#
Quelling the Geometry Factor Effect in Quantum Chemical Calculations of ^{13}C NMR Chemical Shifts with the Aid of the pecG-n (n = 1, 2) Basis Sets

^{*}

## Abstract

**:**

^{13}C NMR shielding constants/chemical shifts in terms of their efficacy in reducing geometry factor errors. The testing was carried out with both large-sized biologically active natural products and medium-sized compounds with complicated electronic structures. The former were treated using the computation protocol based on the density functional theory (DFT) and considered in the theoretical benchmarking, while the latter were treated using the computational scheme based on the upper-hierarchy coupled cluster (CC) methods and were used in the practical benchmarking involving the comparison with experimental NMR data. Both the theoretical and practical analyses showed that the pecG-1 and pecG-2 basis sets resulted in substantially reduced geometry factor errors in the calculated

^{13}C NMR chemical shifts/shielding constants compared to their commensurate analogs, with the pecG-2 basis set being the best of all the considered basis sets.

^{13}C NMR; chemical shift; shielding constant; DFT; coupled clusters; natural products

## 1. Introduction

^{13}C NMR spectrum analysis has now become one of the indispensable physical chemical tools for studying the structure and dynamics of large natural products. However, as is frequently the case for large compounds, the proper assignment of NMR signals is never a simple problem to solve [1,2]. Of utmost importance is precise quantum chemical modeling of the

^{13}C NMR spectra [3]. Indeed, quantum chemical calculations may be of great assistance in resolving the NMR problem, but if and only if they are carried out properly at a sufficient level of electronic structure theory. In this sense, NMR quantum chemical modeling is full of nuances, starting from the quality of the equilibrium geometry, on top of which the NMR calculations are performed, to the computational protocol applied for the calculation of NMR chemical shifts [4,5].

^{13}C NMR shielding constants/chemical shifts, the quality of the equilibrium geometry is one of the most important issues, despite its seemingly inconspicuous influence. Overall, the problem of the equilibrium geometry factor in chemical shift calculations has been recognized for some time now [6,7,8,9] and continues to emerge in modern NMR computational studies every now and then [10,11,12]. As a matter of fact, the quality of the equilibrium geometry is strictly dependent on the level of the electronic structure theory and on the one-electron basis set used at the geometry optimization stage. As for the level of theory, there is a distinct hierarchy of methods with clearly defined computational scaling factors, levels of electron correlation covering, and innate pros and cons [5]. Thus, one can be totally lucid as to what to expect from a particular method. For the basis sets, on the contrary, the issue is far more complicated.

^{31}P chemical shifts on the basis set used on phosphorus atoms in the geometry optimization stage [16]. The study indicated a considerable variation in the average absolute error of calculated phosphorus chemical shifts compared to experimental data due to the geometry factor effect. The effect was thoroughly explored from the standpoint of the most effective polarization of the phosphorus 3p-shell, and new geometry-oriented pecG-n (n = 1, 2) basis sets for phosphorus atoms were developed on the basis of the property-energy consistent (PEC) method that was proposed earlier by Rusakov and Rusakova [17].

^{1}H,

^{13}C,

^{15}N,

^{31}P,

^{19}F, and

^{29}Se NMR shielding constants was based on the coupled clusters singles and doubles method (CCSD) [19,20] and the CCSD equilibrium geometries. Taking into consideration the wide scope of the testing job and its high computational demand, only a limited number of rather small molecules were selected for each nucleus. Therefore, what is really important now for the pecG-n basis sets is to obtain indisputable proof of their effectiveness for NMR chemical shifts calculations with the aid of extensive and austere testing carried out at different levels of electron theory for a wide variety of challenging molecules, starting from very specific compounds with strong electron correlation effects to structurally entangled biologically active naturally occurring species. In this paper, we present the first such study performed for

^{13}C NMR shielding constants/chemical shifts, as these are the most utilized in contemporary NMR analyses, specifically in NMR studies of large compounds of biological interest.

## 2. Results and Discussion

#### 2.1. The PEC Method and the pecG-n (n = 1, 2) Basis Sets

#### 2.2. Theoretical Analysis of the pecG-n (n = 1, 2) Basis Sets

^{13}C NMR shielding constants, we considered a set of ten natural products. These will be referred to as the molecules from set

**1**. Their structures are presented in Figure 1.

**1**includes rather large spatially bulky compounds with up to 90 atoms each. These natural products are produced by living organisms such as marine sponges, fungi, and plants. Practically all of them represent very important compounds with potential biological activity. In particular, 12,28-oxaircinal A was isolated from three collections of an Indonesian sponge of the genus Acanthostrongylophora, together with 13 known manzamine alkaloids, which are known to have activity against infectious, tropical parasitic, and Alzheimer’s diseases [25]. Icajine represents one of the Strychnos icaja alkaloids, which is detected specifically in the stem, root, and collar bark of S. icaja and commonly possesses specific anti-plasmodial activity [26,27]. Physalin D is found in a fraction from the aerial parts of Physalis angulate, known in Brazil as camapu, which is a branched annual shrub that belongs to the Solanaceae family. Extracts from this plant have been used in traditional folk medicine to treat tumors. Physalin D was found to exhibit inhibitory activity against Mycobacterium tuberculosis [28]. Betulinic acid originates from lupane. It is a pentacyclic triterpene, a group characterized by cytotoxic properties, which can be isolated from plants (e.g., Spirostachys africana) or synthesized [29]. The anticancer property of betulinic acid and its derivatives was extensively studied [30,31,32,33]. Anabsinthin is a sesquiterpene lactone that can be extracted from the aerial parts of Artemisia absinthium L., commonly known as wormwood, which is a yellow, flowering, perennial plant that is distributed throughout various parts of Europe and Siberia and is used for its antiparasitic effects, as well as to treat anorexia and indigestion [34]. Itoaic acid or 2β,11β-dihydroxy-3,4-secofriedelolactone-27-oic acid represents a rare naturally occurring triterpenoid with a 3,4-seco-friedelolactone skeleton and potential anti-inflammatory activity against COX-2, which was isolated from Flacouritaceae plants [35]. Matopensine is a symmetrical bisindole alkaloid that can be extracted from the roots of Strychnos matopensis and Strychnos kasengaensis, which are plants from eastern Africa. Matopensine-type alkaloids were found to exhibited potent and selective activities against Plasmodium [36]. Strychnobaillonine is an asymmetrical bisindole alkaloid found in the roots of liana Strychnos Icaja, which is mainly used by local populations of Africa as an arrow or ordeal poison and as a treatment for skin diseases and chronic, persistent malaria [37]. Iguesterine was isolated from the root bark of Catha cassinoides [38]; it has cytostatic activity against HeLa cells [39]. Naucleidinal was isolated from the roots of Nauclea Orientalis and showed significant cytotoxic activity against both HeLa and KB cell lines [40].

^{13}C NMR spectra present a superposition of the individual second- or even higher-order multiplets, forming complex patterns, which are very difficult to analyze. Thus, it is very important to take into account as many factors of accuracy as possible within feasible limits. In this respect, the geometric factor that can change the calculated values of carbon chemical shifts within a couple ppm, depending on the basis set used at the geometry optimization stage, is very important.

**1**, which in total gave us 292 values that resulted in very solid statistics. The equilibrium geometries of the compounds of set

**1**were obtained at the DFT(M06-2X) [41] level of theory while taking into account solvent effects, using the different basis sets listed in Table 1. The solvent effects were calculated using the IEF-PCM model parametrized for chloroform as the solvent for 12-28-oxaircinal, icajine, iguesterin, itoaic acid, matopensine, naucleidinal, and strychnobaillonine; acetonitrile as the solvent for anabsinthin; pyridine as the solvent for betulinic acid; and dimethylsulfoxide as the solvent for physalin D.

**1**contain significant dispersion interactions, the M06-2X functional could be practically useful, as this functional was found to be highly successful at describing dispersion interactions for neutral molecular systems due to its portion of Grimme’s long-range dispersion corrections with an s6 scaling factor of 0.06 [43].

^{1}H,

^{13}C,

^{15}N,

^{17}O, and

^{31}P NMR shielding constant/chemical shift calculations via the PEC algorithm and were presented in our previous papers [47,48]. Overall, the pecS-n (n = 1, 2) basis sets are rather small in size, consisting of only 5/14 and 18/34 functions for hydrogen and the second-row atoms, respectively, and demonstrate a better accuracy compared to the other commensurate shielding-oriented basis sets [49]. Thus, our NMR-oriented basis sets embody a fine balance between size and accuracy, and hence, were our present choice.

**1**that were calculated using the equilibrium geometries obtained using different basis sets against the reference theoretical shielding constants. These are presented in Figure 2.

^{13}C NMR spectra of natural products are modeled, we would not recommend using the cc-pVDZ and 6-311G(d,p) basis sets for geometry optimization of such compounds.

^{13}C NMR spectra is required, and the pecG-2 basis set would apparently be a better choice for the geometry optimization at the expense of the increased computational costs compared to the cc-pVTZ basis set.

**1**was also possible, as their experimental

^{13}C NMR chemical shifts are available [25,28,34,35,37,38,40,53,54,55,56]. Therefore, we evaluated scaled chemical shifts $\stackrel{~}{\delta}\left({\sigma}_{i},\alpha \right)$ from the carbon shielding constants ${\sigma}_{i}$ using a physically meaningful linear regression model. This means that we applied the least squares method (LSM) with a slope equal exactly to −1:

^{6}, with N being the number of basis set functions [59]. The CCSD(T) scheme that, on top of the CCSD, takes into account the triple excitations within the noniterative perturbative treatment, covers the electron correlation already by 99.7%, and has a computational scaling factor of N

^{7}[59]. Ideally, it would have represented a great test if it were possible to apply these methods to the molecules from test set 1, but, unfortunately, these methods are prohibitive for systems of such a large size. Therefore, we introduced a set of systems with moderate sizes that possess very intricate electronic structures and represent a challenge to any computational tool. Eventually, we carried out an analysis based on the reference experimental data that are presented in the Section 2.3.

#### 2.3. Analysis of the Performance of the pecG-n (n = 1, 2) Basis Sets Based on a Comparison of the Theoretical Data with Experimental Data

**2**, which is shown in Figure 3. One can see that set

**2**contains a wide variety of molecules with very difficult electronic structures, including, for example, molecules such as cyclopropane and oxetane, whose specific ring brings about a substantial steric strain. Different hybridization states of carbon atoms in bonding and their unique electron environment in each compound result in a wide variety of experimental

^{13}C NMR chemical shifts, ranging from approx. −3 to 200 ppm. All measurements of carbon chemical shifts were carried out in CDCl

_{3}and referenced to TMS at 0.00 ppm by Cohen et al. [60].

**2**at the CCSD level of theory using different basis sets. This time, in view of the particularly demanding computational tasks, not all the basis sets from Table 1 were used in the geometry optimizations, i.e., apart from our basis sets, pecG-n, we only included into consideration the basis sets with the same size, the 6-31G(2d,2p) and 6-311G(3df,3pd) basis sets, and one very popular basis set, 6-311G(d,p). All the molecules of set

**2**represent highly symmetric rigid structures so there was no need to carry out a conformational analysis. All geometry optimizations were performed taking into account the solvent effects.

**2**were carried out using the GIAO-CCSD(T) method and the pecS-2 basis set on all atoms except for fluorine, for which, the pcS-2 basis set [61] was used. In view of the unavailability of the computational codes for coupled cluster calculations of shielding constants that account for solvent effects, the solvent corrections to the carbon shielding constants were taken into account at the GIAO-DFT(B97-2)/aug-cc-pV5Z level of theory. Unfortunately, the vibrational effects were not taken into account due to the extremely demanding computational costs needed for the computations within the coupled cluster method, while the DFT level was not used deliberately as it cannot be thought of as a trustworthy method for such electronically complicated molecules as those included in set

**2**.

^{13}C NMR chemical shifts were calculated from the corresponding shielding constants in accordance with the LSM method with the slope equal to −1, as was described in the Section 2.2 (see Equations (1)–(3)). The computed

^{13}C NMR chemical shifts and the corresponding experimental data are presented in Table 2 below.

_{1}of DMAc and fluorobenzene. Apparently, these compounds have extremely complicated electron structures with carbon C

_{1}being strongly involved in specific electron interactions. For example, DMAc is a representative of systems where the amino substituent at the C

_{1}atom results in strong n ⟶ π* interaction, implying that the nitrogen atom donates the density of a lone pair (n) of electrons into the empty antibonding π* orbital of the nearby carbonyl group C

_{1}= O, which typically causes a weakening and stretch of the carbon–oxygen bond [62]. At the same time, fluorobenzene is remarkable due to a considerable inductive electron withdrawal effect from the ipso carbon (C

_{1}) by fluorine, making the C

_{1}-F bond highly polarized and the C

_{1}carbon lacking electrons to such an extent that it actually becomes the most de-shielded one in the system. This perturbation propagates along the ring, resulting in a considerable concertation of negative charge at the ortho position, making the C

_{2}carbon the most shielded one. The electronic structure of fluorobenzene is so complicated that its electron-density difference plot is astoundingly similar to that for the phenyl cation with a nearby negative charge [63]! Evidently, a higher level of coupled cluster theory is needed for such complicated systems at all levels of calculation, not to mention the need to take into account explicit solute–solvent interactions.

_{1}carbon of DMAc and fluorobenzene.

**2**against the experimental data is shown in Figure 5.

## 3. Materials and Methods

**1**and set

**2**were carried out accordingly, at the DFT and CCSD levels of theory in the Gaussian program [64]. All equilibrium geometries were obtained while taking into account solvent effects. For this purpose, the integral equation formalism of the polarizable continuum model, the IEF-PCM [65,66], was used. The IEF-PCM was parametrized in accordance with the solvents defined in the experimental papers. For the specification for each compound, please see the “Results and Discussion” section. All equilibrium geometries, obtained at different levels of electron theory and using different basis sets, are presented in the Supplementary Materials.

^{13}C NMR shielding constants, either with or without accounting for the solvent effects, were conducted in the Gaussian program, while all gas phase CCSD(T) calculations of

^{13}C NMR shieldings were carried out in the CFOUR program [67]. All calculated NMR shielding constants are presented in the Supplementary Materials (Tables S1–S15).

## 4. Conclusions

^{13}C NMR shielding constants/chemical shifts calculations. They were able to improve the final accuracy of the calculated NMR data compared to the other basis sets commonly used at the geometry optimization stage, such as Dunning’ cc-pVXZ (X = D, T) and the Pople-style 6-31G(2d,2p), 6-311G(3df,3pd), and 6-311G(d,p) basis sets.

**1**) with the reference data obtained using the equilibrium geometries calculated using the cc-pVQZ basis set. Most importantly, this analysis showed that the pecG-1 and pecG-2 basis sets gave substantially better equilibrium geometries than their direct Pople-style analogs with the same size, the 6-31G(2d,2p) and 6-311G(3df,3pd) basis sets, respectively, and, in fact, they produced a substantially smaller geometry factor error in the calculated carbon shielding constants. Out of all considered basis sets, the pecG-2 basis set was found to be the best, providing an MAE of only 0.11 ppm for the shielding constants of the molecules of test set

**1**.

^{13}C NMR chemical shifts with experimental data. This analysis involved a highly demanding computational protocol using the CCSD level of theory for the geometry optimizations and the GIAO-CCSD(T) level for the shielding constant calculations. This analysis was performed on a set of very electronically complicated systems (set

**2**) and revealed the same pattern that was observed in the theoretical analysis: the pecG-n (n = 1, 2) basis sets showed considerably superior efficacy in quelling the geometry factor error in the

^{13}C NMR chemical shift calculations, with the pecG-2 basis set being the best. The MAE achieved with the pecG-2 basis set for test set

**2**compared to the experimental data was found to be 2.25 ppm.

^{13}C NMR chemical shifts calculations whenever highly precision modeling is required, like in the case of large-sized natural products. On the other hand, if there is a strong limitation on the basis set functions to be treated, the pecG-1 basis set is recommended. Basically, the pecG-1 and pecG-2 basis sets were conceived so as to provide the smallest geometry factor error in molecular property calculations; thus, as they are moderate in size, they are the best for the geometry optimizations performed as part of NMR spectra modeling. In this work, this fact was successfully corroborated using the example of

^{13}C NMR chemical shift calculations.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Compounds used in theoretical analysis (set

**1**). Blue, red, yellow and gray balls represent nitrogen, oxygen, carbon and hydrogen atoms, respectively.

**Figure 2.**MAEs for the

^{13}C NMR shielding constants calculated for set

**1**using the equilibrium geometries obtained using different basis sets (listed along the abscissa) compared to the corresponding reference theoretical data. The red numbers indicate the sizes of the basis sets for the elements of the second period. In these calculations, only the lowest energy conformers were taken into account.

**Figure 4.**MAEs for the

^{13}C NMR chemical shifts calculated for test set

**2**using equilibrium geometries obtained using different basis sets (listed along the abscissa) against the corresponding experimental data. The second bars show the altered statistical figures evaluated without taking into account the chemical shift of C

_{1}of DMAc and fluorobenzene. The red numbers indicate the sizes of the basis sets for the second period elements.

**Figure 5.**Correlation plot for the

^{13}C NMR shielding constants of test set

**2**calculated at the GIAO-CCSD(T)/pecS-2 level using equilibrium geometries obtained at the CCSD/pecG-2 level against the corresponding experimental data.

Basis Set | Element | Contracted Composition | Number of Contracted Basis Functions |
---|---|---|---|

Rusakov’s pecG-n series | |||

pecG-1 | H | [2s2p] | 8 |

B-F | [3s2p2d] | 19 | |

pecG-2 | H | [3s3p1d] | 17 |

B-F | [4s3p3d1f] | 35 | |

Dunning’s cc-pVXZ series | |||

cc-pVDZ | H | [2s1p] | 5 |

B-F | [3s2p1d] | 14 | |

cc-pVTZ | H | [3s2p1d] | 14 |

B-F | [4s3p2d1f] | 30 | |

cc-pVQZ | H | [4s3p2d1f] | 30 |

B-F | [5s4p3d2f1g] | 55 | |

Pople-style K-LMNG(x,y) series | |||

6-311G(d,p) | H | [3s1p] | 6 |

B-F | [4s3p1d] | 18 | |

6-31G(2d,2p) | H | [2s2p] | 8 |

B-F | [3s2p2d] | 19 | |

6-311G(3df,3pd) | H | [3s3p1d] | 17 |

B-F | [4s3p3d1f] | 35 |

Molecule | No. of Carbon Atoms ^{1} | 6-311G(d,p) | 6-31G(2d,2p) | 6-311G(3df,3pd) | pecG-1 | pecG-2 | Exp. ^{2} |
---|---|---|---|---|---|---|---|

Acetaldehyde | 1 | 196.19 | 197.17 | 196.25 | 196.37 | 196.97 | 199.97 |

2 | 34.92 | 34.70 | 35.09 | 34.90 | 35.02 | 30.99 | |

Acetonitrile | 1 | 4.95 | 4.84 | 4.85 | 4.61 | 4.65 | 1.91 |

2 | 115.07 | 115.73 | 115.57 | 115.91 | 115.93 | 116.33 | |

Cyclopropane | −1.61 | −1.66 | −1.37 | −1.63 | −1.57 | −3.15 | |

DMAc | 1 | 160.74 | 161.03 | 161.06 | 160.69 | 161.50 | 170.66 |

2 | 25.58 | 25.25 | 26.12 | 25.34 | 26.10 | 21.58 | |

3 | 40.35 | 39.75 | 40.60 | 39.86 | 40.61 | 38.05 | |

4 | 34.32 | 33.74 | 34.61 | 33.88 | 34.60 | 35.20 | |

Fluorobenzene | 1 | 155.98 | 156.06 | 155.61 | 156.09 | 155.93 | 162.86 |

2 | 111.85 | 111.65 | 111.51 | 111.72 | 111.49 | 115.32 | |

3 | 131.15 | 130.89 | 131.09 | 131.00 | 130.94 | 129.96 | |

4 | 127.11 | 126.88 | 126.52 | 126.89 | 126.55 | 123.98 | |

Isoxazole | 1 | 158.00 | 158.72 | 157.55 | 158.28 | 157.62 | 157.64 |

2 | 106.04 | 105.96 | 105.78 | 105.95 | 105.65 | 103.47 | |

3 | 148.72 | 149.18 | 148.16 | 148.83 | 147.96 | 149.02 | |

Norbornadiene | 1 | 145.04 | 145.27 | 145.17 | 145.05 | 144.97 | 143.43 |

2 | 48.28 | 48.27 | 48.29 | 48.63 | 48.27 | 50.26 | |

3 | 74.67 | 74.19 | 75.34 | 75.13 | 75.50 | 75.32 | |

Oxetane | 1 | 73.80 | 74.05 | 73.96 | 73.87 | 73.94 | 72.55 |

2 | 23.41 | 23.09 | 23.46 | 23.41 | 22.91 | 22.35 | |

Pyridine | 1 | 150.18 | 150.07 | 149.91 | 150.29 | 149.98 | 149.74 |

2 | 125.15 | 124.97 | 124.99 | 124.95 | 124.69 | 123.78 | |

3 | 137.53 | 137.41 | 137.21 | 137.33 | 137.11 | 136.09 | |

α | 196.12 | 196.82 | 197.81 | 197.18 | 198.54 | ||

MAE | 2.43 | 2.34 | 2.39 | 2.31 | 2.25 |

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**MDPI and ACS Style**

Rusakov, Y.Y.; Semenov, V.A.; Rusakova, I.L.
Quelling the Geometry Factor Effect in Quantum Chemical Calculations of ^{13}C NMR Chemical Shifts with the Aid of the pecG-*n* (*n* = 1, 2) Basis Sets. *Int. J. Mol. Sci.* **2024**, *25*, 10588.
https://doi.org/10.3390/ijms251910588

**AMA Style**

Rusakov YY, Semenov VA, Rusakova IL.
Quelling the Geometry Factor Effect in Quantum Chemical Calculations of ^{13}C NMR Chemical Shifts with the Aid of the pecG-*n* (*n* = 1, 2) Basis Sets. *International Journal of Molecular Sciences*. 2024; 25(19):10588.
https://doi.org/10.3390/ijms251910588

**Chicago/Turabian Style**

Rusakov, Yuriy Yu., Valentin A. Semenov, and Irina L. Rusakova.
2024. "Quelling the Geometry Factor Effect in Quantum Chemical Calculations of ^{13}C NMR Chemical Shifts with the Aid of the pecG-*n* (*n* = 1, 2) Basis Sets" *International Journal of Molecular Sciences* 25, no. 19: 10588.
https://doi.org/10.3390/ijms251910588