J. Phys. Chem. 1995, 99, 17822-17824
17822
Chemical Potential and Hardness for Open Shell Radicals: Model for the Corresponding Anions? Ram Kinkar Roy and Sourav Pal* Physical Chemistry Division, National Chemical Laboratory, Poona 411008, India Received: April 7, 1995; In Final Form: August 29, 1995@
In this article, we have obtained theoretical values of chemical potential and hardness for open shell free radicals using a wave function approach. The calculated values of the hardness are useful particularly for rank ordering of the corresponding anions. A ASCF procedure is used to obtain these values, and the results are compared with corresponding experimental values. The procedure also leads us to the reliability of the ASCF procedure for these quantities.
1. Introduction The concept of hardness or softness was first introduced by Mulliken, who defined hardness or softness in terms of exchange repulsion energy between the Lewis acids and bases,’ and was later used by Pearson to give a detailed qualitative explanation of hard and soft behavior of chemical species.* The concept has been of recent interest due to the proposition of the principle of maximum hardness (PMH) by Pearson3 and a proof of this by Parr and Chattaraj.4 The first quantitative definition of hardness was given by Parr and P e a r ~ o n who , ~ defined hardness 11 as
where p is the chemical potential of the system concerned; the subscripts v and T imply the constancy of the potential due to nuclei plus any other external potential and temperature, respectively. A finite difference approximation of this using EN,EN+’,and EN- 1 has often been taken as a working definition of hardness and chemical p ~ t e n t i a l . ~
7 = (IP - EA)/2 p = - (IP
(24
+ EA)/2
(2b)
where IP and EA are the first vertical ionization potential and electron affinity, respectively. Applying Koopmans’ approximation to eq 2, the expression of 7 and p can be rewritten in terms of frontier orbitals as,6 ?, = I (EL
- EH)/2
p = (EL
+ EH)/2
(3)
where and E L are the energies of the highest occupied and lowest unoccupied molecular orbitals, respectively. Both eqs 2 and 3 have been used to support the PMH by different workers.’ It is clear from these studies that a simple expression (3) can also give a reliable trend of hardness. Using similar expressions, a linear relation between hardness and the cube root of polarizability was obtained in polyatomic molecules and atomic clusters.8 Most of the systems where eq 2 or 3 has been used to give a reliable trend of 7 or p are closed shell systems. However, a problem arises for the open shell species when electrons are distributed over the degenerate or quasi-degenerate orbitals. In
* Author for correspondence.
’ NCL communication No. 6260. Abstract published in Advance ACS Abstracts, December 1, 1995.
particular, the use of expression 3 in terms of frontier orbitals would lead to unrealistic values of hardness and chemical potential. At the minimum level the ionization potentials and electron affinities have to be obtained by ASCF procedures. It has been recently realized that the values of chemical hardness of open shell systems (particularly free radicals) are important for rank ordering of the corresponding anions. Evaluation of the hardness values of anions is very difficult, as the EA values of anions are either unrealistic or of little physical or chemical significance. Only the HOMO-LUMO gap will be the important quantity to rate anions in order of increasing chemical p~larizability.~ But this HOMO-LUMO gap is also difficult to determine from either vis-UV spectra or optical polarizabilities. It is obvious that the hardness of any particular neutral species would be higher than that of the corresponding anions. This is due to the fact that hardness has a reciprocal relation to the polarizability, which is larger for anions (because of larger volume). But this enhancement of the hardness value for the anions is true for all the chemical systems. Thus, the hardness values for neutral systems will provide a correct trend of hardness for the anions, as confirmed by the experimental values. However, so far there are no theoretically calculated values of the isolated free radicals in their relaxed geometries, which can be used to test the above argument. Proft et a1.I0 have evaluated the hardness values of some common organic groups by an ab initio wave function formalism using a geometry which these groups adopt in a molecule and not the equilibrium geometry of the radicals themselves. Therefore, the 7 values obtained for such geometry cannot be used with confidence for rank ordering of the corresponding anions. We want to present in this article the calculated 7 and p values of some common free radicals (Le. open shell systems) in their relaxed geometries and also correlate these with those obtained from experimental IPS and EAs. Section 2 presents the methodology and basis set used, and in section 3 we discuss the reliability and stability of our results.
2. Methodology and Basis Set Used In our calculation we have considered the open shell species as restricted open shell systems with spin 1/2. The values of IP and EA and consequently of 7 and p are obtained at the experimental relaxed geometries of the radicals. The methodology used is the ASCF procedure; that is, the IP and EA values are evaluated by carrying out separate SCF calculations for
0022-3654/95/2099-17822$09.00/0 0 1995 American Chemical Society
Chemical Potential and Hardness for Open Shell Radicals
TABLE 1: Calculateda and Experimental Values (in au) of Ionization Potential (IP) and Electron Affinity (EA) of Some Common Open Shell Species (Mostly Radicals) radical exptl IP calcd IP exptl EA calcd EA F 0.640 168 0.629 584 0.124 947 0.037 OOO OH 0.483 985 0.550 701 0.067 251 -0.020 893 0.027 194 -0.051 044 NH2 0.418 939 0.427 858 0.002 940 -0.070 912 CH3 0.360 876 0.325 640 c1 0.478 105 0.466 261 0.133 032 0.084 957 SH 0.382 557 0.423 025 0.084 523 0.032 645 SiH3 0.299 137 0.309 130 0.051 816 -0.017 281 H 0.499 419 0.499 810 0.027 194 -0.033 312 OOH 0.423 716 0.318 831 0.043 731 -0.092 410 NO2 '0.371 165 0.443 832 0.084 523 0.030 729 CH3S 0.296 197 0.377 007 0.069 823 0.014 729 CF3 0.339 929 0.440 190 >0.040 424 -0.029 654 At TZP basis sets. radicals and the ions at the geometry of the neutral systems. The cations (except in two cases, F and Cl) and anions of the open shell systems considered are closed shell with spin value 0. Thus, for neutral species and for ionic species ROHF and RHF levels of calculations have been performed, respectively. For F+ and C1+ a ROHF wave function corresponding to the spin value 1 has been used. F- and C1- are, of course, closed shell systems for which a RHF method is adequate. The calculations have been performed by using the TURBOMOLE package." The basis set used is of TZP quality. For H the contraction used is (5s)/[3s], and for first-row atoms 10s6p primitive Gaussians are contracted to 6s3p. For second-row atoms the contraction used is (12s9p)/[7s5p]. These basis sets are as contained in the TURBOMOLE program system. The exponents of the polarization functions used are given in ref 12a. To test the basis set effects, calculations have been performed with another TZ2P basis set, in which contractions are similar to TZP but two sets of polarization functions which are quoted in ref 12b are present. It is usually felt that the diffuse functions have a much more important role to play, particularly for the anionic systems. For this we have made a separate set of calculations in which we have treated some systems in a basis consisting of extended DZP (EDZP) and then added diffuse functions separately (EDZP diffuse (sp)). The EDZP basis and the exponents for the diffuse functions are quoted in ref 13. We will discuss this aspect in the Results and Discussion section in more detail.
+
3. Results and Discussion The results of the calculation using the TZP basis are given in Tables 1 and 2. Table 1 contains the values of the calculated vertical IP and EA and the experimental IP and EA taken from ref 9. In Table 2 the values of 77 and p from our calculated IP and EA values as well as the experimental ones are reported. We have checked that in the more extensive TZ2P basis IP and EA values and consequently 7 and p values do not change significantly from the TZP values (typical difference is lo-'lo-* au only for open shell atoms and about 10-3-10-4 au for other radicals). To check the effects of diffuse functions, the results of separate calculations in EDZP and EDZP diffuse (sp) basis sets are presented in Table 3. One can see from Table 3 that the additional diffuse functions do not change the values of 9 and p significantly, particularly the trends of the values, which is the most important purpose of this study. Thus, our TZP basis set can be taken to be fairly extensive, and we will restrict our discussion to the results obtained from the TZP basis only. As seen from Table 2, the calculated values of 9 are larger than the experimental values in all cases. The same trend
+
J. Phys. Chem., Vol. 99,No. 51, 1995 17823 TABLE 2: Calculated and Experimental Values of Chemical Hardness ( q ) and Chemical Potential (Ir) of the Common Open Shell Species Mentioned in the Previous Table radical exptl (r) calcd (7) exptl @) calcd @) F 0.257 61 1 0.296 292 -0.382 557 -0.333 292 OH 0.208 367 0.285 797 -0.275 618 -0.264 904 -0.223 067 -0.188 407 0.195 872 0.239 45 1 NHz -0.182 275 -0.127 364 0.178 968 0.198 276 CH3 c1 0.172 720 0.190 652 -0.305 384 -0.275 609 SH 0.150 671 0.195 190 -0.235 194 -0.227 835 SiH3 0.123 844 0.163 206 -0.175 660 -0.145 924 H 0.235 929 0.266 561 -0.263 491 -0.233 249 OOH 0.189 993 0.205 621 -0.233 724 -0.113 211 NO2 20.143 321 0.206 552 '-0.227 844 -0.237 280 CH3S 0.113 922 0.181 139 -0.183 745 -0.195 868 CF3 '0.149 936 0.234 922 '-0.190 360 -0.205 268 The 7 and p are calculated on the basis of IP and EA values given in the previous table. TABLE 3: Values (in au) of Chemical Hardness (q) and Chemical Potential (Ir) Calculated at EDZP and EDZP diffuse (sp) Basis Sets radical VEDZP qEDZP+diff(sp) PEDZP PEDZP+diff(sp) H 0.266 1 1 0.256 34 -0.233 70 -0.243 48 0.276 72 OH 0.270 90 -0.416 70 -0.423 00 -0.281 66 0.245 01 0.239 38 -0.275 57 NH2 -0.135 04 0.197 15 0.191 72 -0.129 61 CH3 SH -0.234 47 0.193 21 0.189 17 -0.230 00 -0.151 10 0.161 07 0.158 23 SiH3 -0.148 02 OOH -0.1 14 07 -0.121 57 0.145 72 0.140 00 0.206 24 0.201 57 -0.237 47 -0.242 59 N0z 0.179 61 CH3S 0.176 30 -0.197 70 -0.201 40
+ ~
follows in the case of the chemical potential also, except for CH$, for which the calculated p is lower. For NO2 and CF3, the experimental values are not known exactly; only bounds to these are known. Our calculation in a given basis takes care of the relaxation effects only. The effects of correlation are known to increase the IP values. Hence, in a given basis our calculated IP values will be less than the full configuration interaction (CI) IP values in that basis. Similarly, the correlation effects will increase the EA values where the system has a positive EA. The systems considered in our examples have positive EAs. Thus, our computed values will be less than the full CI values. However, for the difference of IP and EA (Le. 77) one cannot specify whether the computed values will be larger or smaller than the full CI results. Our results throw light on how the ASCF method performs to extract the difference of IPSand EAs for free radicals. No trend was known for this difference previously. In the absence of full CI numbers, our computed values are compared with those obtained from experimental IPS and EAs. Since our basis is extensive, this may be a reasonable comparison. We find that the hardness values are obtained with a reasonable reliability. About the chemical potential values, one can argue that correlation will depress those obtained via the ASCF procedure in our example cases. In the results reported in Table 2, our calculated p values will be higher in comparison with the full CI p values in most examples. In comparison, the 17 values should be better represented by the ASCF procedure than the p values. However, the agreement of values of chemical potential with the experimental ones is as good as that of hardness values. This only shows that the experimental results are quite different from the possible full CI numbers. The reason for this may be attributed to the fact that the experimental numbers are derived from adiabatic IPS and EAs and the geometry of the ions is quite different from the geometry of the neutral radical.
Roy and Pal
17824 J. Phys. Chem., Vol. 99, No. 51, 1995
We see that our results offer proper ordering of the anions according to their hardness sequence. Chemical evidence as well as the experimental values confirm the following hardness sequence of the common anions:
r
Acknowledgment. The authors thank Prof. Carl Trindle of the Virginia University for many useful discussions on the problem of chemical hardness for open shell systems. Extra Mural Research Grants from the CSIR, India, are acknowledged. References and Notes
From Table 2 we see that the ordering of r values obtained by the ASCF method matched the above ordering. This gives us confidence that at least qualitatively ASCF results can be considered sufficiently reliable. Looking at our result, it seems that the H- ion should be a hard species, whereas actually it is very soft. However, this contradictory situation exists even if we also look at the expeirmental 17 value of the H- ion.I0
4. Conclusion Our calculations of chemical hardness and potential for open shell systems to use them as a model for rank ordering of the corresponding anions are the first of their kind. The ASCF level of calculations reported here gives reasonably good values of chemical hardness and potential at least for a qualitative trend. However, by performing a correlated level of calculation, a quantitative improvement of the results is expected. In this context it should be mentioned that Proft et a1.I0 have shown that there is lowering in the intrinsic group hardness values when calculated at the CISD level. The CISD values thus show better correlation with the experimental results. It is difficult to compare the trend of the values of hardness and chemical potential obtained by the ASCF procedure with that from Koopmans’ like approximationfor open shell systems. The definition of the eigenvalues of an open shell Fock operator is ambiguous, and these will differ depending on whether the ROHF or UHF procedure was adopted. Thus while for closed shell systems such comparison has been done, the same cannot be done unambiguously for the open shell cases. It may be interesting to check how far hardness and chemical potential values, using calculated adiabatic IP and EA, match the experimental results quoted in this paper. However, the definitions of hardness and chemical potential use vertical IP and EA. It is gratifying to note that for qualitative rank ordering these hardness values appear to be trustworthy.
(1) (2) (3) (4)
Mulliken, R. S. J. Am. Chem. SOC. 1952, 74, 811. Pearson, R. G. J . Am. Chem. SOC. 1963, 65,3533. Pearson, R. G. J . Chem. Educ. 1987, 64, 561. Parr, R. G.; Chattaraj, P. K. J . Am. Chem. SOC.1991, 113, 1854. ( 5 ) Parr, R. G.; Pearson, R. G. J . Am. Chem. SOC. 1983, 64, 7512. (6) Pearson, R. G. J . Am. Chem. SOC. 1985, 107, 6801. (7) Pearson, R. G.; Palke, W. E. J . Phys. Chem. 1992,96, 3283. Pal, S.; Vaval, N.; Roy, R. K. J . Phys. Chem. 1993, 97, 4404. (8) Roy, R. K.; Chandra, A. K.; Pal, S. J . Phys. Chem. 1994,96, 10447. Ghanty, T. K.; Ghosh, S. K. J . Phys. Chem. 1993,97,4951 and references therein. (9) Pearson, R. G. lnorg. Chem. 1988, 27, 734. (10) Proft, F. De.; Langenaeker, W.; Geerlings, P. J . Phys. Chem. 1993, 97, 1826. (1 1) TURBOMOLE, an ab initio quantum chemistry program system developed by Ahlrichs, R. and co-workers; see: Haser, M.; Ahlrichs, R. J . Comput. Chem. 1989, 10, 104. (12) The exponents (a)of the polarization functions are as follows. (a) For TZP, ap(H) = 0.8; ad values of C, N, 0, and F are 0.8, 1.0, 1.2, and 1.4, respectively. For second-row atoms Si, S, and C1 ad values are 0.35, 0.55, and 0.65, respectively. (b) For TZ+2P, a l p (H) = 1.39, a 2 p (H) = 0.46. The a l d and a 2 d values of the other atoms are tabulated below. atom
ald
a2d
atom
ald
a2d
C N 0 F
1.58
0.44 0.58 0.69 0.81
Si S
2.00 2.00 2.00
0.50 0.55 0.58
1.73 2.08 2.42
c1
(13) For H the contraction used is (5s)/[3s] and for the first row atoms (C, N, and 0) 1ls6p primitive Gaussians are contracted to 5s3p. For Si and S the contractions used are (13s9p)/[6s5p] and (13slOp)/[6s5p], respectively. The exponents (a)of the polarization functions are as follows. (a) For EDZP, ap(H) = 1.00; ad values of C, N, and 0 are 0.72, 0.98, and 1.28, respectively. For second-row atoms Si and S the ad values are 0.388 and 0.542, respectively. (b) The exponent (a)of the diffuse s function of H is 0.036. The shared exponents of the diffuse (sp) functions of other elements are tabulated below.
atom
a,
a P
C N 0
0.0438 0.0639 0.0845
0.0438 0.0639 0.0845
JP9509929
atom Si S
a,
a,
0.033 1 0.0405
0.033 1 0.0405
