J. Phys. Chem. 1988, 92, 5688-5693
5688
least-squares method in such a way that the relaxation curve calculated by the integration of eq 20 coincided with the experimental curve in Figure 2. Now all the parameters and functions necessary for the calculation of the tl-dependent ESR spectrum have been obtained. The t,-dependent ESR spectrum calculated from eq 14 is shown in Figure 8 with the experimental one. The calculated spectrum compares well with the experimental one, except for the central part of the magnetic field, where the color centers in the irradiated quartz are known to give the ESR spectrum. The reasonable agreement with the experimental spectrum certifies the validity of the applied model composed of the following assumptions: (1) The broadening of the ESR spectrum is caused by the modulation of the hfcc but not by the fluctuation of the nuclear spin quantum
number. (2) The modulation of the hfcc induces the longitudinal relaxation or the mixing of on-resonant A spins and off-resonant B spins. (3) The motion of the spins in the spectral space is expressed by a diffusion equation, and the spectral shape of each hyperfine line a t f l = 0 is equivalent to the distribution of the diffusion time. In conclusion, the resolution enhancement is caused by the spectral diffusion within each hyperfine line, and the diffusion model is applicable for explaining the tl dependency of the ESR spectrum.
Acknowledgment. This work was supported by a Grant-in-Aid for Developmental Scientific Research from the Japanese Ministry of Education, Science and Culture under Contract No. 62840012. Registry No. CH3CH2CH(CH2)2CHS, 19414-29-0;hexane, 110-54-3.
Extended Hiickei Parameters from Density Functional Theory Albert0 Vela and Jose L. Glzquez* Departamento de Qdmica, Diuisibn de Ciencias Bdsicas e Ingenieria, Uniuersidad Autbnoma Metropolitana, Zztapalapa, A.P. 55-534, Mexico D.F. 09340, Mexico (Received: February 22, 1988)
The Kohn-Sham approach to density functional theory is used together with the transition-state method to show that the atomic parameters of the extended Hiickel and the single-configuration iterative extended Hiickel methods can be expressed as Hii = -Ci = ein, Bi = '/2(r-')im,and Ai = I/*$, where qm, (r-')im,and n, are the transition-state eigenvalue, the expectation value of r-', and the principal quantum number of the ith atomic orbital, and the valence orbital ionization energies are given by Ii = -Hii in the extended Hiickel method and by li = Ci + B,q + A,$ (q is the charge of the atom in the molecule) in the iterative extended Huckel method. The values of tim and (r-')imcan be determined from a single self-consistent field Kohn-Sham transition-state calculation. The results obtained for atoms with s and p valence orbitals and for the transition-state elements are, in general, in very good agreement with the usual values for these parameters. The procedure can be readily extended for any valence state of any atom or ion, providing a systematic and reliable method to generate the atomic parameters in these semiempirical theories.
Introduction The extended Hiickel (EHM) and the iterative extended Huckel (IEHM) methods provide a very useful approach to understand many aspects of molecular electronic structure and chemical reactivity.'-s Over the years these methods have become a very important tool to correlate and to analyze experimental information. Besides, the flexibility in some of the approximations introduced in these methods allows adoption of them for particular problems through the use of the appropiate set of parameters, leading not only to a qualitative description but also to an accurate prediction of some molecular properties. In the EHM, the diagonal matrix elements are given in terms of valence orbital ionization energies that do not depend on the charge of the atom in the molecule:' -Hii
= Ii(0)
(1)
In contrast, in the IEHM, the diagonal matrix elements are given by' -Hi,'EHM(q)= li(q) = Ci B g + An2 (2)
+
where q is the net charge of the atom in the molecule, and the (1) McGlynn, S.P.; Vanquickenborne, L.G.; Konoshita, M.;Carroll, D. G. In Introduction to Applied Quuntum Chemistry; Holt, Rinehart and Winston: New York, 1972; pp 97-156. (2) Hoffmann, R. J . Chem. Phys. 1963, 39, 1397-1412. (3) Sung, S.S.;Hoffmann, R. J . Am. Chem. Soc. 1985,107, 578-584. (4) Zheng, C.; Hoffmann, R. J . Am. Chem. Soc. 1986,108,3078-3088. ( 5 ) Gavezzotti, A.; Simonetta, M.In Quantum Theory of Chemical Reactionr; Daudel, R., Pullman, A., Salem, L., Veillard, A., Eds.; Reidel: Dordrccht, Holland, 1980; Vol. I, pp 145-159. 0022-365418812092-5688$01.50/0
constants -Hii in the E H M and Ai, Bi, and Ci in the I E H M are the atomic parameters. Since eq 2 may be interpreted in terms of a Taylor series expansion of li around the neutral atom value, that is Ii(q) = Ii(0)
- q(azi/aN)ZIN=z + Y2q2(aZzi/ap!ZIN=z + ... (3)
(Zis the nuclear charge, N is the number of electrons, and q = 2 - N), then from eq 1-3 one has that -Hjj = Cj = Zj(0)
(4)
Bi = -(azi/aN)ZlN=Z
(5)
which show that the atomic parameters are related not only to the isolated atom value of Zi but also to the derivatives of Zi with respect to N . This analysis also shows that in principle the parameter Ci of the I E H M is equal to the parameter Hii of the EHM. However, in practice the usual values are generally different,' because since in the E H M the valence orbital ionization energy is independent of the charge of the atom in the molecule, the parametrization takes into account somehow a more realistic value for the atom in the molecule. Thus, one can see from the values in Table I11 that, for example, for Li, the value of -Hii is slightly larger than Ci, corresponding to a tendency for this system to be positively charged in the molecule. On the other hand, for F the value of -Hii is slightly smaller than Ci, corresponding to a tendency for this system to be negatively charged in the molecule. 0 1988 American Chemical Society
E H Parameters from Density Functional Theory
In principle, the values of Ci, Bi, and Ai can be fixed by numerical differentiation of the experimental values of Zi for the system and two or more of its positive and negative ions.' Unfortunately such information is not currently available for any valence state of any atom. On the other hand, theoretical values of ri determined from wave function theory can also be used in this context; however, because of the computational difficulties of this procedure it has never been fully implemented. A more attractive route to a systematic calculation of Ci, Bi, and Ai from eq 4-6 is provided by density functional theory: which allows for the number of particles to be varied continuously so that the differentiation can be carried out explicitly. In fact, extended Huckel parameters derived from local density approximations had already been used by Grodzicki' to construct an essentially EHM-like variational local density scheme, by Larsson and Pyykk@" and Rosch and PyykkeSbto parametrize the iterative relativistic EHM, and more recently by Tatsumi and Nakamurag in an iterative quaqirelativistic scheme. The object of the present work is to analyze the expressions for the atomic parameters of the E H M and IEHM within the Kohn-Sham (KS) approach to density functional theoryIo and to present the results for atoms with s and p valence orbitals and for the transition-metal elements, to compare with the values reported in the (6) Parr, R. G. Annu. Rev.Phys. Chem. 1983,34,631-656. (7) (a) Grodzicki, M. J . Phys. B 1980,13,2683. (b) Grodzicki, M. In Local-Density Approximations in Quantum Chemistry and Solid-State Physics; Dahl, J. P., Avery, J., Eds.; Plenum: New York, 1984; pp 785-794. (c) Grodzicki, M.; Walter, H.; Elbel, S.Z . Naturforsch. B: Anorg. Chem., Org. Chem. 1984,398, 1319. (8) (a) Larsson, S.;Pyykko, P. Chem. Phys. 1986,101,355. (b) Rkch, N.; Pyykko,P. Mol. Phys. 1986,57, 193. The transition-state parameters used in this work for F, S,Se, Te, and Xe aie given in: (c) Pyykkij, P. Report HuKI 1-86, University of Helsinski, 1986. (d) Pyykk8, P. In Methods in Computational Chemistry, Wilson, S.,Ed.; Plenum: London and New York; Vol. 2, id press. (9) Tatsumi, K.;Nakamura, A. J. Am. Chem. SOC.1987, 109, 3195. Sham, L. J. Phys. Rev. 1965,140,A1133-AI138. (10) Kohn, W.; (1 1) Anderson, A. B.; Hoffmann, R. J . Chem. Phys. 1974, 60,4271. (12) Canadell, E.;Eisenstein, 0. Inorg. Chem. 1983,22,3856. (13) Trong Anh, N.; Elian, M.; Hoffmann, R. J. Am. Chem. SOC.1978, loo, 110. (14) Summerville, R. H.; Hoffmann, R. J. Am. Chem. Soc. 1976,98,7240. (15) Chen, M. M. L.; Hoffmann, R. J . Am. Chem. Soc. 1976,98,1647. (16) Hinze, J.; Jaff6, H. H. J . Chem. Phys. 1963,67, 1501. (17) Canadell, E.; Eisenstein, 0.;Rubio, J. Organometallics 1984, 3, 759. (18) Thorn, D. L.; Hoffmann, R. Inorg. Chem. 1978,17,126. (19) Hoffmann, R.; Shaik, S.; Scott, J. C.; Whangbo, M. H.; Foshee, M. J. J . Solid State Chem. 1980,34,263. (20) Hughbanks, T.; Hoffmann, R.; Whangbo, M. H.; Stewart, K. R.; Canadell, E. J. Am. Chem. SOC.1982,104,3876. Eisenstein, 0.; (21) Whangbo, M. H.; Gressier, P. Inorg. Chem. 1984,23, 1228. (22) Canadell, E.; Eisenstein, 0. Inorg. Chem. 1983,22, 2398. (23) Charkin, 0.P. Russ. J. Inorg. Chem. (Engl. Trans/.)1974,19,1590. Hughbanks, T. Inorg. Chem. 1984,23, (24) Canadell, E.; Eisenstein, 0.; 2435. (25) Lohr, L. L.; PyykkB, P. Chem. Phys. Lett. 1979.62,333. (26) Munita, R.;Letelier, J. R. Theoret. Chim. Acta (Berlin) 1981, 58, 167. (27) Least-squares fit of data in ref 20. (28) Saillard, J. Y.; Hoffmann, R. J. Am. Chem. SOC.1984,106,2006. (29) Lauher, J. W.; Hoffmann, R. J. Am. Chem. SOC.1976,98, 1729. (30) KuMcek, P.; Hoffmann, R.; Havlas, Z. Organometallics 1982,I , 180. (31) Elian, M.; Hoffmann, R. Inorg. Chem. 1975,14,1058. (32) Lauher, J. W.; Elian, M.; Summerville, R. H.; Hoffmann, R. J. Am. Chem. SOC.1976,98, 3219. (33) Hay, P. J.; Thiebeault, J. C.; Hoffmann, R. J. Am. Chem. Soc. 1975, 97,4884. (34) Silvestre, J.; Albright, T. A. Isr. J . Chem. 1983,23, 139. (35) Ballhausen, C. J.; Gray, H. B. Molecular Orbital Theory; Benjamin: New York, 1964; p 120. (36) Tatsumi, K.; Hoffmann, R. J . Am. Chem. SOC.1981,103, 3340. (37) Hoffmann, R.; Minot, C.; Gray, H. E.J. Am. Chern. Soc. 1984,206, 2001. (38) Tatsumi, K.; Hoffmann, R.; Yamamoto, A.; Stille, J. K. Bull. Chem. Soc. Jpn. 1981.54, 1857. (39) Hoffmann, D. M.; Hoffmann, R.; Fiesel, C. R. J . Am. Chem. SOC. 1982,104,3858. (40) Dedieu, A.; Albright, T. A.; Hoffmann, R. J . Am. Chem. Soc. 1979, 101, 3141. (41) Dubois, D. L.; Hoffmann, R. Nouv. J . Chim. 1977,I , 479. (42) Komiya, S.; Albright, T. A.; Hoffmann, R.; Kochi, J. K. J . Am. Chem. Soc. 1977,99,8440. (43) Brown, D. A.; Owens, A. Inorg. Chim. Acta 1971,5, 675.
The Journal of Physical Chemistry, Vol. 92, No. 20, 1988 5689
Method The KS approach to density functional theory in the local density approximation is based on the self-consistent solution oflo
(7) where V e ~= - _z r
+
-dr' + Vxc(p)
(8)
The electronic density p is given by
(9) and V & J ) is the local exchange-correlation potential. The ground-state energy is then determined from
EbI = Cinti - J[PI + E x c b l
(10)
where n, is the occupation number of the ith orbital, J[p] is the Coulombic energy, and Exc[p] is the exchange-correlation energy. The calculation of the orbital ionization energies within this framework is usually performed through the use of the transition-state (TS) concept." This approach makes use of a wellknown property of the Lagrange multipliers e,, namely4' (awanihl,,",
=
ti
(11)
This relation has been applied to calculate energy differences, since from the mean-value theorem, it is known that AE/Ani is equal to aE/ani evaluated a t some point that lies in the interval Ani. For simplicity, in the TS method the point selected is the one that corresponds to the electronic configuration 'that lies half-way between the initial and final states, and the eigenvalue thus obtained gives directly the specified energy d i f f e r e n ~ e . ~ ~ Such an approach implies that the physical significance of the TS eigenvalues depends on the initial and final states selected. For example, in the calculation of absolute electronegativities by the KS method, the initial and final states correspond to the positive and negative ions, respectively, and the TS corresponds to the neutral atom ground-state configuration, so that according to eq 11 the eigenvalue of the highest. occupied atomic orbital for this configuration would be interpreted as an estimate of the electronegativity for the isolated atom.4gs2 For the case of orbital ionization energies, the initial state corresponds to the neutral atom ground-state configuration, and the final state corresponds to a positive ion whose configuration is that of the neutral atom, with one electron removed from the orbital under consideration; thus the TS is obtained by removing half an electron from the ith orbital, and the eigenvalue for this orbital would be interpreted as an estimate of the orbital ionization energy for the isolated a t 0 r n . 4 ~ * ~ ~The ' ~ good agreement with experimental results suggests that the TS procedure takes care of relaxation and even correlation effect^.'^ Thus, from the point of view of the KS method, the parameters Ci or -Hii can be calculated directly from a self-consistent fiekd calculation with the electronic config0ration for the TS. However, the calculation Bi and Ai as given by eq 5 and 6 requires additional information. One possible approach would be to perform the TS calculation for several values of q around q = l/z arid to carry (44) Slater, J. C. Quantum Theory of Molecules and Solids; McGrawHill: New York, 1974; Vol. 4. (45) Janak, J. F. Phys. Rev.B: Condens. Matter 1978,18,7165-7168. (46) Schwarz, K.J . Phys. E 1978,11, 1339-1351. (47) Sen, K. D. J . Chem. Phys. 1979,71,1035-1036; 1980,73, 47044705; 1981,75, 5971-5972. (48) Bartolotti, L. J.; Gadre, S.R.; Parr, R. G. J . Am. Chem. SOC.1980, 102,2945-2948. (49) Robles, J.; Bartolotti, L. J. J. Am. Chem. Soc. 1984,106,3723-3727. (50) BBzquez, J. L.; Ortiz, E. J. Chem. Phys. 1984, 82, 2741-2748. (51) Ggzquez, J. L.; Vela, A.; GalvBn, M. Srruct. Bonding (Berlin) 1987, 66 79-97 . - . (52) Manoli, S.;Whitehead, M. A. J . Chem. Phys. 1984,81,841-846. (53) GBzquez, J. L.; Ortiz, E.; Robles, J. Chem. Phys. Lett. 1984,109, 394-397. (54) Beebe, N . H. Chem. Phys. Lett. 1973,19, 290-293.
__.
5690 The Journal of Physical Chemistry, Vol. 92, No. 20, 1988
Vela and GIzquez 18
out a numerical differentiation of the eiTs versus q data. Such a procedure is simpler in the KS method than in a conventional wave function method, because since the number of particles can be varied continuously, one can select values of q that lie close to 0.5 to avoid the convergence problem that arises frequently in the case of negative ions. Nevertheless, this procedure implies a great computing effort, since it requires several atomic calculations to determine each parameter. To establish a simpler approach, let us consider first the Taylor series expansion of the KS eigenvalue for an specified valence state, around the point q' = 0, that is
1
I .
I
4 J
- . 2
where ~ ( 0 is ) the eigenvalue of the ith orbital corresponding to the ground-state configuration of the isolated atom. At this point it is important to mention that to make use of eq 12, one must distinguish between the ionization energy for the isolated neutral atom and the ionization energy for the atom in the molecule, which, in general, will have a net charge. In the first case, according to the TS method for ionization energies q' = Z - N = 1 / 2 and from eq 12, one has that
+ X ( a e i / a q ? ~ l d =+~ !h(a2'i/aqR)zl~=o+ ... (13)
In the second case, for the atom in the molecule, according to the TS method for ionization energies q' = Z - N = q + l I 2 , and therefore ti(q
+ !4 = Ci(0) + (4 + 5/2)(aci/aq?zId=o + 1/2(4 + 1/2)2(a2~i/aqR)zI
