Structures and energies of isolated compounds

To test the accuracy of the DFT approach and the CP-PAW package we performed geometry optimizations of four small molecules and two small complexes (water dimer and water-chloride complex) relevant to reaction (i).

We optimized the structures of Cl$_2$, HCl, H$_2$O, and CH$_3$Cl with the CP-PAW program and compared these with results obtained with the ADF package and literature values. The results of the geometry optimizations are compiled in table 3.1. The first two columns show our CP-PAW results with two different plane wave basis sets.

Table 3.1: Comparison of bonding energies (kcal/mol) and geometries (Ångströms and degrees) calculated with the CP-PAW program and the ADF program with other calculations and experiment.

Mol.	Prop.	PAW$^a$	PAW$^a$	ADF$^a$	B3LYP$^b$	G2$^c$	Exp.$^d$
		30Ry	50Ry	TZDP	6-31G*	MP2/6-31G*
Cl$_2$	$\Delta E_{\mathrm{Bond}}$	-58.2	-58.2	-62.3		-56.56	58.0
	$R_{\mathrm{ClCl}}$ [Å]	2.065	2.065	2.023	2.042	2.015	1.988$^e$
HCl	$\Delta E_{\mathrm{Bond}}$	-105.7	-106.0	-107.8		-106.7	106.3
	$R_{\mathrm{HCl}}$ [Å]	1.308	1.304	1.293	1.290	1.280	1.275$^f$
H$_2$O	$\Delta E_{\mathrm{Bond}}$	-232.59	-234.46	-237.88		-232.5	232.2
	$R_{\mathrm{OH}}$ [Å]	0.985	0.979	0.971	0.969	0.969	0.957$^g$
	$\angle_{\mathrm{HOH}}[^{\circ}]$	103.77	104.09	104.07	103.6	104.0	104.4$^g$
CH$_3$Cl	$\Delta E_{\mathrm{Bond}}$	-401.9	-404.2	-402.2		-394.8	393.8
	$R_{\mathrm{CH}}$ [Å]	1.102	1.096	1.093	1.090	1.088	1.08$^h$
	$R_{\mathrm{CCl}}$ [Å]	1.827	1.827	1.803	1.803	1.777	1.78$^h$
	$\angle_{\mathrm{HOCl}}[^{\circ}]$	107.9	107.8	108.3	108.5	108.9	108.2$^h$

$^a$ Our work, using the BP functional. $^b$ B3LYP 6-31G* geometries from ref G2benchmark. $^c$ G2 energies (effectively QCISD(T)/6-311+G(3df,2p) level) based on MP2/6-31G* geometries from ref [79]. $^d$ Experimental bonding energies minus the zero-point energy correction from refs G1benchmark1,G1benchmark2. $^e$ Ref structuretables1. $^f$ Ref structuretables2. $^g$ Ref structuretables3. $^h$ Ref structuretables4.

The geometry appears to be practically converged at the plane wave cutoff of 30 Ry. The largest bond distance discrepancy between the 30 Ry and the large 50 Ry basis set are found for $R_{\rm OH}$ and $R_{\rm CH}$, namely 0.006 Å. Comparing the CP-PAW and ADF results there are larger differences. The largest discrepancy is found for the $R_{\rm CCl}$ (0.024) and $R_{\rm ClCl}$ (0.042) distances. After the simulations had been done, we have been able to trace the differences to the limited number of projector functions for Cl in the PAW method. Increasing the set of projector functions leads to very good agreement with the ADF results. Given the experimental uncertainties, the accuracy of the present CP-PAW results with the smaller set of projectors are satisfactory for our purposes. Bond lengths are slightly overestimated, with the largest errors for $R_{\rm ClCl}$ of 0.077 Å and $R_{\rm CCl}$ of 0.047 Å. Angles are correct within one degree.

Atomization energies obtained with CP-PAW are converged within 0-2 kcal/mol at a plane wave cutoff of 30 Ry. The differences between the CP-PAW energies and the ADF ones are somewhat larger, from 1-4 kcal/mol.

Table 3.2: Water dimer bonding energy (kcal/mol) and geometry (deg and Å) calculated with PAW (two different plane wave cutoffs) and ADF compared to other Car-Parrinello calculations and accurate DFT, ab initio and experimental results.

	$\Delta E_{\mathrm{intermol.}}$	$R_{\mathrm{OO}}$	$R_{\mathrm{OH_b}}$	$R_{\mathrm{H_bO}}$	$\angle_{\mathrm{OHO}}$
PAW-BP 30Ry	-4.49	2.954	0.997	1.962	173.3
PAW-BP 50Ry	-4.35	2.938	0.990	1.955	171.6
ADF-BP/TZDP	-4.94	2.893	0.982	1.916	172.8
CPMD-BP 70Ry$^a$	-4.5	2.95			177
CPMD-BP 150Ry$^a$	-4.3	2.94			177
DFT-BP/aug-cc-pVDZ*$^b$	-4.69	2.886	0.985	1.908	172
DFT-BP/TZVP$^c$	-4.69	2.885
DFT-BLAP3/TZVP$^c$	-4.63	2.979
DFT-PLAP3/TZVP$^c$	-4.68	2.950
DFT-HCTH38/TZ2P$^d$	-4.60	2.952
DFT-B3LYP/aug-cc-pVTZ$^b$	-4.57	2.917	0.970	1.953	172
DFT-B3PWa$^e$	-3.629	2.950	0.962
MP2$^f$	-4.995	2.917	0.966	1.958	172
CASSCF/aug-cc-pVDZ$^g$		3.084	0.948	2.143	172
CCSD(T)$^h$	-4.98	2.925			175.7
CCSD(T)$^i$	-4.96	2.895
Exp.	-5.4$\pm$0.7$^j$	2.946$^k$			174$^l$
	-5.4$\pm$0.2$^m$	2.952$^n$

$^a$ CP simulations with a norm conserving pseudo-potential and 70 Ry and 150 Ry plane wave cutoff [51] $^b$ Kim and Jordan [86]. $^c$ Proynov, Sirois and Salahub [41]. $^d$ GGA fitted to a set including the water dimer, BSSE corrected [87]. $^e$ Beckes hybrid exchange + Perdew-Wang correlation parameterized also for the water dimer, BSSE corrected [40]. $^f$ $\Delta E$ from ref [88] (MP2, 444 AOs, CP corrected) and geometry from ref [89](frozen core, counterpoise corrected); one of the many accurate MP2 results (see text). $^g$ Complete active space of 16 electrons in 12 orbitals (35793 configuration state functions) computation from ref xandun93. $^h$ Schütz et al. [91]. $^i$ Halkier et al. [92](aug-cc-pVTZ, frozen core, rigid monomers). $^j$ ref BeBaTh. $^k$ Experimental microwave result for R $_{0,\mathrm{OO}}$ (=2.976 Å) corrected for anharmonicy, ref oddy80. $^l$ ref CuFrBl. $^m$ ref ReWaKl82. $^n$ Another anharmonicy correction[97] based on the experiment of ref oddy80.

The water dimer H$_2$O-H$_2$O and the H$_2$O-Cl$^-$ complex served as a second validation of CP-PAW. The water dimer has been extensively used as a test model for the hydrogen bond description. A small selection of literature data together with our results are compiled in table 3.2. Large basis set MP2 calculations yield an interaction energy of $\Delta E = -5.0$ kcal/mol[98] and an oxygen-oxygen distance of $R_{\mathrm{OO}}$ = 2.92 Å. The best theoretical estimates are probably given by the CCSD(T)/aug-cc-pVTZ calculation of Halkier et al. [92] extrapolating for the CCSD(T) limit to $R_{\mathrm{OO}} = 2.90$ Å and $\Delta E = -5.0~\pm 0.1$ kcal/mol and also the CCSD(T) result by Schütz et al. ( $R_{\mathrm{OO}} = 2.925$ Å and $\Delta E = -4.98 \pm 0.02$ kcal/mol) and Klopper et al. ( $\Delta E = -5.057$ kcal/mol). The discrepancy of the experimental results ( $\Delta E^{\mathrm{exp}} = -5.4\pm 0.2$ kcal/mol, $R_{\mathrm{OO}}^{\mathrm{exp}}$ = 2.95 Å) with these results is attributed by Schütz et al. to an underestimation of the anharmonicy corrections in the experimental result.

Compared to the high-level ab initio results, the computationally less demanding DFT methods yield similar results. Our CP-PAW results agree very well with the work of Sprik, Hutter and Parrinello [51], who recommended the BP (and BLYP) functional for water simulations. If the larger plane wave basis set of 50 Ry is used, the CP-PAW result for $\Delta E$ is 0.34 kcal/mol less negative than the BP/aug-cc-pVDZ result of Kim and Jordan [86] and the BP/TZVP work of Proynov, Sirois and Salahub [41]. Our ADF computation results in a 0.25 kcal/mol stronger interaction. The advanced BLAP3 functional (which combines Beckes GGA exchange functional [36] with the 4-parameter LAP3 correlation functional which includes also second order derivatives of the density) returns virtually the same interaction energy as the BP functional. Note that the oxygen-oxygen bond length is overestimated with BLAP3 even though the water dimer was included in the parameter fitting set. Using the Perdew-Wang exchange functional [99] in combination with LAP3 gives $R_{\mathrm{OO}} = 2.950$ Å [41]. We conclude that for the water dimer CP-PAW gives satisfactory results for our purposes.

A molecular simulation of the S$_\mathrm{N}$2 reaction (i) involves the solvation of CH$_3$Cl, Cl$^-$ and [Cl$\cdots$CH$_3 \cdots$Cl]$^-$. An accurate simulation requires therefore a good description of the strong hydrogen bonds between water and the electronegative chlorine compounds. Combariza and Kestner[100] have pointed out that proposed empirical force fields for this interaction seem to have serious deficiencies. This is reflected in inaccurate geometries for small clusters Cl$^-$(H$_2$O)$_n$ when compared to experimental evidence and correlated quantum chemical calculations on the MP2 or DFT level.

Table 3.3: Water-chloride bonding energy (kcal/mol) and geometry (Å and deg) calculated with CP-PAW and ADF and compared to other methods.

	$\Delta E_{\mathrm{intermolec.}}$	$R_{\mathrm{HCl}^-}$	$R_{\mathrm{OH}}$	$R_{\mathrm{OH^{\prime}}}$	$R_{\mathrm{OCl}^-}$	$\angle_{\mathrm{OHCl}^-}$	$\angle_{\mathrm{HOH^{\prime}}}$
PAW-BP/30Ry	-15.48	2.090	1.028	0.984	3.115	174.9	101.6
ADF-BP/TZDP	-16.12	2.075	1.014	0.970	3.084	173.0	100.9
DFT-P86/DZVP$^a$		2.15	1.01	0.98
DFT-B3LYP$^b$	-14.2	2.16	0.99	0.96	3.15	168.66	101.39
MP2/aug-cc-pVTZ$^c$	-14.6	2.116	0.991	0.961	3.094	168.9	100.6
MP4/aug-cc-pVTZ$^c$	-14.54	2.125	0.991	0.963	3.103	168.7	100.7
Exp.	-15.0$^d$ -15.2$^{e}$

$^a$ Ref dunbar95 (the type of exchange functional remains unclear in the article). $^b$ DFT results using a 6-31++G(3d,p) basis on water and for Cl$^-$ the basis set of McLean and Chandler augmented with diffuse sp-function and d-functions, ref comkes95. $^c$ BSSE corrected results from Xantheas [102]. $^d$ $\Delta H$ = -14.7 $\pm$ 0.6 kcal/mol from the mass spectroscopy results from ref himiya88 minus the zero point energy correction of $\Delta E^{{\rm ZPE}}=0.3$ kcal/mol from ref xantheas96. $^e$ $\Delta H=-14.9$ kcal/mol both found by Sieck [104] and Yamabe et al. [105] using mass spectroscopy minus $\Delta E^{{\rm ZPE}}=0.3$ kcal/mol[102].

Here we will just consider the simplest (n=1) H$_2$O$\cdots$Cl$^-$ complex. Results are listed in table 3.3. The CP-PAW result for the interaction energy is 0.64 kcal/mol less than the prediction of ADF and the largest geometry difference is found for the $R_{\mathrm{OCl}^-}$ equal to 0.031 Å. DFT-BP slightly overestimates the interaction when compared to experiment and performs on the same level as DFT-B3LYP and MP2/MP4. The angle $\angle_{\mathrm{OHCl}^-}$ found with DFT-BP agrees within 1 degree with other theoretical methods methods.

The overall conclusion is that for the water-water and water-anion interactions CP-PAW provides a sufficiently accurate DFT-BP result. In turn, DFT-BP performs as well as MP2/MP4/B3LYP, which results are all close to the experimental data.