5 August 1994
CHEMtCAL PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 225 ( 1994) 462-466
Density-functional-theory calculations of isotropic hyperfine coupling constants of radicals Nobuhiko Ishii ‘, Tatsuo Shimizu b ’ DepartmentofElectricalEngineering, Fukui Universityof Technology,Gakuen3-6-l. Fukui 910, Japan b DepartmentofElectrical and ComputerEngineering, KanazawaUniversity,Kanazawa920, Japan Received 28 March 1994; in tinal form 8 June 1994
Isotropic hypefine coupling constants for the radicals BH2, CH? , NH2, OH:,
BHr , CH,, NH:,
H*CN, HzCO+ and CH&H
using density-functional theory with a gradient-corrected local-spin-density approximation and a Slater-type-orbital basis set. The agreement between the calculated and observed results is fairly good. The effect of rare-gas atoms surrounding radicals on the isotropic hyperfine coupling constants is also discussed.
1. Introduction Reliable prediction of isotropic hyperfke coupling constants (IHFCCs) is difficult. The reason is that the IHFCC depends on the spin density at the site of the nucleus as follows. The IHFCC for nucleus X at site r, is calculated from the formula a(X) =
and p,(r) are the electron densities with spin up and spin down, respectively, at site r, g, and g, are the g-factors of the free-electron and nucleus X, and BB and jJ,, are the Bohr and the nuclear magnetons, respectively. n, is the number of unpaired electrons. In conventional many-body theories such as the configuration interaction method, the coupled cluster method, and so on, the wavefunction plays a central role and P, (r) and pl.(r) are calculated from it. It is difficult to calculate the many-electron wavefunction precisely. In contrast to these conventional theories, in density-functional theory (DFT), pt (r) and pJ (r) are basic variables, instead of the many-elecp,(r)
tron wavefunction [ 1,2]. If the exact exchange-correlation energy functional were used, we can obtain exact pt (r) and pi (r). This fact is a great advantage of the DFI in the calculation of IHFCCs. Although the exact expression for the exchange-correlation energy functional is unknown, useful approximations exist [ 3-5 1. The exchange-correlation energy in the DFf corresponds to the effects of antisymmetrization of the wavefunction and configuration interaction in the conventional theories. Recently, we calculated the IHFCCs of CHJ, NH2, HCO, NOz and H&O+ [ 61 using DFT [ 3-51 and a Slater-type orbital (STO) basis set. The calculated results are in good agreement with the observed ones. It is found from these calculations that the gradient correction to the local-spin-density approximation (LSDA) is essential to calculate the IHFCCs of A radicals [ 6 1. Several studies of IHFCCs by the DFT and Gaussian-type orbital basis sets were also reported [ 7- 111. The effect of a Ne matrix on the a( ‘H) of CHZ was examined and it is found that this effect is negligible when Ne-H distance is larger than 2.7 8, .
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDZ0009-2614(94)00663-B
N. Ishii, T. Shimizu /Chemical Physics Letters 22s (1994) 462-466
In order to examine the influence of basis sets on the results and illustrate the utility of DFI, we calculated the IHFCCs of radicals using DFT with a gradient-corrected (CC) LSDA and a new ST0 basis set. In addition, we examined the effect of rare-gas atoms on the observed IHFCCs of the radical isolatedly incorporated in a rare-gas matrix.
2. Calculation method There are two major factors in DFI, which are the exchange-correlation energy functional and the basis set. In the present study we have employed the expression for a homogeneous electron gas in the Hartree-Fock approximation plus Becke’s gradient correction [ 41 as the exchange-energy functional and Vosko-Wilk-Nasair’s expression [ 31 plus Perdew’s gradient correction [ 51 as the correlation-energy functional. In the previous work [ 61, we used the STOs with the following radial part as the basis set of each atom: R,(r)=r’exp(-air), where I= 0, 1 and 2 for s-, p- and d-type STOs, respectively. Namely, we used only Is-type STOs as stype STOs. A new basis set used in the present work includes 2s-type STOs for B, C, N, 0 and Ne atoms as follows: Rls(T)=exp(--(YiT),
i = 4-6 for the first-row atoms , Rzs(r)=rexp( i=
l-3 for the first-row atoms ,
i= l-5 for the first-row atoms , RJd(r)=?exp(-~~ir), i = l-4 for the first-row atoms except for Ne .
We used the same values of ai for s-, p- and d-type SIG for simplicity. Table 1 shows values Of (Ti.These values were roughly determined referring to the Hartree-Fock wavefunctions of atoms [ 12 1. Numerical calculations were carried out using the method by Becke and Dickson [ 13 1.
Table 1 Values of orbital exponents al (in atomic units). The same value of ai is used for s-, p and d-type STOs. !Seethe text
H B C(I) C(U) N 0 Ne
0.8 0.9 1.0 0.8 1.1 1.2 1.5
1.1 1.2 1.3 1.2 1.5 2.0 2.5
1.6 2.0 2.5 2.3 3.0 3.5 4.5
3.0 3.5 5.0 4.0 6.0 7.5 8.0
6.0 5.0 6.0 5.5 7.0 8.5 9.5
10.0 11.0 9.0 11.0 16.0 16.0
3. Results and discussion First we calculated the IHFCCs of BHI, CH, , NH2 and OH:. These geometries used in the calculations (1) BHI:BH=1.192 A, are as follows. ~HBHz128.6” [ 141. (2) CH,- :CH=1.139 A, ~HCH=100.4” . (3) NH2:NH=1.024 A, .~HNH=103.3” [ 161. (4) OH; :OH=O.999 A, ~HOH=110.5” . The calculated IHFCCs are shown in Table 2. Two calculations for CH, using basis sets (I) and (II) for the C atom lead to similar results. The difference between the previous and present results for NH2 is small. These facts show that the calculated results are not sensitive to the present change of basis sets. The results are in good agreement with the observed ones except for a( 13C) of CH? . Although the calculated IHFCCs of NH2 and OHf are within about 10% of the observed ones, the discrepancy between the calculated and observed a ( 13C) of CHF is about 30%40% of the observed value. Since CH?, NH2 and OH; are isoelectronic radicals, it is not likely that the reliability of the calculation method used here becomes particularly lower for CH? than for NH2 and OH:. Since the wavefunctions of valence electrons in CH? are more extended than those in NH2 and OH,+, it is probable that a( 13C) observed for CH, is affected by the presence of Ne atoms surrounding it. In order to examine the effect of the Ne atoms surrounding CH, on a ( 13C), we carried out the calculation for a cluster Ne...CH2 whose geometry is as follows. One Ne atom is located on the twofold rotationaxis of CHF so as to be Ne...C = 2.7 A without changing the geometry of CH1. The reason for Ne...C= 2.7
N. Ishii, T. Shin&u /Chemical PhysicsLetters225 (1994) 462-466
Table 2 Isotropic hyperllne coupling constants (in G) calculated by GCLSDA. (I) and (II) for CHr are the results obtained using the basis sets (I) and (II) shown in Table 1 for the C atom. The previous results for NH2 calculated using a different basis set are also shown. Experimental results are shown in the last column Radical
a( “B) a(‘H)
a( “C) a( ‘W
a( 14N) a( ‘W
a( “0) a( ‘H)
127.7 [ 181 13.6 [ 181
126.1 12.5 (I) 29.0; (II) 27.1 (I) - 15.6; (II) - 16.6
A is as follows. Solid Ne is a face-centered-cubic structure with lattice constant 4.5 A. The Ne vacancy is a sphere-like space with diameter 6.4 A. We think CH,- is trapped in a Ne vacancy and adopt Ne...C=2.7 A tentatively. The calculations were carried out using basis set (I) for the C atom. The resultsarea(‘3C)=24.3Ganda(‘H)=-15.2G.The Ne atom mainly influences a ( i3C) in this geometry. a( 13C) observed for CH, in the Ne matrix is closer to that calculated for Ne ... CH,- than for isolated CH,-. This result shows that a( i3C) observed for CH? in the Ne matrix is affected by the presence of Ne atoms surrounding CH,-. We hope the experiments are carried out for CH? in the gas phase. Next we carried out the calculations for isoeleo tronic radicals, BH,- , CH3 and NH:. These have planar equilibrium geometries but low resistance to the out-of-plane bending motions [ 2 11. It is important to take into account the effect of the out-of-plane bending motions accordingly. The expectation value of the out-of-plane angle in the ground vibrational state is about 5” for CH3 [ 221. In the present calculations the out-of-plane angles for these three radicals were set equal to 5” tentatively though the expectation values of the out-of-plane angles for them should be different from one another. The bond lengths are as follows . (5) BH::BH=1.216 A. (6) CH3:CH=1.079A. (7)NH::NH=l.O20A. The calculated IHFCCs are shown in Table 3. The calculated results are in good agreement with the observed ones. The change of the out-of-plane angle from 0” to 5” increases the s-components in the un-
21  -16  10.0  -22.9 
10.3 [ 191 -23.9 [ 191 -29.7  -26.1 
paired Kohn-Sham orbitals for these radicals . These effects increase a( liB) by 3.9, a( i3C) by 6.5, a(14N)by2.5andu(1H)by0.7-1.4G.0ntheother hand, the changes in the contributions to the IHFCCs from the valence and core polarizations are less than 10% of those from the unpaired Kohn-Sham orbitals. Here we also found that the calculated results are not sensitive to the basis set as in the case of the results shown in Table 2. The last examples are H&N, H&O+ and CHzCH which include P-protons. These geometries are as follows. (8) H&N:HC=1.084 A, CN=1.243 A, ~HcH~118.6” . (9) H&O+:HC=1.117 A, CO= 1.198 A, LHCH=123” . (10) CH&H: The geometry is shown in Fig. 1. C&J,= 1.311 A, CaH,=1.089 A, C$I,i=l.l04 A, CsH,= 1.098 A, LC&Ha= 138.1”, LH,&,,C,,= 122.1”, ~H&,C,=121.9” . The calculated IHFCCs are shown in Table 4. The agreement between the calculated and observed results is fairly good except for a( 13C) of H&N and H&O+. The difference between the two values of a( “0) calculated for H&O+ is due to a large difference between the two basis sets for the 0 atom. We think the basis set of the 0 atom used here is better than the previous one since the optimized basis set [ 12 ] resembles the present one overall, rather than the previous one. The a( ‘H) calculated for H&N and H&O+ are in particularly good agreement with the observed ones. On the other hand, the discrepancy between the
N. Ishii, T. Shim&u /Chemical Physics L&ten 225 (1994) 462-466
Table 3 Isotropic hypertine coupling constants (in G) calculated by GC-LSDA. The basis set (I) shown in Table 1 is used for the C atom. The previous results for CHx calculated using a different basis set are also shown. Experimental results are shown in the last column Radical
a( “B) a( ‘H)
a( ‘C) a(‘W
a( “N) a(‘W
H-w Fig. 1. Geometry of CH,CH.
calculated and observed results for a( “C) is about 25% of the observed ones for both H&N and H*CO+. a( ‘H) and a( 14N) observed for H&N in the gas phase [ 28 ] are different from those in aqueous solution [ 261 by 4 and 1 G, respectively. There is no observed value of a( 13C) for these radicals in the gas phase. Judging from this fact, we think the effect of the matrix atoms surrounding these radicals is a possible origin of this discrepancy. It is important to ex-
Exp. 19.9  -15.2 1231
38.9  -22.8 
38.3  -23.0  19.5  -27.4 
amine the effect of the matrix atoms on a( i3C) of these two radicals in order to clarify the origin of this discrepancy. The calculation method used here is different from that in Ref.  in the basis set, the functional form of the gradient correction to the exchange energy and the numerical recipe. These two methods, however, lead to similar results for CH*CH, which are in good agreement with the observed ones as shown in Table 4. In conclusion, we have shown that density functional theory with the gradient-corrected local-spindensity approximation is useful in the calculation of isotropic hyperllne coupling constants of radicals. In some cases, the influence of the matrix atoms surrounding radicals is found to be not negligible.
Table 4 Isotropic hypertime coupling constants (in G) calculated by GCLSDA. The basis set (I) shown in Table 1 is used for the C atom. The previous calculated results are also shown for comparison. Experimental results are shown in the last column Radical H2CN
a( “N) a( “C) a( ‘H)
7.8 -21.9 84.9
a( “0) a( 13C) a( ‘H)
- 16.0 -29.2 133.3
-11.3  -29.9  131.5 
112.0 -4.4 16.5 66.5 42.2
101.3  -4.0 17) 16.2  60.4  36.2 
a( a( a( a( a(
“C,) “ce) ‘Ha) ‘H.,.ti) ‘I&r.)
Exp. 9.2  -28.9  83.2  -38.8  132.7  107.6  -8.6 1241 13.4  65  37 
N. Ishii, T. Shimizu /Chemical Physics Letters 225 (1994) 462-466
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