Experimental and theoretical spin-spin coupling ...

MOLECULAR PHYSICS, 2002, VOL. 100, N O. 17, 2807±2814

Experimental and theoretical spin±spin coupling constants for ‰15NŠ formamide M. J. HANSEN, M. A. WENDT, F. WEINHOLD and T. C. FARRAR* Department of Chemistry and Theoretical Chemistry Institute, University of Wisconsin-Madison, 1101 University Avenue, Madison, Wisconsin 53706, USA (Received 30 May 2001; revised version accepted 5 March 2002) The 15 spin±spin coupling constants (SSCCs) of formamide have been calculated using a new method that combines ®nite perturbation theory (FPT) and density functional theory (DFT). To test the reliability of this method, di€ erent hybrid density functionals and basis sets were used to calculate the Fermi contact (FC) term which was then used to calculate the SSCCs. All three of the hybrid density functionals (B3LYP, B1LYP, and MPW1PW91) gave very similar results; however, the SSCCs changed markedly with basis set. For the standard basis sets, (3-21G, 6-31G, 6-31‡G* and 6-311‡‡G**) the agreement with experiment is improved as the core orbital description is improved, but the basis set convergence is slow. To improve the basis set convergence, new basis sets were constructed with decontracted s functions. These basis sets showed much faster convergence for the FC term. To test the accuracy of the DFT/ FPT method, four experimental values for the SSCCs of formamide were measured in carbon tetrachloride to closely approximate the monomer environment. Comparing this experimental value with the SSCC calculated at the largest basis set (6-311‡‡G**) resulted in a mean absolute deviation of 2.6 Hz for B3LYP, 0.8 Hz for B1LYP, and 4.4 Hz for MPW1PW91.

1. Introduction Recent advances in NMR technology have made possible the observation of spin±spin coupling constants (SSCCs) across hydrogen bonds [1±7]. With this new experimental capability, many scientists have become interested in calculating SSCCs using theoretical methods. A relatively new method for calculating SSCCs is to combine ®nite perturbation theory (FPT) and density functional theory (DFT) [8]. This method has been used successfully to calculate SSCCs for carboranes [9], and for a number of small molecules containing ®rst- and second-row elements [8, 10±12]. The goal of this work is to test this theoretical method using new experimental data that correspond to the theoretical conditions. In order to test the reliability of the DFT/FPT method, a molecule should be chosen that provides a wide range of measurable SSCCs, and formamide is a good choice for many reasons. All of the nuclei in formamide have NMR active isotopes: 1 H, 13C, 17 O, and 15 N, which lead to 15 di€ erent couplings. These couplings also sample a variety of chemical environments involving lone pairs and multiple bonds. Also, formamide has been extensively studied in the past as a model * Author for correspondence. e-mail: [email protected]. edu

for NÐH¢ ¢ ¢OÐC hydrogen bonds in peptides, and hence much experimental and theoretical data exists in the literature. Finally, formamide is a small molecule that allows exploration of a wide range of methods and basis sets at reasonable computational cost. The SSCC between two nuclei can be written as a sum of four contributions: the Fermi contact (FC) term, the paramagnetic spin±orbit (PSO) term, the diamagnetic spin±orbit (DSO) term, and the spin±dipole (SD) term. The FC term is often the dominant contribution to the SSCC [13] (especially for one-bond couplings containing light atoms), and the FC term is easy to calculate with the Gaussian 98 [14] suite of programs. The next two terms (PSO and DSO) often are equal in magnitude but opposite in sign for lighter molecules which leads to a cancellation of terms [10]. The last term, SD, is usually neglected since it is smaller than the error in the DFT calculation of the FC term [10]. For these reasons, only the FC contribution to the SSCCs is studied in this work. It has been found that to calculate SSCCs the relativistic, rovibrational and solvation e€ ects need to be included [13]. However, the relativistic e€ ects are small for light atoms (a few per cent [10]) and the rovibrational e€ ects are probably of the same order of magnitude. Solvation of a molecule will also a€ ect the SSCC. The solvent has a direct e€ ect on the SSCC by partial charge transfer or polarization of the electronic

Molecular Physics ISSN 0026±8976 print/ISSN 1362±3028 online # 2002 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/00268970210142602

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structure of the solute. In addition, solvation has an indirect e€ ect changing the molecular geometry of the solute [13], and this change in geometry can be especially evident for systems that form hydrogen bonds [15, 16]. This indirect solvent e€ ect can result in a change in the FC term because the FC term has been found to be sensitive to changes in geometry [10, 17]. For hydrogen bonded systems these solvation e€ ects can complicate the comparison of theory with experiment. To compare liquid phase experimental values with theoretical calculations, the liquid phase geometry is needed. Typically, a theoretically optimized geometry for a monomer or gas phase experimental geometry is used, since the liquid phase geometry is not known. For this study, the SSCCs of formamide were calculated using a theoretically optimized geometry. It would then be ideal to compare these theoretical SSCCs with experimental values from the same geometry. Some measurements of gas phase SSCCs have been made for formamide [18, 19]; however, these measurements were made at high temperatures. At high temperatures, the amide proton signals coalesce due to rapid rotation around the CÐN bond. Thus, to measure SSCCs for the individual amide protons, other ways to simulate the monomer environment must be found. To measure experimental values for a formamide monomer, a dilute solution of formamide in a weakly interacting solvent was used. Carbon tetrachloride was chosen for the solvent because it has a very small interaction energy with formamide (see below). In this monomer-like formamide solution, individual SSCCs for the amide protons could be observed at room temperature. These new experimental data allow for direct evaluation of the accuracy of the calculated values. To study the performance of the DFT/FPT method, di€ erent hybrid density functionals and di€ erent basis sets were compared. The calculation of SSCCs has been limited to mostly small molecules because the inclusion of electron correlation e€ ects is necessary, especially for the FC contribution [9]. Since DFT accounts for some electron correlation e€ ects [20], this method seems aptly suited to the calculation of the FC term. The basis set dependence was investigated using a set of four standard basis sets and also new ¯exible-core sets. These new basis sets are based on the work of Helgaker et al. [21], which considered improving the core ¯exibility by creating correlation consistent basis sets with decontracted s functions. These correlation consistent basis sets showed smooth convergence of the FC term, and have been used to study SSCCs across hydrogen bonds [15]. For this study, basis sets were developed with decontracted s functions, and the contraction coe cients were set to 1. Keeping the notation used by Helgaker et al., these new basis sets are referred to as

3-21G-su0, 6-31‡G*-su0, and 6-311‡‡G**-su0. The extension refers to the s functions being decontracted, and the numeral 0 re¯ects that no extra functions were added to the basis set.

2. Theoretical methods All theoretical calculations were performed with the Gausian 98 electronic structure program [14]. The formamide molecule geometry is given in table 1 and was calculated at a high level of theory (B3LYP/6311‡‡G**). This geometry was used for all of the SSCC calculations. The FC term was evaluated using spin-unrestricted DFT methods by applying a ®nite ®eld to the calculation. The FIELD keyword accomplished this for Gaussian 98. A value of 0.003 au was used for the perturbation; however the calculation of the SSCC is not sensitive to the value of the perturbation parameter using FPT [8]. The equation used to calculate the SSCC is JAB ˆ

Á

! X 16h·2b ®A ®B ¶¡1 »B·¸ …·A †h¿· j¯…rB †j¿¸ i; …1† 9 ·¸

where h is the Planck’s constant, ·b is the Bohr magneton, and ®A is the gyromagnetic ratio for nucleus A. The gyromagnetic ratios used for the calculations are 26:7519 £ 107 for 1 H, 6:7283 £ 107 for 13C, ¡2:712 £ 107 for 15 N and ¡3:6278 £ 107 for 17 O, all in units of rad T¡1 s¡1 . The FC term was calculated using open-shell calculations where the FC perturbation, ¶h¿· j¯…rA †j¿¸ i is added to the ·¸th element of the aspin matrix elements and subtracted from the b-spin elements. »B·¸ …·A † is the ·¸th element of the spin density matrix element evaluated at nucleus B. For further details of this method, see [9, 22, 23].

Table 1. Geometry of the formamide monomer calculated at B3LYP/6-311‡‡G** level of theory. Parameter CÐ N CÐ O NÐH2 NÐH3 CÐH1 OCN NCH1 CNH3 H2 NH3

¯

A /deg 1.361 AÊ 1.212 AÊ 1.007 AÊ 1.009 AÊ 1.106 AÊ 124.98 112.48 119.48 119.28

Spin±spin coupling constants for ‰15 NŠ formamide 3. Experimental methods The 1 H NMR spectra were recorded on a home-built spectrometer operating at 7.00 T using a 5 mm probe. The magnet is an eighth-order corrected solenoid manufactured by Cryomagnet Systems. The magnet and its environment are su ciently stable that no internal lock is necessary for periods up to several hours. All spectra were acquired at room temperature. ‰15 NŠformamide (98%‡, Cambridge Isotope Laboratories, Inc.) and carbon tetrachloride (99%‡, Aldrich) were used without further puri®cation. The sample tube was cleansed with nitric acid, followed by EDTA, to remove any trace metals. The tube was then rinsed with deionized water and dried in vacuo for at least 24 h before use. The sample was a saturated solution of ‰15 N]formamide in CCl4 . The concentration of this saturated solution is less than 0.1 mol % based on solubility studies performed in this laboratory. The peaks in the spectra were assigned based on the assumption that 3 JHH SSCCs from protons at 1808 dihedral angles are larger than couplings from 0 dihedral angles [24, 25].

4.

Results and discussion 4.1. DFT methods Recently, the hybrid density functional B3LYP has been shown to provide accurate SSCC results [11, 12], signi®cantly better than other non-hybrid density functionals such as BLYP [11]. In this study, the SSCCs of formamide are calculated using three di€ erent hybrid density functionals: B3LYP [20, 26], B1LYP [27, 28] and MPW1PW91 [29]. The B3LYP method uses Becke’s three-parameter functional and the non-local correlation of Lee, Yang and Parr, the B1LYP method uses Becke’s one-parameter functional and the correlation of Lee, Yang and Parr. The MPW1PW9 method uses Barone and Adamo’s Becke-style one-parameter functional, the modi®ed Perdew±Wang exchange and the Perdew±Wang 91 correlation. The calculated SSCCs are shown in ®gure 1(a-o), and each part contains data for the three di€ erent hybrid density functionals. Four of the parts also contain a horizontal line representing the experimental data. The formamide protons are labelled as shown in ®gure 2. As the ®gures show, the di€ erent hybrid density functionals produce the same general trend with respect to basis set, and the e€ ect of changing the hybrid density functional is small compared with changing the basis set. However, the range of values for the FC term increases as the basis set increases, as can be seen by comparing the range of values using the 3-21G basis set with those using the 6-311‡‡G** basis set. This means that the di€ erences in the hybrid density functionals become more apparent with large basis sets.

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In order to compare the accuracy of the three hybrid density functionals, reliable experimental values are needed. Since the geometry used for the calculations is a monomer geometry, the experimental data must also have monomer-like geometry. Therefore, the theoretical SSCCs values are compared with gas phase SSCCs where available. Unfortunately, to the best of our knowledge only three SSCC gas phase values for formamide have been published in the literature, 1 JNH , 2 JNH1 and 3 JHH [18, 19]. Most of these values were measured at elevated temperatures and so 1 JNH and 3 JHH are average values for the amide protons due to rapid rotation about the CÐN bond. Hence, the only gas phase value to compare with our calculated values is 21.5 Hz for 2 JNH1 [18]. To make approximate monomer-like comparisons for the other SSCCs of formamide, a solution of ‰15 NŠformamide in carbon tetrachloride was employed. This solvent was chosen because it has a very small interaction with formamide. To test this, the interaction energy between a monomer of formamide and a monomer of carbon tetrachloride was calculated using Gaussian 98 at the B3LYP/6-31‡G* level of theory. The interaction energy was found to be 1.5 kJ mol¡1 , less than kT at room temperature. The chemical shift has also been found to indicate the extent of interaction between solvent and solute [30]; the stronger the interaction, the larger the shift down®eld. Since the chemical shifts of the amide protons in CCl4 are far up®eld from the neat values (about 5.3 ppm compared with about 7.5 ppm [31]), CCl4 is found to have only minimal interaction with formamide. To measure the SSCCs of ‰15 NŠformamide in CCl4 , a saturated sample was used. The sample was still very dilute (less than 0.1 mol %), as was evidenced by the large number of scans needed to accumulate su cient data (25 000). From these data, three more monomerlike SSCCs for formamide were measured, 1 JNH3 ˆ 87:4 Hz, 1 JNH2 ˆ 88:2 Hz, and 3 J H1H3 ˆ 13:7 Hz. These data are shown in table 2. 2 JH2H3 and 3 JH1H2 are small couplings and could not be resolved. The relative signs of the SSCCs were measured by Randall and coworkers [32] for formamide±water solutions, and their results (shown in table 2) agree with our experimental and calculated values. The four measured dilute-solution values and the experimental gas phase value for 2 JNH1 can be used to compare the accuracy of the three density functionals. Comparing the experimental values with the SSCC using the largest standard basis set (6-311‡‡G**), the mean absolute deviations were 2.6 Hz for B3LYP, 0.8 Hz for B1LYP and 4.4 Hz for MPW1PW91. Of course these results are only for the FC term; if all four terms of the SSCC were calculated, the results might di€ er.

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Spin±spin coupling constants for ‰15 NŠ formamide

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Figure 1. DFT (B3LYP, *; B1LYP, &; MPW1PW91, ~) and basis set dependence (standard ÐÐ; decontracted ± ± ± ) for the SSCCs of formamide: (a) 1 JH1C ; (b) 1 JCO ; (c) 1 JNC ; (d) 2 JH2H3 ; (e) 1 JH3N ; ( f ) 2 JH1N ; (g) 1 JH2N ; (h) 2 JH1O ; (i) 2 JNO ; ( j) 2 JH3C ; (k) 2 JH2C ; (l) 2 JH1H2 ; (m) 3 JH1H3 ; (n) 3 JH2O and (o) 3 JH3O . The dashed horizontal lines in (e), (g), and (m) refer to values measured in dilute CCl4 solution. The dashed horizontal line in ( f ) refers to a gas phase experimental value from [18].

However, just calculating the FC term for this molecule gives reliable results with this large basis set. From the above error calculations, B1LYP gives the values closest to experiment on average; however, all three hybrid density functionals give similar results.

M. J. Hansen et al.

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Table 2. Comparison of the convergence of the decontracted basis sets versus the standard basis sets. Experimental values are also included. All couplings are in units of Hz. B3LYP/decontracted JAB

B3LYP/standard

6-31G

6-311‡‡G**

% Dev.

6-31G

6-311‡‡G**

% Dev

JH1C JNC 1 JCO 1 JH3N

162 715.5 27.3 775.7

173 716.0 16.7 780.8

6.3 3.1 63 6.3

194 718.1 11.1 785.8

178 717.7 16.0 783.6

8.8 2.6 31 2.7

1

JH2N

777.0

781.6

5.7

785.7

784.5

1.5

2

JH2H3 JH1N

4.8 718.4

5.0 719.9

5.0 7.8

2.3 719.4

4.8 720.0

53 3.2

1 1

2

2

JH1O JNO 2 JH3C 2 JH2C 3 JH1H2 3 JH1H3

76.5 1.2 70.51 5.7 0.76 12.3

77.6 1.6 70.44 6.3 0.76 12.6

14 25 14 11 0 2.0

75.8 3.2 71.3 5.5 0.76 14.6

77.5 1.1 70.51 6.0 1.0 12.6

22 200 150 8.4 25 16

3

71.5 70.34

72.1 0.55

31 38 12

70.99 0.41

72.0 70.58

50 170 51

2

JH2O JH3O Average 3

a b c

Exp.

787.4a 786.93b 788.3a 791.30b +2.30b 721.5c 714.47b

+2.10b +13.7a +13.90b

Current work; dilute formamide in CCl4 ; theoretical signs used. 15 mole % formamide in water from [32]. Gas phase values from [19].

Figure 2.

Labelling of the formamide protons.

4.2. Basis sets Relatively few basis set comparisons have been made for SSCCs; however, as SSCC calculations become more feasible, interest in the basis set requirements has increased. The FC term in particular has been shown to converge poorly with increasing basis set [13]. One reason for this may be that the basis sets generally used do not adequately describe the electron density at the nucleus. To investigate this question, four basis sets were chosen to calculate the SSCCs of formamide: 321G, 6-31G, 6-31+G*, and 6-311++G** [33]. The 321G basis set was chosen to see if qualitative agreement is possible for this small, split balance basis set. The 631G basis set was chosen to see how much improvement is gained from increasing the number of functions to describe the core and valence orbitals. The 6-31+G* basis set adds polarization and di€ use functions to the heavy atoms. Since this basis set mainly improves on

the valence orbitals only, not much improvement is expected; however, 6-31‡G* is a much used basis set in the literature. The 6-311++G** basis set is a large split valence basis set that also includes di€ use functions and polarization functions on the hydrogen atoms as well as the heavy atoms. Again this basis set improves on the valence orbitals and not on the core, and so the results should be similar to 6-31G and 6-31‡G*. The results for the basis set dependence of the FC term are shown in ®gure 1(a-o). The results for the standard contracted basis sets (solid lines) are compared with the largest contracted basis set studied, 6311‡‡G**. The decontracted basis sets (dashed line) will be discussed at the end of this section. The relatively poor convergence with basis set is evident for the four standard basis sets studied; however, some trends can be found. For almost every SSCC, the small split valence basis set, 3-21G, grossly underestimates the absolute value of the FC term. If the basis set is enlarged to 6-31G, many of the SSCCs improve dramatically, for example, 1 JH1C (®gure 1 (a)), 1JH3N (®gure 1 (e)), 2 JH1N (®gure 1 ( f )), 1 JH2N (®gure 1 (g)); 1 JNO (®gure 1 (i)), and 2 HH2C (®gure 1(k)). One of the reasons for the increased agreement with experiment may be the increased number of functions to describe

Spin±spin coupling constants for ‰15 NŠ formamide the core region (6 versus 3). Increasing the basis set further to 6-3111‡‡G* does not necessarily increase the accuracy of the SSCC where experimental values are known, for example, 1 JH3N (®gure 1 (e)) and 1 JH2N (®gure 1 (g)). Other studies [21] have found that even larger basis sets (up to quadruple zeta) may be needed for accurate results. As mentioned previously, the poor basis set convergence of the FC term has been attributed to the inability of the basis set to describe adequately the electron density near the nucleus. Helgaker et al. [21] showed that by decontracting the s functions in the correlation consistent basis sets one obtains systematically improved convergence. Therefore we also studied the e€ ect of decontracting the s functions to see if smooth convergence results. These data are presented in ®gure 1; the dashed lines are the SSCCs calculated using the decontracted basis sets and the B3LYP hybrid density functional. Comparison of the contracted versus the decontracted basis sets (both using B3LYP) shows clearly that the decontracted basis sets tend to converge much more smoothly, the only obvious exception being 1 JCO (®gure 1 (b)). This coupling constant involves two heavy nuclei bonded with a double bond and two lone pairs of the oxygen atom. Problems calculating SSCCs through multiple bonds have been reported previously, and it has been found that the accuracy of the SSCC decreases as the number of lone pairs increases [8]. Therefore the poorer convergence seen for 1 JCO may be expected. It has been suggested [8] that new exchange±correlation functionals are needed to describe more accurately the spin density for these types of couplings. With the decontraction of the s functions, the values for many of the SSCCs at 6-31G-su0 are very similar to the corresponding values for 6-311‡‡G**-su0. Table 2 shows the average relative deviations for 6-31G compared with 6-311‡‡G** and 6-31G-su0 compared with 6-311‡‡G**-su0 for all 15 couplings. If the convergence of 1 HCO is ignored, the average relative deviation decreases from 51% for the standard basis sets to 12% for the decontracted basis sets. The mean relative deviation between 6-31‡G*-su0 and 6-311‡‡G**-su0 is 7.5%. The results demonstrated clearly the more rapid convergence of the SSCCs with the decontracted basis functions. These results also show that the decontraction of the s functions is at least as important as adding di€ use functions and polarization functions. Indeed, for many of the SSCCs, the addition of di€ use and polarization functions on the heavy atoms to the 6-31G-su0 basis set (6-31‡G*-su0) does not signi®cantly increase the agreement with the largest basis set, especially for the one-bond couplings. The most notable of these cases are 1 JH1C (®gure 1 (a)), 1 JNC (®gure 1 (c)),

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1

JH2H3 (®gure 1 (d)), 1 JH3N (®gure 1 (e)), 1 JH2N (®gure 1 (g)), 2 JH3C (®gure 1 ( j)), and 3 JH1H3 (®gure 1 (m)). Since the 6-31G-su0 basis set greatly improves the agreement with the largest basis set, this type of basis set may be useful for larger biological systems. 5. Conclusion It is well known that for accurate SSCC calculations electron correlation must be included, and DFT provides a relatively inexpensive way to incorporate electron correlation. Recently, the hybrid density functional B3LYP has shown its e€ ectiveness in calculating SSCCs [11, 12], but comparisons between di€ erent hybrid functionals have not been made. In this study, three hybrid density functionals were considered: B3LYP, B1LYP, and MPW1PW91. The calculated SSCCs of formamide were compared with experimental monomer values, and the B1LYP/6-311‡‡G** method was found to give the most accurare values, with a mean absolute deviation of 0.8 Hz. However, all three hybrid density functionals gave fairly similar results. Studies on the basis set dependence of SSCCs are rare, but it is known that the FC term is especially sensitive to basis set di€ erences [13]. We have con®rmed recent ®ndings [21] that improvements are gained by decontracting the s functions, thereby allowing for a more ¯exible and accurate description of the electron density close to the nucleus. This work focused on the popular Pople-style basis sets: 3-21G, 6-31G, 6-31‡G* and 6-311‡‡G**. The 3-21G basis set underestimates nearly all of the SSCCs, primarily due to the lack of core orbital ¯exibility. By increasing the number of functions representing the core orbitals, the 6-31G basis set greatly improves the agreement with experiment. The 6-31‡G* and 6-311‡‡G** sets mainly improve the valence orbitals description and not much improvement in agreement with experiment is gained with these basis sets. The decontracted basis sets show much smoother and rapid convergence of the FC term compared with the standard basis sets. Indeed, the 6-31G-su0 basis set greatly improves the agreement with 6-311‡‡G**. Note that the molecular geometry for these calculations was calculated at a high level of theory (B3LYP/6311‡‡G**), and so the e€ ect of SSCC on molecular geometry is completely negligible. It is entirely reasonable to apply a well tested method such as B3LYP/631‡G* for the molecular geometry and a less common method such as B1LYP/6-31G-su0 for the SSCC calculations at that geometry. The present calculations test a range of SSCC values varying over about three orders of magnitude, and include one-, two-, and three-bond couplings. This wide variety of SSCC values was then used to test basis sets that range from 33 basis functions for 3-21G

Spin±spin coupling constants for ‰15 NŠ formamide

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to 108 for 6-311‡‡G**-su0. To further improve on the basis sets studied here, a more accurate description of the core orbitals is needed. Additionally, development of functionals designed to provide a ¯exible description of the electron density near the nucleus should improve the results. Finally, we advocate the use of B1LYP/6-31Gsu0 as a practical and e€ ective set for the routine calculation of SSCC values in larger molecules. We thank the National Science Foundation, Grant Number CHE-9500735, for the support of this research. M.H. and M.W. thank the National Science Foundation for ®nancial support. References [1] Dingley, A. J., and Grzesiek, S., 1998, J. Amer. chem. Soc., 120, 8293. [2] Pervushin, K., Ono, A. Ferna ndez, C., Szyperski, T., Kainosho, M., and WuÈ thrich, K., 1998, Proc. Natl Acad. Sci. USA, 95, 14147. [3] Dingley, A. J., Masse, J. E., Peterson, R. D., Barfield, M., Feignon, J., and Grzesiek, S., 1999, J. Amer. chem. Soc., 121, 6019. [4] Cordier, F., and Grzesiek, S., 1999, J. Amer. chem. Soc., 121, 1601. [5] Cornilescu, G., Hu, J.-S., and Bax, A., 1999, J. Amer. chem. Soc., 121, 2949. [6] Shenderovich, I. G., Smirnov, S. N., Denisov, G. S., Gindin, V. A., Golubev, N. S., Dunger, A., Reibke, R., Kirkepar, S., Malkina, O. L., and Limbach, H.-H., 1998, Ber. Bunsenges. phys. Chem., 102, 422. [7] Golubev, N. S., Shenderovich, I. G., Smirnov, S. N., Denisov, G. S., and Limbach, H.-H., 1999, Chem. Eur. J., 5, 492. [8] Malkin, V. G., Malkina, O. L., and Salahub, D. R., 1994, Chem. Phys. Lett., 221, 91. [9] Onak, T., Jaballas, J., and Barfield, M., 1999, J. Amer. chem. soc., 121, 2850. [10] Malkina, O. L., Salahub, D. R., and Malkin, V. G., 1996, J. chem. Phys., 105, 8793. [11] Helgaker, T., Watson, M., and Handy, N. C., 2000, J. chem. Phys., 113, 9402. [12] Sychrovsky , V., GraÈ fensten, J., and Cremer, D., 2000, J. chem. Phys., 113, 3530. [13] Helgaker, T., Jaszun ski, M., and Ruud, K., 1999, Chem. Rev., 99, 293. [14] Frisch, M. J., Trucks, G. W., Schlegel, H. B., Scuseria, G. E., Robb, M. A., Cheesman, J. R.,

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