J. Phys. Chem. 1994,98, 8648-8652
8648
Density Functional Calculations of Isotropic Hyperfine Coupling Constants in fl-Ketoenolyl Radicals Carlo Adamo,? Vincenzo Barone,* and Alessandro Fortunelli'*5 Dipartimento di Chimica, Universith della Basilicata, Via Nazario Sauro 85, I-851 00 Potenza, Italy, Dipartimento di Chimica, Universifh"Federico IF di Napoli. Via Mezzocannone 4, I-801 34 Napoli, Italy, and Istituto di Chimica Quantistica ed Energetica Molecolare del CNR, Via Risorgimento 35, I-561 26 Pisa, Italy Received: February 16, 1994; I n Final Form: June 8, 1994"
Density functional calculations using both local and gradient-corrected functionals have been performed on the organic radicals CH3, CH3CH2, CHzCHCH2, CHsCHCOO-, H C O C H C O H , CH3COCHCOH, CH3COCHCOCH3, and CH3COC(CH3)COCH3. The former four radicals are used as benchmarks. The latter four radicals are derived from common organic ligands and have been observed in recent experimental work on tris(&ketoenolato)cobalt(III) complexes. Their geometry has been optimized both at the unrestricted H a r t r e e Fock level using a double-t basis set and a t the unrestricted K o h n S h a m level using specifically optimized basis sets. From these calculations, values for the isotropic hyperfine coupling constants a t the hydrogen atoms are predicted and compared with experimental data and previous results from ab initio calculations. The agreement is found to be fairly good while the computational cost remains modest, if compared with other post-HartreeFock methods. The dependence of the results on the geometry and on the various approximations utilized in the solution of the Kohn-Sham equations is also briefly discussed.
Introduction The isotropic hyperfine coupling constants (hfcc) of freeradicals are one of the most challenging one-electron properties for ab initio quantum mechanical methods. It is now well recognized that the Hartree-Fock (either restricted, ROHF, or unrestricted, UHF) and perturbative correlation methods are not able to provide quantitative results.1-3 In a series of recent studies, promising results have been obtained using purposely tailored basis sets with multireference configuration intera~tion'~ or coupled ~luster3,~-6 methods. In the latter case, it has also been shown that there is little difference between different implementations of the exponential ansatz or between ROHF and UHF zeroorder wave functions.' Coupling of these methods with proper treatment of large amplitude vibration^^-^ and solvent effects' provides a very reliable theoretical tool for small systems. Unfortunately, the scaling of these refined post-HF methods with the number of active electrons is so heavy (7th or 8th power) that their extension to relatively large systems is prohibitive. A possible way out of this problem was recently proposed by one of uslo through semiempirical extrapolation of results obtained by relatively cheap ab initio methods and/or basis sets. Here we investigate another possibility, through the application of the density functional formalism. Only qualitative trends have been obtained in a number of studies with the unrestricted KohnSham (UKS) formalismin the frameworkof thelocal spindensity (LSD)approximation.Il Much improved results were recently found by two of us12 and by other groups13J4when gradient corrections were introduced into the approximate exchange correlation potentials. These findings prompted us to start a systematic investigation of organic free radicals as a prerequisite for the investigation of organometallic systems. In the present study we will be concerned with well-characterized hydrocarbon n-radicals (methyl, ethyl, allyl) and with radicals derived from 8-ketoenolato organic ligands. Recent experimental work has succeeded in producing these latter radicals directly attached to a transition metal atom.15 In particular, tris(j3-ketoeno1ato)cobalt(111) chelates have been considered as generating complexes t Universitti della Basilicata.
t Universitti "Federico 11" di Napoli. 1 Istituto di Chimica Quantistica ed Energetica Molecolare del CNR. Abstract published in Advance ACS Abstracts, July 15, 1994.
where the '8-ketoenolato" fragment represents, for example, formylacetonate ion H-CO-CH-CO-CH3-, the acetylacetonate ion CH+2O-CH-CO-CH3-, etc. A large variety of such organic ligands has been considered in ref 15, to which we refer for more details. The radicals so obtained have been characterized through electron spin resonance (ESR) measurements, so that values of the hfcc are available. In the present paper, results concerning the isolated ligands will be presented and compared with experimental data and the results of extensive ab initio calculations.10 Work on the complexes is in progress.
The Approach The experimental spectra give the isotropic hyperfine coupling constants (hfcc) for the various hydrogen atoms in the organic ligands. The quantities to be evaluated at the theoretical level are therefore the electron spin densities at the hydrogen atoms, which-through the Fermi contact interaction-determine the isotropic hfcc aH (through a proportionality factor):
where gc/go is the ratio of the isotropic gvalue for the radical to that of the free electron (and will be taken as unity hereafter), and gH and BH are the nuclear magnetogyric ratio and nuclear magneton, respectively, for the hydrogen atom. The ps(rH)are obtained as expectation values of the spin density operator over the electronic wave function:
where the index v runs on all electrons and S, is the quantum number of the total electron spin ( I 1 2 for radicals). The approach we use to evaluate these quantities is the density functional theory in the unrestricted Kohn-Sham (UKS) version.1+18 As it is well known, the main problem within this approach is that the exact functional dependence of the exchange and correlation energies, E, and E,, on the density is not known, and one has to resort to approximate expressions for them in practical calculations. The reliability of the UKS calculation
QO22-3654/94/2Q98-8648s04.5Q/Q 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 35, 1994 8649
hfcc of @-KetoenolylRadicals
TABLE 1: hfcc Values (in C) Obtained by UKS Computations for the & Structure of the Methyl Radical (See Text for Details) LSD C-H
(A)
ac (GI aH (G)
orb-func. aux.funcC EX,
1.094” 8.75 -18.34 DZP‘ A
NLSD 1.094“ 8.13 -18.95 DZP’
B
1.094‘ 26.32 -21.45 DZP’ A PWP
1.094” 28.77 -24.01 DZP’
1.094“ 27.15 -21.97 DZP’
B
B
BP
PWP
1.O9lb 26.75 -21.96 DZP’ B PWP
1.074c 26.48 -2 1.92 DZP’
1.079d 26.79 -21.93 DZP’
B
B
PWP
PWP
ex9 1.079 27.0 -23.0
a Optimized LSD/DZP’ geometry. Optimized NLSD(PWP)/DZP’ geometry. Optimized UHF/dz geometry. Experimental geometry, ref 39; the experimental hfcc are from ref 1. e Auxiliary functions A: H (3,1;3,1); C (4,4;4,4). Auxiliary functions B: H(5,1;5,1); C(5,2;5.2).
+
thus strictly depends on the approximation for E, E,. In the present work, the UKS calculations have been performed using a modified version of the DeMon package,l9 running on IBMRS-6000 machines. In particular, an option of the program has been selected which utilizes the functional of Perdew and Wang20 (alternatively, of Becke)21 for exchange and the functional of Perdew22 for correlation. These belong to the set of so-called nonlocal or (better) gradient-corrected functionals, and the corresponding computations will be labeled as NLSD. Some results will also be reported, obtained with the simplest approximation for E, + E,, namely the local-spin-density approximation (LSD), which is based on the theory of the homogeneous electron gas. In the DeMon program, the LSD approximation to exchangecorrelation employs the Vosko, Wilk, and Nusair parametrization of the correlation energy of the homogeneous electron gas.23 DeMon is a typical density functional program that uses Gaussian basis sets to expand orbitals and to fit the density and exchange+orrelation potentials. In particular, the charge density is fitted analytically, while the exchange-correlation potential is fitted numerically on a grid. The numerical integration procedure applied in the calculations was developed by B e ~ k e . 2The ~ grid comprises 32 radial shells and 26 randomized angular points per shell, giving rise to a total of 832 grid points per atom. In the context of conventional ab initio methods, Chipman showedZ5that the popular HuzinagaDunning double-(basis set (hereafter referred to as DZ) can give reliable results if augmented by polarization functions (DZP basis set) and further decontracted in the outer core-inner valence region. Addition of a very tight s-function on hydrogen also has a beneficial effecL26 In previous papers,12 the resulting (3,1,1/ 1) (H), (5,1,1,1,1/4,1/ 1) (second-row atoms) basis set (hereafter referred to as DZP’) has been tested in evaluating hfcc in the framework of the UKS model. The results obtained for both the geometrical parameters and the physicochemical properties of a series of small a-radicals were encouraging and induced us to further proceed in this direction. Some tests of the convergence of basis sets have been performed using a triple-( (TZ) contraction of thep-functions of the DZP’ basis set, adding s- andp- diffuse functions on heavy atoms and doubling polarization functions. The resulting (3,1,1/ 1,l) (H), (5,1,1,1,1,1/3,1,1,1/ 1,l) (secondrow atoms) basis set (TZ2P’) is larger than the TZ2P basis set, utilized e.g. in refs 27 and 28, where it seemed to give essentially converged results in the UKS computations of a number of properties. Moreover, we tested the influence of the auxiliary basis on hfcc. To this end, we employed two different sets of auxiliary functions: the first has the form (3,1;3,1) for H and (4,4;4,4) for C and 0 (set A),29while the second is (5,1;5,1) for H and (5,2;5,2) for C and 0 (set B).30 Finally, to investigate the influence of geometrical factors on hfcc, calculations have been performed on two sets of geometries: (a) one taken from ref 10, in which the geometry was optimized at the unrestricted Hartree-Fock (UHF) method using the standard LANL1DZ31basis set (in the following, thesegeometries will be referred to as UHF/dz); (b) another obtained through UKS calculations employing the DZP‘basis set and the auxiliary basis set B (referred to in the following as NLSDIDZP‘). The full geometry optimizations have been carried out, at the NLSD
level, by the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method.32 An interface program33 has been used to transfer DeMon orbitals to the GAUSSIAN-92 package for the evaluation of one-electron properties. For comparison, some results will also be reported as obtained with the best available geometries (see below).
The Problem What are the reasons for the good results one expectsI2-l4from the UKS method? If one considers organic a-radicals, one sees that the restricted HartreeFock method gives null values for the electron spin density at a-hydrogen atoms (Le., the hydrogens directly attached to the radical center), whereas the U H F method largely overestimates them.’ The latter fact can be understood by considering that the U H F wave function gains correlation energy by allowing the a- and @-spin orbitals to differ (spin polarization of thedoublyoccupied orbitals). In the UKS method, on the contrary, the correlation functional E,[p] provides a separate estimation of the correlation contribution to the energy; there is thus no need to artificially increase the spin polarization of the doubly occupied orbitals and, by the inclusion of the correlation potential into the Fock operator, the electron spin density is predicted with much greater accuracy.11J4J5 This is also proved by the S2-value over the Slater determinant of noninteracting electrons which comes out to be much closer to the theoretical value for a pure doublet state (see below the discussion of the results of Table 4). There is, however, a further problem with respect to energyonly UKS calculations. In the evaluation of the isotropic hyperfine coupling constants in organic a-radicals, there seems to exist a dependence of the theoretical results on the choice of the basis set and on the choice of the functional which is more pronounced than u s ~ a l :the ~ ~LSD , ~ ~approximation does not always furnish quantitative agreement with experiment” (see also our results below), while in some cases even different gradient-corrected functionals3*produce different numbers (L. A. Eriksson, private communication, and our results below). The problem is essentially that the isotropic hfcc are tiny, pointwise quantities, which are extremely sensitive to basis set, etc. effects.l In the following section, these problems will be taken into account through test calculations employing different orbital basis sets, auxiliary basis sets, and exchange-correlation functionals.
The Results In the present section, the results of UKS calculations will be discussed. In Tables 1 and 2 the results of selected calculations on the methyl and allyl radicals at different geometries, using different exchange functionals and auxiliary basis sets are reported. In Table 3 the optimized U H F parameters (taken from ref 10) and the NLSDIDZP’ ones are reported. The nomenclature is the following: LSD stands for UKS calculations employing a local approximation, NLSD for UKS calculations employing gradientcorrected functionals; PWP (respectively, BP) means that the functional of Perdew and Wang2O (respectively, Becke)Z1is used
8650 The Journal of Physical Chemistry, Vol. 98, No. 35, 1994
Adamo et al.
TABLE 2 hfcc Values (in G) Obtained by UKS Computations at the UHF/dz Optimized Geometry of the Allyl Radical (See Text for Details) LSD
atom CB C. HR Ha' Ha2
orb.func. aux.func.6
-11.05 4.16 0.89 -10.48 -9.97 DZP'
-10.96 3.44 0.79 -9.82 -9.32 TZ2P'
A
A
&E
NLSD -11.49 4.09 0.88 -10.86 -10.44 DZP' B
-13.76 14.87 3.09 -13.52 -12.55 DZP' A PWP
-13.26 14.58 3.15 -12.11 -11.79 TZ2P' A PWP
-13.75 12.82 3.10 -13.85 -12.82 DZP'
A BP
-14.91 16.85 3.22 -14.86 -14.32 DZP' B BP
exP -15.59 18.05 3.50 -14.19 -13.20 DZP' B PWP
-15.88 16.69 3.06 -13.15 -12.12 TZ2P' B PWP
-17.2 21.9 4.2 -14.8 -13.9
Reference 1 . Auxiliary functions A: H (3,1;3,1); C (4,4;4,4). Auxiliary functions B: H(5,1;5,1); C(5,2;5,2).
TABLE 3: UHF/dz Optimized Bond Lengths (A) and Valence Angles (deg) for the Radicals Examined in This work. C H 3 symmetry D3h C,H, = 1.074 (1.091) CH3CH2 symmetry C, C,H, = 1.076 (1.093) CpHp = 1.085 (1.108), CpCp = 1.518 (1.502) HpC,Cp = 121.06 (121.02), HpCpC,= 109.59 (110.23) CH2CHCHz symmetry Cb 1.075 (1.094), C,Ha2 = 1.074 (1.093), CpHp 1.076 (1.099), C,Cp = 1.400 (1.402) C,H,' H,'C,Cp = 121.28 (121.52), Hm2C,Cp 121.27 (121.01), HpC& = 117.70 (117.52) CH3CHCOO- symmetry C, C,H, = 1.077, CpHp 1.088, C,Cp = 1.510, C,C(O2) = 1.509, C(O2)O = 1.270 HpC,Cp a 121.06, H,C,Cp = 120.00, C&,C(Oz) = 120.00, 01C(02)C, 116.76,01C(02)02 = 127.74 HCOCHCOH symmetry C b C,H, = 1.074 (1.087), C@Hp= 1.079 (l.llO), C,Cp = 1.430 (1.417), (2.0 = 1.280 (1.264) 115.83 (123.60), H,qC,qC,= 117.05 (119.42), C,C& = 124.85 (123.59) H,C& CH&OCHCOH symmetry C, C,H, 1.075, C,Cp2 = 1.445, C,Cp' 1.428, Cp'O1 = 1.286, C ~ O =Z 1.278 Cp'Hp = 1.080, C,H, = 1.082, cp2c,= 1.518 H,C,Cp' = 115.26, H,C,CpZ = 115.23, C,Cp'Oi = 125.46, C,Cp202 121.80 C,Cp2C, = 118.36, Hp'Cp'C, = 116.73 CH3COCHCOCH3 symmetry Cb C,H, = 1.075, C,H, = 1.082, C,Cp = 1.440, C& = 1.275, C& = 1.517 122.40 H,C,Cp = 115.61, C.CpC, = 118.09, C,C&l CHJCOC(CH~)COCH~ symmetry Cb 1.082, CpC, = 1.522 Cp'Ol 1.289 C,Cpz 1.526, C,Cp' = 1.457, Cp'Hp = 1.080, C,H, Cp2C,Cp' = 118.40, C,Cp'C, = 120.62, C,Cp'O = 122.05
In parenthesis are given the NLSD/DZP' parameters for the test molecules. Note that the r-skeleton is always planar and that, unless otherwise specified, the three hydrogen atoms of methyl . .~groups . are symmetrically disposed with respect to the fourth carbon bond with fixed tetrahedral angles (1
(109.4712O).
for exchange, the functional of Perdew22 for correlation, and the other acronyms as above. From the results of Tables 1 and 2, one first notes that the UH values are severely underestimated at the LSD level. The same is true for the values of the hfcc a t the carbon atoms (reported here for comparison: in the following, only the H-hfcc will be reported, which have been accessible to experimental measurements).I5 This confirms that, as shown in ref 13, gradientcorrected functionals are mandatory to obtain quantitative results on organic ?r-radicals. The second point worth noting is that the hfcc values exhibit a more pronounced dependence on the choice of the functional form and the auxiliary basis set than usually f o u n d 3 6 ~for ~ ~other ground state properties, notably the ground state energy. Such adependence, however, doesnot prevent theapproach of predicting some gross qualitative features of the experimental data, such as the differentiation of the two a-hydrogens in the allyl radical. Furthermore, the corresponding uncertainty-of the order of 1-2 G-is not larger than that found in ab initio calculations,lo so that it should not possibly preclude the validity of semiempirical extrapolations of the results to the experimental values. In more detail, BP with auxiliary basis set B underestimates the difference between the two a-hydrogens of the allyl radical, whereas with auxiliary basis set A it severely underestimates the hfcc at the CY (and to a lesser extent 8) carbons. Furthermore, PWP with auxiliary basis set A and/or with the T2ZP' basis set gives generally too small hfcc values.
Note furthermore that the UHF/dz geometry seems to be superior to that fully optimized a t the NLSD level, at least as far as the carbon-hydrogen bond distances are concerned (see also ref 40). The situation, however, is not so clear for the carboncarbon and carbon-xygen distances, for which the errors of the U H F and NLSD methods seem to be comparable (see the controversial experimental values for the allyl radical in ref 41). As a final choice, we decided to perform all computations reported in the following on the UHF/dz geometry, utilizing the PWP functional with the DZP' orbital basis set and the auxiliary basis set B: it is our opinion that this choice should minimize the deviation of the calculated hfccvalues with respect to the observed ones. For comparison, we also report the values of the H-hfcc on the CH3, CH3CH2, and CH2CHCH2 radicals obtained at the best available geometries utilizing the PWP functional with the DZP' orbital basis set and the auxiliary basis set B: H, = -2 1.93 (CH3, geometry from ref 3); H, = -21.54, HB= +26.76 (CH3CHz, geometry from ref 42); HA = -13.20, H : = -14.05, Hp = +3.41 (CH2CHCH2, geometry from ref 41). It is apparent (see Table 4 below) that the U H F geometry gives results very close to those obtained with the best experimental ones in these cases. In Table 4 the results of UKS calculations on the full set of radicals are reported, together with (a) the UKS values semiempirically extrapolated as suggested in ref 10, (b) the experimental values (when available), and (c) those obtained through semiempirical extrapolation of extensive ab initio calculations."J The
The Journal of Physical Chemistry, Vol. 98, NO.35, 1994 8651
hfcc of @-KetoenolylRadicals
TABLE 4 hfcc Values (in G) Obtained by NLSD/DZP'
r
Computations Com ared with UQCISD/dza Results and Experimental Data NLSDIDZP'cd radical
MeCHCOO HCOCHCOH MeCOCHCOH MeCOCHCOMe Me-acac
atom
hfcc -21.9 [1.05] -21.3 (-22.4) 26.5 -13.1 (-13.8) -14.2 (-14.9) 3.5 -18.8 (-19.7) 21.4 -15.1 (-15.8) -0.4 -15.4 (-16.2) -0.5 0.010.9 -15.8 (-16.6) 0.010.9 14.9 -0.911.7
(St) UQCISDldP4.d
0.754 0.754 0.766 0.755 0.756 0.756 0.755 0.756
-35.7 [.644] -35.4 (-22.8) 23.6 [1.098] -21.7 (-14.0) -22.6 (-14.6) 6.0 (6.8) -3 1.9 (-20.6) 20.8 (23.7) -24.3 (-15.7) 1.9 (2.2) -24.8 (-16.0) 2.0 (2.3) -1.9 -25.5 (-16.4) -1.9 17.6 (20.1) -2.0
exp. -23.0 -22.5 26.9 -13.9 -14.8 4.2 -19.2 25.0
0 UQCISD stands for unrestricted quadratic configuration interaction with single and double excitations. All the quantum mechanical computations have been performed at UHF/dz optimized geometries. Values obtained through the semiempirical extrapolation are given in parentheses; the values of the semiempirical factors are given in square brackets (see text for details). For the three hydrogen atoms of a methyl group, only the mean value is given; if the three aH do not have the same sign, the arithmetic mean and the mean of absolute values are separated by a slant (/).
semiempirical extrapolation procedure involves extrapolating the predicted hfcc to the experimental values through scaling factors obtained from the simplest possible molecule which contains the kind of atom of interest (i.e., CH3 for a-hydrogen atoms): in Table 4, this procedure has been applied to the hfcc at the a-hydrogen atoms alone. The first four molecules have been chosen as test cases, because they are well characterized from the experimental point of view (the experimental values are taken from refs 1 and 43) and are structurally strictly related to those of interest in the present paper. The other four molecules (Schiff bases) are among the ligands considered in ref 15 and therefore of direct interest in the present context. From the analysis of Table 4 the following conclusions can be drawn. (1) The Sz-value (where S represents the electronic spin) over the reference Slater determinant of noninteracting electrons is close to the theoretical value for a doublet state (0.75), being always smaller than 0.756 (with the only exception of the allyl radical, for which it reads 0.766). This is to be compared with the corresponding results of U H F calculations, in which the S2values for the same radicals range from 0.760 to 0.765 for the methyl, ethyl, or CH3CHCOO- radicals up to 1.1 1.4 for the Schiff base radicals (the allyl radical gives an intermediate value: 0.97-0.99). The UKS method thus seems to reach a reasonable compromise between the request that the reference state be non-spin-contamined (a pure spin state) and the need for a slight spin-contamination, necessary in order to describe the spin-polarization phenomenon at the single-determinant level (a recently proved theorem44asserts that a single determinant with an Sz-value equal to a "pure" value cannot be spin-contamined). Note, however, that such a need might be overcome if one did not evaluate the Sz-value on the reference determinant of noninteracting electrons but considered it to be a functional of the electron density. (2) The results obtained for the a-hydrogen atoms are quite good, especially after the semiempirical extrapolation. The predicted values closely parallel the experimental and/or the best a b initio ones, thus furnishing a reliable way of making predictions on these systems at a very low computational cost.
-
(3) Theresultsobtainedforthe@-hydrogenatomswithinmethyl groups are worse. The predicted value is correct for the ethyl radical, so that no semiempirical extrapolation has been applied to the theoretical values. However, the agreement with the experimental and/or the ab initio values is here less impressive: in particular, it seems that the NLSD calculations tend to underestimate the hfcc values of @-hydrogen atoms of methyl groups directly bound to a conjugated mystem. (4) The results obtained for the y-hydrogen atoms are reasonable: this type of hydrogen atom has not been experimentally observed on these radicals,ls and the corresponding UH have been assumed to be negligible ( I 1 G). Correspondingly, the hfcc are predicted to be very small at the UKS level, with the only possible exception of the C H ~ C O C ( C H S ) C O C H molecule, ~ whose y-hydrogen atom might be observable in proper conditions. ( 5 ) The results obtained for the @-hydrogenatoms lying on the symmetry plane are good: the value is correct for the allyl radical, whereas it is predicted to be not observable for the HCOCHCOH and CHsCOCHCOH radicals, in accordance with experimental measurements (C. Pinzino, private communication). In conclusion, one observes that the overall description of the hfcc in these most elusive radicals seems to be rather good: with the only possible exception of the @-hydrogenatoms of methyl groups directly bound toa conjugated mystem, oneobtains values which are very close to the experimental ones (when available) or to thoseobtained through extensiveab initiocalculations, which, however, require a much larger amount of CPU and/or diskspace facilities.
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