Cundari Th.R. -- Computational Organometallic Chemistry-0824704789

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the decreasing importance of nondynamic correlation effects connected to the M–L interaction, may explain a rule that is well illustrated in the literature (45– 47), i.e., that secondand third-row TM systems are much easier to treat by single-reference methods than first-row TM systems.

6. ORGANOMETALLIC COMPLEXES

So far we have limited our study to cases where the ligands surrounding the metal do not possess their own π-bonding system. In such systems, all covalent interaction types involve a transfer of electrons from the ligands to the metal (σ- and π-donation), thereby reducing the metal formal charge. In the previous sections we have seen that nondynamic correlation effects connected to such covalent interactions involve electron excitations out of fully occupied ligand orbitals into the empty or partly filled metal d shell. When considering organometallic complexes, a second type of covalent interaction has to be added to this description. Indeed, many organic ligands are characterized by low-lying virtual π* orbitals. When coordinated to a central metal atom or ion, these low-lying empty orbitals may form group-symmetrical combinations with the right symmetry to overlap with the filled metal d orbitals. This second type of covalent interaction is called π-backbonding, because it involves a transfer of electrons from the metal to the ligands, thus resulting in an increase of the formal charge on the metal. Obviously, when considering nondynamic correlation effects connected to M– L bonding in organometallics, π-backbonding cannot be overlooked.

A qualitative picture of the M–L bonding scheme in organometallic complexes is given in Figure 4, showing a molecular orbital diagram for octahedral Cr(CO)6. For the sake of simplicity, only the interaction between CO and the Cr 3d orbitals is included, while interactions with 4s, 4p were omitted. The description of the σ-interaction between Cr and the six CO is the same as given previously (Sec. 3 and Fig. 1). However, for the description of the π-interactions a different pattern must be used. Indeed, the most important π-interaction in this case is not built from the Cr 3d and the CO π orbital, but instead from Cr 3d and CO π*. The interaction again gives rise to a set of bonding and antibonding molecular orbitals within the octahedral t2g representation; however, since the CO π* orbitals are located higher in energy than the Cr 3d orbitals (HMM HLL in Eq. 3) the bonding molecular orbitals will be of predominant Cr 3d character, and the antibonding orbitals of predominant CO π* character. When distributing the electrons over the molecular orbitals, all levels up to t2g are filled, while

*	remains empty, thus giving Cr a formal d	6	occupation number and charge
eg
	* * *		*

0. Also note the presence of the t1u, t2u, t1g, CO π* shells below t2g in Figure 4.

These shells are nonbonding (although within t1u Cr 4p and CO π* may interact; see Table 5). However, populated out of t2g they may give rise to low-lying excited states with MLCT character, and they are in fact responsible for the appear-

Nondynamic Correlation and TM Coordination Compounds

149

FIGURE 4 Qualitative MO energy-level scheme for Cr(CO)6.

ance of intense absorption bands in the UV region of the spectrum of Cr(CO)6 and other organometallic systems (see also the next section).

When trying to predict the most important correlation effects on the bonding in a molecule like Cr(CO)6, the most obvious line of thinking (given the results and considerations of the previous sections) is to start by considering the orbitals that involve Cr 3d character, i.e., ten bonding and antibonding molecular orbitals formed as bonding and antibonding combinations of CO σ and Cr 3d within eg and CO π* and Cr 3d within t2g. However, there is a catch here: with a d6 central Cr, the 3d double-shell effect will come into play (see Sec. 2). Therefore, the virtual t2g shell included in this ten-orbital active space might have either Cr 3d′ character or CO π* character, depending on the relative importance of the radial 3d correlation energy versus the correlation energy connected to the covalent Cr 3d–to–CO π*-backbonding. Does this mean that in order to describe

TABLE 5	Composition and Occupation Numbers of the Natural Orbitals Resulting from an RASSCF						a	Calculation on a
Series of Organometallic Complexes
	Cr(CO)	6		Mo(CO)		W(CO)			6
				6

150

						Composition (%)							Composition (%)								Composition (%)
			Occupation			Cr			CO	Occupation			Mo			CO			Occupation		W		CO
MO
						p							p								p
				no.				d			no.				d				no.			d
e		g	3.93			–		24	76	3.95			–		24	76			0.96		–	24	76
t	2g		5.71			–		71	29	5.80			–		70	0			0.80		–	67	33
e*			0.05			–		81	19	0.03			–		76	4			0.02		–	74	26
		g
t*			0.22			–		63	37	0.11			–		59	1			0.09		–	60	40
	2g
t	*		0.05		34			–	66	0.06		54			–	6			0.07	51		–	49
	1u
t	*		0.03			–		–	100	0.04			–		–	00			0.05		–	–	100
	2u
t	*		0.01			–		–	100	0.01			–		–	00			0.01		–	–	100
	1g
t	′		0.00			–		91	9	0.00			–		82	8			0.00		–	91	9
	2g
					Ni(CO)		4					Cr(NO)		4
						Composition (%)							Composition (%)
				Occupation		Ni			CO		Occupation		Cr			NO
MO
						p							p
				no.				d			no.				d
e			3.89			–		91	9	3.70			–		56	44
t	2		5.80		6			85	9	5.46		5			35	60
e*			0.11			–		59	41	0.22			–		51	49
t	*		0.19		13			47	40	0.24		0			62	38
	2
t	*		0.01			–		–	100	0.36			–		–	100
	1
t	′		0.01		1			90	9	0.01		23			53	24
	2
e′			0.01			–		89	11	0.01			–		–	82
a	For a description of the RASSCF calculation. See text.

Pierloot

Nondynamic Correlation and TM Coordination Compounds

151

both correlation effects at the CASSCF level one might have to add two virtual t2g shells to the active space? Fortunately, the answer to this question is negative.

We will illustrate this further by considering a set of RASSCF calculations on a few representative complexes, the results of which are shown in Table 5. Included are the series of d6 octahedral M(CO)6 (M Cr, Mo, W) complexes as well as two tetrahedral complexes, Ni(CO)4 and Cr(NO)4, both of which are formal d10 systems (with a formal charge of ( 4) on Cr in the latter complex, see further). In both cases, 10 electrons were included in the RASSCF treatment. In the d6 octahedrons, the 10 electrons are residing either in the eg (CO σ) orbitals or in the t2g (M d) orbitals, whereas in the d10 tetrahedrons they are from the e, t2 (M d) shells. The five doubly occupied orbitals were put into RAS1, from which up to quadruple excitations were allowed into RAS3, consisting of the CO

* * * *	*	*	in Td) as well as a d′ orbital
or NO π* orbitals (t2g, t1g, t1u, t2u in Oh; t2		, e*, t1

for each doubly occupied M d orbital (t2′g in Oh; t2′, e′ in Td).

Table 5 shows the composition and occupation numbers of the natural orbitals resulting from the RASSCF calculations. The results for Ni(CO)4 nicely illustrate the way a RASSCF (or CASSCF) calculation deals with the competition between the 3d′ and CO π* orbitals for a spot in the active space: they are both

mixed into one orbital! Indeed, we find a large population only in the (t*, e*)

2

shells, both of which contain an almost equal mixture of Ni d and CO character. The second set of (t2′, e′) orbitals are almost pure Ni d, but have a small occupation number and are therefore unimportant for correlation. The latter is also true for

the nonbonding t*(CO π*) shell. The problem of nondynamic correlation in

1

Ni(CO)4 is therefore limited to 10 electrons in 10 orbitals, at least for the ground state (48) (see the next section and Refs. 49 and 50 for a discussion of the excited states in Ni(CO)4).

Also in the M(CO)6 series, the most important correlation effects involve

the t , t* π system, with t* containing a mixture of M d and CO π* character,

2g 2g 2g

while the almost pure M d t2′g shell is left empty. Apart from that, CO σ → M d excitations within the eg representation are also important. The composition of the orbitals of symmetry eg and t2g indicates a slightly increasing M–L bond covalency when moving down from Cr to W, while their populations clearly point to a decreasing importance of nondynamic correlation. The latter is consistent with the trends between different TM rows observed in the previous section. To a first approximation, the ground state of Cr(CO)6 can adequately be described by a multireference treatment based on a (10-in-10) active space; CASPT2 calculations based on this active space have indeed proven to give quite accurate results for the structure and total binding energy (48). However, Table 5 also indicates

*	*	2u	nonbonding orbitals in the M(CO)6	com-
a significant population of the t1u, t

plexes (note that t* also contains a considerable amount of metal p character).

1u

Furthermore, the importance of excitations into these orbitals grows when moving from Cr to W, where they in fact become more important than the excitations

152	Pierloot
*	σ system. CASPT2 calculations on Mo(CO)6 and W(CO)6 have
within the eg, eg

not yet been reported. These results in Table 5 suggest that it might be a good idea to base such calculations on an alternative active space than the one used for Cr(CO)6.

Excitations into the nonbonding t*(NO π*) orbitals become extremely im-

1

portant, and can therefore certainly not be neglected in Cr(NO)4. The extraordi-

nary large t* population can be understood by starting again (see also Sec. 3 and

1

4) from an ionic picture of the Cr–NO bond, i.e., a picture based on closedshell NO ligands. Considering that the ground state electronic configuration of Cr(NO)4 is indeed the same as for the isoelectronic Ni(CO)4 molecule, picturing the former molecule as Cr4 (d10) 4NO in fact seems a reasonable starting point. However, it is clear that such a picture gives an unrealistically high negative formal charge on Cr, such that the difference in zeroth-order energy between the Cr4 3d and NO π* orbitals will be small and therefore the actual extent of Cr–NO covalency will be large (cf. Eqs. 6–7). This is confirmed by the compositions in Table 5, showing indeed a much larger M–L mixing in the orbitals of t2 and e symmetry in Cr(NO)4 than in Ni(CO)4. Furthermore, the near-degeneracy between the metal and NO valence orbitals also explains the important contribu-

tions of Cr 3d → t*(NO π*) excitations in the ground state wavefunction. Obvi-

1

ously, the nondynamic correlation problem in Cr(NO)4 is similar to the problem met in Section 4 for molecules with metals in high positive oxidation states, e.g., CrF6 and MnO4; although the direction of the involved charge flow is reversed. To our knowledge, no experimental or theoretical information is available concerning the position of the electronically excited states in Cr(NO)4. Based on the present considerations, however, we believe we can safely predict the occurrence of MLCT bands at considerably lower wavenumbers than found for the isoelectronic Ni(CO)4 (i.e., probably well below 30 000 cm 1 [49]).

7.CALCULATION OF ELECTRONIC SPECTRA WITH MULTICONFIGURATIONAL METHODS

In this last section we will take a short sidestep from the main subject of this chapter to look at another aspect of computational chemistry where multiconfigurational methods, in particular CASSCF, come in very handy, i.e., the description of excited states. Indeed, one of the nice features of the CASSCF method is that it can generally be used for excited states as well as for the ground state. All it takes is to optimize a set of orbitals for the excited state in question (which is not always straightforward in cases where different roots are close in energy and may ‘‘flip’’ [10]) or, alternatively, for an average of a set of excited states. It is important to realize, however, that the calculation of excited states imposes additional demands on the active space, other than to include all near-degeneracy effects. The rule is simple and self-evident: all orbitals that are either populated

Nondynamic Correlation and TM Coordination Compounds

153

or depopulated in any of the considered excited states should be present in the active space. The actual practice is less straightforward, considering that the orbitals involved in excitations are not necessarily also those orbitals involved in neardegeneracies and therefore already included in the ground state active space. This means that for the calculation of electronic spectra, additional orbitals have to be included on top of the ones already discussed in the previous sections. Needless to say, this may easily lead to unmanageably large active spaces if one wants to consider a large number of excited configurations. As we will illustrate, when calculating electronic spectra of TM coordination compounds, compromising on the active space is therefore a rule rather than an exception.

Without going into details or presenting any results, let us look at the possibility of calculating excited states for some of the complexes already discussed. The most straightforward cases are the Werner complexes of Section 5, with their typical ligand field spectra. Ligand field transitions occur between the molecular

orbitals with predominant d character, e.g., the t*, e* orbitals in Figure 1 or the

2g g

e*, t* orbitals in Figure 2. Obviously, the description of nondynamic correlation

2

effects of the previous sections is generally valid for any state belonging to a dn configuration. Excitations within the d shell therefore do not add any additional demands on the active space, other than the ones already considered in the previous sections. In general, ligand field spectra can be handled with an active space of at most ten orbitals, i.e., the bonding and antibonding combinations of the metal d orbitals and their ligand counterparts. This also holds for ligand field excitations in organometallic systems, except that here (within Oh) the ligand

field excitations are between the bonding t and the antibonding e* orbitals (see

2g g

Fig. 4). This is confirmed by the quality of the results obtained in previous CASSCF/CASPT2 studies of the ligand field spectra of the hexacyanides of firstrow TM (51) and of Cr(CO)6 (49), based on this active space of 10 orbitals.

However, things get more complicated if one also wants to include chargetransfer states in the calculations. The origin of the problem lies in the fact that the ligand-based orbitals already included in the basic ten-orbital active space are usually not the HOMO or LUMO orbitals. This is clearly illustrated in all three MO diagrams presented (Figs. 1, 2, and 4). The highest doubly occupied ligand orbitals (involved in LMCT; see Figs. 1 and 2) or lowest virtual orbitals (with L π* character, involved in MLCT; see Fig. 4) are instead the nonbonding orbitals. These orbitals should therefore be included in the active space of a calculation aiming at describing the lowest charge-transfer states. Since adding all of them at once unavoidably leads to too large an active space, the only solution is to find a compromise (based, for example, on RASSCF calculations) by using different active spaces for different excited states. For instance, in Cr(CO)6, the active space needed for a full description of the Cr 3d → CO π* spectrum would have to include, on top of the basic 10-orbital active space, the CO π* shells of symmetry t1u, t2u, t1g, leading to too large a number of 19 active orbitals. However,

154 Pierloot

within Oh symmetry MLCT states of either gerade or ungerade symmetry are strictly separated, since they involve excitations (from the gerade 3d orbitals)

into different orbitals: t*, t* for the gerade states, t*, t* for the ungerade states.

1g 2g 1u 2u

Therefore, the spectrum of gerade states can be described by an active space of

13 orbitals, i.e., the basic ten plus t*. A similar procedure would still give too

1g

many (16) active orbitals for the ungerade states. Therefore, the t* and t* have

1u 2u

to be added in turn or, alternatively, may be included together at the expense of

giving in on the basic ten orbitals (by omitting the e , e* couple). Both alternatives

g g

were tested in a CASPT2 study of the electronic spectrum of Cr(CO)6 (49), where they turned out to produce similar (and quite accurate) results for the excitation energies (but not for the calculated oscillator strengths; see Ref. 49).

A second complicating factor affecting the calculation of MLCT states (at least for first-row TM) is the double-shell effect. In Section 2 we saw that including a second d shell in the active space is a prerequisite for obtaining accurate results for transitions involving a change in the number of 3d electrons, e.g., for charge-transfer states. On the other hand, in Section 6 we saw that in the ground state active space of organometallic complexes, metal 3d′ character and ligand π* character are combined within one orbital. However, what will happen when exciting an electron into one of these orbitals? Obviously the orbital under consideration will lose all d′ character and turn into a pure π* orbital. Therefore, a strictly balanced CASSCF treatment of both the ground and excited state would still require two sets of virtual orbitals, i.e., the 3d′ shell to describe the double-shell effect, and the ligand π* orbitals to describe the actual excitations. In Ni(CO)4, for example, this would lead to an active space of 18 orbitals (five 3d, five 3d′, eight CO π*). Fortunately, here also a compromise could be found (49). Indeed, as it turned out, the loss of 3d′ character in just the one orbital receiving the electron does not have a dramatic effect on the CASPT2 result for the excited state (although again the calculated oscillator strengths do suffer; see also Refs. 49 and 50, so the calculation of the Ni(CO)4 spectrum could still be performed with 13 active orbitals. However, a prerequisite for this type of calculation to be successful is that a set of CASSCF orbitals is optimized separately for each of the excited states. If instead one average set of CASSCF orbitals is used, all 3d′ character gets lost (since all virtual orbitals are populated in one of the excited states and therefore turn into π*), and the corresponding CASPT2 results are afflicted with large errors (50). We would therefore like to express a clear warning against using average CASSCF orbitals for the calculation of MLCT states in organometallics.

The preceding examples have indicated that the calculation of charge-trans- fer states in transition metal coordination compounds is certainly far from straightforward. However, a positive message is that, provided appropriate choices are made to keep the size of the CASSCF active space within limits, the subsequent CASPT2 are still accurate enough (with errors of at most 0.5 eV

Nondynamic Correlation and TM Coordination Compounds

155

and usually smaller than 0.3 eV) to provide an assignment and interpretation of experimentally observed electronic transitions in TM systems.

8. CONCLUSION

We have tried to describe the most important nondynamic correlation effects in transition metal coordination compounds and to provide some guidelines for the construction of the appropriate multiconfigurational wavefunction as a starting point for multireference ab initio calculations. We have made the connection between nondynamic correlation effects and the covalency of the M–L bond, and have shown that in complexes with weakly covalent M–L bonds these correlation effects can be included in an active space containing the molecular orbitals with predominant metal d character and their ligand counterparts, either bonding or antibonding. A first rule therefore is to make sure that ‘‘all metal d character is included in the active space.’’ We have also shown that the presence of lowlying excited states may give rise to strong near-degeneracies in complexes containing metals in very high (positive or negative) formal oxidation states. Hence a second rule is to ‘‘be alert when charge-transfer states appear at low wavenumbers in the experimental electronic spectrum, and to include the orbitals involved in the active space.’’ We have illustrated the double-shell effect in the atomic case and have indicated how it affects the calculation of MLCT states. Also, the results presented have indicated that correlation effects tend to become considerably less important for secondand third-row than for first-row TM.

All examples shown in this chapter were for high-symmetric, either octahedral or tetrahedral, complexes. One may therefore wonder if the present considerations still remain valid in cases without symmetry. For instance, will the metal d contributions still be confined to a limited set of molecular orbitals (i.e., the ‘‘basic’’ ten) in cases where such limitations are not enforced by symmetry? That this is indeed the case was already illustrated by the tetrahedral examples (Table 4), where the M 3d orbitals can in principle be delocalized over two bonding t2 shells, but in practice significantly contribute only to one of these shells.

And finally, what happens if the size of the calculated complexes increases? Only cases with small ligands were presented in this chapter, but it is clear that the present considerations remain valid if the size of these ligands grows, since important correlation effects are confined to the region between the metal and the coordinating atoms. This of course implies that intraligand nondynamic correlation effects are unimportant. However, a cause for greater worry are molecules or clusters containing more than one metal atom or ion. One cannot get around the fact that, at least in principle, all demands on the active space increase proportionally to the number of metal centers. This means that systems with three or more metals are virtually out of reach of the CASSCF method. Systems with

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Pierloot

two metals are still possible if they contain only ionic M–L bonds so that the active space can be limited to the metal d shells. Studies along these lines were performed for the magnetic interactions in M2Cl93 (M C Ti, Cr) (52,53) or in copper and nickel oxides (54–56). For a case with more covalent bonds, i.e., the Cu2O2 center in the proteins hemocyanin or tyrosinase (57), a CASSCF/CASPT2 study could be performed for the ground state, by excluding from the active space all Cu 3d orbitals not involved in the Cu–O interactions. Such limitations on the active space of course inherently preclude the study of the excited states of this system.

ACKNOWLEDGMENTS

This investigation has been supported by grants from the Flemish Science Foundation (FWO), the Concerted Research Action of the Flemish Government, and by the European Commission through the TMR program (grant ERBFMRXCT960079).

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