Origin of ¹⁷O NMR chemical shifts based on molecular orbital theory: paramagnetic terms of the pre-α, α and β effects from orbital-to-orbital transitions, along with the effects from vinyl, carbonyl and carboxyl groups

Abstract

¹⁷O NMR chemical shifts (δ(O)) were analysed based on the molecular orbital (MO) theory, using the diamagnetic, paramagnetic and total absolute magnetic shielding tensors (σ^d(O), σ^p(O) and σ^t(O), respectively). O²⁻ was selected as the standard for the analysis. An excellent relationship was observed between σ^d(O) and the charges on O for O⁶⁺, O⁴⁺, O²⁺, O⁰ and O²⁻. The data from H₂O, HO⁺, HO⁻ and H₃O⁺ were on the correlation line. However, such relationship was not observed for the oxygen species, other than above. The pre-α, α and β effects were evaluated bases on σ^t(O), where the pre-α effect arises from the protonation to a lone pair orbital on O²⁻, for an example. The 30–40 ppm and 20–40 ppm (downfield shifts) were predicted for the pre-α and β effects, respectively, whereas the values for the α effect was very small in magnitude, where the effect from the hydrogen bond formation should be considered. Similarly, the carbonyl effect in H₂C=O and the carboxyl effects in H(HO)C=O were evaluated from MeOH, together with H₂C=CHOH from CH₃CH₂OH. Very large downfield shifts of 752, 425 and 207 ppm were predicted for H₂C=O*, H(HO)C=O* and H(HO*)C=O, respectively, together with the 81 ppm downfield shift for H₂C=CHO*H. The origin of the effect were visualized based on the occupied-to-unoccupied orbital transitions. As a result, the origin of the ¹⁷O NMR chemical shifts (δ(¹⁷O)) can be more easily imaged and understand through the image of the effects. The results would help to understand the role of O in the specific position of a compound in question and the mechanisms to arise the shift values also for the experimental scientists. The aim of this study is to establish the plain rules founded in theory for δ(¹⁷O), containing the origin, which has been achieved through the treatments.

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Introduction

NMR spectra are commonly measured and analysed on a daily basis to determine the structures and/or follow up the reactions. Indeed, ¹H and ¹³C NMR spectroscopy is the most important tool for the purposes, but NMR spectra other than above are also measured on a daily basis.^1–4 NMR spectroscopy of ¹⁵N, ¹⁷O and ¹⁹F atoms in the second period, has also been a very important technology in current chemical science research.^5–9 Among the nuclei, oxygen is the most abundant chemical element and it will form compounds with any other element, except for some atoms of the Group 18 element. It seems somewhat difficult to form compounds between them. Oxygen is also involved in the various biologically important species, such as amino acids and nucleoacid bases,^10–16 together with the materials of high functionalities.^17,18 Measurements of ¹⁷O NMR spectra in the natural abundance are now much easier by the advances in the spectrometer, irrespective of the very low natural abundance with the spin number of 5/2. As a result, lots of ¹⁷O NMR chemical shifts (δ(O)) of oxygen species have been reported thus far, of which values spread over 2500 ppm.

The importance of the NMR spectroscopy is widely recognized, as mentioned above. Experimental chemists usually analyse NMR spectra with the guidance of empirical rules.^1,2,9 The empirical rules are very useful for assigning the spectra, however, it is difficult to understand the origin of chemical shifts based on the rules. Indeed, only the chemical shift of the reference species is usually provided in such NMR analysis, but any concept and/or data, that help us to image the origin of the chemical shifts, are not provided. As a result, it is very difficult to visualize the origin of the NMR chemical shifts, especially for experimental scientists, who are not the specialists in this field, including the authors. (They are originally experimental chemists, who use calculations extensively to confirm the causality in the experimental results.) This must be the extreme contrast to the cases of the electronic spectra and the infrared spectra, for example. It is easily come to mind the image of the origin for the spectra. They correspond to the electronic transitions between the occupied and unoccupied energy levels and the transitions between the energy levels of internal vibrations, respectively, in molecules and/or atoms.

Our research interested, therefore the aim of this study, is to establish the plain rules founded in theory for the origin of the ¹⁷O NMR chemical shifts for the better understanding of the phenomena. The origin should be visualized based on the specific concepts, such as molecular orbitals (MOs). The plain rules with the origin should be easily imaged and understood by the experimental scientists who are not the specialists. This purpose is given more importance, in this work, than the usual calculations of the NMR parameters, reproducing the observed values accurately and/or to predict well the shift values of unknown target compounds. The results should help to understand the role of O in the specific position of a compound in question and the mechanisms to arise the shift values.

Scheme 1 shows the axes in ROR, used for the analysis, together with some MOs and/or AOs (atomic orbitals). The direction of the p-type lone pair orbital (n_p(O)) in the symmetric ROR was set to the z-axis, which was perpendicular to the molecular plane, the bisected ∠C_ROC_R direction is set to the x-axis, and that perpendicular to the two is set to the y-axis. In the case of unsymmetric ROR′ (R > R′), the z-axis is set to the direction of n_p(O), while the y- and x-axes are set appropriately in the plane of O–C_R and O–C_R′. The axes for the species other than above are shown in the individual figures.

Scheme 1 Axes in ROR and ROR′, analysed in this work, along with some orbitals. The atomic orbitals (AOs) of 1s (O) and 2s (O) are not drawn, since they overlap 2p_z (O), if illustrated.

The α, β, γ and δ effects are well known as the experimental rules, which correspond to the methyl substitutions in the processes of –O–H → –O–CH₃, –O–CH₃ → –O–CH₂–CH₃, –O–CH₂–CH₃ → –O–CH₂–CH₂–CH₃ and –O–CH₂–CH₂–CH₃ → –O–CH₂–CH₂–CH₂–CH₃, respectively. The α, β and γ effects in the ¹⁷O NMR chemical shifts are typically found at −40 ppm (upfield shifts), +30 ppm (downfield shifts), −6 ppm (upfield shifts), respectively, with the δ effect being negligibly small, based on the observed values. The α, β and γ effects are analysed based on the MO theory. We have proposed the “pre-α effect” to establish the plain rules and understand the mechanisms in a unified form.¹⁹ The “pre-α effect” is defined to originate from the protonation to a lone pair orbital of O (O²⁻ → OH⁻, for example). The pre-α, α, β and γ effects are discussed for δ(¹⁷O) in R–¹⁷O–R′, where R and R′ are the saturated hydrocarbons. The values for the effects are calculated per unit group (per Me or H). The effects on δ(¹⁷O) in the unsaturated moieties are also be discussed, exemplified by the vinyl, carbonyl and carboxyl groups, in this paper. The plain rules, established based on the theory, need to be as simple and easily understood.

The chemical shifts of the respective structures can be theoretically calculated. The origin will be elucidated based on the MO theory. The total absolute magnetic shielding tensors (σ^t) are used for the analysis, since σ^t can be calculated with satisfactory accuracy. As shown in eqn (1), σ^t is decomposed into the diamagnetic and paramagnetic shielding tensors (σ^d and σ^p, respectively).^20–22 The magnetic shielding tensors consist of three components: σ_xx^m, σ_yy^m and σ_zz^m (m = d, p and t). Eqn (2) shows the relationship. As shown in eqn (3), σ^d is simply expressed as the sum of the contributions over the occupied orbitals (ψ_i, so is ψ_j), where the contribution from each ψ_i to σ^d (σ^d_i) is proportional to the average inverse distance of electrons from nuclei in ψ_i, <r_i⁻¹> (eqn (4)).²³ σ^p is evaluated by the Coupled-Hartree-Fock (CPHF) method. σ^p can be decomposed into the contributions from the occupied orbitals or the orbital-to-orbital transitions,²⁴ under the DFT levels. σ^p is shown in eqn (5), where the contributions from the occupied-to-occupied orbital transitions are neglected.^19,23 The process to evaluate σ^p is highly complex, therefore, σ^p will be discussed based on the approximate image derived from eqn (6),²⁴ where (ε_a − ε_i)⁻¹ is the reciprocal orbital energy gap, ψ_k is the k-th orbital function, L̂_z,N is orbital angular momentum around the resonance nucleus N, and r_N is the distance from N.

σ^t = σ^d + σ^p (1)

σ^m = (σ_xx^m + σ_yy^m + σ_zz^m)/3 (m = d, p and t) (2)

The NMR chemical shifts of the atoms in the higher periods are predominantly controlled by the σ^p term. The origin and the mechanisms have been thoroughly analysed, such as for δ(Se).¹⁹ Contrary to the atoms in the higher period, the NMR chemical shifts of the atoms in the second period are controlled by both the σ^d and σ^p terms. Therefore, the mechanisms such as for δ(O) will be more complex. Here, we discuss the origin and mechanisms for δ(O) based on the MO theory, employing the pre-α, α and β effects, together with the effects from the vinyl, carbonyl and carboxyl groups. Our explanation is intended to clarify the shift values, mainly based on the orbital-to-orbital (ψ_i→ψ_a) transitions, as aforementioned. The earlier investigations on δ(Se) will help to understand δ(O) easier, we believe, due to the similarities in the basic structures of the species consisted of the atoms.¹⁹

Methodological details in calculations

Calculations were performed using the Gaussian 09 program package, including GaussView.²⁵ The structures were optimized for various oxygen species with the 6–311++G(3df,3pd) (6D10F) basis set (BSS-A). The structural optimizations were performed at the DFT^26–29 (L1) and/or MP2 ^30–32 (L2) levels (L = L1 + L2), after some pre-optimizations. The gauge-independent atomic orbital (GIAO) method^33–37 was applied to calculate the absolute magnetic shielding tensors of O [σ(O)]. To examine the level dependence on the σ(O), the σ(O) values were calculated at the various L1 levels of B3LYP,^26–29 CAM-B3LYP,³⁸ PBE,³⁹ PBE0,⁴⁰ LC-ωPBE⁴¹ and ωB97X-D⁴² with BSS-A (L1/BSS-A) and the L2 level. The basis set of def2TZVP^43,44 was also applied at the B3LYP level (B3LYP/def2TZVP). The solvent effect of CHCl₃ was evaluated with the polarizable continuum model (PCM),⁴⁵ if necessary. The 6-311+G(3d,3p) (6D10F) basis set (BSS-B) operates similarly well to BSS-A, but the results are not discussed.

A utility program⁴⁶ was applied to evaluate the contributions from each ψ_i and/or ψ_i→ψ_a transition. The procedure is explained in Appendix of the ESI.† The charge on O (Q(O)) was obtained with the natural population analysis (NPA).⁴⁷

Results and discussion

Search for suitable level in the calculations: setting the standard for the calculated σ^t(O) values versus the observed δ(O) values

We will tentatively use σ^t(O: S) and δ(O: S) as the calculated and observed values, respectively, in this paper, to avoid confusing the discussion, although this notation might not be completely theoretically appropriate. In this case, σ^t(O: S) and δ(O: S), respectively, stand for the shift values of oxygen species, S.

Before detailed discussion to determine the suitable calculation level in this work, it is necessary to set up the appropriate standard for σ^t(O: S). The δ(O: H₂O) value is taken as the standard for δ(O: S). Therefore, it seems good idea, at first glance, that the σ^t(O: H₂O) value is also taken as the standard for σ^t(O: S), when the σ^t(O: S) values are compared directly with the δ(O: S) values. However, this choice will not give good results, since the observed and calculated conditions are very different especially for H₂O. Water forms poly-clusters through hydrogen bonds (HBs) in liquid,⁴⁸ but a single molecule in the gas phase is assumed in the calculation conditions.

To avoid large differences in the chemical shifts, due to the differences between the observed and calculated conditions in water, we selected the δ(O: Me₂O) value of −52.50 ppm for the common standard of δ(O: S) and σ^t(O: S). Namely, δ(O: Me₂O) = σ^t(O: Me₂O) = −52.50 ppm is chosen at the common standard for both, where σ^t(O: Me₂O) should be denoted by Δσ^t(O: Me₂O), so σ^t(O: S) is by Δσ^t(O: S). The treatment leads Δσ^t(O: H₂O) = 0.00 ppm, fictionally. However, the sign of Δσ^t(O: S) is basically just the opposite to that of δ(O: S). Therefore, −Δσ^t(O: S) should be used, instead of Δσ^t(O: S), for the direct comparison between the calculated and observed values, where δ(O: Me₂O) = −52.50 ppm is used as the common standard of both observed and calculated values.

It is now possible to search for the suitable level in this work, after setting up the initial research conditions. The σ^t(O: S) values for various oxygen species S (ROR + ROR′) were calculated at the DFT levels of B3LYP,^26–29 CAM-B3LYP,³⁸ PBE,³⁹ PBE0,⁴⁰ LC-ωPBE⁴¹ and ωB97X-D⁴² (L1) with BSS-A (L1/BSS-A//L1/BSS-A), together with σ^d(O: S) and σ^p(O: S). The MP2 level (L2) is also applied for the calculations. However, only σ^t(O: S) were obtained at the MP2 level (MP2/BSS-A//MP2/BSS-A). The results are collected in Tables S1–S8 of the ESI.† The calculated values are very close with each other.

The −Δσ^t(O: S) values calculated at the L (=L1 + L2) levels are plotted versus the corresponding δ(O: S), respectively. Fig. 1 shows the plots for S of (ROR + ROR′: the 31 species) at B3LYP. The plot is analysed assuming the linear relationship (y = ax + b: R_c² (the square of the correlation coefficient)), where (a, b, R_c²) = (0.936, 2.88, 0.982) for the plot in Fig. 1. Similar calculations were performed at various L. Table 1 collects the correlations. Judging from the (a, b, R_c²) values in Table 1, B3LYP, CAM-B3LYP and PBE levels seem suitable for our purpose together with others, the b value seems somewhat larger at PBE, and the a values are less than 0.90 at PBE0, LC-ωPBE and ωB97X-D. The MP2 level gave similar results but R_c² = 0.934, the poorest value in Table 1. The a value amounts to 0.960 at B3LYP, if the solvent effect of CHCl₃ is considered. The results with B3LYP/def2TZVP are shown in entry 9 of Table 1. The a and b values seem very good, whereas R_c² = 0.926. The differences between observed and calculated values are around 20 ppm in magnitudes for s-BuOMe and s-BuOEt. The B3LYP/BSS-A method is selected for the calculations based on the results. Our aim of this work can be achieved even without the solvent effect in the calculations. The level is most popularly accepted also by the experimental researchers, which is significant for our purposes. Not so different results will be obtained when other levels in Table 1 are applied to the calculations.

Fig. 1 Plots of the calculated −Δσ^t(O: S) versus the observed δ(O: S) (S: ROR + ROR′) at the B3LYP level, with (●) and without (○) the solvent effect of CHCl₃.

Table 1 Correlations in the plots of calculated –Δσ^t(O: S) versus observed δ(O: S) for the ether type oxygen species, S (ROR + ROR′)^a,b

Entry	Level (L)	a	b	Rc^2	N
a Calculated with the GIAO method under L/BSS-A.b Observed data are used for the corresponding species in the plot.c Under the solvent effect of CHCl3.d Calculated with B3LYP/def2TZVP.
1	B3LYP	0.936	2.88	0.982	31
2	CAM-B3LYP	0.911	2.30	0.979	31
3	PBE	0.976	5.43	0.982	31
4	PBE0	0.894	1.55	0.978	31
5	LC-ωPBE	0.845	−2.09	0.979	31
6	ωB97X-D	0.886	−0.07	0.982	31
7	MP2	0.933	1.24	0.934	31
8^c	B3LYP	0.960	3.36	0.984	31
9^d	B3LYP	0.929	1.22	0.926	31

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Analysis of ¹⁷O NMR chemical shifts and the standard species

To determine the suitable standard for the analysis of ¹⁷O NMR chemical shifts based on σ^d(O), σ^p(O) and σ^t(O), the values were calculated for O⁶⁺, O⁴⁺, O²⁺, O⁰ and O²⁻ with B3LYP/BSS-A and MP2/BSS-A. Table 2 summarizes the results. The σ^t(O) values for O⁶⁺, O⁴⁺ and O²⁻, calculated with the two methods, were very close to each other. O²⁻ was selected as the standard among the three, after the case of σ^p(Se).¹⁹ It is very favourable to use σ^p(O: O²⁻) = 0.0 ppm as a standard, especially for our purpose, although σ^p(O: O⁴⁺) and σ^p(O: O⁶⁺) are also 0.0 ppm. The electronic ¹S_o state of O²⁻ with eight valence electrons by the octet rule and its spherical electron distribution are also favourable for the purpose.

Table 2 Absolute shielding tensors for ¹⁷O* (* = 6+, 4+, 2+, 0 and 2−) in the singlet state^a

Nuclear	Configuration	σ^dB3LYP(O: 1s)	σ^dB3LYP(O: 2s)	σ^dB3LYP(O: 2p)	σ^dB3LYP(O)	σ^pB3LYP(O)	σ^tB3LYP(O)	σ^tMP2(O)
a Calculated by applying the GIAO method under B3LYP/BSS-A and MP2/BSS-A.
O^6+	(2s)^0(2p)^0	272.70	0.00	0.00	272.70	0.00	272.70	272.82
O^4+	(2s)^2(2p)^0	271.45	55.54	0.00	327.00	0.00	327.00	327.09
O^2+	(2s)^2(2p)^2	270.87	49.87	46.41 (×1)	367.15	8382.15	8749.31	6551.47
O^0	(2s)^2(2p)^4	270.67	45.42	39.18 (×2)	394.45	6794.55	7189.01	6010.58
O^2−	(2s)^2(2p)^6	270.66	43.73	31.31 (×3)	408.33	0.00	408.33	407.67

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Table 3 collects the σ^d(O), σ^p(O), σ^t(O), Δσ^d(O), Δσ^p(O) (=σ^p(O) (since σ^p(O): O²⁻ = 0 ppm)) and Δσ^t(O) values for various oxygen species of 1–36, calculated with B3LYP/BSS-A, together with the Q(O) values with NPA. The Δσ*(O: S) (* = d, p and t) values are calculated from O²⁻, according to Δσ*(O: S) = σ*(O: S) – σ*(O: O²⁻). The extended conformers are selected for the calculations, since they are less three-dimensionally crowded than others, although others would contribute in some cases (Table S9 of the ESI†).

Table 3 The σ^d(O), σ^p(O), σ^t(O), Δσ^d(O)_e, Δσ^p(O)_e and Δσ^t(O)_e values for various oxygen species, 1–36, along with the pre-α, α, β, γ and δ effect and the effects from the vinyl, carbonyl and carboxyl groups, based on Δσ^t (O)_e^a,b

Species (sym)	Q(O)	σ^d(O)	(Δσ^d(O))	σ^p(O)^c	σ^t(O)	(Δσ^t(O))	Δσ^d(O)e^d	Δσ^p(O)e^d	Δσ^t(O)e^d	Effect
a Calculated with the GIAO method under B3LYP/BSS-A.b Δσ*(O: S) = σ*(O: S) – σ*(O: O^2−) (* = d, p and t).c Δσ^p(O) = σ^p(O), since (σ^p(O: O^2−) = 0 ppm).d Δσ*(O: S)e = (1/n)(Δσ*(O: S) – Δσ*(O: Se)), see text for n, S and Se.e From EtOH.f From H2C=CHOH.g From MeOH.
O^2− (1: Oh)	−2.000	408.33	(0.00)	0.00	408.33	(0.00)	0.00	0.00	0.00	—
OH^− (2: C∞v)	−1.372	396.59	(–11.74)	−19.56	377.03	(–31.29)	−11.74	−19.56	−31.29	Pre-α
MeO^− (3: C3v)	−0.976	415.05	(6.72)	−133.83	281.22	(–127.11)	18.46	−114.27	−95.81	α
EtO^− (4: Cs)	−0.938	419.47	(11.15)	−290.56	128.91	(–279.41)	4.42	−156.73	−152.31	β
i-PrO^− (5: Cs)	−0.942	423.43	(15.11)	−297.32	126.12	(–282.21)	4.19	−81.75	−77.55	β
t-BuO^− (6: Cs)	−0.970	427.54	(19.22)	−227.30	200.24	(–208.08)	4.17	−31.16	−26.99	β
H2O (7: C2v)	−0.929	392.85	(–15.47)	−66.72	326.13	(–82.19)	−7.74	−33.36	−41.10	Pre-α
MeOH (8: Cs)	−0.740	395.07	(–13.26)	−72.87	322.20	(–86.13)	2.21	−6.15	−3.94	α
EtOH (9: Cs)	−0.751	398.40	(–9.93)	−108.33	290.07	(–118.25)	3.33	−35.46	−32.13	β
i-PrOH (10: C1)	−0.752	402.81	(–5.52)	−152.24	250.57	(–157.75)	3.87	−39.68	−35.81	β
t-BuOH (11: Cs)	−0.759	406.99	(–1.34)	−180.47	226.52	(–181.81)	3.97	−35.87	−31.89	β
n-PrOH (12: Cs)	−0.747	401.99	(–6.33)	−110.17	291.82	(–116.51)	3.59	−1.85	1.74	γ
n-BuOH (13: Cs)	−0.747	405.29	(–3.03)	−112.68	292.62	(–115.71)	3.30	−2.50	0.80	δ
Me2O (14: C2v)	−0.599	396.12	(–12.21)	−73.37	322.75	(–85.58)	1.63	−3.32	−1.69	α
EtOMe (15: Cs)	−0.604	397.46	(–10.86)	−105.43	292.04	(–116.29)	1.35	−32.06	−30.71	β
i-PrOMe (16: C1)	−0.614	401.79	(–6.53)	−128.26	273.53	(–134.79)	2.84	−27.45	−24.61	β
t-BuOMe (17: Cs)	−0.622	405.31	(–3.01)	−141.91	263.41	(–144.92)	3.07	−22.85	−19.78	β
n-PrOMe (18: Cs)	−0.603	400.83	(–7.49)	−105.99	294.84	(–113.48)	3.37	−0.57	2.80	γ
n-BuOMe (19: Cs)	−0.600	405.13	(–3.20)	−110.00	295.13	(–113.19)	4.30	−4.00	0.29	δ
Et2O (20: C2v)	−0.618	396.85	(–11.47)	−136.13	260.72	(–147.60)	0.37	−31.38	−31.01	β
i-Pr2O (21: C2)	−0.631	401.23	(–7.10)	−177.41	223.82	(–184.50)	−1.95	−33.70	−35.65	β
t-Bu2O (22: C2)	−0.656	393.82	(–14.50)	−196.90	196.92	(–211.41)	−3.77	−28.97	−32.74	β
n-Pr2O (23: C2v)	−0.610	397.47	(–10.86)	−132.00	265.47	(–142.86)	0.31	2.07	2.37	γ
n-Bu2O (24: C2v)	−0.609	407.03	(–1.29)	−140.01	267.02	(–141.30)	4.78	−4.01	0.78	δ
H3O^+ (25: C3v)	−0.748	397.19	(–11.13)	−93.28	303.92	(–104.41)	−3.71	−31.09	−34.80	Pre-α
MeH2O^+ (26: Cs)	−0.624	400.40	(–7.93)	−94.92	305.48	(–102.85)	3.21	−1.64	1.56	α
EtH2O^+ (27: C1)	−0.646	408.30	(–0.02)	−132.51	275.80	(–132.53)	7.90	−37.59	−29.68	β
Me3O^+ (28: C3v)	−0.407	403.21	(–5.12)	−106.15	297.05	(–111.27)	2.01	−4.29	−2.29	α
Et3O^+ (29: C3)	−0.457	397.04	(–11.29)	−158.79	238.24	(–170.08)	−2.06	−17.55	−19.60	β
OH^+ (30: C∞v)	0.480	386.73	(–21.60)	1138.35	1525.08	(1116.76)	−21.60	1138.35	1116.76	Pre-α
H2C=CHOH (31: Cs)	−0.695	402.75	(–5.58)	−193.80	208.95	(–199.38)	4.35^e	−85.47^e	−81.12^e	C=C
H2C=CHOMe (32: Cs)	−0.561	402.34	(–5.99)	−173.97	228.36	(–179.96)	−0.41^f	19.83^f	19.42^f	C=C
PhOH (33: Cs)	−0.700	391.76	(–16.57)	−183.66	208.10	(–200.23)	−3.31^g	−110.79^g	−114.10^g	C6H5
H2C=O (34: C2v)	−0.499	404.50	(–3.82)	−833.77	−429.27	(–837.59)	9.44^g	−760.90^g	−751.46^g	C=O
H(HO)C=O* (35: Cs)	−0.582	404.48	(–3.84)	−506.77	−102.29	(–510.62)	9.42^g	−433.90^g	−424.48^g	OC=O*
H(HO*)C=O (36: Cs)	−0.687	399.82	(–8.51)	−284.37	115.44	(–292.88)	4.75^g	−211.50^g	−206.75^g	*OC=O

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Scheme 2 explains the method to calculate the effects, exemplified by the pre-α, α and β effects. The effects are calculated as Δσ^t(O: S)_e = (1/n)[σ^t(O: S) – σ^t(O: S_e)], where S_e are the starting species to give the effects and n is the factor to make Δσ*(O: S)_e per unit group. In the case of the β effect from Me₂O to Et₂O, Et₂O, Me₂O and 2 correspond to S, S_e and n, respectively, in the equation. The difference of Δσ^t(O: S) between S = Et₂O (σ^t(O) = 261 ppm) and Me₂O (σ^t(O) = 323 ppm) is −62 ppm, which correspond to the 2β effect (=Δσ^t(O: S) = σ^t(O: S) – σ^t(O: O²⁻)). The Δσ*(O: S) values are abbreviated by Δ in Scheme 2. Therefore, the β effect in this process is evaluated to be 31 ppm (=Δ/2), for example. The Δσ^d(O: S)_e and Δσ^p(O: S)_e values for the effect are calculated similarly.

Scheme 2 Evaluation of the pre-α, α and β effects. The σ^t(O: S) values in ppm are given in red bold and the differences between the two are by Δ.

The pre-α, α, β, γ and δ effects are calculated, according to the method, so are the vinyl, carbonyl and carboxyl effects. The pre-α, α, β, γ and δ effects are calculated for R-O-R′ (R, R′: saturated hydrocarbons), while the unsaturated moieties of the vinyl, carbonyl and carboxyl effects are calculated from EtOH, MeOH and MeOH, respectively. Table 3 collects the values. Scheme 3 visualizes the effects with the values.

Scheme 3 Pre-α, α, β, γ and δ effects, along with the effects from the vinyl, carbonyl and carboxyl groups, on the ¹⁷O NMR chemical shifts, calculated with the GIAO method under B3LYP/BSS-A.

Behaviour of σ^d(O)

The behaviour of the calculated σ^d(O) values can be understood by considering the two factors derived from eqn (3) and (4). If the number of occupied AOs on O increases, the σ^d(O) values become larger, whereas the magnitude of each σ^d_i(O: AO) becomes smaller, especially that for the outer AOs. The average distance of the electrons from the nucleus O (r_i) in each AO becomes larger due to the increase in electron–electron repulsion if the number of occupied AOs increases. In this case, each < r_i⁻¹> (and therefore σ^d_i(O)) in eqn (4) decrease. The σ^d_i(O) values in Table 3 are well understood as the total effect of the two.

To examine the effect of the charge on O (Q(O)), the σ^d(O) values are plotted versus Q(O) for O⁶⁺, O⁴⁺, O²⁺, O⁰ and O²⁻ (1), as shown in Fig. 2; an excellent correlation by a quadratic function was obtained (y = −1.673x² − 10.24x + 394.5: R_c² = 1.000). The results show that the σ^d(O) values are excellently correlated to Q(O) if the oxygen species has no ligands. The σ^d(O) values for H₂O (7), HO⁺ (30), HO⁻ (2) and H₃O⁺ (25) are also plotted versus Q(O) (see Table 3 for the data). The data points appear on or slightly below the regression curve. The data for HO⁺ (30) and H₃O⁺ (25) are basically located on the regression curve, and those for H₂O (7) and HO⁻ (2) are located slightly below the curve. The results show that the H atom(s) on O affect somewhat on σ^d(O), in addition to the effect on Q(O), although the Q(O) value may change depending on the calculation method.

Fig. 2 Plot of σ^d(O) versus Q(O) for O⁶⁺, O⁴⁺, O²⁺, O⁰ and O²⁻ (1), together with H₂O (7), HO⁺ (30), HO⁻ (2) and H₃O⁺ (25).

Analysis of δ(O) based on the MO theory

The behaviour of σ^d(O: S), where S has at least one alkyl group, is examined, next. Fig. 3 shows the plot of σ^d(O: S) versus Q(O) for 1–36, other than those in Fig. 2. The σ^d(O: S) values are analysed separately by the types of S: RO⁻ (3–6), ROH (8–13), ROMe (14–19) and ROR (20–24), RH₂O⁺ (26 and 27), and R₃O⁺ (28 and 29), along with others (31–36). Each plot for a type of S appears almost the y-direction, except for S of 31–36. The ranges of σ^d(O) amount to 15 ppm, while those of Q(O) are very small in each group. The σ^d(O: S) values become larger in the order of R = Me < Et < i-Pr < t-Bu for RO⁻, ROH and ROMe. The structural dependence appears to control the σ^d(O: S) values.

Fig. 3 Plots of σ^d(O) versus Q(O) for various oxygen species 1–36, other than those in Fig. 2.

As mentioned above, the magnitudes of Δσ^d(O: S) are less than 15 ppm for most species in each group of species (see Table 3). However, the magnitudes of Δσ^d(O: S) are larger than 15 ppm for i-PrO⁻ (5: Δσ^d(O) = 15.1 ppm), t-BuO⁻ (6: 19.2 ppm), H₂O (7: −15.5 ppm), OH⁺ (30: −21.6 ppm) and PhOH (33: −16.6 ppm). The first two are the RO⁻ type, and the last three are H₂O, OH⁺ and PhOH. The results for OH⁺ are effectively understood based on Q(O), where the larger magnitude in Δσ^d(O: S) for OH⁺ (30) potentially comes from the larger positive Q(O) value (=0.482). The magnitudes of Δσ^d(O: S) are much smaller than those of Δσ^p(O: S). The contributions from Δσ^d(O: S) to Δσ^t(O: S) are less than 10%, except for OH⁻ (2: 37.5%), H₂O (7: 18.8%), MeOH (8: 15.4%), Me₂O (14: 14.3%) and H₃O⁺ (25: 10.7%). Specifically, Δσ^p(O: S) contributes predominantly to Δσ^t(O: S), relative to the case of Δσ^d(O: S). As a result, ¹⁷O NMR chemical shifts can be analysed mainly by Δσ^p(O: S); however, Δσ^d(O: S) should be considered when necessary.

Effect of hydrogen bonds on ¹⁷O NMR chemical shifts

What is the effect from the hydrogen bonds (HBs) on σ^d(O), σ^p(O) and σ^t(O)? The effect is to be clarified before the detailed discussion of the values. The σ^d(O), σ^p(O) and σ^t(O) values for the various ether monomers (ROR + ROR′), calculated with B3LYP/BSS-A, are collected in Table S1 of the ESI† (see also Fig. 1). The σ^t(O) value of the Me₂O dimer is calculated to be only 1.4 ppm downfield of that of the monomer, therefore, the effect of the dimer formation in ROR + ROR′ on δ(O) is considered to be negligible. Namely, the data of the monomers can be used for those of ROR + ROR′.

The σ^t(O) values are calculated for the monomers and the dimers of ROH and RCOOH, together with the differences in σ^t(O) between the dimers and the monomers Δσ^t(O)_dm [=σ^t(O: dimer) – σ^t(O: monomer)]. The solvent effect of CHCl₃ on the σ^t(O) and Δσ^t(O)_dm values are also calculated. The values are collected in Table S10 of the ESI.† Fig. 4 illustrates the monomers and dimers, exemplified by H₂O (a) and CH₃COOH (b) with the σ^t(O) (in plain) and Δσ^t(O)_dm (in bold) values in ppm, for the better understanding of the discussion. The dimer formation leads to a downfield shift of 7 ppm for H₂O (up to 8 ppm for ROH as shown in Table S10 of the ESI†) and a upfield shift of 49 ppm for C=O* and a downfield shift of 19 ppm for C–O*–H (totally upfield shift by 15 ppm on average) in RCOOH. The analysis for RCOOH would be more complex, since only the averaged data are available due to the interconversion between topological isomers of RCO*OH and RCOO*H. The contribution from HB formation to δ(O) is well demonstrated, although the direction of the effect may depend on the structures (conformers) of the monomers and dimers.

Fig. 4 Illustration of monomers and dimers for H₂O (a) and CH₃COOH (b). The σ^t(O) (in plain) and Δσ^t(O)_dm (in bold) values are also shown in ppm.

Fig. 5 shows the plot of −Δσ^t(O: S) versus δ(O: S) for the monomers and the dimers of ROH, with and without considering the solvent effect of CHCl₃. Table 4 collects the correlations (entries 1N, 2N, 3Y and 4Y). The correlations seem (very) good. They are very similar with each other, especially for the dimers, with and without considering the solvent effect. The apparent solvent effect on δ(O: S) seems very small, especially for the dimers. The results may show that the monomers and dimers exist (as in equilibrium) in solutions, which controls δ(O: S) and the solvent effect in ROH. Similarly, −Δσ^t(O: S) are plotted versus δ(O: S) for the RCOOH monomers and the dimers, with and without considering the solvent effect, although not shown n a figure. The correlations are shown in Table 4 (entries 5N, 6N, 7Y and 8Y). The correlations become better in the order of (RCOOH monomer: with the solvent effect) ≈ (RCOOH monomer: without the solvent effect) ≪ (RCOOH dimer: without the solvent effect) ≈ (RCOOH dimer: with the solvent effect). The dimer formation seems very important in RCOOH, relative to the case of ROH, together with the considering the solvent effect.

Fig. 5 Plots of calculated −Δσ^t(O: S) versus observed δ(O: S) for some monomeric (I) and dimeric (II) alcohols with and without the solvent effect of CDCl₃: I with (○) and without (●) the solvent effect and II with (■) and without (□) the solvent effect.

Table 4 Correlations in the plots of calculated −Δσ^t(O: S) versus observed δ(O: S) for the monomers and dimers of ROH and ROOH, with and without considering the solvent effect of CHCl₃ under B3LYP/BSS-A^a

Entry^b	Plot for	a	b	Rc^2	N
a Observed data are used for the corresponding species in the plot.b The solvent effect is specified by N (no solvent effect) or Y (solvent effect) after the entry number.
1N	ROH monomers	0.921	−19.26	0.988	9
2N	ROH dimers	0.920	−16.65	0.991	9
3Y	ROH monomers	0.914	−21.44	0.989	9
4Y	ROH dimers	0.921	−16.92	0.991	9
5N	RCOOH monomers	0.939	27.29	0.929	5
6N	RCOOH dimers	1.072	−19.76	0.968	5
7Y	RCOOH monomers	0.829	48.25	0.928	5
8Y	RCOOH dimers	0.966	4.03	0.960	5

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After confirming the basic behaviour of σ^t(O) for ROR + ROR′, ROH and RCOOH, next extension is to clarify the origin of δ(O) based on the MO theory. The pre-α, α and β effects, along with the vinyl, carbonyl and carboxyl effects, are analysed using an approximated image, derived from eqn (6).²⁴

Origin of the pre-α effect

How are the ¹⁷O NMR chemical shifts originated? electrons around a nucleus ¹⁷O shield the external magnetic field at the nucleus. The spherical component of the electron distribution arises the diamagnetic terms σ^d(O), whereas the paramagnetic terms σ^p(O) are originated from the unsymmetrical component of the electron distribution. In the case of O²⁻, only σ^d(O) occurs, since the ten electrons in O²⁻ spherically distribute.

The protonation of O²⁻ yields HO⁻, which introduces the σ(O–H) and σ*(O–H) orbitals, resulting in the unsymmetrical distribution of electrons in HO⁻. The spherical electron distribution of O²⁻ changes to an unsymmetrical distribution in HO⁻, in this process. As a result, the unsymmetrical component produces σ^p(O), although the spherical component arises σ^d(O) in HO⁻. The σ^p(O) terms are caused through the orbital-to-orbital transitions, such as the ψ_i→ψ_a transition, where σ(O–H) and σ*(O–H) operate as the typical ψ_i and ψ_a, respectively, in the ψ_i→ψ_a transition.

We focused our attention to the protonation process on O²⁻ in the NMR analysis as the factor to originate σ^p(O). We proposed to call this process the pre-α effect, when the origin of the ⁷⁷Se NMR chemical shifts were discussed based on σ^p(Se).¹⁹ The pre-α effect is very important, since it is the starting point to image the origin of all NMR chemical shifts.

As shown in Scheme 3, the pre-α effect is evaluated by the (Δσ^d(O)_e, Δσ^p(O)_e, Δσ^t(O)_e) values, which are (−11.7, −19.6, −31.3 ppm), (−7.7, −33.4, −41.1 ppm) and (−3.7, −31.1, −34.8 ppm) for the processes from O²⁻ to HO⁻, H₂O and H₃O⁺, respectively. The values are calculated per unit group (per H in this case). The Δσ^t(O)_e values are all negative, along with Δσ^d(O)_e and Δσ^p(O)_e; therefore, the pre-α effect is theoretically predicted to be the downfield shifts of 31–41 ppm (Δσ^t(O)_e) (see also Table 3). The saturation effect in the pre-α effect on σ^d(O), σ^p(O) and σ^t(O) by the increase of the H atoms seems not so severe in this case. Table 5 lists the σ^d_i(O), σ^p_i(O) and σ^t_i(O) (=σ^d_i(O) + σ^p_i(O)) values for O²⁻, HO⁻, H₂O and H₃O⁺, which are separately by ψ_i. The 1s (O) AO, in the MOs, predominantly contribute to σ^d(O) for each species, whereas the 2s (O), 2p_x (O), 2p_y(O) and 2p_z(O) AOs do much smaller to σ^d(O), as expected. As shown in Table 5, ψ₃ greatly contributes to σ^p(O) (σ^p₃(O) = −85.2 ppm) for HO⁻, along with ψ₄ (−21.9 ppm) and ψ₅ (−21.9 ppm). For H₂O, ψ₃ (σ^p₃(O) = −50.2 ppm), ψ₄ (−57.2 ppm) and ψ₅ (−63.6 ppm) greatly contribute to σ^p(O). In the case of H₃O⁺, ψ₃ (σ^p₃(O) = −43.3 ppm), ψ₄ (−43.3 ppm) and ψ₅ (−61.9 ppm) greatly contribute to σ^p(O). The three orbitals must mainly be constructed by the 2p_x(O), 2p_y(O) and 2p_z(O) AOs.

Table 5 The σ^d(O), σ^p(O) and σ^t(O) values contributed from each MO of O²⁻ (1), HO⁻ (2), H₂O (7) and H₃O⁺ (25)^a

MO (i in ψi)	σ^di(O)	σ^pi(O)	σ^ti(O)
a Calculated with the GIAO method under B3LYP/BSS-A.b The ψ1, ψ2, ψ3, ψ4 and ψ5 MOs of O^2− correspond to 1s (O), 2s (O), 2px (O), 2py (O) and 2pz (O) AOs, respectively.c The σ^d1(O), σ^d2(O), σ^d3(O) and σ^d4(O) values of O^0 are evaluated to be 270.67, 45.42, 39.18 and 39.18 ppm, respectively.
O^2− (1: Oh)^b^,^c
1	270.67	0.00	270.67
2	43.73	0.00	43.73
3	31.31	0.00	31.31
4	31.31	0.00	31.31
5	31.31	0.00	31.31
Total	408.33	0.00	408.33
HO^− (2: C∞v)
1	270.64	0.00	270.64
2	39.36	−4.99	34.37
3	17.78	−85.15	−67.36
4	34.41	−21.92	12.49
5	34.41	−21.92	12.49
ψocc to ψocc		114.41
Total	392.85	−19.56	377.03
H2O (7: C2v)
1	270.61	0.00	270.61
2	38.82	−5.24	33.58
3	18.85	−50.23	−31.38
4	27.35	−57.23	−29.65
5	37.22	−63.55	−26.33
ψocc to ψocc		109.30
Total	392.85	−66.72	326.13
H3O^+ (25: C3v)
1	270.60	0.00	270.60
2	40.44	−1.93	38.50
3	23.69	−43.31	−19.62
4	23.69	−43.30	−19.61
5	38.77	−61.90	−23.13
ψocc to ψocc		57.17
Total	397.19	−93.28	303.91

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Table 6 shows the ψ_i→ψ_a transitions predominantly contributing to σ^p_i→a:xx(O), σ^p_i→a:yy(O) and/or σ^p_i→a:zz(O) for HO⁻ and H₂O, where the three components yield σ^p_i→a(O), according to eqn (2). The magnitudes larger than 6 ppm for σ^p_i→a(O) are provided in Table 6. (The border value for the positive σ^p_i→a(O) values to list the table is usually not specified, since the positive values contribute to the diamagnetic direction.) The ψ₃→ψ₉ (σ^p_3⃑9:xx(O) = −83.9 ppm), ψ₃→ψ₁₀ (σ^p_3⃑10:zz(O) = −83.9 ppm), ψ₄→ψ₈ (σ^p_4⃑8:xx(O) = −61.1 ppm) and ψ₅→ψ₈ (σ^p_5⃑8:zz(O) = −61.1 ppm) transitions greatly contribute to σ^p_i→a(O) in HO⁻. In the case of H₂O, the ψ₃→ψ₈ (σ^p_3⃑8:zz(O) = −28.1 ppm), ψ₃→ψ₁₁ (σ^p_3⃑11:xx(O) = −32.2 ppm), ψ₄→ψ₉ (σ^p_4⃑9:zz(O) = −40.9 ppm), ψ₅→ψ₈ (σ^p_5⃑8:yy(O) = −33.4 ppm) and ψ₅→ψ₉ (σ^p_5⃑9:xx(O) = −58.8 ppm) transitions greatly contribute to σ^p(O) (see Table 6).

Table 6 Main contributions from the occupied-to-unoccupied orbital transitions on σ^p(O) for HO⁻ (2) and H₂O (7)^a

i → a^b	σ^pi→a:xx(O)	σ^pi→a:yy(O)	σ^pi→a:zz(O)	σ^pi→a(O)
a Calculated with the GIAO method under B3LYP/BSS-A. The magnitudes of σ^pi→a(O) larger than 6 ppm are shown.b In ψi→ψa.
HO^− (2: C∞v)
3 → 9	−83.87	0.00	0.00	−27.96
3 → 10	0.00	0.00	−83.87	−27.96
3 → 22	0.00	0.00	−18.14	−6.05
3 → 23	−18.14	0.00	0.00	−6.05
4 → 6	21.65	0.00	0.00	7.22
4 → 7	40.70	0.00	0.00	13.57
4 → 8	−61.09	0.00	0.00	−20.36
4 → 14	−35.47	0.00	0.00	−11.82
5 → 6	0.00	0.00	21.65	7.22
5 → 7	0.00	0.00	40.70	13.57
5 → 8	0.00	0.00	−61.09	−20.36
5 → 14	0.00	0.00	−35.47	−11.82
H2O (7: C2v)
3 → 8	0.00	0.00	−28.12	−9.37
3 →11	−32.22	0.00	0.00	−10.74
4 → 9	0.00	0.00	−40.87	−13.62
4 → 11	0.00	−21.09	0.00	−7.03
4 → 13	0.00	0.00	−18.69	−6.23
4 → 17	0.00	0.00	−25.65	−8.55
5 → 6	0.00	−23.27	0.00	−7.76
5 → 8	0.00	−33.37	0.00	−11.12
5 → 9	−58.81	0.00	0.00	−19.60
5 → 17	−32.49	0.00	0.00	−10.83
5 → 18	0.00	−18.91	0.00	−6.30
5 → 21	−18.69	0.00	0.00	−6.23

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Fig. 5 and 6 illustrate the selected ψ_i→ψ_a transitions for HO⁻ and H₂O, respectively, along with the characteristics of ψ_i and ψ_a and the orbital energies. Fig. 5 shows the ψ₃→ψ₉ and ψ₃→ψ₁₀ transitions in HO⁻, which correspond to the transitions from the occupied σ(O–H) orbital to the vacant 3p_z and 3p_x orbitals, respectively, where 3p_z and 3p_x are equivalent in HO⁻. The ψ₄→ψ₈ and ψ₅→ψ₈ transitions correspond to the transitions from the occupied 2p_z and 2p_x orbitals to the vacant orbitals containing the σ*(O–H) character, respectively. The occupied σ(O–H) and vacant σ*(O–H) orbitals operate as the typical donor and acceptor orbitals, respectively, in the transitions to produce the σ^p_i→a(O) terms.

Fig. 6 Main contributions from each ψ_i→ψ_a transition to the components of σ^p(O) in HO⁻ (2).

The σ(O–H) and σ*(O–H) orbitals in H₂O similarly act as the typical donor and acceptor orbitals, respectively, according to the C_2v symmetry of H₂O, as shown in Fig. 6. The ψ₃ (B2)→ψ₈ (A1) and ψ₃ (B2)→ψ₁₁ (B1) transitions correspond to the occupied σ(H–O–H) orbital to the vacant orbitals containing the σ*(H–O–H) and 3p_z(O) characters, respectively. While the ψ₄ (A1)→ψ₉ (B2) transition corresponds to the occupied n_s(O) orbital to the vacant orbital containing the σ*(H–O–H) character, ψ₅ (B1) in the ψ₅ (B1)→ψ₈ (A1) and ψ₅ (B1)→ψ₉ (B2) transitions has the characters of the occupied n_p(O) (2p_z(O)) orbital. As observed, the σ(O–H) orbitals in H₂O act as the typical donors in the combined form of C_2v, together with 2p_z(O), while the σ*(O–H) orbitals operate as the typical acceptors in the transition, although the character seems to fractionalize to some vacant orbitals, containing the higher 3p_z(O) orbital (Fig. 7).

Fig. 7 Main contributions from each ψ_i→ψ_a transition to the components of σ^p(O) in H₂O (7).

Origin of the α effect

The α effect is evaluated for MeOH, Me₂O and MeO⁺H₂, using (Δσ^d(O)_e, Δσ^p(O)_e, Δσ^t(O)_e), of which values are (2.2, −6.2, −3.9 ppm), (1.6, −3.3, −1.7 ppm) and (3.2, −1.6, −1.6 ppm), respectively. The magnitudes of Δσ^t(O)_e are small in magnitudes (less than 4 ppm). The signs of Δσ^d(O)_e and Δσ^p(O)_e are just inverse, where Δσ^t(O)_e = Δσ^d(O)_e + Δσ^p(O)_e; this is likely the reason for the small α effect predicted based on Δσ^t(O)_e. In the case of MeO⁻ from OH⁻, the (Δσ^d(O)_e, Δσ^p(O)_e, Δσ^t(O)_e) values are (18.5, −114.3, −95.8 ppm). The large magnitude for Δσ^t(O)_e comes from the large magnitude of Δσ^p(O)_e, where the negative charge on MeO⁻ would contribute to the results.

The large upfield shifts observed in ROH as the α effect appear to be difficult to explain based on the calculated Δσ^t(O)_e values, under the calculation conditions employed in this work. The contribution from HB formation and/or the solvent effect under the observed conditions would be responsible for this.

Table 7 lists the σ^d_i(O), σ^p_i(O) and σ^t_i(O) (=σ^d_i(O) + σ^p_i(O)) values, separately by ψ_i, for Me₂O. The inner orbital of ψ₁ is constructed by the 1s (O) AO; therefore, it greatly contributes to σ^d(O) but does not contribute to σ^p(O). Those of ψ₂ and ψ₃ are constructed by the two 1s (C) AOs; therefore, the contributions to σ^d(O) and σ^p(O) are very minimal. ψ₅, ψ₆, ψ₁₀ and ψ₁₁ are mainly constructed by the 2s (C) and 2p (C) AOs; therefore, the contributions to σ^d(O) and σ^p(O) are also minimal. The contributions from ψ₇–ψ₉ and ψ₁₂ to σ^p_i(O) are large (−31 to −69 ppm), where ψ₇–ψ₉ and ψ₁₂ are mainly formed by the 2p (O) AOs. The contributions from ψ₄ and ψ₁₃ to σ^p_i(O) are −13.1 and −17.9 ppm, respectively, where ψ₄ and ψ₁₃ are mainly constructed by both 2s (O) and 2p (O) AOs. As shown in Table 8, the ψ₈→ψ₃₄ (σ^p_8⃑34:zz(O) = −44.9 ppm), ψ₉→ψ₃₄ (σ^p_9⃑34:xx(O) = −48.0 ppm), ψ₁₂→ψ₃₇ (σ^p_12⃑37:zz(O) = −69.4 ppm) and ψ₁₂→ψ₅₁ (σ^p_12⃑51:zz(O) = −51.6 ppm) transitions greatly contribute to the components of σ^p(O) in Me₂O.

Table 7 The σ^d(O), σ^p(O) and σ^t(O) values contributed from each MO of Me₂O (14: C_2v)^a

MO (i in ψi)	σ^di(O)	σ^pi(O)	σ^ti(O)
a Calculated with the GIAO method under B3LYP/BSS-A.
1	270.62	0.00	270.62
2,3	0.06	0.12	0.18
4	33.03	−13.07	19.96
5	9.48	4.29	13.77
6	9.14	1.85	10.99
7	11.54	−31.13	−19.59
8	9.46	−40.02	−30.56
9	10.83	−47.49	−36.66
10	−1.00	−2.70	−3.71
11	0.33	7.79	8.11
12	13.25	−68.58	−55.33
13	29.39	−17.93	11.46
ψocc to ψocc		133.50
Total	396.11	−73.37	322.75

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Table 8 Main contributions from occupied-to-unoccupied orbital transitions on σ^p(O) of Me₂O (14: C_2v)^a,b

i→a^b	σ^pi→a:xx(O)	σ^pi→a:yy(O)	σ^pi→a:zz(O)	σ^pi→a(O)
a Calculated with the GIAO method under B3LYP/BSS-A. The magnitudes of σ^pi→a(O) larger than 6 ppm are shown.b In ψi→ψa.
7 → 30	0.00	0.00	−25.63	−8.54
8 → 34	0.00	0.00	−44.88	−14.96
8 → 37	0.00	0.00	−22.04	−7.35
9 → 34	−47.96	0.00	0.00	−15.99
11 → 28	0.00	0.00	18.47	6.16
11 → 30	0.00	0.00	25.08	8.36
12 → 15	0.00	0.00	−22.81	−7.60
12 → 26	0.00	0.00	−37.56	−12.45
12 → 29	0.00	−21.18	0.00	−7.06
12→37	0.00	0.00	−69.38	−23.13
12 → 38	0.00	0.00	−18.72	−6.24
12 → 51	0.00	0.00	−51.56	−17.19
13 → 14	0.00	−29.14	0.00	−9.71
13 → 15	−45.67	0.00	0.00	−15.22
13 → 18	0.00	24.35	0.00	8.12
13 → 23	34.67	0.00	0.00	11.56
13 → 26	−51.91	0.00	0.00	−17.30
13 → 28	0.00	60.24	0.00	20.08
13 → 30	0.00	−51.05	0.00	−17.02
13 → 34	67.28	0.00	0.00	22.43
13 → 39	0.00	−25.91	0.00	−8.64
13 → 51	−39.27	0.00	0.00	−13.09
13 → 55	38.17	0.00	0.00	12.72

---

Fig. 8 shows the selected ψ_i→ψ_a transitions in Me₂O; these are considered to be the effective transitions. Both occupied and vacant orbitals extend over the entire molecule. Whereas ψ₁₂ (HOMO−1) of the n_s(O) type acts as a good donor in Me₂O, the vacant orbitals around ψ₁₄ (LUMO) do not operate as the effective acceptors in the transitions. The high electronegativity of O, relative to C, potentially prevents the contribution of 2p (O) in the vacant orbitals around the LUMO. AOs on the higher electronegative atoms are tend to contribute in the occupied MOs but not in the vacant MOs. The large contributions from the vacant orbitals around LUMO to Δσ^p(O)_e are predicted for the formation of MeO⁻ from HO⁻, where the high electronegativity of O would be relaxed by the negative charge.

Fig. 8 Main contributions from each ψ_i→ψ_a transition to the components of σ^p(O) in Me₂O (14: C_2v).

Origin of the β effect

The β effect is discussed first for ROH, ROMe and ROR, where R changes from Me to Et, then i-Pr, and then t-Bu. The calculated (Δσ^d(O)_e, Δσ^p(O)_e, Δσ^t(O)_e) values are (−0.4 ∼ 4.7, −43.4 ∼ −20.6, −38.8 ∼ −19.8 ppm) for the processes (see Table 3 and Scheme 3). The magnitudes of Δσ^d(O)_e are less than 5 ppm (usually positive), while the Δσ^p(O)_e and Δσ^t(O)_e values are approximately −40 ∼ −20 ppm. Specifically, the β effect is recognized as the downfield shift of 40∼20 ppm, based on the calculations; this effectively explains the observed effect. Similar results are predicted for the processes from MeOH₂⁺ to EtOH₂⁺ (Δσ^t(O)_e = −29.7 ppm) and from Me₃OH⁺ to Et₃OH⁺ (Δσ^t(O)_e = −19.6 ppm). In the case of the processes from MeO⁻ to EtO⁻, then i-PrO⁻ and then t-BuO⁻, the (Δσ^d(O)_e, Δσ^p(O)_e, Δσ^t(O)_e) values are (4.4, −156.7, −152.3 ppm), (4.2, −81.8, −77.6 ppm) and (4.2, −31.2, −27.0 ppm), respectively. The magnitudes of Δσ^p(O)_e and Δσ^t(O)_e decrease in the order of CH₃ > CH₂ > CH, of which H is substituted by Me. The large negative values of Δσ^p(O)_e (and Δσ^t(O)_e) lead to the large β effect in EtO⁻ and i-PrO⁻, while the effect for t-BuO⁻ appears normal.

Table 9 lists the σ^d(O), σ^p(O) and σ^t(O) values, separately by ψ_i, exemplified by Et₂O (C_2v). The contributions from ψ₁₁, ψ₁₂, ψ₁₄, ψ₂₀ and ψ₂₁ to σ^p_i(O) are large (−29.1 – −59.3 ppm). Table 10 shows the main ψ_i→ψ_a transitions, contributing to σ^p_i→a:xx(O), σ^p_i→a:yy(O), or σ^p_i→a:zz(O). The main transitions are ψ₁₄→ψ₅₇ (σ^p_14⃑57:zz(O) = −31.2 ppm), ψ₁₄→ψ₈₈ (σ^p_14⃑88:zz(O) = −43.8 ppm), ψ₂₀→ψ₈₃ (σ^p_20⃑83:zz(O) = −37.8 ppm), ψ₂₀→ψ₈₅ (σ^p_20⃑85:zz(O) = −37.5 ppm), ψ₂₁→ψ₂₂ (σ^p_21⃑22:yy(O) = −53.9 ppm), ψ₂₁→ψ₃₇ (σ^p_21⃑37:yy(O) = −36.6 ppm), ψ₂₁→ψ₄₄ (σ^p_21⃑44:yy(O) = −30.1 ppm) and ψ₂₁→ψ₅₇ (σ^p_21⃑57:xx(O) = −35.0 ppm), together with ψ₂₁→ψ₅₄ (σ^p_21⃑54:yy(O) = 52.0 ppm) and ψ₂₁→ψ₅₈ (σ^p_21⃑58:xx(O) = 49.1 ppm), which contribute to the diamagnetic direction.

Table 9 The σ^d(O), σ^p(O) and σ^t(O) values contributed from each MO of Et₂O (20: C_2v)^a

MO (i in ψi)	σ^di(O)	σ^pi(O)	σ^ti(O)
a Using the GIAO method under B3LYP/BSS-A.
1	270.61	0.00	270.61
2–5	0.11	0.25	0.35
6	33.53	−8.62	24.91
7	8.26	3.14	11.39
8	5.77	−0.96	4.81
9	3.90	−6.62	−2.71
10	8.46	−0.72	7.74
11	6.58	−59.26	−52.68
12	8.50	−34.89	−26.40
13	0.79	−2.06	−1.28
14	4.31	−36.72	−32.41
15	0.08	−12.83	−12.74
16	2.96	−20.13	−17.17
17	4.94	−13.49	−8.56
18	−0.22	−0.06	−0.28
19	−2.57	−4.12	−6.69
20	12.47	−52.83	−40.36
21	28.38	−29.12	−0.74
ψocc to ψocc		142.90
Total	396.85	−136.13	260.72

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Table 10 Main contributions from the ψ_occ→ψ_unocc transitions on σ^p(O) in Et₂O (20: C_2v)^a

i→a^b	σ^pi→a:xx(O)	σ^pi→a:yy(O)	σ^pi→a:zz(O)	σ^pi→a(O)	i → a^b	σ^pi→a:xx(O)	σ^pi→a:yy(O)	σ^pi→a:zz(O)	σ^pi→a(O)
a Calculated with the GIAO method under B3LYP/BSS-A. The magnitudes of σ^pi→a(O) larger than 6 ppm are shown.b In ψi→ψa.
11 → 34	0.00	0.00	−19.97	−6.66	20 → 65	0.00	0.00	−19.34	−6.45
11 → 64	0.00	0.00	−18.68	−6.23	20 → 83	0.00	0.00	−37.75	−12.58
12 → 57	−24.38	0.00	0.00	−8.13	20 → 85	0.00	0.00	−37.45	−12.48
12 → 58	−21.43	0.00	0.00	−7.14	21 → 22	0.00	−53.89	0.00	−17.96
14 → 57	0.00	0.00	−31.15	−10.38	21 → 37	0.00	−36.58	0.00	−12.19
14 → 58	0.00	0.00	−29.21	−9.74	21 → 44	0.00	−30.09	0.00	−10.03
14 → 88	0.00	0.00	−43.76	−14.59	21→54	0.00	52.02	0.00	17.34
15 → 58	0.00	0.00	−23.00	−7.67	21 → 55	0.00	−19.12	0.00	−6.37
16 → 58	−30.60	0.00	0.00	−10.20	21 → 57	−34.98	0.00	0.00	−11.66
17 → 51	−17.84	0.00	0.00	−5.95	21 → 58	49.06	0.00	0.00	16.35
17→ 54	0.00	0.00	−24.15	−8.05	21 → 64	0.00	−28.92	0.00	−9.64
20 → 51	0.00	−29.16	0.00	−9.72	21 → 83	−29.31	0.00	0.00	−9.77

---

Fig. 9 draws the selected ψ_i→ψ_a transitions in Et₂O, together with the characters of ψ_i and ψ_a and the orbital energies. It is expected to clarify the mechanisms for the β effect. Similar to the case of Me₂O, the occupied and vacant orbitals in Et₂O extend over the whole molecule. It is also curious that the vacant orbitals around ψ₂₂ (LUMO) do not operate effectively as acceptors in the transitions. However, the ethyl groups in Et₂O seem to play an important role in the (large) β effect, contrary to the case of the Me groups in Me₂O, which seem not to play an important role in the α effect, for example.

Fig. 9 Main contributions from each ψ_i→ψ_a transition to the components of σ^p(O) in Et₂O (20).

In the case of Et₂O, ψ₁₁, ψ₁₂, ψ₁₄, ψ₂₀ and ψ₂₁ contribute to σ^p(O), over ∼30 ppm in magnitude and the σ^p(O) value is −136.1 ppm, as the total contribution. The contributions in Et₂O are compared with those in Me₂O and H₂O. The ψ₇, ψ₈, ψ₉ and ψ₁₂ orbitals in Me₂O contribute to σ^p(O), over 30 ppm in magnitude and the σ^p(O) values −73.4 ppm, as the total contribution. In the case of H₂O, ψ₃, ψ₄ and ψ₅ contribute to σ^p(O), over 50 ppm in magnitude, which leads to the total contribution of σ^p(O) of −66.7 ppm. The σ^p(O) values of (Me₂O from H₂O) and (Et₂O from Me₂O) are calculated to be −3.3 and −31.4 ppm (per Me), respectively. The values correspond to the minimal α effect in Me₂O and the large β effect in Et₂O, in magnitudes, based on the calculations. The minimal α effect potentially originates from the cancelling of many (complex) transitions to produce σ^p(O), while this cancelling would be avoided in the β effect.

Origin of the γ and δ effects

The upfield shifts of 1.7, 2.8 and 2.4 ppm by Δσ^t(O)_e were predicted for the γ effect in the formation of n-PrOH from EtOH, n-PrOMe from EtOMe and n-Pr₂O from Et₂O, respectively. Similarly, the upfield shifts of 0.8, 0.3 and 0.8 ppm by Δσ^t(O)_e are for the δ effect in n-BuOH formed from n-PrOH, n-BuOMe from n-PrOMe and n-Bu₂O from n-Pr₂O, respectively. The predicted magnitudes of Δσ^t(O)_e are very small. The magnitudes of Δσ^d(O)_e and Δσ^p(O)_e are also very small, and the signs are the inverse to each other. The mechanisms for the γ and δ effects are not analysed further, due to the negligibly small magnitudes.

Effect from the vinyl group

Large downfield shifts in δ(¹⁷O) (∼80 ppm) are reported for vinyl ethers.⁴⁹ The effect from the vinyl group is calculated, exemplified by the process from EtOH to H₂C=CHOH (C_s), although the process can be diversely described. The (Δσ^d(O)_e, Δσ^p(O)_e, Δσ^t(O)_e) values for the process are calculated to be (4.4, −85.5, −81.1 ppm), which effectively reproduced the observed results.

Table 11 lists the σ^d(O), σ^p(O) and σ^t(O) values of H₂C=CHOH (C_s), separately by ψ_i. The contributions from ψ₇, ψ₁₀ and ψ₁₁ to σ^p_i(O) are (very) large, of which values are −48.0, −67.3 and −56.9 ppm, respectively. As shown in Table 12, the ψ₁₀→ψ₃₀ (σ^p_10⃑30:xx(O) = −21.3 ppm and σ^p_10⃑30:yy(O) = −28.7 ppm) and ψ₁₁→ψ₁₄ (σ^p_11⃑14:xx(O) = −175.5 ppm) transitions provide great contributions.

Table 11 The σ^d(O), σ^p(O) and σ^t(O) values of H₂Ce001CHOH (31: C_s), given separately by each ψ_i^a

MO (i in ψi)	σ^di(O)	σ^pi(O)	σ^ti(O)
a Using the GIAO method under B3LYP/BSS-A.
1	270.61	0.00	270.61
2	0.02	0.05	0.08
3	0.01	0.01	0.02
4	36.10	−6.49	29.61
5	5.35	4.45	9.80
6	10.22	−14.77	−4.55
7	9.55	−48.00	−38.45
8	9.56	−23.28	−13.71
9	6.00	−18.04	−12.04
10	28.45	−67.27	−38.82
11	17.59	−56.91	−39.32
12	9.29	4.93	14.22
ψocc to ψocc		31.50
Total	402.75	−193.80	208.95

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Table 12 Main contributions from occupied-to-unoccupied orbital transitions on σ^p(O) of H₂C=CHOH (31: C_s)^a

i→a^b	σ^pi→a:xx(O)	σ^pi→a:yy(O)	σ^pi→a:zz(O)	σ^pi→a(O)
a Calculated with the GIAO method under B3LYP/BSS-A. The magnitudes of σ^pi→a(O) larger than 6 ppm are shown.b In ψi→ψa.
6 → 14	−0.15	−34.97	0.00	−11.70
7 → 29	−0.29	−21.78	0.00	−7.36
7 → 30	0.00	0.00	−18.56	−6.19
8 → 14	19.02	−54.77	0.00	−11.92
10 → 30	−21.27	−28.67	0.00	−16.65
10 → 31	0.25	−23.28	0.00	−7.68
11 → 13	0.00	0.00	−23.14	−7.71
11 → 14	−175.47	1.34	0.00	−58.04
11 → 30	0.00	0.00	44.83	14.94
11 → 46	0.00	0.00	−27.56	−9.19
12 → 13	5.81	−23.50	0.00	−5.90
12 → 30	15.83	32.73	0.00	16.18

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Fig. 10 illustrates the ψ_i→ψ_a transitions in H₂C=CHOH (C_s) with the axes. The main characters of ψ₁₀, ψ₁₁, ψ₁₄ and ψ₃₀ are the occupied π(C=C–O), occupied n_s(O), vacant π*(C=C–O) and vacant σ*(C=C–O) orbitals, respectively, and they extend over the entire molecule. For the large σ^p(O) values in H₂C = CHOH (C_s), ψ₁₀ (π(C=C–O)) and ψ₁₁(n_s(O)) act as excellent donors, while ψ₁₄ (π*(C=C–O)) and ψ₃₀ (σ*(C=C–O)) operate as good acceptors. In particular, the ψ₁₁ (HOMO−1)→ψ₁₄ (LUMO+1) transition greatly contributes to σ^p_11⃑14(O) of −58.0 ppm.

Fig. 10 Main contributions from each ψ_i→ψ_a transition to the components of σ^p(O) in H₂C=CHOH (31), with the axes.

Effect from the carbonyl group

Very large downfield shifts in δ(¹⁷O) (200–400 ppm) are reported for the species containing the carbonyl group.⁵⁰ The mechanisms are discussed, exemplified by the formation of H₂C=O from MeOH, first.

Table 13 lists the σ^d(O), σ^p(O) and σ^t(O) values of H₂C=O, separately by ψ_i. The contributions from ψ₆ and ψ₈ on σ^p_i(O) are very large, which amount to −264.8 and −480.4 ppm, respectively. The (Δσ^d(O)_e, Δσ^p(O)_e, Δσ^t(O)_e) values are (9.4, −760.9, −751.5 ppm) for H₂C=O from MeOH. As shown in Table 14, the ψ₆→ψ₉ (σ^p_6⃑9:yy(O) = −647.6 ppm) and ψ₈→ψ₉ (σ^p_8⃑9:xx(O) = −1385.1 ppm) transitions are the predominant contributors.

Table 13 The σ^d(O), σ^p(O) and σ^t(O) values of H₂C=O (34: C_2v), given separately by each ψ_i^a

MO (i in ψi)	σ^di(O)	σ^pi(O)	σ^ti(O)
a Using the GIAO method under B3LYP/BSS-A.
1	270.61	0.00	270.61
2	0.03	0.05	0.08
3	32.51	−25.86	6.65
4	8.91	8.95	17.87
5	10.38	9.93	20.31
6	25.20	−264.75	−239.55
7	28.07	−41.88	−13.81
8	28.79	−480.42	−451.63
ψocc to ψocc		−39.79
Total	404.50	−833.77	−429.27

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Table 14 Main contributions from occupied-to-unoccupied orbital transitions on σ^p(O) of H₂C=O (34: C_2v)^a

i→a^b	σ^pi→a:xx(O)	σ^pi→a:yy(O)	σ^pi→a:zz(O)	σ^pi→a(O)
a Calculated with the GIAO method under B3LYP/BSS-A. The magnitudes of σ^pi→a(O) larger than 10 ppm are shown.b In ψi→ψa.
3 → 9	0.00	−37.45	0.00	−12.48
5 → 9	93.99	0.00	0.00	31.33
6 → 9	0.00	−647.61	0.00	−215.87
6 → 18	0.00	0.00	−38.06	−12.69
7 → 33	0.00	−31.32	0.00	−10.44
8 → 9	−1385.05	0.00	0.00	−461.68
8 → 10	0.00	0.00	−40.47	−13.49
8 → 13	−43.55	0.00	0.00	−14.52
8 → 19	−31.54	0.00	0.00	−10.51
8 → 33	0.00	0.00	38.09	12.70

---

Fig. 11 illustrates the selected ψ_i→ψ_a transitions of ψ₆→ψ₉ and ψ₈→ψ₉ in H₂C=O, along with the molecular axes. The ψ₆, ψ₈ and ψ₉ orbitals mainly have the occupied σ(C=O), occupied n_py(O) and vacant π*(C=O) characters, respectively. The ψ₆ (σ(C=O)) and ψ₈ (n_py(O)) orbitals act as excellent donors, while ψ₉ (π*(C=O)) does as an excellent acceptor to produce the very large σ^p (O) in H₂C=O. However, ψ₇ (π(C=O)) seems not a good donor in the transitions, relative to the case of ψ₆ and ψ₈.

Fig. 11 Main contributions from each ψ_i→ψ_a transition to the components of σ^p(O) in H₂C=O (34), together with the axes.

The origin for the very large downfield shift for σ^p(O: H₂C=O) is effectively analysed, along with the mechanism.

Effect from the carboxyl group

The carboxyl effect is closely related to the carbonyl effect, which is discussed for the formation of H(HO*)C=O* from MeOH.

Table 15 lists the σ^d(O), σ^p(O) and σ^t(O) values of H(HO)C=O* and H(HO*)C=O, separately by ψ_i. The contributions from ψ₁₀ and ψ₁₂ to σ^p_i(O) are very large for H(HO)C=O*, which amounts to −130.4 and −234.1 ppm, respectively, while that from ψ₁₀ to σ^p_i(O) is also very large for H(HO*)C=O, which amounts to −120.2 ppm. Table 16 shows the ψ_i→ψ_a transitions, mainly contributing to σ^p_i→a:xx(O), σ^p_i→a:yy(O) and/or σ^p_i→a:zz(O), in H(HO)C=O* and H(HO*)C=O. In the case of H(HO)C=O*, the ψ₁₀→ψ₁₃ (σ^p_10⃑13:yy(O) = −279.3 ppm; σ^p_10⃑13:xx(O) = −45.9 ppm) and ψ₁₂→ψ₁₃ (σ^p_12⃑13:xx(O) = −626.0 ppm; σ^p_12⃑13:yy(O) = −31.5 ppm) transitions predominantly contribute to σ^p(O); additionally, the ψ₁₀→ψ₁₃ (σ^p_10⃑13:yy(O) = −198.3 ppm; σ^p_10⃑13:xx(O) = −53.0 ppm) transition predominantly contributes to σ^p(O) of H(HO*)C=O.

Table 15 The σ^d(O), σ^p(O) and σ^t(O) values of H(HO)C=O (C_s), given separately by each ψ_i^a

MO (i in ψi)	σ^di(O)	σ^pi(O)	σ^ti(O)
a Using the GIAO method under B3LYP/BSS-A.
H(HO)C=O* (35: Cs)
1,3	0.01	0.00	0.01
2	270.61	0.00	270.61
4	9.56	−6.12	3.44
5	23.30	−21.95	1.35
6	4.29	3.74	8.03
7	7.16	15.69	22.85
8	12.87	−83.26	−70.39
9	12.53	−10.06	2.47
10	15.65	−130.36	−114.71
11	18.37	−18.43	−0.07
12	30.15	−234.06	−203.91
ψocc to ψocc		−21.96
Total	404.48	−506.77	−102.29
H(HO*)C=O (36: Cs)
1	270.61	0.00	270.61
2,3	0.02	0.02	0.04
4	22.33	−10.63	11.70
5	13.44	−6.15	7.29
6	15.46	−10.24	5.23
7	11.01	−40.96	−29.95
8	10.18	−8.84	1.35
9	17.85	−33.89	−16.04
10	9.22	−120.24	−111.02
11	18.74	−29.95	−11.22
12	10.95	−42.16	−31.21
ψocc to ψocc		18.67
Total	399.81	−284.37	115.44

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Table 16 Main contributions from occupied to unoccupied orbital transitions on σ^p(O) of H(HO)C=O (C_s)^a

i→a^b	σ^pi→a:xx(O)	σ^pi→a:yy(O)	σ^pi→a:zz(O)	σ^pi→a(O)
a Calculated with the GIAO method under B3LYP/BSS-A. The magnitudes of σ^pi→a(O) larger than 10 ppm are shown.b In ψi→ψa.
H(HO)C=O* (35)
5 → 13	0.56	−30.94	0.00	−10.13
7 → 13	85.27	−2.96	0.00	27.44
8 → 13	−0.68	−159.30	0.00	–53.33
10 → 13	−45.89	−279.32	0.00	−108.40
12 → 13	−625.99	−31.46	0.00	−219.15
H(HO*)C=O (36)
7 → 13	−64.11	−35.48	0.00	−33.20
8 → 13	2.20	70.95	0.00	24.38
10 → 13	−53.03	−198.27	0.00	−83.77
10 → 20	0.00	0.00	−45.92	−15.31
12 → 13	−57.30	−32.99	0.00	−30.10

---

Fig. 12a and b illustrate the selected ψ_i→ψ_a transitions in H(HO)C=O* and H(HO*)C=O, respectively, along with the molecular axes. While ψ₁₀ and ψ₁₂, in the ψ₁₀→ψ₁₃ and ψ₁₂→ψ₁₃ transitions of H(HO)C=O*, have the main character of occupied σ(C=O) and n_p(O), respectively, ψ₁₃ has the vacant π*(C=O) character. Therefore, ψ₁₀ (σ(C=O)) and ψ₁₂ (n_p(O)) act as excellent donors, while ψ₁₃ (π*(C=O)) operates as an excellent acceptor in H(HO)C=O*. However, ψ₁₁ (π(C=O)) seems not a good donor in the transitions, again, if compared with ψ₁₀ and ψ₁₂. Similarly, ψ₁₀ and ψ₁₃, in the ψ₁₀→ψ₁₃ transition of H(HO*)C=O, have the main character of the occupied σ(C=O) and the vacant π*(C=O), respectively. Thus, ψ₁₀ (σ(C=O)) acts as a good donor and ψ₁₃ (π*(C=O)) operates as a good acceptor in the transitions to produce (large) σ^p(O) of H(HO*)C=O.

Fig. 12 Main contributions from each ψ_i→ψ_a transition to the components of σ^p(O) in H(HO)C=O* (35) (a) and H(HO*)C=O (36) (b), together with the axes.

The (Δσ^d(O)_e, Δσ^p(O)_e, Δσ^t(O)_e) values for the process from H₂C=O to H(HO)C=O* are also of interest. The values are (0.0, 327.0, 327.0 ppm), which means that the H(HO)C=O* signal will appear at much higher field of 327.0 ppm from that of H₂C=O*.

The specific π-type O–C=O interaction is responsible for the results. The charge on H(HO)C=O* is less positive than that on H₂C=O, due to the donation from HO to C=O in H(HO)C=O, which leads to the upfield shift. The wider extension of the MOs over the entire molecule in H(HO)C=O needs to be considered, again, although it would be complex. The smaller occupancy of an important orbital in H(HO)C=O*, relative to that in H₂C=O*, would not effectively operate to produce a larger σ^p(O) in magnitude. The energy differences in the transitions also affect on σ^p(O), along with the charge on O. A more upfield σ^p(O) shift is predicted if the charge on O becomes less positive, although the energy term would show the inverse direction from the factor of the charge.

The much larger downfield shift for H₂C=O relative to H(HO)C=O* is effectively reproduced in the calculations. The large upfield shifts in RC(=O)NHR′ and ROC(=O)OR′, relative to H₂C=O, can also be understood based on the structural similarities to H(HO)C=O, relative to H₂C=O. Specifically, the analysis of H₂C=O and H(HO)C=O can aid in the understanding of the ¹⁷O NMR chemical shifts of similar structures. However, further investigations are needed to understand the much higher downfield shifts of the nitroso species and ozone.

Visualization of Δσ^d(O), Δσ^p(O) and Δσ^t(O) in some oxygen containing species

The Δσ^d(O), Δσ^p(O) (=σ^p(O)), Δσ^t(O) values and the components are plotted for Me₂O, Et₂O, H₂C=CHOH, H(HO)C=O* and H(HO*)C=O. Fig. 13 shows the plot. The contributions from the occupied-to-occupied orbital (ψ_i→ψ_j) transitions, shown in green in Fig. 13, are all positive, except for H(HO)C=O*. The σ^p(O) values are all negative, of which magnitude is small for Me₂O but very large for H(HO)C=O*. The contributions from the ψ_i→ψ_j transitions seem to decrease as the magnitudes of σ^p(O) increase. MOs, mainly constructed by the 2p_x(O), 2p_y(O) and 2p_z(O) AOs, should contribute much on σ^p(O). The contributions to σ^p(O) are well visualized, which helps us to understand the rule and the mechanism of σ*(O: * = d, p and t).

Fig. 13 Plots of Δσ^d(O), Δσ^p(O) (=σ^p(O)), Δσ^t(O) and the components, for Me₂O, Et₂O, H₂C=CHOH, H(HO)C=O* and H(HO*)C=O. Each MO contributing to σ^p(O) is shown by -n in HOMO-n.

Contributions from occupied-to-occupied orbital transitions to σ^p(O)

The occupied-to-occupied orbital (ψ_i→ψ_j) transitions are usually not considered to be important; therefore, they are often neglected in the discussion. However, they contribute to σ^p(O) more than expected, in some cases. Such transitions should arise through the redistribution of electrons in a species under an applied magnetic field. Table 17 summarizes the values, again, which are shown in some Tables in the text.

Table 17 The contributions from the occupied-to-occupied orbital (ψ_i→ψ_j) transitions to σ^p(O) in some oxygen containing species^a

Species	σ^p(O)o–o	Species	σ^p(O)o–o
a Using the GIAO method under B3LYP/BSS-A.
HO^− (2: C∞v)	114.41	H2C=CHOH (31: Cs)	31.50
H2O (7: C2v)	109.30	H2C=O (34: C2v)	−39.79
H3O^+ (25: C3v)	57.17	H(HO)C=O* (35: Cs)	−21.96
Me2O (14: C2v)	133.50	H(HO*)C=O (36: Cs)	18.67
Et2O (20: C2v)	142.90

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The paramagnetic contributions from the occupied-to-occupied transitions, σ^p(O)_o-o, are larger than 100 ppm for HO⁻ (C_∞v), H₂O (C_2v), Me₂O (C_2v) and Et₂O (C_∞v), which form group A (g(A)). The σ^p(O)_o–o values are less than 60 ppm for H₂C=CHOH (C_s), H₂C=O (C_2v), H(HO)C=O* (C_s), H(HO*)C=O (C_s) and H₃O⁺ (C_3v), which belong to g(B). According to the discussion about the data in Fig. 13, σ^p(O)_o–o are plotted versus σ^p(O), to examine the relationship between the two. The plot is shown in Fig. S11 of the ESI.† While the correlation was poor for g(A) (y = 105.70–0.261x: R_c² = 0.625), whereas a very good correlation was obtained for g(B), if analysed with a quadric function (y = 82.87 + 0.288x + 0.00017x²: R_c² = 0.996). Indeed σ^p(O)_o–o will change depending on σ^p(O), but the behaviour seems complex.

Conclusions

The ¹H NMR chemical shifts are controlled predominantly by the σ^d term; therefore, δ(¹H) are explained mainly by Q(H), especially for saturated organic species, although other terms, such as the aromatic ring current effect, become important in some cases. For the atoms of the third and higher periods, the spectra can be analysed based on the σ^p term, predominantly, neglecting the σ^d term as the relative values. In the case of the atoms of the second period, the NMR chemical shifts are controlled by both σ^d and σ^p terms, where the contribution ratios will change depending on the atoms and the species containing the atoms. Namely, the analysis of the spectra for the atoms of the second period will be essentially more complex relative to other cases. NMR spectra, containing δ(¹⁷O), are usually analysed with the guidance of empirical rules. Indeed, the empirical rules are useful for assigning the spectra, but the origins of chemical shifts are difficult to understand based on such rules. Then, our research interested is to establish the plain rules founded in theory with the origin of the ¹⁷O NMR chemical shifts for the better understanding of the phenomena, which is the aim of this study. The origin should be visualized based on the specific concepts, such as molecular orbitals. This purpose is given more importance than the usual one in NMR calculations to reproduce the observed values accurately and/or to predict well the shift values of unknown target compounds.

NMR chemical shifts of ¹⁷O are analysed employing the calculated σ^d, σ^p and σ^t terms. The contributions from σ^d(O) to σ^t(O) are approximately one tenth of those from σ^p(O), although the ratio changes depending on the oxygen containing species. The plots of σ^d(O) versus Q(O) for O^x (x = −2, 0, 2, 4 and 6) effectively follow a quadratic regression curve, and those for H₂O, HO⁺, HO⁻, and H₃O⁺ are located (very) near the curve. Therefore, the σ^d(O) values can be understood based on Q(O) for the species. However, σ^d(O) values of ROH and ROR′ (R, R′: alkyl group) change depending on R and R′ but not on Q(O). The σ^p values were analysed based on the occupied-to-unoccupied orbital (ψ_i→ψ_a) transitions, which arose σ^p(O). The relationship between σ^p(O) and Q(O) was not examined, which would be hidden in the complex combinations of in the ψ_i→ψ_a transitions, as shown in eqn (6). Specifically, a broad (but not so strong) relationship between δ(O) and Q(O) has been reported, as expected, if the conditions are satisfied; however, an explicit relationship is not observed for most cases. The occupied-to-occupied orbital (ψ_i→ψ_j) transitions are also examined, of which contributions to σ^p(O) are denoted by σ^p(O)_o–o. The good proportionality between σ^p(O)_o–o and σ^p(O) was confirmed in some cases, but not widely. The treatments provided useful information for σ^p(O), where the contributions from the ψ_i→ψ_j transitions are usually neglected. The relationships between Q(O) and between σ^p(O) and the orbital–orbital transitions (interactions) are widely clarified, in this work.

The origin of the effects is visualized based on the occupied-to-unoccupied orbital (ψ_i→ψ_a) transitions, where σ^p(O) arises from the transitions. As a result, the plain rules with the origin can be more easily imaged and understood through the contributions of transitions to the effects also by the experimental scientists, including the authors. The results will help to understand the role of O in the specific position of a compound in question and the mechanisms to arise the shift values. This work also has the potential to provide an understanding of the δ(O) values of unknown species and facilitate new concepts for the strategies to create highly functional materials based the observed δ(O) values, along with the calculated σ^d(O) and σ^p(O) values.

Author contributions

W. N. and S. H. formulated the project. K. M. contributed to investigation and writing. W. N. and S. H. organized the data, contributed to supervision, data curation, resources and writing – review & editing the paper. All authors have read and agreed to the published version of the manuscript.

Conflicts of interest

The authors declare no conflicts of interest.

Acknowledgements

The authors are very grateful to Prof. Masahiko Hada and Dr Daisuke Yamaki of Tokyo Metropolitan University for the utility programs.

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Footnote

† Electronic supplementary information (ESI) available: Additional tables and figures and the fully optimized structures given by Cartesian coordinates, together with total energies. See DOI: https://doi.org/10.1039/d4ra00843j

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