Simplified box orbitals for molecules containing atoms beyond Ar

Molecular Physics An International Journal at the Interface Between Chemistry and Physics ISSN: 0026-8976 (Print) 1362-3028 (Online) Journal homepage: http://www.tandfonline.com/loi/tmph20 Simplified box orbitals for molecules containing atoms beyond Ar Victor García, David Zorrilla, Jesús Sánchez-Márquez & Manuel Fernández To cite this article: Victor García, David Zorrilla, Jesús Sánchez-Márquez & Manuel Fernández (2018): Simplified box orbitals for molecules containing atoms beyond Ar, Molecular Physics, DOI: 10.1080/00268976.2018.1481543 To link to this article: https://doi.org/10.1080/00268976.2018.1481543 View supplementary material Published online: 06 Jun 2018. Submit your article to this journal Article views: 16 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tmph20 MOLECULAR PHYSICS https://doi.org/10.1080/00268976.2018.1481543 RESEARCH ARTICLE Simplified box orbitals for molecules containing atoms beyond Ar Victor García , David Zorrilla , Jesús Sánchez-Márquez and Manuel Fernández Departamento de Química-Física, Facultad de Ciencias, Campus Universitario Río San Pedro, Universidad de Cádiz, Cádiz, Spain ABSTRACT ARTICLE HISTORY Simplified Box Orbitals (SBOs) are a kind of spatially restricted basis functions with a similar use to Slater functions, but fulfilling an exact version of the zero-differential overlap approximation. These functions also allow for a drastic reduction in the number of bielectronic integrals when dealing with huge systems, and can be adapted to study confined systems such as molecules in solution. In previous studies, the necessary SBOs parameters to be used for different elements were defined. However, the accuracy of those basis functions decreases with the atomic number of the atoms under study, and therefore their use was discouraged beyond the Ar atom. In the present study, we verify that slightly increasing the terms of SBOs for a better definition, is enough to correctly handling atoms beyond Ar. This, together with other improvements exposed in this work, allowed obtaining accurate SBOs for K–Kr atoms. To make possible the use of SBOs in standard quantum chemistry calculation software, Gaussian expansions to the proposed basis functions–were achieved. Then, simple formulas for directly obtaining those expansions were deduced. Finally, the results of an SZ basis set of the proposed SBOs are analysed and compared with a similar STO basis set. Received 14 March 2018 Accepted 18 May 2018 1. Introduction In previous studies [1,2] we introduced a new type of spatially restricted functions named Simplified Box Orbitals (SBOs): SBO(r, θ , ϕ) = RSBO (r) · ϒm (θ , ϕ) (1) with the radial part defined as a piecewise function: ⎧ N ⎪ ⎨  Cj rn (r0 − r)3j for 0 ≤ r < r0 , RSBO (r) = j=1 (2) ⎪ ⎩0 for r ≥ r0 . These functions have led to excellent results in hydrogen-like and helium-like systems [1] as well as in molecules containing H–Cl atoms [2]. The evaluation CONTACT Victor García [email protected] Universidad de Cádiz, Puerto Real, 11510 Cádiz, Spain Spatially restricted basis functions; box orbitals; Gaussian expansion; ab-initio calculations; confined systems of molecular integrals problem was tackled in [2] via Gaussian expansions of the radial part of the SBOs, like Gaussian expansions of the Slater Type Orbitals: STO-nG [3,4]. One of the main benefits of SBOs is the fact that they are rigorously zero beyond a certain distance from the centre to which they are referred: r ≥ r0 ⇒ χ (r, θ , ϕ) = 0. (3) Whenever the distance between the centres P(χ p ) and Q(χ q ) is greater than the sum of the radii associated with χ p (rp ) and χ q (rq ), the integrals involving the product χ p χ q are exactly zero: PQ > (rp + rq ) ⇒ (χp χq |χr χs ) = 0. Departamento de Química-Física, Facultad de Ciencias, Campus Universitario Río San Pedro, Supplemental data for this article can be accessed here. https://doi.org/10.1080/00268976.2018.1481543 © 2018 Informa UK Limited, trading as Taylor & Francis Group KEYWORDS (4) 2 V. GARCÍA ET AL. Therefore, the use of SBOs provides advantages analogous to the zero-differential overlap approximation [5] used in semi-empirical methods, but in an ab-initio context. At this phase of our research, we have selected the values for the r0 and Cj parameters in Equation (2) by making each SBO as similar as possible to an adequate STO. In other words, by applying the least squares condition:  ε = (φ − χ )2 dτ = MINIMUM, (5) where φ = STO and χ = SBO. In our previous study, focused on H–Ar atoms [2], it was proved that by using three-terms SBO [N = 3 in Equation (2)], the STOs results can be properly reproduced for atoms with atomic number not too high. However, the quality of the results obtained with three-terms SBO (SBO3) decrease as the atomic number grows (see Tables S1(a) and S1(b) in the Supporting Information). The mean value of the difference between the energies achieved by using the SBOs and the STOs used in their generation, for H–Kr atoms, is 0.032% for SBO3 and 0.0055% for SBO4 (four terms SBOs). Figure 1 compares the results obtained by using SBO3 and SBO4. The relative error ε(SBO4) regarding the ε(STO) is 0.008% for H–Ar atoms and 0.005% for K–Kr atoms. Consequently, in the present study we have used N = 4 instead of N = 3 in Equation (2) to define the SBOs. The four-term SBOs (SBO4) allow for a more accurate definition of the atom orbitals, as verified for all atoms with 1 ≤ Z ≤ 36 (see exhaustive results achieved in the Supporting Information). Finally, we have obtained SBO4-3G expansions of the exact SBOs and we have found extremely simple formulas to directly obtain those expansions. In this way, we have extended the SBO functions usefulness beyond the Ar atom and, in addition, we have substantially improved the results for systems with H–Ar atoms. 2. Methodology The fit of a four-term SBO function: RSBO (r) = 4  Cj rn (ro − r)3j (0 ≤ r ≤ ro ) (6) j=1 to the radial part of a given STO: φ= rm exp (−ar) (7) has been done by using a proprietary script made for Mathematica [6] named ‘STOtoSBO.nb’ [7] and described in [2]. This software makes it possible to obtain Figure 1. Differences between the energies calculated by using SBO3 and SBO4 and those calculated by using the STO employed to generate them. It can be observed that the error when using SBO3 increases very quickly from the Ti atom, while the error achieved by using the SBO4 is kept always small, and for Z > 15 it is always negative (energy achieved by SBO4 is lower than the achieved by the original STO). MOLECULAR PHYSICS the values of ro and Cj coefficients of the SBO corresponding to each STO of the considered atoms. The STOs with exponents of Clementi and Raimondi [8] for isolated atoms were chosen as starting points of our new four terms SBOs. The SBO4 obtained can be seen in Tables S2–S9 of the Supporting Information. In a second step, we achieved the SBO4-3G expansions by using our software UCA-GSS [9], which generates Gaussian expansions for spatially restricted basis functions directly working with GAUSSIAN [10]. The complete results can be seen in Tables S10–S17 of the Supporting Information. Tables S18–S21 of the Supporting Information show a comparison of the properties: E (Total Energy), eHOAO (Highest Occupied Atomic Orbital Energy), < R**2 > (Electronic spatial extent) and < −V/T > (Virial ratio), calculated for all the atoms with 1 ≤ Z ≤ 36 by using STOs, SBOs and Aug-cc-pVxZ basis sets. The improvement achieved by using SBO4 instead of SBO3 can be appreciated in Figures 1–3. 3 The SBO4-3G expansions for the atoms with 1 ≤ Z ≤ 36, ready to be used with GAUSSIAN [10] software (by using the ‘GEN’ command), are provided at the end of the Supporting Information. In addition, they can easily be built up by using the formulae introduced in the following sections. Each orbital includes the appropriate radius and scale factor for both: isolated atom (SF = 1.00) and the optimal factors to be used in molecular calculations. Tables S18–S21 in the Supporting Information include the results of atomic properties for H–Kr atoms calculated by using STOs and SBOs, compared with those providing results close to the Hartree–Fock limit. 3. Four-term SBOS radii and coefficients From the data shown in the Tables S2–S9 of the Supporting Information, it is easy to see that the radii and coefficients for SBO4 satisfy very accurately the Figure 2. Error % of the HOAO energy calculated for the H–Kr atoms with SBO3 and SBO4, with respect to the energy calculated with the STO that has been employed to generate them. It is observed that the use of 3d orbitals is ‘problematical’ even if four-term SBOs are used. Figure 3. % Error of the < R**2 > calculated for the H–Kr atoms by using SBO3 and SBO4 basis, with regard to the result obtained with the STO that has been employed to generate them. The abnormally good result of the Ti atom with SBO3 is clearly accidental. 4 V. GARCÍA ET AL. relationships: r0 = Cj = 4. Exponents and coefficients of the SBO4-3G expansions r0 (α = 1) , α Cj (r0 = 1) 3j r0 (8) . (9) In Table 1, the ro (α = 1) and Cj (ro = 1) values achieved in [2] for SBO3 are compared with those achieved in this study for SBO4. The use of Equations (8) and (9), with the parameters of Table 1, perfectly reproduces the data shown in Tables S2–S9 of the Supporting Information (obtained by using the STOtoSBO.nb software [7]). The coefficients Cj shown in Table 1 correspond to non-normalised rn (ro −r)3j functions. For normalised functions: χ= 4  Dj [Nj rn (ro − r)3j ] (0 ≤ r ≤ ro ), In our previous study about SBO3 [2], we verified that the ‘-nG’ Gaussian expansions achieved for those orbitals fit some simple formulas very well. If the Gaussian expansion is: G= (2n + 6j + 3)! 2n+6j+3 (2n + 2)! (6j)! r0 . (11) It should be remarked that Equation (9) only works for coefficients of unnormalised rn (ro −r)3j functions, that is for Cj , not for Dj coefficients. Therefore, it is recommended to select the adequate ‘normalisation’ option when using UCA-GSS software [9] ([use of nonnormalised components for F(r)]) to obtain correct SBO4-nG basis. 2 (12) then: βSBO−nG [r0 ] ≈ βSBO−nG [r0 = 1] r02 (13) and the coefficients Cj are independent of the ro values: CSBO−nG [r0 ] = CSBO−nG [r0 = 1]. j=1 Nj = Cj e−βj r j=1 (10) where the coefficients Dj are Cj /Nj with: n  (14) For the SBO4-3G, we found that these formulas remain valid, but the β k [ro = 1] and Ck [ro = 1] parameters take different values than for SBO3-3G. The β k [ro = 1] exponents and Ck [ro = 1] coefficients needed to write the SBO4-3G of any atom from H to Kr can be obtained from the data shown in Tables S10–S17 of the Supporting Information, and the values achieved are summarised in Table 2. The remarkable improvement achieved by using SBO4-3G instead of SBO3-3G can be appreciated in Table 3. But it can be further improved by taking into account the arguments of the next section. Table 1. Radii and coefficients of three and four-term SBOs. Orbitals SBO ro (α = 1) C 1 (ro = 1) C 2 (ro = 1) C 3 (ro = 1) C 4 (ro = 1) 1s SBO3 SBO4 8.6658461 10.4563649 7.81940E−02 1.6079729E−02 1.41649E−01 1.8674980E−01 7.46509E−01 1.8708317E−01 – 5.9746216E−01 2s,2p SBO3 SBO4 9.7401304 11.5390701 3.93496E−02 7.3714048E−03 7.92569E−02 1.1703473E−01 7.69873E−01 1.3132295E−01 – 6.9460936E−01 3s,3p,3d SBO3 SBO4 10.8023019 12.6099831 1.91444E−02 3.32276111E−03 4.01567E−02 6.98427239E−02 7.07428E−01 8.34986361E−02 – 7.22775667E−01 4s,4p SBO3 SBO4 – 13.6718439 – 1.47630802E−03 – 4.00379874E−02 – 4.83819907E−02 – 6.87172142E−01 Note: Coefficients are for unnormalised rn (ro − r)3j terms. (râĂŇo is given in atomic units). Table 2. Exponents and coefficients for SBO4-3G with ro = 1. Orbital 1s 2s 2p 3s 3p 3d 4s 4p β 1 (ro = 1) β 2 (ro = 1) β 3 (ro = 1) C1 C2 C3 244.4442 300.8649 125.1898 82.2504 276.0701 87.1156 42.9388 66.2452 45.0166 22.1381 32.9071 14.4639 24.0175 28.3451 9.2886 18.2559 12.0197 8.3759 10.9011 7.2980 9.1638 10.6544 6.1049 8.1503 0.1497813 −0.0673117 0.1479261 −0.2058041 −0.0163397 0.1389403 −0.3487410 −0.1069783 0.5359077 0.5545264 0.5576594 0.4749263 0.5181134 0.5636616 0.6993747 0.4380977 0.4479972 0.5050481 0.4435213 0.6344980 0.5515174 0.4528855 0.4876878 0.6537731 Note: Exponents for ro = 1 can be obtained by using formula (9). Coefficients are independent of the ro value. MOLECULAR PHYSICS Table 3. RMSE of some H–Kr atomic properties for SBO3-3G and SBO4-3G (with respect to non-standard STO-3G results). Basis set SBO3-3G (H–Kr atoms) SBO4-3G (H–Kr atoms) SBO3-3G (H–Ar atoms) SBO4-3G (H–Ar atoms) SBO3-3G (K–Kr atoms) SBO4-3G (K–Kr atoms) Total energy HOAO energy < R**2 > 0.39 0.05 0.08 0.01 0.54 0.07 0.04 0.01 0.01 0.00 0.05 0.01 0.27 0.09 0.12 0.03 0.36 0.12 Notes: The great improvement obtained by using SBO4 instead of SBO3 is evident. All the properties are in a.u. 5 Table 4. Cusp errors (%) for the STO → SBO3, STO → SBO4, STO-nG, SBO3-nG and SBO4-nG approximations for a 1s orbital with exponent α = 1.00. cusp (STO → SBO) STO1s (α = 1) 0% SBO3 (r0 = 8.666 a.u.) 3.4% SBO4 (r0 = 10.456 a.u.) 1.3% cusp STO-3G 19.4% SBO3-3G 20.7% SBO4-3G 19.8% (STO → nG, SBO → nG) STO-4G 12.7% SBO3-4G 13.4% SBO4-4G 13.0% STO-5G 8.6% SBO3-5G 9.3% SBO4-5G 8.9% STO-6G 6.0% SBO3-6G 6.8% SBO4-6G 6.3% Note: The cusp errors for STO → SBOs approximations are very small while the errors for SBO-nG expansions are very similar to the STO-nG ones. 5. Some properties of the SBO4-3G functions As part of the study of the properties of the SBO-3G basis sets, we have examined the SBO ‘cusp error’, which in this case it can be of two types: (1) The well-known cusp error of the Gaussian expansions. (2) A supplementary cusp error related with the difference in the value at r = 0, that is, in the starting point of the STO and the fitted SBO. Table 4 shows that ‘SBO to STO’ cusp error is much lower than the -nG expansion cusp error, especially if SBO4 functions are used instead of SBO3 ones. Also, it can be noticed that the cusp error of the SBO-nG expansions is very similar to the STO-nG ones, and it could be reduced (if necessary) by increasing the number of Gaussian functions of the expansion, or by adding to the SBOs other functions like the ‘mixed ramp’ of McKemmish et al. [11,12]. Another remarkable advantage of the SBOs regarding the STOs is that for systems big enough, the number of bielectronic integrals to be calculated drastically decrease [1,2]. Since the radius ro is a bit larger for SBO4 than for SBO3, the reduction of the number of non-zero integrals is slightly lower for SBO4 than for SBO3 functions. But it is easy to check that this difference is irrelevant when big systems are studied. Figure 4 shows the variation in the number of non-zero (pq|rs) integrals regarding the number of carbon atoms (n), in successive linear molecules Cn H2 taken as an example, and studied by using STO, SBO3 and SBO4 as basis functions. When STO basis set is used, the number of integrals (N) can be related to the Figure 4. Graphical representation of the number of integrals needed to be evaluated in HF calculation for Cn H2 molecules with minimal basis set, regarding the number of carbon atoms (n). The polynomials shown have been fitted from n = 6 to n = 20. 6 V. GARCÍA ET AL. number of basis functions (m) through the well-known relation: 1 N = (m4 + 2m3 + 3m2 + 2m). 8 (15) Since, for a minimal basis set, m = 5n + 2 in Cn H2 molecules, it leads to the next relation with n: N= 625 4 625 3 975 2 175 n + n + n + n + 6. 8 4 8 4 (16) Equation (16) is exactly fulfilled when the STO data are fitted to a polynomial of grade 4 (see Figure 4). In this same figure can be seen, however, that the equation is fulfilled when the SBO3 or SBO4 data are fitted to polynomials of grade 2. This allows estimating, very precisely, the reduction in the number of integrals needed to calculate properties of any size systems. This can be done by using the relations shown in Figure 4. For example, for n = 100 in the molecule Cn H2 (which, by using a minimal basis set, involves the use of 500 basis functions), the necessary number of integrals (N) for the calculation when STO basis functions are used is 100 times greater than when SBO4 functions are used: N STO = 100·N SBO4 . This difference rapidly grows as the number of carbon atoms (n) increases. For example, for n = 103 , it would be N STO = 104 ·N SBO4. Evidently, the use of SBOs opens the door to the possibility of studying atomic clusters much greater than the STOs. 6. The anomalous behaviour of 3D shells The preliminary study performed by using the proposed SBOs in atomic calculations, allowed underscoring a problem with the 3d orbitals, that is also present when STOs are used, but that can be much more easily fixed when SBOs are used instead: The energy error of the HOAO achieved by using SBO3 or SBO4 regarding the STO employed to generate them, is very small for atoms with Z < 20 (see Figure 2). Nevertheless, it rapidly increases from the Sc atom, for which the last occupied orbital begins to be the 3d orbital. Figure 2, again, shows that although the results obtained by using SBO4-3G are better than those obtained by using SBO33G, they are still unacceptable. A check was also performed of whether an improvement is achieved by using the SBO5-3G instead of the SBO4-3G basis set to define the 3d orbital, but it was not good enough. Just as it is not worth increasing the number of terms in the SBOs, addressing the problems of the 3d orbitals by increasing the number of Gaussian functions in the expansions used to manage them, is not a satisfactory solution, because -as it can be seen in Figure 5 (or Figure S1 in the Supporting Information)-, the problem was Figure 5. Comparison of the HOAO energies calculated with SBO, STO and Aug-cc-pVQZ basis sets. Although the energies calculated with SBO4 are identical to those calculated with the original STOs, both are unacceptable for Fe, Co, Ni, Cu and Zn atoms. already present in the STOs employed to generate the SBOs. Therefore, the anomalous behaviour of the 3d orbitals is related to something more basic than a simple fitting problem. Indeed, Figure 5 shows that the behaviour of the eHOAO energies calculated by directly using STO-3G orbitals, compared with the results achieved by using a basis set close to the Hartree–Fock limit, is clearly abnormal for Fe, Co, Ni, Cu and Zn atoms (see Table 5 and Figure 5). The energies eHOAO obtained by using SBO33G or SBO4-3G basis sets for the quoted atoms are similar to those achieved by using STOs (They provide absurd ionisation energies (eHOAO > 0)). To clarify the origin of this problem, we have compared the energies eHOAO calculated with the SZ orbitals of Clementi–Raimondi [8] (used as starting point to Table 5. Comparison of the HOAO energies calculated with SBO, STO and Aug-cc-pVQZ basis. Atom Z SBO3-3G SBO4-3G K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 −0.13851 −0.18053 −0.14769 −0.19286 −0.16306 −0.15287 −0.16828 0.14122 0.16328 0.19020 0.29333 0.31157 −0.08053 −0.22823 −0.30447 −0.24034 −0.31436 −0.39352 −0.13910 −0.18149 −0.16808 −0.19257 −0.19722 −0.18715 −0.20526 0.09450 0.11141 0.13297 0.23006 0.21026 −0.15680 −0.23479 −0.31244 −0.25120 −0.32702 −0.40907 STO-3G −0.13921 −0.18164 −0.17815 −0.17262 −0.19749 −0.20354 −0.20562 0.07237 0.08690 0.15046 0.20027 0.21026 −0.16761 −0.23789 −0.31622 −0.25639 −0.33293 −0.41489 pVQZ −0.14764 −0.19553 −0.20452 −0.21003 −0.21626 −0.22143 −0.22648 −0.22939 −0.23234 −0.23623 −0.28106 −0.29251 −0.21569 −0.23758 −0.37017 −0.34909 −0.43426 −0.5242 Note: The absurd results (eHOAO > 0) have been highlighted. See, in this context, Figure 5. MOLECULAR PHYSICS obtain our SBOs), with the results obtained by substituting some of these SZ orbitals by DZ orbitals of Roetti–Clementi [13]. In Figure 6, we can see that duplicating the 3d or 4s shells separately, hardly improves the results. However, by duplicating both, the 3d and 4s shells together, the energies eHOAO achieved for the Cr–Zn atoms become almost equal to those calculated with Aug-cc-pVQZ basis. Additionally, we have realised that in order to get good energies eHOAO for Ga–Kr atoms, the 4p shell should be also duplicated (see Figure 7). Then, once stablished that the anomalies detected for the Fe–Zn atoms are due to the inaccuracy of the 3d orbitals of Clementi–Raimondi [8] used to generate the SBOs, we can consider generating the SBO-3d from better atomic orbitals, such as those of Bunge–Barrientos [14] or those of Roetti–Clementi [13]. In this work, we have solved the problem in an acceptable way by generating the SBO-3d orbitals (as SZ functions) from the DZ-3d orbitals of Roetti–Clementi [13]. This way we have obtained a set of improved SZ3d orbitals that, besides keeping the basis set as small as possible, enable us to avoid the balancing problems that could possibly arise by mixing SZ and DZ valence orbitals. In order to carry out the generation of those SBOs, we have developed a new software: 2STOto1SBO.nb [15] which is able to fit an SBO to a combination of two STOs with equal principal quantum number and different exponents: ϕ = C1 rm exp(−a1 r) + C2 rm exp(−a2 r). Figure 6. Comparison with the Aug-cc-pVQZ results of the energies of the last occupied orbital, calculated with the STO-SZ of Clementi and Raimondi (C–R), and with SBOs using DZ only for 3d, only for 4s and for both. 7 (17) Replacing the SBOs achieved from the Clementi– Raimondi 3d orbitals by those obtained from the Roetti–Clementi ones, allow achieving HOMO energies surprisingly close to those obtained by using Aug-ccpVQZ basis sets (see Figure 8). The DZ 4s or 4p type orbitals are functions with nodes (obtained by combination of several Slater functions with different principal quantum numbers). Therefore, they are Figure 7. Comparison of the eHOAO energies calculated for Ga–Kr atoms by using the very large Aug-cc-pVQZ basis set and by using Clementi–Raimondi SZ STOs (C–R), Roetti–Clementi DZ for 3d orbitals (DZ-3d), DZ for 3d and 4s orbitals (DZ 3d 4s), DZ for 3d, 4s and 4p orbitals (DZ 3d 4s 4p). To get results near to those obtained with the Aug-cc-pVQZ basis set, it is essential to improve the Clementi–Raimondi SZ 3d, 4s or 4p shells before obtaining the SBO orbitals. 8 V. GARCÍA ET AL. Figure 8. Comparison of the HOAO energies, calculated with: (1) STO-SZ of Clementi and Raimondi (‘simple STO’), (2) SBO4 obtained from the Clementi and Raimondi orbitals, but using SBOs-SZ for 3d orbitals obtained by fitting the Roetti and Clementi 3d orbital and (3) the Aug-cc-pVQZ basis used as a reference. too different to the STOs which gave rise to the SBOs. Consequently, a direct fitting of SBOs to this kind of orbitals is not realistic, and frequently concludes without convergence. Since the software to fit SBOs to combinations of STOs only works fine for combinations of two STOs with an equal principal quantum number, the ‘SBO contraction’ DZ → SZ performed for the 3d orbitals have not been extended to the 4s or 4p orbitals in this work. Due to the complexity of this problem, and the fact that good enough results are obtained by improving solely the 3d orbitals (see Figure 8), it will be treated in a future research. 7. Molecular calculations In order to test the suitability of the SBOs to carry out molecular calculations, including atoms from K to Br, molecular properties calculations were performed for a collection of representative systems (X2 , XH, with X from H to Br; and linear molecules Xn H2 with X being C, Si, Ge and n = 2, 4, 6, 8 and 10), by using basis sets that provide results close to the HF limit, and these results were compared with those achieved: (A) By using SBO4-3G as basis functions (obtained as described above, with scale factors for valence orbitals variationally optimised). (B) By using the well-known STO-3G molecular basis set of GAUSSIAN [10]. All calculations have been carried out for multiplicity M = 1 or M = 2. In several cases (as for the O2 molecule) the minimum energy multiplicity is different, but we have considered this fact irrelevant to compare SBOs results with those achieved by using basis sets close to the HF limit. As found in [2], by adding scale factors to the atomic SBO3-3G functions, the results in molecular properties calculation by using minimal basis sets are significantly improved. Since the same effect can be expected with MOLECULAR PHYSICS 9 Table 6. Comparison of RMSE by using SBO-3G and by using standard STO-3G in the calculation of molecular properties. Bond length (Å) X2 from H to Cl X2 from K to Br SBO4-3G 0.13 0.16 HOMO energy (a.u.) STO-3G 0.28 0.55 SBO4-3G 0.09 0.08 STO-3G 0.16 0.10 Vibration frequency (cm−1 ) SBO4-3G 205 106 STO-3G 305 258 Electronic spatial extent < R**2 > SBO4-3G 19 38 STO-3G 41 87 Notes: Molecules X2. Reference basis set Aug-cc-pVQZ. The superiority of the SBO basis set is remarkable, especially beyond the K atom. Table 7. Comparison of RMSE by using SBO-3G and by using standard STO-3G in the calculation of molecular properties. Bond length (Å) X2 from H to Cl X2 from K to Br SBO4-3G 0.06 0.12 HOMO energy (a.u.) STO-3G 0.09 0.17 SBO4-3G 0.09 0.06 STO-3G 0.13 0.10 Vibration frequency (cm−1 ) SBO4-3G 181 384 STO-3G 527 504 Electronic spatial extent < R**2 > SBO4-3G 0.82 1.6 STO-3G 0.80 3.5 Notes: Molecules XH. The superiority of the SBO basis set is again remarkable, especially beyond the K atom. Table 8. RMSE of molecular properties for several Xn H2 (n = 2, 4, 6, 8 and 10), with respect to QZVP basis set results. Basis X=C X = Si X = Ge X = All HOMO energy SBO4-3G STO-3G 0.029 0.058 0.024 0.190 0.057 0.117 0.040 0.133 Atomisation energy SBO4-3G STO-3G 0.146 0.185 0.239 0.238 0.239 0.590 0.213 0.383 R(X–X) bond length SBO4-3G STO-3G 0.021 0.015 0.035 0.154 0.050 0.219 0.038 0.155 R(X–H) bond length SBO4-3G STO-3G 0.016 0.014 0.042 0.046 0.045 0.105 0.037 0.066 Stretch CC frequency SBO4-3G STO-3G 137 273 119 185 66 150 111 209 Stretch CH (SGG) SBO4-3G STO-3G 247 351 353 320 228 484 83 184 Property Note: QZVP basis set was used instead Aug-cc-pVQZ due to convergence problems with the larger basis set. the SBO4-3G functions, we verified that the scale factors obtained in [2] for the three-term SBOs in atoms with 1 ≤ Z ≤ 18 are basically valid for the four-term SBOs on those atoms. The SBO4 basis functions and the scale factors to be used in molecules can be found at the end of the Supporting Information. For their use in atomic calculations, the scale factors of all basis functions should be fixed as 1.0. As can be seen in Tables 6 and 7 (and Tables S22–S25 of the Supporting Information), the SBO4-3G functions allow obtaining much better results than the standard STO-3G basis functions for all the studied properties. All the molecular properties studied are detailed in the Supporting Information (Tables S22–S29 and Figures S2–S5). The Root mean squared errors (RMSEs) shown in 6 and 7 underscore the superiority of the SBO4-3G over the standard STO-3G. Considering the limitations of minimal basis sets, this superiority is remarkable and only can be attributed to a better behaviour of SBO4 functions over STO ones (Table 8). In Figure S2 of the Supporting Information, it can be seen that the n-dependence of the atomisation energies achieved for the Cn H2 systems is reasonably reproduced by using SBO4-3G. However, although the use of SBO4-3G basis set improves the results achieved by the STO-3G standard basis set for Sin H2 and Gen H2 systems, the dependence with n of the atomisation energies is not properly described. At any rate, the results achieved by using SBO4-3G are much better than those obtained Figure 9. Comparison between the hydrogen atom Scale Factor, variationally obtained from XH molecules, and estimated from the electronegativity of the atom X, SF(H) ≈ 2.034·Elneg + 0.728. Electronegativity of each atom calculated as −½(HOAO + LUAO) and HF / Aug-cc-pV5Z. 10 V. GARCÍA ET AL. Table 9. Electronegativities (elneg), χ = (I + A) / 2 ≈ -(HOAO + LUAO) / 2, for H–Kr atoms. Atom H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar Elneg (a.u.) SF(H) in XH Estimate atom Elneg (a.u.) SF(H) in XH Estimate 0.2357 0.4132 0.0923 0.1457 0.1486 0.2114 0.2412 0.2537 0.3024 0.3471 0.0863 0.1197 0.1045 0.1524 0.1665 0.1725 0.2106 0.2464 1.194 – 1.001 1.143 1.161 1.173 1.179 1.239 1.29 – 0.897 0.907 0.932 1.049 1.103 1.11 1.154 – 1.209 – 0.916 1.025 1.031 1.159 1.22 1.246 1.345 – 0.904 0.972 0.941 1.039 1.068 1.08 1.158 – K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr 0.0673 0.0402 0.0961 0.098 0.048 0.1025 0.1091 0.1111 0.1131 0.1155 0.1174 0.1345 0.1029 0.152 0.1735 0.1777 0.2416 0.2237 0.872 0.829 0.904 0.839 0.84 0.845 0.923 0.875 0.881 0.881 0.963 1.002 1.015 1.064 1.111 1.163 1.21 – 0.865 0.809 0.924 0.927 0.825 0.937 0.95 0.954 0.958 0.963 0.967 1.002 0.938 1.038 1.082 1.09 1.221 – Note: Calculated with HF/Aug-cc-pZ5Z method, and scale factors for 1s(H) optimised in XH molecules. by using STO-3G. This is also confirmed for the bond length and vibrational frequency as shown in Figures S3(a)–S4(b) in the Supporting Information. The hydrogen atom is a special case: the very limited quality of a basis set formed with a single 1s orbital implies that optimising the exponent is more important in this case than for atoms in which more basis functions are used. In a hydrogen atom, the variationally optimised scale factor takes a relatively high value and varies greatly depending on the atom that it is bonded to. We have determined the optimal values of the scale factors for the 1sH basis functions in XH molecules (see Table S30 in the Supporting Information) and we have detected a clear relationship between the scale factors obtained and the electronegativity of the X atom (see Figure 9). The relationship obtained: SF(H in XH) = 2.017 elneg(X) + 0.7301, exists independently of the electronegativity scale adopted (Pauling [16], Mulliken [17], Allen [18] or Rham and Hoffmann [19]). We have preferred obtaining the electronegativity as –(eHOAO + eLUAO )/2 with eHOAO and eLUAO calculated with HF/Aug-cc-pV5Z (See Table 9). In this way, we are able to manage systems (atoms, molecules or ions) not included in the most usual electronegativity scales (see Figures S6 of the Supporting Information). 8. Conclusions • The atomic properties calculated with the four-term SBOs (SBO4), for all the atoms from H to Kr, are practically indistinguishable from the properties calculated with the STOs [8] used to generate them, except for Mn, Fe, Co, Ni, Cu and Zn atoms. In addition, the SBOs have the advantages mentioned in the introduction paragraph for large or confined systems. • The use of SBO4 makes it possible to calculate the properties of systems with atoms of Z > 18 (beyond Argon), without the loss of accuracy detected previously for three-term SBOs (SBO3) [2]. • For atoms with Z ≥ 21 (from Sc atom), the use of the SZ basis set leads to inconsistencies in the calculation of orbital energies, that become very serious from Mn to Zn atoms. It is important to emphasise that these inconsistencies also appear in the same way when SZ Slater orbitals are used as basis functions. • The problems detected for atoms with Z ≥ 21 and minimal basis sets (SBO as well as STO), are completely corrected by substituting the SZ basis set with a DZ basis set applied for the 3d and 4s orbitals for Sc–Zn atoms or 3d, 4s and 4p orbitals for Ga to Br atoms. The parameters for the SBO-DZ functions can be achieved, with enough precision, by using the Roetti–Clementi orbitals [13]. • Because using DZ-3d orbitals in conjunction with the rest of the SZ orbitals could be inadequate (balancing problems may arise), these DZ-3d orbitals can be substituted by SZ-SBO-3d achieved through the fit of a single SBO to the corresponding combination of two STOs of Roetti–Clementi [13]. By this way, most of the problems detected for atoms with Z ≥ 21 disappeared. • For SBO minimal basis sets, there is a clear relationship between the optimal scale factor for 1s-3G basis functions of hydrogen and the electronegativity of the atom to which it is bonded to. This relation is valid for all the atoms from H to Br. • The SZ SBO4-3G finally proposed allows calculating any kind of molecular property, such as eHOMO , EAtomisation , bond lengths, vibrational frequencies . . . etc., with an error level very much lower than a standard STO-3G basis set [20]. MOLECULAR PHYSICS Acknowledgments Calculations were performed through CICA (Centro Informático Científico de Andalucía). Disclosure statement No potential conflict of interest was reported by the authors. Funding We thank the Ministerio de Economía y Competitividad of the Spanish Government [grant number ENE2014-58085-R]. ORCID Victor García http://orcid.org/0000-0002-9868-8194 David Zorrilla http://orcid.org/0000-0003-1673-5274 Jesús Sánchez-Márquez http://orcid.org/0000-0001-9498-1361 Manuel Fernández http://orcid.org/0000-0002-0965-9974 References [1] V. García, D. Zorrilla and M. Fernández, Int. J. Quant. Chem. 114, 1581 (2014). [2] V. García, D. Zorrilla, J. Sánchez and M. Fernández, Int. 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