New pecJ-n (n = 1, 2) Basis Sets for Selenium Atom Purposed for the Calculations of NMR Spin–Spin Coupling Constants Involving Selenium

We present new compact pecJ-n (n = 1, 2) basis sets for the selenium atom developed for the quantum–chemical calculations of NMR spin–spin coupling constants (SSCCs) involving selenium nuclei. These basis sets were obtained at the second order polarization propagator approximation with coupled cluster singles and doubles amplitudes (SOPPA(CCSD)) level with the property-energy consistent (PEC) method, which was introduced in our previous papers. The existing SSCC-oriented selenium basis sets are rather large in size, while the PEC method gives more compact basis sets that are capable of providing accuracy comparable to that reached using the property-oriented basis sets of larger sizes generated with a standard even-tempered technique. This is due to the fact that the PEC method is very different in its essence from the even-tempered approaches. It generates new exponents through the total optimization of angular spaces of trial basis sets with respect to the property under consideration and the total molecular energy. New basis sets were tested on the coupled cluster singles and doubles (CCSD) calculations of SSCCs involving selenium in the representative series of molecules, taking into account relativistic, solvent, and vibrational corrections. The comparison with the experiment showed that the accuracy of the results obtained with the pecJ-2 basis set is almost the same as that provided by a significantly larger basis set, aug-cc-pVTZ-J, while that achieved with a very compact pecJ-1 basis set is only slightly inferior to the accuracy provided by the former.

In this paper, we present new compact J-oriented basis sets for the selenium atom that have been obtained within the property-energy consistent (PEC) method, introduced in our previous studies [98][99][100]. The PEC method is based on the consistent optimization of all exponents using the Monte Carlo (MC) simulations [101][102][103], with respect to the property under consideration and the total molecular energy. It gives compact basis sets that are capable of providing accuracy comparable to that achieved with the other property-oriented basis sets of larger sizes generated with a standard even-tempered technique [104].
This work has been called for by the needs of compact and accurate basis sets for highly computationally demanding calculations of the spin-spin coupling constants involving selenium in large-and medium-sized selenium compounds. Only a few J-oriented basis sets for selenium exist at the moment, and all of them are rather large to be efficiently used in large-scale calculations. These are the acvXz-J (X = 2, 3, 4) [93], the aug-cc-pVTZ-J (the first version) [86], and the aug-cc-pVTZ-J (the second version) [87]. The aug-cc-pVTZ-J (the first version) basis set for selenium was obtained from the energy-optimized uncontracted Dunning's basis set of triple-zeta quality, aug-cc-pVTZ(uc) (21s14p10d2f ) [105], by the addition of two tight s-type functions and the subsequent application of the contraction scheme based on the molecular orbital coefficients of the selenium hydride, (23s14p10d2f ) -> [12s9p6d2f ]. The f -and g-angular spaces were not considered in the saturation procedure in the mentioned paper. The acvXz-J (X = 2, 3, 4) basis sets [93] were designed from the augmented core-valence Dyall's basis sets dyall.acvXz (X = 2, 3, 4) [106][107][108] of double-(16s12p8d2f ), triple-(24s17p11d5f 1g), and quadruple-zeta (31s22p14d6f 4g1h) quality by the consecutive even-tempered saturation of the s-, p-, d-, f -, and g-angular functional spaces. As a result, it was found that the double-zeta quality basis set dyall.acv2z is not complete in the f -space for the correct calculation of the selenium SSCCs; i.e., two f -functions are not enough. For example, the addition of only one diffuse f -function to the f -space of dyall.acv2z resulted in the increase of the 1 J( 77 Se, 1 H) of MeSeH molecule by about 5 Hz. Thus, the proposed acvXz-J basis sets were profoundly expanded in the f -space as compared to the corresponding dyall.acvXz basis sets. Therefore, the acvXz-J basis sets have the following configurations: [19s12p8d4f |11s8p5d4f ] for X = 2, [27s17p11d5f 1g|14s10p6d5f 1g] for X = 3, and [35s22p14d6f 4g|18s12p7d6f 4g] for X = 4. In this sense, the aug-cc-pVTZ-J for the selenium atom has also been revisited recently [87] because the former version of this basis set contained only two f -functions. Thus, the contemporary configuration of the aug-cc-pVTZ-J for the selenium atom is [26s16p12d5f |17s10p7d5f ]. Although the acvXz-J and aug-cc-pVTZ-J (the second version) basis sets are complete enough in all important functional spaces to provide a flexible description of the selenium SSCCs, the problem is that these basis sets are very large. To be more precise, the number of functions for the uncontracted/contracted forms of these basis sets is as follows: 123/88 (acv2z-J), 177/118 (acv3z-J), 249/167 (acv4z-J), and 169/117 (aug-cc-pVTZ-J). On the other hand, we have recently proposed the PEC method [98] for generating property-oriented basis sets that reoptimizes all angular spaces and gives very efficient compact basis sets, providing more accurate results as compared to the property-oriented even-tempered basis sets of similar sizes. Thus, in this paper, we present new pecJ-n (n = 1, 2) basis sets for the calculation of selenium SSCCs that embody both desirable features: they are moderate in size and provide very accurate results.

Creation of pecJ-n (n = 1, 2) Basis Sets for Selenium
The PEC method [98] consists of the optimization of basis sets in relation to a certain molecular property, provided that the least possible total molecular energy is achieved. Exponents are randomly generated around the starting basis set via Monte Carlo simulations. The generated arrays are verified whether they give the property under interest within a desired diapason or not. In the end, only one set is selected-the one that provides the property value within the desired range and the lowest energy. For the detailed description of the PEC algorithm and its peculiarities as applied to the generation of the J-oriented basis sets, we refer the reader to our earlier works presenting the basics of the method [98,99]. Although, it should be mentioned that the PEC method is unique in the sense of dealing with two target functions (the property and the energy) simultaneously. The PEC optimization procedure of the exponents {ζ i } for the property under consideration with respect to the "ideal" values under the energetic constrain represents a nonlinear problem with multiple solutions. In this case, the PEC optimization procedure can be thought of as approaching the isoline of the "ideal" property value that is formed by the intersection of the "ideal" property plane with the property surface representing the multi-argument function of the varying exponents f (ζ 1 , ζ 2 , . . . , ζ n ), both determined in the multidimensional exponential space. All points of this isoline give the same "ideal" value of property but different molecular energies. The PEC method is aimed at selecting the basis set that provides the lowest molecular energy among those which belong to the "ideal" isoline, since the lower the energy, the better the description of the molecular wave-function. The optimization problem in the multidimensional exponential space with multiple solutions can hardly be solved using standard procedures based on the directed search, like numerical Newton-like methods [109]. This is connected with the natural limitations of such techniques due to the fact that they are aimed at finding a single extremum of a property in the vicinity of the starting point. In the case of Newton-type optimization, one can imagine the process as a gradual ascending or descending to a certain single point. In that way, such algorithms lapse the other solutions, unless initiated many times from different starting guesses, and, even if they were started from various guesses, they would find only a limited number of the solutions. Based on this reasoning, one can conclude that a conventional directed Newton-like optimization is not fully suitable for the problem with multiple solutions; in this sense, the PEC method is unique and totally justified for generating the J-oriented basis sets.
In this work, all optimizations of exponents were carried out using the SOPPA(CCSD) method for SSCC calculations and the CCSD method for the energy calculations. We used two fitting molecules, SeH 2 and Se=C, in which the FC contributions to 1 J( 77 Se, 1 H) and 1 J( 77 Se, 13 C) SSCCs were considered as the arguments of the target function. The target function that was minimized represents the mean absolute error of these J FC against their "ideal" values: This minimization was performed under the energetic constrain: ∑ 2 n=1 E n → min , which guarantees that the least possible total molecular energy of two molecules is achieved. The energy tolerance threshold was set to 10 −4 Hartree. The optimization that involves two fitting molecules provides more robustness of the generated basis sets towards the diversity of the electronic systems as compared to the basis sets obtained with only one fitting molecule.
The "ideal" values were evaluated at the SOPPA(CCSD) level using the extended dyall.aae4z basis set (dyall.aae4z + ) on all atoms in both fitting molecules. The dyall.aae4z + was obtained by adding 3, 2, and 1 tight s-functions to the dyall.aae4z basis set for hydrogen, carbon, and the selenium atom, respectively, in the even-tempered manner. Accordingly, the corresponding configurations of the dyall.aae4z + basis set for these atoms are as follows: (15s4p3d2f ), (21s11p6d4f 2g), and (32s22p14d9f 5g1h). In this respect, dyall.aae4z + provides overwhelming flexibility in all angular spaces, including the most important tight s-region to gain the values of SSCCs that are very close to the CBS limit.  77 Se-13 C SSCCs are of different signs. The sign of the J FC (A,B) can be deduced from the signs of the nuclear magnetogyric ratios of the coupled nuclei (γ A , γ B ) and that of the reduced SSCC, K FC (A,B), in accordance with the following relationship: J FC (A,B) = (h/4π 2 )·(γ A γ B )·K FC (A,B). As the magnetogyric ratios of the isotopes 1 H, 13 C, and 77 Se are of the same positive sign, the difference in the sign of the 1 J FC ( 77 Se, 1 H) and 1 J FC ( 77 Se, 13 C) stems from different signs of the FC contributions to the corresponding reduced SSCCs. In this respect, the latter is totally determined by the details of the electronic structure of the considered molecules and can be deduced based on the fundamental rules proposed by Gil and von Philipsborn [110]. The authors proved that, if one of the coupled nuclei has lone electron pair(s) (LEP(s)), they always give the contributions of a negative sign to 1 K FC (A,B). As a consequence, the removal of a lone electron pair, namely by protonation, alkylation, oxide formation, or complexation, leads to an increase in 1 K FC (A,B). The negative contributions from LEP(s) ( 1 K FC LP ) compete with the always positive contributions from the bonding orbital between coupled nuclei ( 1 K FC BD ) [111]. In the case of the SeH 2 molecule, the 1 K FC BD is very large [111] and, apparently, is not quite suppressed by the competing 1 K FC LP , giving (in total with minor rest contributions) a relatively small positive 1 K FC (Se,H) [43], while for most 1 J(Se,C) SSCCs, the FC contribution has a negative sign [32,43], which can be related to the total domination of the negative 1 K FC LP over the positive 1 K FC BD . It was shown earlier [87,93] that the expansion of the f -angular space of a standard energy-optimized basis set of double-zeta quality leads to a dramatic change in the onebond SSCCs involving selenium and other NMR-active half spin nuclei. A weaker but no less important sensitivity of the selenium SSCCs has been found when varying the tight s-region. Therefore, before starting the basis set optimization process, we carried out the investigation of the sensitivity of 1 J( 77 Se, 1 H) of SeH 2 and 1 J( 77 Se, 13 C) of Se=C to the expansion of different angular spaces of basis sets of double-and triple-zeta quality on the selenium atom. For that purpose, we have set the uncontracted pecJ-1 or pecJ-2 basis sets on hydrogen and carbon atoms [98], while the cc-pVDZ+2f or cc-pVTZ+2f basis sets were set on the selenium atom. The cc-pVDZ+2f represents the uncontracted cc-pVDZ basis set [105] augmented with both f-functions of aug-cc-pVTZ basis set [105], while the cc-pVTZ+2f is the uncontracted cc-pVTZ basis set [105], whose single f-function was replaced with two f-functions of the aug-cc-pVTZ basis set. These basis sets were considered the starting sets for further expansion. The behaviors of the 1 J( 77 Se, 1 H) and 1 J( 77 Se, 13 C) SSCCs upon the saturation of the cc-pVDZ+2f or cc-pVTZ+2f basis sets on selenium atom are shown in Figure 1a,b.
follows: (15s4p3d2f), (21s11p6d4f2g), and (32s22p14d9f5g1h). In this respect, dyall.aae4z + provides overwhelming flexibility in all angular spaces, including the most important tight s-region to gain the values of SSCCs that are very close to the CBS limit. Thus, , = Se , H = 74.65 Hz and , = Se , C = −154.25 Hz. It is interesting to note that the ideal values for the FC contributions to one-bond 77 Se-1 H and 77 Se-13 C SSCCs are of different signs. The sign of the JFC(A,B) can be deduced from the signs of the nuclear magnetogyric ratios of the coupled nuclei (γA, γB) and that of the reduced SSCC, KFC(A,B), in accordance with the following relationship: JFC(A,B) = (h/4π 2 )·(γAγB)·KFC(A,B).
As the magnetogyric ratios of the isotopes 1 H, 13 C, and 77 Se are of the same positive sign, the difference in the sign of the 1 JFC( 77 Se, 1 H) and 1 JFC( 77 Se, 13 C) stems from different signs of the FC contributions to the corresponding reduced SSCCs. In this respect, the latter is totally determined by the details of the electronic structure of the considered molecules and can be deduced based on the fundamental rules proposed by Gil and von Philipsborn [110]. The authors proved that, if one of the coupled nuclei has lone electron pair(s) (LEP(s)), they always give the contributions of a negative sign to 1 KFC(A,B). As a consequence, the removal of a lone electron pair, namely by protonation, alkylation, oxide formation, or complexation, leads to an increase in 1 KFC(A,B). The negative contributions from LEP(s) ( 1 KFC LP ) compete with the always positive contributions from the bonding orbital between coupled nuclei ( 1 KFC BD ) [111]. In the case of the SeH2 molecule, the 1 KFC BD is very large [111] and, apparently, is not quite suppressed by the competing 1 KFC LP , giving (in total with minor rest contributions) a relatively small positive 1 KFC(Se,H) [43], while for most 1 J(Se,C) SSCCs, the FC contribution has a negative sign [32,43], which can be related to the total domination of the negative 1 KFC LP over the positive 1 KFC BD . It was shown earlier [87,93] that the expansion of the f-angular space of a standard energy-optimized basis set of double-zeta quality leads to a dramatic change in the onebond SSCCs involving selenium and other NMR-active half spin nuclei. A weaker but no less important sensitivity of the selenium SSCCs has been found when varying the tight s-region. Therefore, before starting the basis set optimization process, we carried out the investigation of the sensitivity of 1 J( 77 Se, 1 H) of SeH2 and 1 J( 77 Se, 13 C) of Se=C to the expansion of different angular spaces of basis sets of double-and triple-zeta quality on the selenium atom. For that purpose, we have set the uncontracted pecJ-1 or pecJ-2 basis sets on hydrogen and carbon atoms [98], while the cc-pVDZ+2f or cc-pVTZ+2f basis sets were set on the selenium atom. The cc-pVDZ+2f represents the uncontracted cc-pVDZ basis set [105] augmented with both f-functions of aug-cc-pVTZ basis set [105], while the cc-pVTZ+2f is the uncontracted cc-pVTZ basis set [105], whose single f-function was replaced with two f-functions of the aug-cc-pVTZ basis set. These basis sets were considered the starting sets for further expansion. The behaviors of the 1 J( 77 Se, 1 H) and 1 J( 77 Se, 13 C) SSCCs upon the saturation of the cc-pVDZ+2f or cc-pVTZ+2f basis sets on selenium atom are shown in Figure 1a  The saturation of basis sets on the selenium atom was carried out in the tight region of each angular space by means of applying the geometrical progression or even-tempered recurrent ratio ζ i = αβ i, with α representing the largest exponent ζ n in the original functional set and β being the ratio of two largest exponents, β = ζ n /ζ n−1 . The saturation of basis sets has been made in the consecutive manner, that is, we passed to the saturation of the next angular space only if the current space was totally saturated and provided the converged value.
From Figure 1a,b, one can see that the addition of two tight s-functions to both cc-pVDZ+2f and cc-pVTZ+2f basis sets provides the converged values in both cases. Thus, two s-functions have been added to their initial configurations. The modifications made in the p-space did not cause any effect, while the expansion of the d-space slightly affected both SSCCs only in the case the of cc-pVDZ+2f basis set. Thus, it was decided to add one d-function to the cc-pVDZ+2f basis set and to remove one d-function from the cc-pVTZ+2f basis set, for the sake of lowering the least needed size of the d-space in the latter case. The expansion of the f -region resulted in very substantial changes in both SSCCs calculated with both double-and triple-zeta basis sets. As can be seen from Figure 1, the decrease of the 1 J( 77 Se, 1 H) and 1 J( 77 Se, 13 C) upon adding four additional tight functions to the f -space of both basis sets is about 25 and 12-14 Hz, respectively. Figure 1 tells us that the least needed number of additional f -functions to the original two functions is three in all cases, thus giving five f -functions in total. In that way, five f -functions provide sufficiently converged values within the f -space. However, we decided to deal with only four and three f -functions in the f -spaces of the pecJ-2 and pecJ-1 basis sets, respectively. This seemingly controversial decision can be explained by the very essence of our PEC method. This method is not a mere even-tempered augmentation of the basis set but an optimization of the exponents that implies dealing with a fewer number of exponents providing the same or even better accuracy than the larger even-tempered basis set. Thus, leaning on our experience of the PEC method, three additional even-tempered exponents can be replaced with one and two optimized exponents for the double-and triple-zeta levels, respectively, resulting in three and four f -functions in total. That is enough for the correct representation of the first polarization shell. This also can be thought of as an explanation of our decision of removing one d-function in the final configuration of the pecJ-2 basis set. We also decided to introduce one g-function to improve the description of the second polarization shell of the selenium atom. The resulting configurations and the modifications made are represented in Table 1. Thus, having the resulting configurations for the pecJ-n basis sets, we have set the obtained trial basis sets on the selenium atom and pecJ-1 or pecJ-2 basis sets on the hydrogens and carbon atoms in the fitting molecules SeH 2 and Se=C, and commenced the PEC optimization of the exponents. It is worth mentioning once again that, in the current case, we considered only the FC terms of the selenium SSCCs. In this sense, it could be said that we have developed the pecJ-n basis sets for the FC-dominating selenium SSCCs, covering by that the majority of cases [32,47,48,50,112]. It is also very important that all exponents in all shells were varied during the optimization process, which is different from what was conducted in our first work on the PEC method [98], where for the nonhydrogen atoms of the second period, only the s-, d-, and f -shells were varied to converge the target FC term to the ideal value. The final deviations of the FC contributions to the SSCCs of both fitting molecules from their ideal values amounted to ca. 0.01 Hz for both pecJ-1 and pecJ-2 basis sets. The resulting optimized exponents of the pecJ-1 and pecJ-2 basis sets can be found in Tables S2-S5 of Supplementary Materials, where the final selenium pecJ-n basis sets are presented in the format of the Dalton [113] and CFOUR [114] programs.
To reduce the sizes of the obtained uncontracted basis sets (for that we will use the notation "(uc)" throughout the text), pecJ-n(uc), we have applied a general contraction scheme [115]. However, this time, it was not a mere gain of the contraction coefficients from the molecular energy Self-Consistent Field (SCF) calculations of the simplest hydrides but was the application of the PEC algorithm to minimize the contraction error, providing the least possible molecular energy. In more detail, to obtain the contraction coefficients for the selenium pecJ-n basis sets, we consecutively (in relation to shells) minimized the sum of the absolute differences between the values of the 1 J( 77 Se, 1 H) of SeH 2 and 1 J( 77 Se, 13 C) of Se=C (this time, it was the total values consisting of the four Ramsey contributions) and the corresponding contemporary reference values. The pecJ-1 or pecJ-2 basis sets were set on the hydrogen and carbon atoms, depending on the cardinal number of the basis set used on the selenium atom. This routine can be thought of as the following: 1.
The PEC optimization of the contraction coefficients for the s-shell with respect to the target function: The PEC optimization of the contraction coefficients for the p-shell starts; the reference values are those that were recalculated in the previous step. When the optimized coefficients for the p-shell are obtained, the J re f i are redefined once again: the new ones are calculated with the pecJ-n basis set, in which the s-shell is contracted with the coefficients obtained in the first step and p-shell is contracted with the newly optimized coefficients, while the remaining shells are kept uncontracted; 3.
The PEC optimization of the contraction coefficients for the d-shell starts; the reference values are those that were recalculated in the previous step. In the end, we arrive at the final pecJ-n basis set with the contracted s-, p-, and d-shells and uncontracted f -shell, plus 1g function in the pecJ-2 basis set.
During this algorithm, the contraction patterns with different contraction depths were considered for each shell. A decision for the choice of the number of contracted functions was made based on the resulting contraction error ∆ for each step. Table 2 shows different contraction patterns with the final averaged absolute errors (∆) in relation to the current reference values. Table 2. Contraction patterns with varying depths 1 . Step

Basis Set Contraction Pattern Detailed Contraction Pattern Final Contraction Error ∆ (in Relation to Current Reference Values), in Hz
Step 2 (p-shell) pecJ-1 11p → 6p (7,7,1,1,1,1) 1.080 11p → 7p (6,6,1,1,1,1,1 The final configurations of the pecJ-n basis sets for the selenium atom together with the mean absolute percentage errors (ε), which they provide at the SOPPA(CCSD) level against the values obtained with the totally uncontracted pecJ-n(uc) basis set used on the selenium atom, are compiled in Table 3. The final contraction coefficients for the pecJ-n basis sets are presented in Tables S2-S5 of Supplementary Materials. Table 3. Final configurations of pecJ-n (n = 1, 2) for selenium atom. It is worth mentioning that the contracted aug-cc-pVTZ-J basis set for the selenium atom has the configuration [17s10p7d5f ], thus providing the N bas of 117. Our contracted second-level basis set, pecJ-2, has a pronounced benefit in size as compared to the aug-cc-pVTZ-J basis set, being 16 basis set functions fewer than the latter. If we recall the formal computational scaling [62] for the most popular high-quality computational methods applied to the calculation of SSCCs, namely N 5 for SOPPA, N 5 for SOPPA(CC2), N 6 for SOPPA(CCSD), N 6 for CCSD, and N 4 for DFT, with N being the total number of basis set functions participating in the calculation, the additional 16 functions on selenium result in a significant increasing of the operations needed to evaluate characteristic operators in a molecular orbital basis. To roughly exemplify the operational costs provided by the pecJ-n and aug-cc-pVTZ-J basis sets, we show the assessment of N k (with k being the scaling factors of different methods) for the simplest selenium hydride, SeH 2 ; see Table 4. We set the basis set of the same type on the hydrogen atom. From Table 4, it can be seen that the pecJ-n basis sets are beneficial compared to the aug-cc-pVTZ-J basis set. For example, for the most popular SOPPA and CCSD methods, the calculations with the pecJ-1 and pecJ-2 basis sets are, respectively, ten and two times less computationally demanding than that with the aug-cc-pVTZ-J basis set.

Testing New Basis Sets
In this section, we demonstrate the accuracy provided by our new basis sets against the experiment. All couplings were calculated at the CCSD level of theory, taking into account vibrational, solvent, and relativistic corrections to be properly compared with the experimental data. Overall, we calculated 13 selenium SSCCs of different types in seven representative molecules. The basic CCSD values were calculated with the pecJ-n and aug-cc-pVTZ-J basis sets being set on all atoms with the exclusion of chlorine; for the chlorine, atom we used the Dunning's correlation consistent basis sets of different qualities depending on the basis sets used on the rest of atoms, namely, the cc-pVDZ on Cl with the pecJ-1 on the rest, and cc-pVTZ on Cl with the pecJ-2 or aug-cc-pVTZ-J on the rest. All three types of corrections were calculated using the SVWN5 exchangecorrelation functional [116,117]. The SVWN5 represents the local density approximation (LDA) [118,119], in which the exchange is uniquely defined analytically in the form of the exchange energy of a homogeneous electron gas, while its correlation term is defined through several parameterizations, mostly relying on the highly accurate quantum Monte Carlo simulations. We have chosen this function based on the fact that the LDA model is particularly stable towards triplet instabilities [120], reflecting a balanced description of exchange and correlation. The latter is very important for the calculation of triplet excitation properties such as FC and SD contributions to the spin-spin coupling constants.
The solvent corrections to SSCCs were calculated as the differences between their values obtained in the gas and liquid phases. The IEF-PCM scheme [121,122] for a particular solvent (specified in accordance with the experimental data) was used for each compound in the liquid phase simulation. In the calculations of the solvent corrections, the same basis set schemes as those applied in the CCSD calculations were used.
The vibrational corrections to the SSCCs were calculated at zero temperature (zeropoint vibrational corrections, ZPVC) within the vibrational second-order perturbation theory (VPT2) [123] as applied in combination with the effective geometry approach of Ruud et al. [124]. The effective geometry represents the vibrationally averaged molecular geometry to the second-order in perturbation theory (involving the cubic force-field tensor). Thus, by means of calculating the property at the effective geometry, one partially takes into account the contribution to a vibrationally averaged property due to the anharmonicity of the potential, while the inclusion of the contribution from the averaging of the molecular property over the harmonic oscillator requires the additional step where the second derivative of the property surface is calculated at the effective geometry. In all vibrational calculations, we used the pecJ-1 basis set on all atoms except for the chlorine atom, on which the cc-pVDZ basis set was used.
The relativistic corrections to coupling constants were evaluated as the differences between the SSCCs obtained using the relativistic four-component Dirac-Kohn-Sham-Hamiltonian and those obtained within the "10c limit scheme". The "10c limit scheme" implies the increase of the speed of light by 10 times in the relativistic four-component calculations, resulting in a sufficiently accurate approximation of nonrelativistic values.
The "10c limit scheme" is now a widely used approximation [24,25,93,94,[125][126][127][128][129] that is applied to exclude the basis set inequivalence in passing from the four-component relativistic to the one-component nonrelativistic framework. The convergences of the selenium SSCCs with the increasing of the speed of light are shown in Figures S1 and S2 in the Supplementary Materials for the examples of 1 J(Se,H) in SeH 2 and 1 J(Se,C) in Se=C, respectively, calculated at the four-component DFT(SVWN5)//pecJ-2(uc) level. The speed of light was increased from one (totally relativistic calculation) to ten ("10c limit scheme") times. It follows that, for both molecules, the convergence within ca. 0.1-0.2 Hz already appears at the increasing of the speed of light by seven to eight times. Thus, the "10c limit scheme" can be regarded as appropriate for the calculations of the selenium SSCCs considered in this work.
In the four-component calculations, the same basis sets as in the CCSD calculations were used, though applied in the uncontracted form. The four-component calculations were performed under the unrestricted kinetic balance condition (UKB) [130,131].
The results are compiled in Tables 5 and 6, with the former presenting the roles of four Ramsey contributions to total CCSD values of SSCCs and the latter showing the comparison of the corrected CCSD values with the experiment. Table 5. Different contributions to SSCCs 1 involving selenium in compounds 1-7 calculated at the CCSD level of theory with pecJ-n (n = 1, 2) basis sets.  As can be seen from Table 5, the FC contribution by far dominates the other three contributions for most of the considered SSCCs. Only in the case of Se-P SSCCs are the PSO contributions significant on an absolute scale, though, as compared to the FC terms of the Se-P SSCCs, these are not so significant, being only about 20% of the magnitude of the FC terms. It is also worth mentioning that in some one-bond Se-C and Se-H SSCCs, the SD contribution is noticeable on an absolute scale (for example, in compounds 2, 5, 7), though being substantially inferior to the FC terms.
As can be seen from Table 6, the agreement of total theoretical values obtained with introduced basis sets with the experimental data is rather good. The mean absolute percentage errors (MAPEs) evaluated for all final theoretical selenium SSCCs against the experiment are presented in Figure 2. the FC terms. It is also worth mentioning that in some one-bond Se-C and Se-H SSCCs, the SD contribution is noticeable on an absolute scale (for example, in compounds 2, 5, 7), though being substantially inferior to the FC terms. As can be seen from Table 6, the agreement of total theoretical values obtained with introduced basis sets with the experimental data is rather good. The mean absolute percentage errors (MAPEs) evaluated for all final theoretical selenium SSCCs against the experiment are presented in Figure 2. From Figure 2, one can see that the total accuracy provided by the pecJ-2 basis set (MAPE = 6.8%) is practically the same as that provided by a rather larger aug-cc-pVTZ-J basis set (MAPE = 6.5%). In terms of formal operational costs, for the most popular SOPPA or CCSD methods, we can arrive at a 50% reduction (as was roughly estimated above in the example of selenium hydride) of the number of operations without any noticeable loss of accuracy by using the pecJ-2 basis set instead of the aug-cc-pVTZ-J basis set. Meanwhile, the calculations with the pecJ-1 basis set are several times less computationally demanding than those with the pecJ-2 basis set, and the accuracy is only slightly (MAPE = 8.9 vs. 6.8%) inferior to that provided by the pecJ-2 basis set. Thus, we can conclude here that our compact pecJ-1 basis set may be very useful in large-scale calculations or in very demanding calculations like those with highly correlated methods involving triple-or higher excitations or in the problem of vibrational averaging.

Materials and Methods
Geometry optimizations were carried out at the DFT level of theory with the Minnesota M06-2X exchange-correlation functional [137] using the pc-3 basis set [138][139][140][141]. Media effects were taken into account within the IEF-PCM solvation model when optimizing the geometrical parameters for the testing compounds 1-7, while the optimization of the geometry of both fitting molecules, SeH2 and Se=C, has been performed in the gas phase. All optimizations were performed in the Gaussian program [142]. Final equilibrium geometries are presented in Table S1 of Supplementary Materials. The SOPPA(CCSD) and CCSD calculations of SSCCs were carried out in the Dalton [113] and CFOUR [114] programs, respectively. Solvent and vibrational corrections to SSCCs were calculated using the Dalton program. Relativistic values were calculated within the DIRAC program [143]. From Figure 2, one can see that the total accuracy provided by the pecJ-2 basis set (MAPE = 6.8%) is practically the same as that provided by a rather larger aug-cc-pVTZ-J basis set (MAPE = 6.5%). In terms of formal operational costs, for the most popular SOPPA or CCSD methods, we can arrive at a 50% reduction (as was roughly estimated above in the example of selenium hydride) of the number of operations without any noticeable loss of accuracy by using the pecJ-2 basis set instead of the aug-cc-pVTZ-J basis set. Meanwhile, the calculations with the pecJ-1 basis set are several times less computationally demanding than those with the pecJ-2 basis set, and the accuracy is only slightly (MAPE = 8.9 vs. 6.8%) inferior to that provided by the pecJ-2 basis set. Thus, we can conclude here that our compact pecJ-1 basis set may be very useful in large-scale calculations or in very demanding calculations like those with highly correlated methods involving triple-or higher excitations or in the problem of vibrational averaging.

Materials and Methods
Geometry optimizations were carried out at the DFT level of theory with the Minnesota M06-2X exchange-correlation functional [137] using the pc-3 basis set [138][139][140][141]. Media effects were taken into account within the IEF-PCM solvation model when optimizing the geometrical parameters for the testing compounds 1-7, while the optimization of the geometry of both fitting molecules, SeH 2 and Se=C, has been performed in the gas phase. All optimizations were performed in the Gaussian program [142]. Final equilibrium geometries are presented in Table S1 of Supplementary Materials. The SOPPA(CCSD) and CCSD calculations of SSCCs were carried out in the Dalton [113] and CFOUR [114] programs, respectively. Solvent and vibrational corrections to SSCCs were calculated using the Dalton program. Relativistic values were calculated within the DIRAC program [143].

Conclusions
We presented new compact and accurate pecJ-n (n = 1, 2) basis sets for the selenium atom purposed for the quantum-chemical calculations of NMR spin-spin coupling constants involving selenium nuclei. These basis sets were obtained with the property-energy consistent method, which is efficient in generating compact property-oriented basis sets due to its peculiarity to fully reoptimize all angular spaces of the trial basis sets. In this way, the final PEC-generated basis sets are significantly smaller than the other specialized basis sets of the same zeta-quality obtained with the even-tempered technique. Thus, new pecJ-1 and pecJ-2 basis sets consist of only 78 and 101 basis functions, respectively. This gives a significant benefit as compared to the other existing J-oriented selenium basis sets, acvXz-J (X = 2, 3, 4) and aug-cc-pVTZ-J, with the total number of functions being as much as 88,118,167, and 117, respectively. In particular, for the SeH 2 molecule, the operational costs of the SOPPA and CCSD calculations can be said to be significantly reduced by using the pecJ-1 and pecJ-2 basis sets by approximately ten and two times, respectively, as compared to the costs of the calculations with the aug-cc-pVTZ-J basis set.
New basis sets were tested on the CCSD calculations of thirteen SSCCs involving selenium in the representative series of seven molecules carried out with taking into account relativistic, solvent, and vibrational corrections. The comparison with the experiment revealed that the accuracy of the results obtained with a compact pecJ-2 basis set is almost the same as that provided by an essentially larger basis set, aug-cc-pVTZ-J (the MAPEs are 6.8% and 6.5% for the former and the latter, respectively), while the accuracy achieved with a significantly smaller basis set, pecJ-1, is only slightly inferior (MAPE = 8.9%) to the accuracy provided by the pecJ-2 basis set. Overall, we recommend resorting to the pecJ-2 basis set in large-scale calculations of selenium SSCCs; this would provide accuracy comparable to that of the aug-cc-pVTZ-J basis set and give significant CPU time savings. The pecJ-1 can be recommended for highly demanding calculations, such as those involving the coupled cluster methods of higher hierarchy that treat triple-and higher excitations or in the problem of vibrational averaging.