J. Phys. Chem. A 1997, 101, 9391-9398
9391
Density Functional Theory Predictions of Second-Order Hyperpolarizabilities of
Metallocenes
Nobuyuki Matsuzawa* and Jun’etsu Seto
SONY Corporation Research Center, 174 Fujitsuka-cho, Hodogaya-ku, Yokohama 240, Japan
David A. Dixon*,†
DuPont Central Research and DeVelopment, Experimental Station, P.O. Box 80328,
Wilmington, Delaware 19880-0328‡
ReceiVed: August 23, 1995X
The geometries in the staggered and eclipsed conformations of the metallocenes, M(C5H5)2 with M ) Mn,
Fe, Co, Ni, and Ru, have been calculated at the local and nonlocal density functional theory (LDFT and
NLDFT) levels. The M-C distance is predicted to be too short at the LDFT level and too long at the NLDFT
level. The doublet low-spin states for M ) Mn and Co show distortions away from the idealized fivefold
symmetries. The low-spin state for M ) Mn is predicted to be lower in energy than the high-spin state in
contrast to the observed experimental results. The size of the splitting is strongly dependent on the
computational level. The values of R and γ were calculated for the various metallocenes. The highest value
of γ was found for M ) Co.
Introduction
There is significant interest in the development of nonlinear
optical (NLO) materials because of their potential applications
in electronic and optical devices. The NLO properties of organic
materials have been extensively studied because of the ease of
synthesis of such materials and the potential ease of processability.1 Many of these materials have extended π electron
systems, as this seems to be the easiest way to introduce large
nonlinearities into organic systems.1
Nonlinearities for molecular systems can be expressed in
terms of the hyperpolarizabilities defined in eq 1
µi ) µi° +
βijkFjFk/2 + ∑ γijkFjFkFl/6 + ...
∑j RijFj + ∑
j,k
j,k,l
(1)
where µi is the dipole moment of a molecule under an applied
field of F, µi° is the dipole moment without the applied field,
Rij is the polarizability, βijk is the hyperpolarizability, and γijkl
is the second-order hyperpolarizability. The subscripts i, j, k,
and l denote Cartesian axes.
Because of the success of inorganic nonlinear optical materials
such as KTiOPO4 and LiNbO3 for applications where the firstorder hyperpolarizability is important, interest in organic
nonlinear optical materials has focused more recently on the
second-order hyperpolarizability (γ) for actual device applications. We have been using computational methods to predict
the nonlinear optical behavior of a variety of molecules.2,3
Although one can use semiempirical molecular orbital methods4,5 for organic molecules in either finite field2,4 or sum of
states5 approaches to predict β and γ, only the sum of states
approach can be used for molecules containing transition metals,
as there are few semiempirical molecular orbital parameters for
transition metal systems outside of those implemented for
†
Current address: Pacific Northwest National Laboratory, Environmental
Molecular Sciences Laboratory, P.O. Box 999, MSK1-83, Richland, WA
99352.
‡ Contribution No. 7183.
X Abstract published in AdVance ACS Abstracts, January 1, 1996.
S1089-5639(95)02465-0 CCC: $14.00
spectroscopy predictions in computer programs such as ZINDO.6
The sum of states approach is probably limited to studies of β
as compared to γ because of slow convergence of the sum for
the latter. Thus, one must rely on other methods to predict γ
for molecules which contain transition metals. Although
traditional ab initio molecular orbital methods can be used to
provide benchmark studies for nonmetallic systems,7 molecular
orbital methods based on a single configuration often have
difficulties in treating the electronic structure of molecules
containing transition metals.8 Furthermore, ab initio molecular
orbital methods scale at least as N5, where N is the number of
molecular orbitals if correlation corrections are included and
thus become computationally intractable for large systems.
Density functional theory9 (DFT) has proven to be an extremely
useful tool for modeling transition metal systems, as its
computational effort shows much better scaling with increasing
molecular size (order N3) and molecules with transition metals
can be treated with a reasonable degree of accuracy.10 We have
previously reported DFT calculations on nonlinear optical
properties of a number of systems including metalloporphines.3
In order to further investigate the ability of DFT to predict the
NLO properties of molecules containing transition metals, we
describe below the results of DFT calculations on metallocenes.
This allows us to further test the impact of the metal on the
organic π electron system in a different type of geometry than
found in the metalloporphines which we have previously
studied.3e
There has been one previously reported calculation11 of the
value of γ for ferrocene at the CNDO level. Extended basis
sets were used in a coupled Hartree-Fock approach. The
calculated value (CHF-PT-EB-CNDO level) of 24.58 × 10-36
esu is significantly less than the experimental12 value of (96.2
( 10.8) × 10-36 esu.
Calculations
The DFT calculations described below were done with the
program system DMol13 both at the local (LDFT) and nonlocal
levels (NLDFT). The calculations have been described in detail
previously.3 The form for the exchange-correlation energy of
© 1997 American Chemical Society
9392 J. Phys. Chem. A, Vol. 101, No. 49, 1997
Matsuzawa et al.
the uniform electron gas for the LDFT calculations is that
derived by von Barth and Hedin (BH).13c The NLDFT
calculations were done with the gradient-corrected exchange
potential of Becke14 and the gradient-corrected correlation
potential of Lee, Yang, and Parr (BLYP).15 The calculations
were done with a double numerical basis set augmented by
polarization functions (DNP) and with a basis set (DNP+)
obtained by augmenting the DNP basis set with the field-induced
polarization (FIP) basis functions for C (spd) and H (p) given
by Guan et al.16 The spin-unrestricted open-shell formalism
was used for open-shell molecules and a spin-restricted formalism was used for closed-shell molecules. Geometries were
optimized by using analytic gradient methods with the DNP
basis set.13b The density was converged to 10-6 for the
geometry optimizations, and the FINE mesh was used.
The calculations of the nonlinear optical properties were done
at the LDFT and NLDFT levels, except for the doublet state of
manganocene, where we had significant convergence problems
at the local level. A finite field approach1 was used with the
XFINE grid and the DNP and DNP+ basis sets; the density
was converged to 10-8. In this approach, the response of the
ground state charge distribution to an external electric field at
zero frequency is investigated. A molecule in an applied electric
field will exhibit an induced dipole moment. This induced
moment can be expanded in a Taylor series in powers of the
electric field as shown in eq 1. The equations given by Sim et
al.17 for obtaining R, β, and γ from dipole moments calculated
in an applied electric field (field strength3 ) 0.005 au unless
otherwise noted) were used. The scalar values for R, β, and γ
calculated from their vector or tensor components are defined
as follows:
R)
∑i Rii/3
(2)
∑i βiµk/|µ|
(3)
β ) (3/5)
γ)
γiijj/5
∑
i,j
(4)
where |µ| and βi are given as
µ ) |µ|
βi )
∑j βijj
(5)
(6)
Results and Discussions
Calculations were done on the following metallocenes:
magnanocene, ferrocene, cobaltocene, nickelocene, and ruthenocene. The numbering system of the atoms and the
orientation of the molecule along the Cartesian axes are shown
in Figure 1. For the metallocenes, the HOMO and LUMO
consist predominantly of metal d orbitals (see discussion below),
in contrast to the case of the metalloporphines previously
studied,3e where the HOMO and LUMO are predominantly
composed of π orbitals in the organic fragment. The order of
the metal d orbitals in metallocenes in terms of energy is e2g
(dxy, dx2-y2) ∼ a1g (dz2) < e1g (dxz, dyz).18 The electron
configuration at the metal for ferrocene and ruthenocene is thus
(e2g)4(a1g)2 if we make the usual assumption that the metal is in
the M2+ charge state, so the ground state of ferrocene and
ruthenocene should be a closed-shell singlet. For cobaltocene
the configuration is (e2g)4(a1g)2(e2g)1, and the resulting ground
state should be a doublet. For nickelocene the electron
configuration is (e2g)4(a1g)2(e2g)2 and the ground state is a triplet.
Figure 1. Numbering system of the atoms and the orientation along
the Cartesian axes for the metallocene.
The description of the electronic structure of magnanocene,
which has 5 d electrons for Mn2+, is more complicated. A
simple orbital-filling model yields configurations which are
either (e2g)4(a1g)1 or (e2g)3(a1g)2, and the lowest energy doublet
state is thought to be 2E2g, which obviously will undergo a
Jahn-Teller distortion. However, it is possible that the ligand
field splitting is small enough so that all of the electrons are in
separate orbitals, yielding a sextet state, 6A1g. Both states were
considered for manganocene. Besides the complexities introduced by the metal orbital occupancies, there are two possible
conformers, staggered (D5d) and eclipsed (D5h), for the metallocenes, and both geometries were considered in the structural
part of this study.
Geometries, Energies, and Orbital Levels. The LDFT (BH/
DNP) optimized geometry parameters are shown in Table 1
together with experimental geometry parameters obtained from
gas-phase electron diffraction measurements.19 For the lowspin state of manganocene, we also include crystal structure
results for decamethylmanganocene, which is of low spin.20 The
calculation predicted structures of D5d or D5h symmetry, except
for the two doublet structures. The two doublet structures
exhibit Jahn-Teller distortions away from the idealized D5d or
D5h symmetry. The cyclopentadienyl (Cp) rings in cobaltocene
and the doublet low-spin state of manganocene are no longer
planar, yielding structures of C2h and C2V symmetry for the
staggered and eclipsed conformers, respectively. This result
for cobaltocene differs from the gas-phase electron diffraction
results measured at ∼120 °C, where it was suggested that any
distortion from D5d or D5h symmetry is small.19c However, the
electron diffraction measurements probably cannot distinguish
the difference in symmetry at this temperature, as the rings are
rapidly rotating about the approximate fivefold axis to give a
structure with nominal fivefold symmetry. This is consistent
with another experimental study of cobaltocene which reported
unusually large Co-C vibrational amplitudes in the gas-phase
structure consistent with the above discussion.21 Furthermore,
the crystal structure of decamethylmanganocene, which is lowspin, shows deviations from fivefold symmetry.20
The LDFT calculated geometry parameters for the eclipsed
and staggered conformers do not differ significantly. The LDFT
C-H bond lengths are shorter than the experimental values and
fall within the experimental error limits, except for ruthenocene,
manganocene (sextet), and nickelocene. For the former two
metallocenes, the calculated values are shorter by ∼0.03 Å than
the experimental values, and we note that the experimental
values are clearly too long. For nickelocene, the LDFT C-H
bond lengths are longer than the experimental values, and the
values approximately fall within the experimental error limits.
The NLDFT C-H bond lengths are always shorter than the
LDFT values by 0.005-0.007 Å.
Second-Order Hyperpolarizabilities of Metallocenes
J. Phys. Chem. A, Vol. 101, No. 49, 1997 9393
TABLE 1: BH/DNP and BLYP/DNP Calculated Geometry Parametersa
calc (eclipsed)
BH/DNP
calc (staggered)
BLYP/DNP
BH/DNP
BLYP/DNP
expt
b
r(C-H)
r(C-C)
d(C-Mn)
d(Cp-Cp)
τ(H-C-C-C)
1.094,f 1.094,g 1.095h
1.430,i 1.420,j 1.412k
2.043,l 2.082,m 2.148n
3.233,o 3.371,p 3.599q
-176.3,r -178.1,s 179.2t
Manganocene(Doublet)
1.089,f 1.089,g 1.089h
1.096,f 1.096,g 1.096h
1.446,i 1.433,j 1.424k
1.431,i 1.420,j 1.413k
2.117,l 2.157,m 2.233n
2.049,l 2.080,m 2.135n
3.415,o 3.540,p 3.777q
3.423u
-177.3,r 180.0,s 177.6t 175.3,r 177.6,s 180.0t
r(C-H)
r(C-C)
d(C-Mn)
d(Cp-Cp)
τ(H-C-C-C)
1.095
1.418
2.337
4.004
180.0
1.090
1.432
2.474
4.306
-179.0
r(C-H)
r(C-C)
d(C-Fe)
d(Cp-Cp)
τ(H-C-C-C)
1.094
1.428
2.013
3.211
177.7
1.088
1.439
2.097
3.406
178.9
r(C-H)
r(C-C)
d(C-Co)
d(Cp-Cp)
τ(H-C-C-C)
1.095,f 1.093,g 1.094h
1.430,i 1.411,j 1.441k
2.034,l 2.100,m 2.052n
3.278,o 3.414,p 3.326q
177.0,r 177.2,s 180.0t
Cobaltocened
1.089,f 1.088,g 1.088h
1.095,f 1.093,g 1.094h
1.443,i 1.423,j 1.453k
1.430,i 1.410,k 1.441k
2.127,l 2.184,m 2.149n
2.039,l 2.105,m 2.060n
3.493,o 3.607,p 3.545q
3.367u
178.3,r 178.6,s 180.0t
177.1,r 178.1,s 180.0t
r(C-H)
r(C-C)
d(C-Ni)
d(Cp-Cp)
τ(H-C-C-C)
r(C-H)
r(C-C)
d(C-Ru)
d(Cp-Cp)
τ(H-C-C-C)
1.094
1.421
2.143
3.539
178.3
1.094
1.428
2.185
3.631
179.8
Manganocene(Sextet)b
1.095
1.418
2.338
4.004
180.0
Ferrocenec
1.094
1.426
2.017
3.222
177.1
1.089,f 1.089,g 1.089h
1.444,i 1.433,j 1.424k
2.129,l 2.163,m 2.231n
3.610u
177.5,r 180.0,s -177.4t
1.418
2.112 (2.144) 2.130
1.090
1.432
2.474
4.307
-179.1
1.125 ( 0.010
1.429 ( 0.008
2.380 ( 0.006 (2.433)
1.088
1.439
2.101
3.405
178.7
1.104 ( 0.006
1.440 ( 0.002
2.064 ( 0.003
1.089,f 1.088,g 1.088h
1.443,i 1.423,j 1.453k
2.130,l 2.185,m 2.151n
3.564u
178.4,r 178.3,s 180.0t
1.095 ( 0.008
1.430 ( 0.0015
2.113 ( 0.0015
176.40 ( 1.66
1.089
1.434
2.238
3.752
180.0
Nickelocenee
1.094
1.422
2.139
3.528
178.0
1.089
1.434
2.239
3.754
180.0
1.083 ( 0.0095
1.430 ( 0.0015
2.196 ( 0.004
1.088
1.440
2.267
3.815
180.0
Ruthenocenec
1.094
1.428
2.185
3.631
179.3
1.088
1.440
2.268
3.818
180.0
1.130 ( 0.006
1.439 ( 0.002
2.196 ( 0.003
179.72 ( 1.45
a
Distances in Å, angles in degrees. d(Cp-Cp) is the distance between the two pentagons consisting of five carbon atoms. b Experimental values
for low-spin manganocene from ref 20 obtained from the X-ray crystal structure of decamethylmanganocene. Experimental values for high-spin
manganocene from ref 19a in the gas-phase. Value in parentheses is for gas-phase 1,1′-dimethylmanganocene from ref 19e, and the italicized value
is for gas-phase decamethylmanganocene from ref 19f. c Experimental values from ref 19b. d Experimental values from ref 19c. e Experimental
values from ref 19d. f r(C1-H1). g r(C2-H2) ) r(C5-H5). h r(C3-H3) ) r(C4-H4). i r(C1-C2) ) r(C1-C5). j r(C2-C3) ) r(C4-C5). k r(C3-C4).
l
r(M-C1). m r(M-C2) ) r(M-C5). n r(M-C3) ) r(M-C4). o r(C1-C1′). p r(C2-C2′) ) r(C5-C5′). q r(C3-C3′) ) r(C4-C4′). r τ(H1-C1-C5-C4).
s
τ(H5-C5-C4-C3). t τ(H4-C4-C3-C2). u Averaged value.
Reasonable agreement for the C-C bond lengths was found
for the manganocene doublet (averaged values of 1.423 and
1.422 Å for the staggered and eclipsed conformers, respectively,
at the LDFT level), cobaltocene (averaged values of 1.424 and
1.425 Å for the staggered and eclipsed conformers, respectively,
at the LDFT level), and nickelocene with differences between
theory and experiment less than 0.008 Å at the LDFT level.
For manganocene (sextet), ferrocene, and ruthenocene, the
agreement at the LDFT level is still reasonable with the
calculated values shorter than the experimental values by 0.0110.014 Å. If nonlocal corrections are included, the C-C bond
lengths are longer than the LDFT values, and the differences
between the theoretical and experimental values are now less
than 0.004 Å expect for the doublets. For the doublets, the
agreement between theory and experiment becomes worse when
nonlocal corrections are included, with the calculated values
longer by 0.007-0.018 Å as compared to experiment.
A larger disagreement was found for the metal-carbon
distances. Consistent with previous LDFT calculations, the
LDFT calculated values are shorter than the experimental values
by ∼0.01 Å,22 ∼0.04 Å, ∼0.05 Å, ∼0.04 Å,22 ∼0.04 Å, and
∼0.01 Å for manganocene (doublet), manganocene (sextet),
ferrocene, cobaltocene, nickelocene, and ruthenocene, respec-
tively. The τ(H-C-C-C) torsion angles calculated at the
LDFT and NLDFT levels are in agreement with the experimental values, although the experimental error is quite large
and all of the experimental values are not available. The LDFT
and NLDFT calculations predict the Cp hydrogen atoms to be
located closer to the metal atom as compared to the carbons in
agreement with experimental studies19,23 except for ruthenocene
and doublet manganocene at the NLDFT level. For doublet
manganocene, the Cp rings are not planar because of the JahnTeller distortion.
The inclusion of nonlocal corrections can lead to an improvement in the prediction of the M-C distance, as shown by Bérces
et al., who reported r(Fe-C) ) 2.048 Å for ferrocene.24 The
inclusion of nonlocal corrections does little to change the
geometries of the cyclopentadienyl rings but does significantly
affect the M-C bond distances. For ferrocene, the NLDFT
Fe-C distance (for the eclipsed conformer) is 0.033 Å too long
whereas, for ruthenocene, the NLDFT Ru-C distance is now
0.071 Å too long, but this error could be in part due to the
need for relativistic corrections for Ru which are not present in
the calculations. For cobaltocene the Co-C distance is too long
by 0.046 Å at the NLDFT level, and for nickelocene the Ni-C
distance is 0.042 Å too long. For the 6A1g state of manganocene,
9394 J. Phys. Chem. A, Vol. 101, No. 49, 1997
Matsuzawa et al.
TABLE 2: BH/DNP and BLYP/DNP Calculated Relative Energies of the Spin States and Conformers of the Metallocenes
(C5H5MC5H5) in kcal/mol
relative energy of the spin states of manganocene
eclipsed
doublet
sextet
staggered
BH
BLYP
BH
BLYP
0.00
36.84
0.00
12.13
0.00
35.96
0.00
11.77
relative energy of the conformers
M ) Mn
doublet
eclipseda
staggered
sextet
M ) Fe
M ) Co
M ) Ni
M ) Ru
BH
BLYP
BH
BLYP
BH
BLYP
BH
BLYP
BH
BLYP
BH
BLYP
0.00
0.84
0.00
0.34
0.00
-0.04
0.00
-0.01
0.00
1.05
0.00
0.53
0.00
0.49
0.00
0.18
0.00
0.13
0.00
0.01
0.00
0.42
0.00
0.21
a
BH/DNP calculated energies of the eclipsed conformer are -1532.488 957, -1532.430 254, -1645.151 877, -1764.068 100, -1889.482 777,
and -4822.899 381 au for manganocene(doublet), manganocene(sextet), ferrocene, cobaltocene, nickelocene, and ruthenocene, respectively. BLYP/
DNP calculated energies of the eclipsed conformer are -1538.109 014, -1538.089 690, -1650.875 510, -1769.925 763, -1895.475 796, and
-4830.922 119 au for manganocene(doublet), manganocene(sextet), ferrocene, cobaltocene, nickelocene, and ruthenocene, respectively.
TABLE 3: Relative Energies (kcal/mol) and Selected Geometry Parameters (Å) for Distorted and Undistorted Doublet Statesa
manganocene (doublet)
relative energy (kcal/mol)
r(C-Mn) (Å)
r(C-C) (Å)
eclipsed
BH
staggered
BH
eclipsed
BH
staggered
BH
eclipsed
BH
staggered
BH
0.00
0.71
0.00
0.44
2.101
2.072
2.096
2.075
1.425
1.422
1.423
1.425
distorted
undistorted
cobaltocene (doublet)
relative energy (kcal/mol)
eclipsed
distorted
undistorted
r(C-Co) (Å)
staggered
eclipsed
r(C-C) (Å)
staggered
eclipsed
staggered
BH
BLYP
BH
BLYP
BH
BLYP
BH
BLYP
BH
BLYP
BH
BLYP
0.00
1.18
0.00
0.97
0.00
1.11
0.00
0.98
2.068
2.071
2.159
2.163
2.074
2.075
2.160
2.166
1.425
1.424
1.437
1.436
1.424
1.423
1.437
1.436
a Distorted geometries obtained without smearing of the charge distribution. Undistorted geometries obtained by smearing of the charge distribution.
See text.
the Mn-C distance is 0.094 Å too long at the NLDFT level.
For the doublet state of manganocene, the Mn-C distance can
be compared to that reported in the gas phase19e for 1,1′dimethylmanganocene, 2.144(12) Å, or to that reported in the
crystal20 for decamethylmanganocene, 2.112 Å. The other
reported Mn-C distance is 2.130 Å from the gas-phase electron
diffraction measurement for decamethylmanganocene.19f The
average calculated value of 2.179 Å for the eclipsed conformer
is 0.035 or 0.049 Å longer than the gas-phase values19e,f and
0.067 Å longer than the crystal structure value.20 The agreement
with experiment in either case is better than found for the sextet
state.
The energies of the staggered conformer relative to those of
the eclipsed conformer are shown in Table 2. The calculations
predict that the eclipsed conformer is more stable than the
staggered conformer with the difference in energies being quite
small, < ∼1 kcal/mol. The only exception is for the sextet state
of manganocene, where the staggered structure is of essentially
the same energy as the eclipsed one. This is not surprising, as
the M-C distance is almost 0.3 Å longer than in the other
metallocenes with first-row transition metals and the steric
interactions between the rings should be smaller. In all cases
the difference in energies is smaller at the NLDFT level as
compared to the LDFT level consistent with the longer M-C
distances predicted at the NLDFT level which separate the rings
by a larger amount leading to reduced steric effects. The
experimental result for ferrocene showed that the equilibrium
conformation is eclipsed,19b consistent with our result. It was
also reported that the barrier to the internal rotation of ferrocene
is 0.9 ( 0.3 kcal/mol,19b in excellent agreement with our
calculated values and with the nonlocal DFT value of 0.69 kcal/
mol previously reported.24 It has been suggested19b that the
equilibrium conformation of ruthenocene is eclipsed, consistent
with the DFT results. Hedberg et al. reported that their gasphase electron diffraction result for cobaltocene is consistent
with a free-rotation model for the Cp rings and suggested that
the rotational barrier height would be in the order ferrocene >
cobaltocene > nickelocene.19c The calculations are consistent
with this, and for the latter two, essentially free rotation is
predicted at the NLDFT level.
The electronic ground states of manganocenes are either high
spin or low spin depending on the substituent.25 The parent
manganocene has been shown to be high spin by a variety of
measurements. However, the calculations predict the low spin
state to be of lower energy. The energy difference between
the two states is predicted to be more than 30 kcal/mol at the
local level. This difference is improved by including nonlocal
corrections to give a value of 12.1 kcal/mol for the staggered
geometry.
If integer occupation numbers are used, the doublet states of
manganocene and cobaltocene exhibit a Jahn-Teller distortion.
We reoptimized these structures with the electrons smeared at
the Fermi level, i.e., noninteger occupation numbers, so that a
symmetric structure could be examined. The BH/DNP and
BLYP/DNP energies and selected geometry parameters are
shown in Table 3. The optimized geometries at the LDFT and
NLDFT levels for the doublets with the smeared charge
distribution do not exhibit Jahn-Teller distortions, having D5h
and D5d symmetry for the eclipsed and staggered conformers,
respectively, as expected. For cobaltocene, the HOMO consists
Second-Order Hyperpolarizabilities of Metallocenes
J. Phys. Chem. A, Vol. 101, No. 49, 1997 9395
The energy difference between the symmetric and asymmetric
structures gives the energy needed to reach the conical intersection, and it is very low, less than 0.75 kcal/mol for manganocene
and less than 1.25 kcal/mol for cobaltocene. These results are
consistent the observation in an electron diffraction experiment
of an averaged structure with D5d symmetry for cobaltocene.
The calculated average structural parameters with or without
smearing are essentially the same for cobaltocene, whereas, for
manganocene, the metal carbon lengths calculated with smearing
are shorter by 0.02-0.03 Å than those without smearing.
The molecular orbital energies at the BLYP/DNP level for
the metallocenes are shown in Figure 2. The orbitals are
essentially the same for the eclipsed and staggered conformers.
The DFT calculations show that the HOMO for the metallocenes
is localized on the metal. The order of the highest occupied d
orbitals is a1g (dz2) < e2g (dxy, dx2-y2) < e1g (dxz, dyz) except for
the manganocene doublet, although we note that the difference
in energy between the e2g and a1g levels is very small for
ruthenocene. For the manganocene doublet, this ordering holds
for the β-orbitals, whereas, for the R-orbitals, it is e2g (dxy, dx2-y2)
< a1g (dz2) < e1g(dxz, dyz). The order a1g < e2g < e1g has
previously been predicted for cobaltocene at the DFT level.26
In contrast, ab initio molecular orbital calculations suggest that
the highest occupied molecular orbitals reside on the Cp rings
and that Koopmanns’ theorem cannot be used.27 If one performs
SCF calculations on the ion, the orbital orderings then given
are properly consistent with the experimental observations for
ferrocene.28
The calculated charges and spin populations on the metal are
shown in Table 4. The charge distributions are somewhat
different between the local and nonlocal levels. The charges
Figure 2. BLYP/DNP calculated energy levels of the metallocenes.
Levels with marks of v, V, vV, and d correspond to d orbitals, with the
first three marks (v, V, and vV) showing the occupation of d-orbitals. E
) eclipsed. S ) staggered. D ) doublet. Sxt ) sextet.
of a degenerate e2g orbital occupied by one electron. If smearing
is not applied, one of the degenerate orbitals is singly occupied
whereas, if the charge distribution is smeared, fractional
occupation occurs with half of an electron in each of the two
e2g orbitals. A similar situation is found for manganocene with
the configuration (e2g)3(a1g)2. The e2g orbitals split apart with
one electron in one orbital and two in the other if smearing is
not used. If smearing is used, approximately 1.7 electrons are
placed in each of the a1g and e2g orbitals, giving a symmetric
structure. The energies of the symmetric (smeared) structures
are higher than those of the distorted structures (no smearing).
TABLE 4: BH/DNP and BLYP/DNP Calculated Charges and Spin Densities of the Metallocenesa
manganocene
doublet
sextet
eclipsed
staggered
eclipsed
staggered
orbital
BH
BLYP
BH
BLYP
BH
BLYP
BH
BLYP
3d
dz2
dxy, dx2-y2
dxz, dyz
4s
4p
total on M
5.97 (1.04)
1.78 (0.03)
2.57 (0.91)
1.62 (0.10)
0.49 (0.00)
0.55 (0.03)
-0.03 (1.07)
5.91 (1.23)
1.79 (0.05)
2.62 (0.99)
1.50 (0.19)
0.44 (0.01)
0.48 (0.04)
0.15 (1.28)
5.98 (1.03)
1.78 (0.03)
2.58 (0.90)
1.62 (0.10)
0.49 (0.00)
0.55 (0.03)
-0.04 (1.06)
5.92 (1.22)
1.80 (0.04)
2.63 (0.99)
1.49 (0.19)
0.45 (0.01)
0.48 (0.04)
0.14 (1.27)
5.57 (4.37)
1.01 (0.92)
1.96 (1.93)
2.60 (1.52)
0.40 (0.08)
0.54 (0.10)
0.48 (4.54)
5.46 (4.56)
1.01 (0.94)
1.99 (1.97)
2.45 (1.65)
0.32 (0.09)
0.46 (0.12)
0.75 (4.77)
5.57 (4.37)
1.01 (0.92)
1.97 (1.93)
2.60 (1.52)
0.40 (0.08)
0.54 (0.10)
0.48 (4.54)
5.46 (4.56)
1.01 (0.94)
1.99 (1.97)
2.46 (1.65)
0.32 (0.09)
0.46 (0.12)
0.75 (4.77)
ferrocene
cobaltocene
eclipsed
staggered
eclipsed
staggered
orbital
BH
BLYP
BH
BLYP
BH
BLYP
BH
BLYP
3d
d z2
dxy, dx2-y2
dxz, dyz
4s
4p
total on M
7.03
1.70
3.45
1.89
0.52
0.61
-0.15
7.00
1.88
3.37
1.75
0.47
0.53
-0.01
7.04
1.71
3.47
1.87
0.52
0.60
-0.17
7.01
1.82
3.43
1.74
0.47
0.53
-0.02
7.94 (0.67)
1.81 (0.01)
3.62 (0.04)
2.50 (0.61)
0.51 (0.00)
0.60 (0.01)
-0.07 (0.67)
7.92 (0.80)
1.87 (0.03)
3.65 (0.05)
2.40 (0.72)
0.45 (-0.01)
0.53 (0.01)
0.09 (0.79)
7.95 (0.68)
1.80 (0.03)
3.65 (0.03)
2.50 (0.62)
0.51 (-0.01)
0.60 (0.01)
-0.07 (0.67)
7.93 (0.80)
1.90 (0.02)
3.63 (0.07)
2.40 (0.72)
0.45 (-0.01)
0.53 (0.01)
0.08 (0.79)
nickelocene
ruthenocene
eclipsed
staggered
eclipsed
staggered
orbitalb
BH
BLYP
BH
BLYP
BH
BLYP
BH
BLYP
3d
dz2
dxy, dx2-y2
dxz, dyz
4s
4p
total on M
8.80 (1.02)
1.90 (0.02)
3.81 (0.03)
3.10 (0.97)
0.49 (-0.02)
0.60 (0.00)
0.12 (1.00)
8.80 (1.08)
1.94 (0.02)
3.82 (0.03)
3.04 (0.98)
0.43 (-0.03)
0.53 (-0.01)
0.24 (1.04)
8.80 (1.02)
1.90 (0.02)
3.81 (0.03)
3.11 (0.97)
0.50 (-0.01)
0.60 (0.00)
0.09 (1.00)
8.80 (1.09)
1.91 (0.02)
3.85 (0.04)
3.04 (1.03)
0.43 (-0.03)
0.53 (-0.01)
0.24 (1.04)
6.98
1.70
3.44
1.86
0.47
0.48
0.09
7.06
1.84
3.42
1.80
0.46
0.44
0.07
7.00
1.70
3.44
1.86
0.47
0.48
0.08
7.07
1.84
3.43
1.80
0.46
0.44
0.06
a
Charges in units of electrons. Numbers in parentheses are spin densities. b For ruthenocene, 3d, 4s, and 4p should be read as 4d, 5s, and 5p,
respectively.
9396 J. Phys. Chem. A, Vol. 101, No. 49, 1997
Matsuzawa et al.
TABLE 5: LDFT and NLDFT Calculated Scalar Values of Polarizability and First- and Second-Order Hyperpolarizability of
the Eclipsed Conformer of the Metallocenesa
β
R
BH/
DNP
0.005b
BH/DNP+
0.0025b
0.005b
2.002
2.042
1.894
1.999
c,d
manganocene(D)
manganocene(S)c,d
ferrocenee
cobaltocene(C2V)d,f
cobaltocene(D5h)d,g
nickelocened
ruthenocenee
1.871
1.946
1.988
2.028
2.039
2.038
2.052
BLYP/
DNP+
0.005b
2.065
2.108
1.992
2.087
2.093
2.149
2.158
BH/
DNP
0.005b
γ
BH/DNP+
0.0025b
0.005b
BLYP/
DNP+
0.005b
BH/
DNP
0.005b
BH/DNP+
0.0025b
0.005b
96.80
52.99
26.82
130.76
0.871
0.129
0.318
0.519
0.316
19.57
38.52
24.50
18.61
52.83
61.39
26.54
BLYP/
DNP+
0.005b
58.12
44.58
34.01
95.77
98.08
49.07
37.67
a
Units are ×10-23 cm3 for R, ×10-30 esu for β, and ×10-36 esu for γ. BH/DNP and BH/DNP+ values calculated at the BH/DNP optimized
geometry. BLYP/DNP+ values calculated at the BLYP/DNP+ optimized geometry. b Applied field strength in au. c D ) doublet. S ) sextet.
d Calculated at the spin-unrestricted level. e Calculated at the spin-restricted level. f Calculated at the Jahn-Teller distorted geometry without smearing.
g
Calculated at the D5h geometry with smearing.
on the metal at the nonlocal level are generally more positive
(less negative) except for ruthenocene. This is consistent with
less backbonding from the ligands due to the longer M-C
distances at the NLDFT level. The difference between the two
computational levels is about 0.15 e except for the high-spin
state of manganocene, where a larger difference of 0.27 e is
found. The charge distribution for Fe in ferrocene at the
nonlocal level shows that there are 7 electrons in the d orbitals,
0.5 electrons in the 4s orbitals, and 0.5 electrons in the 4p
orbitals. Overall, the charge on the Fe is essentially neutral. A
similar charge distribution is found for ruthenocene except that
there is less charge in the valence 5p orbital, about 0.1 e, so
that the Ru is now slightly positive with charges of 0.06 e and
0.07 e for the two conformers at the NLDFT level. For
cobaltocene, the additional electron goes predominantly into the
dxz and dyz orbitals with the dz2 orbitals gaining -0.01 e
(eclipsed) and 0.08 e (staggered) and the dxy and dx2-y2 orbitals
gaining 0.28 e (eclipsed) and 0.20 e (staggered) at the NLDFT
level. The remainder of the charge distribution is like that in
ferrocene, although the Co is somewhat less negative than the
Fe with charges of 0.09 e and 0.08 e on the Co for the two
conformers. As would be expected, most of the spin is on the
metal, 0.80 e, with this spin residing in the dxz and dyz orbitals
(0.72 e). According to the spin densities on Ni, nickelocene is
a triplet with one spin on the metal and one spin localized on
the rings. The changes in the electron distribution found in
substituting Co for Fe are the same as those for substituting Ni
for Co. For nickelocene, the Ni is actually positive with a
charge of 0.24 e. Again the spin is localized in the dxz and dyz
orbitals.
For sextet manganocene, there are an additional 0.5 electrons
in the dxz and dyz orbitals over that predicted from a simple
orbital-filling model. There are also 0.32 e in the valence 4s
orbital and 0.46 e in the 4p orbitals, consistent with the results
for the other metallocenes, although the value in the 4s orbital
is somewhat smaller than those for the other compounds. There
is a significant difference in the charge on the metal of almost
0.3 e at the local and nonlocal levels. Most of the spin resides
on the metal with 0.46 e of spin on the ligand at the local level
and 0.23 e on the ligand at the nonlocal level. The doublet
state of manganocene is predicted to have a slightly positive
metal at the NLDFT level in contrast to the much higher positive
charge predicted for the sextet state. The orbital populations
suggest that the dz2 orbital is mostly occupied, but there are
only 1.5 electrons in the dxz and dyz orbitals and 2.6 e in the dxy
and dx2-y2 orbitals. All of the positive spin is on the metal in
cobaltocene with a negative spin density on the ligands (-0.3
e). The spin is predicted to be mostly localized in the dxy and
dx2-y2 in-plane orbitals.
Polarizabilities and First- and Second-Order Hyperpolarizabilities. The scalar values for the polarizabilities (R), firstorder hyperpolarizabilities (β), and second-order hyperpolarizabilities (γ) at the BH/DNP, BH/DNP+, and BLYP/DNP+
levels for the metallocenes are shown in Table 5, and the tensor
components are in Table 6. We calculated the values for the
eclipsed conformer (D5h or C2V), as this conformer is usually
more stable than the staggered conformer (D5d or C2h). Values
of β are only given for doublet manganocene and cobaltocene,
where a Jahn-Teller distortion is found, and for the others, the
value of β is zero by symmetry.
For R, the predicted values do not strongly depend on the
computational parameters, such as the applied finite field
strength and basis sets, and are in the order ferrocene <
manganocene(doublet) < cobaltocene < manganocene(sextet)
< nickelocene < ruthenocene at the NLDFT level. The BH/
DNP+ values are larger by 1-3% as compared to the BH/DNP
values, as found in our previous study on substituted benzenes.3b
The BH/DNP+ values at 0.0025 au are slightly larger than the
BH/DNP+ values at 0.005 au, although the difference is less
than 1%. These results clearly show that these values are well
converged. The inclusion of nonlocal corrections increases the
value of R by about 3-5%, comparable to the improvement
due to adding diffuse functions to the basis set.3 The calculated
value of R for ferrocene of 1.99 × 10-23 cm3 is in excellent
agreement with the experimental value29 of 1.90 × 10-23 cm3.
At the CHF-PT-EB-CNDO level,11 the value of R is calculated
to be slightly too high at 2.16 × 10-23 cm3.
The values of β for doublet manganocene and cobaltocene
are quite small. This is consistent with the small dipole
moments and with the very low energies needed to reach the
symmetric structure from the distorted structure. Thus, the
doublets do not strongly deviate from the ideal D5d symmetry
and the value of β would be expected to be near zero. The
various components of the β tensor are consistent with this result
and are small. The scalar value of β for manganocene is larger
than that for cobaltocene, consistent with the larger distortion
from D5h symmetry predicted for the former as compared to
the latter, as discussed above. For example, the difference in
the distances between the carbons in the two Cp rings can be
as large as 0.366 Å for manganocene, whereas, for cobaltocene,
it is only as large as 0.136 Å. The difference in the degree of
the distortion could account for the difference in the calculated
β values for these two species.
The BH/DNP+ value for γ at 0.005 au is larger by about
40% for ferrocene and ruthenocene as compared to the BH/
DNP value at 0.005 au, consistent with our previous calculations
on benzene derivatives.3b A significantly larger enhancement
due to the addition of diffuse functions is predicted for
Second-Order Hyperpolarizabilities of Metallocenes
J. Phys. Chem. A, Vol. 101, No. 49, 1997 9397
TABLE 6: LDFT and NLDFT Calculated Tensor Components Polarizability and Second-Order Hyperpolarizability of the
Eclipsed Conformer of the Metallocenesa
BH/DNP+
BLYP/DNP+
BH/DNP+
BLYP/DNP+
BH/DNP+
BLYP/DNP+
b,d
µx
Rzz
βxzz
γyyyy
γxxzz
-0.34405
2.4976
-0.469
42.19
25.66
Manganocene(Doublet)
Rxx
1.8578
βxxx
-0.693
βx
-1.452
γzzzz
69.16
γyyzz
26.96
1.8387
-0.289
49.88
12.08
Ryy
βxyy
τxxxx
γxxyy
Rxx ) Ryy
γxxxx ) γyyyy
γxxzz ) γyyzz
1.9387
24.93
27.10
1.9933
26.03
20.93
Rzz
γzzzz
Manganocene(Sextet)b
2.2477
90.71
2.3387
69.27
γxxyy
7.99
8.93
Rxx ) Ryy
γxxxx ) γyyyy
γxxzz ) γyyzz
1.7324
24.16
10.84
1.7778
27.51
14.64
Rzz
γzzzz
Ferrocenec
2.2164
27.27
2.4193
39.67
γxxyy
7.57
8.41
-0.02637
2.3717
-0.412
106.75
25.57
-0.01697
2.5666
-0.337
80.72
24.31
Rxx
βxxx
βx
γzzzz
γyyzz
Cobaltocene (C2V)b,d
1.7708
0.398
-0.865
166.10
134.22
1.8167
0.388
-0.527
107.45
92.05
Ryy
βxyy
γxxxx
γxxyy
1.8557
-0.851
22.42
19.47
1.8764
-0.579
26.76
15.59
Rxx ) Ryy
γxxxx ) γyyyy
γxxzz ) γyyzz
1.8390
31.60
34.23
1.8780
29.50
25.42
Rzz
γzzzz
Nickeloceneb
2.4370
87.20
2.6899
68.45
γxxyy
9.82
8.12
Rxx ) Ryy
γxxxx ) γyyyy
γxxzz ) γyyzz
1.7900
17.27
11.69
1.8368
24.36
15.09
Rzz
γzzzz
Ruthenocenec
2.5756
40.26
2.7991
56.63
γxxyy
5.58
11.32
µx
Rzz
βxzz
γyyyy
γxxzz
a Units are ×10-18 esu, ×10-23 cm3, ×10-30 esu, and ×10-36 esu for µ, R, β, and γ, respectively. Only non-zero components are shown.
Applied finite field strength ) 0.005 au. BH/DNP and BH/DNP+ values calculated at the BH/DNP optimized geometry. BLYP/DNP+ values
calculated at the BLYP/DNP+ optimized geometry. b Calculated at the spin-unrestricted level. c Calculated at the spin-restricted level. d Calculated
at the Jahn-Teller distorted geometry with integral occupation numbers.
cobaltocene and nickelocene. The BH/DNP+ value is 3.4 and
2.5 times larger than the BH/DNP value for cobaltocene and
nickelocene, respectively. We also calculated the BH/DNP+
value at a field strength of 0.0025 au for these two compounds,
and this yielded values which are smaller by 14-26% than those
at 0.005 au, exhibiting a somewhat larger dependence of γ on
the applied field strength than usually observed.3b Inclusion of
nonlocal corrections leads to different effects. For the closedshell metallocenes, the value for ferrocene is raised by 27%
whereas that for ruthenocene is raised by 42%. For the openshell species, where a comparison can be made, the effect of
nonlocal corrections is to lower the value of γ by about 20%.
The best calculated value for ferrocene at the DFT level of
34.01 × 10-36 esu is about a factor of 3 less than the
experimental12 value of (96.2 ( 10.8) × 10-36 esu and is about
50% higher than the CHF-PT-EB-CNDO value11 of 24.58 ×
10-36 esu. Two possibilities exist for the discrepancy between
the DFT value and experiment. The basis set for the DFT
calculations may not be adequate enough, and the experimental
value may not have been extrapolated back to zero frequency.
The calculated γ values at the BLYP/DNP+ level are in the
order ferrocene ∼ ruthenocene < manganocene(sextet) <
nickelocene < manganocene(doublet) < cobaltocene. The
values of γ for all of the metallocenes fall in the range (3060) × 10-36 esu except that for cobaltocene, which is almost
100 × 10-36 esu. The doublet states have the highest values
for γ. We also calculated γ for doublet cobaltocene in D5h
symmetry at the BYLP/DNP optimized geometry. This geometry was obtained by smearing the electrons at the Fermi level,
and the calculation of γ was obtained with this technique. This
calculation yielded a value of 98.08 × 10-36 esu for γ,
essentially the same as the value of 95.77 × 10-36 esu obtained
for the distorted geometry where there are no fractional
occupancies. This shows that the enhanced values of γ for the
doublet species cannot be attributed to the distortion in geometry.
This is consistent with the small values of β as discussed above.
This suggests that the presence of an unpaired spin helps to
increase the value of γ. We note that the HOMO levels of the
doublets are predicted to have the highest values for the
metallocenes that we studied (Figure 2).
Except for the doublets, the ordering of the magnitude of γ
does not correlate with the ordering of the HOMO levels. For
example, the HOMO of ferrocene is higher than that for
ruthenocene, whereas the value of γ for ferrocene is smaller
than that for ruthenocene. Furthermore, the order of the
magnitude for γ does not correlate with the order of the
calculated charges on the metal (Table 4), which are related to
the degree of charge transfer between the metal and the Cp rings.
The quantity which does correlate with the magnitude of γ
except for the doublets is the electron density in the dxz and dyz
orbitals. The order of the charges is Ni (3.04) > Mn (sextet,
2.45) > Ru (1.80) > Fe (1.50), the same as the ordering of the
magnitudes of γ. If the unpaired spin is placed in the dxz or dyz
orbitals (e1g molecular orbital), which are nominally antibonding,
there is a larger increase in the magnitude of γ than found by
placing the unpaired spin in the in-plane dxy and dx2-y2 orbitals
or in the dz2 orbital. The HOMO of the (Cp)2 system should
be an e1g (or e1u) orbital, whereas the LUMO is an e2g (or e2u)
orbital. Thus the dx2-y2, dxy, and dz2 orbitals (e2g or a1g) will not
interact significantly with the HOMO of (Cp)2 (e1g or e1u), and
the charge distribution in the Cp rings in the ground state will
be less impacted by changes in the electron population in these
orbitals. The dxz and dyz orbitals (e1g) can interact with the
HOMO of (Cp)2, and changes in the electron population in these
orbitals can affect the charge distribution in the Cp rings, leading
to changes in the value of γ.
9398 J. Phys. Chem. A, Vol. 101, No. 49, 1997
Conclusions
The geometries in the staggered and eclipsed conformations
of the metallocenes, M(C5H5)2 with M ) Mn, Fe, Co, Ni, and
Ru, were calculated at the LDFT and NLDFT levels. The M-C
distance is predicted to be too short at the LDFT level and too
long at the NLDFT level. The doublet low-spin states for
manganocene and cobaltocene show distortions away from the
idealized fivefold symmetries. For manganocene, the doublet
low-spin state is predicted to be lower than the sextet highspin state in contrast to the observed experimental results. The
size of the splitting in energy of the doublet and sextet
manganocene is strongly dependent on the computational level,
with NLDFT values being smaller than LDFT values but still
too large as compared to experiment.
The values of R and γ were also calculated for the various
metallocenes. The calculated values of R are essentially
identical for the metallocenes, falling in the range (1.99-2.16)
× 10-23 cm3, whereas the values of γ differ significantly. The
order of the calculated γ values is ferrocene < ruthenocene <
manganocene(sextet) < nickelocene < manganocene(doublet)
< cobaltocene, with the highest values found for the doublet
species (95.77 × 10-36 esu and 58.12 × 10-36 esu for the
cobaltocene and manganocene doublets, respectively). Except
for the doublets, it is found that the magnitude of γ is affected
by the electron density in the dxz and dyz orbitals (e1g molecular
orbital), which can interact with the HOMO of the (Cp)2 system
(e1g or e1u molecular orbital).
References and Notes
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