J. Phys. Chem. 1993,97, 10948-10951
10948
Atomic Fukui Indices from the Topological Theory of Atoms in Molecules Applied to Hartree-Fock and Correlated Electron Densities Jerzy Cioslowski,' Martin Mertinov, and Stacey T. Mixon Department of Chemistry and Supercomputer Computations Research Institute, Florida State University, Tallahassee, Florida 32306-3006 Received: April 19, 19936
Rigorous definitions of the atomic Fukui indices, based on the topological theory of atoms in molecules, are proposed. Spin contributions to the electron density responses to electron detachment/attachment are quantified with the intra- and interspin indices. The atomic Fukui indices are calculated within five different levels of approximation for a series of six oxygen-containing molecules. Results of these test calculations suggest that in order to obtain reliable estimates of the indices at least the R O H F level of theory has to be employed.
Introduction The Fukui functions have been introduced by Parr and Yangl as partial derivatives (under a constant external potential) of electron density with respect to the number of electrons. Using a finite-differencescheme,the Fukui functions can beconveniently written as linear combinations of changes in electron density that accompany electron attachment or detachment. Because of the well-known derivative discontinuities at the integer values of the number of electrons2,there are several different types of Fukui functions. Among them, of particular interest are the functions
f + ( i )= p + ( i ) - p ( i )
objectives,we present and carefully analyze the results of relevant calculations carried out for several molecules at varying levels of theory. Atomic Fukui Indices
In the past, we have demonstrated5 how the topological theory of atoms in molecules6 can be used to rigorously define such important chemical concepts as covalent bond orders? steric hindrances! and electronegativities,bond hardnesses, and chargetransfer components of bond energies9 in terms of quantummechanical observables. All of these definitions stem from the basic equation6
and
(3)
that are measures of reactivity toward nucleophiles and electrophiles, respectively.' In the above equations, p ( i ) , p + ( i ) , and p-(i) are the ground-state electron densities of the system under study, M, and the systems M- and M+ with one more and one less electron, respectively. Like electron densities, the Fukui functions depend on three Cartesian coordinates. Although such reactivity measures are suitable for graphical displays, a more convenient description of site reactivities is provided by atomic indices. The atomic Fukui indices are usually obtained from atomic charges obtained with the Mulliken population analysis.3 However, in light of the notorious unreliability of population analyses based on Hilbert space partitioning: it is obvious that such definitions yield atomic Fukui indices that have little or no relevance for the determination of site reactivities. Despite the potential importance of the Fukui indices and functions to research on molecular structures and reactivities, thesubject of their accurateevaluation has not attractedadequate attention in the chemical literature. In articular, the question of how the drastic approximations tof ( i )andf-(i) given by the HOMO and LUMO densities1 compare with more accurate estimtes has never been addressed. The influence of electron correlation effects on the Fukui functions and indices has not been examined either. The research presented in this paper aims at providing rigorous definitions of the atomic Fukui indices and finding the computationally least expensive level of electronic structure theory that assures reliable evaluation of their values. To achieve these
D
0
To whom correspondence should be addressed. Abstract published in Advunce ACS Abstracts, October 1, 1993.
0022-365419312097- 10948$04.00/0
that expresses the first-order atomic property described by a oneelectron operator 8 in terms of the spinless one-electron density matrix r and the atomic basin S2a of atom A. Taking into account that the choice of 8 being the unit operator yields the number of electrons associated with a particular atom, we propose that the Fukui indices of an atom in a molecule be defined as (4)
and (5) where 52; and S2, are the atomic basins of the atom A in M- and M+, respectively. The so-defined indices can be rigorously calculated as expectation values and satisfy the sum rules
where the summations run over all the atoms A of molecule M. The changes in electron densities that accompany the process of electron detachmentlattachment are superpositions of responses of the electrons with the same spin (intraspin responses) and those with the opposite spin (interspin responses). In order to quantify these phenomena, we propose the following definitions of the intraspin atomic Fukui indices (7)
and their interspin counterparts 0 1993 American Chemical Society
Atomic Fukui Indices
The Journal of Physical Chemistry, Vol. 97, No. 42, 1993 10949 (9)
In eqs 7-10, p,(i) and p g ( i ) stand for the densities of the a and
0 electrons, respectively, and the electron detachment (or attachment) is assumed to involve the a electron. In analogy with eq 6, such indices satisfy the sum rules
and
The indices, eqs 7-10, can be derived from the spin Fukui funtions introduced by Galvan, Vela, and Gazquez.lo In most systems, the interspin indices that describe (among others) the spin polarization effects are expected to be much smaller in magnitude than their intraspin counterparts. Approximations to the Atomic Fukui Indices While the aforedescribed definitions are capable of providing accuratevalues of theatomic Fukui indices, practical calculations involve approximations associated with a given level of electronic structure theory. In particular, one can apply Koopmans' approximation" and assume that the spin orbitals of the parent system M remain unchanged upon electron detachment/attachment. Such an approach, in which the electron densities are approximated by
pi(?) = pa(?) -+~oMo(~)rc~HoMo(i); P ~ ( V = P S ( ~ ) (14)
where $HOMO(?)and $LUMO(?) are the HOMO and LUMO orbitals of M, has been applied in conjunction with the original definition of the Fukui functions' in order to relate it to the frontier orbital theory.12 With the atomic Fukui indices, one can go even further and freeze the atomic surfaces i2: and i2, at their QA shapes. Such a drastic simplification, which we call here the Koopmans- 1 approximation, yields
in the spirit of the frontier orbital theory.12 In eqs 15 and 16, ($HOMO I $HOMO)Aand ( ' h U M 0 I rhUMO)Aaretherespectiveatomic overlap matrices (AOMS)'~that refer to the parent system M. This means that calculations within the Koopmans-1 approximation do not require separate computations for M, M+, and M-. In the Koopmans-2 approximation, the electron densities are still obtained from eqs 13 and 14, but the atomic basins of M+ and M- are allowed to relax. The calculations carried out within this approximation are more costly than those involving the Koopmans-1 approach as the atomic properties (but not the electronic wave functions) have to be recalculated for M+ and M-. Thecomputed indices allow for atomic basin relaxation, but completely neglect the orbital relaxation, spin polarization, and electron correlation effects.14 The three other approximations discussed in this paper involve atomic surfaces and electron densities calculated separately for M, M+, and M- a t a given level of theory. The R O H F approximation employs quantities calculated within the restricted
TABLE I: Calculated Prowrties of the Parent Svstems energy of M (au) atomic charges in M molecule (M) HF MP2 HF MP2 co -112.7681 -113.1151 C 1.319 1.105 0 -1.319 -1.105 co2 -187.6864 -188.2631 C 2.524 2.133 @ -1.262 -1.066 HzCO -113.9001 -114.2792 C 1.184 1.OOO 0 -1.218 -1.041 Hb 0.017 0.020 HCOOH -188.8228 -189.4189 C 1.851 1.583 OC -1.305 -1.129 od -1.265 -1.129 He 0.077 0.076 I-Lf 0.642 0.599 CH3OH -115.0791 -115.4824 C 0.697 0.521 0 -1.211 -1.098 I-Lf 0.593 0.561 Hg 0.000 0.030 Hh -0.040 4.007 C2H40 -152.9098 -153.4624 0 -1.095 -0.902 C' 0.522 0.377 HI 0.013 0.037 a Two symmetry-equivalentoxygen atoms. Two symmetry-equivalent hydrogen atoms. The0 atom of the CO group. The 0 atom of the OH group. The H atom of the CH group. /The H atom of the OH group. 8 The H atom of the in-plane CH group. Two symmetry-equivalent H atoms of the out-of-planeOH group. Two symmetry-equivalent carbon atoms. 1 Four symmetry-equivalent hydrogen atoms. ~~
open-shell Hartree-Fock level of theory. The resulting atomic Fukui indices reflect atomic basin and orbital relaxation effects, but neglect both spin polarization and electron correlation. Thanks to the different-orbitals-for-different-spins (DODS) approach, the unrestricted Hartree-Fock (UHF) approximation yields indices that include atomic basin, orbital relaxation, and spin polarization effects, but still neglect electron correlation. Finally, the UMP2 approximation, in which the electron densities are calculated with the unrestricted second-order Maller-Plesset perturbation theory, approximately accounts for the electron correlation effects in the resulting indices.
Test Calculations and Discussion In order to assess the reliability and relative accuracy of the aforediscussed approximations, test calculations on six oxygencontaining molecules, namely CO, C02, H F O , HCOOH, CH30H, and C2H4O (ethylene oxide), have been carried out in conjunction with the 6-31 l++G** basis set. The use of such a basis set minimizes the errors in computed quantities due to the basis set incompleteness. The MP2/6-3 1l++G** optimized geometries of the parent systems were used throughout the calculations. The total energies and atomic charges of the parent systems are listed in Table I. In all of the molecules under study, the electronic states of the M+ and M- species used in calculations of the atomic Fukui indices were those with the lowest UMPZ energies. In agreement with the usual trend, the calculated ionization potentials obtained within Koopmans' approximation are larger than those calculated a t the UMP2 level, whereas the ASCF values computed a t both the R O H F and U H F levels are consistently too small. The U H F wave functions of the cations are moderately spin-contaminated, except for those of CO+ and C02+. On the other hand, the UHF wave functions that describe the molecular anions have only marginal spin contaminations. Because of the spin contamination problem, thef-atomic Fukui indices of the CO molecule were not calculated at either the U H F or UMPZ level. Comparison of the Koopmans-1, K o o p mans-2, and ROHF values of thef' and f - indices (Table 11) reveals that for C O both levels of Koopmans' approximation fail to provide adequately accurate results. This is especially pronounced for the intenpinf -indices. Concerning thef' indices,
10950 The Journal of Physical Chemistry, Vol. 97, No. 42, 1993
Cioslowski et al. TABLE I V Fukui Indices of Atoms in the H2CO Molecule
TABLE Ik Fukui Indices of Atoms in the CO Molecule approximation atom(A) index Koopmans-1 Koopmans-2 ROHF
c
approximation UHF
UMPZ
atom(A) index Koopmans-1 Koopmans-2 ROHF
0.862 0.847 0.837 0.010 0.024 0.007 0.872 0.871 0.844 0.138 0.153 0.163 -0.010 -0.024 -0.007 0.128 0.129 0.156 0.787 a a -0.129 a a 0.658 a a 0.213 a a 0.129 a a 0.342 a a
C
Spin contamination is too large for reliable values of Fukui indices.
0
0
C
0
0.726 0.000 0.726 f t(uu) 0.274 f t(Q8) 0.000 0.274 f-fzu) 0.845 f-pb) 0.000 0.845 f-fzu) 0.155 f i40.000 0.155 f tb3
ft'4 ft
ft fa fa
0.910 0.019 0.929 0.090 -0.019 0.071 0.875 0.040 0.915 0.125 -0.040 0.085
f;(uR)
0
atom(A) index Koopmans-1 Koopmans-2 ROHF
c
f ;(mu)
Ha
ft(u8)
011
f; f
;(a3
f ;(am
ft
fi"'
c
f+8)
@
fa j-2")
fi4
fa
UHF
UMPZ
0.320 0.003 0.323 0.340 -0.001 0.338
0.284 0.286 0.331 -0.010 -0.014 -0.008 0.274 0.272 0.323 0.358 0.357 0.335 0.005 0.007 0.004 0.363 0.364 0.339
0.006 0.000 0.006 0.497 0.000 0.497
-0.020 -0.027 -0.047 0.509 0.014 0.524
-0.019 -0.027 -0.046 0.509 0.014 0.523
b b b b b b
b b b b b b
Two symmetry-equivalent oxygen atoms. Spin contamination is too large for reliable values of Fukui indices.
their ROHF values are in good agreement with those calculated at either the UHF or UMPZ level. Similar observations are pertinent to the COZmolecule (Table 111), with the exception that in this case the Koopmans-2 approximation appears to work better than for CO. Both thef andf-atomic Fukui indicesof the H&O molecule (Table IV) are obtained with reasonable accuracy at the ROHF level. Interestingly, the Koopmans- 1 and Koopmans-2 values of thef indices are in good agreement with those afforded by more sophisticated approximations. The indices of the HCOOH molecule (Table V) are more difficult to calculate accurately, with the ROHF approximation required for thef- indicesand the UHF approximation necessary for thef' ones. Finally, for both CH30H (Table VI) and CzH40 (Table VII), accurate values of f a r e obtained at the ROHF level, whereas even the Koopmans- 1 +
+
approximation yields satisfactory accuracy for t h e f
+
indices.
Conclusions Like many other atomic and group quantities of the density functional theory? the atomic Fukui indices can be defined in a rigorous manner within the framework of t h e topological theory of atoms in molecules. In addition, spin-specific indices can be calculated within the same approach. There are several approximationsthat one could use in practical computations of t h e indices. Although computationally c h e a p and conceptually simple, the Koopmans- 1 approximation cannot be recommended for such calculations, as it affords atomic Fukui indices that are too often grossly inaccurate. This failure can be attributed t o t h e fact (compare eqs 15 and 16) that neither negative
f; f;(uu) f
cam
fi C
fp) f-p'
0.096 0.000 0.096 0.084
O.Oo0 0.084 0.410 0.000 0.410
fa f-64
0.097 0.000 0.097 0.637 o.Oo0 0.637 0.133
fi4
O.oClO
fa fp) fi"'
Ha
0.120 0.000 0.120 0.440 0.000 0.440
ff i (iu u ) f ;(am
TABLE IIk Fukui Indices of Atoms in the COz Molecule approximation
f;(4
fa
0.133
UHF
UMPZ
0.039 0.008 0.047 0.026 -0.001 0.025 0.468 -0.004 0.464
0.045 0.057 -0.002 0.013 0.002 0.055 0.058 0.059 0.053 0.051 0.040 0.110 0.026 0.037 -0.018 0.077 0.077 0.091 0.453 0.452 0.446 -0.020 -0.020 -0.019 0.432 0.432 0.427
0.037 -0.075 -0.038 0.636
-0,083 -0,122 -0.205 0.754 -0.135 0.619 0.165 0.128 0.293
0.OOO 0.636 0.164 0.037 0.201
-0.172 -0.038 -0.210 0.801 -0.175 0.626 0.186 0.106 0.292
-0.071 -0.037 -0.108 0.684 -0,158 0.525 0.194 0.098 0.292
Two symmetry-equivalent hydrogen atoms.
TABLE V
Fukui Indices of Atoms in the HCOOH Molecule approximation
atom(A) index Koopmans-1 Koopmans-2 ROHF 0.039 O.Oo0
0.039 0.154 0.000 0.154 0.286 0.000 0.286 0.338 0.000 0.338 0.183 0.000 0.183 0.043 0.000 0.043 0.738 0.000 0.738 0.125 O.Oo0
0.125 0.091
O.Oo0 0.091 0.003 0.000 0.003
UHF
UMPZ
0.01 1 0.001 0.01 1 0.059 -0.002 0.057 0.159 0.001 0.160 0.409 -0.002 0.407 0.363 0.001 0.365
0.043 0.015 0.057 0.070 0.014 0.084 0.113 -0.001 0.111 0.673 -0.035 0.637 0.102 0.008 0.110
-0,008 -0.013 -0.021 0.056 0.007 0.063 0.156 0.004 0.159 0.088 0.022 0.110 0.709 -0.019 0.690
0.003 -0.008 -0.006 0.078 0.008 0.086 0.202 -0.001 0.200 0.106 0.019 0.125 0.614 -0.018 0.595
0.004 -0.042 -0.038 0.748 0.012 0.760 0.133 0.007 0.140 0.112 0.021 0.135 0.003 -0.001 0.002
-0.052 -0.072 -0.124 0.793 -0.118 0.674 0.096 0.053 0.149 0.136 0.109 0.245 0.028 0.028 0.056
-0.102 -0.028 -0.130 0.813 -0.139 0.673 0.112 0.044 0.156 0.151 0.094 0.246 0.027 0.029 0.056
-0.054 -0.017 -0.071 0.732 -0.158 0.572 0.142 0.057 0.199 0.149 0.085 0.233 0.033 0.033 0.066
The 0atom of the CO group. The 0atom of the OH group. The H atom of the CH group. The H atom of the O H group.
values of any of the indices nor non-zero values of the interspin indices can be obtained within this approximation. It should be noted (Tables 11-VII) t h a t the latter indices often have quite large magnitudes. Some i m p r o v e m e n t is provided by the
Atomic Fukui Indices
The Journal of Physical Chemistry, Vol. 97, No. 42, 1993 10951
TABLE VI: Fukui Indices of Atoms in the CHsOH Molecule approximation atom(A) index Koopmans-1 Koopmans-2 ROHF UHF UMPZ C
0
H'
Hb
HC
C
0
H'
Hb
HC
0.033 O.Oo0 0.033 0.149 O.Oo0 0.149 0.143 0.000 0.143 0.133 0.000 0.133 0.27 1 0.000 0.271
0.026 0.007 0.033 0.063 -0.001 0.063 0.213 O.Oo0 0.213 0.142 -0.001 0.140 0.278 -0.002 0.275
-0.019 0.006 -0.014 0.073 0.010 0.083 0.209 -0.006 0.204 0.119 0.012 0.131 0.309 -0.011 0.298
-0.016 0.034 0.005 0.002 -0,012 0.037 0.074 0.102 0.010 0.011 0.084 0.113 0.209 0.225 -0.006 -0.007 0.203 0.219 0.119 0.122 0.012 0.011 0.131 0.133 0.308 0.258 -0.011 -0.009 0.297 0.250
0.120 O.Oo0 0.120 0.648 0.000 0.648 0.004 0.000 0.004 0.004 0.000 0.004 0.112 0.000 0.112
0.079 -0.048 0.032 0.650 0.003 0.654 0.001 -0.004
-0.151 -0.184 -0.335 0.772 -0.143 0.629 0.072 0.069 0.141 0.080 0.080 0.160 0.113 0.089 0.203
-0.181 -0.153 -0.334 0.788 -0.161 0.627 0.061 0.078 0.139 0.082 0.080 0.160 0.126 0.078 0.204
-0,004 0.001 -0.004 -0.003 0.134 0.026 0.161
-0.107 -0.123 -0.230 0.704 -0.159 0.545 0.062 0.073 0.135 0.071 0.073 0.144 0.135 0.068 0.203
a The H atom of the OH group. The H atom of the in-plane CH group. Two symmetry-equivalentH atoms of theout-of-planeOHgroup.
Koopmans-2 approximation, which occasionally affords quite accurate results. However, it fails in an equally large number of the cases tested and it is therefore not recommended for reliable evaluation of the atomic Fukui indices. Interestingly, the results of the ROHF and U H F approximations are usually only marginally different, the one exception being the f + indices for HCOOH. Moreover, trends in the interspin indices calculated at the U H F level of theory are in most cases well reproduced within the R O H F approximation. Thismeans that, in thesystems tested, the spin polarization effects do not contribute substantially to the atomic Fukui indices. Finally, the results of our test calculations show that inclusion of electron correlation at the MP2 level does not change qualitative trends in Fukui indices. On the basis of these observations, we conclude that the R O H F approximation is usually adequate for computations of the atomic Fukui indices that are intended for use in gaining qualitative understanding of chemical reactivity. For highly accurate values of the indices, the UMPZ level of theory should be used unless large spin contaminations of the wave functions of the positive and negative ions are expected. Finally, one should mention that the interspin Fukui indices, introduced in this paper, are likely to find diverse applications in assessment of the reactivity of molecules in processes that involve spin polarization and/or changes of multiplicity, such as photochemical reactions proceeding through the triplet excited states.
TABLE VII: Fukui Indices of Atoms in the C&O Molecule approximation atom(A) index Koopmans-1 Koopmans-2 ROHF UHF UMPZ 0
p u )
0.039 o.oO0 0.039 0.063
ft(u8)
O.oO0
f:
0.063 0.209 O.Oo0 0.209
f ;(a4
fp8'
C"
Hb
ft f
f;(uu)
f t(@
ft 0
C'
Hb
fp' f d4
fa f;;'"'
f-p'
fa fp)
0.313 O.Oo0 0.313 0.309 0.000 0.309 0.0 17
f id)0.000
fa
0.017
0.015 -0.001 0.014 -0.072 0.007 -0,065 0.283 -0.003 0.279
0.031 0.030 0.034 0.016 0.016 0.016 0.047 0.046 0.050 -0.022 -0.085 0.033 -0,059 0.000 0.002 -0.081 -0.085 0.035 0.254 0.285 0.226 0.025 -0.004 -0.005 0.279 0.281 0.220
0.527 0.210 0.737 0.154 -0.156 -0,003 0.042 0.025 0.067
0.753 -0,159 0.593 -0.065 -0.086 -0.151 0.095 0.082 0.177
0.786 -0.189 0.595 -0,093 -0,059 -0.152 0.101 0.076 0.177
0.693 -0.201 0.492 -0.042 -0.034 -0.076 0.098 0.067 0.165
a Twosymmetry-equivalentcarbon atoms. Four symmetry-equivalent hydrogen atoms.
Acknowledgment. This work was partially supported by the National Science Foundation under Contract CHE-9224806, the Florida State University through time granted on its Cray Y-MP digital computer, the U.S. DOE through its Supercomputer Computations Research Institute, and the donors of the Petroleum Research Fund, administered by the ACS (Grant PRF 25076G6). References and Notes (1) Parr, R. G.; Yang, W. J. Am. Chem. Soc. 1984,106,4049. Yang, W.; Parr, R. G. Proc. Natl. Acad. Sci. U.S.A.1985, 82, 6723. (2) Perdew, J. P.; Parr, R. G.; Levy, M.; Balduz, J. L., Jr. Phys. Reu. Lett. 1982, 49, 1691.
(3)
Yang, W.; Mortier, W. J. J . Am. Chem. Soc. 1986, 108, 5708.
(4) See,for example: Cioslowski, J.; Hay, P. J.; Ritchie, J. P. J. Phys. Chem. 1990, 94, 148.
(5) For a review, see: Cioslowski, J.; Surjan, P. R. J. Mol. Struct. (THEWHEM) 1992, 255, 9. (6) Bader, R. F. W. Atoms in Molecules: A Quantum Theory; Clarendon Press: Oxford, U.K., 1990. (7)
Cioslowski, J.; Mixon, S.T. J . Am. Chem. Soc. 1991, 113, 4142.
(8) Cioslowski, J.;
(9)
Mixon, S . T. J. Am. Chem. SOC.1992, 114, 4382.
Cioslowski, J.; Mixon, S . T. J . Am. Chem. Soc. 1993, 115, 1084.
(10) Galvan, M.; Vela, A.;
Gazquez,J. L. J . Phys. Chem. 1988,92,6470.
(11) Koopmans, T. Physica 1933, I , 104. (12) Fukui, K. Science 1982,218, 747. (13)
Biegler-KBnig, F. W.; Bader, R. F. W.; Tang, T. H. J . Comput.
Chem. 1982,3, 317. ( 14) For a discussion on deficiencies of the Koopmans' approximation,see for example: Pickup, B. T.; Goscinski, 0. Mol. Phys. 1973, 26, 1013.
