5.6: Adiabaticity criterion

Now let’s investigate more generally the factors that determine whether dynamics proceed adiabatically and when we need to consider nonadiabatic couplings. Since the adiabatic states \(\Psi_{\alpha}(t) \equiv|\alpha\rangle\) are orthogonal for all times, the adiabatic time propagator, eq. (5.5.1), can be used to express the time-dependent wavefunction as

\[\Psi(t) = \sum_{\alpha} b_{\alpha}(t) \exp \left[-\frac{\mathrm{i}}{\hbar} \int_{0}^{t} E_{\alpha}(t) \, d t' \right] |\alpha\rangle\]

Although these are adiabatic states, we recognize that the expansion coefficients \(b_{\alpha}\) can be time-dependent in the general case. So, we would like to investigate the factors that govern this time-dependence. To make the notation more compact, let’s define the time-rate of change of the eigenfunction as

\[|\dot{\alpha}\rangle=\frac{\partial}{\partial t}\left|\Psi_{\alpha}(t)\right\rangle\]

If we substitute the general solution eq. (5.6.1) into the TDSE, we get

\[\mathrm{i} \hbar \sum_{\alpha}\left(\dot{b}_{\alpha}|\alpha\rangle+b_{\alpha}|\dot{\alpha}\rangle-\frac{\mathrm{i}}{\hbar} E_{\alpha} b_{\alpha}|\alpha\rangle\right) \exp \left[-\frac{\mathrm{i}}{\hbar} \int_{0}^{t} E_{\alpha}(t) \, d t' \right]=\sum_{\alpha} b_{\alpha} E_{\alpha}|\alpha\rangle \exp \left[-\frac{\mathrm{i}}{\hbar} \int_{0}^{t} E_{\alpha}(t) \, d t' \right]\]

Note, the third term on the left-hand side equals the right-hand term. Acting on both sides from the left with \(\langle\beta|\) leads to

\[-\dot{b}_{\beta} \exp \left[-\frac{\mathrm{i}}{\hbar} \int_{0}^{t} E_{\beta}(t) \, d t' \right]=\sum_{\alpha} b_{\alpha}\langle\beta | \dot{\alpha}\rangle  \exp \left[-\frac{\mathrm{i}}{\hbar} \int_{0}^{t} E_{\alpha}(t) \, d t' \right]\]

We can break up the terms in the summation into one for the target state \(|\beta\rangle\) and one for the remaining states.

\[-\dot{b}_{\beta}=b_{\beta}\langle\beta | \dot{\beta}\rangle+\sum_{\alpha \neq \beta} b_{\alpha}\langle\beta | \dot{\alpha}\rangle \exp \left[-\frac{\mathrm{i}}{\hbar} \int_{0}^{t} d t' E_{\alpha \beta} (t')\right]\]

where \(E_{\alpha \beta}(t)=E_{\alpha}(t)-E_{\beta}(t)\).

The adiabatic approximation applies when we can neglect the summation in eq. (5.6.5), or equivalently when \(\langle\beta | \dot{\alpha}\rangle \ll \langle\beta | \dot{\beta}\rangle\) for all \(|\alpha\rangle\). In that case, the system propagates on the adiabatic state \(|\beta\rangle\) independent of the other states: \(\dot{b}_{\beta}=-b_{\beta}\langle\beta | \dot{\beta}\rangle\). The evolution of the coefficients is

\[\begin{aligned}
 b_{\beta}(t) & =b_{\beta}(0) \exp \left[-\int_{0}^{t}\left\langle\beta (t') | \dot{\beta} (t')\right\rangle d t'\right] \\
& \approx b_{\beta}(0) \exp \left[\frac{\mathrm{i}}{\hbar} \int_{0}^{t} E_{\beta}(t')  \, d t'\right]
\end{aligned}\]

Here we note that in the adiabatic approximation \(E_{\beta}(t)=\langle\beta(t) |H(t)| \beta(t)\rangle\). Equation (5.6.6) indicates that in the adiabatic approximation the population in the states never changes, only their phase.

The second term on the right in eq. (5.6.5) describes the nonadiabatic effects, and the overlap integral

\[\langle\beta | \dot{\alpha}\rangle=\left\langle\Psi_{\beta} \left| \frac{\partial \Psi_{\alpha}}{\partial t}\right\rangle\right. \]

determines the magnitude of this effect. \(\langle\beta | \dot{\alpha} \rangle\) is known the nonadiabatic coupling (even though it refers to couplings between adiabatic surfaces), or as the geometrical phase. Note the parallels here to the expression in eq. (5.3.7) for the nonadiabatic coupling in evaluating the validity of the Born-Oppenheimer approximation, however, here the gradient of the wavefunction is evaluated in time rather than the nuclear position. We can make some connections between these two results by linking the gradient variables through the momentum of the particles involved.

So, when can we neglect the nonadiabatic effects? We can obtain an expression for the nonadiabatic coupling by expanding

\[\frac{\partial}{\partial t}\left[H|\alpha\rangle=E_{\alpha}|\alpha\rangle\right]\]

and acting from the left with \(\langle\beta|\), which for \(\alpha \neq \beta\) leads to

\[\langle\beta | \dot{\alpha}\rangle=\frac{\langle\beta|\dot{H}| \alpha\rangle}{E_{\alpha}-E_{\beta}}\]

For adiabatic dynamics to hold \(\langle\beta | \dot{\alpha}\rangle\ll\langle\beta | \dot{\beta}\rangle\), and so we can say

\[\frac{\langle\beta|\dot{H}| \alpha\rangle}{E_{\alpha}-E_{\beta}}\ll-\frac{\mathrm{i}}{\hbar} E_{\beta}\]

So how accurate is the adiabatic approximation for a finite time-period over which the systems propagates? We can evaluate eq. (5.6.5), assuming that the system is prepared in state \(|\alpha\rangle\) and that the occupation of this state never varies much from one. Then the occupation of any other state can be obtained by integrating over a period \(\tau\) as

\[\begin{aligned}
\dot{b}_{\beta} & =\langle\beta \mid \dot{\alpha}\rangle \exp \left[-\frac{\mathrm{i}}{\hbar} \int_{0}^{\tau} d t' E_{\alpha \beta} (t' )\right] \\
b_{\beta} & \approx \mathrm{i} \hbar \frac{\langle\beta|\dot{H}| \alpha\rangle}{E_{\alpha \beta}^{2}}\left\{\exp \left[-\frac{\mathrm{i}}{\hbar} E_{\alpha \beta} \tau\right]-1\right\} \\
& =2 \hbar \frac{\langle\beta|\dot{H}| \alpha\rangle}{E_{\alpha \beta}^{2}} e^{-\frac{\mathrm{i}}{\hbar} E_{\alpha \beta} \tau} \sin \left(\frac{E_{\alpha \beta} \tau}{2 \hbar}\right)
\end{aligned}\]

Here I used \(e^{\mathrm{i} \theta}-1=2 \mathrm{i} e^{\mathrm{i} \theta / 2} \sin (\theta / 2)\). For \(\left|b_{\beta}\right|\ll1\), we expand the \(\sin\) term and find

\[\left\langle\Psi_{\beta}\left|\frac{\partial H}{\partial t}\right| \Psi_{\alpha}\right\rangle \ll \frac{E_{\alpha \beta}}{\tau}\]

This is the criterion for adiabatic dynamics. It says that the rate at which the non-adiabatic coupling matrix elements change must be slow compared to the energy splitting between adiabatic states over that interaction period \(\tau\).

---

References

\[\tag {1} \label {Cite1}\]	Truhlar, D. D. Potential Energy Surfaces. Encyclopedia of Physical Science and Technology; 3rd ed.; Meyers, R. A., Ed.; Vol. 13; Academic Press, 2001; pp 9-17.
\[\tag {2} \label {Cite2}\]	Tully, J. C. Perspective: Nonadiabatic Dynamics Theory. Journal of Chemical Physics 2012, 137, 22A301. DOI: 10.1063/1.4757762

---