Wavepacket dynamics fixes a particular initial packet. A Kubo-transformed correlation function instead asks for thermal real-time memory. The example here is the position-position correlation \(K_{xx}(t)\), computed by combining DVR eigenstates with a spectral Kubo weight.

Definition

For two operators \(A\) and \(B\),

\[ K_{AB}(t)= \frac{1}{\beta Z} \int_0^\beta d\lambda\, \mathrm{Tr}\!\left( e^{-(\beta-\lambda)H} A e^{-\lambda H} B(t) \right), \]
\[ B(t)=e^{iHt}B e^{-iHt}, \qquad Z=\mathrm{Tr}(e^{-\beta H}). \]

For \(A=B=x\), the function measures how much a thermal position fluctuation at time zero is remembered at time \(t\).

Energy-basis derivation

Insert energy eigenstates \(H|n\rangle=E_n|n\rangle\) into the trace. The time dependence gives

\[ \langle j|B(t)|i\rangle = e^{-i(E_i-E_j)t}B_{ji}. \]

The imaginary-time integral contains the energy gap \(\Delta_{ij}=E_i-E_j\):

\[ \frac{1}{\beta}e^{-\beta E_i} \int_0^\beta e^{\lambda(E_i-E_j)}\,d\lambda. \]

For \(i\ne j\), this becomes

\[ \frac{e^{-\beta E_j}-e^{-\beta E_i}} {\beta(E_i-E_j)}. \]

For \(i=j\), the limit is \(e^{-\beta E_i}\). Define

\[ W_{ij}= \begin{cases} \dfrac{e^{-\beta E_j}-e^{-\beta E_i}}{\beta(E_i-E_j)},& i\ne j,\\ e^{-\beta E_i},& i=j. \end{cases} \]

Spectral formula

The Kubo-transformed correlation is then

\[ K_{AB}(t)= \frac{1}{Z} \sum_{ij} W_{ij}A_{ij}B_{ji}e^{-i(E_i-E_j)t}. \]

For position autocorrelation, \(x_{ji}=x_{ij}^\ast\), so

\[ K_{xx}(t)= \frac{1}{Z} \sum_{ij}W_{ij}|x_{ij}|^2 \cos\!\left[(E_i-E_j)t\right]. \]

The expression is explicitly real and even in time.

DVR matrix elements

After solving the one-dimensional DVR eigenproblem, the position matrix is approximated by

\[ x_{mn}\approx \sum_i c_m(x_i)\,x_i\,c_n(x_i)\,\Delta x. \]

Only a finite number of low-energy states are retained in the code. That is a temperature-dependent truncation: as \(\beta\) decreases, higher-energy states matter more.

Position-position Kubo correlation curve
Example \(x-x\) Kubo correlation. The horizontal axis is time and the vertical axis is \(K_{xx}(t)\). The black curve is the real spectral-sum result for the chosen one-dimensional anharmonic potential; its oscillation frequencies come from energy-level gaps \(E_i-E_j\).

From position Kubo to flux-side

This note reuses the operator-matrix viewpoint from Parts II-IV, but the initial object is now the thermal Kubo kernel rather than a selected incoming packet. Part VI applies the same spectral weighting to a rate-like flux-side observable, and Part VII adds local adiabatic projectors for population-resolved transfer diagnostics.

Code used in this note

DVR series map

  1. Part I: Wavepacket Dynamics Methods - discretization choices and the harmonic-oscillator benchmark.
  2. Part II: Operator Matrices in the DVR Basis - matrix structure behind kinetic, side, and flux operators.
  3. Part III: Wavepacket Dynamics with DVR - exact finite-dimensional propagation of a prepared wavepacket.
  4. Part IV: Density-Matrix Ensemble Evolution - thermal ensembles, filtering, and absorber bookkeeping.
  5. Part V: Kubo-Transformed Position Correlation - spectral thermal correlation functions.
  6. Part VI: Flux-Side Kubo Correlation - rate-like observables from side and flux operators.
  7. Part VII: Excited-Ground Kubo Population Correlation - local adiabatic projectors and population-resolved transfer diagnostics.