A comparison of the two algorithms used to determine the adiabatic to diabatic transformation matrices . Solid black lines represent the ab initio values of the potential energies (upper panels) and norm of the derivative coupling between the two states (lower panels). Dotted red lines denote the potential energies (upper left) and norm of the derivative coupling (lower left) obtained from the two diabatic expansions (with reference points at −0.50 bohr and +0.36 bohr). Dashed blue and solid blue lines (right panels) denote the results from interpolation using the 0th order algorithm (detailed in Sec. III A) and first order algorithm (detailed in Sec. III B), respectively.
Influence of cusps in ab initio potential energy surfaces on interpolators. This example shows a cut of the potential energy surface in ethylene using a small active space CASSCF wavefunction—SA-3-CAS(2/2). The ab initio S1 potential energy curve has a small cusp arising from orbital rotation into and out of the active space (at the center of the x axis which corresponds to rigid torsion about the C=C bond). This is a distorted geometry where one of the H atoms is “migrating” across the C=C bond. The ab initio potential energy curve is shown as a solid black line. In the upper panel, the curve predicted from a single Taylor expansion centered far from the cusp (expansion center at −60° denoted with red dot) is shown (red dotted line). In the lower panel, we show the curve which results when a Taylor expansion is centered near the cusp (at −4°).
The energies (upper panels) and norms of the derivative couplings (lower panels) for the low-lying excited states of butadiene are plotted as a function of the dihedral torsion of the terminal CH2 group (see inset, middle panel). The ab initio calculations (plotted as asterisks) were calculated using SA4-CASSCF(4/4) with the 6-31G* basis set. Taylor expansions for the diabatic surfaces were constructed at 15° and 75° (left and middle panels, respectively) and the energies and derivative coupling norms predicted from these Taylor expansions are plotted as solid lines in the left and middle panels. The rightmost panels show the predicted energies and derivative coupling norms from the weighted sum of the two reference points.
The error in the interpolated energies and gradients as a function of the number of reference points in the interpolated potential. The 50th, 90th, and 99th percentiles of the errors are plotted together with fits to the theoretical power law described in the text ( for energies and for gradients). Note that these errors are for an unbiased test set of geometries (obtained from direct dynamics simulations). The error at the reference geometries used to construct the interpolator is identically zero for both the potential energy and energy gradient of all states included in the calculation.
A comparison of the norm of the ab initio and interpolated derivative couplings for 5000 test geometries as a function of the number of reference points used to construct the interpolation. The dashed line is a visual guide corresponding to perfect agreement between ab initio and interpolated values. The “fit error,” which is commonly used as a quality measure for single and multi-surface potential energy fitting is shown as red dots. This corresponds to the error in the interpolated quantities at the input data points. Because the method we describe here is a true interpolator the fit error should vanish, however the derivative couplings have small translational and rotational components that cannot be described using internal coordinates.
The percentage of trajectory basis functions that failed to conserve energy or wavefunction normalization during dynamics simulations as a function of the number of reference points used to construct the interpolated potential surfaces. An initial condition was classified as “failed” if either the classical energy of any of the TBFs drifted by more than 1 mH, or if the wavefunction norm of the set of coupled TBFs drifted from unity by more than 5%. The dotted line is the percentage of direct dynamics intial conditions that failed.
Projected probability distributions on S1 for three important internal coordinates as a function of time (upper, middle, and lower panels correspond to the dihedral, pyramidalization, and HCC angles, respectively). From left to right, the panels correspond to results obtained from dynamics on interpolated surfaces constructed from 10, 100, and 1000 reference points and direct dynamics.
The population of S0 as a function of time for photoexcited ethylene. The population is calculated as an average over 1000 intial conditions. The black line is calculated from direct ab initio dynamics and the colored lines are results from interpolated surfaces constructed with the indicated number of reference points.
Comparison of the convergence of the S0 population calculated using direct dynamics with increasing number of initial TBFs and interpolated dynamics with increasing number of Taylor expansion reference points and fixed number (1000) of initial TBFs. The direct dynamics results are plotted as error bars corresponding to an uncertainty of one standard deviation in the calculated population for that sample size. The error bars for direct dynamics were calculated by a bootstrap procedure using 1000 samples (with replacement) from the dataset.
Convergence of the RMS error (in the S0 population sampled at eight time points, as described in the text) in direct dynamics (red +'s labelled by the number of initial TBFs) and interpolated dynamics (blue ×’s, labelled by the number of reference points used to create the interpolated surface) as a function of the number of ab initio gradient calculations required. The expected asymptotic convergence behavior is also shown for both direct dynamics (red dashed line) and interpolated dynamics (blue dashed line).
Article metrics loading...
Full text loading...