The equilibrium N–H bond length

Chemical Physics 260 (2000) 65±81

www.elsevier.nl/locate/chemphys

The equilibrium N±H bond length J. Demaison a,*, L. Margules a, James E. Boggs b a b

Laboratoire de Physique des Lasers, Atomes et Mol ecules, UMR CNRS 8523, Centre d'Etude et Recherches Lasers et Applications, B^ at. P5, Universit e de Lille 1, 59655 Villeneuve d'Ascq Cedex, France Institute for Theoretical Chemistry, Department of Chemistry and Biochemistry, University of Texas at Austin, Austin, TX 78712, USA Received 10 May 2000; in ®nal form 29 July 2000

Abstract The equilibrium structures of NH3 , NH2 F, NHF2 , NF3 , HNC, HNO, HN3 , HNCO, and NH‡ 4 were calculated at the CCSD(T) level of theory with basis sets of at least quadruple zeta quality and with a correction for the core correlation calculated with the cc-pCVQZ basis set. Then, the possibility of calculating accurate ab initio N±H bond lengths is examined using a sample of 13 molecules whose equilibrium structures are known. Three di€erent correlated methods are compared: CCSD(T), MP2, and DFT using the hybrid functional B3LYP. As expected, the CCSD(T)/cc-pVQZ method gives the most accurate results but, except for HNO, the MP2/6-31G** and B3LYP/6-311++G(3df,2pd)  provided a small o€set correction is methods are also satisfactory, their standard deviation being smaller than 0.002 A taken into account. Ó 2000 Published by Elsevier Science B.V.

1. Introduction Although the N±H bond is quite common in chemistry and biochemistry, there are very few accurate data on its length (see for instance Ref. [1]) and on its variation with the environment. This is due to several reasons. First, many molecules containing the N±H bond are dicult to study experimentally (they are either unstable, or dicult to obtain in the gas phase, . . .). Even when the rotational spectrum can be measured, there is still a long way to go to determine an accurate structure. As for other X±H bonds (see for instance Ref. [2]), the isotopic substitution H ! D considerably modi®es the vibration±rotation interactions and makes r0 or rs structures inaccurate. Further-

*

Corresponding author. Fax: +33-3-20-43-40-84. E-mail address: [email protected] (J. Demaison).

more, the vibration of the N±H bond is strongly anharmonic and, ®nally, the inversion motion, when present, further complicates the problem. Although ammonia, NH3 , is one of the molecules most studied by spectroscopy, there is not yet an accurate experimental equilibrium structure available. See Table 1 which gathers a few structure determinations of NH3 . The range of values is  for the distance and 2° for the bond large: 0.014 A angle. For instance, Spirko and Kraemer [3] propose three di€erent bond distances (1.0180, 1.0141  depending on the approximations and 1.0149 A) made in the potential function. Furthermore, as far as the experimental structures are concerned, there is no improvement of accuracy with time, the old equilibrium structure of Benedict and Plyler [4], published in 1957 being very close of the best recent ab initio calculations. It is also worth noting that the most accurate value for re (N±H) is the smallest value in the table.

0301-0104/00/$ - see front matter Ó 2000 Published by Elsevier Science B.V. PII: S 0 3 0 1 - 0 1 0 4 ( 0 0 ) 0 0 2 5 3 - 6

66

J. Demaison et al. / Chemical Physics 260 (2000) 65±81

Table 1 Evolution of estimates of the structure of ammonia with time  and angle in degree) (distance in A Year

r(N±H)

\(HNH)

Type

1940 1945 1951 1957 1957 1964 1968 1969 1971 1971 1972 1974 1981 1982 1989 1989 1989 1992 1998 1999

c1.014 1.014 1.016 1.0144 1.0124 1.0116 1.011 1.0136 1.0156 1.0156 1.025 1.0128 1.0178 1.0162 1.018 1.0141 1.0149 1.0124 1.0126 1.0111

107.3 106.78 107 107 106.67 106.7 108.2 107.05 107.28 107.28 107.00 107.03 106.75 107.47 107.28 107.49 107.40 106.18 106.61 106.75

re re r0 r0 re re re rs r0 r0 re rs ±a r0 re b re c re d re re e re f

Range

0.014

2.02

Ref. [80] [81] [82] [83] [4] [84] [85] [86] [87] [88] [89] [90] [91] [92] [3] [3] [3] [93] [34] [94]

a

From spin±spin interaction. b With ab initio force constants constrained. c Better for the prediction of energies of excited states. d The best for the reproduction of inversion splittings. e CCSD(T)/aug-cc-pVQZ results. f CCSD(T)/cc-pV1Z+aug(N) with core correlation correction.

The goal of this paper is to critically review the data on the N±H bond length, to determine the N±H bond length in a few further molecules using high level ab initio methods and, ®nally, to see which ab initio methods may be used to accurately predict an N±H bond length. It is indeed well established that it is easy to obtain a reliable ab initio C±H bond length, the CCSD(T)/cc-pVTZ method being accurate [5] and the simple MP2/6-31G** method giving satisfactory results in most cases [2]. Therefore, it is interesting to establish at which price an accurate N±H bond length can be obtained because it is now well known that ab initio methods can be quite accurate for small molecules but at a high computational cost. In most cases, the method of convergence (using a high level of electron correlation and increasing the size of the basis set up to convergence), which should be the method of choice, cannot be used. Di€erent ap-

proximate methods have to be used, often in combination. The result from an in®nite basis size can be estimated using an empirical extrapolation formula. The method of calibration is also quite useful: if the electron correlation is correctly taken into account and if the basis size is large enough (and well chosen), the remaining error is almost constant for a given bond and can hence be corrected using o€sets obtained from molecules whose structure is accurately known. Finally, the linear combination method is also often used: it combines a calculation at a high level of electron correlation with a relatively small basis set and calculations at a lower level of electron correlation with a larger basis set. These di€erent approximate methods will be used and compared. Although this work is mainly limited to closedshell molecules, it should be noted that accurate equilibrium structures have been recently determined for NH [6], NH2 [7±9] and NH‡ 2 [10].

2. Computational methods The calculations with all electrons correlated (ae) were performed with a local version of the A C E S I I program system [11] whereas for the frozen core (fc) approximation either the G A U S S I A N 1 [13] programs were used. 9 4 [12] or M O L P R O 9 6 Numerous studies have shown that while SCF calculations yield reasonable geometries, their accuracy is signi®cantly worse than that of most experimental structures. For this reason two correlated methods were used: second-order Mùller± Plesset perturbation theory (MP2) [14] and coupled cluster theory with single and double excitations [15] augmented by a perturbational treatment of connected triple excitations (CCSD(T)) [16]. In most cases, the CCSD(T) method is believed to yield results close to the exact n-particle solution within the given basis set [5,17±21] while the MP2 geometries are remark1 M O L P R O 9 6 is a package of ab initio programs written by H.-J. Werner, P.J. Knowles, with contributions from J. Alml of, R.D. Amos, M.J.O. Deegan, S.T. Elbert, C. Hampel, W. Meyer, K.A. Peterson, R.M. Pitzer, A.J. Stone, P.R. Taylor.

J. Demaison et al. / Chemical Physics 260 (2000) 65±81

ably accurate although much less expensive than the CCSD(T) ones [17]. The choice of the basis set is important. Ideally, one should use the correlation consistent polarized core±valence basis sets with di€use functions (labelled aug-cc-pCVnZ with n ˆ D, T, Q, 5, . . .) of Dunning and coworkers [22] all electrons being correlated, and increase the size of the basis set (i.e. n) up to convergence. The problem is that these basis sets are huge and it is still dicult to go beyond n ˆ Q except for very small molecules. A remedy is to extrapolate the results n ˆ D, T, Q to n ˆ 1 using an empirical extrapolation formula [23]. A further simpli®cation may be obtained if one notices that the e€ect of the di€use function is additive and, furthermore, can be calculated at a lower level (MP2 for instance) without much loss of accuracy [24]. Even with these simpli®cations, the cc-pCVnZ basis sets can only be used for rather small molecules because of the large memory and disk storage needed for the calculations. A much cheaper alternative is to use the well-known Dunning's correlation-consistent polarized valence basis sets labelled cc-pVnZ (with n ˆ D, T, Q) [25] which are signi®cantly smaller. Usually, the core electrons are frozen during the calculations. In this case, even at in®nite basis size, the results may lie rather far from the true equilibrium structure, due mainly to the neglect of core correlation. This can be rather easily solved by treating the core correlation e€ects as an additive correction and calculating this correction with the cc-pCVQZ basis set at the CCSD(T) level or at a lower level such as MP2. As this correction is not too large for ®rst row atoms  using the MP2 (a few thousandths of an A), method does not lead to a great loss of accuracy and makes the calculations much easier [26]. It is also possible to assume that the core correlation is constant for a given bond; the loss of accuracy is still acceptable in most cases. One problem which is more dicult to solve is that the variation of the parameters might not be monotonic with basis set enlargement [24]. In some cases, the problem may be solved by taking into account the e€ect of diffuse functions on the electronegative atoms [27]. But, as we will see below, this extrapolation method is purely empirical and it is not certain

67

that it will always converge toward the right value. The only safe solution is to go up to n ˆ 5 to achieve convergence. It is also possible to use a mixed basis set composed of cc-pV5Z at all nonhydrogen atoms and cc-pVQZ at H, which is denoted as cc-pV(5,Q)Z. Such a basis set is supposed to lead to little loss in accuracy compared to use of the full cc-pV5Z set at all atoms while reducing the computation times signi®cantly [28]. An interesting alternative is to use the cc-pVnZ basis sets but with all electrons correlated. The main advantage of this method is that the results are often closer to the equilibrium values, once basis set convergence is achieved. Generally, the cc-pVQZ basis set slightly exaggerates the e€ect of the core correlation but this exaggeration is quite  or less) and it seems small (one thousandth of an A justi®ed to estimate it at the MP2 level [26] or by determining an o€set from molecules whose structure is known as was done for instance for the C±F bond [24]. Density functional theory (DFT) has also been used to calculate the N±H bond length. During the past few years, DFT has become a popular approach to chemical problems because it includes some of the e€ects of electron correlation much less expensively (less memory and less disk space) than traditional correlated methods [29,30]. Furthermore, analytic derivatives are available at the DFT level. There are many di€erent variants of the density functional. One of the most successful is the Becke-style hybrid B3LYP functional (Becke's three parameter exchange functional (B3) [31], as slightly modi®ed by Stephen et al. [32] in combination with the Lee±Yang±Parr (LYP) functional [33]). It has the reputation of giving results that are comparable to MP2 (or better) [29]. We have carried out geometry optimizations for the same sample of molecules as above using the B3LYP method as implemented in G A U S S I A N 9 4 [12] with the 6-311++G(3df,2pd) basis set. 3. Structures In this section we review and, if necessary, recalculate the structures of simple molecules containing the N±H bond. We ®rst compare the

68

J. Demaison et al. / Chemical Physics 260 (2000) 65±81

di€erent methods on ammonia and the ¯uoramines.

in®nite basis size was tried using the following empirical exponential function [23]: r…n† ˆ r1 ‡ b eÿcn ;

3.1. Ammonia Previously reported structures of NH3 are gathered in Table 1 and the results of the di€erent methods used in this work for ammonia are given in Table 2. It is to be noted that high level ab initio calculations of the structure of NH3 already exist. Particularly relevant to this work is the recent paper by Peterson et al. [34] where the structure was calculated up to the CCSD(T)/aug-cc-pVQZ and CCSD(T)/cc-pV5Z levels. The N±H bond length is practically converged at the quadruple f level but for the \(HNH) bond angle there is still a signi®cant change of 0.33° when going from CCSD(T)/cc-pVQZ to CCSD(T)/ cc-pV5Z. The cc-pV(5,Q)Z and cc-pV5Z results are very close as expected. An extrapolation to

…1†

where n is an index associated with each basis set, 2 ˆ DZ, 3 ˆ TZ, 4 ˆ QZ. For the N±H bond length, the CCSD(T)(fc)/cc-pV5Z and CCSD(T)(fc)/cc-pV1Z results are almost identical, but for the bond angle, the extrapolated value is smaller by 0.15° than the cc-pV5Z value, indicating that, in this case, the extrapolation is not accurate. This is not surprising because there is still rather a large change of 0.54° upon going from the CCSD(T)(fc)/ cc-pVTZ to the CCSD(T)(fc)/cc-pVQZ level. This conclusion remains true when all electrons are correlated. Using either the fc approximation, or correlating all electrons with the cc-pVnZ basis sets, or using the cc-pCVnZ basis set with all electrons correlated gives almost identical results provided a  or 0.17°) is made in small core correlation (0.001 A

Table 2  and angles in degree)a Ab initio CCSD(T) structures of ammonia (distances in A n

a

cc-pVnZ

cc-pCVnZ

fc

aug (fc)

r(N±H) D T Q 5,Q 5 1 Core correction re

1.0273 1.0141 1.0124 1.0125 1.0121 1.0121 ÿ0.0012c 1.0109e

1.0238 1.0147 1.0128 1.0128b 1.0124b 1.0123 ÿ0.0012c 1.0111e

\(HNH) D T Q 5,Q 5 1 Core correction he

103.54 105.64 106.18 106.53 106.51 106.37 0.17c 106.68f

105.92 106.38 106.52 106.59b 106.58b 106.58 0.17c 106.75e

ae

aug (ae)

ae

1.0264 1.0113 1.0101

1.0229 1.0119 1.0105

1.0261 1.0126 1.0111

1.0228 1.0133 1.0115

1.0100 0.0010d 1.0110e

1.0104 0.0010d 1.0113e

1.0109

1.0111

1.0109

1.0111

103.59 106.06 106.36

105.97 106.80 106.70

103.59 105.80 106.37

105.96 106.52 106.71

106.40 0.01d 106.41e

106.71 0.01d 106.72e

106.56

106.80

106.56

106.80

fc ˆ frozen core approximation, ae ˆ all electrons correlated, aug ˆ diffuse functions on N. Contribution of the di€use functions calculated at the MP2 level. c CCSD(T)(ae)/cc-pCVQZ-CCSD(T)(fc)/cc-pCVQZ. d CCSD(T)(ae)/cc-pCVQZ-CCSD(T)(ae)/cc-pVQZ. e CCSD(T)/cc-pV1Z ‡ core correction. f CCSD(T)/cc-pV5Z ‡ core correlation. b

Median aug (ae)

1.0110

106.70

J. Demaison et al. / Chemical Physics 260 (2000) 65±81

the ®rst two cases. At the extrapolated basis set limit, the e€ect of di€use functions on N is almost negligible for the bond length but remains signi®cant for the bond angle (about 0.2°). We take the median of the di€erent determinations (see Table 2, median) as our best theoretical estimate of the equilibrium geometry. The accuracy is probably  for the bond length and 0.1° better than 0.0005 A for the bond angle. 3.2. Mono¯uoramine Christen et al. [35] analyzed the microwave spectrum of mono¯uoramine, NH2 F, and its mono- and di-deuterated species. They determinated r0 and rZ structures. They also calculated the structure ab initio [36], but the basis sets used were rather small; furthermore the calculations were limited to the MP2 level. We have calculated the structure at the CCSD(T) level using the cc-pVnZ basis sets with n ˆ D, T, Q, either correlating all electrons or using the fc approximation. To check whether convergence is achieved at the quadruple zeta level, an extrapolation to in®nite basis size was tried with Eq. (1). The e€ect of the di€use functions was estimated with the aug-cc-pVnZ basis set on N and F and the standard cc-pVQZ basis set on H. Their e€ect becomes almost negligible at the extrapolated basis set limit and it is additive, but only in the frame of the fc approximation or with all electrons correlated. In other words, it is not possible to mix the results of the two methods. This is due to the fact that the rate of convergence of the two approximations is di€erent. The e€ect of the core correlation was estimated at the MP2 level with the cc-pCVQZ basis set. The cc-pVQZ basis set, all electrons being correlated, exaggerates  for the e€ect of core correlation by about 0.001 A the N±H bond whereas this overestimation may be  The difneglected for the N±F bond (0.0002 A). ferent methods of calculation give results in very good agreement (Table 3). 3.3. Di¯uoramine In 1963, Lide [37] analyzed the microwave spectrum of di¯uoramine, NHF2 , and determined

69

an r0 structure from the ground state rotational constants of the H and D isotopic species. More recently, the structure was calculated ab initio but, as for NH2 F, the basis sets used were rather small; furthermore the method was limited to the MP2 level [36]. The results of the new ab initio calculations are presented in Table 4. The cc-pVQZ basis set, all electrons being correlated, exaggerates the e€ect of  for the N±H core correlation by about 0.001 A bond whereas this overestimation may be ne The value of glected for the N±F bond (0.0002 A). the T1 diagnostic, T1 ˆ 0:0129, indicates that the CCSD(T) method should perform well [38]. However, it is obvious that there are problems. With the cc-pVnZ basis sets, all electrons being correlated, the convergence is not monotonic. This problem is solved, either by adding di€use functions on N and F (except for the \(FNF) angle), or by using the fc approximation, or by using the cc-pCVnZ basis sets. At the quadruple f level, convergence is almost achieved for the N±H bond length and the \(HNF) angle, but the N±F bond length is still too long as can be checked by extrapolating to in®nite basis size: the correction is  when the di€use functions are neonly 0.001 A  when they are glected, but it increases to 0.003 A taken into account. On the other hand, the e€ect of the di€use functions is almost negligible for the N±H bond length and the \(HNF) angle. The situation of the \(FNF) bond angle is perhaps worse. Although the variation from the cc-pVDZ level to the cc-pVQZ level is rather small, 0.15°, the convergence is not exponential, the variation from cc-pVTZ to cc-pVQZ being larger than the variation from cc-pVDZ to cc-pVTZ. Thus, it is not possible to extrapolate to in®nite basis size. Still worse, the di€erent basis sets give di€erent results at the quadruple f level. However, as the CCSD(T)(fc)/cc-pVQZ+aug(N,F) and the CCSD (T)(ae)/cc-pCVQZ+aug(N,F) results are close, the best estimate of the \(FNF) angle is probably obtained at the CCSD(T)(ae)/cc-pCVQZ+aug (N,F) level, \…FNF† ˆ 103:1°. For both ¯uoramines, the ab initio equilibrium structure is in fair agreement with the experimental r0 structure (taking into account the limited accuracy of the latter). However, it is to be noted that

70

J. Demaison et al. / Chemical Physics 260 (2000) 65±81

Table 3  and angles in degree) Ab initio structures of mono¯uoramine (distances in A Method

Basis set

a

r0 [35] MP2(fc) MP2(fc) MP2(ae) MP2(fc) MP2(ae) CCSD(T)(ae)

re b CCSD(T)(ae)

re b;d CCSD(T)(fc)

re e CCSD(T)(fc)

re e;d

cc-pVQZ cc-pVQZ+aug(N,F) cc-pVQZ cc-pCVQZ cc-pCVQZ cc-pVDZ cc-pVTZ cc-pVQZ cc-pV1Z cc-pVDZ+aug(N,F) cc-pVTZ+aug(N,F) cc-pVQZ+aug(N,F)c cc-pV1Z+aug(N,F) cc-pVDZ cc-pVTZ cc-pVQZ cc-pV1Z cc-pVDZ+aug(N,F) cc-pVTZ+aug(N,F) cc-pVQZ+aug(N,F)c cc-pV1Z+aug(N,F)

r(N±H)

r(N±F)

\(HNH)

\(HNF)

1.0225(3) 1.0157 1.0162 1.0134 1.0158 1.0145 1.0311 1.0169 1.0160 1.0159 1.0171 1.0295 1.0184 1.0167 1.0164 1.0175 1.0319 1.0198 1.0184 1.0182 1.0169 1.0304 1.0212 1.0189 1.0182 1.0169

1.4239(3) 1.4173 1.4196 1.4144 1.4171 1.4146 1.4375 1.4223 1.4217 1.4217 1.4219 1.4508 1.4260 1.4225 1.4219 1.4221 1.4381 1.4271 1.4247 1.4241 1.4216 1.4520 1.4327 1.4270 1.4245 1.4220

106.27(8) 105.10 105.27 105.27 105.11 105.26 103.01 104.76 105.02 105.07 105.01 104.51 105.20 105.18 105.18 105.17 102.97 104.43 104.85 105.03 105.18 104.47 104.87 105.02 105.10 105.25

101.08(7) 101.63 101.58 101.76 101.64 101.76 100.56 101.37 101.43 101.43 101.42 100.50 101.34 101.46 101.48 101.47 100.54 101.16 101.30 101.33 101.44 100.45 101.08 101.25 101.31 101.43

a

The rZ angles are: \Z (HNH) ˆ 105.50(3)°; \Z (HNF) ˆ 100.88(2)°. CCSD(T)(ae)/cc-pV1Z+MP2(ae)/cc-pCVQZ-MP2(ae)/cc-pVQZ. c Contribution of di€use functions calculated at MP2 level. d Preferred structure, e€ect of di€use functions taken into account. e CCSD(T)(fc)/cc-pV1Z+MP2(ae)/cc-pCVQZ-MP2(fc)/cc-pCVQZ. b

the experimental bond angles are less accurate than stated. It is interesting to notice that, at the CCSD(T)(ae)/cc-pVQZ level, the o€set (re ÿ rcalc: ) for the N±F bond length does not seem constant when we compare the results for NH2 F and NHF2 . The situation is not improved when the e€ect of di€use functions is taken into account. 3.4. Tri¯uoramine Although tri¯uoramine does not contain an NH bond, its structure is also considered here in order to investigate the variation of N±F bond length with substitution. For this molecule, there is an experimental equilibrium structure available [39]. But this structure was determined from the equilibrium rotational constants Be of the 14 N- and

15

N-isotopic species, not from the Be and Ce constants of one species. For this reason, the derived structure is highly sensitive to small errors in the rotational constants and it was indeed recently shown that this structure might not be accurate [40]. We have calculated the structure using the same procedure as for NHF2 and the results are presented in Table 5. The e€ect of di€use functions is small but not completely negligible. At the quadruple f level, convergence is almost achieved for the N±F bond length but with the cc-pVnZ basis sets, the convergence of the \(FNF) angle is not monotonic. This problem is solved using the augcc-pVnZ basis sets but, as for NHF2 , the convergence is still not exponential and it is thus not possible to extrapolate the \(FNF) angle to in®-

J. Demaison et al. / Chemical Physics 260 (2000) 65±81

71

Table 4  and angles in degree) Ab initio structures of di¯uoramine (distances in A Method r0 [37] MP2(fc) MP2(fc) MP2(ae) MP2(ae) MP2(fc) CCSD(T)(ae) CCSD(T)(ae)

re b CCSD(T)(fc)

re b CCSD(T)(fc)

re b CCSD(T)(ae)

re (best estimate)c

Basis set cc-pVQZ cc-pVQZ+aug(N,F) cc-pVQZ cc-pCVQZ cc-pCVQZ cc-pVDZ cc-pVTZ cc-pVQZ cc-pVDZ+aug(N,F) cc-pVTZ+aug(N,F) cc-pVQZ+aug(N,F)a cc-pV1Z+aug(N,F) cc-pVDZ cc-pVTZ cc-pVQZ cc-pV1Z cc-pVDZ+aug(N,F) cc-pVTZ+aug(N,F) cc-pVQZ+aug(N,F)a cc-pV1Z+aug(N,F) cc-pCVDZ cc-pCVTZ cc-pCVQZ cc-pCV1Z

r(N±H)

r(N±F)

\(HNF)

\(FNF)

1.026(2) 1.0214 1.0224 1.0191 1.0203 1.0216 1.0355 1.0197 1.0218 1.0359 1.0252 1.0229 1.0222 1.0234 1.0364 1.0254 1.0241 1.0240 1.0227 1.0368 1.0276 1.0251 1.0242 1.0229 1.0354 1.0239 1.0230 1.0230 1.0232

1.400(2) 1.3882 1.3894 1.3854 1.3856 1.3880 1.4058 1.3860 1.3897 1.4164 1.3943 1.3901 1.3891 1.3893 1.4063 1.3954 1.3925 1.3915 1.3891 1.4174 1.3995 1.3937 1.3910 1.3886 1.4055 1.3932 1.3899 1.3887 1.3889

99.8(2) 100.02 100.03 100.14 100.13 100.03 99.47 100.06 100.09 99.35 100.00 100.13 100.16 100.16 99.45 99.85 99.97 100.03 100.14 99.32 99.85 99.99 100.03 100.14 99.45 99.95 100.09 100.15 100.14

102.9(2) 103.49 103.33 103.53 103.53 103.49 103.39 103.68 103.28 102.81 102.98 103.15 103.38 103.32 103.23 102.80 102.96 103.08 103.38 103.33 103.28 103.15

a

Contribution of the di€use functions calculated at the MP2 level. With core correlation correction estimated at the MP2 level. c Median of the re values from the previous lines. b

nite basis size. It was thus assumed that the convergence is almost achieved at the quadruple f level. The ab initio structure is in good agreement with our recent experimental structure [40]. The huge decrease of the N±F bond length with increasing ¯uorination is well known and has already been discussed [36]. 3.5. Hydrogen isocyanide The millimeterwave spectra of HNC and DNC were measured by Okabayashi and Tanimoto [41]. The excited states were obtained in a discharge plasma and it was thus possible to determine the equilibrium rotational constants of the two isotopic species. However, only the fundamental states

were analyzed and the magnetic correction to the rotational constants was neglected. If we assume that neglect of the higher order vibration±rotation interaction constants leads to a systematic error of about 5% (which seems to be a reasonable order of magnitude [42]), the N±H bond length is too large  and the N@C bond length too small by 0.0005 A  The rotational g-factor has not been by 0.0002 A. experimentally determined but it has been recently calculated ab initio, g ˆ ÿ0:0968 [43]. The magnetic correction is negligible. Thus, it seems that the experimental equilibrium structure is highly accurate, unless unaccounted anharmonic resonances have a sizable e€ect. Furthermore, the cubic force ®eld was determined both experimentally [44] and ab initio [45,46] permitting calculation of independent equilibrium structures. There is also a

72

J. Demaison et al. / Chemical Physics 260 (2000) 65±81

Table 5  and angles Ab initio structures of tri¯uoramine (distances in A in degree) Method re [39] re [40] MP2(ae) MP2(fc) MP2(ae) MP2(ae) MP2(ae) MP2(ae) CCSD(T)(ae) CCSD(T)(ae) CCSD(T)(ae) CCSD(T)(ae) re a CCSD(T)(ae) CCSD(T)(ae) CCSD(T)(ae) re c (best estimate)

Basis set

cc-pCVQZ cc-pCVQZ cc-pVTZ aug-cc-pVTZ cc-pVQZ aug-cc-pVQZ cc-pVDZ cc-pVTZ cc-pVQZ cc-pV1Z aug-cc-pVDZ aug-cc-pVTZb aug-cc-pVQZb aug-cc-pV1Z

r(N±F)

\(FNF)

1.365(2) 1.3670 1.3620 1.3643 1.3633 1.3647 1.3618 1.3618 1.3836 1.3677 1.3659 1.3657 1.3661 1.3913 1.3691 1.3658 1.3653 1.3655

102.367(33) 101.945 101.996 101.951 102.036 101.794 101.998 101.959 101.930 101.977 101.945 101.943 101.577 101.735 101.906 101.904

a CCSD(T)(ae)/cc-pV1Z with core correlation correction estimated at the MP2 level. b Contribution of the di€use functions calculated at the MP2 level. c CCSD(T)(ae)/aug-cc-pV1Z with core correlation correction estimated at the MP2 level.

very recent semiempirical determination of the equilibrium structure, the so-called rm structure [47], see Table 6. Strictly speaking, these di€erent structures are not compatible. However, this is mainly due to an underestimation of the uncertainties. If we assume that the accuracy is not  the agreement is satisfactory. better than 0.0005 A, The results of our new ab initio calculations are also presented in Table 6. The N@C bond length is not fully converged at the quadruple f level in the fc approximation and there is a small e€ect of the di€use functions on the N atom, but this e€ect decreases with the size of the basis set and becomes vanishingly small after extrapolation to in®nite basis size. When all electrons are correlated, the convergence of the cc-pVnZ basis set is not monotonic for the N@C bond. It is worth noting that calculating the e€ect of the core correlation either at the MP2 level or at the CCSD(T) level gives almost identical results, as anticipated. Our ®nal structure is quite close to the previous ones,

but it nevertheless indicates that the accuracy of the experimental equilibrium structure [41] is not as high as quoted. 3.6. Nitrosyl hydride We have recently determined the ab initio bond angle of HNO [40]. We report here more complete results. HNO is very interesting because it is a very light and nonrigid molecule with electronegative atoms. For these reasons, the previous determinations of the structure are inaccurate. The results of the ab initio calculations are presented in Table 7. The CCSD(T)/cc-pV(5,Q)Z results indicate that convergence is almost achieved at the quadruple f level. For the N±H bond, there is a small increase of the length when going from cc-pVQZ to ccpV(5,Q)Z. A similar behaviour was found for NH3 (see Table 2) and it is probably an artefact of the mixed cc-pV(5,Q)Z basis set. Extrapolation to in®nite basis size using the double, triple and quadruple f results give absurd results for the N@O bond length and the \(HNO) angle (underlined values in Table 7). This is easily explained by noticing that the convergence is not exponential, the variations from cc-pVDZ to cc-pVTZ and from ccpVTZ to cc-pVQZ being of the same order of magnitude. The value of the T1 diagnostic, T1 ˆ 0:0134, indicates that the CCSD(T) method should perform well [38]. Furthermore, extrapolation using the triple, quadruple, and quintuple f results gives signi®cantly better results. This indicates that the extrapolation method cannot be used without precaution and that it is always better to use basis sets as large as possible. The equilibrium structure is obtained, either from the CCSD(T)/ccpVQZ+aug(N,O) (or from the cc-pV(5,Q)Z+ aug(O)) with a correction for core correlation or from the CCSD(T)(ae)/cc-pCVQZ+aug(N,O). The di€erent methods give compatible results and it is to be noted that the e€ect of the di€use functions is small. 3.7. Hydrazoic acid The rotational spectrum of HN3 has been studied several times (see Ref. [48] for a list of references) and an accurate substitution structure

J. Demaison et al. / Chemical Physics 260 (2000) 65±81

73

Table 6  and angles in degree) Ab initio structures of hydrogen isocyanide (distances in A Method

Basis set

fc approximation r(H±N)

r(N@C)

r(H±N)

r(N@C)

MP2 MP2 MP2 CCSD(T) CCSD(T)

cc-pVQZ cc-pVQZ+aug(N) cc-pCVQZ cc-pCVQZ cc-pVDZ cc-pVTZ cc-pVQZ cc-pV1Z re cc-pVDZ+aug(N) cc-pVTZ+aug(N) cc-pVQZ+aug(N) cc-pV1Z+aug(N) re c

0.9959 0.9964 0.9960 0.9962 1.0075 0.9961 0.9962 0.9962 0.9952a 1.0084 0.9978 0.9966 0.9964 0.9954a

1.1731 1.1736 1.1729 1.1718 1.1916 1.1755 1.1720 1.1711 1.1686a 1.1928 1.1761 1.1724 1.1713 1.1688a

0.9939

1.1703

0.9950 0.9952 1.0068 0.9932 0.9941

1.1702 1.1693 1.1909 1.1712 1.1693 1.1691 1.1692b 1.1921 1.1719 1.1697 1.1694 1.1692b

0.9956 0.9940(8) 0.9953(11) 0.996064(3) 0.9954

1.1686 1.1689(2) 1.1686(3) 1.168351(2) 1.1688

CCSD(T)

Previous structures re (mixed) [45] re (from force cubic ®eld) [44] rm [47] re [41] Best estimatea;c

All electrons correlated

0.9952b 1.0076 0.9949 0.9945 0.9945 0.9956b

a

CCSD(T)(fc)/cc-pV1Z+CCSD(T)(ae)/cc-pCVQZ-CCSD(T)(fc)/cc-pCVQZ. CCSD(T)(ae)/cc-pV1Z+MP2(ae)/cc-pCVQZ-MP2(ae)/cc-pVQZ. c Di€use functions on N included. b

has been determined [49]. There are also several ab initio studies of this molecule. Particularly, the potential energy function has recently been calculated using either the CCSD(T) method or various density functional approaches [50]. The problem is that there are rather large discrepancies between the experimental rs values and the ab initio results as shown in Table 1 of Ref. [50], and in Table 8. The ab initio structure of HN3 is reported in Table 8. The basis set convergence is monotonic and, furthermore, achieved at the cc-pVQZ level. At the quadruple f level, the e€ect of the di€use functions (on N) increases the N±H and N±Nc  and decreases the bond lengths by about 0.001 A \(NNN) angle by about 0.17°. At extrapolated basis set limit, the e€ect of di€use functions is almost negligible for the bond lengths but reduces the \(NNN) bond angle by as much as 0.29°. Taking correctly into account the e€ect of the core correlation shows that the result for the cc-pVQZ basis set, all electrons being correlated, is too small

 for the N±H and N±Nc bonds. It by about 0.001 A is to be noted that the ab initio structure is in very good agreement with the substitution structure. 3.8. Isocyanic acid As HNCO is a quasilinear molecule, its rotational spectrum exhibits several peculiarities. For this reason, it has been extensively studied and a substitution structure [51] as well as an average structure [52] have been determined. This molecule has also been extensively studied ab initio and, recently, an ab initio anharmonic force ®eld was calculated permitting estimation of the equilibrium rotational constants from the experimental ground state constants and derivation of a semiexperimental equilibrium structure [53]. However, these results are not yet completely satisfactory and it was recently conjectured that the value of the \(HNC) angle, 123.3(2)° [40] could be inaccurate.

74

J. Demaison et al. / Chemical Physics 260 (2000) 65±81

Table 7  and CCSD(T) structures of nitrosyl hydride (distances in A angles in degree)a Basis set

r(N±H)

r(N@O)

\(HNO)

cc-pVDZ(fc) cc-pVTZ(fc) cc-pVQZ(fc) cc-pV(5,Q)Z (fc) cc-pV1Z re b aug0 -cc-pVDZ(fc) aug0 -cc-pVTZ(fc) aug0 -cc-pVQZ(fc) cc-pV(5,Q)Z+aug(O)(fc)c aug0 -cc-pV1Z re b cc-pCVDZ(ae) cc-pCVTZ(ae) cc-pCVQZ (ae) cc-pCVQZ(fc) cc-pV1Z(ae) aug0 -cc-pCVQZ(ae) cc-pVDZ(ae) cc-pVTZ(ae) cc-pVQZ(ae) aug0 -cc-pVQZ(ae) re d Best estimatee

1.0720 1.0554 1.0535 1.0539 1.0533 1.0526 1.0655 1.0556 1.0537 1.0540

1.2196 1.2144 1.2107 1.2099 1:2016 1.2079 1.2259 1.2152 1.2111 1.2101 1:2086 1.2081 1.2184 1.2119 1.2085 1.2105 1:2050 1.2086 1.2191 1.2103 1.2082 1.2083 1.2085 1.2086

107.66 107.86 108.01 108.09 108:43 108.16 107.71 108.01 108.11 108.10 108.16 108.17 107.69 107.93 108.08 108.01 108:34 108.18 107.68 108.03 108.09 108.19 108.18 108.18

1.0527 1.0710 1.0542 1.0524 1.0537 1.0522 1.0528 1.0710 1.0523 1.0510 1.0514 1.0527 1.0528

a

aug0 means di€use functions on the N and O atoms. cc-pV(5,Q)Z ‡ core correlation calculated with the cc-pCVQZ basis set. c E€ect of the di€use functions on O calculated at the MP2 level. d aug0 -cc-pVQZ ‡ core correlation correction. e CCSD(T)(ae)/aug0 -cc-pCVQZ. b

For this reason, the structure of this molecule has been reinvestigated. The ab initio structure of HNCO was calculated in the same way as for the ¯uoramines and HN3 (see preceding sections) and the results are reported in Table 9. The convergence is monotonic and almost achieved at the cc-pVQZ level, except for the r(C@O) bond length which decreases by  when going from cc-pVQZ to cc-pV1Z. 0.001 A At the cc-pV1Z level, the e€ect of di€use functions on N and O is extremely small, except for the \(HNC) angle which increases by 0.15°. The exaggeration of the core correlation by the cc-pVQZ  for the N±H bond length basis set is only 0.001 A and still smaller for the other bonds, but it is as large as ÿ0.4° for the \(HNC) angle. It has already been found that the e€ect of core correlation

on angles may be important for nonrigid molecules such as HNCO [40]. The ®nal ab initio structure is in very pleasing agreement with the mixed structure [53] provided that the large e€ect of the core correlation is taken into account. Also to be noted is the good agreement with the semi…2† empirical rm structure [47], except for the N±H bond length. This deviation is not surprising because, in order to determine this bond length, a rather arbitrary Laurie correction (variation of the bond length upon deuteration) had to be taken into account. 3.9. cis-Thionylimide There are few studies of HNSO. In 1969, Kirchho€ [54] analyzed the microwave spectra of ®ve isotopic species and determined a substitution structure. Later, Dal Borgo et al. [55] improved the accuracy of the rotational constants of the parent and deuterated species and determined a new substitution value for the N±H bond length. The structure and the harmonic force ®eld were also calculated ab initio, but only with very modest basis sets [56]. Quite recently, the structure of this molecule was determined using the CCSD(T) method as well as several experimental methods q …2† , rm , . . .) [57]. The equilibrium N±H bond (rZ , rm  length was estimated to be 1.0203 A. 3.10. Iminosilicon We complete this review on closed-shell neutral molecules with two further molecules whose structures are accurately known. The unstable species HNSi has been extensively studied because of its astrophysical interest. Particularly, an anharmonic force ®eld was calculated ab initio and used to correct the ground state rotational constants of HN28 Si, HN29 Si and HN30 Si which permitted derivation of an accurate mixed equilibrium structure [58]. The correction Be ÿB0 was calculated variationally as well as by means of standard perturbation theory. Both methods are found to give very similar results. Later, the emission spectra of fundamental and hot vibration±rotation bands were observed by high resolution Fourier transform spectroscopy [59]. The derived equilib-

J. Demaison et al. / Chemical Physics 260 (2000) 65±81

75

Table 8  and angles in degree) Ab initio structures of hydrazoic acid, HNNc Nt (distances in A Method rs [49] re [50] MP2(ae) MP2(fc) MP2(ae) MP2(fc) MP2(fc) CCSD(T)(ae) CCSD(T)(ae) CCSD(T)(ae) CCSD(T)(ae) re b

re b;d

Basis

cc-pCVQZ cc-pCVQZ cc-pVQZ cc-pVQZ aug0 -cc-pVQZa cc-pVDZ cc-pVTZ cc-pVQZ cc-pV1Z aug0 -cc-pVDZa aug0 -cc-pVTZc aug0 -cc-pVQZc aug0 -cc-pV1Z Preferred value

r(N±H)

r(N@Nc )

r(Nc @Nt )

\(NNN)

\(NNH)

1.015(15) 1.0176 1.0148 1.0162 1.0136 1.0161 1.0168 1.0292 1.0149 1.0144 1.0144 1.0156 1.0278 1.0169 1.0152 1.0149 1.0162

1.243(5) 1.2477 1.2355 1.2385 1.2344 1.2386 1.2393 1.2566 1.2415 1.2403 1.2401 1.2412 1.2638 1.2419 1.2408 1.2408 1.2419

1.134(2) 1.1357 1.1401 1.1423 1.1398 1.14244 1.1427 1.1505 1.1326 1.1301 1.1298 1.1300 1.1513 1.1320 1.1302 1.1300 1.1302

171.3(50) 171.43 171.560 171.487 171.723 171.522 171.355 171.174 171.890 171.927 171.929 171.766 170.127 171.524 171.635 171.644 171.481

108.8(40) 108.06 110.264 109.968 110.381 109.976 110.164 107.534 108.749 109.020 109.097 108.980 107.925 109.034 109.217 109.253 109.136

a

Di€use functions on N. CCSD(T)(ae)/cc-pV1Z-MP2(ae)/cc-pVQZ+MP2(ae)/cc-pCVQZ. c E€ect of di€use functions on N calculated at MP2 level. d E€ect of di€use functions on N taken into account. b

Table 9  and angles in degree) Ab initio structures of isocyanic acid, HNCO (distances in A Method rs [51] rZ [52] re [53] …2† rm [47] MP2(ae) MP2(fc) MP2(ae) MP2(fc) MP2(fc) CCSD(T)(ae)

re b CCSD(T)(ae)

re b;d a

Basis set

a

cc-pCVQZ cc-pCVQZ cc-pVQZ cc-pVQZ aug0 -cc-pVQZ cc-pVDZ cc-pVTZ cc-pVQZ cc-pV1Z aug0 -cc-pVDZ aug0 -cc-pVTZc aug0 -cc-pVQZc aug0 -cc-pV1Z Best estimate

r(N±H)

r(N@C)

r(C@O)

\(HNC)

\(NCO)

0.995(6)

1.214(2)

1.1664(8)

1.003(2)

1.2145(6)

1.1634(4)

123.9(17) 124.0(1) 123.3(2)

172.6(27) 172.1(1) 172.2(2)

1.013(2) 1.0017 1.0031 1.0004 1.0030 1.0036 1.0173 1.0027 1.0018 1.0017 1.0031 1.0161 1.0040 1.0025 1.0023 1.0037

1.2130(3) 1.2135 1.2168 1.2129 1.2169 1.2174 1.2360 1.2165 1.2140 1.2137 1.2143 1.2371 1.2169 1.2144 1.2141 1.2148

1.1637(2) 1.1673 1.1694 1.1672 1.1696 1.1701 1.1777 1.1668 1.1640 1.1630 1.1630 1.1803 1.1667 1.1644 1.1640 1.1641

123.4(2) 124.752 124.140 125.149 124.221 124.411 119.276 123.141 123.608 123.672 123.275 120.628 123.359 123.769 123.841 123.444

172.6(3) 172.088 171.969 172.098 171.968 171.929 171.855 172.268 172.388 172.437 172.427 171.916 172.254 172.357 172.403 172.393

aug0 means di€use functions on N and O. CCSD(T)(ae)/cc-pV1Z-MP2(ae)/cc-pVQZ+MP2(ae)/cc-pCVQZ. c E€ect of the di€use functions calculated at the MP2 level. d E€ect of the di€use functions included. b

76

J. Demaison et al. / Chemical Physics 260 (2000) 65±81

rium rotational constants were found to be in good agreement with the ab initio ones of Ref. [58] con®rming the accuracy of the structure. The ®nal value of the N±H bond length is: re …N±H† ˆ  (mean value). 1:00015 A 3.11. trans-Diazene trans-Diazene was recently extensively studied. Particularly, an experimental rZ structure was calculated using new experimental ground state rotational constants of N2 H2 , N2 D2 , and N2 DH and a newly re®ned harmonic force ®eld [60]. This structure was used to estimate an approximate, but accurate, re structure. In parallel, the structure was calculated at the CCSD(T) level using basis sets of quadruple f quality or using the ab initio anharmonic force ®eld to determine a mixed structure [61]. The di€erent results are in good agreement and the best value of the equilibrium bond length  with an accuracy likely to is re …N±H† ˆ 1:0283 A  be considerably better than 0.001 A. 3.12. Linear ions As molecular ions containing the N±H bond are of astrophysical interest, the structures of the simplest ones are accurately known, although a purely experimental equilibrium structure could be obtained only for the linear triatomic molecule: hydrogendinitrogen (1‡) ion, HN‡ 2 . The rotational spectrum of HN‡ 2 was studied by Owrutsky et al. [62] who determined the equilibrium rota‡ tional constants for HN‡ 2 and DN2 from an analysis of the three fundamental vibrational states. The equilibrium N±H bond length was given with an extremely small uncertainty (smaller  but, as only the fundamental states than 0.0005 A), were analyzed and the magnetic correction to the rotational constants was neglected, this high precision might be expected to be too optimistic. However, the accuracy of this structure was con®rmed by high level ab initio calculations [63]. An independent semiexperimental structure (mixed structure) was also obtained using the experimental ground state rotational constants and the theoretical rovibrational constants calculated from an ab initio anharmonic force ®eld [64]. It was found

to be in perfect agreement with the experimental structure. Furthermore, Kabbadj et al. [65] recently measured many hot bands and were able to determine an improved equilibrium rotational constant for HN‡ 2 . This new value is compatible with the previous one, although slightly smaller. The mixed structure of several other ions have also been calculated by Botschwina et al. The ions considered in this study are: protonated hydrogen cyanide, HCNH‡ [66], and protonated cyanogen, HNCCN‡ [67]. These structures are believed to  Larger have an accuracy of about 0.0005 A. ‡ protonated cyanopolyynes, HCn NH with n ˆ 3±9 were also studied by Botschwina et al. (see Refs. [68,69] and references therein) but they are not retained in our study because these species are similar to HCNH‡ , they are rather exotic, and, furthermore they are already quite large. 3.13. Ammonium ion, NH‡ 4 This important ion of symmetry Td has been the subject of many investigations, particularly since its gas phase infrared spectrum was detected in 1983 almost simultaneously by Crofton and Oka [70] and Schafer et al. [71] (a list of references on the spectroscopy of this molecule may be found in Ref. [72]). Nakagawa and Amano [73] estimated the Be equilibrium rotational constant of NH3 D‡ by using the a constants of CH3 D and arrived at  Crofton and Oka [74] obre …N±H† ˆ 1:021(3) A.  by assuming the same tained re …N±H† ˆ 1:0208 A r0 ÿ re as for CH4 . There are also numerous ab initio calculations on this species. Particularly relevant to this work are the results of Botschwina [75] who calculated the structure with a basis set of 117 contacted Gaussians and the CEPA-1 method. His recommended value is re …N±H† ˆ 1:0205(10)  More recently, Thomas et al. [76] calculated the A. structure at the CCSD(T)/DZP level. Their result,  is too large, as is expected for a re …N±H† ˆ 1:025 A basis set of double zeta quality. We have calculated the ab initio structure of NH‡ 4 at the CCSD(T)/cc-pVnZ (n ˆ D, T, Q) level and the results are reported in Table 10. The convergence is monotonic and, furthermore, achieved at the cc-pVQZ level. At the quadruple f level, the e€ect of the di€use functions (on N) is

J. Demaison et al. / Chemical Physics 260 (2000) 65±81 Table 10  Ab initio structures of NH‡ 4 (in A) Method

Basis set

r(N±H)

MP2(fc) MP2(fc) CCSD(T)(fc) CCSD(T)(ae) CCSD(T)(fc)

cc-pVQZ aug0 -cc-pVQZa cc-pwCVQZ cc-pwCVQZ cc-pVDZ cc-pVTZ cc-pVQZ cc-pV1Z

1.0199 1.0202 1.0214 1.0205 1.0316 1.0226 1.0214 1.0212 1.0203

re (best estimate)b a

aug0 means di€use functions on N. CCSD(T)(fc)/cc-pV1Z+CCSD(T)(ae)/cc-pwCVQZ-CCSD(T)(fc)/cc-pwCVQZ. b

 and can safely be neextremely small, 0.0003 A glected at the extrapolated basis set limit. The e€ect of core correlation is calculated at the CCSD(T) level with the newly optimized cc-pwCVQZ basis set 2 where the core±valence correlation energy is heavily weighted over the core±core correlation energy. Our ®nal result,  is in extremely good agreere …N±H† ˆ 1:0203 A ment with the previous value of Botschwina [75].

4. The N±H bond length The experimental equilibrium values of the N±H bond lengths of a few molecules are summarized in Table 11 together with the ab initio values calculated using three di€erent methods: CCSD(T) and MP2 with di€erent basis sets and B3LYP/6-311++G(3df,2pd). The mean, stan2 cc-pCVQZ basis sets were obtained from the Extensible Computational Chemistry Environment Basis Set Database, Version 1.0, as developed and distributed by the Molecular Science Computing Facility, Environmental and Molecular Sciences Laboratory which is part of the Paci®c Northwest Laboratory, P.O. Box 999, Richland, Washington 99352, USA, and funded by the US Department of Energy. The Paci®c Northwest Laboratory is a multi-program laboratory operated by Battelle Memorial Institute for the US Department of Energy under contract DE-AC06-76RLO 1830. Contact David Feller or Karen Schuchardt for further information.

77

dard deviation, median, and extremal values of the residuals are also given at the bottom of the table. The e€ect of the di€use functions (on N) is small, its largest contribution being only 0.0010  in NHF2 . The core correlation correction A (MP2(ae)/cc-pCVQZ-MP2(fc)/cc-pCVQZ) is al most constant, its median value being ÿ0.0013 A. It is interesting to note that it is smaller for the ‡ ions HN‡ 2 and HNCCN where it is only ÿ0.0008  A. When the calculations are made with the ccpVQZ basis set, all electrons being correlated, the o€set to be applied is almost constant, its value  see Table 11. being 0.0011(1) A, With the B3LYP/6-311++G(3df,2pd) method, HNO is de®nitely an outlier, the di€erence exp:  With this value omitted, ÿcalc: being ÿ0.009 A. the equilibrium values are smaller than the B3LYP  values, the mean di€erence being ÿ0.0023(9) A  and the range of di€erences only 0.0031 A. The MP2/6-31G** also gives satisfactory results, except or HNO. The mean di€erence is rather simi but the standard deviation is lar: ÿ0.0031(15) A slightly larger as well as the range: 0.0015 and  respectively. Going to the MP2/cc-pVTZ 0.0050 A, level decreases the mean di€erence but increases the standard deviation as well as the range: 0.0018  respectively with HNO omitted. and 0.0064 A, Going to the MP2/cc-pVQZ level still reduces the mean di€erence (which is now zero) but HNO is still an outlier and neither the standard deviation, nor the range signi®cantly decrease. The ®rst conclusion is that at the MP2 level of theory the modest basis set 6-31G** gives results at least as good as the larger basis sets, cc-pVTZ or even ccpVQZ. This, of course, re¯ects a cancellation of errors. Going to the CCSD(T) level signi®cantly improves the situation . With the cc-pVTZ basis set, all electrons being correlated, the mean dif the standard deviference is only ‡0.0010(12) A,  ation is rather good and the range small, 0.0038 A. Going to the cc-pVQZ level improves the situation  further: the mean deviation is still ‡0.0015(5) A but the standard deviation and the range are very  respectively and, still small: 0.0005 and 0.0019 A, more important, the mean deviation and median deviation are equal, indicating that there is no longer any outlier.

ÿ0.0028 0.0020 ÿ0.0024 ÿ0.0086 ÿ0.0002

1.0614 1.0326 1.0262 1.0365 1.0215 1.0180 1.0136 1.0150 1.0152 1.0049 0.9972 1.0004 1.0198 1.0239

B3LYPa

ÿ0.0029 0.0015 ÿ0.0028 ÿ0.0061 ÿ0.0006

1.0534 1.0321 1.0253 1.0372 1.0265 1.0193 1.0126 1.0178 1.0164 1.0047 0.9976 1.0032 1.0197 1.0228 ÿ0.0009 0.0019 ÿ0.0007 ÿ0.0057 0.0025

1.0503 1.0286 1.0223 1.0350 1.0260 1.0173 1.0111 1.0157 1.0142 1.0040 0.9960 1.0019 1.0171 1.0210

b

0.0001 0.0019 0.0000 ÿ0.0039 0.0040

1.0488 1.0272 1.0214 1.0344 1.0242 1.0161 1.0098 1.0149 1.0135 1.0030 0.9959 1.0009 1.0157 1.0199 0.0011 0.0011 0.0008 ÿ0.0003 0.0033

1.0523 1.0279 1.0197 1.0314 1.0196 1.0149 1.0113 1.0127 1.0113 1.0027 0.9932 0.9976 1.0168 1.0204 0.0016 0.0008 0.0014 0.0007 0.0038

1.0126 1.0031 0.9952

1.0115 1.0018 0.9941 0.9964 1.0160 1.0196 0.0003 0.0003 0.0003 ÿ0.0002 0.0009

1.0171 1.0204

1.0522 1.0283 1.0230 1.0331 1.0199 1.0156 1.0111

Best estimateb

1.0510 1.0271 1.0218 1.0319 1.0189 1.0144 1.0101

cc-pVQZ

cc-pVTZ

cc-pVQZ

6-31G**

cc-pVTZ

CCSD(T)(ae)

MP2(fc)

6-311++G(3df,2pd) basis set. CCSD(T)(ae)/cc-pVQZ+MP2(ae)/cc-pCVQZ-MP2(ae)/cc-pVQZ. c MP2(fc)/cc-pVQZ+aug(N)-MP2(fc)/cc-pVQZ. d MP2(ae)/cc-pCVQZ-MP2(fc)/cc-pCVQZ. e MP2(ae)/cc-pCVQZ-MP2(ae)/cc-pVQZ.

a

1.053 1.028 1.023 1.034 1.020 1.016 1.011 1.013 1.013 1.004 0.995 1.000 1.018 1.020

HNO HNNH NHF2 HN‡ 2 HNSO HN3 NH3 HNCCN‡ HCNH‡ HNCO HNC HNSi NH2 F NH‡ 4

Residuals Mean Std. dev. Median Min. Max.

re

Method

Table 11  Experimental and ab initio N±H bond lengths (in A)

0.0005 0.0002 0.0005 0.0003 0.0010

0.0004 0.0006 0.0010 0.0004 0.0003 0.0007 0.0003 0.0004 0.0003 0.0006 0.0005 0.0007 0.0005 ÿ0.0007

Di€usec

ÿ0.0012 0.0002 ÿ0.0013 ÿ0.0014 ÿ0.0008

ÿ0.0013 ÿ0.0010

ÿ0.0013 ÿ0.0013 ÿ0.0013 ÿ0.0008 ÿ0.0013 ÿ0.0014 ÿ0.0012 ÿ0.0008 ÿ0.0012 ÿ0.0014 ÿ0.0010

Core correlationd

0.0011 0.0001 0.0011 0.0010 0.0013

0.0011 0.0009

0.0013 0.0012 0.0012 0.0012 0.0010 0.0012 0.0010 0.0011 0.0011 0.0013 0.0011

Core correlatione

78 J. Demaison et al. / Chemical Physics 260 (2000) 65±81

J. Demaison et al. / Chemical Physics 260 (2000) 65±81

The standard deviation gives a rough estimate  of the accuracy of each method. A goal of 0.002 A seems relatively easy to reach. The CCSD(T) method can give a signi®cantly better accuracy  at least when basis set con(better than 0.001 A), vergence is achieved and when the e€ect of the di€use functions and the core correlation correction are taken into account. It is tempting to check whether open-shell molecules follow the same behaviour, but reliable equilibrium structures have been determined for very few of them (NH, NH2 , and NH‡ 2 ). Furthermore, a calculation with the UMP2/6-31G** and B3LYP/6-311++G(3df,2pd) methods indicate that the o€set is not the same as for the closed-shell molecules and that it doesn't seem to be constant. Thus, before drawing any conclusion, it would be necessary to determine the equilibrium structures of more open-shell molecules. It is interesting to check whether there is a correlation between the re (N±H) bond length and the corresponding isolated stretching frequency as has been found for the C±H and Si±H bonds [77,78]. The isolated frequency is taken from the infrared spectrum of the isotopic species where all H but one have been replaced by D (e.g. ND2 H for NH3 ). This is not sucient to warrant that the stretching frequency is isolated because some Fermi resonances are still possible (for instance, in the particular case of DNNH, strong Fermi resonances may exist between the vibrational states m1 ˆ 1 and m3 ˆ 2, m3 ˆ m4 ˆ 1, and m4 ˆ 2. In addition, a Darling±Dennison resonance may be present between m1 ˆ 1 and m5 ˆ 3 [79]) but this is enough for a rough check. Taking into account the small number of experimental data summarized in Table 12, there seems to be a rather good correlation between the re (N±H) bond length and the corresponding isolated stretching frequency, see Fig. 1. It is interesting to note that there are actually two correlations, one for the neutral molecules and another one for the cations. This is not surprising because in the cations the bonding differs from all other species as in HCO‡ [2] because their lowest dissociation channels are di€erent. The two open-shell molecules for which the isolated stretching frequency may be estimated (NH and NH2 ) are also included in this correlation.

79

Table 12  Isolated stretching frequency (cmÿ1 ) and equilibrium length (A) for the N±H bond Molecule

m(N±H)

Ref.

re (N±H)

HNO HNNH NH NHF2 HN‡ 2 NH2 HNSO HN3 HND2 HNCCN‡ DCNH‡ HNCO HNSi HNC

2684.0 3109.6 3125.6 3193.0 3234.0 3258.2 3308.5 3339.9 3403.8 3448.3 3473.1 3533.0 3588.4 3652.7

[95] [79] [6] [96] [97] [7] [98] [99] [100] [101] [102] [52] [59] [103]

1.053 1.028 1.036 1.023 1.034 1.025 1.020 1.016 1.011 1.013 1.013 1.004 1.000 0.995

Fig. 1. Correlation between isolated stretching frequencies,  m(N±H) (cmÿ1 ) and equilibrium distances re (N±H) (A).

Their behaviour seems to be close to this of the cations.

Acknowledgements We acknowledge the Robert A. Welch Foundation for support of the work done in Austin. The Institut du Developpement et des Ressources en Informatique Scienti®que (CNRS) is thanked for a grant in computer time. We also thank K.A. Peterson and T.H. Dunning, Jr., for providing the cc-pwCVQZ basis sets prior to publication.

80

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References [1] G. Graner, E. Hirota, T. Iijima, K. Kuchitsu, D.A. Ramsay, J. Vogt, N. Vogt, in: K. Kuchitsu (Ed.), Structure Data of Free Polyatomic Molecules, LandoltB ornstein, Numerical Data and Functional Relationships in Science and Technology (New series), group II, vol. 25, Springer, Berlin, 1999. [2] J. Demaison, G. Wlodarczak, Struct. Chem. 5 (1994) 57. [3] V. Spirko, W.P. Kraemer, J. Mol. Spectrosc. 133 (1989) 331. [4] W.S. Benedict, E.K. Plyler, Can. J. Phys. 35 (1957) 1235. [5] T.J. Lee, G.E. Scuseria, in: S.R. Langho€ (Ed.), Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy, Kluwer, Dordrecht, 1995, p. 47. [6] J.M.L. Martin, Chem. Phys. Lett. 292 (1998) 411. [7] W. Gabriel, G. Chambaud, P. Rosmus, S. Carter, N.C. Handy, Mol. Phys. 81 (1994) 1445. [8] P. Jensen, R.J. Buenker, G. Hirsch, S.N. Rai, Mol. Phys. 70 (1990) 443. [9] K. Kobayashi, H. Ozeki, S. Saito, M. Tonooka, S. Yamamoto, J. Chem. Phys. 107 (1997) 9289. [10] G. Osmann, P.R. Bunker, P. Jensen, W.P. Kraemer, J. Mol. Spectrosc. 186 (1997) 319. [11] J.F. Stanton, J. Gauss, J.D. Watts, W.J. Lauderdale, R.J. Bartlett, Int. J. Quantum Chem. Symp. 26 (1992) 879. [12] M.J. Frisch, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G. Johnson, M.A. Robb, J.R. Cheeseman, T. Keith, G.A. Petersson, J.A. Montgomery, K. Raghavachari, M.A. Al-Laham, V.G. Zakrzewski, J.V. Ortiz, J.B. Foresman, J. Cioslowski, B.B. Stefanov, A. Nanayakkara, M. Challacombe, C.Y. Peng, P.Y. Ayala, W. Chen, M.W. Wong, J.L. Andres, E.S. Replogle, R. Gomperts, R.L. Martin, D.J. Fox, J.S. Binkley, D.J. Defrees, J. Baker, J.P. Stewart, M. Head-Gordon, C. Gonzalez, J.A. Pople, G A U S S I A N 9 4 , Inc., Revision E.2, Gaussian, Inc., Pittsburgh PA, 1995. [13] C. Hampel, K.A. Peterson, H.-J. Werner, Chem. Phys. Lett. 190 (1992) 1. [14] C. Mùller, M.S. Plesset, Phys. Rev. 46 (1934) 618. [15] G.D. Purvis III, R.J. Bartlett, J. Chem. Phys. 76 (1982) 1910. [16] K. Raghavachari, G.W. Trucks, J.A. Pople, M. HeadGordon, Chem. Phys. Lett. 157 (1989) 479. [17] T. Helgaker, J. Gauss, P. Jùrgensen, J. Olsen, J. Chem. Phys. 106 (1997) 6430. [18] J.M.L. Martin, Chem. Phys. Lett. 242 (1995) 343. [19] D.A. Feller, K.A. Peterson, J. Chem. Phys. 108 (1998) 154. [20] K.A. Peterson, T.H. Dunning Jr., J. Chem. Phys. 106 (1997) 4119. [21] A. Halkier, P. Jùrgensen, J. Gauss, T. Helgaker, Chem. Phys. Lett. 274 (1997) 235. [22] D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 103 (1995) 4572.

[23] D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 99 (1993) 1914. [24] L. Margules, J. Demaison, J.E. Boggs, J. Phys. Chem. A 103 (1999) 7632. [25] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007. [26] L. Margules, J. Demaison, J.E. Boggs, Struct. Chem. 11 (2000) 145. [27] B.J. Persson, P.R. Taylor, J.M.L. Martin, J. Phys. Chem. A 102 (1998) 2483. [28] K.A. Peterson, R.A. Kendall, T.H. Dunning Jr., J. Chem. Phys. 99 (1993) 1930. [29] C.W. Bauschlicher, A. Ricca, H. Partridge, S.R. Langho€, in: D.P. Chong (Ed.), Recent Advances in Density Functional Methods, Word Scienti®c, Singapore, 1997, p. 165. [30] R.G. Parr, W. Yang, Density Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989. [31] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [32] P.J. Stephens, F.J. Devlin, C.F. Chabalowski, M.J. Frisch, J. Phys. Chem. 98 (1994) 11623. [33] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [34] K.A. Peterson, S.S. Xantheas, D.A. Dixon, T.H. Dunning Jr., J. Phys. Chem. A 102 (1998) 2449. [35] D. Christen, R. Minkwitz, R. Nass, J. Amer. Chem. Soc. 109 (1987) 7020. [36] H.-G. Mack, D. Christen, H. Oberhammer, J. Mol. Struct. 190 (1988) 215. [37] D.R. Lide, J. Chem. Phys. 38 (1963) 456. [38] T.J. Lee, P.R. Taylor, Int. J. Quant. Chem. Symp. 23 (1989) 199. [39] M. Otake, C. Matsumura, Y. Morino, J. Mol. Spectrosc. 28 (1968) 316. [40] L. Margules, J. Demaison, J.E. Boggs, J. Mol. Struct. Theochem. 500 (2000) 245. [41] T. Okabayashi, M. Tanimoto, J. Chem. Phys. 99 (1993) 3268. [42] J. Demaison, G. Wlodarczak, H.D. Rudolph, in: I. Hargittai, M. Hargittai (Eds.), Advances in Molecular Structure Research, vol. 3, JAI Press, Greenwich, 1997, p. 1. [43] S.M. Cybulski, D.M. Bishop, J. Chem. Phys. 106 (1997) 4082. [44] R.A. Creswell, A.G. Robiette, Mol. Phys. 36 (1978) 869. [45] P. Botschwina, S. Seeger, M. Horn, J. Fl ugge, M. Oswald, M. Mladenovic, U. H oper, R. Oswald, K. Schick, in: I. Nenner (Ed.), Proceedings of the Meeting on Physical Chemistry of Molecules and Grains in Space, Conference Proceedings No. 312, American Institute of Physics, New York, 1994, p. 321. [46] T.J. Lee, C.E. Dateo, B. Gadzy, J.M. Bowman, J. Phys. Chem. 97 (1998) 8937. [47] J.K.G. Watson, A. Roytburg, W. Ulrich, J. Mol. Spectrosc. 196 (1999) 102. [48] M.C.L. Gerry, N. Heineking, H. M ader, H. Dreizler, Z. Naturforsch 44a (1989) 1079. [49] B.P. Winnewisser, J. Mol. Spectrosc. 82 (1980) 220.

J. Demaison et al. / Chemical Physics 260 (2000) 65±81 [50] M. Rosenstock, P. Rosmus, E.-A. Reinsch, O. Treutler, S. Carter, N.C. Handy, Mol. Phys. 93 (1998) 853. [51] K. Yamada, J. Mol. Spectrosc. 79 (1980) 323. [52] L. Fusina, I.M. Mills, J. Mol. Spectrosc. 86 (1981) 488. [53] A.L.L. East, C.S. Johnson, W.D. Allen, J. Chem. Phys. 98 (1993) 1299. [54] W.H. Kirchho€, J. Amer. Chem. Soc. 91 (1969) 2437. [55] A. Dal Borgo, G. Di Lonardo, F. Scappini, Chem. Phys. Lett. 63 (1979) 115. [56] K. Raghavachari, J. Chem. Phys. 76 (1982) 3668. [57] L. Margules, J. Demaison, J.E. Boggs, H.D. Rudolph, in press. [58] P. Botschwina, M. Tommek, P. Sebald, M. Bogey, C. Demuynck, J.L. Destombes, A. Walters, J. Chem. Phys. 95 (1991) 7769. [59] M. Elhanine, B. Hanoune, G. Guelachvili, J. Chem. Phys. 99 (1993) 4970. [60] J. Demaison, F. Hegelund, H. B urger, J. Mol. Struct. 413± 414 (1997) 447. [61] J.M.L. Martin, P.R. Taylor, Mol. Phys. 96 (1999) 681. [62] J.C. Owrutsky, C.S. Gudeman, C.C. Martner, L.M. Tack, N.H. Rosenbaum, R.J. Saykally, J. Chem. Phys. 84 (1986) 605.  Heyl, R. Oswald, [63] P. Botschwina, M. Oswald, J. Fl ugge, A. Chem. Phys. Lett. 209 (1993) 117. [64] S. Schmatz, Dissertation, G ottingen, 1996. [65] Y. Kabbadj, T.R. Huet, B.D. Rehfuss, C.M. Gabrys, T. Oka, J. Mol. Spectrosc. 163 (1994) 180.  Heyl, M. Horn, J. Fl [66] P. Botschwina, A ugge, J. Mol. Spectrosc. 163 (1994) 127. [67] P. Botschwina, J. Fl ugge, S. Seeger, J. Mol. Spectrosc. 157 (1993) 494.  Heyl, Mol. Phys. 97 (1999) 209. [68] P. Botschwina, A.  Heyl, P. Botschwina, T. Hirano, J. Chem. Phys. 107 [69] A. (1997) 9702. [70] M.W. Crofton, T. Oka, J. Chem. Phys. 79 (1983) 3157. [71] E. Schafer, M.H. Begemann, C.S. Gudeman, R.J. Saykally, J. Chem. Phys. 79 (1983) 3159. [72] J. Park, C. Xia, S. Selby, S.C. Foster, J. Mol. Spectrosc. 179 (1996) 150. [73] T. Nakagawa, T. Amano, Can. J. Phys. 64 (1986) 1356. [74] M.W. Crofton, T. Oka, J. Chem. Phys. 86 (1987) 5983. [75] P. Botschwina, J. Chem. Phys. 87 (1987) 1453. [76] J.R. Thomas, B.J. DeLeeuw, G. Vacek, H.F. Schaefer III, J. Chem. Phys. 98 (1993) 1336. [77] D.C. McKean, Chem. Soc. Rev. 7 (1978) 399.

81

[78] J.L. Duncan, J.L. Harvie, D.C. McKean, J. Mol. Struct. 145 (1986) 225. [79] F. Hegelund, H. B urger, O. Polanz, J. Mol. Spectrosc. 181 (1997) 151. [80] D.M. Dennison, Rev. Mod. Phys. 12 (1940) 175. [81] G. Herzberg, in: D. van Nostrand (Ed.), Infrared and Raman Spectra of Polyatomic Molecules, New York, 1945. [82] M.T. Weiss, M.P.W. Strandberg, Phys. Rev. 83 (1951) 567. [83] G. Erlandsson, W. Gordy, Phys. Rev. 106 (1957) 513. [84] J.L. Duncan, I.M. Mills, Spectrochim. Acta 20 (1964) 523. [85] K. Kuchitsu, J.P. Guillory, L.S. Bartell, J. Chem. Phys. 49 (1968) 2488. [86] P. Helminger, W. Gordy, Phys. Rev. 188 (1969) 100. [87] P. Helminger, F.C. De Lucia, W. Gordy, J. Mol. Spectrosc. 39 (1971) 94. [88] A.F. Krupnov, L.I. Gershtein, V.G. Shustrov, Opt. Spektr. 30 (1971) 790. [89] A.R. Hoy, I.M. Mills, G. Strey, Mol. Phys. 24 (1972) 1265. [90] P. Helminger, F.C. De Lucia, H.W. Morgan, P.A. Staats, Phys. Rev. A 9 (1974) 12. [91] M.D. Marshall, J.S. Muenter, J. Mol. Spectrosc. 85 (1981) 322. [92] E.A. Cohen, H.M. Pickett, J. Mol. Spectrosc. 93 (1982) 83. [93] J.M.L. Martin, T.J. Lee, P.R. Taylor, J. Chem. Phys. 97 (1992) 8361. [94] J.M.L. Martin, private communication. [95] J.W.C. Johns, A.R.W. McKellar, E. Weinberger, Can. J. Phys. 61 (1983) 1106. [96] J.J. Comeford, D.E. Mann, L.J. Schoen, D.R. Lide, J. Chem. Phys. 38 (1963) 461. [97] C.S. Gudeman, M.H. Begemann, J. Pfa€, R.J. Saykally, J. Chem. Phys. 78 (1983) 5837. [98] M. Carlotti, G. Di Lonardo, G. Galloni, A. Trombetti, J. Mol. Spectrosc. 84 (1980) 155. [99] A.S.C. Cheung, A.J. Merer, J. Mol. Spectrosc. 127 (1988) 509. [100] A. Louteiller, J.P. Perchard, J. Mol. Struct. 198 (1989) 51. [101] H.E. Warner, T. Amano, J. Mol. Spectrosc. 145 (1991) 66. [102] T. Amano, J. Chem. Phys. 81 (1984) 3350. [103] J.B. Burkholder, A. Sinha, P.D. Hammer, C.J. Howard, J. Mol. Spectrosc. 126 (1987) 72.