553
THEORETICAL STUDY OF HYPERFINE COUPLING CONSTANTS sumed to be tet!ahedral with carbon-hydrogen bond lengths of 1.09 A. Actually, some of the molecules have asymmetric methyl groups while in others the methyl rotate about axes which are not collinear . groups with the axial bonds? All of these effects will lend unAs accurate certainty to the structural and barrier data become available, this ap-
proach should lead to improved intramolecular nonbonded potential functions. Such functions are important for predicting the conformations of proteins and other polymers.’ (6) J. Wollrab and V. W. Laurie, J. Chem. Phys., 48,5058 (1968). (7) H. A. Scheraga, J. J. Leach, R. A. Scott, G. Nemethy, Discuss.
Faraday SOC., 40,268 (1965).
A Theoretical Study of Hyperfine Coupling Constants of Some u
Radicals Based on the INDO Method by M. F. Chiu, B. C. Gilbert,* and B. T. Sutcliffe Department of Chemistry, University of York, Heslington, York, United Kingdom Y O 1 600. (Received July 9, 1971) Publication costs borne completely by The Journal of Physical Chemistry
A study based on the INDO method has been made of a variety of Q radicals. The desirability of minimizing the energy with respect to molecular geometry for a number of small radicals (e.g., .CONH2, H2C=N., H2C= CH) is discussed. INDO calculations have also been performed for a series of iminoxy radicals (RaC=NO.), the appropriate geometry around the radical center being chosen as that which minimizes the energy for the smallest radical in the series. An alternative method of determining the proportionality constants for relating esr hyperfine splittings to spin density matrix elements is suggested. Spin density distributions in the space around some of the radicals studied are expressed in the form of spin density contour maps.
1. Introduction Organic radicals exhibit a wide variation in their isotropic esr hyperfine splittings, but it is possible, broadly speaking, to divide them into two classes. These are the 9 radicals, typified by aromatic anions and cations, by semiquinones, and by methy1,I and the so-called u radicals, such as phenyl.2 From a theoretical point of view it is possible to rationalize the observed hyperfine splittings in many 9 radicals by means of calculations made within the context of the so-called “9-electron approximation,” e.g., a Huckel Moaor a Pariser-Parr-Pople3 110 calculation. I n these approaches, certain 140 coefficients are related to proton splittings by means of a McConnell-type relationship. For u radicals, however, calculations made within the T-electron approximation are inadequate to account for the observed splittings. In a pioneering VB calculation on the vinyl r a d i ~ a l Barplus ,~ and Adrian6 were able to obtain good agreement with experiment and were able to interpret their calculations in terms of an unpaired electron in a hybrid orbital in the molecular plane a t the radical center. This result suggests that for a calculational method to be successful for u radicals it must take
into account the electronic “core” of the molecule, which is specifically excluded from consideration in Telectron methods. With the development of the INDO procedure by Pople, Beveridge, and Dobosh,’ a semiempirical method has become available which can, to some extent, allow the presence of an electronic “core.” The details of this method are well known and the relevant parts are summarized in the Appendix to this paper. Pople, Beveridge, and Dobosh (PBD)* and Beveridge and Dobosh (BD)9 have already carried out calculations (1) R. W. Fessenden and R. H. Schuler, J . Chem. Phys., 42, 3670 (1965). (2) J. E. Bennett, B. Mile, and A. Thomas, Proc. Roy. SOC.Ser. A , 293, 246 (1966). (3) See, for instance, R. G. Parr, “Quantum Theory of Molecular Electronic Structure,” Benjamin, New Y o r k , N. Y . , 1964. (4) H. M. McConnell, J . C h m . Phys., 28, 1188 (1958). (5) (a) R. W. Fessenden, J . Phys. Chem., 71, 74 (1967); (b) R. W. Fessenden and R. H. Schuler, J . Chem. Phys., 39, 2147 (1963). (6) M.Karplus and F. J. Adrian, ibid., 41, 56 (1964). (7) J. A. Pople, D. L. Beveridge, and P. A. Dobosh, ibid., 47, 2026 (1967). (8) J. A. Pople, D. L. Beveridge, and P. A. Dobosh, J . Amer. Chem. Soc., 90, 4201 (1968). (9) D. L. Beveridge and P. A. Dobosh, J . Chem. Phys., 48, 5532 (1968).
The Journal of Physical Chemistry, Vol. 76, No. 4 , 1973
554 using thc UHF’O approach within the INDO mcthod and have mct with considerablc succcss for a largc number of radicals. Esscntially, thcir proccdure is to postulate a “reasonablc” idcalized geometry for the radicals in question (though othcr investigation^^^'^^^^ have found minimum encrgy geomctrics for somc small radicals and molecules), thcn to calculatc approximate spin densities a t various nuclei in thc radical, and, by correlating these values with experimental splittings, to derive parameters to relate spin densities and splittings in subsequent calculations. The radicals employed for parametrization included fcw u radicals and in consequence it is perhaps not surprising that, in our pilot calculations on radicals in the iminoxy (u) serics, neither thc PBD nor the B D parametrization (the latter bcing appropriate to spin densities after annihilation of the quartet contaminant) gave good agreement with expcrimcnt. We therefore decided to rcinvestigate the parametrization for spin densities in the INDO method in the hope of arriving a t values more appropriate for u radicals and also in an attempt to find some internal critcrion within the INDO calculation with which to decide, for a given radical under investigation, whether or not the original PBD (or BD) scheme was appropriate.
2. The Parametrization for u Radicals The calculated properties of molecules under study may depend critically on the disposition of the nuclei in space, and to avoid ambiguity, we attempted to refer, wherever possible, to the calculated equilibrium geometry. To do this for some small u radicals, we proceeded to vary the geometry until the minimum energy configurations were found. For such radicals, it was found that variations in the relative positions of atoms adjacent to the radical center lead to quite sharp variations in thc spin densities, in the energy, and in the value of (S2>av.Broadly speaking, however, the value of (SZ)av approached 0.75 (the correct value for a pure doublet), the more closely a minimum energy configuration was achieved. It was of course not possible to perform such extensive variations of geometry on larger u radicals but some fairly simple investigatiqns showed that in these cases the energy, (S2).v,and the spin densities were much less sensitive to variations in the positions of atoms remote from the radical center. As a compromise, therefore, we determined the minimum energy configuration of the radical center for the simplest member of the series (e.g., for the iminoxy series, the CNO group in HZC=NO.) and used this conformation, with idcalized geometry for other substituents, for other radicals in the scries (see “Details of Computation,” below). To calculate the actual parameters for relating spin densities to splittings we did not have enough experimental data to perform an effective correlation like that The Journal of Physical Chemistry, Vol. 78, No. 4 , 1072
M. F. CHIU,B. C. GILBERT, AND B. T. SUTCLIFFE possible for x radicals. Instead, we opted for a Direct Parametrization Schcme (DP) the details of which are outlined in the Appendix; this treatment is based essentially on thc very good approximation that Slatertype 2s orbitals (which give zero elcctron density at the nucleus) on first row atoms can be replaced by 2s-like orbitals, orthogonalizcd to postulated 1s orbitals on the same center (such 2s orbitals have finite, nonzero electron density at the nuclcus) . The D P parameters are shown in Table I together with the PBD and B D parameters for comparison. In this kind of approach, one parameter set must be considered appropriate for calculations both involving and not involving annihilati~n’~ of the quartet spin component since no correlation with experiment is involved in the parameter determination. I n the PBD and BD work an approximation to the spin density was used (the “diagonal” approximation) for the purposes of parametrization, but in the D P approach it is not necessary to make this approximation and the second term in the expansion [see Appendix, equation A51 may easily be included. We report investigations of the effect of its inclusion in a number of cases.
3. Previous Investigations into u Radicals A . Simple Treatments. Several approaches, including many of the “Extended Huckel” type (EH), have been employed. The calculation of Karplus and Adrian for vinyl6 involved an ingenious choice of hybrid orbitals for the a carbon and enabled the authors to obtain a qualitative angle of description of the dependence on the H$,C, the two P-proton coupling constants. Using an E H treatment coupled with the theory of spin polarization, Sutcliffe’4 obtained very reasonable agreement with experiment for all three proton splittings when the H,6,CB angle was chosen to be 140”. A feature of this and all E H methods is that the choice of the WolfsbergHelmholz parameter, K,I6 is arbitrary and therefore “adjustable.” Other investigations based on the EH technique were carried out by Drago and Peterson,lB but as they developed no theory of spin polarization (or some other mechanism for obtaining nonzero values of the 2s function a t the nucleus), they were obliged to develop somewhat ad hoc rules for correlating experimental results with calculated quantities. The best agreement with experiment from an E H calculation was (10) J. A. Pople and R. K. Nesbet, J . Chem. Phys., 22, 571 (1954). (11) C.Thomson, Theor. Chim. Acta, 17, 320 (1970). (12) M.S. Gordon and J. A. Pople, J. Chem. Phys., 49,4643 (1968). (13) A. T.Amos and L. Snyder, ibid., 39, 362 (1963). (14) B.T.Sutcliffe, International Colloquia of CRNS (Paris), 19667 p 253ff. (15) M. Wolfsberg and L. Helmholz, J . Chem. Phys., 20,837 (1952). (16) R.S. Drago and H. Peterson, J . Amer. Chem. Soe., 89,3978,6774 (1967).
THEORETICAL STUDYOF HYPERFINE COUPLING CONSTANTS
555
Table I : Values of A" Obtained by Different Methods (in Gauss) Atom n
PBD (before annihilation)" BD (after annihilation)" Direct Parametrisation, D P a
By least-squares regression.
1H
IQB
IlB
'BC
1 4 N
170
18F
539.86 711.25 875. 78b
*..
*..
820.10 828.97 1109.8
379.36 1126.80 549.95
888.69 2604.37 - 1649.22
44829.2 47884,O 17962.74
...
...
719.25
240.90
* Value obtained when Is&=)
has exponent 1.2.
that obtained by Petersson and Ril~Lachlan~~ who used 2s orbitals orthogonalized to 1s orbitals on the same center. The H,6,Cp angle a t which splittings were best reproduced was 146". Simple treatments of iminoxy radicals and the nitrosobenzene cation have also appeared. For example, Cramer and Drago, using the methods of ref 16, obtained reasonable agreement with experiment for the iminoxy radicals (I, R1 = H, R Z = CH,), (I, RI = phenyl, Rz = H), and (I, RI = H, RZ = phenyl)18 and for (11, R = phenyl).1g However, to obtain
ton and Hinchliffez4 has indicated that qualitative agreement with experiment may be obtained by this method for 120" < LH,C,Cp Qrlt(r>
(A2)
where ~ ( ris) a row matrix of the basis orbitals (AO's) and Q =
p" - p @
(A3)
where (42) R.McWeeny and B.T.Sutcliffe, Mol. Phys., 6,493 (1963). The Journal of Physical Chemistry, Vol. YO, No. 4, 197.8
M. F. CHIU,B. C. GILBERT,AND B. T.SUTCLIFFE
564
and T u is the matrix of occupied orbital coefficients for the in terms of the vc and T@a similar matrix for the 4@. I n cases where the UHF function is not far from being an eigenfunction of E2 it is appropriate to use an annihilated form of Q(r) (see ref 13). I n the INDO approach a zero differential overlap approximation is assumed for basis functions (taken to be Slater functions vc’(r)) on different centers: i.e. vt*(r)vj(r> =
aij
’v2’ *(r>v,’(r)
where 6$,’ is zero unless q r and 9, are on the same center. Noting that pm,pv, and pr orbitals give vanishing contributions to the spin density a t the nucleus on which they are centered, we can write Q(rn) in this approximation as Q(rJ =
~ n vn,-s(rn) l
1
+ C Qij~r*(rnhj(rn)atj’ (A51 *,3
where qn,s denotes the 2s orbital if n is a first row atom or the 1s orbital if n is a hydrogen atom, pn is the appropriate diagonal elemcnt of Q. For a general point in the space about thc molecule, the spin density is
Q (r)
=
C QUV t * ( r > d r >6u ’ w
(A6)
In the usual INDO calculation of spin densities the is treated second term in (A5) is ignored and lvn,8(rn)12 as a parameter to be assigned by least-squares regression of expcrimental values of an against calculated pa. I n our work we chose to use the values of I@(0)\2of Morton, Rowlands, and Whiffen48to assign Ivn,s(rn)l and the relationship (derivable from (hl)) that
A” =
S/,TYnfiJ v n , s ( m )
I
(-47)
in order to calculate A” directly; for points away from the nucleus, the vc were approximated by Slater orbitals. The values of Y~ were taken from ref 44 or, in the case of YB, calculated from niiclear data in ref 45. The re-
The Journal of Physieal Chemktry, Val. 76,No. 4, 1972
sulting parametrization of A” is the “direct parametrization” (DP) of this paper. We made a number of trial calculations in which we included the second term in (A5), but except for the a hydrogen in vinyl (as discussed in section 4A), the remaining terms amounted to less than 10% of the first term. Thus, (Al) can for most purposes be written as an
!=!Anpn
(A81
Details of Computation. The original version of the INDO was kindly supplied by QCPE.46 This was modified for use on a much smaller computer with only 16K of free store. Extra subroutines were written to perform the following: (i) annihilation of the quartet spin state by the method of Amos and Snyder,13 (ii) calculation of the additional terms, as given in equation (A5), (iii) calculation of spin density maps (output to the line printer). All results quoted in this paper, except for the iminoxy radical from fluorenone, used a convergence criterion of 6 decimal places in the electronic energy. For the larger radicals, this often required more than the 20 cycles allowed by the original INDO program. If the convergence criterion was reduced to 5 decimal places, usually not a great deal of difference was noted in the coupling constants obtained, though for one or two centers (C, notably), the alteration in coupling constants was about 1 G. The “idealized geometry” referred to is similar to that used by Pople, et a1.,8 in which the main interbond angles are taken to be 120’ (methyl groups are tetrahedral) and the following bent lengths are -a! sumed: r C 5 = 1.08 1,r2c = 1.40 A, rcN = 1.35 A, rco = 1.36 A, ?“OF = 1.35 A, r N O = 1.30 A. (43) J. R. Morton, J. R. Rowlands, and D. H. Wiffen, Nut. Phys. Lab. Pub., NO.3PR,13 (1962). (44) A. Carrington and A. D. McLachlan, “Introduction to Magnetic Resonance,” Harper and Row, New York, N. Y.,1967. (45) E. R. Andrew, “Nuclear Magnetic Resonance,” Cambridge University Press, New York, N. Y.,1955. (46) Program CNINDO; Quantum Chemistry Program Exchange No. QCPE 141.
