Possibly the simplest case of a time-dependent interaction is the Zeeman interaction with the magnetic component of a polarized monochromatic radiofrequency radiation field. The periodic nature of such an oscillating magnetic field may be exploited in order to simplify computations, and such methods are considered in the sections below. One may also introduce e.g. pulse sequences or random broadband radiation but in either case the nice periodicity property is lost and one may need to rely on numerical integration. In addition to the methods described here, note that a method for treating radiofrequency radiation, based on Floquet theory, was developed recently.
The Hamiltonian
Interactions between a spin system and radiofrequency radiation are just special cases of the Zeeman interaction with a time-dependent magnetic field:
\begin{equation} \require{newcommand} \newcommand{\vc}[1]{\mathbf{#1}} \newcommand{\desc}[1]{\begin{quote}\emph{#1}\end{quote}} \newcommand{\p}{\partial} \newcommand{\bra}[1]{\langle #1 |\,} \newcommand{\ket}[1]{\,| #1 \rangle} \newcommand{\braket}[2]{\langle #1 | #2 \rangle} \newcommand{\matel}[3]{\langle #1 | #2 | #3 \rangle} \newcommand{\mexp}[1]{\langle #1 \rangle} \newcommand{\kb}[2]{| #1 \rangle\langle #2 |} \DeclareMathOperator{\Tr}{Tr} \vc{H} = g\mu_B \vc{S} \cdot \vc{B}(t), \end{equation}where typical forms of the magnetic field are:
\begin{equation} \vc{B}_1(t) = B_0 \begin{pmatrix} \cos(\omega t) \\ \sin(\omega t) \\ 0 \end{pmatrix}, \qquad \vc{B}_2(t) = B_0\begin{pmatrix} 0 \\ 0 \\ \cos(\omega t) \end{pmatrix} \, . \end{equation}Here \(B_0\) is the field strength, or amplitude, and \(\omega\) the angular frequency. \(\vc{B}_1\) describes circularly polarized radiation in the \(xy\)-plane while \(\vc{B}_2\) describes linearly polarized radiation with the magnetic field oscillating along the \(z\)-axis. Note that the Hamiltonian for circularly polarized radiation, i.e. using \(\vc{B}_1\), may be put in a different form:
\begin{equation} \vc{H} = \frac{g\mu_B B_0}{2} \left( e^{-i\omega t} \vc{S}_{+} + e^{i\omega t}\vc{S}_{-} \right) \, . \end{equation}This form of the Hamiltonian reveals that the action of the radiation field changes the magnetic quantum number of the spin by \(\pm 1\) as might be expected - absorption or emission of a photon with a spin (or helicity) of 1 should affect the angular momentum of the spin system accordingly.
The rotating reference frame approach
When the only time-dependent interaction is a Zeeman term with a circularly polarized oscillating magnetic field one may sometimes apply a transformation that makes the Hamiltonian time-independent. Let the unit vector \(\hat{\vc{n}}\) denote the rotation axis for the oscillating field, and assume that the total spin Hamiltonian (except the time-dependent term) has rotational symmetry about \(\hat{\vc{n}}\), then the rotating frame approach is simply a coordinate transformation of the system to a reference frame that rotates together with the oscillating magnetic field, such that the oscillating magnetic field is constant in this reference frame, and the methods for working with a time-independent Hamiltonian may be applied. Of course the assumption that the total Hamiltonian has rotational symmetry is quite severe and makes the rotating reference frame approach significantly less useful in practice. Still the method may be useful for gaining insights about how radiation affects radical pairs.
Transformation of the spin system to a rotating reference frame is done using the unitary operator \(\vc{U}\) defined as:
\begin{equation} \vc{U} = \exp\left(-i\frac{\omega}{\hbar}\vc{J}\cdot\hat{\vc{n}} \, t\right) , \label{eq_TDSpinDyn_rotframe_U} \end{equation}where \(\omega\) defines the angular frequency of the rotation and \(\vc{J}\) is the total spin operator for both electrons and nuclei. With this transformation, the Liouville-von Neumann equation reads as:
\begin{equation} \frac{\p}{\p t}(\vc{U}\rho\vc{U}^\dagger) = -\frac{i}{\hbar} [\vc{U}\vc{H}\vc{U}^\dagger,\vc{U}\rho\vc{U}^\dagger] + \left(\frac{\p\vc{U}}{\p t}\right)\rho\vc{U} + \vc{U}\rho\left(\frac{\p\vc{U}^\dagger}{\p t}\right) . \label{eq_TDSpinDyn_transformed_Liouville} \end{equation}There are two new terms compared to the Liouville-von Neumann equation, both containing \(\frac{\p\vc{U}}{\p t} = -\frac{i}{\hbar}\omega \vc{J}\cdot\hat{\vc{n}} \, \vc{U}\), and it is therefore enlightening to rewrite Eq. (\ref{eq_TDSpinDyn_transformed_Liouville}) in the form:
\begin{equation} \frac{\p \tilde{\rho}}{\p t} = -\frac{i}{\hbar} [\tilde{\vc{H}},\tilde{\rho}], \label{eq_TDSpinDyn_rotframe_Liouville} \end{equation}where \(\tilde{\rho}\) denotes the transformed density operator, \(\tilde{\rho} = \vc{U}\rho\vc{U}^\dagger\), and \(\tilde{\vc{H}} = \vc{U}\vc{H}\vc{U}^\dagger - \omega \vc{J}\cdot\hat{\vc{n}}\) stands for the transformed Hamiltonian. Note that the transformation to the rotating reference frame adds the term \(-\omega\vc{J}\cdot\hat{\vc{n}}\) to the Hamiltonian, in addition to the transformation of the Hamiltonian itself. This extra term in the Hamiltonian is not a real interaction but a fictitious interaction just like the centrifugal force when also considering accelerated reference frames in classical mechanics. Note that the condition \(\frac{d \tilde{\vc{H}}}{dt} = 0\) should be satisfied since the point of making the transformation was to remove the time-dependence from the Hamiltonian - this condition is just an explicit statement of the symmetry requirement mentioned above.
No reaction operators were included in Eq. (\ref{eq_TDSpinDyn_rotframe_Liouville}) but they may also be transformed using \(\vc{U}\), and are in general invariant under the transformation to the rotating reference frame.
Quantum yield calculation with time-dependent interactions
Sometimes the rotating reference frame method may be applied, but when that is not the case one needs to look for other ways to approach the problem.
Exploiting periodicity
When the Hamiltonian has a periodic time-dependence, \(\vc{H}(t + T) = \vc{H}(t)\) for a period \(T\), it may be significantly faster to calculate the time-evolution of a spin system ensemble using numerical integration. If only the density operator at a given time \(\tau\) is needed, a total propagator for a period may be assembled:
\begin{equation} \vc{U}_p = \vc{U}(t + \Delta t)\vc{U}(t + 2\Delta t) \ldots \vc{U}(t + T) \, . \label{eq_TDSpinDyn_PeriodPropagator} \end{equation}Applying \(\vc{U}_p\) takes \(\rho(t)\) to \(\rho(t + T)\), i.e. it propagates the system by a period in one go, and once \(\vc{U}_p\) is computed it may be used several times or even multiplied by itself to create an operator that propagates the system by several periods in one go. Unfortunately this is not useful for calculating quantum yields since the density operator at all intermediate times must be known rather than just obtaining \(\rho(\tau)\) in order to calculate the time-integral (although creating such propagators is part of the \(\gamma\)-COMPUTE algorithm described in the next section). Still, all of the individual propagators corresponding to the smallest numerical integration steps, \(\vc{U}(t + \Delta t)\), \(\vc{U}(t + 2\Delta t)\), \(\ldots\), \(\vc{U}(t + T)\), need only be calculated once and then reapplied. Thus an improvement of the runtime may be achieved at the cost of more memory usage, and this improvement might be significant if the period is short compared to the lifetime of the spin system.
The \(\gamma\)-COMPUTE algorithm
A very efficient way for obtaining quantum yields for radical pairs subject to an oscillating magnetic field is using the \(\gamma\)-COMPUTE algorithm, which additionally performs averaging over the phase of the oscillating field. Several steps are involved in the derivation of the method:
- Propagators similar to Eq. (\ref{eq_TDSpinDyn_PeriodPropagator}) are obtained, using \(n\) discretization steps for a period.
- The propagators are diagonalized in order to diagonalize the average Hamiltonian.
- Use propagators and diagonalized average Hamiltonian to form correlation functions.
- Fourier transform on correlation functions to obtain spectral densities, \(J_{rs}\).
- Form the quantum yield using the spectral densities.
The final result after application of these steps is the following formula for calculation of the singlet quantum yield:
\begin{equation} \mexp{\phi_S} = \frac{\sqrt{2\pi}}{Mn^2}\sum_{r,s=1}^{4M} \sum_{q=-\frac{n}{2}}^{\frac{n}{2} - 1} |J_{rs}(q)|^2 \frac{k^2}{k^2 + (\overline{\omega}_{rs}' - q\omega_{RF})^2} \, . \label{eq_TDSpinDyn_GammaCompute} \end{equation}Here \(M\) is the dimension of the nuclear spin space, \(n\) is the number of discretization steps, \(k\) the decay rate, \(\omega_{RF}\) the frequency of the oscillating magnetic field. The quantity \(\overline{\omega}_{rs}'\) is related to the eigenvalues of the average Hamiltonian. A detailed account of the quantities used in Eq. (\ref{eq_TDSpinDyn_GammaCompute}) as well as a detailed derivation is found in the PhD theses of C. T. Rodgers and J. C. S. Lau (Physical & Theoretical Chemistry Laboratory, University of Oxford).