Electronic structure calculations yield the electronic energy and the wave function of a molecular system in a particular electronic state. The wave function itself is not very suitable for interpret- ation, since it is a function of the coordinates of all electrons. Yet we need simplified characteristics
of the wave function in order to gain insight into the electronic structure of molecules and to predict their chemical reactivities and other properties.
Much work has been done in the field of population analyses to assign a discrete charge to an atom in a molecule. Electron population analysis can be also applied to molecular systems to derive atomic orbital (AO) contributions to MOs.
Within the LCAO–MO formalism, the wave function for theitheigenstate of the molecule/ion can be written as
i ẳ X
ci ð5ị
i ẳ X
ci ð6ị
for an atom localized basis set {}.39
Figure 1 Isocontour surfaces (0.025 A˚) of the HOMO (b1 symmetry) of [Ru(terpy)2]2þ. DFT(B3LYP/
LanL2DZ) calculation (upper picture, Gaussian 98), INDO/S calculation (lower picture, HyperChem 5.11).57 MO Description of Transition Metal Complexes by DFT and INDO/S 653
If the MOs are obtained with zero differential overlap(ZDO) methods (see Chapter 2.38), then the overlapintegrals, S, between different AOs are neglected, and the contribution of theth AO to the ithMO is equal to the square of the corresponding LCAO coefficient,ðciị2 for spin and ðciị2 for spin. As a result, the electron population of the atomAequals to
X
i
ẵiX
2A
ðciị2 ỵi X
2A
ðciị2 ð7ị
whereiruns over all MOs. This is no longer the case if the overlapbetween AOs is not neglected.
To analyze wave functions with non-zero AO overlapit is necessary to include the so-called overlap populations, 2ciciS and 2ciciS, in calculations. Several ways have been proposed in the literature to deal with overlap populations.
2.51.2.1 Mulliken Population Analysis
The most popular and widely used procedure is the Mulliken population analysis (MPA).51–55In MPA, the overlap population 2ciciS (2A,2B) is split equally between atomsAandB, so the contribution of thethAO of atomAto the ithMO is equal to
X
ciciSðspin orbitalị ð8ị
X
ciciSðspin orbitalị ð9ị
where the sum includes all AOs in the molecule. Two objections to MPA are frequently cited:56 1. Mulliken populations can have non-physical negative values or be in excess of two. The
calculated AO contributions to MOs can exceed 100% or be negative.
2. Mulliken populations are sensitive to a basis set, particularly as the basis set is enlarged to get higher accuracy and includes polarization and diffuse functions. Two different basis sets could give identical properties (energy, electron density, etc.) but entirely different Mulliken populations.
The reason for these problems is the imbalance of overlap populations and the net atomic populations, and this imbalance is due primarily to the arbitrary equal distribution of the overlap population between atoms involved. In spite of these two problems, which have to be remembered when analyzing populations of orbitals and atoms, MPA is still frequently used for electron population analysis.
2.51.2.2 Frontier Molecular Orbitals of [Ru(terpy)2]2+
To compare FMOs from DFT and INDO/S, let’s consider the [Ru(terpy)2]2þ complex (terpyẳ2,20;60,200-terpyridine,Figure 1) as an example.Table 1shows irreducible representations, energies, and compositions (obtained using MPA) of the FMOs of this complex.57,58 It can be seen that the FMO compositions derived from INDO/S calculations are very similar to those from DFT. Of course, different XC functionals have different accuracy and produce results of different quality. The local functional SVWN, with a well-known overbinding tendency,25,59 produces more covalent structure and short RuN bonds (1.960 A˚ and 2.031 A˚). The more complex BP86 functional gives more ionic character to the Ru–ligand interactions and accurate RuN bond distances (2.517 A˚ and 2.086 A˚; X-ray:60 1.96–1.98 A˚ and 2.05–2.09 A˚). The widely used hybrid B3LYP functional, which mixes a fraction of HF exchange with KS exchange, gives even higher ionic character to the Ru–ligand interactions and longer than experimental RuN bonds (2.011 A˚ and 2.110 A˚). The INDO/S calculations using these DFT-optimized structures show the same trend: the structure optimized with the SVWN functional displays the greatest
d(Ru)-,*(terpy) orbital mixing, and the structure obtained using the B3LYP functional shows the least amount of the d(Ru)-,*(terpy) mixing.
The agreement between DFT and INDO/S for FMO compositions holds for a large number of coordination compounds (Figure 2). This, however, cannot always be the case. First of all, due to significant differences between DFT and INDO/S, it can be expected that the further away we are from the HOMO and the LUMO the higher the differences between the MOs obtained from DFT calculations and those from INDO/S. Secondly, in order to obtain the agreement between DFT and INDO/S results, both methods should produce a correct description of the electronic structure of the particular molecular system. Since these computational methods do have their limitations, this may not be achievable for all systems under investigation. One such case involves the halide and pseudo-halide complexes of transition metals where DFT calculations with common XC functionals give a ground state MX bonding description that is usually too covalent.45,61,62 The problem appears to be more pronounced in complexes of late transition metals.
There is another difference in description between DFT and INDO/S. Unoccupied MO energies and HOMO–LUMO gaps differ very significantly in DFT and INDO/S. The HOMO–LUMO gaps Table 1 Irreducible representations, energies, and compositionsa of the FMOs of [Ru(terpy)2]2þ.57 INDO/S calculations (results are shown in parentheses) were performed on the corresponding DFT
optimized structures.
MO b SVWN BP86 B3LYP
(D2d) "(eV) Ru(%) "(eV) Ru(%) "(eV) Ru(%) LUMOþ3 b1 8.92 (6.02) 5 (5) 8.31 (6.05) 5 (4) 7.57 (6.01) 3 (4) LUMOþ2 a2 9.05 (6.19) 0 (0) 8.41 (6.18) 0 (0) 7.66 (6.12) 0 (0) LUMOþ0,1 e 9.14 (6.35) 11 (8) 8.54 (6.37) 11 (7) 7.79 (6.31) 8 (6) HOMO b1 11.24 (12.52) 62 (58) 10.58 (12.56)c 66 (62) 11.20 (12.63)c 70 (65) HOMO-1,2 e 11.36 (12.54) 65 (64) 10.68 (12.52)c 68 (68) 11.31 (12.55)c 72 (71) HOMO-3 a2 12.44 (13.68) 0 (0) 11.76 (13.59) 0 (0) 12.36 (13.63) 0 (0) HOMO-4 b1 12.5 (13.82) 3 (6) 11.81 (13.71) 3 (6) 12.41 (13.74) 3 (6)
a AO contributions derived using MPA.57 b Irreducible representations. c The order of these MOs is different in DFT and INDO/S calculations. HOMO and HOMO-1,2 are interchanged.
Figure 2 Ruthenium character of the LUMO of [Ru(NH3)5py]2þ (1), [Ru(NH3)5NO]3þ (2), [Ru(NH3)4
(LL)]2ỵ(LLẳbpy (3), bpz (4), dioxolene (5), benzoquinonediimine (6)), [Ru(terpy)2]2ỵ(10) and ruthenium character of the LUMOỵ1,2 of [Ru(LL)3]2ỵ(LLẳbpy (7), bpz (8), benzoquinonediimine (9)), as derived
from DFT (B3LYP/LanL2DZ) and INDO/S calculations using MPA.57,58
MO Description of Transition Metal Complexes by DFT and INDO/S 655
obtained from DFT calculations are always smaller than those calculated usingab initioor INDO/S methods. Let’s take the [Ru(terpy)2]2þ complex as an example again. The HOMO–LUMO gap (Table 1) from the DFT calculations is 2.0–2.1 eV (SVWN and BP86), or 3.4 eV (B3LYP), reason- ably close to the lowest energy MLCT band observed in the electronic spectrum of [Ru(terpy)2]2þ (2.62 eV63,64). On the other hand, INDO/S gives a much higher value of the HOMO–LUMO gap (6.2 eV). The reason for such a discrepancy between DFT and INDO/S is explained below.