathrm{diag}\left(Cov\right)} \end{equation*}\]