J . Phys. Chem. 1990, 94, 148-151
148
Charge Distributions and Effective Atomic Charges in Transition-Metal Complexes Using Generalized Atomic Polar Tensors and Topological Analysis J. Cioslowski,*.t P. J. Hay,+ and J. P. Ritchief Los Alamos National Laboratory, Theoretical Division, Los Alamos, New Mexico 87545 (Received: March 17, 1989; In Final Form: June 15, 1989) Advantages and shortcomings of three different definitions of the atomic charges, namely, the Mulliken, the generalized atomic polar tensors (GAPT), and the topological ones, are judged by applying them to the results of ab initio calculations on the TiF,, Ni(C0)4, and FeH2- molecules. In agreement with previous reports, we find that the Mulliken charges vary widely with the choice of basis sets and therefore their utilization for the analysis of electronic structure of the transition-metal complexes is of little practical importance. On the other hand, both the GAPT and Bader's charges show a remarkable insensitivity to the quality of the basis sets.
Introduction The analysis of molecular wave functions in terms of chemical concepts such as net atomic charges, bond order, and hybridization has played a central role in theoretical chemistry. In the case of LCAO-based methods, such as the Hartree-Fock calculations now routinely applied to molecules, the analysis proposed by Mullikenl remains the most widespread for providing net atomic populations, bond overlap populations, and effective charges. Various alternative approaches2-8have been proposed by numerous investigators in efforts to put such analyses on a more rigorous footing. Numerous ab initio Hartree-Fock calculations on transitionmetal compounds have been carried out in recent years. While investigators typically do report Mulliken populations, they invariably stress that the quantities themselves lack inherent physical significance and that the trends between different molecules are more meaningful. For example, negative orbital populations are often observed and negative bond overlap populations can also be obtained for stable chemical species9 It has been generally recognized that one of the inherent problems lies in the difference in spatial extent between the relatively diffuse 4s and 4p orbitals, on the one hand, and the comparatively smaller 3d orbitals, on the other. This, in turn, leads to difficulties in apportioning charge to the metal and ligand centers according to a general prescription. These problems have led us to investigate two alternative approaches: the first is a scheme recently developed by one of the authors for population analysis based on atomic polar tensors;I0 the second is the Bader topological analysis of molecular charge densities.l"18 To our knowledge neither approach has been applied previously to transition-metal compounds. Theoretical Approaches
The theoretical approaches employed in this study are summarized briefly below. More detailed references to earlier work are given in each section. Mulliken Charges. The conventional Mulliken charge assigned to atom A (the gross population at atom A) is given as fo1lows:l
The first sum extends over the basis set functions centered at atom A. The second sum contains overlap contributions from the basis set functions centered at other atoms. In eq 1 P and S are the bond-order and overlap matrices, respectively, and ZAis the atomic number of A. One has to point out that the Mulliken charges are not true atomic properties: they cannot be computed from the electronic wave function itself. *To whom correspondence should be addressed. Present address: Department of Chemistry and Supercomputer Computations Research Institute, Florida State University, Tallahassee, FL 32306-3006. 'Group T-12. Mail Stop 5569. 'Group T-14. Mail Stop B214.
0022-3654/90/2094-0148$02.50/0
Generalized Atomic Polar Tensors. It has been shown that an expectation value of any origin-independent operator can be partitioned into atomic contributions by using the notion of generalized atomic polar tensors (GAPT).l0 In particular, for the atomic charge one obtains
This expression can be identified with the one-third of a trace of the atomic polar tensor." The atomic polar tensor of atom A is defined by the components of the first derivatives of the dipole moment of the molecule with respect to the Cartesian coordinates of center A. Calculation of GAPT charges requires solving the CPHF equations as the atomic polar tensor is a second-order response property. An alternative route involves computation of the geometry gradients in the presence of an electric field followed by a numerical differentiation. Topological Analysis of the Electron Density. Atomic charges in the topological method are obtained as the difference between the respective nuclear charge and the number of electrons contained within the volume flA (called the atomic basin) traced out by all the paths of steepest descent in the electron density emanating from the nucleus in question.12
Thus, the vector quantity ?p(?), the negative of which defines these paths, is of central concern. Also of interest is the scalar quantity V2p(?). It has been shown that these quantities can be used to define molecular structure, to characterize atomic interactions, and to identify reactive sites in molecules. Some atoms are joined by gradient paths emanating from the respective nuclei and terminating at a common critical point, where Vp = 0. If this critical point is found to be a minimum in p along the path linking the atoms and a maximum in the other two orthogonal directions, then the paths are called bond paths and the common critical point ( 1 ) Mulliken, R. A. J . Chem. Phys. 1955, 23, 1833, 1841, 2338, 2743, 3428. (2) Lowdin, P.-0. J. Chem. Phys. 1950, 18, 365; 1953, 21, 374. (3) McWeeny, R. J. Chem. Phys. 1951, 19, 1614; 1952, 20, 920. (4) Davidson, E. R. J. Chem. Phys. 1967, 46, 3320. ( 5 ) Pollack, M.; Rein, R. J . Chem. Phys. 1967, 47, 2045. (6) Cusachs, L. Ch.; Politzer, P. Chem. Phys. Lett. 1968, I , 529; 1968, 2,
I.
(7) Christofferson, R. E. Chem. Phys. Lett. 1971, 8, 4. (8) Noell, J. 0. Inorg. Chem. 1982, 21, 11. (9) Ammeter, J. H.; Burgi, H.-B.; Thibeault, J. C; Hoffmann, R. J . Am. Chem. SOC.1978, 100, 3686. (10) Cioslowski, J. Phys. Rev. Letr. 1989, 62, 1469. (1 1 ) See for example: Person, W.; King, W. In Vibrational Intensities in Vibrational and Raman Spectroscopy; Person, W., Zerbi, G.. Eds.; Elsevier: New York, 1982. (12) Bader, R. F. W.; Nguyen-Dang, T. T.; Yal, Y. J . Chem. Phys. 1979, 70,4316. Bader, R. F. W.; Anderson, S. G.; Duke, A. J. J . Am. Chem. SOC. 1979, 101, 1389.
0 1990 American Chemical Society
Atomic Charges in Transition-Metal Complexes TABLE I: "Large" Basis Sets for Ti and Fe Ti
r 206 082 3 I 226.8 7 199.32 2048.75 670.790 243.650 95.925 0 39.810 1 12.220 5 5.000 82 1.28569 0.5 12 806 0.209485 0.085 576 0.033 302 1264.70 301.230 96.977 7 36.372 7 14.781 4 6.27465 2.478 78 1.016 18 0.398 I62 0.156 009 0.061 128
The Journal of Physical Chemistry, Vol. 94, No. 1, 1990 149
Fe
C
C
s Functions 0.000 25 257 539 0.001 93 38 636.9 0.009 88 8 891.44 0.039 94 2 544.01 0.12737 844.717 1 .o 1.o
1 .o
1 .o 1 .o 1 .o
1.o 1 .o 1 .o 1 .o
31 2.527 125.593 53.498 7 17.715 1 7.376 77 2.01 8 47 0.779 935 0.298 470 0.1 I4220 0.041 889
p Functions 0.002 14 1678.40 0.01725 396.392 0.081 17 128.588 0.241 43 49.1 15 8 1 .o 1 .o 1 .o
1 .o 1.o 1 .o 1 .o
20.503 5 8.987 12 3.68249 1.521 75 0.592 684 0.230837 0.089 906
25.992 4 7.086 34 2.348 71 0.800 198
d Functions 0.022 53 41.452 6 0.11961 11.5403 0.327 00 3.885 43 0.466 26 1.323 80
0.262 049 0.072 000
1 .o 1 .o
0.416 680 0.113300
0.000 29 0.002 26 0.01 1 52 0.045 66 0.14035 1.o 1 .o 1.o 1 .o 1 .o 1.o 1.o 1 .o 1.o 1 .o 0.002 49 0.020 15 0.091 99 0.259 91
1 .o 1 .o 1 .o 1 .o 1 .o 1.o
TABLE 11: Basis Sets Used in Calculations basis set central atom ligands TiF, 3-21G S/S 3-21G (5D) S/L 3-21G (5D) 6-31G L/S (15sllp6d/ll~8p3d) 3-21G L/L ( 1 5 ~ l l p 6 d / l l ~ 8 p 3 d ) 6-31G
S/S S/L L/S
L/L
Ni(CO), 3-21G (5D) 3-21G 3-21G (5D) 6-3 1G (1 5sl lp6d/ 1ls8p3d) 3-2 1G ( 1 5 ~ l l p 6 d / l l ~ 8 p 3 d ) 6-31G
S/S S/L L/S L/L
FeH,& 3-21G (5D) 3-21G (5D) ( 1 5 ~ lp6d/l 1 ls8p3d) ( 1 5 ~ lp6d/l 1 ls8p3d)
EHF,hartrees -1240.164858 -1 242,164926 -1244.405604 -1246.367623 -1947.729678 -1950.013 902 -1 955.143 487 -1957.400314
3-2 1G -1 258.1 18 200 ( 5 ~ 1 3 ~ ) -1258.166865 3-21G -1264.259909 ( 5 ~ / 3 ~ ) -1264.391 736
TABLE 111: Results of Various Population Analyses of the TiF, Molecule basis set
s/s S/L LIS L/L
atom Ti F Ti F Ti F Ti F
QMUll 1.948 -0.487 2.028 -0.507 1.501 -0.375 1.536 -0.384
QGAn
2.998 -0.766 3.102 -0.776 2.900 -0.725 2.98 1 -0.745
Qtop
2.708 -0.677 2.706 -0.676 2.804 -0.701 2.772 -0.693
1.o
0.025 11 0.136 26 0.353 23 0.468 67
1 .o 1.o
is called a bond point. The set of all bond paths in a molecule provides a topological definition of molecular structure by indicating the major pairwise interaction^.'^ Characterization of these interactions is accomplished by calculating values of p ( 7 ) and V2p(?)at the bond point, denoted pcritand V2pht. For example, bond order and the value of pcritare closely related.14 In addition, the value of V2pCri,serves to indicate the bond type.I5 Positive values are frequently found when the bonding is characterized by closed-shell interactions, as in ionic and hydrogen bonds. Negative values are characteristic of shared interactions, as in covalent bonds. Finally, V2p(7) may also serve to identify sites of nucleophilic and electrophilic attack,16 as well as to provide a basis for understanding molecular geometry.I7 Calculations reported in this paper were performed using the PROAIMprogram,18 which was modified to handle the basis sets used herein. Details of the Calculations Molecular geometries for the theoretical calculations were taken from experiment. For the tetrahedral molecules TiF, and Ni(CO), (13) Bader, R. F. W.; Nguyen-Dang, T. T.; Yal, Y. Rep. Prog. Phys. 1981, 44,893. Bader, R. F. W.; Nguyen-Dang, T. T. Adv. Quantum Chem. 1981, 14.63. Bader. R. F. W . In The Force Concept in Chemistry; Deb, B. M., Ed.; Van Nostrand: New York, 1981; p 39. (14) Bader, R. F. W.;Slee, T. S.; Cremer, D.; Kraka, E. J . Am. Chem. SOC.1983, 105, 5061. (IS) Bader, R. F. W.; Essen, H. J . Chem. Phys. 1984,80, 1943. (16) Carroll, M. T.;Chang, C.; Bader, R. F. W. Mol. Phys. 1988,63,387. Bader, R. F. W.; MacDougall, P. J. J. Am. Chem. SOC.1985, 107, 6788. (17) Bader, R. F. W.; Gillespie, R. J.; MacDougall, P. J. J . Am. Chem. SOC.1988, 110, 7329. Bader, R. F. W.;MacDougall, P. J.; Lau, C. D. H. J . Am. Chem. SOC.1984, 106, 1594. (18) Biegler-Koenig, F. W.; Bader, R. F. W.; Tang, T. H. J . Comput. Chem. 198Z3.3 17. The authors wish to thank Professor Bader for providing them with a copy of this program.
TABLE I V Topological Analysis of the Electron Density in the TiF, Molecule" basis set SIS
SIL
LIS
LIL
1.7047 0.1552 0.8659
1.6913 0.1515 0.9873
1.6848 0.1534 0.81 11
1.6724 0.1491 0.9545
5333.8 337.28
5333.7 420.59
6943.2 337.40
6943.2 420.57
Ti-F bond rri-crit Pcrit
V2Pcrit
atoms PTi PF
OAll entries in atomic units.
bond lengths were as follow^:^^^^^ Ti-F, 1.745 A; Ni-C, 1.838 A; C-0, 1.141 A. In the octahedral complex FeH6' a Fe-H bond length of 1.609 A was taken from the crystal structure of FeH6Mg4Br,,SClo,5(THF)B.Z1 At the level of theory employed here, FeHt- is not a stable species itself but represents an element of the crystal structure. The metal and ligands in these complexes were each described by two different levels of basis sets. For the ligands the smaller (S) basis set consisted of the 3-21G basis22developed for first-row atoms and hydrogen; for the metal the smaller set was the 3-21G basisz3derived for third-row metals. The more accurate 6-3 1G basis24was used as the larger (L) set for first-row atoms, and a [5s]/(3s) contracted basis set25for hydrogen was employed. The (14s9p5d) primitive basis of Wachters26provided the starting point for the larger basis sets of the transition-metal atoms. Following (19) Spectroscopic Properties of Inorganic and Organometallic Compounds; Royal Society of Chemistry, The Garden City Press Limited: London, 1984; Vol. 16, p 362. (20) Hedberg, L.; Ijima, J.; Hedberg, K. J . Chem. Phys. 1979, 70, 3224. (21) Bau, R.; Chiang, M. Y.; Ho, D. M.; Gibbins, S. G.; Emge, T. J.; Koetzle, T. F. Inorg. Chem. 1984, 23, 2823. (22) Binkley, J. S.; Pople, J. A.; Hehre, W.J. J . Am. Chem. SOC.1980, 102, 939. (23) Dobbs, K. D.; Hehre, W. J. J . Comput. Chem. 1987, 8, 861. (24) Hehre, W. J.; Ditchfield, R.; Pople, J. A. J . Chem. Phys. 1972, 56, 2251. (25) Huzinaga, S. J. Chem. Phys. 1965, 42, 1293. (26) Wachters, A. J. H. J . Chem. Phys. 1970, 52, 1033.
Cioslowski et al.
150 The Journal of Physical Chemistry, Vol. 94, No. 1 , 1990 TABLE V: Results of Various Population Analyses of t k Ni(C0). Molecule basis set atom QM~II QGAF-T Qtop Ni 0.103 -1.369 0.427 SIS C 0.459 1.210 0.942 -1.049 -0.485 -0.868 0 Ni -0.524 -1.382 0.441 Slf1.303 1.003 C 0.541 -0.957 -1.113 -0.410 0 Ni -1.166 0.214 0.339 LIS 1.034 1.008 C 0.398 -0.743 -0.482 0 -1.062 -1.228 0.295 -0.420 Ni LJL 1.144 1.054 0.454 C -0.837 -0.350 0 -1.128
a similar procedure employed in a series of calculations on Ni complexes by Schaefer et al.,27the basis was augmented by an interior s orbital intermediate between 3s and 4s in character and by two diffuse p orbitals appropriate for the 4p atomic orbital. In each case the exponents of the added functions were chosen in a geometric progression relative to the original basis. An additional diffuse d orbital was also added to allow for expansion of the 3d orbitals.28 The resulting (15sl lp6d) basis was contracted to [ 1ls8p3dI; the basis sets for Ti and Fe appear in Table I. Calculations on the metal complexes are denoted as (S/S), (S/L), (L/S), and (L/L) depending on the choice of basis for the (metal/ligand) under consideration (Table 11).
Results Both the GAPT and topological charges are compatible with the ionic character of the TiF4 molecule (Table 111). They are nearly invariant to the different basis sets used. The variation of Q(Ti)amounts to 0.2023 and 0.0960 for the GAPT and topological charges, respectively. The Mulliken charges exhibit a much larger spread of 0.5267 and are considerably smaller in value. The values shown in Table IV provide further evidence of the ionic interactions in TiF4. Values of V2pcit are relatively large and positive, indicating closed-shell interactions of the Ti and F. It is also interesting to note that values of petit are quite insensitive to the choice of basis set. Values of V2pcit show larger, but still relatively small, variations. Since V2pd, is a derivative quantity, actually second derivative, it is not surprising that it shows more sensitivity to basis set than the electron density function itself, pcrit.Magnitudes of the electron density at the nuclei show quite large variations due to the Gaussian basis sets used, but this is more closely related to the behavior of core electrons rather than valence electrons. Thus, the descriptions of atoms and their interactions in the topological theory are relatively insensitive to (reasonable) choice of basis set for TiF4. The Ni(C0)4 system is an example of a molecule with nonpolar metal-ligand bonds. Unlike the charges in the TiF4 molecule, the results of the GAPT and topological analyses differ significantly for Ni(CO)+ This can be explained as follows: the nonzero effective atomic charges arise as a result of both charge transfer between atoms and the distortion from the spherical symmetry of the electron cloud assigned to a particular atom. These effects are weighted differently in the GAPT and Bader's definitions. In the ionic systems, there is almost no distortion and the resultant atomic charges arise from chargetransfer effects. Therefore, the GAPT and topological charges are very similar in these systems. However, both mechanisms are operative in nonpolar and/or covalent systems resulting in different magnitudes of effective atomic charges. Despite these differences, both the GAPT and topological charges do not vary much from one basis set to another. The are 0.2166 and 0.2277. The analogous respective spreads of QNi) figure for Mulliken charges is much higher: 0.6265. In fact, the Mulliken charge on the iron atom even changes sign going from (27) Spangler, D.; Wendoloski, J.; Dupuis, M.; Chen, M. M. L.; Schaefer H.F. J . Am. Ckem. Soc. 1981, 103, 3985. (28) Hay, P J. J . Ckem. Phys. 1977, 66, 4377.
TABLE VI: Results of Various Population Analyses of the CO Molecule basis set 3-21G 6-31G
atom C 0 C 0
QMull 0.437 -0.437 0.346 -0.346
QGam
Qtop
0.334 -0.334 0.406 -0.406
1.092 -1.092 1.148 -1.148
TABLE VII: Topological Analysis of the Electron Density in the Ni(CO), Molecule" basis set
SJS
SJL
LIS
LJL
1.7785 0.1283 0.5504
1.7799 0.1301 0.5367
1.7365 0.1250 0.6462
1.7382 0.1262 0.6226
0.7493 0.4410 0.5092
0.7325 0.4467 0.4732
0.7490 0.4400 0.5169
0.7317 0.4467 0.4775
11201 94.753 234.05
11199 119.20 291.78
14535 94.943 234.11
14535 119.21 291.77
Ni-C bond rNi-crit
Pmt
VZPcrit
C-0 bond rc-crii Peril
VZPCrit
atoms PNi PC PO
"All entries in atomic units.
TABLE VIII: Topological Analysis of the Electron Density in the CO Molecule" basis set 3-21G
6-31G
0.7337 0.4429 1.0392
0.7 197 0.45 11 0.8770
94.808 234.00
119.41 291.82
C-O bond rcsrit Pnit
V2Pcrit
atoms PC
Po
"All entries in atomic units.
the (S/S)and (L/S) basis sets to the (L/S) and (L/L) ones (Table V). Similar conclusions (differences between the GAPT and topological charges and relative insensitivity to the choice of the basis set) can be drawn from the results of calculations on the CO molecule itself (Table VI). Results of the topological analysis, summarized for Ni(C0)4 in Tables V and VI1 and for CO in Tables VI and VIII, are consistent with nonpolar closed-shell Ni-CO interactions, as suggested by standard models of r complexes. The topological results shown in Table V show relatively small charges on the metal, especially in the larger and presumably better basis sets. In the L / S and L/L basis metal charges of 0.21 and 0.29, respectively, are found. These correspond to charge transfer to each CO molecule of about 0.05 and 0.07. In addition, comparison of the topological charges for CO as a free molecule, presented in Table VI, with those for CO in Ni(CO), demonstrates the charges on C and 0 to be only slightly perturbed by binding with the metal atom. Thus, the m e t a l 4 0 interaction is relatively nonpolar and the ligands are clearly identified as perturbed CO molecules. Table VI1 contains values of pEti, and V2pEtitfor the Ni-C and C-O bonds. Again it is found that pEtit is remarkably insensitive to the choice of basis set and that V2pcrit,while showing larger variations, behaves at least qualitatively similarly. The small values of pait along with moderately large and positive values of V2pdt for the Ni-C bond are consistent with a dominant contribution of closed-shell metal-ligand interactions. These reflect , ~ ~ are the character of u donation and 7~ b a ~ k - d o n a t i o nwhich
111,
(29) Bauschlicher, Jr., C. W.; Bagus, P. S . J. Chem. Phys. 1984, 44, 893.
Atomic Charges in Transition-Metal Complexes
The Journal of Physical Chemistry, Vol. 94, No. 1 , 1990 151
TABLE IX: Results of Various Population Analyses of the FeHf Ion
basis set
atom
QMull
QOAm
Qw
SJS
Fe H
SIL
Fe H Fe H Fe H
-1.513 -0.415 -2.202 -0.300
-0.724 -0.546 -0.382 -0.603 -0.269 -0.622 0.405 -0.734
0.956 -0.826 1.021 -0.837 0.951 -0.825 0.756 -0.793
LIS LIL
-0.110
-0.648 2.209 -1,035
TABLE X: Topological Analysis of the Electron Density in the FeHf Ionu basis set SIS S IL LIS L/L Fe-H bond 1.8119 1.7828 1.7886 1.8142 ‘Ftcrit 0.0729 0.0696 0.0657 0.0628 Pcrit 0.3603 0.3774 0.3806 0.4187 V2Pcrit ~~~~
~
atoms PFc
PH
8919.7 0.2890
8919.6 0.3530
11572 0.2910
11572 0.3670
“All entries in atomic units.
typically envisioned as an electron pair interacting with a vacant orbital. It can also be noted that, comparing the values of pdt(C-O) and V2pc,it(C-O) for CO bound or free, the ligated CO has a lesser C-O bond order and is less dominated by closed-shell C-0 interactions than the free CO. This is again expected from ?r complex models. Finally, we present the results for the FeH2- ion. In this case we have deliberately used basis sets that are not large enough to describe the H- ions correctly. This allows us to investigate the differences in atomic charges in situations where the basis sets are of a poor quality. The Mulliken analysis fails completely to provide even qualitative results (Table IX). The gFe) varies from -2.20 to +2.21 which corresponds to a spread of 4.41! The GAPT charges fare much better (a spread of 1.13), but not well enough to be of quantitative value. Finally, the topological charges show a remarkable stability (spread of 0.26). This stability, however, is not shared by other properties partitioned within the Bader scheme. While the atomic charges of the H atoms vary within only 5.5%, the atomic dipole moments change by more than 50% (from ca. 1.4 D for the L/S basis to ca. 2.6 D for the S / S basis). This clearly demonstrates a severe incompleteness of the basis sets used. The topological values shown in Table X are in accord with those obtained in the other systems. pWitand V2pd, are relatively insensitive to basis set choice, while the electron density at the nuclei is much more sensitive. The small values of pdt along with the positive values of V2pcritindicate closed-shell interactions as expected of the ionic Fe-H bond indicated by the calculated charges.
Discussion As mentioned in the Introduction, many methods of obtaining charges of atoms in molecules have been proposed. These methods can be divided into two distinct categories. The first one comprises the basis set based charge distribution analyses. The Mulliken’ and the recently proposed31 natural population analyses belong to this category. The second category includes the molecular property based charge analyses. Methods of the first variety attempt, by one means or another, to assign populations to atomic orbitals input as basis functions. The Mulliken analysis is typical of this approach. These methods, although quite popular, map the molecular electron density onto a set of basis functions, which
is an essentially arbitrary exercise. The assignment of “overlap populations” is but one example of the choices to be made for which there is no fundamental logic by which this decision can be made. Hence, these methods measure whether the basis set being used is properly balanced in terms of describing an atom’s charge, and these methods are frequently quite basis set dependent. As pointed out a long time ago,m the second shortcoming of these methods can be illustrated by imagining a set of basis functions for ammonia placed only on the N atom. Even though such a basis set is perfectly capable of providing a relatively good description of ammonia, provided the expansion is carried to high enough order, basis set based charge analyses will never get any charge on the hydrogens-a result which is entirely unrealistic. Likewise, basis functions placed somewhere other than a nucleus will also be assigned finite charges belonging to the basis set center. On the other hand, molecular property based charge analyses attempt to make charge assignments derived from molecular properties. Since the molecular properties are usually observables, such schemes can, at least in principle, be compared with experiments. The topological theory analyzes the electron density directly and provides a compact mathematical description of the charge distribution. This description has elements similar to traditional descriptions of bonding in molecules. The GAPT analysis provides a model of molecular charges based upon the response of the molecular dipole moment to changes in the positions of the nuclei. It is also worth noting that molecular property based charge analyses are not so vulnerable to basis set choices as the basis set based methods, since they do not attempt mapping of the electron density into basis sets. While the property based approaches do require significantly more computational effort (typically of the same order of magnitude as the S C F calculation necessary to generate the wave function) than basis set based methods, they do not suffer from the sensitivity to changes in basis set as discussed above. Caution should be exercised, however, in using these approaches in the interpretation of results based on calculations with very poor basis sets which can be quite misleading. Of the two approaches, the GAPT generally requires significantly less effort and can be carried out routinely by using standard ab initio programs. The results described in this paper corroborate these views. It was found that the Mulliken analysis was quite sensitive to basis sets, sometimes giving different bond polarizations in different basis sets. On the other hand, the GAPT and topological analyses were frequently in agreement and showed much less sensitivity to the basis sets.
Conclusions No population analysis can be better than the wave function used to describe the molecule. Obviously, if the properties of the molecule are poorly reproduced, no population analysis can produce good charges. Nevertheless, the use of incomplete basis sets is a practical necessity and it is generally believed that calculations using finite basis can be at least qualitatively reliable. We thus recommend the use of molecular property based charge analysis because they typically contain few or no arbitrary assumptions and depend upon a molecular property, which presumably can be quantitatively reproduced. We have shown that such analyses are relatively insensitive to basis set choice for some transitionmetal complexes. Acknowledgment. This research has been pursued under the auspices of the US.D.O.E. Registry No. TiF,, 7550-45-0; Ni(CO),, 13463-39-3; FeH:-, 91208-66-1. (30) Politzer, P.; Milliken, R. S. J . Chem. Phys. 1971, 55, 5135. (31) Reed, A. E.; Weinhold, F. J . Chem. Phys. 1983, 78, 4066. Reed, A. E.; Weinstock, R. D.; Weinhold, F. J . Chem. Phys. 1985, 83, 735.
