Setting the SR parameters

In this thesis we have determined $\Delta t$ by verifying the stability and the convergence of the SR algorithm for fixed $\Delta t$ value.

The simulation is stable whenever $1 / \Delta t > {\rm\Lambda_{cut} }$ , where $\Lambda_{cut}$ is an energy cutoff that is strongly dependent on the chosen wave function and it is generally weakly dependent on the bin length. Whenever the wave function is too much detailed, namely has a lot of variational freedom, especially for the high energy components of the core electrons, the value of $\Lambda_{cut}$ becomes exceedingly large and too many iterations are required for obtaining a converged variational wave function. In fact a rough estimate of the corresponding number of iterations $P$ is given by $P \Delta t >> 1/G$ , where $G$ is the typical energy gap of the system, of the order of few electron Volts in small atoms and molecules. Within the SR method it is therefore extremely important to work with a bin length rather small, so that many iterations can be performed without much effort.

Figure 2.1: Example of the convergence of the SR method for the variational parameters as a function of the number of stochastic iterations. In the upper(lower) panel the Jastrow (geminal) parameters are shown. For each iteration, a variational Monte Carlo calculation is employed with a bin containing $15000$ samples of the energy, yielding at the equilibrium a standard deviation of $\simeq 0.0018 H$ . For the first 200 iteration $\Delta t= 0.00125 H^{-1}$ , for the further 200 iterations $\Delta t= 0.0025 H^{-1}$ , whereas for the remaining ones $\Delta t= 0.005 H^{-1}$ .

In a Monte Carlo optimization framework the forces $f_k$ are always determined with some statistical noise $\eta_k$ , and by iterating the procedure several times with a fixed bin length the variational parameters will fluctuate around their mean values. These statistical fluctuations are similar to the thermal noise of a standard Langevin equation:

$$\partial_t \alpha_k = f_k +\eta_k ,$$ (2.14)

where

$$\langle \eta_k (t) \eta_{k^\prime} (t^\prime) \rangle= 2 T_{noise} \delta (t-t^\prime) \delta_{k,k^\prime}.$$ (2.15)

The variational parameters $\alpha_k$ , averaged over the Langevin simulation time (as for instance in Fig.2.1), will be close to the true energy minimum, but the corresponding forces $f_k= -{ \partial_{\alpha_k} E }$ will be affected by a bias that scales to zero with the thermal noise $T_{noise}$ . Within a QMC scheme, one needs to estimate $T_{noise}$ by increasing the bin length, as clearly $T_{noise} \propto 1/ {\rm Bin~length }$ , this noise being directly related to the statistical fluctuations of the forces. Thus there is an optimal value for the bin length, which guarantees a fast convergence and avoid the forces to be biased within the statistical accuracy of the sampling. Moreover in the fluctuation around the minimum also non Gaussian correction will be present, but in analogy to the an-harmonic effects in solids, this error is expected to vanish linearly with the temperature $T_{noise}$ . An example is shown in Fig. 2.1 for the optimization of the Be atom, using a basis two exponentials for each orbital both for the geminal and the three-body Jastrow part. The convergence is reached in about 1000 iteration with $\Delta t= 0.005 H^{-1}$ . However, in this case it is possible to use a small bin length, yielding a statistical accuracy in the energy much poorer than the final accuracy of about $0.05 mH$ . This is obtained by averaging the variational parameters in the last $1000$ iterations, when they fluctuate around a mean value, allowing a very accurate determination of the energy minimum which satisfies the Euler conditions, namely with $f_k=0$ for all parameters. Those conditions have been tested by an independent Monte Carlo simulation about $600$ times longer than the bin used during the minimization.

Figure 2.2: Calculation of the derivative of the energy with respect to the second $Z$ in the $2p$ orbital of the geminal function for the Be atom. The calculation of the force was obtained, at fixed variational parameters, by averaging over $10^7$ samples, allowing e.g. a statistical accuracy in the total energy of $0.07 mH$ . The variational parameters have been obtained by an SR minimization with fixed bin length shown in the x label. The parameter considered has the largest deviation from the Euler conditions.

As shown in Fig. 2.2 the Euler conditions are fulfilled within statistical accuracy even when the bin used for the minimization is much smaller than the overall simulation. On the other hand if the bin used is too small, as we have already pointed out, the averaging of the parameters is affected by a sizable bias.

Whenever it is possible to use a relatively small bin in the minimization, the apparently large number of iterations required for equilibration does not really matter, because a comparable amount of time has to be spent in the averaging of the variational parameters, as shown in Fig. 2.1.

It is easy to convince oneself that for high enough accuracy the number of iterations needed for the equilibration becomes negligible from the computational point of view. In fact, in order to reduce, e.g. by a factor of ten, the accuracy in the variational parameters, a bin ten times larger is required for decreasing the thermal noise $T_{noise}$ by the same factor. Whereas to reduce the statistical errors by the same ratio, it has to be done average on $100$ times steps more. This means that the fraction of time spent for equilibration becomes ten times smaller compared with the less accurate simulation.