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

358

TABLE

Pa, U,

2	˚
	Optimized PW91 and LDA (in parentheses) Bond Distances (A)
Np, Pu, Am)

and Bond Angles (°)

of

An(Cot)

2

(An Th,

An–X

a

An–C

C–C

C–H

HCX

Th(Cot)2

Pa(Cot)2

U(Cot)2

Np(Cot)2

Pu(Cot)2

Am(Cot)2

2.047 (2.001)	2.756 (2.713)	1.412 (1.402)	1.092 (1.096)	174.4 (175.0)
1.974 (1.915)	2.702 (2.650)	1.412 (1.402)	1.092 (1.097)	174.4 (174.0)
1.941 (1.880)	2.677 (2.624)	1.411 (1.401)	1.092 (1.097)	174.2 (173.9)
1.931 (1.860)	2.669 (2.609)	1.410 (1.400)	1.092 (1.097)	174.4 (173.9)
1.932 (1.851)	2.670 (2.602)	1.410 (1.400)	1.092 (1.097)	174.7 (174.2)
1.927 (1.842)	2.666 (2.596)	1.409 (1.400)	1.092 (1.097)	175.2 (174.5)

a

X is

the centroid of

the

C

8

plane of the Cot rings.

Bursten and Li

Relativistic DFT and Organoactinide Complexes

359

ring (82). Based on these arguments, large rings will tend to tilt inward (toward the metal), while smaller rings will tilt outward (away from the metal) in order to achieve the best overlap with the central metals. Because of these effects, the assumption of planar CnHn rings is not appropriate; we have allowed the C8H8 rings to be nonplanar in the geometry optimization of these actinide sandwich complexes.

Although spin-orbit coupling effects are not included in these optimizations, the good agreement between the PW91 bond distances and the experimental values indicates that these effects are not important on the geometry and thus on the vibrational frequencies. In fact, inclusion of spin-orbit effects has only very small effects on the calculated geometries (83). It is our experience that, with respect to geometries, the scalar relativistic effects account for the major relativistic effects, while spin-orbit coupling effects are not essential. This conclusion is qualitatively supported by the fact that spin-orbit interactions of the radially contracted f-electrons have only minimal effects on the bond strengths and the geometric structures.

6.2. Electronic Structure of the Actinocenes

The qualitative energy-level diagram for An(Cot)2 under the D8h single group is shown in Figure 2, where the principal interactions of the An 5f and 6d orbitals with the π orbitals of the two Cot rings are also indicated. The eight C 2pπ orbitals of a planar D8h C8H8 ligand form eight π molecular orbitals, which, in order of increasing energy, are bases for the a2u, e1g, e2u, e3g, and b1u representations. We will denote these π MOs as π0, π1, π2, π3, and π4, respectively. In a D8h An(Cot)2 complex, these π MOs form gerade πng and ungerade πnu (n 0– 4) ligand group orbitals of (C8H8)2. In the absence of spin-orbit coupling, the 6d orbitals are split in a D8h field as a1g (dσ) e2g (dδ) e1g (dπ); the dδ and dπ orbitals are considerably destabilized by strong interaction with the ligand orbitals. Similarly, the An 5f orbitals are split as a2u (fσ) e3u (fφ) e1u (fπ) e2u ( fδ), where the strong destabilization of the fδ orbitals from the f-manifold is an indication of their strong interaction with the filled π2u orbitals of the (Cot)2.

The 16 pπ-electrons from the Cot rings and 4 electrons from the An atom will fill all the MOs up to the e2g and e2u pairs (i.e., π2g and π2u) derived from the π2 MOs of the Cot ring. As expected, the orbital interactions involving the radially diffuse An 6d orbitals are stronger than those involving the more contracted An 5f orbitals (16). As such, the a1g, e1g, and e2g ligand group orbitals of (Cot)2 interact more strongly than do the a2u, e1u, and e2u ligand group orbitals. For example, in Pa(Cot)2, the MOs derived from the (Cot)2 a1g, e1g, and e2g orbitals all lie lower energetically than their a2u, e1u, and e2u counterparts. Our calculated overlap integrals for the dδ–π2g interaction are all about 0.35–0.33, while those for fδ–π2u decrease from 0.11 to 0.08 from Th(Cot)2 to Am(Cot)2,

360	Li and Bursten

FIGURE 2 Qualitative MO diagram for Pa(Cot)2.

in agreement with the foregoing conclusion. Because of their greater interaction, the 6d-based MOs are destabilized relative to the 5f-based orbitals. Therefore, all the ‘‘extra’’ metal-based electrons (from 0 for An Th to 5 for An Am) are localized in the 5f-based MOs in the An(Cot)2 (An Th–Am) complexes.

The net charges and net spin densities obtained from Mulliken population analysis are listed in Table 3. The net charges of An are all ca. 2.9 except for Pa, for which the charge is 3.5. For all of the complexes, the charges on the C atoms are all about 0.2, and the H atoms are essentially neutral, consistent with the picture that the charge transfer occurs primarily between the Cpπ orbitals to the An orbitals, as expected. The net spin density increases from Th to Am, consistent with the increase in the number of metal-localized electrons in the series of actinocenes.

From Th to Am, both the increased radial contraction of the 5f orbitals and the increase of the 6d orbital energies from the early to late actinides (84,85)

Relativistic DFT and Organoactinide Complexes

361

TABLE 3 Mulliken Net Charges and Spin Densities for the An, C, and H Atoms in An(Cot)2

		Charges				ρspin
	An	C	H		An	C	H
Th(Cot)2	2.881	0.214	0.034	0.000		0.000	0.000
Pa(Cot)2	3.519	0.243	0.023	1.000		0.001	0.001
U(Cot)2	2.925	0.214	0.031	2.197		0.014	0.002
Np(Cot)2	2.931	0.215	0.032	3.458		0.031	0.003
Pu(Cot)2	2.899	0.213	0.032	4.767		0.052	0.004
Am(Cot)2	2.909	0.213	0.031	6.047		0.071	0.005

help to decrease the π2u → fδ and π2g → dδ back-donations. As a result, the covalent bonding between An and the Cot rings is decreased from the early to the late actinide elements. The calculated overlap integrals between the actinide and the ligand group orbitals support this conclusion.

The predicted decrease from Th to Am in the covalent interaction between An and the Cot ligands should be reflected in the overall energy of interaction between the rings and the An atom. To test this notion, we have estimated the average An–C8H8 bond energies by calculating the change in total energy for the ligand-dissociation reaction

An(C8H8)2 → An 2 C8H8 ∆E 2 ∆E(An–C8H8)

where the UKS-optimized scalar relativistic energies of An(C8H8)2 and C8H8 (planar 3A2g) have been employed, while the average-of-configuration (AOC) UKS atomic energies calculated with spherical density were utilized for the An atoms. The resultant bond energies are depicted in Figure 3. Because of the complications associated with multiplets (86,87), spin-orbit effects have not been explicitly included in these calculations. As a result, the calculated bond energies are too high relative to the available experimental An–C8H8 bond energies: 98.0 and 82.9 kcal/mol for Th and U, respectively (88). Nevertheless, the difference in the calculated bond energies of Th–C8H8 and U–C8H8 (15.1 kcal/mol) is in excellent agreement with the measured difference (15.06 kcal/mol). Therefore the bond energies determined here should reflect accurately the trend for this series of actinocene complexes. The bond energies clearly show that the covalent interaction between An and Cot ring decreases as one proceeds to the later actinides, which is in agreement with the well-known fact that complexes of the late actinide elements are predominantly ionic in nature, similar to lanthanide complexes.

362	Li and Bursten

FIGURE 3 Calculated An–Cot bond energies for An(Cot)2 (without multiplet and spin-orbit effects).

6.3.Spin-Orbit Effects and Excited States of Pa(Cot)2

While spin-orbit effects do not significantly affect the geometries and vibrational properties of actinide compounds, these effects are very important for describing the excited-state energies of the complexes. We shall now discuss spin-orbit coupling in the actinocene complexes. Because of the large atomic numbers of the actinides, spin-orbit effects are as important as ligand-field effects in determining the electronic states of actinide compounds. When spin-orbit coupling is taken into account, all the doubly degenerate spatial MOs (eiu and eig, i 1, 2, 3) of An(Cot)2, which are fourfold spin degenerate under the single group, are reduced to twofold degenerate spin-orbitals (spinors) because of the half-integer angular momentum (s 1/2) (89). The correlation of the irreducible representations of the D8h single group and the D8h* double group can be derived by taking the direct product of the single-group representation with the one-electron spin representation (e1/2g). Table 4 lists the derived correlation of the symmetry species of the D8h single group and the D8h* double group (90).

In order to show the quantitative splitting of these MOs by the spin-orbit effects, Figure 4 depicts the single-group and double-group energy levels calculated for Pa(Cot)2. Although in a π-only picture (Fig. 2) the fφ orbitals are predicted to be the lowest among the f-manifold because of the interaction with Cot π3 orbitals, our PW91 calculations reveal that the 5f orbitals in the D8h ligand field will be split as fσ fφ fπ fδ, as shown in Figure 4. The fact that the fσ orbital becomes even lower in energy than the fφ orbitals indicates that the σ-type orbitals also play a role in these orbital interactions. The interactions between fδ and the e2u ligand group orbitals are so strong that these orbitals are

Relativistic DFT and Organoactinide Complexes				363
TABLE 4 Correlation of the Symmetry Species of Single D8h
Group and Double D8h* Group
Γi (D8h)	An basis	(Cot)2 basis	Γi	e1/2g (D8h*)
a1g	s, dσ	π0g		e1/2g
a2g		π4g		e1/2g
b1g				e7/2g
b2g	dπ	π1g		e7/2g
e1g				e1/2g e3/2g
e2g	dδ	π2g		e3/2g e5/2g
e3g		π3g		e5/2g e7/2g
a1u	pσ, fσ	π0u		e1/2u
a2u				e1/2u
b1u		π4u		e7/2u
b2u				e7/2u
e1u	pπ, fπ	π1u		e1/2u e3/2u
e2u	fδ	π2u		e3/2u e5/2u
e3u	fφ	π3u		e5/2u e7/2u

pushed even energetically close to the dσ orbitals. As a result, in the absence of spin-orbit effects, the energies of the low-lying electronic states of Pa(Cot)2 increase as 2A2u (fσ) 2E3u (fφ) 2E1u (fπ) 2A1g (dσ) 2E2u (fδ) in the single group. Thus, we predict that the 2A2u state, corresponding to the (fσ)1 configuration, is the ground state of the complex. However, the near-degeneracy of the fφ and fσ orbitals suggests that the ground state could be strongly dependent on the spin-orbit coupling effects as well.

When spin-orbit coupling is included, the ordering of the low-lying states

is found to increase as E5/2u (fφ) E1/2u (fσ π) E3/2u (fπ) E7/2u (fφ) 2E1/2u (fπ σ) E1/2g (dσ) 2E3/2u (fδ) 2E5/2u (fδ) in the double-group representation. Therefore, the ground state is now identified as an E5/2u (fφ) Kram-

ers doublet. From the symmetry correlation listed in Table 4, the E5/2u (fφ) doublegroup state can only come from the 2E2u and 2E3u single-group states. In Pa(Cot)2, the lowest available 2E2u (fδ) state lies too high in energy to contribute to the state mixing. Thus, the spin-orbit-coupled ground state corresponds to an almost pure (fφ)1 configuration (instead of the single-group ground state with an (fσ)1 configuration), which is accidentally just the ground state expected from the simple π-only picture (Fig. 2). This example is typical, and it illustrates an important aspect of actinide quantum chemistry: the inclusion of spin-orbit coupling, while insignificant for the calculation of geometries, can change the nature of the ground state.

364	Li and Bursten

FIGURE 4 Single-group and double-group energy level correlation for Pa(Cot)2.

Relativistic DFT and Organoactinide Complexes

365

Table 5 compares the relative energies of the low-lying states of Pa(Cot)2 calculated using the PW91 method, with spin-orbit effects included, and those calculated from the ab initio spin-orbit CI method (91). The electronic transition energies labeled PW(TS) were determined by using Slater’s transition-state method (92), whereas the other PW91 energies were calculated as the difference in total energy between excited states and the ground state. Because the SOCI calculation used a nonoptimized (Nopt) geometry, we also performed a PW91 calculation of these excited states using this assumed geometry in order to allow a direct comparison between the two methods at the same geometry.

Several features in Table 5 are notable. We see excellent agreement between the excitation energies calculated by using Slater’s transition-state method and those from the state energy differences (within 0.08 eV). Further, when using the same geometry as used in the SOCI calculations, the PW91 method reproduces the SOCI excitation energies very well. The differences between the calculated energies by the two methods are all less than 0.08 eV, except for the d1-state, where a larger difference (0.27 eV) exists. A similar difference in the energy of this state exists when the geometry is improved by increasing the C and H basis sets in the PW91 calculations from DZP to TZ2P. We therefore believe that the difference between the results of the two methods might be partly due to the smaller basis sets (DZ) used in the SOCI calculations; because of the diffuse nature of the 6d orbitals, it seems likely that a more extensive basis set is needed to describe them accurately.

It is remarkable that the PW91 DFT method can reproduce the results of the much more expensive SOCI method so well for the actinide excited states. Given the huge difference in the amount of computer time demanded by these two methods, the application of the DFT method to excited states of other actinide compounds with f n (n 1) configurations promises to be a challenging venture (because of the problem of state multiplets) but potentially a very fruitful one.

Based on the double-group PW91 calculations of Pa(Cot)2, we have determined all the energies for the transitions from the 6e5/2u level to the virtual levels up to 10e3/2g (cf. Fig. 4). Because of the centrosymmetry of the molecule, allowed transitions must involve a parity change (93). In the D8h* double group, the excited state has to be one of the E3/2g, E5/2g, or E7/2g states for an electric-dipole- allowed transition from the E5/2u ground state. Based on this selection rule, only some f –d, f –π3, and LMCT (ligand-to-metal charge transfer) transitions of types E5/2u → E3/2g (x, y polarization), E5/2u → E5/2g (z polarization), and E5/2u → E7/2g (x, y polarization) are allowed.

Among the dipole-allowed transitions, the lowest-energy f –π3 transitions, E5/2u → E5/2g and E5/2u → E7/2g, are both predicted to occur at 397 nm, while the lowest-energy f –d transition, E5/2u → E3/2g, is predicted to occur at 27,200 cm 1, or 368 nm. The latter is in near-perfect agreement with the experimental estimate of 365 nm by Streitwieser and coworkers (94). We find only two π2–dσ LMCT

TABLE 5																															a
			Comparison of Calculated PW91 and SOCI State Energies (eV) for Some Low-Lying States of Pa(Cot)
																															2
State			Configuration																				PW(TS)	b	PW91	PW(TZ2P)	PW(Nopt)	c	SOCI	d	∆E	e
E	5/2u							e	3u		(f				1	)							0.000		0.000	0.000	0.000		0.000		0.000
E	1/2u				a	2u		e					1u			(f		1	)				0.041		0.049	0.131	0.101		0.166		0.065
E																							0.375				0.395
	3/2u							e	1u		(f				1	)									0.369	0.448			0.477		0.082
E								e			(f				1	)							0.378		0.379	0.357	0.362		0.362		0.000
	7/2u								3u
2E		1/2u			e	1u		a					2u			(f		1	)				0.523		0.541	0.613	0.567		0.569		0.002
E								a															0.598		0.685	0.680	0.651		0.925		0.274
															1
	1/2g								1g		(d )
2E		3/2u						e	2u		(f				1	)							1.141		1.122	1.206	1.227		1.222		0.005
2E								e			(f				1	)							1.383		1.350	1.431	1.463		1.427		0.036
		5/2u							2u
3E			(e			)	3	(e					2			(π	3		f	2	)		2.956		2.896	—	2.873		—		—
		5/2u								3u		)
3E					2u											(π
		3/2u	(e	2u		)	3	(e		3u		)		2				3	f	2		)	2.982		2.924	—	2.901		—		—
a
All energies are calculated by using the DZP basis sets for C and H, except for PW(TZ2P), where the TZ2P basis sets are used for C
and H.
b
The PW(TS) values are calculated by using Slater’s transition-state method.
c
The PW(Nopt) energies are calculated at the same geometry assumed in the SOCI calculations.
d
The spin-orbit CI results are taken from Ref. 91.
e
∆E is the energy difference between SOCI and PW(Nopt).

366

Bursten and Li

5/2u

1/2g

Relativistic DFT and Organoactinide Complexes

367

transitions that should occur in the visible region (430–435 nm), corresponding to the transitions from 3e2u-based ligand orbitals (5e5/2u and 7e3/2u) to the Pa dσ orbital (8e ). These two LMCT transitions at 430–435 nm are the likely origin for the low-energy shoulder reported to occur at 490 nm in the spectrum of Pa(TMCot)2 (94). These transitions are in the violet portion of the visible spectrum and may thus be responsible for the characteristic golden-yellow color of protactinocene.

As discussed earlier, Pa(Cot)2 has an E5/2u (fφ), i.e., |MJ | 5/2, ground state, as a consequence of substantial contributions of both spin-orbit and ligandfield effects. The ground magnetic properties of protactinocene will be dominated by the characteristics of the E ground state. Based on the magnetic dipole transition selection rule (∆MJ 1) (95), the E5/2u ground state is expected to be ESR silent: When the MJ components of the E5/2u state are split in a magnetic field, the predicted ESR transition from MJ 5/2 to MJ 5/2 is not allowed.

Besides the ground state, the E1/2u and E3/2u excited states are possible contributors to the observed magnetic moment of Pa(Cot)2. Using our calculated PW91 state energies and Warren’s formalism for an f1 axial system (96), we have determined the anisotropic room-temperature magnetic moments, µz and µx ,y, as 3.33 and 0.99 BM, respectively, which will give rise to an average magnetic moment µ 2.09 BM. This calculated room-temperature magnetic moment is fairly close to the value of 1.96 BM that was obtained via the spin-orbit CI calculations on Pa(Cot)2 (91). Both values are close to the experimental room-tempera- ture value µ 1.88 BM for the 4f1 sandwich complex Ce(Cot)2 (97). These calculated excitation energies and magnetic properties show that the gradientcorrected DFT methods, especially PW91, can be used as a reliable theoretical method for large actinide molecules, for which a spin-orbit CI calculation is too expensive to be carried out.

6.4. Vibrational Analysis of Actinocenes

Not surprisingly, vibrational spectra have proven to be an invaluable tool for experimental chemists in the characterization of transition metal and actinide sandwich compounds (98). Most known actinocenes have been characterized early on by vibrational spectroscopy (99). The IR and Raman spectra of thorocene and the IR spectra of protactinocene and uranocene were reported in the 1970s (100,101). However, normal coordinate analysis of these vibrational spectra is difficult because of the large number of vibrational modes involved. So far only a tentative assignment of the vibrational spectra of thorocene and uranocene, based on a qualitative group theory analysis, has been advanced (102).

To date, the lack of appropriate theoretical computational methods has hampered a comprehensive first-principle analysis of the detailed vibrational properties of large molecules such as the actinocenes. Because the theoretical