DOI 10.1007/s00214-006-0164-7
R E G U L A R A RT I C L E
Isomerization reactions of RSNO (R
=H, C
n
H
2n
+1
n
≤ 4)
Chin-Hung Lai · Elise Y. Li · Pi-Tai Chou
Received: 21 March 2006 / Accepted: 2 June 2006 / Published online: 23 August 2006 © Springer-Verlag 2006
Abstract We have applied various theoretical meth-ods to gain detailed insights into the isomers as well as the transition states (TSs) along the corresponding reac-tion pathways for RSNO (R=H, CnH2n+1n≤ 4). On the
basis of G2 and G2MP2 results, the relative order of stability for R=H is estimated to be trans-HSNO > cis-HSNO > HNSO > cis-HONS≈ trans-HONS, while it is cis-CH3SNO ≈ trans-CH3SNO > CH3NSO >
trans-CH3ONS > cis-CH3ONS for R=CH3. A similar trend
is also obtained from the B3P86 method with consid-erably less computing effort if the nearly isoenergetic isomers cis-HONS and trans-HONS are ignored. Based on the results of B3P86, cis-RSNO is more stable than trans-RSNO when R=H is replaced by alkyl groups ex-cept for R=t-Bu. Natural bond orbital analyses allow us to explore whether the high reactivity of S-nitros-othiols is due to the strong negative hyperconjugation (nπO ↔ σ ∗N−S). The mesomeric effect of
S-nitrosothi-ols, although non-negligible, does not cause the break-age of N–O bond due to the compensation of columbic attraction between N and O.
Keywords RSNO· G2 · B3P86 · Negative
hyperconjugation· Mesomeric effect
Electronic supplementary material Supplementary material is
available to authorised users in the online version of this article at http://dx.doi.org/10.1007/s00214-006-0164-7 .
C.-H. Lai (
B
)· E. Y. Li · P.-T. Chou Department of Chemistry,National Taiwan University, 106, Taipei, Taiwan, R.O.C. e-mail: chinhunglai@ntu.edu.tw P. -T. Chou
e-mail: chop@ntu.edu.tw
1 Introduction
S-nitrosothiols (RSNOs) have attracted much attention because species containing an -SNO functional group have been found in vivo as part of the metabolism of nitric oxide (NO), [1–3] an important biological messen-ger. For a long time, the S-nitrosothiols have been pro-posed to play a key role in transporting and storing NO within the organism [1–6]. The S-nitrosothiols also ex-hibit many biological properties similar to those of NO, including vasodilatation of arteries, inhibition of plate-let aggregation, smooth muscle cell proliferation, etc. [5, 7–11]. Furthermore, the derivatives of S-nitrosothi-ols, with a form of S-nitrosate cysteine thiS-nitrosothi-ols, were found to be involved in Zn2+ complexation to disrupt cys-teine–Zn2+ linkages via a transnitrosation mechanism [12–17]. Unfortunately, the S-nitrosothiols (RSNOs) readily release NO•via thermal agitation [18], radiation [19–21], or reactions catalyzed by certain metal ions [22– 33], superoxide [34, 35] and seleno compounds [36]. Al-though the heterolytic cleavage might as well take place, the biological activity of RSNOs has been mostly attrib-uted to the homolytic cleavage of the S–NO bond with the release of NO•[37]. Accordingly, experimental ap-proaches on the RSNOs are rare [38, 39], and our current understanding on the physical and chemical properties of this important class of biomolecules is still unsatisfac-tory.
Alternatively, computational chemistry seems to of-fer a reasonable access to investigate the reaction path-ways of RSNOs. Houk and others have performed some calculations to probe the bond dissociation energy of the N–S bond as well as the energetics of trans–cis con-formers [40–43]. In a study of primarily photolyzing cis-HSNO, forming the unexpected trans-HONS,
Nonella et al. have performed an HF calculation in or-der to extract some possible isomers that can correlate with the experimental results [44]. In another approach, Javakumar and Kolandaivel [45] have applied the max-imum hardness principle in an attempt to explain the relative order of stability for various isomers, including HSNO, HONS, HNOS and HOSN. In this paper, we re-port a comprehensive computational approach on the isomers as well as the corresponding transition states (TSs) along the reaction pathways for RSNOs (R=H, CnH2n+1 n≤ 4). On the optimization of theoretical methods, details on structures and thermodynamics of various isomers and TSs are resolved. To the best of our knowledge, this study appears to be the first approach to discuss the corresponding R-shifted isomers regarding S-nitrosothiols.
2 Theoretical methodology
All calculations are done with the Gaussian 03 program [46]. MP2 (full) and various DFT methods are used, including LSDA (SVWN), GGA (BLYP, BP86 and BPW91), and hybrid DFT (B3LYP, B3P86 and B3PW91) methods with the basis set 6-31++G**, except for tran-sition metal [47–53]. For trantran-sition metal (Cu+), we use the relativistic effective core potential (RECP) LANL2 plus DZ basis set, i.e. LANL2DZ [54]. The calculated minima and transition states (TSs) have been carefully checked by frequency analysis to examine whether the number of the imaginary frequency is zero or one. To gain more accurate relative energies among various isomers, G2 and G2MP2 calculations were further performed [55]. All mentioned energetic values are cor-rected for zero-point vibrational energy (ZPVE). Furthermore, NBO 6.0, implemented in Gaussian 03, is used to perform the natural bond orbital (NBO) analysis [56, 57].
3 Results and discussion
We would first like to discuss the optimization of theoret-ical methods in performing RSNO calculations. Tables S1 and S2 summarize the relative energies and isomerization barriers for cis-HSNO and cis-MeSNO on various theoretical methods. Note that the optimized structures are similar among different methods applied. If it is assumed that the approaches on higher levels, such as G2 and G2MP2, are more accurate, so that the results can be used as a criterion, the results of the DFT/B3P86 method seem to be better than that of the computationally more expensive method such as MP2 (full) (see Tables S1 and S2). For example, if both
results of DFT/B3P86 and MP2 (full) refer to that of the G2 method, the MP2 (full) overemphasizes the stability of RNSO by 5.2 kcal/mol for HNSO and 4.6 kcal/mol for CH3NSO.
On the above basis, the results and discussion here-after will mainly rely on B3P86 with the assistance of the NBO analysis. The energetic values and structures of each isomer and TS calculated on B3P86/6-31++G** are listed in Fig. 1 and Fig. S1. Comparing our B3P86/6-31++G** results with respect to those reported by Gauld et al. [21] based on B3P86/6-31++G(3df, 3pd) (see Table S3), it is found that the calculated bond length of N–S (r (N–S)) and N–O (r (N–O)) varies only by 0.02 and 0.01 Å, respectively, while the homolytic bond dis-sociation energy (BDE) on average differ by as little as 1 kcal/mol for both trans-HSNO and cis-MeSNO. It is thus not necessary to use 6-311+G(2df,p) or an even larger basis set, to obtain reliable structures or other properties for the cases of RSNO with R groups at least up to C4H9.
Thionitrous acid (HSNO) can be viewed as the simplest S-nitrosothiol, the results of which may also provide supplementary information for its alkyl ana-logues such as CH3SNO. The calculated results of HSNO,
along with those obtained from previous experimental and theoretical works, are collected in Table 1. Based on the G2 results, the relative energies with respect to cis-HSNO are -1.1, 4.4, 4.4 and 2.7 kcal/mol for trans-cis-HSNO, cis-HONS, trans-HONS, and HNSO, respectively. In comparison, a similar trend could also be found in pre-vious reports [21, 22, 24, 31, 32], except for Bharatam’s work [32], who predicted an opposite stability order be-tween cis- and trans-HSNO. Applying the Hartree-Fock method with various basis sets, Nonella and Huber [44] predicted the relative stability to be on the order of HNSO > HOSN > HSNO > HONS, while a relative en-ergy order of HSNO > HONS > HNOS > HOSN was estimated by Kolandaivel et al. [45] based on the HF/6-31G method. In addition, trans-HSNO was more stable than cis-HSNO by 0.6 kcal/mol. Although their results vary, depending on the basis sets applied, the predic-tion that HSNO is more stable than HONS is in mutual agreement and is also consistent with this study (see Table 1).
Table 2 summarizes the calculated results on CH3SNO.
For a fair comparison, data obtained from previous re-sults are also collected in Table 2. Based on the G2 results, the relative energies with respect to cis-MeSNO are 0.6, 14.4, 13.7 and 4.1 kcal/mol for trans-MeSNO, cis-MeONS, trans-MeONS and MeNSO, respectively. This tendency is qualitatively in agreement with previ-ous reports based on B3LYP and CBS-4M [41–43, 60]. Comparing Tables 1 and 2, it can be promptly perceived
Fig. 1 Structures and relative
energetics of various HSNO isomers and the
corresponding transition states (H in white, N in ashy, O in dark ashy and S in black
colors) calculated by
B3P86/6-31++G** (energies in kcal/mol, bond lengths in Å, and angles in◦)
that both relative energies of isomers and the asso-ciated reaction barriers increase as R=H is replaced by a methyl group, despite the relative stability orders
being slightly different in these two cases. For R=H, the order of stability is calculated to be trans-HSNO > cis-HSNO > HNSO > cis-HONS≈ trans-HONS, while
Table 1 The selected geometric parameters, Wiberg-type bond
orders and relative energies of isomers and transition states of HSNO calculated by B3P86/6-31++G** (energies in kcal/mol, bond lengths in Å)
r r BO BO Relative
(N–S) (N–O) (N–S) (N–O) energy
cis-HSNO 1.84 1.18 1.07 1.93 0.0 1.75a 1.17a trans-HSNO 1.86 1.18 1.06 1.94 −0.9 1.76a 1.17a −0.3a 1.91e 1.18e −0.6b −1.1c 0.9f cis-HONS 1.59 1.34 1.75 1.17 6.9 1.56a 1.31a 5.3a 4.8b 7.0d trans-HONS 1.58 1.37 1.80 1.12 6.9 1.55a 1.33a 5.1a 4.8b HNSO 1.63 1.23 1.37 1.45 2.6 TS1 2.00 1.16 9.8 1.85a 1.16a 10.4f TS2 1.71 1.21 47.3 TS3 1.65 1.26 28.7 33.5d TS4 1.57 1.42 20.7 1.54a 1.38a TS5 1.58 1.34 50.8 BO Bond order
aBased upon HF/basis (c), see ref. [44] bBased upon HF/6-31G, see ref. [45] cBased upon CBS-4M, see ref. [41]
dBased upon B3LYP/6-311G**, see ref. [56, 57] eBased upon UB3LYP/6-31G*, see ref. [41] fBased upon B3LYP/6-31+G*, see ref. [58, 59]
it is cis- CH3SNO ≈ trans- CH3SNO > CH3NSO >
trans- CH3ONS > cis- CH3ONS for R=CH3. This
re-sult might be due to the fact that both N–S and N–O bonds possess a double bond character in NOS, whereas the strength ofπN−S is weaker than that of πN−O. As a result, substituents R (R=H, Methyl) prefer to at-tach to the N–S bond, especially on the S center. For R=H, there might exist another important factor to influ-ence the relative order, namely the strength of hydrogen bonding from e.g. O–H. . . .S, O–H. . .N, etc. Our results indicate that the RSNO appears to be the most stable form among its isomers, regardless of the trans or cis form. Table 3 lists the results obtained from B3P86/6-31++G**, so that a fair comparison can be made of the relative energies between cis- and trans-RSNO, in which R=ethyl, n-propyl and iso-propyl, n-butyl, iso-butyl and tert-butyl. In all cases (R=H∼ t-C4H9), r (N–S) in
trans-RSNO is longer than that in cis-trans-RSNO, whereas the r (N–O) in trans-RONS is longer than that in cis-RONS.
Table 2 The selected geometric parameters, Wiberg-type bond
orders and relative energies of isomers and transition states of MeSNO calculated by B3P86/6-31++G** (energies in kcal/mol, bond lengths in Å)
r r BO BO Relative
(N–S) (N–O) (N–S) (N–O) energy
cis-MeSNO 1.82 1.19 1.10 1.86 0.0 1.87b 1.19b 1.86d 1.18d trans-MeSNO 1.82 1.19 1.09 1.89 1.0 1.86d 1.17d 0.5a 1.0c 0.8e cis-MeONS 1.60 1.34 1.73 1.17 17.5 trans-MeONS 1.58 1.35 1.77 1.13 16.9 MeNSO 1.64 1.23 1.30 1.49 5.3 TS1 1.96 1.17 14.7 13.2c TS2 1.71 1.21 67.8 TS3 1.61 1.23 53.4 TS4 1.57 1.41 31.2 TS5 1.60 1.30 76.5
aBased upon CBS-4M, see ref. [41] bBased upon UB3LYP/6-31G*, see ref. [41] cBased upon B3LYP/6-31+G*, see ref. [58, 59] dBased upon B3LYP/6-311+G*, see ref. [42, 43] e Grel in ref. [42,43]
These tendencies could be qualitatively rationalized by the repulsion of lone pair electrons between sulfur and oxygen in trans-RSNO as well as in trans-RONS (see Scheme 1, trans-MeSNO and trans-MeONS were taken as examples). Furthermore, as shown in Table 3, ex-cept for R= t-Bu, the cis form is more stable than the trans form when R=H is replaced by alkyl groups. The anomalous tendency for R=t-Bu could be attributed to the significant steric hindrance between tert-butyl and oxygen atom in the cis form.
As supported by Table 3, the quantity of the relative energy is around 0.6 kcal/mol for primary S-nitr-osothiols (R=Et, n-Pr, n-Bu), and 1.2 kcal/mol for the tertiary one (R=t-Bu), while it is wider for secondary ones, being 0.2 kcal/mol for i-PrSNO and 0.6 kcal/mol for i-BuSNO. Furthermore, the rotational barriers of cis-EtSNO, cis-i-PrSNO and trans-t-BuSNO have been calculated by B3P86/6-31++G**, and the results are summarized in Table 4. Note that these three reactions are all endothermic. Interestingly, independent of the quantity of endothermicity, the rotational barriers are close in these three reactions, being calculated to be 14.4, 13.7, 13.2 kcal/mol for cis-EtSNO, cis-i-PrSNO and trans-t-BuSNO, respectively. Based on the method developed by Wiberg [33] the bond orders for some crit-ical bonds are listed in Table 2. The bond order for N–S
Scheme 1 Simple illustration of the repulsion in tran-RONS and
trans-RSNO, respectively (H in white, C in light ashy, N in ashy,
O in dark ashy and S in black colors)
is calculated to be 1.09 and 1.10 for trans-MeSNO and cis-MeSNO, respectively, the result of which agrees with the conclusion (single bond character) made by Gauld and co-workers [40]. Furthermore, the calculated N–S bond orders (BO (N–S)) are 1.73 and 1.77 for cis-MeONS and trans-MeONS, respectively. The BO (N–O) is 1.86 and 1.89 for cis-MeSNO and trans-MeSNO, respectively. The results lead us to conclude that both N–S in MeONS and N–O in MeSNO possess double bond characters. These are in consistence with that of the previous re-ports [40, 41]. Note that a similar double bond character is found by HSNO isomers. For example, BO (N–S) is calculated to be 1.80 and 1.75 for trans- and cis-HONS. Likewise, the bond order of N–O (BO (N–O)) for trans-and cis-HSNO is estimated to be 1.94 trans-and 1.93.
As for the transition state (TS), as shown in Fig. 1 and Table 4, r (N–S) is lengthened and r (N–O) is short-ened among cis and trans forms analogues in all TS1, the structures of which are depicted in Fig. S1. The energy difference between cis and trans conformers, i.e. |E(trans)–E(cis)|, was∼0.6 kcal/mol for 1◦ S-nitrosothi-ols and 1.2 kcal/mol for 3◦ S-nitrosothiol (t-BuSNO).
Table 3 The relative energies and bond lengths of cis- and
trans-RSNO for R=Et, n-Pr, i-Pr, n-Bu, i-Bu, t-Bu (energies in kcal/mol, bond lengths in Å) R= r r Relative (N–S) (N–O) energy Et cis 1.81a 1.19a 0.0 1.86b 1.18b trans 1.82a 1.19a 0.5a 1.86b 1.18b 0.5c n-Pr cis 1.81a 1.19a 0.0 trans 1.82a 1.19a 0.6a i-Pr cis 1.81a 1.19a 0.0 trans 1.82a 1.19a 0.2a 0.1c n-Bu cis 1.81a 1.19a 0.0 trans 1.82a 1.19a 0.6a i-Bu cis 1.81a 1.19a 0.0 trans 1.82a 1.19a 0.6a t-Bu cis 1.78a 1.20a 0.0 1.82b 1.19b trans 1.80a 1.19a −1.2a 1.84b 1.18b −1.2c aBased upon B3P86/6-31++G**
bBased upon B3LYP/6-311+G*, see ref. [42, 43] c Grel in ref. [42,43]
Moreover, the rotational barrier is calculated to be 14.4 and 13.9 kcal/mol for cis-EtSNO and trans-EtSNO, respectively. Thus, trans- and cis- forms of primary S-nitrosothiols should interconvert quickly and revers-ibly at ambient temperature. Conversely, the rotational barrier is 13.2 and 12 kcal/mol for trans-t-BuSNO and cis-t-BuSNO, respectively. From both thermodynamic and kinetic viewpoints, 3◦S-nitrosothiols is less prone to undergo trans⇐⇒ cis isomerization reaction. Accord-ing to the above comparison, the relative stability order is 3◦S-nitrosothiols > 1◦S-nitrosothiols. This viewpoint is in agreement with two experimental evidences. Firstly, it has been reported that primary S-nitrosothiols are typ-ically unstable and could be characterized only by tran-sient spectroscopy [62, 63], whereas tertiary ones have been successfully isolated and have indefinite stability [64, 65]. Secondly, 1◦ S-nitrosothiols typically produce shorter-lived biological effects than 3◦[66]. The results of relative energetics based on B3P86/6-31++G** are very close to those obtained by Houk et al. based on the B3LYP/6-311+G* method [42, 43].
In yet another approach, Bharatam has proposed that the high reactivities of S-nitrosothiols are due to a negative hyperconjugation, i.e. nπO ⇐⇒ σ∗N−S (see Scheme 2, cis-MeSNO taken as example) [60, 67], and has accordingly proven this viewpoint by calculat-ing CH3SN=X (X=O, NH, CH2) systematically. Their
Table 4 Comparison of the
structures and energies of cis-,
trans- and TS of RNSO for
R = Et, i-Pr, t-Bu calculated by B3P86/6-31++G** (energies in kcal/mol, bond lengths in Å and angles in◦)
R r r r
(N–S) (N–O) (S–C) (C–S–N) (S–N–O) (C–S–N–O) Energy
Et cis 1.81 1.19 1.81 102.5 117.1 −0.70 0.0 trans 1.82 1.19 1.82 94.75 116.4 178.68 0.5 TS 1.96 1.17 1.83 94.00 113.2 88.55 14.4 i-Pr cis 1.81 1.19 1.82 103.2 117.5 −0.11 0.0 trans 1.82 1.19 1.83 95.03 116.4 −179.99 0.2 TS 1.95 1.17 1.85 93.42 114.1 83.85 13.7 t-Bu cis 1.78 1.20 1.86 109.1 118.9 −0.19 0.0 trans 1.80 1.19 1.86 97.31 115.7 −179.97 −1.2 TS 1.94 1.17 1.87 98.10 113.8 84.27 12.0
Scheme 2 The illustration of negative hyperconjugation in
S-nitrosothiols (H in white, C in light ashy, N in ashy, O in dark
ashy and S in black colors)
NH and CH2 step by step, the number of lone pair
electrons decrease along with an increase of the BDE of the N–S bond, hence the decrease in r (N–S). By NBO analysis, we also find this orbital interaction in either cis-RSNO and trans-RSNO (R=H, Me). For example, the E(2) values regarding this interaction are 44.0 and 42.8 kcal/mol for trans-MeSNO and cis-MeSNO, respec-tively. On comparing HF and B3P86 [44] as shown in Table 1, it could be promptly found that electron corre-lation is necessary to describe the negative hyperconju-gation interaction. Otherwise, a shorter r (N–S) would be gained. Note that if the bond length correlates well with BDE, a higher BDE for N–S bond is predicted without electron correlation.
From the NBO analysis, it appears that there ex-ists another strong orbital interaction between nπS and π∗N−O, namely the mesomeric effect [69] and E(2) val-ues regarding this interaction are calculated to be 25.5 and 32.3 kcal/mol for cis-HSNO and cis-MeSNO, respec-tively. Bharatam et al. [60] also noted this interaction but without discussing whether this effect facilitate or in-hibit the negative hyperconjugation [67]. To clarify this, we calculate the energetics and N–S bond length of the complexes formed by cis-HSNO and Lewis acid, BH3, at
O and S sites, namely the O complex and S complex (see Table 5). Doctorovich and coworkers synthesized and
Table 5 The comparison of O complex and S complex by BH3or Cu+of cis-HSNO (energies in kcal/mol, bond lengths in Å)
BDEa r (N–S)b Complexation energyc BH3 O complex 38.4 1.70 −8.0 S complex 13.6 1.98 −10.4 N complex 35.4 1.76 −22.0 Cu+ O complex 50.2 1.66 −33.9 S complex 30.9 2.22 −42.2 N complex 42.2 1.721.71d −35.6
aOriginal BDE of N–S = 30.6 kcal/mol bOriginal r (N–S) = 1.84 Å
cComplexation energy = E (O, S complex, or N-complex)- E
(cis-HSNO)- E (BH3or Cu+)
dBased upon B3P86/6-311+G(2df,p), see ref. [31]
calculated a stable Ir–PhSNO complex [70]. According to their reported X-ray crystal structure and theoretical result, they found that the Ir center was coordinated by the N atom. They proposed that this stabilization was probably due to the electrophilicity of the Ir center, as shown by their DFT calculations. Likewise, we also con-sider this condition, namely N complex. As revealed in Table 5, the formation of O and N complex stabilizes cis-HSNO and shortens the N–S bond, whereas the for-mation of S complex lengthens the N–S bond as well as lowers the BDE of the N–S bond. This result can be rationalized by the basicity of N atom and the rival-ing between the negative hyperconjugation effect and the mesomeric effect depicted in Scheme 3. Note that the O-atom and S-atom induce a negative hyperconju-gation (destabilization) and mesomeric (stabilization) effects, respectively, to RSNO. The addition of BH3to
the O-site, in part, removes the negative hyperconju-gation, resulting in the stabilization of the O complex. Conversely, mesomeric effect is ridden by accommo-dating the S-atom lone pair electron (via the Lewis acid BH3) destabilizes the S complex. Note that although the
Scheme 3 The possible site of RSNO attacked by Lewis acid (H in white, C in light ashy, N in ashy, O in dark ashy and S in black colors)
mesomeric effect obviously weakensπN−O, there exists a columbic attraction between N and O in the form of N+-O−. The mutual compensation does not cause the breakage of N–O bond in S-nitrosothiols. On the con-trary, the electronegativities of N and S are close to each other, so that one would expect their columbic interac-tion to be weaker than that of N and O. Once the N–S bond is weakened, decomposition takes place sponta-neously.
We believe that a similar mechanism is also appli-cable to other Lewis acids. For example, if ions, such as Cu+ or Fe2+, acting as a Lewis acid, attack the S site (see Scheme 3), the net result should cause the destabilization of S-nitrosothiols and consequently accelerate the release of NO. To prove this, we chose Cu+ ion as an instance. The results are also listed in Table 5. According to our results, we find Cu+has sim-ilar effect as BH3. This gives an explanation why some
ions could catalyze the decomposition of S-nitrosothiols on attaching RSNOs [22–33]. However, an intriguing thought then comes up to be what if the Lewis acid attacks the O-site, the net effect should increase the BDE of the N–S bond and consequently inhibit the NO formation.
4 Conclusion
In conclusion, we have carefully investigated the energetic minima and TSs on isomeric potential en-ergy surfaces of S-nitrosothiols (RSNOs). Based on the results of B3P86, cis-RSNO is more stable than trans-RSNO when R = H is replaced by alkyl groups (CnH2n+1
n≤ 4) except for R = t-Bu. Considering relative com-putational resources and accuracy, B3P86 turns out to be the most suitable method to describe the relative thermodynamics as well as the isomerization process among various theoretical methods used. We could thus summarize our results as follows:
1. The energy of TS1 of the studied S-nitrosothiols is independent of the endothermicity of the reaction. 2. The order of stability is concluded to be 3◦> 1◦
S-nitrosothiols from both thermodynamic and kinetic viewpoints.
3. The S-nitrosothiols are possibly unstable due to the negative hyperconjugation (nπO ⇐⇒ σ∗N−S) ef-fect that strongly weakens theσN−Sbond [58, 59, 67]. Accordingly, a plausible mechanism as for the role played by the metal ions in stabilizing or desta-bilizing the RSNOs is proposed and depicted in Scheme 3.
Acknowledgements We are grateful to the National Center for High-Performance Computing of Taiwan for allowing us gener-ous hours of computing time. We also thank the National Science Council for financial support.
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