ARTICLE pubs.acs.org/JPCA
Calculations of Bond Dissociation Energies. New Select Applications of an Old Method Edward A. Boudreaux*,† Department of Chemistry, University of New Orleans, 102 Chemical Sciences Annex, 2000 Lakeshore Drive, New Orleans, Louisiana 70148, United States ABSTRACT: Application of Sanderson’s definition of electronegativity2-10 as stability ratios (SRs), which had been applied in the past to a wide variety of organic and nontransitional metal inorganic compounds with very good success, has been revived, modified so as to be applied to any types of molecules, including those containing transition metals, lanthanides, and actinides. This paper is limited to a demonstration of the method which is applied to a few metal cyclopentadienyl compounds, plus specific emphasis on the U(III) metallocene (CpSiMe3)3U-AlCp* recently prepared by Arnold and co-workers15 having no experimental bond energies available. It is shown that computed bond energies of pertinent metallocyclopentadieneyls are in excellent agreement with the available experimental data. Calculated bond energies for all essential bonds in the uranium metallocene cited above are provided together with a further analysis of the bonding and magnetic properties of this unique compound.
’ INTRODUCTION The most widely used approaches for investigating molecular electronic structures, bonding, and bond dissociation energies are through the application of molecular theory. Among the methods currently in vogue are density functional theory (DFT) and ab initio methods when applicable. However, it has been recently pointed out by Paul von Rague Schleyer,1 University of Georgia, that while DFT methods are good in attaining reliable geometries, they are poor in providing accurate bond dissociation energies. Since the “exchange correlation energy” presents great difficulty for DFT methods, this must be accounted for by empirically adjusting the functionals. In general, it may be accurately stated that when it comes to bond dissociation energies, reliable charge distributions, and associated properties, DFT methods are by and large not very consistent in providing reliable results. A case in point is the HgF bond dissociation energy, D(HgF), in HgF2. The results obtained from a variety of DFT and ab initio methods2 give D(HgF) ranging from 744 to 493 kJ/ mol, while the reported experimental value3 is 380 kJ/mol. Note that the very best result (QRPP-CCSDFT) yields 493 kJ/mol, which is still some 1.3 greater than the experimental value. The discrepancy appears to be to be due to an overestimate of atomic charges, thus causing the electrostatic contribution to the bond energy to be greater than it should be. On the other hand, the same report2 provides a relativistic DFT HgCl bond dissociation energy of 439 kJ/mol for HgCl2,, in good agreement with the 346 kJ/mol reported experimentally.3 Dolg has also conducted bond dissociation energy calculations using quasi-relativistic pseudopotentials on a variety of halogen dimers and their hydrides, which provided varying results. Consequently, it is the purpose of the work to resurrect (with modification) a previously applied alternative for computing reliable bond dissociation energies and related properties, which is relatively simple in principle and systematic in approach. r 2011 American Chemical Society
Many years ago R. T. Sanderson derived an electronegativity concept which he defined as the stability ratio (SR).4-12 SR was defined as the ratio of an atom’s (electronic charge/volume) to that of the (electronic charge/volume) of an isoelectronic noblegas-like atom. Sanderson demonstrated that the most definitive application of the SR concept was in determining electronic charges and dissociation energies of bonded atoms.10-12The bond energies of many hundreds of compounds ranging from highly ionic to totally covalent were calculated with no less than 90% accuracy, and in most instances the results were accurate within a range of 95-100%.10-12 However, the majority of metal-metal bonded systems and compounds containing transition metals, lanthanides, and actinides were not treated by Sanderson. While he did provide some good results for main group metals bonded to saturated hydrocarbons, no data were given for metal cyclopentadienyl compounds, which is the focus of this study. But prior to proceeding to specific examples of interest, an overview of Sanderson’s approach with some modifications included will be presented. The stability ratio is SR = EDA/EDI, where EDA is the atomic electron number density of a specific atom, A, and EDI is the atomic electron number density of an isoelectronic noble-gas-like atom, I. This latter quantity is obtained by interpolation from a plot of EDI for real noble gas elements, as a function of their numbers of electrons, Ze. In any case, ED = Ze/(4/3)πrc3, in which rc is the nonpolar covalent radius of the particular atom, normally expressed as half the homonuclear diatomic bond distance. The SRM, stability ratio of a molecule (or formula unit), is defined as the geometric mean of SRs of all atoms in the formula unit. According to Sanderson, the charge acquired by any Received: September 17, 2010 Revised: December 13, 2010 Published: February 14, 2011 1713
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bonded atom, qA, is given by qA ¼
(relative to volume units) has the following form SR M - SR A pffiffiffiffiffiffiffiffi 2:08 SR A
However, eq 1 has been modified to eq 2, which allows for the total number of atoms, n, bonded to a given atom, A, since the net charge, qA, will vary in proportion to the “n” numbers of atoms bonded to it in different environments. qA ¼
X 1 pffiffiffiffiffiffiffiffi ½ðSR M - SR A Þ þ ðSR n - SR A Þ 3:12ðnT Þ SR A n ð2Þ
where nT is the total numbers of atoms bonded to A and SRn is the stability ratio of each nth atom. But eq 2 applies best when two or more bonds to a given atom are involved. The simpler version provided in eq 1, with the constant changed from 2.08 to 1.56, is better, when only one bond is involved. The fractional ionic character, IC, of atom A bonded to n atoms is P jqA nqn j ð3Þ ICðAÞ ¼ 1þn The bonding radius, rb, of an atom in a heteronuclear bond is obtained from the relation rb = rc Bq, where Bq is an empirical charge/size parameter for the shrinking of a þq charged atom, or the expansion of a -q charged atom. Pertinent values are listed by Sanderson.13 Sanderson’s original approach was to partition the bond energy into ionic (Ei) and covalent (Ec) contributions, wherein only the former incorporates atomic charge and is determined in terms of the following fixed relationship scaled by IC (see eq 3). The ionic contribution is given by Ei ¼
332 IC RAB
e2 - ID þ EA þ - ERes RAB
# ð5Þ
in which IC and CC are fractional ionic and covalent characters, respectively, SAB is the D-A overlap, ID the donor ionization energy, EA the electron affinity of the acceptor, and ERes the resonance energy between ionic and covalent contributions to the bond energy. The EDA is finally multiplied by the IC. ERes was originally expressed by Pauling as the square of the electronegativity difference between bonded atoms A and B.14 This concept when applied to the SR definition of electronegativity
14:43 R 4 AB
ðeVÞ
ð6Þ
where RAB is in angstroms and the SR values are unitless. Consequently, the total electrostatic contributions to the bond energy are ð7Þ Eels ¼ ½EDA þ Ei ðICÞ According to Sanderson, the covalent portion of the bond energy may be expressed as Rc ðEAA EBB Þ1=2 CC ð8Þ Ecov ¼ R0 in which Rc is the sum of the homopolar covalent radii of A and B, R0 is the observed bond distance, EAA and EBB are the bond energies for molecules A2 and B2, and CC is the fractional covalent character. But eq 8 is modified to include overlap effects based on the following. The bond energy, BE, of a homonuclear diatomic molecule is expressible as15 Rþβ ð9Þ BE ¼ 1þS where R and β are the Coulomb and exchange energies, respectively, and S is the interatomic overlap. Thus the covalent contribution to the bond energy should be multiplied by (1 þ S), so eq 8 becomes Ecov ¼ ð1 þ SÞðRc =R0 ÞðEAA EBB Þ1=2 CC
ð10Þ
In the case of the H2 molecule, R, β, and S may be evaluated in closed form, involving the internuclear bond distance, R.15 1 1þR R ¼ - þ expð - 2RÞ ð11aÞ R R
ð4Þ
in which RAB is the bond distance, in angstrom units, and Ei is in kJ/mol. Equation 4 has been modified to include a charge-transfer contribution in terms of a donor (D) and acceptor (A) format, utilizing the same relations provided in an earlier publication.14 The expression for the D-A contribution to the bond energy " 1=2 # IC IC 2 EDA ¼ 1 - 2 SAB þ S CC CC AB "
ERes ¼ jSR A - SR B j
ð1Þ
β ¼ - ð1 þ RÞ expð - RÞ
ð11bÞ
S ¼ ½1 þ R þ R 2 =3 expð - RÞ
ð11cÞ
Actually the R terms in eqs 11a11a-11c contain the exponent to the wave function, ζ, so each of the R terms is ζR, but ζ = 1 for the H atoms. Note that eq 11c is exact only for overlap between 1s orbitals (ζ = 1). However, since a modified form of eq 11c is employed for interatomic overlaps between atoms regarded to be essentially spherical, the ζ in the ζR terms of eq 11c is an effective value for any overlapping atomic orbitals, regarded as having their Æræ essentially the same as for spherical distributions The relation used for evaluating ζ is ζ = (n0 þ 0.5)/Æræ, where n0 is an effective principal quantum number for overlap; n0 = n for n = 1, 2; n0 = 2 for n g 3; n0 = 3 for f orbitals. In evaluating the interatomic overlap involving all other atoms, eq 11c becomes S ¼ ½1 þ ζR þ ðζRÞ2 =3e - ζ R
ð12Þ
If the two bonded atoms, A and B, are heteronuclear, ζeff = (ζA þ ζB)/2 . In calculations of R and β according to eqs 11a and 11b for a number of homonuclear and heteronuclear diatomics, the author has found (unpublished work) that β ≈ 0.3R. In this work all energies are in electronvolt units and the overlaps in atomic units. The final bond energies are in kilojoules per mole. 1714
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Table 1. Preliminary Data for Al-U Bond Energy Calculations Al
U
Al-U
Table 2. Preliminary Date for MgCp2 Bond Energy Calculations Mg
Cp
Cp2
Mg-Cp2
SR
1.65
3.00
2.22
SR
1.32
3.70
-
2.62
IPa EAa
5.98 0.43
6.19 ∼2.5
-
IPa-i EAa-i
7.65 ∼-0.22
8.9 1.8
-
-
qb
0.15
-0.15
-
qa-i
0.56
-0.28
-
-
IC
-
-
0.15
IC
-
-
-
0.38
-
-
0.85
CC
-
-
-
0.62
1.26
1.91
-
Ra-i (1)
1.36; 0.57
1.8 (see text)
-
-
C.C Rc (1) Rc (2)
-
-
3.17
BEd
3.0
2.3
-
Ra-i (2)
-
AOe Æræ f ζg h
S
ERi
(2þ, see text) -
1.5
2.4
3s 2.08
3p 3.05
5f 1.44
6d 3.22
7s 4.34
-
BEa-i AOa-i
1.33 3s
2p(π)-centroid
4.3 -
(see Table 8) -
1.2
0.82
2.43
0.78
0.58
0.85 (avg)
Æræa-i
3.03
1.80
-
-
0.091
ζa-i
0.83
1.39
1.39
1.11 (avg)
0.24
Sa-i
-
-
0.196
0.085
ERa-i
-
-
-
1.07
-
-
a
Ionization potentials (IP) and electron affinities (EA) in eV, from ref 17. b Atomic charges according to eq 1. c (1) Standard covalent radii and charge adjusted radii are available from numerous sources. Atomic radii taken as rmax from ref 21. (2) Bond distance is sum of covalent or charge adjusted radii. All in angstrom units. d Bond energies in eV as obtained from ref 19. e Valence atomic orbitals of bonding atoms. f Orbital radial expectation values in atomic units from ref 21. g Orbital exponents, ζ, determined as stated in the text. h Interatomic overlap, S, according to eq 12. i Resonance energy in eV according to eq 6.
’ COMPUTATIONS AND RESULTS Sample calculations of bond energies for U-Al, MgCp2 (magnesium dicyclopentadienide), FeCp2 (ferrocene), and UCp4 are given, since these have experimental bond energies with which to compare the calculated values. Final calculations are applied to a U(III);Al(I) system with an unsupported bond, which is found in the organometallic complex (CpSiMe)3U-AlCp* (Cp* = C5Me5-) reported by Arnold and coworkers.16 Required SR data has been extended to include the d-block and f-block elements.17 1. U-Al. All pertinent data are presented in Table 1. Since the electron affinity is not reported in the literature, it must be interpolated. The original Pauling electronegativities of U and W are both listed as 1.7, with the ionization potential and electron affinity reported for W as 7.98 and 0.82 eV, respectively.18 Thus the Mulliken electronegativity of W is (7.98 þ 0.82)eV/2 = 4.4 eV, with transformation to the Pauling electronegativity of - 4.4 eV 0.386/eV = 1.7, which is the reported Pauling electronegativity of W. Since U has the same Pauling electronegativity as W, its Mulliken electronegativity must also be 4.4 eV. The first ionization potential of U is 6.19 eV,18 so according to the Mulliken electronegivity expression the electron affinity is calculated to be 2.6 eV. Another approach is to consider differences in the electron affinities between MF6 molecules and their respective M atoms. The electron affinity of WF6 is 3.5 eV18, and using the electron affinity for W given above, the difference in electron affinities ΔEA = 2.7 eV. Similarly, it is assumed that EA(UF6) EA(U) = 5.1 eV18 - EA(U) = 2.7 and EA(U) ≈ 2.4 eV. Finally, a quasi-relativistic HF-SCF calculation of the EA(U) = 2.2 eV (Boudreaux and Baxter, unpublished work). Taking the average of the two interpolated values, the EA(U) ≈ 2.5 eV, as listed in Table 1. Also reported in Table 1 are the homonuclear bond dissociation energies for U2 and Al2 = 221 kJ/mol (2.3 eV) and
a-i
See footnotes in Table 1.
144 kJ/mol (1.5 eV), respectively.3 The overlap was evaluated via eq 12 at an internuclear distance of 3.17 Å (sum of covalent radii). The required ζ were obtained according to the Cusachs and Corrington recipe for single ζ STOs,19 ζ = (n þ 0.5)/Æræ. Although reference20 contains much essential data for a number of elements, the Æræ values for U were obtained from the Dirac-Fock relativistic wave functions of Declaux.20 According to eq 2 the charges on U and Al are -0.15 and þ0.15, respectively. This is not surprising since the electronegativity (SR) of U is nearly twice as large as that of Al (see Table 1). Note that charges derived from this procedure are very different from those obtained by most MO calculations. As is shown later, if charges from DFT calculations are employed rather than those derived from eq 2 the calculated Eels (eq 7) is much too large in comparison to the Ecov term (eq 8); hence, the calculated bond energies substantially exceed the experimentally reported data. The diatomic overlap at R = 5.99 au with ζeff = 0.85 (the weighted average for U(6d,7s) and Al(3s3p) orbitals) is only 0.91, with a resonance energy, ERes = 0.20 eV, as determined by eq 6. All of the essential data for the bond energy calculation of U-Al are presented in Table 1. The remaining quantities, EDA, Ei, and Ecov and bond energy, BE, are all provided below in Table 8. 2. MgCp2. Magnesium dicyclopentadienide is a fairly ionic compound; hence, it is anticipated that there will be a sizable electrostatic contribution to the Mg-Cp bond dissociation energy. All data required for calculating the bond energy are presented in Table 2. The experimental Mg-Cp bond distance is 4.355 au.21 At this distance and ζeff = 1.11 (average for Mg and Cp, see Table 2) the interatomic overlap is S = 0.094. However, the EDA is negative, - 0.10 eV, but the total Eels = 2.19 eV. In evaluating the homonuclear bond energy for Cp2, the following approach was taken. It has been noted from numerous prior studies (Boudreaux, unpublished work) that the magnitude of the Coulomb integral, R, for a bonding orbital (single or hybrid) is given approximately by a [|1/Ær-1æ][|1/IP|(eV)], for which R ≈ 3.9 eV for a Cp 2p orbital; hence, β ≈ 1.2 eV. At R(Cp-Cp) ≈ 2.84 au, ζ = 1.39, S = 0.196; thus, according to eq 9 the bond energy for Cp2 is BE = 4.3 eV. 1715
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Table 3. Preliminary Data for FeCp2 Bond Energy Calculation Fe
Cp
Cp2
Fe-Cp2
SR
3.19
3.70
-
3.52
IPa-i EAa-i
7.87 0.15
8.9 1.8
-
-
qa-i
0.081
-0.041
IC
-
-
-
0.054
CC
-
-
-
0.946
Ra-i (1)
1.24; 0.82 (2þ)
1.8 (see text)
-
-
Ra-i (2)
-
-
1.5
2.5
BEa-i
≈ 2.13
-
4.3
(see Table 8)
AOa-i Æræa-i
3d 1.67
4s 2.72
ds -
2p(π) 1.80
-
-
ζa-i
1.50
0.92
1.21
1.39
-
1.30 (avg)
Sa-i
-
-
0.196
0.043
ERa-i
-
-
-
0.19
a-i
-
See footnotes in Table 1.
Table 4. Preliminary Data for UCp4 Bond Energy Calculations U
Cp
Cp2
U-Cp4
SR
3.00
3.70
-
3.55
IPa-i
6.19
8.9
-
-
EAa-i
∼2.5
1.8
-
-
qa-i
0.124
-0.031
-
-
IC
-
-
-
0.050
CC
-
-
-
Ra-i (1) 1.55; 1.03 (3þ)
1.8 (see text)
Ra-i (2) 2.3 BEa-i
-
AOa-i
5f
Æræa-i
1.46 3.21 4.31 -
ζa-i
6d
7s
(ds) 2p(π)
0.950 -
1.5 4.3
2.81 (avg) (see Table 8)
-
-
-
-
2.40 0.78 0.58 0.68 1.39
1.39
1.04
Sa-i
-
-
0.196 0.068
ERa-i
-
-
-
a-i
1.8
0.16
See footnotes in Table 1.
Table 8 lists the bond energy data and for the questionable reported experimental value (see footnote c Table 8). 3. FeCp2. Ferrocene is a classic example in the arena of organometallic complexes. The Fe-Cp bond distance is 4.73 au, somewhat longer than that of MgCp2, even though ferrocene is decidedly more covalent according to theory and experiment. As shown in Table 3, the calculated charge on Fe is only þ0.081, thus giving each Cp a charge of -0.041 and IC = 0.054, so that the Fe-Cp bonding is essentially covalent (CC = 0.946). Given the data in Table 3 with ζeff = 1.15, the interatomic overlap is quite small (S = 0.072), which is surprising in light of the predominantly covalent bonding character. To obtain the correct Fe-Fe bond energy, it is noted that the value 1.23 eV is too small to be representative of a true covalent bond.3 But according to ref 3, it is notable that the bond energies of Fen clusters increase up to 3.17 eV at n = 5, subsequently decrease at higher n, but rise again to 3.61 eV for the infinite metallic lattice. If the initial values (n = 2-5) are considered prior to the bonding becoming decidedly metallic, these should be representative of
essentially covalent type bonding. The average bond energy for these cases is 2.13 eV, which is more indicative of covalent bonding and is the value reported in Table 3. So for FeCp2, Ecov = 2.97 eV and the net calculated bond energy, 313 kJ/mol (see Table 8), compares quite favorably (97%) with the reported experimental value23 of 297 kJ/mol. Most experimental bond energy data provide reported values which vary within 3-8%. 4. UCp4. This compound is reported to have variable U-Cp bond distances ranging from 5.27 to 5.35 au, which average to 5.31 ( 0.04 au.24 Such an effect is a consequence of pronounced crowding of the four Cp ligands about the U. Here again the calculated charges are low. Also, as shown in Table 4, ζeff = 0.91 and the overlap is S = 0.109 so that the small EDA (0.031 eV) is for all practicable purposes negligible. In evaluating the Ecov contribution, the ratio of Rc to R0 (see eq 8) has to be considered, since these quantities are expected to differ considerably due to U-Cp bond lengthening. The reported radius of U(III) is 1.947 au, but that of Cp- must be interpolated from radii trends in C0 (1.46 au) to C4- (4.91 au). The interpolated value for C1- is 2.33 au, which is applied to the bonding carbon in the Cp- ligand. Hence, Rc = 1.947 þ 2.33 = 4.28 au and R0 = 5.31 ( 0.04 au, so Rc/R0 = 0.806 ( 0.008, or 0.81. As listed in Table 8, the calculated bond energy (280 kJ/ mol) is higher than the reported value25 of 247 kJ/mol. But the same Cp crowding affecting bond lengthening also causes the bond energy to be lowered. Since the U-Cp bonds are to the center of the Cp centroid, it is expected that Cp/Cp repulsion will be involved. The Cp;Cp distances are about 2(3.21 au) = 6.42 au, and the Cp charges are -0.031 each. Thus the electrostatic repulsion force between any adjacent pair of Cp ligands translates into a bond-destabilization energy of about -0.4 eV. Hence, the net U-Cp bond energy is BE = (2.92 - 0.4) eV = 2.52 eV (243 kJ/mol), thus agreeing very well with the reported experimental value. It is notable that this effect is very similar to that associated with “lone pair” weakening, as discussed by Sanderson.12 5. (CpSiMe3)3U-AlCp*. This last example was recently prepared and characterized by Arnold and co-workers.16 In dealing with this case three separate parts will be considered: (1) AlCp*, (2) (CpSiMe3)3U, and (3) (CpSiMe3)3U-AlCp*. (1). AlCp*. This fragment consists of aluminum in an unusual formal oxidation state of Al(I) binding to the Cp* = (C5Me5)1-. There is not as much information available for Cp* as there is for Cp, but the stability ratios are not very different: SR(Cp) = 3.70 and SR(Cp*) = 3.85. As stated previously, ionization energy and electron affinity of Cp are 8.93 and 1.8 eV, respectively. The Cp* ionization energy according to Meyer and Harrison26 is inferred to be 7.77 eV, but the electron affinity is not available. Returning to the Mulliken electronegativity vs IP and EA relations presented earlier, the Mulliken electronegativity for Cp is (8.93 þ 1.8)eV/2 = 5.37. With SR(Cp) = 3.70, the Mulliken electronegativity is reproduced by 3.70/0.689 = 5.37. Hence, for SR(Cp*) = 3.85 its Mulliken electronegativity should be about 3.85/0.689 = 5.59, and since IP = 7.77 eV the EA = 3.4 eV. The Cp2* bond energy is derived in the same manner as was that of Cp2 previously presented. The Cp*-Cp* bond distance was taken to be 3.0 au vs 2.84 au for Cp;Cp. The presence of the Me groups on the adjacent bonding carbons in Cp2* causes sufficient repulsion to stretch the bond distance by about 0.16 au. At R = 3.0 au and ζ ≈ 1.4, the overlap between Cp*-Cp* is S = 0.166. As described earlier, the Coulomb energy is estimated to be R ≈ 4.5 eV and β ≈ 1.4 eV and according to eq 9 BE = 5.1 eV. All pertinent data are presented in Table 5. 1716
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Table 5. Preliminary Data for AlCp* Bond Energy Calculations Al
Cp*
Cp2*
Table 7. Preliminary Data for (Me3SiCp)3UAlCp* Bond Energy Calculations Al0.21þ
Al-Cp*
U0.097þ
Si*Cpj (Si*Cp)2 (Si*Cp)3U-AlCp*
SR
1.65
3.85
-
2.52
SR
2.21
3.06
3.56
-
2.96
IPa-i EAa-i
5.98 0.43
∼7.8 ∼3.4
-
-
IPa-i EAa-i
8.7 1.3
6.5 2.9
-
-
-
qa-i
0.21
-0.21
-
-
qa-i
0.21
0.097
IC
-
-
-
0.25
IC
-
-
-
-
0.17
CC
-
-
-
0.75
CC
-
-
-
-
0.83
-
Ra-i (1) 1.13
1.52
-
-
-
R a-i(2) -
-
-
1.59
3.12
2.3
-
4.5
(see Table 8)
0.02 (on U)
Ra-i (1) 1.26; 0.57 (þ3)
∼2.0 (see text) -
Ra-i (2) -
-
1.59 (see text) 1.89
BEa-i
1.6
-
3.8
(see Table 8)
BEa-i
∼1.9k
AOa-i Æræa-i
3s 3p 3(sp) 2p(π) 2.08 3.05 1.78
-
-
AOa-i Æræa-i
3s 3p sp 2.08 3.73 -
-
-
-
ζa-i
1.24 (U/Al)
5f 6d fd 1.44 3.22 -
1.20 0.82 1.01
1.40
1.40
1.21 (avg)
ζa-i
1.20 0.67 0.94 1.74 0.78 1.53 -
-
a-i
S
-
-
0.166
0.153
Sa-i
-
-
-
0.166
0.018
ERa-i
-
-
-
2.5
ERa-i
-
-
-
-
0.19
a-i
See footnotes in Table 1. j Si* = SiMe3. k As interpolated from AlþAln cluster data in ref 19 a-i
See footnotes in Table 1.
Table 6. Preliminary Data for U(CpSiMe3)3 Bond Energy Calculations U
CpSiMe3
(CpSiMe3)2
Table 8. Calculated vs Experimental Bond Energies
U-(CpSiMe3)3
compound
bond
EDAa
Eia
Eelsa Ecova BE(calcd)b BE(exptl)b ref
SR
3.00
3.56
-
3.41
UAl
U-Al
0.14
0.68
0.82 2.44
313
326 ( 29 3
IPa-i
6.19
∼8.6
-
-
MgCp2
Mg-Cp -0.10
2.29
2.19 1.61
366c
310c 22
EA
∼2.5
∼1.7
-
-
FeCp2
Fe-Cp -0.028
0.31
0.28 2.97
307
297 23
qa-i
0.097
-0.032
-
-
UCp4
U-Cp
0.031
0.26
0.29 2.63
280
247 25
IC
-
-
-
0.049
AlCp*
Al-Cp*
0.54
1.91
2.45 2.59
476
- -
CC
-
-
-
0.95
USi3*d
U-Si*
0.10
0.55
0.65 3.24
368
- -
-
-
-0.17 -0.13 -0.30 1.52
115
- -
-
∼1.6 (see text) 2.54 (avg) ∼4.2 (see text) (see Table 8)
a-i
Ra-i (1) 1.55; ∼0.87 (3þ) Ra-i (2) 2.3 BEa-i
Si3*UAlCp* U-Al
-
-
-
-
AOa-i
5f
Æræa-i
1.44 3.22 4.34 -
ζa-i
2.43 0.78 0.58 0.68 1.39
-
1.04 (avg)
Sa-i
-
-
0.166
0.097
ERa-i
-
-
-
0.19
a-i
6d
7s
(ds) 2p(π) -
See footnotes in Table 1.
The Al-Cp* bond distance is reported to be 3.578 au16 and ζeff = (1.01 þ 1.4)/2 = 1.21 (see Table 5) and S = 0.212. From Table 8 BE = 5.04 eV (486 kJ/mol), which is substantially higher than that of the Mg-Cp bond in MgCp2. But this is what is anticipated to be the case. (2). U(CpSiMe3). The CpSiMe3 ligand will hereafter be designated as CpSi* which has SR = 3.56, but there is no ionization energy or electron affinity for this entity. But according to the data for Cp, the ratio of the electron affinity to the ionization energy is about 0.2 and the Mulliken electronegativity can be formulated as (x þ 0.2x)/2 = 3.56/0.689 = 5.17, for which x = 8.6 eV = IP and 0.2x = 1.7 eV = EA. The average bond distance reported for U-CpSi* is 4.80 au.16 The U(CpSi*)3 molecule has SR = 3.41, and the calculated charges are 0.097 and -0.032 for U and CpSi*, respectively. Naturally the iconicity is very small, IC = 0.05, while CC = 0.95. All pertinent data are in Table 6. The bond energy for (Si*Cp)2 is derived in the same manner as was applied to Cp2 and Cp2*. As a matter of fact, the Si*Cp;CpSi* bond distance is taken to be
a In eV. b In kJ/mol. c The reported MS appearance potentials in ref 22 are for the process MgCp2 f MgCp2þ f MgCpþ þ Cp. The release of Cp is reported to have the energy 310 kJ/mol, but the Mg-Cp bond energy from MgCp2 in this work is calculated to be 366 kJ/mol. The difference in these two values is rationalized in terms of the following relation: net BE = BE0 - [E0eles - Eþeles þ E0cov þ Eþcov þ EþDA], where the superscripts 0 and þ refer to the neutral molecule and its cation, respectively. Without going into details, the results are the following: net BE (eV) = 3.80 - [2.18 - 2.47 þ 1.61 þ 1.23 - 1.97] = 3.22 eV or 304 kJ/mol, in good agreement with the reported 310 kJ/mol.
about the same as in Cp* (3.0 au). The Coulomb energy for eq 9 was found to be R ≈ 4.1 eV and β ≈ 1.2 eV. With the overlap, S = 0.166, the BE = 4.5 eV. However, there will be some bond destabilization effect owing to the bulky Si* group attached to the carbon adjacent to the one involved in the Cp;Cp bond. The CH3 groups on Si are about 6.43 au from the Cp;Cp bond site. The perturbation on the energy of the Cp;Cp bond, Ebb, may be expressed as Ebp ≈ -(Rp(BE))/R3, where up is the polarizability of CH3 (17.6 au), BE is the energy of the pertinent bond (4.5 eV), and R = 6.43 au. Hence, Ebp ≈ - 0.3 eV and BE0 = 4.5 0.3 = 4.2 eV (404 kJ/mol). Returning to U(CpSi*)3, the ζeff = 1.04 for the U-Cp bond and S = 0.097(3). All pertinent bond energy data are provided in Table 8. (3) . (Si*Cp)3 U Al Cp*. This compound has SR = 2.96 and is considered to be formed from the union of (Si*Cp)3U with AlCp*, which are designated A and B respectively. In A the 1717
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The Journal of Physical Chemistry A
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Figure 1. Occupied σ and π MOs for U-Al in (Me3Si)3UAlCp*.
charge on U is 0.02 and in B the charge on Al is 0.21. But in AB the charges are 0.02 and 0.29 on U and Al, respectively. Note that these charges are very different from the charges 1.9 on U and 0.59 on Al derived from DFT calculations.16 Thus in forming AB the Al undergoes a small increase in charge from 0.21 to 0.29 while the U charges decreases from 0.097 to 0.02. The ionization energy of Alþ0.21 is interpolated to be 8.7 eV, and the electron affinity is 1.3 eV. The Uþ0.097 and interpolated ionization energy and electron affinity are 6.5 and 2.9 eV, respectively. Owing to the repulsive nature of the U-Al bond, its bond energy is anticipated to be substantially lower than that of the isolated U-Al bond presented earlier. Both EDA and Ei are negative, having values of -0.17 and -0.13 eV respectively, so the calculated BE = (118 kJ/mol) is only 38% of that for an isolated U-Al bond. This example an example regarding the nonreliability of DFT calculated atomic charges. If the latter charges, 1.9 on U and 0.59 on Al, were reliable, the electrostatic contribution to the bond energy would be -5.2 eV. When this is combined with the covalent contribution, 1.52 eV, the BE = -3.68 eV, or in other words there would be no bond. Furthermore, even according to chemical considerations, it is untenable to expect a much larger positive charge on U than on Al, when the electronegativity of U is significantly higher than that of Al on any electronegativity scale. 6. Magnetic Properties of (Si*Cp)3 U AlCp*. The magnetic moment reported for (Si*Cp)3UAlCp* is μeff = 3.0 μB, which is stated to be in agreement with a uranium oxidation state U(III), but the magnetic moments reported for U3þ as reported in all trihalides27 are in the range of 3.65 ( 0.07 BM. Hence, it is of interest to examine this reported lower magnetic moment in greater detail. The required g factors are evaluated in terms of the following relation JðJ þ 1Þ þ SðS þ 1Þ - LðL þ 1Þ ð13Þ gJ ¼ 1 þ ð1:002 32Þ 2JðJ þ 1Þ where the factor 1.002 32 is actually ge = 2.002 32 (for the free electron) - 1. From the MO description of the U-Al bond in (Si*Cp)3UAlCp* according to the QR-SCMEH-MO method28 given in
Table 9. Angular Momentum Coupling and geff Factors for U-Ala AO coefficients within MOs orbital configuration 2
U(5f ) U(5f1) U(6d1) U(7s1) Al(3p1)
S
L
J
Jeff
(1 (1/2 (1/2 (1/2 (1/2
3 3 2 0 1
4, 2 7/2, 5/2 5/2, 3/2 3/2, 1/2
2.83 2.96 1.94 0.867
7s
5fπ
6dπ
3pπ
0.99 0.01 0.58 0.26 0.16 0.26 0.02 0.10
geff 1.039 1.020 1.040 1.110 1.114
From Figure 1. MO(2): net geff = 0.99(1.039) þ 0.01(1.114) = 1.040. MO(3): net geff = 0.58(1.020) þ 0.26(1.040) þ 0.10(1.14) þ 0.02(1.110) = 0.994. a
Figure 1, the three electrons in the highest occupied open shells have predominantly U 5f character (75%) with some 6d (13%) and some Al 3p (12%). So eq 13 is evaluated in terms of an average J, L, and S values, which are in terms of the weighted (via MO coefficients) averages of the 5f2(3p1) and the (5f16d1)(3p1) configurations. Hence, eq 13 provides a geff appropriate for the MO description of the unpaired electrons. All essential data are in Table 9. The molar magnetic susceptibility for a free atomic system is given by28 χM ¼
2 Nβ2 geff Jeff ðJeff þ 1Þ 2Nβ2 ðgeff - 1Þðgeff - 2Þ þ ð14Þ 3kT 3λeff
in which N is Avogadro’s number, β is the bohr magneton (BM), k is Boltzmann’s constant, Jeff is the geometric average of allowed J values, T ≈ 300 K, and λeff is the effective total spin-orbit coupling constant. The first term is the spin-free temperature dependent contribution, while the second term is temperature independent and usually several orders of magnitude smaller than the first. Using 14 with the data from Table 9, the susceptibilities for electrons in MO(2) and MO(3) (see Figure 1) are χ(2) = 4.88 10-3 and χ(3) = 3.92 10-3 emu and having magnetic moments μeff(2) = 3.42 and μeff(3) = 1718
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The Journal of Physical Chemistry A 3.33 μB, respectively. Thus, the average moment is μeff = 3.38 μB, which is significantly higher than the reported experimental value15 of 3.01 μB. Apparently there is some degree of spin exchange within the U-Al bond causing a reduction in the magnetic moment. As noted in Figure 1 the three unpaired electrons are confined to π type MOs, which are subject to a three-electron spin exchange between two nuclear centers. In this case, μeff is given by the relation29 3FðTÞ 1=2 μeff ¼ gðTÞ BM ð15Þ 2 in which both g(T) and F(T) are complex exponential functions relating to the spin-exchange coupling constant, Jex, plus other terms. Jex can be evaluated from the following relation: Jex = (Esp-Δ)/2, where Esp is the spin-exchange energy to form the doublet state in MO(2) upon pairing two of the three spins (Figure 1) and Δ is the splitting between the doublet and the open shell quartet state of MOs (2) and (3). The spin pairing energy was evaluated to be Esp = 0.88 eV.25 The weighted average energy of MOs (2) and (3) in the quartet configuration is 1.72 eV, while the energy of MO(3) in the doublet state is 2.70 eV. Hence, Δ = 0.98 eV and 2Jex = -0.10 eV, or Jex = -0.05 eV. Since Jex is negative it describes antiferromagnetic coupling between the spins. At the fixed temperature, g(T) in eq 15 can be replaced by geff and F2(T) = 2/3; thus, |μeff| = geff per electron. For MO(2) μeff = 2(1.039) BM = 2.078 μB and for MO(3) μeff = 0.994 μB, thus yielding a net effective magnetic moment, μeff = 3.07 μB, which is very close to the reported experimental value, μeff = 3.01 μB.15
’ CONCLUSION It has been shown that an application of a revised format of Sanderson’s procedure for calculating bond dissociation energies of a select number of metal cyclopentadieneyl compounds has been just as successful as was Sanderson’s original approach based on a somewhat simpler model, but none the less highly successful as applied to numerous types of inorganic and organic compounds. In this revised model the basis of the definition of electronegativity (SR) remains the same; i.e., SR is the ratio of electron densities in volume units; however, charge transfer and overlap effects have been added as factors affecting the bond energy. Ever since Pauling first proposed the electroneutrality principle in 1948, it has been subjected to much scrutiny. In fact, there are those today who even contend that this principle is invalid. Yet it is obvious that electroneutrality works exceedingly well when applied to the calculation of bond energies, as shown previously by Sanderson and demonstrated in this work. The very concept of what atomic charges actually are has always been an issue of debate for a very long time and still continues unto this day. Various experimental data (dipole and nuclear quadrupole moments, M€ossbauer spectroscopy, etc.) infer atomic charges of one magnitude or another, but even these often do not agree among themselves. MO calculations frequently yield atomic charges very different from those derived experimentally, and these MO results are highly dependent upon the specific computational method. Electronegativity considerations provide yet another set of atomic charges. The point is that no one actually knows what actually constitutes an “accurate set” of atomic charges. However,
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Sanderson has justified his method of atomic charge determination with the following words: “They (atomic charges) have been proven INDISPENSABLE in the CALCULATION OF ENERGIES of 15 000 bonds between more than 200 different pairs involving 38 major group elements in more than ONE THOUSAND molecular and nonmolecular compounds, both INORGANIC and ORGANIC. In 191 examples, partial charges calculated from ELECTRONEGATIVTES differ from those calculated INDEPENDENTLY FROM EXPERIMENTAL DATA ONLY by an average of only 0.007 charge unit. Bond energies based on these partial charges have been used successfully to calculate REORGANIZATIONAL ENERGIES of 27 organic and 154 inorganic radicals, and then 268 organic and 160 inorganic BOND DISSOCIATION ENERGIES.”30 Obviously, the Sanderson concept appears to be a most consistently reliable approach to electronegativity, atomic charges, and bond energy calculations. However, a novice may experience difficulty in correctly judging the application of available data and creating appropriate extrapolation schemes essential to the required applications It has also been shown that the complex (Me3Si Cp)3UAlCp*has magnetic properties which reflect antiferromagnetic spinexchange interaction between three electrons associated with the U-Al bonding, thus accounting for the observed reduced magnetic moment for a U(III) oxidation state.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. Present Addresses †
910 East 121st Court, Thornton, CO 80241.
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