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

138

TABLE 3 from Two

Number of Included Configuration State Functions (CSF) and Calculated Correlation Energy (a.u.) Obtained RASSCF Calculations on a Set of Octahedral and Tetrahedral Complexes

						Metal		RASSCF (all valence)
						formal	Number		Correlation
Complex						charge		of CSF	energy
CrF		6	4			2	32 864		.001473
CrF		6	3			3	127 492		.012747
CrF			2			4	251 015		.070053
		6
CrCl			2			4	251 015		.100166
			6
CrF						5	265 586		.201012
		6
CrF						6	133 981		.401547
		6
VCl						3	101 273		.030320
		4
CrF						4	101 273		.103309
		4
MnO				3		5	101 273		.256321
VO			4			5	53 173		.235151
		3
	4
CrO			2			6	53 173		.396291
		4				7	53 173
MnO									.563461
			4
a					(e, t	, e*, t*) in the tetrahedral complexes.
RASSCF
					2	2

RASSCF (e	, t	2g	, e*, t* )		a
g			g	2g
Number		Correlation
of CSF			energy
560			.001283
1 881			.008668
3 489			.052583
3 489			.108689
3 460			.127509
1 699			.218086
3 428			.027108
3 428			.073406
3 428			.198441
1 699			.145562
1 699			.246475
1 699			.344010

Difference
Number	Correlation
of CSF	energy
32 304	.000190
125 611	.004079
247 526	.017470
247 526	.008593
262 126	.073503
132 282	.183461
97 845	.003212
97 845	.029903
97 845	.057880
51 474	.089590
51 474	.148916
51 474	.219451

Pierloot

Nondynamic Correlation and TM Coordination Compounds

139

the Cr–F bonds. Also note that the admixture is always larger for eg than for t2g, consistent with the larger metal–ligand σ- than π-overlap.

As expected, the decreasing gap between metal and ligand orbital energies with an increasing formal charge on the metal also gives rise to increasing correlation effects. This can be seen from the occupation numbers in Table 2 and from the correlation energy obtained from the RASSCF calculation, shown at the lefthand side in Table 3. The correlation energy is almost insignificant for CrF64 , increases strongly with the formal charge on the metal, and becomes quite important ( 0.4 a.u.) for the neutral CrF6 molecule. The occupation numbers reveal the same trend. For CrF64 , all occupation numbers in Table 2 are equal to the ROHF numbers. This molecule could therefore, without significant loss of accuracy, be described starting from a single reference configuration. On the other hand, for CrF6 all orbitals are either considerably populated or depopulated. Adopting the strict rule that all orbitals with an occupation different by more than 0.01 from its ROHF occupation (or 0.02 for a doubly degenerate and 0.03 for a triply degenerate shell) should be included in a multiconfigurational reference CI, we must conclude that for CrF6 such a calculation would require the entire valence space.

However, it is important to note that when going from CrF64 to CrF6, important correlation effects first become apparent in the t2g, eg shells, and only afterwards in the other bonding and nonbonding orbitals. Thus, as the orbital occupation numbers in Table 2 show, both CrF63 and CrF62 would still be satisfied with a 10-orbital active space, including only the bonding (eg, t2g) and anti-

bonding (e*, t* ) combinations of Cr 3d and F 2p, while excitations out of the

g 2g

other orbitals become important in CrF6 . This is also shown by a comparison (Table 3) of the correlation energy obtained from the full-valence RASSCF calculation with a similar, smaller calculation, containing only eg and t2g in the RAS1 space. As one can see, the first three molecules in the CrF6x series are indeed almost equally well described by the small active space: the (much larger) number of omitted configurations is responsible for a total contribution of less than 0.02 a.u. to the correlation energy. For the other two molecules, CrF6 and CrF6, this contribution is considerably larger and cannot be overlooked. The present considerations are also corroborated by previous, more quantitative treatments of the considered molecules. The ligand field spectra of the complexes CrF6x (x 2– 4) were successfully calculated using either an MRCI (25,26) or CASPT2 treatment (27) based on a limited reference active space. However, a CASPT2 calculation of the relative stability of an octahedral and trigonal prismatic structure for

*	*
CrF6 (28), based on an active space of only 10 orbitals (eg, eg	, t2g, t2g), turned

out to produce results that deviate considerably from similar studies performed using either the coupled-cluster method (29,30) or density functional theory (31). The failure of the CASPT2 treatment for this problem must undoubtedly be traced back to the inadequacy of the employed 10-orbital active space.

140

Pierloot

Looking back at the considerations made at the beginning of this section, we note that CrF64 can be classified as an almost purely ionic complex (case 1), CrF64 and CrF63 both belong to the weakly covalent case (case 2), while CrF6 and CrF6 were classified as a third, strongly covalent, case. As already noted, the increasing extent of covalency within this series is due to a decreasing difference in energy (ionization potential) between the metal and ligand valence orbitals. A second factor determining the extent of covalency of the M–L bonds is the overlap between these orbitals [GML in Eqs. (6) and (7)]. In order to further investigate this second factor, we have added one more example to our series of octahedral test molecules, i.e., CrCl62 . Cl should indeed give a more covalent M–L bond, not because of its valence energy (F and Cl having a similar electron affinity), but because of a stronger M–L overlap. This is confirmed by the composition of the orbitals of eg and t2g symmetry in CrCl62 as compared to CrF62 . And also in this case, stronger covalency leads to stronger correlation effects: the correlation energy obtained from the full-valence RASSCF calculation is indeed larger for CrCl62 than for CrF62 . It is, however, gratifying to see that the correlation effects in CrCl62 are still limited to the orbitals of eg, t2g symmetry. Indeed, the occupation numbers of the other orbitals remain very close to their ROHF values, while the difference in correlation energy between the small and large RASSCF calculation in Table 3 is even slightly smaller for CrCl62 than for CrF62 . This indicates that the dimension of the orbital space involved in nondynamic correlation effects in TM systems is not determined by the covalency of the M–L interactions as such, but rather by the difference in valence orbital energies between the metal and ligand ions (starting from an ionic picture).

In order to show that the foregoing findings are not limited to six-coordinate octahedral complexes, we have also performed a similar set of test calculations on a series of tetrahedral molecules. The molecular orbital scheme for a tetrahedral ML4 complex is shown in Figure 2. The metal 4s orbital is now found in representation a1, 4p transforms as t2, while the tetrahedral ligand environment splits the 3d orbitals into e and t2. On the other hand, the group-symmetrical combinations of ligand orbitals are found in a1, t2 for σ and e, t1, t2 for π. As such, in ML4 only the t1 shell remains nonbonding and purely ligand based. The metal 3d orbitals belonging to representation e (dz2, dx2 y2) may only be involved in π interactions with the ligands, while the t2 orbitals (dxy, dxy, dyz) can form a mixture of σ and π bonds. This means that the latter shell is more strongly destabilized, by both a stronger repulsion and a stronger overlap with the ligands.

In Tables 3 and 4, RASSCF results have been included for three formal 3d2 molecules—VCl4 , CrF4, and MnO43 , containing metals in a formal oxidation state ( 3), ( 4), and ( 5), respectively—and for three formal 3d0 complexes, i.e., the isoelectronic series VO43 (vanadate), CrO42 (chromate), MnO4 (permanganate), with formal charges on the metal of ( 5), ( 6), and ( 7), respectively. The calculations were performed in a similar way as for the octahedral

TABLE 4				Composition and Occupation Numbers of the Natural Orbitals Resulting from an RASSCF (All-Valence)																										a
Calculation on a Series of Tetrahedral Complexes
						VCl											CrF								MnO			3
									4											4							4
							Composition (%)											Composition (%)								Composition (%)
			Occupation						V		Cl			Occupation						Cr		F	Occupation					Mn		O
																	s									s
MO				no.		s			p	d					no.					p	d			no.				p	d
t	1			6.00		–			–	–	100			5.99			–			–	–	100	5.95			–		–	–	100
a		1		2.00	9				–	–	91			1.99		3				–	–	98	1.96		0			–	–	100
t	2			6.00		–			4	–	96			5.98			–			2	0	98	5.95			–		2	2	96
e				3.99		–			–	6	94			3.98			–			–	9	91	3.94			–		–	17	83
t	2			5.95		–			2	14	84			5.90			–			2	22	76	5.71			–		1	36	63
e*				2.00		–			–	96	4			2.02			–			–	91	9	2.07			–		–	84	16
t	*			0.06		–			2	72	26			0.14			–			0	72	28	0.42			–		0	62	38
	2
					VO				3							CrO				2					MnO
								4											4									4
							Composition (%)											Composition (%)								Composition (%)
			Occupation						V		O				Occupation					Cr		O		Occupation				Mn		O
																	s			p						s
MO				no.		s			p	d					no.						d			no.				p	d
t	1			5.91		–			–	–	100			5.85			–			–	–	100	5.77			–		–	–	100
a		1		1.97	0				–	–	100			1.95		0				–	–	97	1.93		1			–	–	99
t	2			5.93		–			1	2	97			5.88			–			3	3	94	5.83			–		5	7	88
				3.94
e						–			–	23	77			3.88			–			–	25	75	3.85			–		–	44	56
t	2			5.86		–			1	28	71			5.79			–			1	35	64	5.76			–		1	39	60
e*				0.15		–			–	66	34			0.28			–			–	64	36	0.40			–		–	57	43
t	*			0.24		–			0	64	36			0.37			–			0	60	40	0.46			–		0	54	46
	2
a	For a description of the RASSCF (all-valence) calculation, see text.

Compounds Coordination TM and Correlation Nondynamic

141

142	Pierloot
*	orbitals with predominant metal 3d character are put in
complexes: the e*, t2

RAS2, while up to quadruple excitations are allowed out of the RAS1 space, including all ligand-based valence orbitals in a full-valence RASSCF calculation and only one set of (bonding) e, t2 orbitals in a smaller RASSCF calculation. Note that the full-valence RASSCF calculation now includes a total of 17 active orbitals, which is still too large to be handled by a regular CASSCF calculation. In a tetrahedral complex, covalent M–L interactions can be formed in the molecular orbitals of symmetry t2 and e. A difference with the octahedral situation is that the ligand valence orbitals within Td symmetry give rise to two group-symmetrical t2 shells. This means that, at least in principle, the metal 3d orbitals may become

delocalized in 13 valence molecular orbitals (two t ,e, t*, e*) as opposed to only

2 2

10 (t , e , t*, e*) in an octahedron. However, the results in Table 4 indicate that

2g g 2g g

in practice only one of the bonding t2 shells, i.e., the one with the lowest occupation number, contains a significant amount of metal 3d character. The other t2 shell remains almost pure L in all cases, with a maximum 3d contribution of 7% in MnO4 . As for the other shells of e, t2 symmetry, Table 4 again indicates an increasing covalent interaction with an increasing formal charge on the metal. A limiting case is permanganate, with a formal charge of ( 7) on Mn. Here, both

the e, e* and t , t* couples contain an almost equal mixture of metal 3dand L

2 2

2p character. As concerns the appearance of nondynamic correlation effects, the results in Tables 3 and 4 are also consistent with the octahedral situation. Both in VCl4 [formal charge ( 3) on V] and CrF4 [formal charge ( 4) on Cr] such

effects are limited to the 10 orbitals (e, e*, t , t*) containing a mixture of ligand

2 2

and metal 3d character: The other natural orbitals, t1, a1, t2, remain close to doubly occupied (Table 4), and excitations out of these orbitals do not contribute much to the correlation energy (Table 3). On the other hand, the latter orbitals become increasingly more important in the complexes MnO43 and the d0 series VO43 CrO42 , MnO4 , where we find a growing formal charge on the metal. The present results are consistent with an earlier study on the permanganate MnO 4 ion (32), painting a detailed picture of the bonding and correlation effects in this system. We would also like to refer to a recent RASSCF study on the bonding and spectroscopy of the tetraoxoferrate(VI) FeO2 4 ion (9), where the presence of strong near-degeneracies between the Fe(3d) and O(2p) levels was also recognized and studied in detail.

On the whole an important, though negative, conclusion is to be drawn from the preceding considerations, i.e., that transition metal complexes containing metals in high formal oxidation states ( 5 or higher) may demonstrate severe nondynamic correlation effects involving a large number of orbitals, and are therefore utterly hard to treat by multireference methods (and of course more so by single-reference methods). The origin of these correlation effects should be brought back to the near-degeneracy of the metal 3d and ligand valence orbitals, which is ultimately due to the high ionization energy (or electron affinity) con-

Nondynamic Correlation and TM Coordination Compounds

143

nected with the high oxidation state of the metal. Obviously, these near-degenera- cies should also be manifested by the presence of low-lying ligand-to-metal charge-transfer (LMCT) states in the experimental optical spectra of the complexes under consideration. Indeed, MnO4 is intensely purple due to the presence of low-lying LMCT transitions starting at 18 000 cm 1 (23,33) (first allowed excitation to 1T2). The CT states are shifted upward by around 5 000 cm 1 in the yellow CrO42 ion and even more so in the colorless VO43 ion (23). It is interesting to note that the lowest CT states in these d0 systems indeed correspond to excitations from the nonbonding t1 shell (23,34), consistent with the qualitative ordering of the orbitals in Figure 2. Looking at the chromium-fluoride compounds, we find LMCT bands (35) at 32 700 cm 1 in CrF4 and from 30 000 to 40 000 cm 1 in CrF62 , while in CrF6 the onset of the charge-transfer band is found even at ca. 20 000 cm 1, with the first prominent band appearing at 26 700 cm 1 (35). On the other hand, going to lower formal charges on the metal, e.g., in CrF63 and CrF64 , the lowest CT bands are shifted strongly upward in energy, e.g., above 45 000–50 000 cm 1 (36).

Fortunately, many transition metal complexes contain metals in rather low [( 4) or lower] formal oxidation states. The results presented in this section have also indicated that for such complexes all important nondynamic correlation effects may be efficiently dealt with in a reference CASSCF calculation with an active space of at most ten orbitals, i.e., the antibonding molecular orbitals of predominantly metal 3d character and their bonding ligand counterparts. In the next section we will look in more detail at these ‘‘weakly covalent’’ systems and further investigate the connection between the extent of covalency of the metal– ligand interactions on the one hand and the importance of nondynamic correlation effects on the other hand.

5. THE CASE OF WEAK COVALENCY

Transition metal complexes containing metals in oxidation states ranging between ( 2) and ( 4), combined with ‘‘classical’’ (i.e., noncarbon) neutral or negatively charged ligands, constitute a class of coordination compounds that are often designated Werner complexes (referring to Alfred Werner, who, at the beginning of the twentieth century, developed the modern picture of coordination complexes) (37). This is the class of complexes that can be described with considerable success by the semiempirical ligand field theory and related methods (e.g., the currently still commonly used angular overlap model [24]). The success of these methods is related to the fact that the metal–ligand interactions in this group of complexes range from ionic to weakly covalent, a range that is within the limits of the approximations assumed by such models (leading, for example, to energy expressions as given by Eqs. 4 and 5).

144

Pierloot

FIGURE 2 Qualitative MO energy-level scheme for regular tetrahedral complexes ML4 of a transition metal M with ligands L that have one σ and two π active orbitals each.

In this section we will take a short walk through the domain of the Werner complexes and look for electronic structure–correlation relationships. We will do this by considering two plots, shown in Figure 3. This figure is based on a series of CASSCF calculations on octahedral ML6 transition metal complexes with different ligands, such as NH3, H2O, and the halides F , Cl , Br , I . Figure

3a includes results obtained for the 4A ((t*)3) ground state of some formal d3

2g 2g

complexes, with M (V, Nb, Ta)2 , (Cr, Mo, W)3 , or (Mn, Tc, Re)4 ; the results

Nondynamic Correlation and TM Coordination Compounds

145

FIGURE 3 Plot of correlation energy (Ecor, obtained from a CASSCF calculation; see text) versus the M d contribution (in terms of percentage) in the bonding eg orbital in a series of octahedral ML6 complexes, with L H2O, NH3, F , Cl , Br , and I , and M a transition metal with a formal d3 (a) or d 6

(b) occupation number. Solid lines connect metals with a formal charge ( 3); dashed lines connect metals with a formal charge ( 4). For simplicity, the formal charges on the metals have been omitted from the plots.

presented in Figure 3b are for formal d6 complexes, with M (Co, Rh, Ir)3 ,

or (Ni, Pd, Pt)		4	1	* 6	) state (which is not the ground state in all cases;
			in the A1g((t2g)
3	has a high-spin			5	*	4	* 2	) ground state [38]). The CASSCF
e.g., CoF6				T2g((t2g)			(eg )

calculations were performed as follows: for the d3 systems 13 electrons were

		*	*	*
correlated in an active space consisting of 10 orbitals: eg			, eg	, t2g, t2g. On the
other hand, in the d	6	*		*
		systems, the t2g shell is fully occupied, so t2g → t2g excitations

cannot contribute to the wavefunction. The t2g shell was therefore left out of the active space, leaving 7 active orbitals, including 10 electrons. The (x, y) plots in

146

Pierloot

Figure 3 combine two sets of results obtained from these CASSCF calculations. The x-axis represents the metal 3d contribution (in terms of percentage) in the bonding eg orbitals, as a measure of the extent of covalency of the M–L(σ) interactions, while the y-axis represents the correlation energy (in a.u.) obtained from the calculations, i.e., the energy lowering with respect to a single-configurational SCF (in case of d6) or CASSCF (in case of d3) treatment. The lines in the plots connect points obtained for metals of the firstto third-row TM series, in the same oxidation state, and coordinated to the same ligands.

A first glimpse at both plots confirms the trend already discussed in the previous section, i.e., that, generally speaking, static correlation energy is an increasing function of the extent of covalency of the M–L bonds. When looking for trends in correlation effects we should therefore start by looking for factors affecting the M–L covalency. Considering both plots in detail we note the following.

1. For the same metal, the M–L covalency and connected correlation effects increase in the following order of ligands:

F OH2 NH3 Cl Br I

(8)

The increasing tendency to form covalent bonds with an increasing ligand polarizability is not unexpected. Actually, the series presented here is closely related to the nephelauxetic series, originating from ligand field theory (23,24,39). The nephelauxetic effect was originally defined by C. K. Jørgensen (39) as the reduction with respect to the free ion of the interelectronic repulsion in the ligand field states of a transition metal coordination compound. This reduction, expressed as the ratio of the Racah parameter B in the complex and in the free ion, β Bcomplex/ Bion, depends on the character of the surrounding ligands. From spectroscopic data the following nephelauxetic series was obtained, ordering the ligands with respect to decreasing β values (for the same TM in the same oxidation state):

F OH2 (NH2)2CO NH3 H2NCH2CH2NH2
(COO)22 (CO3)2 NCS Cl CN	(9)

Br N3 I S2(C2H2O)PS22 diarsine

The fact that the two preceding series are equivalent is by no means surprising. Indeed, the nephelauxetic effect is related directly to covalency: the reduction of B by complex formation is caused by delocalization of the d-electron cloud on the ligands, which is in turn caused by the formation of covalent bonds. Even if not complete, the nephelauxetic series can come in handy when having to construct the reference space of a multireference calculation, since it helps to

Nondynamic Correlation and TM Coordination Compounds

147

decide which ligands are likely to give covalent bonds and hence give rise to important nondynamic correlation effects.

2.All curves systematically appear at higher (x, y) in Figure 3b than in Figure 3a, indicating that more covalent M–L bonds and concomitant correlation effects occur for the d6 than for the d3 complexes. Again, this is only a small representation of a more general trend within a row of TM ions: the increasing polarizing power of the ions (in the same oxidation state) from left to right in the same row (corresponding to a decreasing (HMM HLL) in Eqs. 4–7) gives rise to a growing tendency to form covalent M–L bonds and hence also to increasingly more important nondynamic correlation effects. As such, the most strongly covalent M–L bonds are to be expected for TM ions at the right-hand side of their series, combined with soft and easily polarizable ligands. A typical example is the Cu(II)–cysteine combination in the so-called blue copper proteins (40–

43). The intense blue color of these proteins is due to a strongly covalent Cu(II)– thiolate bond giving rise to the intense ‘‘blue’’ cysteine → Cu LMCT band in the visible region. As was shown recently (44), substituting Cu(II) by Co(II) in these proteins goes together with a considerable weakening of the covalency of the M–cysteine interaction, consistent with the trends predicted in this chapter.

3.The plots also clearly show (see also Sec. 4) the steeply increasing covalency and concomitant correlation effects with an increasing formal charge

on the metal. Thus we find that the MX62 lines are strongly shifted in the ( x,y) direction as compared to the MX63 lines for X F, Cl in the d3 plot and for X F in the d6 plot, while the d3 M(H2O)62 complexes are found at the bottom left side of Figure 3a, below and at the left of the M(H2O6)3 complexes.

4.Finally we can also compare TM ions from the same column but between different rows of the periodic table, i.e., the data connected by the lines in the plots. Considering first the trends in covalency, we see that in all calculated complexes the third-row metals give a more ionic M–L bond than the secondrow metals. However, the observed shifts in M–L covalency between the firstand second-row TM are not unequivocal: in the d6 systems and also in the d3 MCl63 complexes second-row metals give the most covalent bonds, whereas in the other d3 systems the strongest covalency is found for the first-row metals. One thing is clear, however: In all considered series, nondynamic correlation effects become less important when moving down in the periodic table. This is

also the case for those complexes where the first-row TM do not give the most covalent M–L bonds. The general rule ‘‘increasing M–L covalency → increasing correlation effects’’ is obviously not always valid when considering transition metals belonging to different rows of the periodic system.

Before finishing this section, we would like to remind the reader of another trend between different rows of the TM, i.e., the strongly reduced 4d as compared to 3d double-shell effect (see Sec. 2 and Table 1). The latter trend, together with