Structural optimization

In the last few years remarkable progresses have been made to develop Quantum Monte Carlo (QMC) techniques which are able in principle to perform structural optimization of molecules and complex systems (52,29). Within the Born-Oppheneimer approximation the nuclear positions $\vec{R}_i$ can be considered as further variational parameters included in the set $\{ \alpha_i \}$ used for the SR minimization (2.13) of the energy expectation value. For clarity, in order to distinguish the conventional variational parameters from the ionic positions, in this section we indicate with $\{c_i\}$ the former ones, and with $\vec{R}_i$ the latter ones. It is understood that $R_i^\nu =\alpha_k$ , where a particular index $k$ of the whole set of parameters $\{ \alpha_i \}$ corresponds to a given spatial component ($\nu=1,2,3$ ) of the $i-$ th ion.

We computed the forces $\vec{F}$ acting on each of the $M$ nuclear positions $\{\vec{R}_1, \ldots , \vec{R}_M\}$ , being $M$ the total number of nuclei in the system:

$$\vec{F}(\vec{R}_a) = -\vec{\nabla}_{\vec{R}_a} E(\{c_i\},\vec{R}_a)$$ (2.21)

$$= - { \langle \Psi \vert O_R H + H O_R + \partial_R H\vert \Psi \ra... ...ngle \Psi \vert H \vert \Psi \rangle \over \langle \Psi \vert \Psi \rangle^2 },$$ (2.22)

where operator $O_R$ are defined as logarithmic derivatives respect to nuclear position of the trial-function in analogy to the operator $O_k$ 2.4. This generalized forces were than used to perform structural optimization using the the iteration (2.13). In the first part of this thesis we have used a finite difference operator ${ \vec{\Delta \over \Delta R}_a }$ for the evaluation of the force acting on a given nuclear position $a$ :

$$\vec{F} (\vec{R}_a)=- { \vec{\Delta \over \Delta R}_a } E = - { E... ... R}_a ) - E(\vec{R}_a - \vec{\Delta R}_a ) \over 2 \Delta R } + O({\Delta R}^2)$$ (2.23)

where $\vec{\Delta R}_a$ is a 3 dimensional vector. Its length $\Delta R$ is chosen to be $0.01$ atomic units, a value that is small enough for negligible finite difference errors.

In order to evaluate the energy differences in Eq. 2.23 with a finite variance we have used the Space-Warp coordinate transformation (46,53). This transformation was also used in the evaluation of the wave-function derivatives respect to nuclear positions $O_R$ . Even if Space-Warp transformation is a very efficient technique to reduce the variance of the forces, it is very time consuming and so for larger systems we preferred to use Zero Variance forces (29), as it was described in the chapter 1.

The $O_R$ operators are used also in the definition of the reduced matrix $\bar s$ for those elements depending on the variation with respect to a nuclear coordinate. In this way it is possible to optimize both the wave function and the ionic positions at the same time, in close analogy with the Car-Parrinello(54) method applied to the minimization problem. Also Tanaka (48) tried to perform Car-Parrinello like simulations via QMC, within the less efficient steepest descent framework.

Figure: Plot of the equilibrium distance of the $Li_2$ molecule as a function of the inverse bin length. The total energy and the binding energy are reported in Tables 3.3 and 3.2 respectively. For all simulations the initial wave-function is optimized at $Li-Li$ distance $6$ a.u.

An important source of systematic errors is the dependence of the variational parameters $c_i$ on the ionic configuration $\vec{R}$ , because for the final equilibrium geometry all the forces $f_i$ corresponding to $c_i$ have to be zero, in order to guarantee that the true minimum of the potential energy surface (PES) is reached (55,56). As shown clearly in the previous subsection, within a QMC approach it is possible to control this condition by increasing systematically the bin length, when the thermal bias $T_{noise}$ vanishes. In Fig. 2.3 we report the equilibrium distance of the Li molecule as a function of the inverse bin length, so that an accurate evaluation of such an important quantity is possible even when the number of variational parameters is rather large, by extrapolating the value to an infinite bin length. However, as it is seen in the picture, though the inclusion of the 3s orbital in the atomic AGP basis substantially improves the equilibrium distance and the total energy by $\simeq 1mH$ , this larger basis makes our simulation less efficient, as the time step $\Delta t$ has to be reduced by a factor three.

We have not attempted to extend the geometry optimization to the more accurate DMC, since there are technical difficulties (57), and it is computationally much more demanding.