Local analysis and comparative study of the hydrogen bonds in the linear (HCN)n and (HNC)n clusters

Journal of Molecular Structure (Theochem) 630 (2003) 187–204 www.elsevier.com/locate/theochem

Local analysis and comparative study of the hydrogen bonds in the linear ðHCNÞn and ðHNCÞn clusters Cheng Chen*, Min-Hsien Liu, Lung-Shing Wu Department of Applied Chemistry, Chung-Cheng Institute of Technology, Ta-Hsi, Taoyuan, Taiwan, ROC Received 7 August 2002; accepted 2 December 2002

Abstract B3LYP/6-311 þ G(2d,p), the density functional theory method of GAUSSIAN 98 package, is applied to study the hydrogen bonding of a series of linear ðHCNÞn and ðHNCÞn molecular clusters (for n ¼ 1 – 10). By the localization analysis methods we developed, pair-wised s type H-bond orders and bond energies are calculated for each pair of the two near-by molecules in both ðHCNÞn and ðHNCÞn clusters. The calculated results are checked well with the shortening of N– H or C – H distance, the elongation of CH or NH bond distance, and the red shift of stretching frequencies of CH or NH. All pieces of evidence show that the central pair of the two molecules forms the strongest H bond when n of ðHCNÞn or ðHNCÞn is even, and the two middle pairs form the two strongest H bonds when n is odd. Two terminal pairs of HCN or HNC molecules always form the two weakest Hbonds in each molecular cluster. When comparing molecular cluster energies between ðHCNÞn and ðHNCÞn for various values of n; the well-known ðHCNÞn is found more stable than the related ðHNCÞn from energy calculation. However, if outcomes of Hbond local analysis are contrasted, our analysis significantly shows that inter-molecular H-bonds inside of ðHNCÞn clusters are much stronger than the corresponding H-bonds in ðHCNÞn with the same n: In comparing energy differences between these related clusters per monomer, ½EðHNCÞn 2 EðHCNÞn =n is found decreasing monotonically as n increases. All pieces of evidence from this theoretical prediction indicate that ðHNCÞn with large n is probably constructed by its relative strong H-bonds. q 2003 Elsevier Science B.V. All rights reserved. Keywords: B3LYP/6-311 þ G(2d,p); ðHCNÞn and ðHNCÞn molecular clusters; Localization analysis methods of inter-molecular hydrogen bonding

1. Introduction Both HCN and HNC monomers are linear structures with high polar behavior. In these two molecules, the triple-bond-type connection between C and N atoms is constructed by two p orbitals and one sp hybridized type s orbital. The sp type non-bonding s orbital of terminal N or C causes high polarity for the molecule * Corresponding author.

and the significant H-bond acceptor behavior. In addition, such a non-bonding s orbital usually has a high tendency to overlap with the HC or HN s bonding orbital of the other monomer, resulting in an intermolecular H-bond formation. In order to form the maximum overlap between these two different types of s orbitals, the stable bond angle between these two linear molecules with H-bond is usually 1808 for linear ðHCNÞn and ðHNCÞn H-bonded type polymer formations (n is an integer equal to or greater than two).

0166-1280/03/$ - see front matter q 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0166-1280(03)00154-4

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Numerous experimental [1 – 8] and theoretical [9 – 15] studies have been performed on linear ðHCNÞn clusters. With our newly developed localized H-bond analysis theoretical method [16], a set of ðHCNÞn clusters for n from 2 to 10 has been studied by us successfully [17]. This analytical method has been successfully introduced to various kinds of intramolecular H-bond problems [18 – 23]. In a recent work of Grabowski [24], this important analysis was also specially pointed out. From now on we plan to apply this powerful method to various inter-molecular and inter-ionic H-bonding problems [25]. Hydrogen bonding of the inter-molecular problems depends on the following two important factors. The first factor is the H-bond acceptor behavior of the polarized negative end (N of HCN and C of HNC). The second factor is the H atom donor property of the positive end. In comparing these two factors for monomers of HCN and HNC, most of the theoretical and experimental reports [26 – 30] indicate that both H-bonding factors are in favor of H-bond formation of HNC. Unfortunately, in the recent literature, documents with results relating to H-bonding problem of ðHNCÞn with large n ðn $ 3Þ are seldom found. From the experimental point of view, the ground state energies of HNC and ðHNCÞn with small n are significantly higher than the corresponding energies of the related monomer or polymers of HCN. It is particularly difficult to find ðHNCÞn with large n by polymerization reaction from its high-energy monomer or short-chained polymer of HNC. This is due to the unwanted products of side reactions products that are possibly produced in the various experimental procedures. However, from the theoretical points of view, due to a relative stronger H-bond energy and larger cooperative effects, the H-bonding problem of ðHNCÞn with large n is more important than that of the related ðHCNÞn : Consequently, both sets of ðHCNÞn

and ðHNCÞn for n from 2 to 10 are particularly selected together in this work for an H-bonding study.

2. Calculations 2.1. Ordinal systems of ðHCNÞn and ðHNCÞn Since this work aims in performing localization analyses of the various H-bonds in ðHCNÞn and ðHNCÞn clusters, we must first define the ordinal systems of the molecules. All atoms and the related H-bonds are as shown in Fig. 1. Starting from the left side, with H1 atom at the left end, H1C1N1 of ðHCNÞn and H1N1C1 of ðHNCÞn are defined as the first molecules of the corresponding cluster. The last molecule Hn Cn Nn at the other end of ðHCNÞn and Hn Nn Cn of ðHNCÞn are defined as the last molecules of the linear clusters. Under this arrangement, with the view of the localization analysis, the first H-bond of ðHCNÞn is N1 –H2, and of ðHNCÞn is C1 –H2; the last H-bond of ðHCNÞn is Nn21 – Hn ; and of ðHNCÞn is Cn21 – Hn : The rest of the H-bonds are arranged successively in order between these two H-bonds of each cluster. In this work, n in each of the linear molecular clusters ðHCNÞn and ðHNCÞn is selected from 1 to 10. The arrangement and ordinal systems are all arranged according to the linear structures shown in Fig. 1. 2.2. SCF and vibration frequency calculation Various calculation methods of ab initio and density function theory (DFT) types have been tried by us for this work. We find that the results of DFT with B3LYP-type calculation [31,32] are better than Hartree – Fock or Moller– Plesset perturbation calculation of ab initio types. Such calculation costs substantially less than any other traditional correlation technique. When we apply GAUSSIAN 98 program [33]

Fig. 1. The structures and the defined numbered system of ðHCNÞn and ðHNCÞn n ¼ 1 –10.

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189

Table 1 Calibration of vibration frequencies of CH stretching (cm21) for ðHCNÞn Molecules

nobs a

ncal

f ¼ 0:960

n0:960

f ¼ 0:961 Error

RE (%)

n0:961

f ¼ 0:962 Error

RE (%)

n0:962

Error

RE (%)

HCN (HCN)2

3304 3303 3202

3440 3434 3345

3303 3296 3211

21.43 26.60 9.12

20.04 20.20 0.28

3306 3300 3214

2.01 23.16 12.46

0.06 20.10 0.39

3309 3303 3218

5.45 0.27 15.81

0.16 0.01 0.49

(HCN)3

3297 3198 3180

3432 3324 3309

3294 3191 3177

22.52 27.16 22.88

20.08 20.22 20.09

3298 3194 3180

0.91 23.83 0.43

0.03 20.12 0.01

3301 3197 3184

4.34 20.51 3.74

0.13 20.02 0.12

(HCN)4

3291 3187 3176 3167

3431 3312 3309 3277

3294 3179 3176 3146

2.55 27.80 0.24 221.08

0.08 20.24 0.01 20.67

3297 3183 3180 3149

5.98 24.49 3.55 217.81

0.18 20.14 0.11 20.56

3300 3186 3183 3152

9.41 21.18 6.86 214.53

0.29 20.04 0.22 20.46

RMSE a

0.27

0.24

0.26

The observed values are selected from Ref. [5].

to the ðHCNÞn and ðHNCÞn with n # 10 in this work, we find that the B3LYP/6-311 þ G(2d,p) with triple zeta diffused and polarized basis [34 –36] for studying the structure of this hydrogen bonding is the most suitable method, from the viewpoints of both the computer’s memory space and computational time. Vibration frequency calculation is also very successful. Without any imaginary frequency in all calculation cases, the structures of molecular clusters are proved to be stable. In order to testify the reliability of vibration frequencies, a set of observed stretching frequencies of CH of ðHCNÞn with n # 4 selected from Ref. [5] to calibrating the scaling factor, f is shown in Table 1. 0.960, 0.961 and 0.962 were tried for this calibration. The percentage relative errors of these are all less than 0.3%. The best value f ¼ 0:961 was chosen for all of the frequencies of this work. In addition to this calibration, several important experimental physical quantities of HCN and HNC monomers are chosen from various references [5, 37 –40] to check the reliability of our calculation method, shown in Table 2. Most of the relative errors for bond distances and the rest vibration frequencies of these two monomers are # 1%: One exception is that the bending frequency of HNC has a comparatively larger error due to this small frequency. These successful results of monomers and nCH of small polymers of ðHCNÞn indicate that the DFT-type calculation method in this work is quite reasonable

and meaningful for the following calculations and analyses of ðHCNÞn and ðHNCÞn cluster systems. 2.3. Localization analysis of H bonds This work aims in comparing the localized hydrogen bonding effect of various N –H H-bonds in ðHCNÞn and C –H H-bonds in ðHNCÞn : The localized analysis method of H-bonds between each pair of the two directly connected molecules in the various part of molecular cluster is the most important subject for this theoretical work [16 –25]. In our former works, we define H – X in YH – X, instead of YH –X itself, as the localized Hbond. With this definition, H – X is the H-bond and YH is an ordinary s bond, which shares the same H of its related H – X H-bond. There are four hydrogen-bonding related parameters to be selected herein for the purpose of comparison. Between all the nearby X and H atoms (X is N in ðHCNÞn and is C in ðHNCÞn ), the X – H distance (dX – H ) is directly selected from the SCF result, and with the point charge approximation, the Coulomb attraction energy (2EX – H ) calculated by Eq. (1) are the two parameters for the corresponding local H-bonds. EX – Hz ¼ QX QH =dX – H , 0 and 2 EX – H ¼ 2QX QH =dX – H . 0

ð1Þ

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Table 2 Dipole moments, bond distances and vibration frequencies of HCN and HNC molecules Molecules

HCN

Analysis

Calculated

Observed

RE (%)

Calculated

Observed

RE (%)

3.06 1.146 1.067 3306 2108 718

2.98a 1.153b 1.065b 3304d 2089d 722d

2.68 0.61 0.19 0.06 0.91 0.55

3.09 1.165 0.999 3655 2016 403

3.05a 1.168c 0.996c 3620e 2029e 477e

1.31 0.26 0.30 0.97 0.64 15.51

Dipole moments (D) ˚) Bond distances (A Vibration frequencies (cm21, f ¼ 0:961)

a b c d e

m dCN dCH or dNH nCH or nNH nCN Bending

HNC

Taken from Ref. [37]. Taken from Ref. [38]. Taken from Ref. [39]. Taken from Ref. [5]. Taken from Ref. [40].

ð3Þ

and nth atomic orbitals in the molecular orbital method. As shown in our former works concerning the intra-molecular H-bonding [18 –23], the four local analysis parameters, (BE)X – H, PX–H ; dX–H and 2EX–H ; are important to determine the local H-bond strength. In addition to this direct analysis, two important indirect parameters of local H-bond will also be selected herein for comparison. The YH’s bond distance (dYH ) and stretching frequency (nYH ) of the relative H-bond (Y is C in ðHCNÞn and N in ðHNCÞn will also be recorded to verify the relative H-bond strength. As shown in many of our works, the elongation of dYH and red shift of nYH are two effective parameters to determine strengths of the H-bonds.

ð4Þ

3. Results and discussion

In the above equation, QH and QX are the charge densities of DFT optimization calculation results. From the above-mentioned way, the best DFT optimized geometry and semi-empirical type localized analysis methods of hydrogen bond as mentioned in Ref. [16], the localized bond energy (BE)X – H was calculated through the energy breaking procedure. And the localized bond order PX – H was calculated through the semi-empirical typed population analysis ðBEÞX – H ¼ fX £ DEX þ fH £ DEH 2 EX – H fX ¼ EX–H

. all XA A

EX–A ;

fH ¼ EH–X

. all XA

ð2Þ

EH–A

A

PX–H ¼ ðP22s;1s þ P22px;1s þ P22py;1s þ P22pz;1s Þ1=2

In Eq. (2), DEX and DH are the energy differences of the related atoms which have been calculated by the energy differences from before and after the MO calculation. EX–H of Eq. (2) is the energy partitioned from the total molecular energy of the MO calculation. fX and fH of Eq. (2) are the semiempirical partition factors as defined in Ref. [16], in which the partition factor fX and fH are defined in Eq. (3). In Eq. (4), PX–H is calculated by way of the ‘local bond population analysis’ method, in which Pmn ’s are the bond orders of the related mth

3.1. Comparison of energy calculation To apply B3LYP/6-311 þ G(2d,p) type DFT calculation of GAUSSIAN 98 in this work, both the results of the ðHCNÞn and ðHNCÞn molecular clusters(for n ¼ 1 – 10) are successfully obtained. All of them are proved to be stable local minima. With the same size of the basis set, the energy differences between ðHCNÞn and ðHNCÞn were calculated and listed in Tables 3a and 3b for comparison. The calculated DEscf ; DUð0 KÞ with

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191

Table 3a Energy and energy differences of ðHCNÞn and ðHNCÞn (energy in a.u.; energy differences in kJ/mol) n

ESCF (without zero point energy)

1 2 3 4 5 6 7 8 9 10 a

Uð0 KÞ (with zero point energy)

ðHCNÞn

ðHNCÞn

DESCF a

DESCF =n

ðHCNÞn

ðHNCÞn

DUð0 KÞ

DU=nð0 KÞ

293.4562587 2186.9192911 2280.3841575 2373.8496779 2467.3154806 2560.7814236 2654.2474441 2747.713511 2841.1796078 2934.6457248

293.4332891 2186.8772918 2280.3244597 2373.7728218 2467.2217386 2560.6709356 2654.1202894 2747.5697376 2841.0192461 2934.4687948

60.3 110.3 156.7 201.8 246.1 290.1 333.8 377.5 421 464.5

60.3 55.1 52.2 50.4 49.2 48.3 47.7 47.2 46.8 46.5

293.440019 2186.885473 2280.332682 2373.780517 2467.228638 2560.676877 2654.125209 2747.573571 2841.021976 2934.47039

293.417935 2186.844265 2280.273767 2373.704505 2467.135849 2560.567496 2653.999319 2747.431235 2840.863233 2934.295291

58.0 108.2 154.7 199.6 243.6 287.2 22295 330.5 416.8 459.7

58.0 54.1 51.6 49.9 48.7 47.9 47.2 46.7 46.3 46.0

1 a.u. ¼ 2625.5 kJ/mol. D ¼ energy of ðHNCÞn minus energy of ðHCNÞn :

Table 3b Enthalpy and Gibbs free energy of ðHCNÞn and ðHNCÞn at 298 K (energy in a.u.; energy differences in kJ/mol) n

H (298 K)

1 2 3 4 5 6 7 8 9 10 a

G (298 K)

ðHCNÞn

ðHNCÞn

DH a

DH=n

ðHCNÞn

ðHNCÞn

DG

DG=n

293.436524 2186.878133 2280.32152 2373.765554 2467.209872 2560.654334 2654.098874 2747.543462 2840.988081 2934.432719

293.414048 2186.836762 2280.26264 2373.689798 2467.11755 2560.545619 2653.973865 2747.402225 2840.830656 2934.259131

59.01 108.6 154.6 198.9 242.4 285.4 328.2 370.8 413.3 455.8

59.0 54.3 51.5 49.7 48.5 47.6 46.9 46.4 45.9 45.6

293.459381 2186.914083 2280.369687 2373.825711 2467.282142 2560.738262 2654.195048 2747.650994 2841.107559 2934.563938

293.437498 2186.871946 2280.308808 2373.746648 2467.185199 2560.623939 2654.062778 2747.501499 2840.940543 2934.37992

57.5 110.6 159.8 207.6 254.5 300.2 347.3 392.5 438.5 483.1

57.5 55.3 53.3 51.9 50.9 50.0 49.6 49.1 48.7 48.3

1 a.u. ¼ 2625.5 kJ/mol. D means the energy of ðHNCÞn minus energy of ðHCNÞn :

zero point energies correction, DH(298 K) and DG(298 K) with thermal chemical corrected values of the HCN/HNC reaction of monomers are 60.3, 58.0, 59.0 and 57.5 kJ/mol (or 14.4, 13.9, 14.1 and13.7 kcal/mol). These are close to the observed enthalpy value 14.8 ^ 2 kcal/mole of Pau and Hehre [41], and the theoretical value 14.6 kcal/mol of Pearson, Schaefer, and Walgren [42]. In addition to calculating the energy differences, the energy difference between the related clusters per monomer of each n; ½EðHNCÞn 2 EðHCNÞn =n; is also listed in these two tables for comparison. DEscf =n; DUð0 KÞ=n;

DHð298 KÞ=n and DGð298 KÞ=n are all monotonically decreased from 60.3, 58.0, 59.0 and 57.5 kJ/mol to 46.5, 46.0, 45.6 and 48.3 kJ/mol when n increases from 1 to 10. Due to limitations on computational time and memory space, it is impossible for us to study ðHCNÞn and ðHNCÞn with very large n at the present time. However, the above-mentioned monotonically decreasing type energy differences quite probably decrease to a very small value or zero whenever n is very large from the theoretical point of view. If we examine the figures of X-ray probability diagram of solid phase of ðHCNÞn of Ref. [1], it seems

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that the hydrogen atoms in the linear structure is just situated at the center point between N and C atoms. Therefore, with the above-mentioned judgement, (HCN)1 and (HNC)1 in the solid linear structure are probably the identical substance. This monotonic decreasing behavior clearly indicates that the intermolecular H-bond energy of ðHNCÞn is relatively stronger than the corresponding energy of ðHCNÞn for various values of n: 3.2. Dipole moments and rotational constants In order to compare the molecular polarization and geometric behaviors between ðHCNÞn and ðHNCÞn clusters, all the calculated dipole moments, rotational constants and their related difference values are listed in Table 4. In addition to the dipole moments of monomers shown in Table 2 which have been discussed, all the other known dipole moments and rotational constants also have achieved good results. For HCN and its linear dimmer and trimmer, the calculated rotational constants 44,958, 1742 and 469 MHz are very close to the related observed values 44,316 [44], 1746 [4] and 469 [3] MHz, and the calculated dipole moments 3.06, 6.94 and 11.08 D are also close to the corresponding observed results 2.98 [37], 6.50 [43] and 10.6 D [3] indicated in the related references. As for the calculated Be ¼ 45,713 MHz for HNC in this work, it is also reasonably close to the observed value 45,497 MHz [39]. If we compare all the calculated Be ’s of ðHCNÞn (n ¼ 1– 10), most of Table 4 Dipole moment (in Debye) and rotational constant (in MHz) n

Dipole moments

1 2 3 4 5 6 7 8 9 10 a

ðHCNÞn

ðHNCÞn

3.06 6.94 11.08 15.34 19.64 23.36 28.32 32.68 37.04 41.41

3.09 7.40 12.15 17.11 22.19 27.32 32.50 37.70 42.92 48.16

Dm

0.03 0.46 1.07 1.77 2.55 3.96 4.18 5.02 5.88 6.75

Rotational constants ðHCNÞn a

ðHNCÞn

44957.5 1741.6 468.5 192.9 98.0 56.5 35.6 23.8 16.7 12.2

45713.0 1895.2 512.8 212.0 107.9 62.4 39.3 26.4 18.5 13.5

DB

755.5 153.6 44.3 19.1 9.9 5.9 3.7 2.6 1.8 1.3

The most of rotational constants of ðHCNÞn are close to the results of Ref. [14].

them are close to the recently calculated results of Dykstra [14]. For linear-typed molecular clusters, rotational constants are directly related to the geometrical arrangement of various atoms in the cluster systems. To compare the Be ’s of ðHCNÞn and ðHNCÞn ; both sets change with the same pattern. And the differences of them, DB’s, in Table 4 are decrease from 755.5 to 1.3 MHz steadily. For large n; DB’s are about 10% of the related Be ’s of ðHNCÞn : Both sets of dipole moments in these two cluster systems are extensively increased when n increases from 1 to 10. mðHCNÞ10 =10 ¼ 4.14 D and mðHNCÞ10 =10 ¼ 4.82 D are much greater than the 3.06 and 3.09 D of their related monomers. For the linear cluster structure, the dipole moment of the molecular cluster is simply equal to the sum of the dipole moments of these different molecules if there is no special molecular interaction between the related molecules. However, this simple vector sum idea cannot apply to these molecular clusters in their dipole moments due to the strong cooperative effect of polarization created by their Hbond formation. Looking through the two sets of dipole moments as those listed in Table 4, the increment of dipole moments in the ðHNCÞn are significantly larger than that of the corresponding ðHCNÞn clusters. This result may indirectly signify that H-bonding effect of ðHNCÞn is quite probably stronger than that of the ðHCNÞn with the same n: 3.3. Analysis of localized H-bonds in various molecular clusters With the linear-typed structures, there are n 2 1 local H-bonds in both ðHCNÞn and ðHNCÞn molecular clusters. Localized bond energy ((BE)X – H), bond order (PX – H ), bond distance (dX – H ), and Coulomb attraction energy (2EX – H ) of each local H-bond X – H of all the molecular clusters were calculated separately. In addition to this four parameters, the related bond distances of the shared YH bond (dYH ) are also evaluated to show their elongation effect created by H-bond formation. As shown in the linear structures defined in Fig. 1, X is atom N and Y is atom C in ðHCNÞn ; X is atom C and Y is atom N in ðHNCÞn : The above-mentioned five local H-bond related parameters of the ðHCNÞn and ðHNCÞn clusters are separately listed in Tables 5a and 5b for comparison.

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Table 5a ˚) The results of H-bond local analysis of various ðHCNÞn clusters (energies in kJ/mol and distances in A H-bonds

HCN

(No bond)

(HCN)2

(No bond) N1 –H2

18.26a

0.1082

2.2322

1.0681 1.0743

19.52

(No bond) N1 –H2* N2 –H3

21.59(1)b 21.31(2)

0.1214(1) 0.1199(2)

2.1712(1) 2.1809(2)

1.0684 1.0766(1) 1.0762(2)

19.03(2) 31.73(1)**

(No bond) N1 –H2 N2 –H3* N3 –H4

21.67(3) 24.67(1) 21.92(2)

0.1263(2) 0.1365(1) 0.1244(3)

2.1547(2) 2.1162(1) 2.1665(3)

1.0685 1.0773(2) 1.0791(1) 1.0768(3)

19.74(3) 31.47(2) 34.26(1)**

(No bond) N1 –H2 N2 –H3* N3 –H4 N4 –H5

23.10(3) 26.67(2) 26.68(1)** 22.84(4)

0.1267(3) 0.1406(1) 0.1402(2) 0.1248(4)

2.1494(3) 2.0977(1) 2.1000(2) 2.1618(4)

1.0686 1.0776(3) 1.0801(1) 1.0799(2) 1.0770(4)

20.04(4) 32.66(3) 34.33(2) 35.06(1)**

(No bond) N1 –H2 N2 –H3 N3 –H4* N4 –H5 N5 –H6

23.37(4) 27.30(3) 28.20(1) 27.31(2) 23.10(5)

0.1274(4) 0.1425(2) 0.1454(1) 0.1420(3) 0.1255(5)

2.1468(4) 2.0911(2) 2.0814(1) 2.0941(3) 2.1595(5)

1.0686 1.0778(4) 1.0805(2) 1.0810(1) 1.0803(3) 1.0772(5)

20.19(5) 33.18(4) 35.71(1) 35.23(3) 35.41(2)

(No bond) N1 –H2 N2 –H3 N3 –H4* N4 –H5 N5 –H6 N6 –H7

23.64(5) 27.62(4) 28.86(2) 28.87(1)** 27.65(3) 23.38(6)

0.1282(5) 0.1433(3) 0.1473(1) 0.1472(2) 0.1427(4) 0.1262(6)

2.1435(5) 2.0883(3) 2.0747(1) 2.0755(2) 2.0917(4) 2.1562(6)

1.0686 1.0779(5) 1.0807(3) 1.0814(1,2) 1.0814(1,2) 1.0805(4) 1.0771(6)

20.28(6) 23.76(5) 36.29(2) 36.67(1)** 35.62(3) 35.59(4)

(no bond) N1 –H2 N2 –H3 N3 –H4 N4 –H5* N5 –H6 N6 –H7 N7 –H8

22.41(7) 27.03(4) 28.31(3) 28.61(1) 28.36(2) 26.95(5) 22.77(6)

0.1298(6) 0.1452(4) 0.1494(2,3) 0.1503(1) 0.1494(2,3) 0.1442(5) 0.1275(7)

2.1425(6) 2.0853(4) 2.0715(2) 2.0689(1) 2.0723(3) 2.0907(5) 2.1562(7)

1.0686 1.0782(6) 1.0808(4) 1.0817(2) 1.0818(1) 1.0816(3) 1.0805(5) 1.0774(7)

20.35(7) 33.61(6) 36.59(3) 37.28(1) 37.09(2) 35.82(4) 35.70(5)

(No bond) N1 –H2 N2 –H3 N3 –H4 N4 –H5* N5 –H6 N6 –H7 N7 –H8 N8 –H9

22.58(8) 27.08(6) 28.53(4) 29.01(2) 29.03(1)** 28.56(3) 27.11(5) 22.90(7)

0.1299(7) 0.1451(5) 0.1499(3) 0.1514(1,2) 0.1514(1,2) 0.1497(4) 0.1445(6) 0.1278(8)

2.1419(7) 2.0861(5) 2.0700(3) 2.0652(1) 2.0655(2) 2.0712(4) 2.0898(6) 2.1549(8)

1.0686 1.0780(7) 1.0808(5) 1.0817(3,4) 1.0820(1,2) 1.0820(1,2) 1.0817(3,4) 1.0806(6) 1.0774(8)

20.37(8) 33.66(7) 36.71(4) 37.57(2) 37.72(1)** 37.30(3) 35.93(5) 35.77(6)

(HCN)3

(HCN)4

(HCN)5

(HCN)6

(HCN)7

(HCN)8

(HCN)9

(HCN)10

(No bond)

(BE)N – H

2EN – H

Molecules

PN – H

dN – H

dCH 1.0672

1.0686 (continued on next page)

194

C. Chen et al. / Journal of Molecular Structure (Theochem) 630 (2003) 187–204

Table 5a (continued) Molecules

H-bonds

(BE)N – H

PN – H

dN – H

dCH

2EN – H

N1 –H2 N2 –H3 N3 –H4 N4 –H5 N5 –H6* N6 –H7 N7 –H8 N8 –H9 N9 –H10

22.09(9) 28.12(7) 29.66(5) 30.22(3) 30.38(1) 30.24(2) 29.69(4) 28.15(6) 23.72(8)

0.1288(8) 0.1442(6) 0.1491(4) 0.1508(2,3) 0.1512(1) 0.1508(2,3) 0.1489(5) 0.1436(7) 0.1268(9)

2.1411(8) 2.0852(6) 2.0686(4) 2.0631(2) 2.0618(1) 2.0636(3) 2.0698(5) 2.0889(7) 2.1542(9)

1.0779(8) 1.0809(6) 1.0818(4) 1.0822(1,2,3) 1.0822(1,2,3) 1.0822(1,2,3) 1.0817(5) 1.0807(7) 1.0774(9)

20.39(9) 33.73(8) 36.82(5) 37.75(3) 38.03(1) 37.96(2) 37.44(4) 36.02(6) 35.81(7)

* Superscribes are assigned for the strongest H-bonds. ** Superscribes are assigned for the strongest H-bond which contradict to the assignment of the other parameters. a The observed net binding energy of linear (HCN)2 is 18.41 kJ/mol, which reported in Ref. [45]. b Subscribes on all the listed values represent the order of H-bond strength.

As for the other important parameters of characteristic stretching vibration of the shared bond YH of Hbonds including frequency (nYH ), the related red shift, enhancement of IR intensity and detailed local H atom motion coordinates analysis in the normal mode will all be reported separately in Tables 6a and 6b for an additional supporting analysis of local H-bonding effect. 3.3.1. Comparison of the parameter ranges between ðHCNÞn and ðHNCÞn clusters Before the comparing local H-bond strengths in the particular local H-bonds of various clusters in detail, we first compare the ranges of the five sets of parameters defined above for these two types of molecular clusters. From the ðHCNÞn results of Table 5a 18.26 kJ # (BE)N – H # 30.38 kJ, ˚ # dN – 0.1082 # PN – H # 0:1512; 2.2322 A ˚ ˚ ˚, 0.0094 A # D(dCH ) # 0.0136A H # 2.0618 A , and 18.26 kJ # 2 EN – H # 30.38 kJ. Similarly from the ðHNCÞn results of Table 5b 35.43 kJ # (BE)C – H # 72.05 kJ, 0.1660 # PC – ˚ # d C – H # 1.8720 A ˚, 2.0793 A H # 0.2378, ˚ ˚ 0.0171 A # D(d NH) # 0.0435 A, and 32.36 kJ # 2 EC – H # 102.32 kJ. In the above ranges, D(dYH) ¼ dYH 2 dYH(monomer). With very few overlaps between two sets of the ranges of the assigned parameters, all pieces of evidence prove that the Hbonding effect of ðHNCÞn clusters are significantly stronger than the corresponding H-bonding effect of ðHCNÞn clusters. This significant deviation of Hbonding strength between these two sets of clusters

also gives a good explanation why ½EðHNCÞn 2 EðHCNÞn =n is monotonically decreasing when n is increased with the increment of H-bonding effect shown in Tables 3a and 3b. 3.3.2. Assignment of ordinal system of the localized H-bond strength According to our experience in localized analysis treatments [16 – 23], the order of local H-bond strength might be arranged by the increasing behavior of bond energy ((BE)X – H), bond order (PX – H), Coulomb attraction energy (2 EX – H) and YH bond distance (d YH), and also by the decreasing tendency of H-bond distance (dX – H). In Tables 5a and 5b all orders of the ive sets are arranged and marked with subscribed numbers for the related parameters. Due to the above-mentioned dominant assignment, the strongest local H-bond of each cluster is assigned and marked with a superscript sign (*) for such strong Hbond. If there is any exception in any set of these five sets of parameters, the superscript sign (**) is marked to such strong H-bond parameters accordingly. In all the cases of ðHNCÞn except the assigned order of 2 EC – H, the orders of the rest four parameters coincide with each other and matched well with the superscript (*) assignment of the strongest H-bonds in Table 5b. The superscript (*) assignment of the strongest H-bonds of ðHCNÞn in Table 5a is assigned according to the bond order (PN – H), H-bond distance (d N – H) and CH bond distance (dCH). Most of the ordinal systems of local bond energy ((BE)N – H) of ðHCNÞn are closely matched with the other ordinal

C. Chen et al. / Journal of Molecular Structure (Theochem) 630 (2003) 187–204

195

Table 5b ˚) The results of H-bond local analysis of various ðHNCÞn clusters (energies in kJ/mol and distances in A H-bonds

HNC

(No bond)

(HNC)2

(No bond) C1 –H2

35.43

0.1660

2.0793

1.0001 1.0161

32.36

(No bond) C1 –H2* C2 –H3

43.88(1)a 43.00(2)

0.1865(1) 0.1823(2)

2.0099(1) 2.0264(2)

1.0006 1.0227(1) 1.0215(2)

36.55(2) 70.29** (1)

(No bond) C1 –H2 C2 –H3* C3 –H4

47.06(2) 54.89(1) 46.18(3)

0.1934(2) 0.2081(1) 0.1885(3)

1.9893(2) 1.9466(1) 2.0073(3)

1.0008 1.0253(2) 1.0305(1) 1.0235(3)

37.40(3) 86.98(1) 73.21(2)

(No bond) C1 –H2 C2 –H3* C3 –H4 C4 –H5

48.67(3) 59.15(1) 58.94(2) 47.55(4)

0.1965(3) 0.2164(1) 0.2152(2) 0.1908(4)

1.9796(3) 1.9244(1) 1.9282(2) 2.0007(4)

1.001 1.0261(3) 1.0337(1) 1.0333(2) 1.0244(4)

37.8(4) 91.1(1) 90.9(2) 75.0(3)

(No bond) C1 –H2 C2 –H3 C3 –H4* C4 –H5 C5 –H6

50.50(4) 62.41(2) 65.28(1) 62.15(3) 49.43(3)

0.1965(4) 0.2185(2) 0.2232(1) 0.2169(3) 0.1907(5)

1.9762(4) 1.9144(2) 1.9025(1) 1.9200(3) 1.9972(5)

1.0010 1.0269(4) 1.0349(2) 1.0371(1) 1.0346(3) 1.0248(5)

38.1(5) 92.8(3) 95.4(1) 93.4(2) 75.8(4)

(No bond) C1 –H2 C2 –H3 C3 –H4* C4 –H5 C5 –H6 C6 –H7

51.15(5) 63.72(3) 67.61(1) 67.57(2) 63.20(4) 49.87(6)

0.1975(5) 0.2206(3) 0.2272(1) 0.2268(2) 0.2184(4) 0.1913(6)

1.9731(5) 1.9093(3) 1.8932(1) 1.8943(2) 1.9163(4) 1.9956(6)

1.0010 1.0273(5) 1.0360(3) 1.0388(1) 1.0386(2) 1.0351(4) 1.0251(6)

38.2(6) 93.8(4) 97.1(2) 98.0** (1) 94.5(3) 76.2(5)

(No bond) C1 –H2 C2 –H3 C3 –H4 C4 –H5* C5 –H6 C6 –H7 C7 –H8

51.516) 64.38(4) 69.04(2) 70.26(1) 69.05(3) 63.66(5) 50.29(7)

0.19806) 0.2216(4) 0.2294(2) 0.2313(1) 0.2289(3) 0.2188(5) 0.1918(7)

1.9718(6) 1.9071(4) 1.8876(2) 1.8831(1) 1.8890(3) 1.9156(5) 1.9940(7)

1.0011 1.0275(6) 1.0365(4) 1.0397(2) 1.0405(1) 1.0395(3) 1.0353(5) 1.0253(7)

38.3(7) 94.2(5) 98.1(3) 100.0(1) 99.3(2) 94.9(4) 76.5(6)

(No bond) C1 –H2 C2 –H3 C3 –H4 C4 –H5* C5 –H6 C6 –H7 C7 –H8 C8 –H9

51.48(7) 64.45(5) 69.55(3) 71.44(1) 71.40(2) 69.30(4) 63.93(6) 50.14(8)

0.1989(7) 0.2227(5) 0.2314(3) 0.2344(1) 0.2342(2) 0.2303(4) 0.2203(6) 0.1925(8)

1.9705(7) 1.9057(5) 1.8842(3) 1.8769(1) 1.8776(2) 1.8872(4) 1.9129(6) 1.9935(8)

1.0011 1.0276(7) 1.0367(5) 1.0403(3) 1.0415(1) 1.0414(2) 1.0398(4) 1.0357(6) 1.0254(8)

38.3(8) 94.4(6) 98.6(4) 101.0(2) 101.3** (1) 99.8(3) 95.4(5) 76.6(7)

(HNC)3

(HNC)4

(HNC)5

(HNC)6

(HNC)7

(HNC)8

(HNC)9

(HNC)10

(No bond)

(BE)N – H

2EC – H

Molecules

PN – H

dC – H

dNH 0.999

1.0011 (continued on next page)

196

C. Chen et al. / Journal of Molecular Structure (Theochem) 630 (2003) 187–204

Table 5b (continued) Molecules

H-bonds

(BE)N – H

PN – H

dC – H

dNH

2EC – H

C1 –H2 C2 –H3 C3 –H4 C4 –H5 C5 –H6* C6 –H7 C7 –H8 C8 –H9 C9 –H10

50.94(8) 64.10(6) 69.39(4) 71.52(2) 72.05(1) 71.41(3) 68.98(5) 63.37(7) 49.52(9)

0.2000(8) 0.2243(6) 0.2335(4) 0.2370(2) 0.2378(1) 0.2365(3) 0.2320(5) 0.2216(7) 0.1934(9)

1.9702(8) 1.9043(6) 1.8818(4) 1.8737(2) 1.8720(1) 1.8750(3) 1.8859(5) 1.9125(7) 1.9937(9)

1.0277(8) 1.0370(6) 1.0407(4) 1.0422(2) 1.0425(1) 1.0419(3) 1.0401(5) 1.0358(7) 1.0254(9)

38.37(9) 94.64(7) 98.97(5) 101.57(3) 102.32(1) 101.94(2) 100.89(4) 95.53(6) 76.67(8)

* Super scribes are assigned for the strongest H-bonds. ** Super scribes are assigned for the strongest H-bond which contradict to the assignment of the other parameters. a Subscribes on all the listed values represent the order of H-bond strength.

systems of these three parameter set assignments as well. For the exceptional cases with the superscript sign (**) of (HCN)5, (HCN)7 and (HCN)9, the energy differences between the first and second strongest (BE)N – H in all three cases are less than 0.02 kJ/mole. This small uncertainty may have been either created by a computational error of the semi-empirical method or the electrostatic attraction energy effect mixed in the energy-breaking type calculation, which will be explained in Section 3.3.3. In summary, according to all the orders of PX – H, dX – H and dYH or according to the major part of the order of (BE)X – H, the strongest local H-bond of both ðHCNÞn and ðHNCÞn are usually situated in the middle of the linear cluster. If n is even and the number of H-bonds equals to an odd number n 2 1; the unique strongest local H-bond with the superscript sign (*) is formed by Xn/2 –H(n/2)þ1. If n is odd and the number of Hbonds equals to an even number n 2 1; the strongest H-bond is X(n21)/2 – H(nþ1)/2 with the marked superscript sign (*), which is the left one of the two center H-bonds of cluster. The right side of the two central H-bonds is X(nþ1)/2 –H(nþ3)/2, which is the second strongest H-bond. According to the orders of major part of the above-mentioned four parameters, H-bond strength decreases from the center toward the two terminals with the alternating order of -left-right-leftright-of Fig. 1 up to the weakest H-bond at the right end of cluster. Since such order of localized H-bond strength is quite different from the bonding strength assigned by Coulomb attraction energy (2 EX – H) as shown in Section 3.3.3, we define this localized

H-bond strength as ‘the order of covalent type bonding strength’ in this work. 3.3.3. Coulomb attraction energy (2 EX – H) In most of our former intra-molecular H-bonding works, we deal with the cases of O – HY type of local H-bond with strong electrostatic interaction [18 – 23]. The H-bonding strength determined by 2 EO – H and by the above-mentioned four parameters, (BE)O – H, PO – H, dO – H and dYH were almost with the same pattern. It is difficult to determine whether such Hbond is simply created by the electrostatic effect or constructed by the charge transform type partial covalent bond. However, for the inter-molecular Hbonding problem of the linear clusters in this work, the situation is quite different. After a detailed examination of the H-bond strength order by 2 EX – H, in both Tables 5a and 5b significant differences from the former defined ‘the order of covalent type of bonding strength’ are found. Although most of the two types of the H-bond strength orders show that the middle parts of ðHCNÞn and ðHNCÞn are the regions of the strongest H-bonding, when checking these detailed ordinal systems their arrangements of the orders of bonding strength are quite different. Looking through both of the two sets of clusters in the related two Tables, from all orders of the covalent type of bonding strength, stronger strengths usually favor on the left side with an H end of Fig.1. And the rightmost H-bond (Xn21 – Hn) is always the weakest H-bond. On the other hand, the order of 2 EX – H electrostatic type strength usually favor the right side with X end of

Table 6a Vibration frequencies of CH stretching and their vibrational coordinates of the related H atoms in ðHCNÞn Molecules

HCN

nCHa

2Dn

3306

H1

H2

H3

H4

H5

H6

H7

H8

H9

H10

Most important H

Intensity I (km/mol)

0.99

H1 1

H H2*

69.5 356.0*

H1 H3 H2*

72.7 1.1 899.7*

H1 H4 H2 H3*

73.7 343.9 65.5 1134.9*

H1 H5 H2 H4 H3*

74.6 226.4 324.5 0.8 1688.4*

H1 H6 H2 H4 H3 H4*

75.0 265.9 293.3 248.0 7.7 2156.6*

H1 H7 H2 H6 H3 H5 H4*

75.2 280.4 291.4 9.5 477.5 0.0 2646.9*

3300 3214*

6 92*

0.99 0.04

20.04 0.99*

(HCN)3

3298 3194 3180*

8 112 126*

0.99 20.02 0.03

20.04 20.66 0.74*

0.00 0.74 0.66

(HCN)4

3297 3183 3180 3149*

9 123 126 157*

0.99 0.01 0.03 0.01

20.04 0.30 0.90 0.27

0.0 20.32 20.17 0.92*

0.0 0.89 20.37 0.24

(HCN)5

3297 3179 3176 3149 3130*

9 127 130 157 176

0.99 0.00 0.04 20.01 0.01

20.04 20.07 0.96 20.18 0.16

0.00 0.03 20.24 20.65 0.71*

0.00 20.21 20.01 0.71 0.66

(HCN)6

3296 3177 3174 3144 3136 3117*

10 129 132 162 170 189*

0.99 0.00 0.04 20.01 0.01 0.01

20.04 20.02 0.96 0.00 0.16 0.10

0.00 0.00 20.22 20.46 0.71 0.46

0.00 0.01 0.02 0.63 20.08 0.76*

0.00 20.20 20.01 20.58 20.66 0.42

0.00 0.97 0.01 20.13 20.13 0.08

(HCN)7

3296 3176 3172 3141 3137 3125 3108*

10 130 134 165 169 181 198*

0.99 0.00 0.04 0.00 0.01 20.01 0.00

20.03 20.01 0.96 20.10 0.16 20.11 0.06

0.00 0.00 20.22 20.38 0.66 20.51 0.31

0.00 0.00 0.02 0.45 20.39 20.46 0.64*

0.00 0.01 0.00 20.53 20.20 0.53 0.62

0.00 20.19 0.00 0.58 0.56 0.47 0.28

0.00 0.97 0.01 0.12 0.11 0.08 0.05

(HCN)8

3296 3174 3170 3139 3137 3128

10 132 136 167 169 178

0.99 0.00 0.04 0.00 0.01 0.00

20.03 20.01 0.96 0.06 0.17 20.10

0.00 0.00 20.22 0.23 0.68 20.47

0.00 0.00 0.02 20.26 20.53 20.18

0.00 0.00 0.00 0.30 0.18 0.67

0.00 0.01 0.00 20.52 0.16 20.30

0.00 20.19 0.00 0.69 20.38 20.41

0.00 0.96 0.08 0.16 0.13

0.00 0.97 0.00 0.15 20.08 20.08

197

H1 75.5 275.8 H8 H2 289.1 H7 240.6 19.9 H3 H5 471.0 (continued on next page)

C. Chen et al. / Journal of Molecular Structure (Theochem) 630 (2003) 187–204

(HCN)2

67.9

198

H2

H3

H4

0.00 0.00

20.08 0.05

20.39 0.22

20.58 0.50

0.04 0.64*

0.60 0.48

0.34 0.19

0.06 0.04

10 132 135 168 169 177 185 196 209*

0.99 0.00 0.04 0.00 0.01 0.00 0.00 0.00 0.00

20.03 0.00 0.97 20.03 0.17 0.08 0.08 0.06 0.03

0.00 0.00 20.22 20.14 0.72 0.37 0.38 0.29 0.16

0.00 0.00 0.01 0.13 20.55 0.11 0.42 0.56 0.39

0.00 0.00 0.00 20.13 0.23 20.55 20.42 0.30 0.58*

0.00 0.00 0.00 0.23 20.05 0.59 20.35 20.35 0.57

0.00 0.01 0.00 20.53 20.11 20.19 0.47 20.55 0.36

0.00 20.19 0.00 0.75 0.21 20.36 0.35 20.26 0.14

0.00 0.97 0.00 0.16 0.04 20.07 0.06 20.05 0.03

10 132 134 169 170 178 183 191 201 213*

0.99 0.00 0.04 0.00 0.01 0.00 0.00 0.00 0.00 0.00

20.03 0.00 0.97 0.02 0.17 0.07 20.07 20.06 0.04 0.02

0.00 0.00 20.21 0.07 0.73 0.31 20.37 20.32 0.22 0.11

0.00 0.00 0.01 20.06 20.57 0.07 20.30 20.48 0.48 0.29

0.00 0.00 0.00 0.05 0.25 20.45 0.51 0.06 0.46 0.49

0.00 0.00 0.00 20.09 20.09 0.58 20.03 0.55 20.02 0.57*

0.00 0.00 0.00 0.22 0.01 20.48 20.49 0.01 20.48 0.48

0.00 0.01 0.00 20.54 0.05 0.13 0.36 20.49 20.46 0.28

0.00 20.19 0.00 0.77 20.10 0.29 0.35 20.29 20.20 0.10

nCHa

2Dn

3116 3102*

190 204*

(HCN)9

3296 3174 3171 3138 3137 3129 3121 3110 3097*

(HCN)10

3296 3174 3172 3137 3136 3128 3123 3115 3105 3093*

Molecules

a

H1

H5

H6

H7

H8

H9

* Super scribes are assigned for the most important H atom in the vibration mode of the largest red shift and strongest IR intensity. Frequency factor f ¼ 0:961 for B3LYP/6-311 þ G(2d,p) and unit in cm21.

H10

0.00 0.97 0.00 0.16 20.02 0.06 0.06 20.05 20.04 0.02

Most important H

Intensity I (km/mol)

H6 H5 *

0.2 3227.5*

H1 H9 H2 H8 H3 H6 H7 H4 H5 *

75.4 278.1 293.3 110.1 269.2 0.1 633.6 0.0 3621.4*

H1 H10 H2 H9 H3 H6 H5 H6 H4 H6 *

75.3 280.5 301.0 199.0 114.1 196.4 0.1 761.4 0.3 4115.0*

C. Chen et al. / Journal of Molecular Structure (Theochem) 630 (2003) 187–204

Table 6a (continued)

Table 6b Vibration frequencies of NH stretching and their vibrational coordinates of the related H atoms in ðHNCÞn Molecules HNC

nNH a

2Dn

3655

H1

H2

H3

H4

H5

H6

H7

H8

H9

H10

Most important H

Intensity I (km/mol)

1.00

H1 1

H H2*

303.8 1412.8*

H1 H3 H2*

320.9 55.1 3660.6*

H1 H4 H2 H3*

327.5 1093.6 688.4 4792.3*.

H1 H5 H2 H4 H3*

331.5 1065.8 1250.3 5.5 7374.1*

3647 3351*

8 304*

1.00 0.04

20.03 0.99*

(HNC)3

3642 3278 3230*

13 337 425*

1.00 20.02 0.03

20.03 20.59 0.80*

0.00 0.80 0.59

(HNC)4

3640 3234 3215 3107*

15 421 440 548*

1.00 0.00 0.03 0.01

20.03 0.06 0.95 0.28

0.00 20.24 20.25 0.93*

0.00 0.96 20.12 0.22

(HNC)5

3639 3219 3200 3103 3033*

16 436 455 552 622*

1.00 0.00 0.03 20.01 0.01

20.03 20.03 0.97 20.16 0.17

0.00 0.01 20.23 20.65 0.72*

0.00 20.19 0.00 0.72 0.65

(HNC)6

3638 3211 3189 3082 3057 2978*

17 444 466 573 598 677*

1.00 0.00 0.03 20.01 0.01 0.01

20.03 20.01 0.97 20.12 0.16 0.10

0.00 0.00 20.22 20.47 0.71 0.46

0.00 0.00 0.01 0.62 20.08 0.77*

0.00 20.18 20.01 20.59 20.66 0.42

0.00 0.98 0.01 20.11 20.12 0.08

(HNC)7

3638 3207 3183 3067 3054 3008 2937*

17 448 472 588 601 647 718*

1.00 0.00 0.03 0.00 0.01 20.01 0.01

20.03 20.01 0.97 20.06 0.16 20.11 0.07

0.00 0.00 20.21 20.28 0.72 0.50 0.32

0.00 0.00 0.00 0.34 20.48 20.46 0.65*

0.00 0.00 0.00 20.52 20.09 0.56 0.62

0.00 20.17 0.00 0.71 0.45 0.43 0.27

0.00 0.98 0.01 0.13 0.08 0.08 0.05

(HNC)8

3637 3204 3180 3059 3049 3013 2972

18 451 475 596 606 642 683

1.00 0.00 0.03 0.00 0.01 0.00 0.00

20.03 0.00 0.97 0.02 0.17 20.09 20.08

0.00 0.00 20.21 0.07 0.77 20.40 20.38

0.00 0.00 0.00 20.09 20.55 20.24 20.60

0.00 0.00 0.00 0.16 0.19 0.70 0.06

0.00 0.00 0.00 20.48 0.01 20.38 0.62

0.00 20.17 0.00 0.83 20.16 20.35 0.31

0.00 0.98 0.03 0.14 0.13

0.00 0.98 0.00 0.15 20.03 20.06 0.06

H1 H6 H2 H4 H3 H4*

333.5 1154.9 1215.8 1087.6 2.8 9530*

H1 H7 H2 H6 H3 H5 H4*

334.6 1184.6 1248.0 235.8 1832.2 0.1 11875.4*

H1 H8 H2 H7 H3 H5 H6

199

335.2 1194.2 1261.8 1056.6 439.4 1762.8 0.6 (continued on next page)

C. Chen et al. / Journal of Molecular Structure (Theochem) 630 (2003) 187–204

(HNC)2

264.2

200

Molecules

nNH a

2Dn

2905*

750*

H1 0.00

H2 0.05

H3 0.22

H4 0.51

H5 0.65*

H6 0.47

H7 0.18

H8

H9

0.04

(HNC)9

3637 3203 3178 3054 3044 3010 2985 2942 2880*

18 452 477 601 611 645 670 713 775*

1.00 0.00 0.03 0.00 0.01 0.00 0.00 0.00 0.00

20.03 0.00 0.97 0.00 0.18 0.06 0.08 0.06 0.04

0.00 0.00 20.21 20.02 0.79 0.29 0.36 0.28 0.16

0.00 0.00 0.00 0.02 20.53 0.17 0.44 0.56 0.39

0.00 0.00 0.00 20.04 0.18 0.55 20.46 0.30 0.59*

0.00 0.00 0.00 0.13 20.05 0.62 20.31 20.39 0.57

0.00 0.00 0.00 20.49 20.01 20.30 0.50 20.53 0.35

0.00 20.17 0.00 0.84 0.05 20.29 0.31 20.23 0.13

0.00 0.98 0.00 0.15 0.01 20.05 0.06 20.04 0.03

(HNC)10

3637 3203 3177 3052 3041 3005 2990 2960 2918 2862*

28 452 478 603 614 650 665 695 737 829*

1.00 0.00 0.03 0.00 0.01 0.00 0.00 0.00 0.00 0.00

20.03 0.00 0.97 0.00 0.18 20.04 20.07 20.06 20.05 0.03

0.00 0.00 20.20 0.00 0.81 20.20 20.33 20.29 20.22 0.12

0.00 0.00 0.00 0.00 20.52 20.13 20.35 20.49 20.49 0.30

0.00 0.00 0.00 0.00 0.16 0.41 0.56 0.08 20.46 0.50

0.00 0.00 0.00 20.03 20.05 20.53 20.14 0.58 0.06 0.57*

0.00 0.00 0.00 0.12 0.01 0.57 20.41 20.09 0.51 0.46

0.00 0.00 0.00 20.47 0.00 20.29 0.41 20.49 0.44 0.26

0.00 20.17 0.00 0.85 20.01 20.26 0.27 20.24 0.17 0.09

a

H10

0.00 0.98 0.00 0.15 0.00 20.05 0.05 20.04 0.03 0.02

Most important H

Intensity I (km/mol)

H5*

14096.2*

1

H H9 H2 H8 H3 H6 H7 H4 H5*

335.9 1212.3 1267.6 864.5 957.0 7.9 2671.1 0.1 16324.5*

H1 H10 H2 H9 H3 H7 H5 H6 H7 H6*

336.1 1217.7 1274.6 972.3 875.9 733.7 0.1 3158.2 0.7 18569.0*

* Super scribes are assigned for the most important H atom in the vibration mode of the largest red shift and strongest IR intensity. Frequency factor f ¼ 0:961 for B3LYP/6-311 þ G(2d,p) and unit in cm21.

C. Chen et al. / Journal of Molecular Structure (Theochem) 630 (2003) 187–204

Table 6b (continued)

C. Chen et al. / Journal of Molecular Structure (Theochem) 630 (2003) 187–204

Fig. 1. And the leftmost 2 EX1 –H2 is always the smallest Coulomb attraction energy among all 2 EX – H’s in the same molecular cluster. With the relatively stronger H-bond, the differentia of the two types of bonding strength in ðHNCÞn of Table 5b is very clearcut for all values of n from 2 to 10. For even n; taking (HNC) 4 as an example, although C2 – H 3 is the strongest H-bond for both types of determination in the cluster, the order of the other is different. In this cluster, 2 EC1 – H2 # 2 EC3 –H4 on the other hand, all the other four parameters including local H-bond energy (BE)C – H indicate that the left-side C1 – H2 Hbond is stronger than the right side H-bond of C3 –H4. Taking (HNC)7 as an example of odd n; both start from the middle part, bonding strength order is arranged according to the decreasing order of covalent type of the related four parameters: C3 –H4, C4 –H5, C2 – H3, C5 – H6, C1 – H2 and, C6 – H7 without any exception. But for the 2 EX – H electrostatic type determination with the large electrostatic effect on the right side with C end and the completely differently ordinal system, it will be found as C4 – H5, C3 –H4, C5 – H6, C2 – H3, C6 – H7 and C1 –H2. As for the other molecular clusters of Table 5b, in both even and odd cases of n; most are arranged in the above-mentioned orders accordingly. Due to the relative weaker H-bond in the ðHCNÞn system of Table 5a than the related ðHNCÞn with same n in Table 5b, the bonding strength orders of the ðHCNÞn system are a little more complicated than the corresponding problems with ðHNCÞn : However, the properties of electrostatic type assignment preferring the right side and covalent type assignment preferring the left side of cluster are also held in the ðHCNÞn system. Sometimes these differentiated assignments in the ðHCNÞn clusters are even more prominent than that happens in the related ðHNCÞn clusters. The order assignment of covalent type bonding strength of ðHCNÞn which is assigned by all the PN – H, d N – H, d CH, and most of the (BE)N – H are completely the same as such kind of assignment for the related ðHNCÞn clusters. But the order assignment of Coulomb attraction energy (2 EX – H) is quite different. When n # 5; this electrostatic effect monotonically decreases from the right side (N end) of Fig.1 to the left side (the H end). Taking (HCN)5 as an example, this decreasing strength of 2 EN – H is simply in the N4 – H5, N3 –H4, N2 –H3 and N2 –H1 monotonically order. In cases of ðHCNÞn with larger n

201

(n $ 6), although most of the largest 2 EN – H are situated in the central part of the cluster, more of the comparable larger 2 EN – H’s are situated on the right side of the linear structure, creating more deviation between the ordinal systems of the covalent and electrostatic types than that shown in the cases of ðHNCÞn : The enhancement electrostatic term on the left side of ðHCNÞn cluster, sometimes influences the ordinal system of the local bond energy, (BE)N – H. If the covalent type strength is defined by the ordinal systems of PN – H, dN – H and d CH and the electrostatic type strength is defined by order of 2 EN – H, the ordinal system of (BE)N – H is situated between these two types. The weak bond energies (BE)N – H’s of the two sides of cluster are very close to each other as shown in the results of ðHCNÞn with large n in Table 5a. Such ordinal system looks more close to covalent type than the electrostatic type. Since (BE)N – H of this work is calculated by the energy breaking procedure of Eq. (2), this calculated result may be including both covalent and electrostatic energies. The ordinal result of (BE)N – H is closer to the covalent strength than the electrostatic type. One may explain that the major part of this energy is covalent type, and a small amount of minor electrostatic energy is already mixed in the related energy calculation for the H-bonding formation. 3.3.4. Analysis of YH stretching vibration Among the 9n 2 5 normal modes of vibration of the ðHYXÞn linear cluster, n of which with the largest frequencies are the characteristic frequencies of YH stretching vibration, nYH. These n vibration modes are closely related to the n 2 1 X –H type of local Hbonds. For the ðHCNÞn clusters, X is N atom and Y is C atom; for the ðHNCÞn clusters, X is C atom and Y is N atom. In the ordinary case, the red shift and the increment of IR intensity of the vibration mode of this type of YH bond stretching are closely related to the strength of H-bond formation of X –H H-bonds and the cooperation effect whenever n of the (HYX)n cluster increases. In this work, the red shift of nYH is defined as 2 Dn ¼ nYH(monomer) 2 nYH, as shown in Tables 6a and 6b. Since most of these vibration modes are the motion of mixed YH coordinates along the molecular axis, for the purpose of H-bond local analysis, one has to identify which H axes are mixed in such vibration mode. Y is much heavier and less

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active than H in this motion, and the coordinate of Y in this motion is negligibly small. Consequently the coordinates of Y are usually ignored in this coordinate assignment. By means of making a close comparison between H coordinate assignment and above-mentioned localized H-bond analysis, the largest H coordinate in each vibration mode is particularly picked up and defined as ‘the most important H’ in this motion. For the vibration motion with the largest red shift and the strongest IR intensity, the related ‘most important H’ is possibly located at the H atom, forming strong H-bonds. All the recorded quantities for such vibration motion of all the molecular clusters are marked with the subscript (*) as shown in Tables 6a and 6b. Looking at the results of these two tables, several general phenomena are clearly found. With the smallest nYH, the largest red shift and the strongest intensity, ‘the most important H’ with the strongest H-bond is usually located in the middle part of the cluster. The cooperative effect and the stronger H-bonding formation are shown in both ðHCNÞn and ðHNCÞn molecular clusters when n increases. If comparing such effect by the increment of the red shift and ir intensity, it may be easily found that the strength of H-bonding in ðHNCÞn is comparably stronger than that of ðHCNÞn with the same n: The most important subject of such vibration analysis in this work is to find a reliable and efficient way to identify the local H-bonds and the relative bond strength in each molecular cluster. The n vibration modes usually are constructed with the mixed coordinates of various YH motions. By means of clarifying the local analysis procedure, the related motion coordinates of all the H atoms in the vibration mode are assigned for the detailed comparison. For the sake of simplicity, we choose four of the n vibration modes for this type of analysis. For (HYX)n with n from 3 to 10, the largest nYH within the n nYH’s always relates purely to the leftmost non-hydrogen bonded H1 of H1Y1X1. And the corresponding relative coordinate value is greater than or equal to 0.99. The second largest nYH relates mostly to the atom Hn of the nth HnYnXn molecule of the right terminal with the relative coordinate greater than or equal to 0.89. This hydrogen atom Hn is closely related to the weakest localized H-bond Xn21 – Hn among all of the H-bonds in the cluster. The third vibration mode of

nYH relates mostly to the atom H2 of the second HYX from the left side of the cluster with the relative coordinate greater than or equal to 0.90. Such hydrogen atom H2 just connects with the X1 atom of the leftmost molecule H1Y1X1 for the second weakest H-bond X1 – H2 formation. For the rest n 2 3 smaller nYH with larger red shift cases, the motion coordinate is usually constructed with the coordinates of many interior H atoms, which are quite different from the first three purely populated types of motions. With the various types of cooperative mixing among so many different YH motions in the cluster system, it is very difficult to analyze all of the normal modes of YH stretching in the cluster. However, the normal mode of nYH with the largest red shift, the strongest ir intensity and the biggest cooperative effect in the cluster is the most interesting subject for the detailed analysis in this work. As shown in Tables 6a and 6b, quantities related to this important normal mode are all marked with the superscript (*). In this normal mode of vibration, the largest relative H atom coordinate and the H atom belong to which is also marked with the superscript (*). The H atom of each mode marked with the superscript (*) is defined as the ‘most important H’ of this vibration mode. After looking through all the ‘most important H‘ of all the results of both ðHCNÞn and ðHNCÞn molecular clusters from n equaling 2 – 10, the generalized result may be summarized as the following. For the cases of even values of n; the ‘most important H’ in each cluster is H(n/2)þ1; and for the cases of odd values of n; the ‘most important H’ in each cluster is H(nþ1)/2. The most interesting subject of the local H-bonding analysis type comparison is to compare the ‘most important H’ of this section with the strongest local Hbond determined by the ‘the order of covalent type bonding strength’ defined in Section 3.3.2. According to this covalent type strength determination, the strongest H-bond of an even n is X(n/2) 2 H(n/2)þ1 and of an odd n is X(n21)/2 – H(nþ1)/2 for both ðHCNÞn and ðHNCÞn as shown in Tables 5a and 5b. The H atom in such strongest H-bond of each cluster is exactly equivalent to the ‘most important H’, which is assigned in this section without any exception. Therefore, such result of vibration motion analysis gives an additional sound piece of evidence that the local H-bond in this work is constructed with the covalent type in nature.

C. Chen et al. / Journal of Molecular Structure (Theochem) 630 (2003) 187–204

4. Conclusion To compare the localized H-bonding analyses of the ðHCNÞn and ðHNCÞn molecular clusters, several important points are to be summarized and concluded as follows: 1. Due to the large cooperative effect of the Hbonding formation, all the physical quantities with increments related to the increasing of H-bonding strengths are extensively increased when the number of molecules, n; of ðHCNÞn or ðHNCÞn is increased. In comparing the related rates of their increments from the dipole moments of Table 4, local H-bond parameters of Tables 5a and 5b, and the calculated red shift and IR intensity of YH stretching vibrations (Y is atom C in ðHCNÞn and is atom N in (HNC)n) in Tables 6a and 6b, all pieces of the evidence show that the cooperative effect in the ðHNCÞn system is significantly larger than that in the ðHCNÞn system for all n: 2. The orders of the local H-bond strength among n 2 1 H-bonds in ðHCNÞn and ðHNCÞn are quite similar. In most cases the strongest H-bond is located in the central part of the molecular cluster. And the rest of the local H-bonds decrease monotonically form the center to the two terminals of the linear structure. The H-bond strength order determined by the Coulomb attraction energy, 2 EX – H, is quite different from the order determined by the other four important parameters. As defined in this work, the order system determined by 2 EX – H is called the ‘electrostatic type strength’ and the order system determined by the local H-bond order(PX – H), local H-bond distance(dX – H), H-bond energy (BE)X – H and the shared bond distance(d YH) is the ‘covalent type strength’. The ‘covalent type strength’ is in favor of the left side (or the positively charged side with the H atom at the terminal), but the ‘electrostatic type strength’ is in favor of the right side (or the negatively charged side with X atom at the terminal) of the linear chain. In cases of ðHNCÞn as shown in Table 5b, ordinal systems of the ‘covalent type strength’ determined by all of the four parameters are completely identical without any exception. There are several exceptions of (BE)N – H ordinal problems in cases of (HCN)n. But the related energy differences are very small, as shown in Table 5a, which is quite different from the ‘electrostatic type strength’. The most interesting

203

subject is the YH stretching vibration analysis. The largest nYH relates to non-H-bonded H1; the second largest nYH relates to the weakest H-bond Yn 2 1 – Hn at the right end; the third largest nYH relates to the second weakest H-bond Y1 –H2 at the left end of all the molecular clusters. For the vibration with the smallest nYH, the largest red shift and the strongest ir intensity, the most important H with significant cooperative effect in this normal mode is always related to the strongest H-bond determined by the ‘covalent type strength’. All of these matched analyses show that the local H-bonds in both ð HCNÞn and ðHNCÞn are created by the charge transfer type covalent bond in nature. 3. From the energy point of view, ðHCNÞn is more stable than ðHNCÞn with the same n: Even the [E(HNC)n 2 E(HCN)n]/n between them decrease monotonically when n is increased. But the decreasing tendency is not very fast, and the order of the energy between them appears very difficult to change. This is the reason why it is so difficult to find experimental data of ðHNCÞn with large values of n up to the present time. However, the theoretical optimization and frequency calculation show that ðHNCÞn with different values of n (n # 10) are all the stable local minima and with large cooperative effect. The H-bonding in this type of molecular cluster is very prominent. For the very large n both (HCN)1 and (HNC)1 are possibly forming linear type solid structures. As shown in the X-ray probability figure in Ref. [1] of linear polymer of HCN solid phase, the H atom in the linear cluster is situated at the center between N and C atoms. Based on this evidence, we may suggest that (HCN)1 and (HNC)1 are the identical substances whenever n approaches to infinity.

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