8.7 DMC calculations

DMC calculations were performed using the methods and algorithms of chapters 3 and  4. Core electrons were modelled by an accurate norm-conserving LDA pseudopotential, [58]. The guiding wave functions were of the Slater-Jastrow type, consisting of the product of a Slater determinant of single-particle orbitals obtained from the CRYSTAL [130] or Gaussian94 [157] codes multiplied by a ``homogeneous'' Jastrow factor (see section 4.3.2). For the single particle orbitals uncontracted valence Gaussian basis sets of four s, four p and one d function were used. The basis sets were optimised for each structural class of isomer (ring, cage, bowl and sheet), yielding basis sets of high efficiency.

Up to 80 parameters in the Jastrow factor were optimized for every structure using the method of variance minimization described in chapter 5. In excess of 75% of the DMC correlation energy was obtained in all cases. No attempt to obtain ``benchmark'' accuracy VMC results was made: the aim was to obtain wavefunctions of sufficient accuracy for use in DMC calculations.

For the fixed-node DMC calculations, a timestep of 0.003 a.u. and an ensemble of 640 walkers were used. The DMC calculations averaged $\sim 30000$ moves in length.

A special consideration was made for the $\mathrm {C}_{26}$ ring. The $\mathrm {C}_{26}$ ring is ``cumulenic'' whereas the other rings in the study are ``polyacetylenic''^8.2and have two unique bond lengths. MCHF and CI calculations show that the $\mathrm{C}_{4N+2}$, $N=1,2,\ldots$ cumulenic rings have significant multi-determinantal character, unlike the $\mathrm{C}_{4N}$ rings. [161] Due to the likely improvement in the trial wavefunction, a CISD calculation was performed and the configurations with largest weight used to form a multi-determinant trial wavefunction. 43 determinants, including the HF reference determinant were selected. Although only a small reduction in the variance of the trial wavefunction resulted (by $\sim 0.1$ a.u.$^2$ to $\sim 5.4$ a.u.), a small reduction in DMC energy was found due to the improved nodal structure of the wavefunction. Results comparing single and multi-determinant wavefunctions for this isomer are presented in the next section.