RM1 Semiempirical Quantum Chemistry: Parameters for Trivalent Lanthanum, Cerium and Praseodymium.

The RM1 model for the lanthanides is parameterized for complexes of the trications of lanthanum, cerium, and praseodymium. The semiempirical quantum chemical model core stands for the [Xe]4fn electronic configuration, with n =0,1,2 for La(III), Ce(III), and Pr(III), respectively. In addition, the valence shell is described by three electrons in a set of 5d, 6s, and 6p orbitals. Results indicate that the present model is more accurate than the previous sparkle models, although these are still very good methods provided the ligands only possess oxygen or nitrogen atoms directly coordinated to the lanthanide ion. For all other different types of coordination, the present RM1 model for the lanthanides is much superior and must definitely be used. Overall, the accuracy of the model is of the order of 0.07Å for La(III) and Pr(III), and 0.08Å for Ce(III) for lanthanide-ligand atom distances which lie mostly around the 2.3Å to 2.6Å interval, implying an error around 3% only.

Introduction

Lanthanum complexes find their usage as catalysts, for example, in the transesterification of triglycerides to monoesters [1], important in the making of biodiesel fuel, in the synthesis of novel antioxidants with high superoxide scavenging activity [2], in asymmetric epoxidation reactions [3], in P4 activation by lanthanum naphthalene complex [4], etc. Furthermore, lanthanum complexes may serve as extreme pressure lubrication additives in paraffin oil [5], they may display pH sensitivity [6], and are of interest to studies on chelator design [7] and polymer build up [8].

Cerium(III) complexes display low toxicity when compared to other lanthanide ions and are, for example, of interest in the design of new drugs targeting DNA [9]. They, of course, may also be used as catalysts, for example in the catalytic cleavage of phosphate esters, an important reaction which mimetizes the hydrolytic cleavage of DNA [10]. Also, due to the relative ease by which they can convert to Ce(IV), Ce(III) complexes may act as antioxidation agents, for example, as a hydroxyl radical quencher in fuel cell electrolyte membranes [11]. The structure of novel Ce(III) complexes are also thoroughly studied due to their several potential applications [12,13].

The 4f2 electronic configuration of Pr(III) gives rise to a series of electronic states, and therefore luminescence of Pr(III) covers a wide range of wavelengths, ranging from the ultraviolet to the near infrared. However, Pr(III) luminescence is rarely observed in the visible region[14]–the most important use of Pr(III) complexes in luminescent and electroluminescent devices being as near infrared emitters [15]. Biological activities of praseodymium complexes have also been observed due to their substantial affinities for many biomolecules, as are the cases of DNA-binding [16] and their presence in Schiff-base complexes [17]. Further, the large variety of architectures of Pr(III) complexes have been receiving continued interest, as novel complexes are being reported [18], including a complex of Pr(III) with organofluorotitanate ligands with coordination number 12 [19]. Furthermore, larger clusters are being prepared, as is the case of the tetramer [Pr4Cl10(OH)2(thiazole)8(H2O)2], the first thiazole complex of a lanthanide ion reported [20], and of the three-dimensional 5-aminoisophtalate Pr(III) polymeric complex which presents good gas storage capabilities [21].

Therefore, the theoretical modeling of complexes of La(III), Ce(III), and Pr(III) is an open area of research, important for the selection of ligands, of counter ions, of specific coordination geometries, of metal to ligand ratios, of polymer framework topologies, of thermal and photostability, for the fine tuning of optical and magnetic properties, and so on.

Previously, our research group introduced the Sparkle Model [22,23], originally to allow the calculation of Eu(III) complexes within the semiempirical AM1 model [24], together with a prescription to compute the UV-Visible electronic spectra of the complexes using another semiempirical model, INDO/S [25-28]. The model was then successfully applied to the design of luminescent Eu(III) complexes [29-32] and proven useful for the prediction of ligand field parameters [33]. In 2004, we introduced Gaussian functions in the core-core repulsion of our sparkle model in order to make it consistent with AM1, something that greatly improved the accuracy of the model [34]. Thus, in 2005 we introduced the new model, we called Sparkle/AM1 for all trivalent lanthanide ions [35]. Since then, we have further introduced the Sparkle Model for La (III), Ce (III), and Pr (III) for PM3 [36-38], for PM6 [39], for PM7 [40], and RM1 [41].

The sparkle model proved very accurate for more ionic bonds of the hard-hard type, such as the cases of directly coordinated oxygen and nitrogen atoms. However, the sparkle model fails when other types of atoms are directly coordinated to the lanthanide ion, as is the cases of carbon, sulfur, and the heavier halogens.

In order to be able to address all types of bonds between the central trivalent lanthanide ion and its ligands, we introduced in 2013 a new and more perfected model, we called the RM1 model for the lanthanides, and presented parameters for Eu(III), Gd(III), Tb(III) [42], Dy(III), Ho(III), and Er(III) [43]. In this model, we considered the lanthanide ion to be a regular neutral atom within RM1 [44] where the semiempirical core represents the electronic configuration [Xe]4fn, with n varying from 0 for La (III) to 14 for Lu (III), and we add three electrons to a valence shell comprised of the semiempirical atomic orbitals 5d, 6s, and 6p. This new RM1 model for the lanthanides proved to be very general and capable of much more accurately describing the multitude of different bonds that show up in lanthanide chemistry.

In the present article, we further extend the RM1 model for the lanthanides to complexes of La(III), Ce(III), and Pr(III).

Method

As indicated before, the RM1 model for the lanthanides assumes that the electronic configuration [Xe]4fn with n = 0,1,2, for La(III), Ce(III) and Pr(III), respectively, can be correctly described by the semiempirical core of charge +3. In addition, the model attaches a set of semiempirical 5d, 6s, and 6p orbitals to describe the valence shell, which always contains 3 electrons for all lanthanide trications. As a result, 22 parameters need to be optimized for each of the lanthanides.

A usual and recurrent criticism of semiempirical models is that they tend to perform well for systems for which they were parameterized, and tend to perform poorly or even badly for other systems. In order to minimize this, we created, in our research group, a method of parameterization which seeks to obtain much more robust models [34,35,45]. We start by collecting all existing complexes of the lanthanide ion of interest that can be found in the Cambridge Structural Database (CSD) [46-48]. In order to guarantee quality in our parameters, we restrict ourselves to collect only complexes of high crystallographic quality (R < 0.05). Of course, we understand that, due to their unique characteristics, each lanthanide metal has a particular palette of applications, each requiring their own specific type of ligands. Therefore, we assume that the more useful complexes will be naturally more numerous in the universe of high quality structures of the CSD database for each particular metal. Having collected that, we note that there is no point in parameterizing the model for all existing high quality CSD complexes simultaneously because, there could be many repeating ligands, which would be overrepresented in the parameterization set and which could cause an imbalance in the parameters. Therefore, at this point, we need to select sub-sets of complexes to serve as parameterization sets. In addition, this selection must take into account the relative difficulty of predicting, from quantum chemical calculations, the geometries of the complexes. For the purpose of this selection, we assume that a good measure of this difficulty is the difference between the crystallographic geometries and geometries obtained by our previous model Sparkle/AM1 [37,38,49]. Thus, for each complex i, we define the following measure Ri: where d refers to distances, and θ to angles; CSD refers to data obtained from the CSD, and Calc refers to data obtained from our previous model calculation (Sparkle/RM1); j runs over all types of bonds, e.g. Ln-O, Ln-N, Ln-C, etc, and k runs over all bonds of the j type; is the standard deviation of all differences between CSD and Sparkle/AM1 (calc) for all bonds of the j type; l runs over all angles; and is the standard deviation of all angle differences between CSD and Sparkle/AM1 (calc). The set of measures Ri was then used as input for a divisive hierarchical clustering analysis, DIANA [50], from which we selected two parameterization sets from the universes of complexes for each lanthanide metal: one we call the small set, with only 15 complexes for La(III), 8 complexes for Ce(III), and 7 complexes for Pr(III); and another one we call the large parameterization set, with 38 complexes for La(III), 18 for Ce(III), and 16 for Pr(III). The next step, is the optimization of the model where, by means of a combination of a few non-linear optimization techniques, we seek to minimize a response function, which is the sum of all Ris of Eq (1), with the difference that calc will now refer to the particular distance or angle calculated by means of the intermediary set of parameters of the optimization procedure. When the nonlinear optimization process converges for the small set of complexes, we start it all over again with the large set. Finally, we declare the process of nonlinear optimization to be finished when it converges for the large set.

Assessments of the accuracy of the model can be made via the unsigned mean error, UMEi, defined for each complex i as where CSD and Calc are as in Eq (1), and the summation runs over all the n bonds being considered. As before, we use two different measures: UME(Ln-L)i and UMEi. The first contains all j distances between the lanthanide ion and its directly coordinated atoms. The second, includes, in addition, all distances between all directly coordinated atoms and indirectly also reflects a measure of the accuracy of the predicted angles within the coordination polyhedron.

The next step in verifying the robustness of the parameterization is to determine if the distribution of unsigned mean deviations between the predicted and crystallographic geometries can be adequately described by a gamma distribution function. That can be ascertained, by means of the one-sample nonparametric Kolmogorov-Smirnoff test whose p-value must be larger than 0.05, indicating that usage of the mean and variance of the gamma distribution fit as accuracy measures of the models are statistically justified within a 95% level of confidence.

Table 1 presents the three sets of 22 RM1 parameters found for Ln(III), Ce(III), and Pr(III). And Tables 2 and 3 present the mean and variance of the gamma distribution fits for the both types of unsigned mean errors for the universe of complexes, together with the p-value which is larger than 0.05 for all cases. All that indicates that the RM1 models here advanced for La(III), Ce(III), and Pr(III) are capable of predicting the geometries of the corresponding complexes in a reliable manner, and that the eventual deviations from the experiment behave as random around the correct values.

Table 1

Parameters* for the RM1 model for the trications of La, Ce and Pr.

RM1
	La3+	Ce3+	Pr3+
U ss	-14.68043413	-14.71938888	-14.52408063
U pp	-6.73473860	-7.68942949	-7.05682683
U dd	-20.48996706	-20.45157682	-20.68932756
ζ s	1.27267748	1.28102807	1.53803892
ζ p	1.42327584	1.42536636	1.58164715
ζ d	1.41036886	1.41286566	1.37490374
β s	-7.66955512	-7.66878654	-7.94799309
β p	0.47769647	0.44444183	0.85381597
β d	-3.71147661	-3.74493844	-3.83029281
F0SD	7.72081332	7.71512874	7.61830081
G2SD	3.91674532	3.91829281	3.96586318
POC	1.87517566	1.87508416	1.84741576
α	1.28404676	1.28623319	1.28060282
ZSN	0.78452960	0.81941318	0.78194425
ZPN	1.50661174	1.43034873	1.29298193
ZDN	1.17206190	1.19321984	0.98960461
a 11	0.62855906	0.66196096	0.45337932
b 21	7.86084906	7.89025802	7.82319885
c 31	1.30447476	1.26282727	1.56516739
a 12	0.08164180	0.07599228	0.01039415
b 22	10.34685773	10.31610935	10.28836535
c 32	3.24704021	3.24781703	3.26870321

*Parameters are s, p, and d atomic orbital one-electron one-center integrals U , U and U ; the s, p, and d Slater atomic orbital exponents ξ , ξ , and ξ ; the s, p, and d atomic orbital one-electron two-center resonance integral terms β , β , and β ; the core-core repulsion term α; the two-electron integrals F0 SD, G2 SD; and the additive term ρcore needed to evaluate core-electron and core-core nuclear interactions; the second set of exponents to compute the one-center integrals ξ ’, ξ ’, and ξ ’; and the six parameters for the two Gaussian functions: height, ai; inverse broadness, bi; and displacement, ci; as in where R, is the interatomic distance between the lanthanide and the other atom.

Table 2

Means and variances of the γ distribution fits for the UME(Ln-L)s computed for the N complexes for each lanthanide trication.

UME(Ln-L)s
lanthanide ion	N	mean (Å)	variance (Å2)	p-value
La3+	84	0.0710	0.0032	0.1418
Ce3+	57	0.0805	0.0040	0.9630
Pr3+	65	0.0685	0.0037	0.6141

Table 3

Means and Variances of the γ distribution fits for the UMEs computed for the N complexes for each lanthanide trication.

UMEs
lanthanide ion	N	mean (Å)	variance (Å2)	p-value
La3+	84	0.1524	0.0276	0.7502
Ce3+	57	0.1621	0.0373	0.2089
Pr3+	65	0.1790	0.0469	0.5420

Results and Discussion

Table 4 presents unsigned mean errors for each of the specific types of distances between the lanthanide ion and its directly coordinated atoms found in the universe of complexes for La(III), both for the present RM1 model for the lanthanides and for each of the previous sparkle models. In order to facilitate interpretation of the table, the smallest error in each line is being bolded. Clearly, for dinuclear complexes, the La-La bond is more accurately predicted by Sparkle/PM3. However, its error is relatively close to the RM1 error. The same happens for La-O bonds, where Sparkle/PM3 is again the best model. However, its unsigned mean error of 0.0610Å is too close to the RM1 error of 0.0698Å. However, for all other distances, RM1 presents the smallest errors while the previous Sparkle models sometimes display huge errors as is the case of La-S bonds when the average errors of the Sparkle models is 0.4345Å, a value more than 6 times larger than the RM1 error of 0.0680Å. In Table 4, La-L refers to the unsigned mean error of all distances of all types between the central lanthanum ion and its directly coordinated other atoms, whereas L-L includes all interatomic distances between all directly coordinated atoms, and is, indirectly, a measure of the angles within the coordinated polyhedron. Clearly, RM1, with its unsigned mean error of 0.1704Å is 52% smaller than the average of the previous sparkle models, a situation similar to what happens to the next unsigned mean error, which includes all 5315 types of distances for all lanthanum complexes considered: La-L, La-La, and L,L’, when RM1 displays an error which is 56% smaller than the average error of all previous sparkle models.

Table 4

Sparkle/AM1, Sparkle/PM3, Sparkle/PM6, Sparkle/PM7, Sparkle/RM1 and RM1 unsigned mean errors for lanthanum(III) complexes.

Type of distances	unsigned mean errors for specific types of distances (Å)
	N	RM1	Sparkle
			AM1	PM3	PM6	PM7	RM1
La—La	13	0.1807	0.2056	0.1404	0.2153	0.1900	0.2963
La—O	580	0.0698	0.0853	0.0610	0.0767	0.0962	0.1252
La—N	205	0.0487	0.0541	0.0873	0.0685	0.2437	0.0673
La—C	119	0.0980	0.2193	0.2681	0.2120	0.2771	0.2114
La—S	15	0.0680	0.4018	0.4302	0.3847	0.5240	0.4316
La—Cl	26	0.0946	0.3033	0.3441	0.2836	0.3693	0.3517
La—Br	4	0.0957	0.3974	0.4282	0.4009	1.7007	0.4742
La—L	962	0.0710	0.1092	0.1087	0.1044	0.1697	0.2144
L—L	4353	0.1704	0.2334	0.2300	0.2521	0.2857	0.2941
La-L, La—La and L-L’	5315	0.1524	0.2110	0.2081	0.2255	0.2648	0.2809

Tables 5 and 6 show equivalent results for Ce(III) and for Pr(III) complexes. For Ce(III) complexes, RM1 is more accurate when compared to the other sparkle models with respect to all measures except for Ce-O distances. Likewise, for Pr(III) complexes, RM1 is more accurate than the other sparkle models except for Pr-O and Pr-N distances. Even in these cases, the accuracy of RM1 is close to the best accuracy available from the other sparkle models. A noteworthy case, are the distances of the three lanthanide ions and bromine, where Sparkle/PM7 displays enormous unsigned mean errors, larger than 1Å, suggesting there might be perhaps some difficulties with the parameterization of bromine in PM7.

Table 5

Sparkle/AM1, Sparkle/PM3, Sparkle/PM6, Sparkle/PM7, Sparkle/RM1 and RM1 unsigned mean errors for cerium(III) complexes.

Type of distances	unsigned mean errors for specific types of distances (Å)
	N	RM1	Sparkle
			AM1	PM3	PM6	PM7	RM1
Ce—Ce	5	0.1781	0.2003	0.2022	0.2811	0.1271	0.1484
Ce—O	283	0.0900	0.0787	0.0746	0.1662	0.1564	0.0800
Ce—N	111	0.0622	0.0836	0.0708	0.0580	0.0636	0.0753
Ce—C	185	0.0800	0.2307	0.2524	0.1477	0.3390	0.2602
Ce—S	11	0.0434	0.4792	0.4607	0.3801	0.3074	0.4380
Ce—Cl	17	0.0609	0.2647	0.2882	0.1999	0.1604	0.2969
Ce—Br	9	0.0378	0.3757	0.3823	0.3159	1.4764	0.3934
Ce—L	622	0.0805	0.1406	0.1436	0.1641	0.2054	0.1434
L—L	3030	0.1789	0.2421	0.2507	0.2504	0.3078	0.2597
Ce—L, Ce—Ce and L—L’	3652	0.1621	0.2249	0.2326	0.2373	0.2903	0.2399

Table 6

Sparkle/AM1, Sparkle/PM3, Sparkle/PM6, Sparkle/PM7, Sparkle/RM1 and RM1 unsigned mean errors for praseodymium(III) complexes.

Type of distances	unsigned mean errors for specific types of distances (Å)
	N	RM1	Sparkle
			AM1	PM3	PM6	PM7	RM1
Pr—Pr	11	0.1770	0.2278	0.1998	0.2232	0.2673	0.2229
Pr—O	421	0.0736	0.0786	0.0841	0.0732	0.1044	0.0723
Pr—N	119	0.0626	0.1183	0.1014	0.0714	0.0534	0.1223
Pr—C	143	0.0552	0.2446	0.2536	0.2322	0.3591	0.2211
Pr—S	12	0.0787	0.3817	0.3870	0.4304	0.5795	0.3898
Pr—Cl	39	0.0526	0.2549	0.2739	0.2899	0.2975	0.3700
Pr—Br	6	0.0251	0.3460	0.3509	0.3814	1.2262	0.4627
Pr—L	751	0.0685	0.1348	0.1376	0.1248	0.1738	0.1318
L—L	3462	0.2029	0.2621	0.2401	0.2628	0.2893	0.2566
Pr—L, Pr—Pr and L—L’	4213	0.1790	0.2394	0.2218	0.2382	0.2687	0.2345

Figs 1–10 present the data in Tables 3–5 in pictorial form in order to provide the user with an instant comprehension of the relative accuracies of the present RM1 model for the lanthanides (light green bar) and the previous sparkle models. Note that the Sparkle/RM1 green bar lies next, followed by the blue bar of Sparkle/PM7. Thus, the situation highlighted above on the inadequacy of Sparkle/PM7 for any of the parameterized lanthanide ions when directly bonded to bromine can be immediately detected in Fig 7. On the bright side, the good accuracy of all previous sparkle models for directly coordinated oxygen and nitrogen bonds is very clearly manifested (Figs 2 and 3).

Fig 1

UME(Ln-Ln)s obtained using the RM1 model for the lanthanides and all five versions of the Sparkle Model: Sparkle/RM1, Sparkle/PM7, Sparkle/PM6, Sparkle/PM3 and Sparkle/AM1 for all complexes of the universe set for each of the lanthanide trications: La(III), Ce(III) and Pr(III).

The UMEs are calculated as the absolute value of the difference between the experimental and calculated Ln-Ln interatomic distances, summed up for all complexes, for each of the lanthanides.

Fig 10

UMEs obtained using the RM1 model for the lanthanides and all five versions of the Sparkle Model: Sparkle/RM1, Sparkle/PM7, Sparkle/PM6, Sparkle/PM3 and Sparkle/AM1 for all complexes of the universe set for each of the lanthanide trications: La(III), Ce(III) and Pr(III).

Fig 7

UME(Ln-Br)s obtained using the RM1 model for the lanthanides and all five versions of the Sparkle Model: Sparkle/RM1, Sparkle/PM7, Sparkle/PM6, Sparkle/PM3 and Sparkle/AM1 for all complexes of the universe set for each of the lanthanide trications: La(III), Ce(III) and Pr(III).

The UMEs are calculated as the absolute value of the difference between the experimental and calculated Ln-Br interatomic distances, summed up for all complexes, for each of the lanthanides.

Fig 2

UME(Ln-O)s obtained using the RM1 model for the lanthanides and all five versions of the Sparkle Model: Sparkle/RM1, Sparkle/PM7, Sparkle/PM6, Sparkle/PM3 and Sparkle/AM1 for all complexes of the universe set for each of the lanthanide trications: La(III), Ce(III) and Pr(III).

The UMEs are calculated as the absolute value of the difference between the experimental and calculated Ln-O interatomic distances, summed up for all complexes, for each of the lanthanides.

Fig 3

UME(Ln-N)s obtained using the RM1 model for the lanthanides and all five versions of the Sparkle Model: Sparkle/RM1, Sparkle/PM7, Sparkle/PM6, Sparkle/PM3 and Sparkle/AM1 for all complexes of the universe set for each of the lanthanide trications: La(III), Ce(III) and Pr(III).

The UMEs are calculated as the absolute value of the difference between the experimental and calculated Ln-N interatomic distances, summed up for all complexes, for each of the lanthanides.

UME(Ln-C) obtained using the RM1 model for the lanthanides and all five versions of the Sparkle Model: Sparkle/RM1, Sparkle/PM7, Sparkle/PM6, Sparkle/PM3 and Sparkle/AM1 for all complexes of the universe set for each of the lanthanide trications: La(III), Ce(III) and Pr(III).

The UMEs are calculated as the absolute value of the difference between the experimental and calculated Ln-C interatomic distances, summed up for all complexes, for each of the lanthanides.

UME(Ln-S) obtained using the RM1 model for the lanthanides and all five versions of the Sparkle Model: Sparkle/RM1, Sparkle/PM7, Sparkle/PM6, Sparkle/PM3 and Sparkle/AM1 for all complexes of the universe set for each of the lanthanide trications: La(III), Ce(III) and Pr(III).

The UMEs are calculated as the absolute value of the difference between the experimental and calculated Ln-S interatomic distances, summed up for all complexes, for each of the lanthanides. There are no Ho-S distances in the universe of Ho(III) complexes considered.

UME(Ln-Cl)s obtained using the RM1 model for the lanthanides and all five versions of the Sparkle Model: Sparkle/RM1, Sparkle/PM7, Sparkle/PM6, Sparkle/PM3 and Sparkle/AM1 for all complexes of the universe set for each of the lanthanide trications: La(III), Ce(III) and Pr(III).

The UMEs are calculated as the absolute value of the difference between the experimental and calculated Ln-Cl interatomic distances, summed up for all complexes, for each of the lanthanides.

UME(L-L’)s obtained using the RM1 model for the lanthanides and all five versions of the Sparkle Model: Sparkle/RM1, Sparkle/PM7, Sparkle/PM6, Sparkle/PM3 and Sparkle/AM1 for all complexes of the universe set for each of the lanthanide trications: La(III), Ce(III) and Pr(III).

The UMEs are calculated as the absolute value of the difference between the experimental and calculated interatomic distances between the coordinated atoms, L-L’, summed up for all complexes, for each of the lanthanides.

Finally, the raw data used to arrive at the values presented in Tables 4–6, can be found in Tables 7, 8, and 9, which show individual unsigned mean errors for each of the complexes considered, and identifies by a underlined and bolded codes, the complexes used in the small and large parameterization sets.

Table 7

Unsigned mean errors, UME(Eu-L)s and UMEs, for RM1 model for lanthanides, as compared to the respective experimental crystallographic values, obtained from the Cambridge Structural Database, for each of the 84 lanthanum(III) complexes.

Structure	RM1		Structure	RM1
	UME(La-L)s (Å)	UME (Å)		UME(La-L)s (Å)	UME (Å)
ABXALA	0.0684	0.1722	NERQUW	0.1504	0.2095
ALANIC	0.0676	0.1159	NOHNIH	0.0479	0.1353
APBNLA	0.0513	0.1067	OFEGIP	0.0304	0.1433
AXOMOP	0.1060	0.1183	PAFNEP	0.0394	0.1035
AYULUB	0.1187	0.1565	PIBGOW	0.0679	0.2653
BEQPOC	0.0593	0.2330	PIRSEO	0.0879	0.1403
BIZTIN	0.0614	0.2370	POHDUL	0.0640	0.2192
BOKZUX	0.0536	0.1135	PUHYAS	0.0488	0.1787
BUVVIX01	0.0559	0.1049	PUWZIQ	0.0615	0.1252
CABLAS01	0.0603	0.1862	PUZHOH	0.0406	0.1630
CEFQOT	0.0619	0.0820	QAKWEE	0.0499	0.1854
CESRUO	0.0983	0.1775	QAPXAG	0.0590	0.1403
COTDOF	0.1244	0.1221	QUBWIT	0.0432	0.0859
DUBWEC	0.0401	0.1174	RIWQOE	0.0245	0.3574
DUCBOS	0.0688	0.1450	SILWEQ	0.0486	0.3046
EBEGOH	0.0530	0.1483	SIXBIK	0.0515	0.1749
EPAILA	0.0398	0.1493	SUXLIG	0.0435	0.0876
EZIPUY	0.0094	0.1105	SUZXIU	0.0672	0.1007
FABPUT	0.0875	0.1234	TEPSOW	0.0758	0.2000
FICJEG	0.0410	0.2938	TEQBIA	0.0559	0.2320
FIVCIW	0.0262	0.0392	TEQBOG	0.0556	0.1459
FURLOT	0.0976	0.1405	TUPWEG	0.0552	0.0808
GIMMIY	0.0487	0.2507	VUBLIN	0.1069	0.2255
GOJQAX	0.1064	0.1765	WAVNAI	0.0551	0.0889
GOZBEC	0.0660	0.1429	WEHTAE	0.0648	0.2119
GULFOI	0.1129	0.1787	XALSOS	0.0808	0.1125
HAMYUP	0.1216	0.1383	XAWVUM	0.0710	0.1040
HELHOV	0.0690	0.0958	XECQEB	0.0540	0.1164
HELMIU	0.0543	0.2287	XEMNUY	0.1021	0.1723
HETALA11	0.0600	0.1263	XERCAY	0.1141	0.1245
HUQBAX	0.0606	0.1112	XONXUT	0.0427	0.0738
IDAJON	0.1064	0.2217	XUJTOL	0.0587	0.1160
IKUWER	0.0815	0.1222	YUCXAV	0.0512	0.2157
KIXHAA	0.0504	0.1331	ZAMHEA	0.0585	0.2756
LANITA	0.0379	0.0983	ZAZQAS	0.0694	0.1013
LAPTEB10	0.1106	0.1662	ZEHTUB	0.1072	0.1479
LIWQEN	0.0502	0.1240	ZEJFOJ	0.0564	0.1001
MENQOL	0.0297	0.1359	ZEQVUM	0.0997	0.2726
MILWEJ	0.0426	0.1003	ZIDSOX	0.1210	0.1691
NASLUO	0.1577	0.1606	ZIQXIG	0.1029	0.2243
NASTOQ	0.0391	0.0803	ZULFOB	0.0432	0.1534
NEHDAF	0.0656	0.0698	ZUWFOM	0.0585	0.0812

Table 8

Unsigned mean errors, UME(Gd-L)s and UMEs, for RM1 model for lanthanides, as compared to the respective experimental crystallographic values, obtained from the Cambridge Structural Database, for each of the 57 cerium(III) complexes.

Structure	RM1		Structure	RM1
	UME(Ce-L)s (Å)	UME (Å)		UME(Ce-L)s (Å)	UME (Å)
ABETEI	0.0570	0.1694	JOLYAK	0.0966	0.1260
ABETUY	0.0508	0.1141	KEDCAX	0.0507	0.1249
AFURUO	0.0709	0.1398	KIXXOE	0.0223	0.0632
APSBCE	0.1310	0.2886	LELBOT	0.0838	0.1315
BABZIN	0.0659	0.2240	LIFHUD	0.0492	0.1293
CIBSAH	0.0599	0.1059	LIKFUH	0.0649	0.3247
CUMCIW	0.0825	0.1876	MIPTAG	0.0728	0.1203
DESYAC	0.0843	0.1290	NATCIW	0.1034	0.1928
DEWDEP	0.0736	0.1434	NOJTAH	0.0933	0.1923
EJIPES	0.0643	0.1081	OXDACE	0.1151	0.2071
ETOQUZ	0.0904	0.1280	PEKWEH	0.0347	0.1101
ETOROU	0.1048	0.1430	PIDBAF	0.1492	0.1750
FEPKAN	0.0640	0.0734	PUTQAW	0.0986	0.2408
FILKEQ	0.0692	0.0982	QEGVOO	0.0789	0.1403
FOTQOV	0.1048	0.3579	RIWRAR	0.0281	0.3782
FUHFEZ	0.0494	0.0950	SASCEV	0.1177	0.1556
GACJIE	0.0718	0.1022	TIJCIY	0.0779	0.1040
GAPFIM	0.1105	0.2786	UKAPEB	0.0460	0.1627
GEGZEX	0.1674	0.1814	ULUQUN	0.0489	0.1076
GETLOG	0.0535	0.0793	VAKJAS	0.0363	0.0972
GINNUM	0.0786	0.1197	VAPCAQ	0.0409	0.1133
HIDLUB	0.1094	0.1252	WAVTAQ	0.1294	0.1695
HIXWEQ	0.0772	0.1228	WOPHAL	0.0510	0.1331
HUMDOI	0.0890	0.2701	XASROZ	0.1168	0.1635
HURRAN	0.1177	0.2186	XEXCUY	0.1015	0.5062
INDCEP	0.0996	0.1544	XOLMAM	0.0464	0.2421
JAPPUL	0.0932	0.2866	XONYAA	0.0601	0.0871
JEXXOZ	0.0762	0.2089	ZUNMAW	0.0441	0.0663
JOCCUA	0.0937	0.1357

Table 9

Unsigned mean errors, UME(Tb-L)s and UMEs, for RM1 model for lanthanides, as compared to the respective experimental crystallographic values, obtained from the Cambridge Structural Database, [46–48] for each of the 65 praseodymium(III) complexes.

Structure	Method RM1		Structure	Method RM1
	UME(Pr-L)s (Å)	UME (Å)		UME(Pr-L)s (Å)	UME (Å)
ACURLB	0.0503	0.1516	KOBRUO	0.0587	0.1110
BABZOT	0.0460	0.1984	LEJSOI	0.0379	0.1816
BAFYOX	0.1280	0.3584	LIYFIJ	0.0583	0.1196
BIFYUK	0.0454	0.2495	MIPTEK	0.0966	0.0773
BUVWIY01	0.0562	0.1794	MOGFUJ	0.0400	0.2012
CAZGUF	0.0290	0.1338	NEPVAF	0.0481	0.1168
CESROI	0.0458	0.1909	NEPVUZ	0.0438	0.1765
CUMCOC	0.0722	0.1834	NPYPRP10	0.0658	0.1038
DEWDIT	0.0348	0.1324	PEHHIP	0.0649	0.1191
DIYMUT	0.0320	0.1600	PEHXIJ	0.0467	0.3069
DORDIX	0.0594	0.2030	PELGOC	0.0459	0.1514
DUCHAK	0.0707	0.1391	POGWIR	0.1623	0.2897
ECABAL	0.0661	0.0767	POPJAF	0.0536	0.2040
EFUJEU	0.0268	0.0683	PUQNOF	0.1467	0.1709
EJINUG	0.0330	0.0860	QIMRIN	0.1260	0.2671
EWIROR	0.1045	0.3750	QOBBIS	0.0675	0.1509
FAGYIW	0.0857	0.3122	QOVXII	0.0516	0.1505
FATWOM	0.0783	0.2212	QOZVEG	0.0591	0.1323
FEDYAO	0.0764	0.0899	RASROS	0.0498	0.2181
GIWWEO	0.0764	0.1152	RUGQUF	0.0724	0.3452
GUMXIW	0.1048	0.1463	SERWOB01	0.0383	0.1068
HEDBOH	0.0778	0.2625	VELRUZ	0.0787	0.1039
HEDKAC	0.0648	0.1268	VOXJIB	0.0878	0.1415
HERVUV	0.0810	0.1132	WUWXAN	0.0707	0.2963
HODDOT	0.0601	0.2396	WUWXER	0.0677	0.1117
HODFEL	0.0763	0.2376	XASRUF	0.0540	0.0987
JALMEP	0.0982	0.1785	XAVWUM	0.0664	0.3295
JALMUF	0.1256	0.3801	XOKYIF	0.0587	0.1340
JERWOS	0.0433	0.2205	XULNUO	0.0731	0.3470
JEXXUF	0.1778	0.2405	YOTYUB	0.0979	0.2103
JUSBII	0.0399	0.1257	ZAXSEW	0.0617	0.2579
KAHGEF	0.0766	0.1355	ZULRED	0.0815	0.0805
KAWBIT	0.0170	0.0636

Case Study

The new RM1 model was applied to predict the structure of tetramer of praseodymium, [Pr4Cl10(OH)2(thiazole)8(H2O)2][20]. The RM1 structure was calculated using MOPAC 2009 software and keywords used were the following: RM1 (the Hamiltonian used), PRECISE, GNORM = 0.25, SCFCRT = 1.D-10 (in order to increase the SCF convergence criterion) and XYZ (the geometry optimizations were performed in cartesian coordinates).

Fig 11 shows the overlapping of the RM1 and crystallographic structures. The good match observed visually can be confirmed by the low value obtained for the root mean square deviation (RMSD) of 0.034Å, obtained via a RMSD fit and alignment.

Fig 11

Root mean square deviation fit and alignment of crystallographic and RM1 fully optimized structures of the tetramer of praseodymium, [Pr4Cl10(OH)2(thiazole)8(H2O)2][20].

A detailed analysis reveals that for the RM1 structure, the average bond length between the Pr3+ ions is 4.54Å whereas the average obtained from crystallographic structure is 4.58Å. The UME considering all Pr3+—L distances (where L = Pr3+, O, N and Cl) is 0.12Å. It is important to highlight that the CPU time for the full geometry optimization using the RM1 model was very fast, less than 3 minutes using a laptop core i7 with 8GB of RAM memory.

Conclusion

The overall advantage of the RM1 model for the lanthanides presented in this article is that it can perform a full geometry optimization on a complex such as the tetramer of praseodymium, [Pr4Cl10(OH)2(thiazole)8(H2O)2], with relative ease; something that would be exceedingly difficult for an ab initio type calculation. The same can be said of calculations on the three-dimensional 5-aminoisophtalate Pr(III) polymeric complex, which presents good gas storage capabilities [21]. Even if ab initio calculations would be later needed for specific properties that could not be obtained at useful accuracy levels by any other means, they could be carried out on RM1 optimized geometries—something that could save an enormous amount of computing time and resources.

In conclusion, the previous sparkle models seem to be very good models provided the complex has only nitrogen or oxygen directly coordinated to the lanthanide ion. However, if the complex of interest has other types of atoms directly coordinated to the lanthanide ion, then the RM1 model for the lanthanides, introduced in this article, must be the method of choice.

Instructions on how to run the RM1 model for the lanthanides in MOPAC2012, together with sample calculations on complexes of each of the parameterized lanthanide trications: La(III), Ce(III), and Pr(III).

(DOCX)

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