Small diatomic molecules, methane, and water

Except from $Be_2$ and $C_2$ , all the molecules presented here belong to the standard G1 reference set; all their properties are well known and well reproduced by standard quantum chemistry methods, therefore they constitute a good case for testing new approaches and new wave functions.

Table: Total energies in variational ($E_{VMC}$ ) and diffusion ($E_{DMC}$ ) Monte Carlo calculations; the percentages of correlation energy recovered in VMC ( $E^{VMC}_c(\%)$ ) and DMC ( $E^{DMC}_c(\%)$ ) have been evaluated using the ``exact'' ($E_0$ ) and Hartree-Fock ($E_{HF}$ ) energies from the references (1). Here ``exact'' means the ground state energy of the non relativistic infinite nuclear mass Hamiltonian. The energies are in Hartree.

	$E_0$	$E_{HF}$	$E_{VMC}$	$E^{VMC}_c(\%)$	$E_{DMC}$	$E^{DMC}_c(\%)$
$Li$	-7.47806 (59)	-7.432727 (59)	-7.47721(11)	98.12(24)	-7.47791(12)	99.67(27)
$Li_2$	-14.9954 (16)	-14.87152 (16)	-14.99002(12)	95.7(1)	-14.99472(17)	99.45(14)
$Be$	-14.66736 (59)	-14.573023 (59)	-14.66328(19)	95.67(20)	-14.66705(12)	99.67(13)
$Be_2$	-29.33854(5) (16)	-29.13242 (16)	-29.3179(5)	89.99(24)	-29.33341(25)	97.51(12)
$O$	-75.0673 (59)	-74.809398 (59)	-75.0237(5)	83.09(19)	-75.0522(3)	94.14(11)
$H_2O$	-76.438(3) (60)	-76.068(1) (60)	-76.3803(4)	84.40(10)	-76.4175(4)	94.46(10)
$O_2$	-150.3268 (16)	-149.6659 (16)	-150.1992(5)	80.69(7)	-150.272(2)	91.7(3)
$C$	-37.8450 (59)	-37.688619 (59)	-37.81303(17)	79.55(11)	-37.8350(6)	93.6(4)
$C_2$	-75.923(5) (16)	-75.40620 (16)	-75.8293(5)	81.87(10)	-75.8810(5)	91.87(10)
$CH_4$	-40.515 (61)	-40.219 (61)	-40.4627(3)	82.33(10)	-40.5041(8)	96.3(3)
$C_6H_6$	-232.247(4) (62)	-230.82(2) (63)	-231.8084(15)	69.25(10)	-232.156(3)	93.60(21)

The $Li$ dimer is one of the easiest molecules to be studied after the $H_2$ , which is exact for any Diffusion Monte Carlo (FN DMC) calculation with a trial wave function that preserves the node-less structure. $Li_2$ is less trivial due to the presence of core electrons that are only partially involved in the chemical bond and to the $2s-2p$ near degeneracy for the valence electrons. Therefore many authors have done benchmark calculation on this molecule to check the accuracy of the method or to determine the variance of the inter-nuclear force calculated within a QMC framework. In this thesis we start from $Li_2$ to move toward a structural analysis of more complex compounds, thus showing that our QMC approach is able to handle relevant chemical problems.

With our approach more than $99\%$ of the $Li_2$ correlation energy is recovered by a DMC simulation (Table 3.1), and the atomization energy is exact within few thousands of eV ( $0.02~kcal~mol^{-1}$ ) (Table 3.3). Similar accuracy have been previously reached within a DMC approach(16), only by using a multi-reference CI like wave function, that before our work, was the usual way to improve the electronic nodal structure. As stressed before, the JAGP wave function includes many resonating configurations through the geminal expansion, beyond the $1s~2s$ HF ground state. The bond length has been calculated at the variational level through the fully optimized JAGP wave function: the resulting equilibrium geometry turns out to be highly accurate (Table 3.2), with a discrepancy of only $0.001 a_0$ from the exact result.

Table 3.2: Bond lengths ($R$ ) in atomic units; the subscript 0 refers to the ``exact'' results. For the water molecule $R$ is the distance between O and H and $\theta$ is the angle HOH (in deg), for $CH_4$ $R$ is the distance between C and H and $\theta$ is the HCH angle.

	$R_0$	$R$	$\theta_0$	$\theta$
$Li_2$	5.051	5.0516(2)
$O_2$	2.282	2.3425(18)
$C_2$	2.348	2.366(2)
$H_2O$	1.809	1.8071(23)	104.52	104.74(17)
$CH_4$	2.041	2.049(1)	109.47	109.55(6)
	$R^{CC}_0$	$R^{CC}$	$R^{CH}_0$	$R^{CH}$
$C_6H_6$	2.640	2.662(4)	2.028	1.992(2)

The good bond length, we obtained, is partially due to the energy optimization that is often more effective than the variance minimization, as shown by different authors (40,41,42), and partially due to the quality of the trial-function.

Indeed within our scheme we obtain good results without exploiting the computationally much more demanding DMC, thus highlighting the importance of the SR minimization described in Subsection 2.2.

Table 3.3: Binding energies in $eV$ obtained by variational ( $\Delta _{VMC}$ ) and diffusion ( $\Delta _{DMC}$ ) Monte Carlo calculations; $\Delta _0$ is the ``exact'' result for the non-relativistic infinite nuclear mass Hamiltonian. Also the percentages ( $\Delta _{VMC}(\%)$ and $\Delta _{DMC}(\%)$ ) of the total binding energies are reported.

	$\Delta _0$	$\Delta _{VMC}$	$\Delta _{VMC}(\%)$	$\Delta _{DMC}$	$\Delta _{DMC}(\%)$
$Li_2$	-1.069	-0.967(3)	90.4(3)	-1.058(5)	99.0(5)
$O_2$	-5.230	-4.13(4)	78.9(8)	-4.56(5)	87.1(9)
$H_2O$	-10.087	-9.704(24)	96.2(1.0)	-9.940(19)	98.5(9)
$C_2$	-6.340	-5.530(13)	87.22(20)	-5.74(3)	90.6(5)
$CH_4$	-18.232	-17.678(9)	96.96(5)	-18.21(4)	99.86(22)
$C_6H_6$	-59.25	-52.53(4)	88.67(7)	-58.41(8)	98.60(13)

Let us now consider larger molecules. Both $C_2$ and $O_2$ are poorly described by a single Slater determinant, since the presence of the non-dynamic correlation is strong. Instead with a single geminal JAGP wave function, including implicitly many Slater-determinants(15), it is possible to obtain a quite good description of their molecular properties. In both the cases, the variational energies recover more than $80\%$ of the correlation energy, the DMC ones yield more than $90\%$ , as shown in Tab. 3.1. These results are of the same level of accuracy as those obtained by Filippi et al(16) with a multi-reference wave function by using the same Slater basis for the antisymmetric part and a different Jastrow factor. From the Table 3.3 of the atomization energies, it is apparent that DMC considerably improves the binding energy with respect to the VMC values, although for these two molecules it is quite far from the chemical accuracy ($\simeq$ 0.1 eV): for $C_2$ the error is 0.60(3) eV, for $O_2$ is 0.67(5) eV. Indeed, it is well known that the electronic structure of the atoms is described better than the corresponding molecules if the basis set remains the same, and the nodal error is not compensated by the energy difference between the separated atoms and the molecule. In a benchmark DMC calculation with pseudo-potentials (64), Grossman found an error of 0.27 eV in the atomization energy for $O_2$ , by using a single determinant wave function. Probably, pseudo-potentials allow the error between the pseudo-atoms and the pseudo-molecule to compensate better, thus yielding more accurate energy differences. As a final remark on the $O_2$ and $C_2$ molecules, our bond lengths are in between the LDA and GGA precision, and still worse than the best CCSD calculations, but our results may be considerably improved by a larger atomic basis set.

Methane and water are very well described by the JAGP wave function. Also for these molecules we recover more than $80\%$ of correlation energy at the VMC level, while DMC yields more than $90\%$ , with the same level of accuracy reached in previous Monte Carlo studies (65,61,67,66). Here the binding energy is almost exact, since in this case the nodal energy error arises essentially from only one atom (carbon or oxygen) and therefore it is exactly compensated when the atomization energy is calculated. Also the bond lengths are highly accurate, with an error lower then 0.005 $a_0$ .

For $Be_2$ we applied a large Gaussian and exponential basis set for the determinant and the Jastrow factor and we recovered, at the experimental equilibrium geometry, the $90\%$ of the total correlation energy in the VMC, while DMC gives $97.5\%$ of correlation, i.e. a total energy of -29.33341(25) H. Although this value is better than the one obtained by Filippi et al (16) (-29.3301(2) H) with a smaller basis ($3 s$ atomic orbitals not included), it is not enough to bind the molecule, because the binding energy remains still positive (0.0069(37) H). Instead, once the molecular geometry has been relaxed, the SR optimization finds a bond distance of $13.5(5)~a_0$ at the VMC level; therefore the employed basis allows the molecule to have a Van der Waals like minimum, quite far from the experimental value. In order to have a reasonable description of the bond length and the atomization energy, one needs to include at least a $3s 2p$ basis in the antisymmetric part, as pointed out in Ref. (68). Indeed an atomization energy compatible with the experimental result (0.11(1) eV) has been obtained within the extended geminal model (69) by using a much larger basis set (9s,7p,4d,2f,1g) (70). This suggests that a complete basis set calculation with JAGP may describe also this molecule. However our SR method can not cope with a very large basis in a feasible computational time. Therefore we believe that at present the accuracy needed to describe correctly $Be_2$ is out of the possibilities of the approach.

Table: Binding energies in $eV$ obtained by variational ( $\Delta _{VMC}$ ) and diffusion ( $\Delta _{DMC}$ ) Monte Carlo calculations with different trial wave functions for benzene. In order to calculate the binding energies yielded by the 2-body Jastrow we used the atomic energies reported in Ref. (2). The percentages ( $\Delta _{VMC}(\%)$ and $\Delta _{DMC}(\%)$ ) of the total binding energies are also reported.

	$\Delta _{VMC}$	$\Delta _{VMC}(\%)$	$\Delta _{DMC}$	$\Delta _{DMC}(\%)$
Kekule + 2body	-30.57(5)	51.60(8)	-	-
resonating Kekule + 2body	-32.78(5)	55.33(8)	-	-
resonating Dewar Kekule + 2body	-34.75(5)	58.66(8)	-56.84(11)	95.95(18)
Kekule + 3body	-49.20(4)	83.05(7)	-55.54(10)	93.75(17)
resonating Kekule + 3body	-51.33(4)	86.65(7)	-57.25(9)	96.64(15)
resonating Dewar Kekule + 3body	-52.53(4)	88.67(7)	-58.41(8)	98.60(13)
full resonating + 3body	-52.65(4)	88.869(7)	-58.30(8)	98.40(13)