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

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3d3 and 3d24s1 is overestimated by 0.56 eV. The smaller error for this ion as compared to Co is of course related to the reduced number of 3d electrons. When looking instead at the d → d transitions, we find that all excitation energies are overestimated at the lowest, CAS(6), level of calculation, although the absolute errors are much smaller than for the d → s transitions: up to 0.4 eV for Co and Rh , and only 0.12 eV for Ti .

Including correlation effects should take care of the large errors obtained at the CAS(6) level. Two approaches are presented in Table 1: in a first CASPT2 calculation based on CAS(6), all correlation effects are treated by second-order perturbation theory; in a second approach, the important radial correlation effects are first dealt with by the larger CAS(11) calculation, and perturbation theory is used only for the remaining correlation effects. Let us look at the results for Co first. Here we find that the first, perturbative, approach quite strongly overestimates the differential correlation effects on the d → s transitions. Indeed, at the CAS(6)/CASPT2 level all three d7s1 states are calculated about 0.3 eV too high in energy with respect to the d8 ground state. On the other hand, considerable improvements of the d7s1–d8 splitting are obtained even at the CASSCF level when including a second d shell in the active space: all d → s transitions are still calculated too low, but the error has been reduced to less than 0.5 eV. A further improvement is obtained at the CAS(11)/CASPT step. The final CASPT2 results are, however, still too low by up to 0.3 eV. This final error should be traced back to the incapacity of the limited basis set used to describe to its full extent the large 3d correlation effects and their variation with the number of 3d electrons present.

The mere fact that the CASPT2 results are so strongly dependent on the size of the CASSCF active space clearly points to the necessity of including a second d shell in the reference correlation treatment of any transition involving a change of the number of 3d electrons in the considered Co ion. One might argue that the overshooting of correlation effects in the CAS(6)-based calculation is merely a result of the perturbative approach in the second step and that a CI or CC treatment based on the same reference wavefunction should give much superior results. This is of course true; however, the experience with the Ni atom (20) has indicated that at the MRCI level too, including (the most important) 3d–3d′ excitations in the reference wavefunction is a prerequisite for obtaining quantitative accuracy. To our knowledge, no coupled-cluster treatment of 3d → 4s transitions in transition metal atoms or ions has been performed with a basis set that is larger enough to be able to decide whether or not the 3d radial correlation effects can be captured by a single-reference treatment using this method. This is therefore still an open question.

Naturally, the double-shell effect should play a less important role for the Co 3d → 3d transitions. Indeed, the difference between the CASSCF results (Table 1) obtained with or without the 3d′ shell is much more limited for the d8

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excited states: a lowering by 0.15 eV or less is found. The CASPT2 results are also less affected, although the effect is not negligible: up to 0.2 eV. It brings the final CASPT2 results very close to the experimental values, with deviations of less than 0.1 eV for all three d → d transitions. Apparently, the description of 3d → 3d transitions is less basis-set demanding than transitions involving an alteration of the 3d-occupation number. This is a fortunate conclusion, considering that the lowest excited states in many transition metal coordination compounds correspond to 3d → 3d excitations.

With only three valence electrons, the results obtained for Ti are less sensitive to the presence of 3d′ in the active space. Adding 3d–3d′ excitations leads to a stabilization of the 4F (d3) state with respect to the 4F (d2s1) ground state by 0.3 eV at the CASSCF level. The corresponding effect at the CASPT2 level is 0.12 eV in the opposite direction. A moderate effect is also found for the 4F– 2F excitation within the 3d shell. As for Co , the final CASPT2 result for the d → d transition is excellent, while the s → d transition has a persisting error of 0.26 eV, ascribable to basis-set deficiencies.

An important conclusion to be drawn from the data in Table 1 is that the double-shell effect seems to vanish when moving down to the second-row (and presumably also third-row) transition series. Indeed, for Rh the difference between the CASPT2 results obtained with the small and large active space is never larger than 0.1 eV. Of course, this does not mean that radial 4d correlation effects do not affect the relative energies of configurations with a varying number of 4d electrons. The effect is still manifested at the CASSCF level, giving results for the d7d1–d8 splitting that are increased by up to 0.5 eV in CAS(11) as compared to CAS(6). However, the fact that the CASPT2 relative energies are virtually indifferent to the altered reference wavefunctions indicates that the differential 4d correlation effect can in a satisfactory manner be described by second-order perturbation theory. The 4d → 4d transitions care even less about the presence of the 4d′ shell than the 4d → 5s transitions: at both the CASSCF and CASPT2 levels the difference between the CAS(11) and CAS(6) results is less than 0.1 eV. Also note that the CASPT2 results for all three 4d → 5s excitations are significantly closer to the experimental values for Rh than was the case for Co , thus indicating that the basis-set requirements for describing varying d occupations also tend to loosen up when moving down in the transition series. On the other hand, an exceptionally large deviation from experiment is found for the two lowest excited states 1D, 3P, 0.21 eV and 0.14 eV, respectively. A possible explanation comes from the lack of spin-orbit coupling in the calculations, combined with the averaging procedure used to obtain the experimental values in Table 1. The latter procedure may no longer be justified in heavy metals like Rh , where spin-orbit coupling becomes important and may lead to a mixing of the rather close-lying 1D and 3P states (C. Ribbing, personal communication, 1998).

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Finally, some words concerning the relevance of the correlation problem in TM atomic or ionic spectroscopy for the large coordination compounds we will look at in the next sections. In such compounds the ground state is most often built from the dn 2 configuration, and d → s excitations often become highlying and unimportant. Instead, many coordination compounds are characterized by their low-lying d → d (the so-called ligand-field) transitions (23), for which the double-shell effect is of less importance (although not absent; see earlier). However, two cases can clearly be distinguished where one cannot avoid having to deal with the second d shell: 1) In electronic spectroscopy, when describing charge-transfer (CT) excitations: such excitations indeed again involve an increase (ligand-to-metal LMCT) or decrease (metal-to-ligand MLCT) of the d- occupation number, so that correlation effects similar to those for the d ↔ s transitions may be expected; 2) When describing the total bonding energy (e.g., ML6 → M 6L) or the consecutive dissociation of the different ligands (e.g., ML6 → ML5 → ML4 → ML3 . . .), one will certainly at some point be confronted again with a low-lying 4s shell and the 3d double-shell effect.

3. COVALENT VERSUS IONIC METAL–LIGAND BONDS

In order to understand the occurrence of nondynamic correlation effects on the bonding in transition metal coordination compounds it is useful to begin with a short introduction to the basic ideas behind the molecular orbital approach, as applied to such systems. We do this by considering a qualitative molecular orbital diagram of a hypothetical octahedral ML6 complex, as shown in Fig. 1. Here, L is assumed to be a ligand that can form one σ and two π bonds with the metal (e.g., an atom or ion with an np valence shell). At the right-hand side of Fig. 1, symmetry-adapted combinations of the ligand orbitals are constructed that transform according to one of the irreducible representations of Oh. These are a1g, eg, and t1u for the σ orbitals and t1g, t2g, t1u, t2u for the π orbitals. On the other hand, within Oh, the metal 4s orbital belongs to the a1g representation, 4p belongs to t1u, and the five 3d orbitals form two groups: two d(eg) orbitals (dz2 and dx2y2) belonging to the eg representation, and three d(t2g) orbitals (dxy, dxz, and dyz) that belong to t2g. The interaction of the metal atomic orbitals and the ligands is assumed to proceed in two steps. In the first step, the metal is placed at the center of the six L, without allowing any (covalent) interaction. Due to electronic repulsion with the ligands the metal orbitals are destabilized and split. 4s and 4p undergo the largest repulsion and are strongly pushed up in energy with respect to the 3d orbitals, which themselves are split into eg (directed toward the ligands) at higher energy and t2g (not directed toward the ligands) at lower energy. This

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FIGURE 1 Qualitative MO energy-level scheme for regular octahedral complexes ML6 of a transition metal M with ligands L that have one σ and two π active orbitals each.

first step describes the situation as incorporated in the classical crystal field model.

In a second step, overlap between the metal and ligand orbitals is taken into consideration and the molecular orbitals are constructed, taking into account that only group-symmetrical orbitals belonging to the same Oh symmetry representation may interact. Since the ligand t1g and t2u orbitals have no counterpart on the metal, these orbitals remain nonbonding. The same is (to a first approxima-

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tion) true for the t1u ligand π orbitals, due to the virtual absence of π interactions with the 4p(t1u) orbitals on the metal. Within the other representations, bonding (ψA) and antibonding (ψB) combinations of the metal orbitals (φM) and the corresponding group-symmetrical ligand orbitals (φL) are constructed as follows:

ψB CLφL CM φM	(1)
ψA CL′φL CM′ φM	(2)

The coefficients may be obtained from a solution of the appropriate secular equation:

HLL E	HML EGML	0
HML EGML	HMM E		(3)

with GML the so-called group overlap integral (24).

Due to the large zeroth-order energy difference between the metal 3d and

4s, 4p orbitals, the antibonding combinations a* and t* fall outside the valence

1g 1u

region of our ML6 molecule, and will not play an important role in the consideration of near-degeneracy effects in the next sections. At this point we therefore focus on the molecular orbitals involving the metal 3d orbitals, i.e., the bonding and antibonding combinations of eg and t2g symmetry.

For the latter two representations, an approximate solution of Eq. (3) may be obtained by assuming that the difference in zeroth-order energy between the metal orbitals and the ligand valence orbitals, HMM HLL, is large while the interaction between them is relatively small, or, GML 0 and HML (HMM

HLL).

Then one obtains the following approximate solutions (24):

Eb HLL (HML HLLGML)2

HMM HLL

Ea HMM (HML HMMGML)2

HMM HLL

while for the coefficients CM, CL, CM′ , CL′ one finds that:

	CM				GML			1
	CL			HMM HLL
	CL′				GML			1
CM′					HMM HLL

(4)

(5)

(6)

(7)

Or the bonding orbital is of predominantly ligand origin, the antibonding orbital of predominantly metal 3d origin. Furthermore, the ratio CM/CL (or CL′/CM′ ) gives a qualitative measure of the extent of covalency of the M–L interaction. In the

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extreme case that CM CL′ 0 (GML 0), both molecular orbitals entirely retain their atomic character and the bonding is purely ionic. On the other hand, an increasing value of CM/CL is indicative of an increasing degree of covalency. Also notice that the group overlap GML will in general be larger for the σ orbitals belonging to the eg representation than for the π orbitals in t2g, thus giving rise to more covalent σ than π bonds.

In order to determine the ground state electronic structure of the ML6 compound, the available valence electrons have to be distributed over the molecular orbitals. Thereby, the bonding and nonbonding orbitals get first priority, leaving only the remaining electrons to be assigned to the antibonding orbitals. Let us consider CrF63 as an example. Here, 39 electrons have to be distributed over the molecular orbitals in Figure 1. Of these 39, 36 fit into the 18 bonding and

nonbonding fluorine orbitals, leaving three electrons for the antibonding t* orbit-

2g

als with Cr3d character. Even if this MO occupation scheme is independent of the way one imagines the constituent parts on both sides of the complex, i.e., either as neutral atoms Cr 6F ( 3 extra electrons) or as ions Cr3 6F , the latter picture is obviously more consistent with the actual situation presented by the proposed MO diagram. Furthermore, the ionic starting point forms the basis for concepts like ‘‘formal metal oxidation state’’ and ‘‘formal metal 3d occupation number,’’ used throughout the literature. Thus, CrF63 is always described as a 3d3 system, with a formal oxidation state (III) or charge ( 3) on chromium. The same formal charge and 3d occupation number is also found, for example, in Cr(H2O)63 . Whether or not this formal picture indeed corresponds to the actual situation in the complex, depends of course on the extent of covalency of the metal–ligand bonds. As we will show later, the ionic picture is in the case of both CrF63 and Cr(H2O)63 quite close to reality. However, in other TM complexes the M-L interaction may acquire a much more covalent character, and in those cases the ionic picture becomes nothing but a formal starting point. In the next section we will also show that such covalent M-L interactions are responsible for the occurrence of strong nondynamic correlation effects in the latter TM complexes.

4. M–L COVALENCY AND CORRELATION EFFECTS

The ideal starting point of any multireference correlation treatment is a CASSCF calculation including (at least) all valence orbitals and electrons. In the octahedral ML6 complex described in Figure 1 this would mean including 23 orbitals: 18 nonbonding or bonding molecular orbitals with predominant ligand character and five antibonding orbitals with predominant metal 3d character. An active space of this size is, however, not even close to what can be handled by today’s hardware and software. Therefore, restrictions are in order. Such restrictions can be accomplished in two ways: 1) Restrict the number of orbitals included in the active space, 2) Restrict the number of configurations included in the CI space.

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The latter may, for example, be accomplished by performing instead a RASSCF (restricted active space SCF) calculation, in which the active space is divided into three subspaces, RAS1, RAS2 and RAS3. All possible occupations of RAS2 are then still allowed, but excitations out of RAS1 and into RAS3 are restricted to a maximum number of electrons. When designed economically, such calculations can handle a considerably larger total active space than a regular CASSCF calculation. However, to our knowledge no method is currently available that is capable of treating dynamic correlation based on an RASSCF reference treatment. Such RASSCF calculations are, however, often useful by themselves and may help the user to decide which of the active orbitals give rise to important neardegeneracies and should therefore be selected as active in a subsequent CASSCF reference treatment including a more limited number of such orbitals (if such a selection can be made at all). In the present section, we will present the results of a series of RASSCF calculations on some representative octahedral and tetrahedral complexes of first-row TM. These calculations [denoted as RASSCF (all valence)] were designed as follows: The five (antibonding) orbitals with predominant metal 3d character are included in the RAS2 space, while all ligand valence orbitals (bonding and nonbonding) are included in RAS1. Up to quadruple excitations from RAS1 into RAS2 are included in the calculations. This should suffice to provide information as to which of the valence orbitals indeed give rise to strong near-degeneracies; from there we can try to design a more economical CASSCF active space.

However, before looking at the results of these calculations, we believe that a few important points can already be made based on the octahedral MO scheme in Figure 1 and the considerations of the previous section. Thus, as a first case, suppose that we are dealing with the extreme situation of a truly ionic transition metal system. In such a case all molecular orbitals are either entirely ligand or entirely metal based. The ligand valence orbitals are fully occupied and at considerably lower energy than the metal 3d orbitals, with which they do not interact. Therefore, one may in this case expect important correlation effects to occur only within the 3d valence shell. This means that a reference CASSCF calculation on such a system should include only the metal 3d, and possibly a second 3d′ shell (see Sec. 2).

A second case is what we will call the case of weak covalency. This case occurs when there is a distinct overlap between the metal 3d and valence orbitals (GML ≠ 0), which are, however, still well separated [(HMM HLL) 0]. According to Eqs. (6) and (7) one then finds a significant contribution of ligand character

in the antibonding molecular orbitals of e* and t* symmetry, and a corresponding

g 2g

3d contribution in the bonding eg and t2g orbitals. In other words, the metal 3d electrons are delocalized in the ligand valence shell and vice versa. A CASSCF calculation on such a system should include the bonding eg, t2g and antibonding

e*, t* combinations of the metal 3d and ligand valence orbitals in the representa-

g 2g

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tions. The other, bonding and nonbonding, valence orbitals in Figure 1 can be expected to be of minor importance, since they are still well separated from the open-shell antibonding orbitals.

However, suppose that, in a third case, we also give up the restriction that (HMM HLL) 0 and allow the ligand valence orbitals to approach the metal 3d shell. Apart from observing a further strengthening of the covalent interactions within the molecular orbitals of eg and t2g symmetry, we may now also get confronted with important contributions in the ground state wavefunction coming from excitations out of the bonding and nonbonding orbitals of symmetry a1g,

*	*
t1u, t2u, and t1g into the open-shell eg	, t2g orbitals. Such contributions can obviously

be accounted for only by including the entire ligand valence shell into the multiconfigurational treatment of nondynamic correlation. Since a CASSCF calculation with such a large active space is out of the question, systems like this are out of the reach of CASPT2 and other presently available multiconfigurational correlation methods.

The foregoing considerations are further illustrated by the results obtained from the previously described set of RASSCF calculations. A first set of calculations concerns the series CrF6x , with x varying between 4 and 0. CrF64 is a

formal 3d4 complex with a t*3 e*1 quintet ground state and a formal charge of

2g g

( 2) on chromium. Each consecutive withdrawal of an electron from the anti-

bonding molecular orbitals (obviously e* is depopulated first) brings about a

g

reduction in the formal metal 3d occupation and a simultaneous increase in the formal oxidation state on the metal, until in CrF6 we find a formal charge of ( 6) on chromium, with no 3d electrons left. With this ionic picture in mind, one can see how this CrF6x series reflects a decrease in the difference between the zerothorder energies HMM and HLL at both sides of Figure 1. Indeed, by identifying these zeroth-order energies with ionization potentials (Koopman’s theorem) it becomes clear that the higher the formal charge on chromium, the more the energy of the 3d valence shell is pushed down toward the F valence shells.

The results obtained form a series of RASSCF calculations on the CrF6x complexes are included in Tables 2 and 3. Table 2 shows the composition of the valence natural orbitals and the corresponding occupation numbers; Table 3 lists the number of configurations included and the correlation energy obtained from these RASSCF calculations. In Table 2 the first, t2u, and t1g nonbonding shells are 100% fluorine based in all complexes, while in a1g and both t1u shells we find a small and almost constant contribution of Cr s and p character, respectively. However, more important is the composition of the molecular orbitals of eg and t2g symmetry. Here we clearly observe a growing admixture within the series of chromium 3d character in the bonding eg, t2g shells and of fluorine 2p char-

acter in the antibonding e*, t* shells. According to the definition of covalency

g 2g

from the previous section, this growing admixture with an increasing formal charge on chromium is the reflection of a concomitant increase in covalency of

TABLE 2						Composition and Occupation Numbers of the Natural Orbitals Resulting from an RASSCF															(All-Valence)		a
Calculation on a Series of Octahedral Chromium Complexes
							CrF		4					CrF	3					CrF	2
									6						6						6
								Composition (%)						Composition (%)						Composition (%)
					Occupation				Cr		F		Occupation		Cr		F		Occupation		Cr		F
MO						no.		s	p	d			no.	s	p	d			no.	s	p	d
t	2		u			6.00		–	–	–	100	6.00		–	–	–	100	6.00		–	–	–	100
t						6.00		–	–	–	100	6.00		–	–	–	100	6.00		–	–	–	100
	1		g
a		1		g		2.00	0		–	–	100	2.00		2	–	–	100	2.00		3	–	–	97
t						6.00		–	0	–	100	6.00		–	2	–	100	5.99		–	0	–	100
	1		u
t						6.00		–	0	–	100	6.00		–	2	–	98	5.98		–	3	–	97
	1		u
t						6.00		–	–	0	99	6.00		–	–	2	98	5.98		–	–	4	96
	2		g
e						4.00		–	–	4	96	3.99		–	–	11	89	3.94		–	–	15	85
		g
t	*					3.00		–	–	9	91	3.00		–	–	98	2	2.01		–	–	91	9
	2		g			1.00		–	–							60	40
e		*								97	2	0.01		–	–			0.08		–	–	83	17
		g

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							CrF								CrF						CrCl		2
									6							6							6
								Composition (%)							Composition (%)							Composition (%)
						Occupation			Cr		F		Occupation			Cr		F		Occupation			Cr		Cl
	MO					no.		s	p	d			no.		s	p	d			no.		s	p	d
	t	2		u		5.98		–	–	–	100	5.89			–	–	–	100	6.00			–	–	–	100
	t					5.97		–	–	–	100	5.91			–	–	–	100	6.00			–	–	–	100
		1		g
	a					1.99	3		–	–	97	1.98		3		–	–	97	2.00		15		–	–	85
			1		g
	t					5.95		–	2	–	98	5.91			–	3	–	97	6.00			–	4	–	96
		1		u
	t					5.94		–	3	–	97	5.85			–	2	–	98	5.98			–	5	–	95
		1		u
	t					5.92		–	–	16	84	5.84			–	–	26	74	5.93			–	–	10	90
		2		g
						3.90		–	–								38	62
	e		g							34	66	3.88			–	–			3.82			–	–	34	66
	t	*				1.18		–	–	78	22	0.46			–	–	69	21	2.06			–	–	93	7
		2		g
	e		*			0.18		–	–	66	34	0.28		31		–	63	37	0.22			–	–	70	30
			g
	a	For the description of the RASSCF (all-valence) calculation, see text.

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