Quadratic coupling

The second type of coupling considered for the tunnelling benchmark outlined above involves quadratic coupling between the tunnelling coordinate and every environmental oscillator. In this case the potential energy due to the environment is given by Venv(q) = f X j=2 1 2q 2 j +aq1qj2 , (4.9)

where, again following previous investigations of this system, a= 0.05.47,82,118,119 From here on referred to as Model III, this constitutes a far more challenging prob- lem than either Model I or II. Now the tunnelling coordinate is directly coupled to every other degree of freedom, the coupling itself is quadratic instead of linear and the strength of the latter is constant across allf₋1 coordinates. In order to demonstrate the applicability of the adaptive sampling algorithm beyond systems with only a few degrees of freedom, already represented by Model I and II, the number of DOFs for Model III was chosen to be f = 20. While still a synthetic benchmark, Model III is far more representative of real higher-dimensional sys- tems involving quantum tunnelling and consequently a significant challenge for any quantum dynamics method.

As a result of this, a common approach,118 when sampling the basis set for solving this problem, is to sample basis functions not only from the initial wave- function, centred around q1 = −2.5 and qj>1 = 0, but also from the mirrored wavefunction, representing complete tunnelling to the other well, centred around

q1 =−2.5 andqj>1 = 0. This, while resulting in excellent dynamics,118 somewhat reduces the sampling challenge, as the basis set generated by this approach inher- ently supports tunnelling, independent of whether the sampling strategy captures it or not. However, Model III still constitutes a challenging benchmark with re- spect to the accuracy with which a given method captures the dynamics of the wavefunction, irrespective of the basis set’s inclusion of tunnelling effects.

To assess both the performance of the aTSA for this benchmark, as well as comparing to the TSA, two calculations, one for each approach, were run. Both calculations were run for a total sampling time of 120 a.u. and employed sampling and propagation timesteps of ∆tt= ∆p = 0.002 a.u., as well asm = 50 sampling trajectories. The TSA calculation consisted of nt = np = 60000 timesteps of sampling and propagation and a sampling frequency of1_/n

s =1/4000. For the aTSA

nt = np = 1000 timesteps were run, storing basis functions with a frequency of 1_/n

s = 1/200 and employing a MP minimisation and optimisation convergence

criterion of ζ = 0.995. These input parameters are summarised in Table 4.6. Finally, it is worth noting that, in accordance with the sampling strategy set out above and with previous investigations of this benchmark,118 _{the initial burst of} sampling involved selecting basis functions from both wells in the potential energy. As with the two models of linear coupling, the tunnelling autocorrelation func- tion, Ct(t), was calculated for both methods, the results of which are shown in Figure 4.9. Comparing again to exact CI data,119 _{the challenging nature of Model} III is immediately apparent. The TSA completely fails to capture the tunnelling dynamics of the system beyond about 10 a.u., as too large a proportion of the wavefunction tunnels to the opposite well, furthermore failing to return to its ori- ginal configuration, instead remaining effectively trapped. The latter is indicated by the decreasing amplitudes of the oscillations in Ct(t) in Figure 4.9(b).

The aTSA, on the other hand, does not exhibit either of the two shortcomings above, capturing at the very least, the broad qualitative features ofCt(t). While by no means following the CI results119 _{exactly, the extent to which the wavefunction} tunnels is reproduced relatively well, with the exception of the last 25 a.u. of time. Given the relatively small basis sets employed for both calculations, asNtotal = 994 and 884, for the TSA and aTSA respectively (trajectory sampled and MP inherited basis set sizes are shown in Table 4.6), this is extremely encouraging.

The underlying conditions of Model III, that is the high number of DOFs and the complex, quadratic coupling of the tunnelling coordinate to all others, makes this a relatively realistic benchmark for systems exhibiting strong quantum tunnelling. The adaptive sampling method, while, to an extent, addressing the classical-quantum divergence, negatively affecting the TSA, still relies on sampling trajectories driven by purely classical mechanics. Thus, even the limited qualit- ative accuracy achieved here indicates that sufficiently frequent resampling of the

Table 4.6: Input parameters and basis set sizes for calculations of Model III, using the TSA and the aTSA, the results of which are shown in Figure 4.9.

nt ∆tt/a.u. ns np ∆tp/a.u. m Ntotal

TSA 60000 0.002 4000 60000 0.002 50 994

aTSA 1000 0.002 200 1000 0.002 50 884

Nb ζ Nt NM P

4.5. ALGORITHM PARAMETERS AND PERFORMANCE 0 0.05 0.1 (a) Abs [ Ct ( t )] CI aTSA 0 25 50 75 100 0 0.05 0.1 0.15 0.2 (b) time/a.u. CI TSA

Figure 4.9: Tunnelling autocorrelation function for Model III, calculated using (a) the adaptive sampling algorithm and (b) with the trajectory sampling algorithm, both with respect to an exact CI benchmark.119

wavefunction makes strong quantum effects accessible with such relatively simple and computationally inexpensive tools.

4.5

Algorithm parameters and performance

The approach taken in this section follows very closely that used in Section 3.4. The 4-D pyrazine benchmark was again used here, to allow comparison to the results in the previous chapter.