Kinetics and mechanism of the reactions of ground-state Y (4d15s2, 2D) with ethylene and propylene: Experiment and theory

Article abstract of DOI:10.1021/jp010646t

The kinetic isotope effects, bimolecular rate constants, and primary products for the reactions of Y (4d¹5s², ²D) with C₂H₄ and C₂D₄ and with C₃H₆ and C₃D₆ were determined in a fast flow reactor at 300 K, with He/N₂ buffer gas at 0.8 torr. The stationary points on the doublet potential energy surface for the reaction Y + C₂H₄ → YC₂H₂ + H₂ were characterized using density functional theory in its B3LYP and mPW1PW91 forms with a large basis set. A new lowest-energy path involving the concerted elimination of H₂ was identified, consistent with the observation of YC₂H₂ products. Theory provided a set of geometries and vibrational frequencies for use in statistical rate models of the hot metallacyclopropane complex decay. The preferred model required no adjustments to the mPW1PW91 energies. This model was consistent with the collection of Y + C₂H₄ experimental data. The results indicated that nonadiabatic and/or steric effects contribute to the reaction inefficiency at room temperature.

Full text of DOI:10.1021/jp010646t

J. Phys. Chem. A 2001, 105, 6655-6667
6655
2
Kinetics and Mechanism of the Reactions of Ground-State Y (4d15s2, D) with Ethylene and
Propylene: Experiment and Theory
Meredith Porembski and James C. Weisshaar*
Department of Chemistry, UniVersity of WisconsinsMadison, 1101 UniVersity AVenue,
Madison, Wisconsin 53706-1396
ReceiVed: February 18, 2001; In Final Form: May 3, 2001
2
Bimolecular rate constants, primary products, and kinetic isotope effects for the reactions of Y (4d15s2, D)
with C2H4 and C2D4 and with C3H6 and C3D6 are measured in a fast flow reactor at 300 K with He/N2 buffer
gas at 0.8 Torr. The H2 and D2 elimination products and Y(alkene)-stabilized complexes are detected using
single photon ionization at 157 nm and time-of-flight mass spectrometry. We find a small normal isotope
effect (kH/kD ) 1.75 ( 0.12) for the reaction with ethylene but no significant isotope effect (kH/kD ) 1.06 (
0.07) for the reaction with propylene. We use density functional theory in its B3LYP and mPW1PW91 forms
with a large basis set to characterize stationary points on the doublet potential energy surface for the reaction
Y + C2H4 f YC2H2 + H2. Theory finds no energy barrier to the formation of a long-range Y-ethylene
complex. Subsequent steps involving CH bond insertion by metallacyclopropane complexes are consistent
with earlier work. However, a new, low-energy path involves concerted rearrangement of the HYC2H3 insertion
intermediate directly to a weakly bound, product-like complex with no exit channel barrier to elimination
products. Theory also provides a set of geometries and vibrational frequencies for use in statistical rate models
of the hot metallacyclopropane complex decay. The preferred model, consistent with the collection of Y +
ethylene experimental data, requires no adjustments to the mPW1PW91 energies. As in earlier work, B3LYP
places key transition state energies too high by 6-9 kcal/mol. The available evidence suggests that nonadiabatic
and/or steric effects contribute to the reaction inefficiency at room temperature.
I. Introduction
existence of a barrier to the approach of Y with ethylene along
the adiabatic entrance channel, explicitly confirmed the mech-
anism of primary CH insertion or determined the mechanism
of H2 elimination from the insertion intermediate HYC2H3.
In the past decade, considerable experimental and theoretical
effort has been devoted to the study of the reactivity of ground-
state transition metal atoms Y, Zr, and Nb with alkenes.1-17
Such work has the potential to improve our understanding of
the polymerization of olefins by transition metal compounds
via the Ziegler-Natta process.18-20 We and others have used
fast flow reactors equipped with laser-induced fluorescence
(LIF) detection of metal atoms to determine room-temperature
chemical reaction kinetics.1-4 Identification of reaction products
is now possible using photoionization mass spectrometry (PIMS)
at 157 nm.5,6 The chemical reaction dynamics of Y, Zr, and Nb
with ethane, ethylene, or acetylene have been studied in a
crossed-beam apparatus equipped with PIMS detection at higher
collision energies.7-9
More recently, we have performed extensive density func-
tional theory (DFT) calculations for the Zr (4d25s2, 3F) +
ethylene reaction, following the evolution of ground-state
reactants to H2 elimination products on both the triplet and
singlet potential energy surfaces (PESs).10 Statistical rate
modeling suggests that the hybrid density functional B3LYP21
places key transition state energies too high by 6-9 kcal/mol,
while the mPW1PW9122,23 functional gives much more realistic
energies. This is consistent with the performance of B3LYP in
similar studies involving transition metal cations and small
alkanes.24-27
The bulk of the theoretical work on the interaction of second-
row transition metals with small alkenes must be credited to
Siegbahn, Blomberg, and others.11-17 For the Y + ethylene
reaction, the studies identified those stationary points expected
to be involved in primary CH bond insertion, including the
strongly bound metallacyclopropane complex YC2H4, a CH
insertion transition state, and an insertion intermediate
HYC2H3.11,12,15 On the basis of the calculated exothermicity of
the YC2H2 + H2 products (-11.5 kcal/mol) and relatively small
CH insertion barrier (+1.9 kcal/mol), it was suggested that
ground-state Y should effect bimolecular H2 elimination from
ethylene at room temperature;3 however, detailed mechanistic
questions remain. To date, no study has investigated the possible
Toward a comprehensive and consistent mechanistic picture
of TM + alkene systems, we extend the joint experimental and
2
theoretical approach to the reactions of Y (4d15s2, D) with
ethylene and propylene. For the first time, we report the primary
products and kinetic isotope effects at 300 K for the reactions
of Y with C2H4, C2D4, C3H6, and C3D6. Bimolecular H2
elimination products are observed for all four reactions, which
corroborates a significant theoretical prediction.3 Stabilized
Y(alkene) complexes are also detected for the reactions with
C2D4, C3H6, and C3D6. A small positive isotope effect is
measured for Y + ethylene, while undeuterated and deuterated
propylene react with Y at essentially the same rate.
This paper also presents a detailed theoretical study of Y
2
(4d15s2, D) + ethylene using the B3LYP and mPW1PW91
* To whom correspondence should be addressed. E-mail: weisshaar@
chem.wisc.edu.
functionals with a large basis set, including complete charac-
10.1021/jp010646t CCC: $20.00 © 2001 American Chemical Society
Published on Web 06/19/2001

6656 J. Phys. Chem. A, Vol. 105, No. 27, 2001
Porembski and Weisshaar
terization of all intermediates and transition state structures along
the doublet PES. Theory identifies the entrance channel transi-
2
tion state TSent, although both DFT methods find it to lie
essentially isoenergetic to ground-state reactants. We locate the
metallacyclopropane complex YC2H4 and the CH insertion
transition state 2TSins, calculating rotational constants and
vibrational frequencies for use in the statistical rate models.
Similar to our recent investigation of the triplet Zr + ethylene
PES,10 we find that the doublet Y + ethylene PES branches
from the insertion intermediate HYC2H3 into two low-lying
paths. The higher energy, stepwise rearrangement path passes
through a dihydrido intermediate and over a substantial exit
2
barrier TSexit lying 23 kcal/mol above products. The lower
energy path directly connects HYC2H3 to a product-like complex
2
H2sYC2H2 via a multicenter transition state MCTS. In this
concerted rearrangement mechanism, the product-like complex
dissociates to YC2H2 (2A1) + H2 products with no barrier.
Finally, we use RRKM28,29 statistical rate modeling based
on the electronic structure calculations to determine the adjusted
energies of key stationary points involved in primary CH bond
insertion that bring the model results in accord with all of the
experimental data. Our preferred model successfully predicts
the observed Y + ethylene isotope effect and branching ratios,
requiring essentially no adjustments to the mPW1PW91 ener-
2
gies. As before,10 the B3LYP value of TSins is found to be at
least 6-9 kcal/mol too high. This suggests that nonadiabatic
or steric effects play an important role in controlling the overall
reaction efficiency of ground-state Y and Zr with small alkenes.
II. Experimental Section
Figure 1. Time-of-flight mass spectra for Y without ethylene flow
and with C2H4 and C2D4 flow as indicated.
A. Methods. An earlier publication describes our flow tube
apparatus and the PIMS technique in detail.5 Briefly, we use
laser ablation to generate gas-phase transition metal atoms in a
fast flow of predominately He and some N2 (added to quench
He* and Y* metastable states). The flow tube temperature is
300 K, and the pressure may vary from 0.5 to 1.1 Torr, with
N2 partial pressure constant at 120 mTorr. Frequent collisions
with the buffer gas thermalize metal atoms as they travel down
the flow tube. In the reaction zone, hydrocarbon flow is
regulated by a flow controller and monitored by a mass flow
meter (Tylan). Hydrocarbon gases C2H4 (Matheson 99.99%),
C2D4 (Cambridge Isotopes 98%), C3H6 (Matheson 99.6%), and
C3D6 (Cambridge Isotopes 98%) were used directly from the
bottle. Because flow meter response is not linear with gas flow,
we calibrate the flow meters with each reactant gas, measuring
pressure vs time while flowing gas into a calibrated volume.
Calibration curves for deuterated and undeuterated gases are
noticeably different. A skimmer (floated at +8 V) terminates
the reaction and permits neutral species to proceed to the
detection region where a 157 nm, 7.9 eV laser (Lambda Physik,
LPX 210i) ionizes unreacted metal and metal products. The ions
are collected via time-of-flight mass spectrometry (TOF-MS).
By measuring integrated ion currents as a function of
hydrocarbon flow at a fixed mean reaction time, we extract
effective bimolecular rate constants. A variety of complications
such as ion-molecule reactions, excited-state reactions, mul-
tiphoton effects, and photoionization followed by fragmentation
of reaction products and have been considered and were found
to be minimized under our experimental conditions.5 The good
agreement with rate constants derived from state-specific laser-
induced fluorescence (LIF) data2 confirms that we are probing
ground-state reactions with the PIMS technique.
of either C2H4 or C2D4. At zero hydrocarbon flow we observe
Y+ (89 amu, 100% natural abundance) and YO+; the latter is
likely from YO ablated from the oxidized metal surface. As
alkene flow increases, both Y+ and YO+ signals decrease while
the metal-containing hydrocarbon products labeled in Figure 1
emerge. We find that the primary products of the reactions of
Y with C2H4 and C2D4 are:
Y + C2H4 >90%8 YC2H2 + H2
Y + C2D4 67%8 YC2D2 + D2
33%8YC2D4
(1)
(2)
Multiphoton effects are negligible as we observe that neither
the products present in the mass spectra nor their relative
intensities change with laser fluence over at least a factor of
10.
In the absence of experimental complications (see below),
the PIMS ion intensities reflect the partitioning of internally
hot collision complexes between elimination and collisional
stabilization product channels. For the Y + C2H4 reaction, we
estimate that collisionally stabilized YC2H4 complexes are less
than 10% of the primary products over a flow tube pressure
range of 0.5 to 1.1 Torr. For the Y + C2D4 reaction, a fraction
of the collision complexes are stabilized and subsequently
ionized to YC2D4+. Branching ratios, defined as the ratio of
stabilized complexes to elimination products, are listed in Table
1. Our ability to measure the product peak integrals is limited
by experimental signal-to-noise and mass-resolution. The preci-
sion of the resulting branching ratios is reported as (1σ.
B. Results. Figure 1 shows typical photoionization mass
spectra taken with zero hydrocarbon flow and with small flows

2
Reactions of Ground-State Y (4d15s2, D)
J. Phys. Chem. A, Vol. 105, No. 27, 2001 6657
TABLE 1: Primary Reaction Products, Branching Ratios,
Effective Bimolecular Rate Constants, and Resulting Isotope
2
Effects for the Reaction of Y(4d15s2, D) with Alkenes at 0.8
( 0.05 Torr He/N2 and 300 ( 5 K
primary
alkene reaction product
branching
ratiob
c
k (10-12 cm3 s-1
)
kH/kD
C2H4
C2D4
>90% YC2H2 0.10 ( 0.03
67% YC2D2 0.33 ( 0.09
33% YC2D4
7.4 ( 0.3
4.2 ( 0.12
1.75 ( 0.12
C3H6
8% YC3H2 0.79 ( 0.15
122 ( 7
116 ( 2
48% YC3H4
44% YC3H6
10% YC3D2 1.38 ( 0.21
32% YC3D4
C3D6
1.06 ( 0.07
58% YC3D6
a Inferred from photoionization mass spectra. b The ratio of stabilized
Y(alkene) complexes to elimination products, assuming comparable
reactivities and photoionization cross-sections of the primary reaction
products (see text for details). The uncertainties refer to the precision
of experiments; ( 1σ of the mean of five values in the case of Y +
ethylene, and 10 values in the case of Y + propylene. c Uncertainties
refer to the precision of experiments; ( 1σ of the mean of three to
five values. Absolute accuracy of rates is ( 30%. Rate constants
measured at 0.8 Torr He/N2 and 300 K with LIF detection are 8.2 (
0.8 × 10-12 cm3 s-1and 141 ( 14 × 10-12 cm3 s-1 for the Y(2D3/2
)
reaction with C2H4 and C3H6, respectively (ref 2).
Although we do not detect the primary products of the YO +
ethylene reactions, possibly the result of a higher ionization
potential and/or smaller photoionization cross-section, products
from the reaction with a second and third ethylene molecule
are observed (not shown).
Similarly, as shown in Figure 2, we find the primary products
of the reactions of Y with C3H6 and C3D6 to be:
Figure 2. Time-of-flight mass spectra for Y without propylene flow
and with C3H6 and C3D6 flow as indicated.
Y + C3H6 8%8 YC3H2 + 2H2
48%8 YC3H4 + H2
(3)
44%8 YC3H6
Y + C3D6 10%8 YC3D2 + 2D2
32%8 YC3D4 + D2
(4)
58%8 YC3D6
Branching ratios are included in Table 1. Again, we find that
neither the products nor their relative intensities change with
laser fluence. While double dehydrogenation of alkenes by
neutral transition metal atoms has not been previously reported,
double H2 elimination of alkanes and alkenes by the early 3d
transition metal ions is well documented.30-38 The primary
products of the YO + C3H(D)6 reactions are detected, also
labeled in Figure 2.
The secondary reactions of Y and YO with all four hydro-
carbons will not be discussed here. Briefly, Y reacts successively
with four molecules of C2H4 or C2D4 and with three molecules
of C3H6 or C3D6. Alkene addition, rather than elimination of
H2 or D2, is increasingly favored as the number of ligands
surrounding the metal increases.
Representative pseudo-first-order kinetics plots for the Y +
ethylene and Y + propylene reactions are shown in Figure 3
and Figure 4, respectively. The resulting effective bimolecular
rate constants at 300 K and 0.8 Torr total pressure are also
collected in Table 1 for all four reactions. Each rate constant is
the mean of at least three experiments and in most cases five.
Figure 3. Semilogarithmic plots of metal atom density, proportional
to integrated ion current, vs hydrocarbon flow for Y + C2H4 and Y +
C2D4 reactions.
For comparison, we include the rate constants for Y (4d15s2,
2D3/2) + C2H4 and Y (4d15s2, 2D3/2) + C3H6 as measured with
LIF detection2 (footnote c in Table 1). The absolute accuracy
of our measurements is ( 30% due to uncertainties in the mean
reaction time, flow calibrations, etc. However, the typical
precision of the experiments is much higher. Since most
systematic errors should cancel for ratios of rate constants and
we have corrected for different flow calibrations, we estimate

6658 J. Phys. Chem. A, Vol. 105, No. 27, 2001
Porembski and Weisshaar
TABLE 2: Calculated Energies (kcal/mol) of Stationary
Points along the Y + Ethylene Doublet PESa
mPW1PW91/
MCPF+ B3LYP/Stuttgart+
Stuttgart+
species
PCI-80b 6-311++G(d,p) 6-311++G(d,p)c
2TSent
-0.28 (-0.33)
-16.3 (-16.2)
-22.3 (-22.0)
+4.8 (+5.9)
-1.4 (-1.5)
Y-C2H4 (2B2) (1a)
YC2H4 (2A1) (1b)
2TSins
-27.7
+1.9
-31.0 (-30.7)
-3.0 (-1.8)
nonplanar HYC2H3 (2a) -27.4
planar HYC2H3 (2b)
2TSâH
-22.8 (-21.6)
-23.3 (-21.9)
+4.3 (+6.9)
-27.3 (-26.1)
-27.3 (-26.0)
+0.26 (+2.9)
-7.9 (-6.3)
2MCTS
-1.0 (+0.66)
-9.0 (-6.3)
H2Y-C2H2 (3a)
-11.7 (-9.0)
-7.9 (-6.5)
H2-YC2H2 (3b)
-2.2 (-0.84)
+19.9 (+22.7)
+19.3 (+21.8)
-3.2 (-0.99)
2TSexit
+15.0 (+17.9)
Y-C2H2 (2B2) + H2 (4a)
YC2H2 (2A1) + H2 (4b) -11.5
-7.6 (-5.4)
YH2 (2A1) + C2H2
+19.1e +17.3 (+20.1)
a Energies relative to free reactants, corrected for differential zero-
point energy effects. Parentheses denote energetics for Y + C2D4.
b MCPF+PCI-80 calculations by Blomberg and Siegbahn from ref 3,
unless otherwise indicated. Geometries of these species were not
described. c Single-point energies for all structures optimized at the level
of B3LYP/Stuttgart 6-311++G(d,p). e Estimated exothermicity using
the heat of formation of C2H2 and the calculated binding energy of
YH2 (2A1) from Siegbahn, P. E. M. Theor. Chim. Acta 1994, 87, 441.
Figure 4. Semilogarithmic plots of metal atom density, proportional
to integrated ion current, vs hydrocarbon flow for Y + C3H6 and Y +
C3D6 reactions.
that the ratios kH/kD in Table 1 are accurate to within ( 7% for
the reactions with ethylene and propylene. There is a small
normal deuterium isotope effect with ethylene (kH/kD ) 1.75 (
0.12) and no isotope effect with propylene within experimental
uncertainty (kH/kD ) 1.06 ( 0.07).
do not change even as the laser intensity decreases by a factor
of 10. In summary, it remains difficult to quantify the possible
impact of secondary reactions and photoionization efficiencies
on the reported branching ratios. Ultimately, we find that the
statistical rate models can account for the observed Y + ethylene
branching ratios, suggesting that the complications described
above are minimal.
For purposes of our statistical rate modeling efforts (Section
IV), we assume that the chemical branching between collisional
stabilization and H2 elimination is accurately measured by the
experimental PIMS integrated intensity ratios reported in Table
1. However, several factors could produce systematic errors in
the ratios, including differences in the reactivity and/or photo-
ionization cross-sections of the primary products. First, the
primary products of all four reactions are observed to decrease
as a function of hydrocarbon flow, while secondary products
appear. The branching ratios in Table 1 are derived from mass
spectra collected at low hydrocarbon flows, when primary
products dominate. In fact, for the Y + propylene reactions,
branching ratios were determined from low-flow mass spectra
for which no secondary products were detected. However, for
Y + ethylene the secondary reaction is much faster than the
primary reaction step. The steady-state concentration of primary
products is quickly achieved even at low ethylene flows, and is
much smaller than the concentration of secondary products. In
this regime, for example, if YC2D2 elimination products react
less efficiently with C2D4 than do stabilized YC2D4 complexes,
the reported branching ratios will underestimate the fraction of
YC2D4 complexes produced in the first reaction step. However,
we find that all branching ratios are essentially independent of
hydrocarbon flow and the different products qualitatively appear
to react with similar efficiencies.
Second, it remains possible that the photoionization cross-
section and hence the detection efficiency of the products varies,
affecting the accuracy of calculated branching ratios. Comparing
the total decay of reactant Y+ signal to the total accumulated
product signal at low hydrocarbon flows for which primary
products dominate, we estimate that the photoionization cross-
sections of the primary products are about a factor of 2 smaller
than the bare metal. Intuition suggests that the cross-sections
of the primary product species, e.g., YC2H2 and YC2H4, are
likely to be more similar and largely unaffected by isotopic
substitution. Furthermore, as noted above, the branching ratios
III. Theoretical: Y + Ethylene
A. Methods. A series of theoretical studies by Blomberg and
Siegbahn have identified several stationary points along the Y
+ C2H4 PES, primarily those involved in primary CH bond
insertion.11,12,15,3 Potential minima and transition states were
optimized at the Hartree-Fock (HF) level and energies were
computed using modified coupled pair functional (MCPF)
theory.11,12,15 Geometry optimizations used double-ú quality
basis sets, while larger sets including diffuse and polarization
functions on Y were used for energy calculations. The most
recent study repeats these calculations, correcting the MCPF
energies for differential effects of configuration interaction using
the PCI-80 approximation (Table 2).3 When possible, energies
and structures calculated at the current level of theory will be
explicitly compared to those from these earlier studies.
The density functional theory approach used here has been
recently described.10 Briefly, for geometry optimizations of
stationary points, potential energy surface scans and intrinsic
reaction coordinate (IRC) searches from computed transition
states, we combine the B3LYP21 density functional with the
Stuttgart ECP + valence set39 (triple-ú quality) for Y and the
6-311++G(d,p) basis set for C and H. We call this hybrid set
Stuttgart+6-311++G(d,p) and carry out the electronic structure
calculations using the GAUSSIAN-98 (G98) program.40 All
open-shell doublet calculations were performed at the spin-
unrestricted level and deviations of the Sˆ2 expectation values
from (3/4)p2 were less than 1% for all optimized structures
except three, as noted below. After optimization at the level of
B3LYP/Stuttgart+6-311++G(d,p), we compute single-point
energies for most stationary points using the mPW1PW9122,23

2
Reactions of Ground-State Y (4d15s2, D)
J. Phys. Chem. A, Vol. 105, No. 27, 2001 6659
hybrid functional. As found earlier, reoptimization of species
at the level of mPW1PW91/Stuttgart+6-311++G(d,p) does not
significantly lower energies or alter geometries and vibrational
frequencies.10 Electronic stability tests confirmed that the lowest
energy solution to the SCF equations was in fact found for each
stationary point. IRC following established the connection of
all transition states to the expected minima. Unless otherwise
indicated, all energies of stationary points are reported relative
along the reaction coordinate, R. Calculated relative to the
lowest energy atomic configuration described above,
(dx1zdz02.01 dx02.0-3y2s1.96), E(2TSent) ) +0.29 kcal/mol. However, the
atomic configuration with good spin, (d0z2.25 dx02.7-5y2s2), places
2TSent at -0.28 kcal/mol (below ground-state reactants). Al-
though our choice of reference ground-state leads to a reported
“barrier height” below zero, we include the latter energy in Table
2 since the saddle point is characteristic of the complex forming
process. Furthermore, the transition state provides a set of
vibrational frequencies and rotational constants for the RRKM
model 2 described in Section IV-B. Ultimately, the energy of
2TSent will be adjusted in accord with the experimental results.
2
to ground-state Y (4d15s2, D) + C2H4 reactants, including
corrections for zero-point energy (ZPE) effects.
Ideally, the true Y (4d15s2, 2D) ground state should be
represented by a computed atomic configuration with integral
s and d occupations that correspond to a single Slater determi-
nant with a definite value of L.41,42 However, because we cannot
apply symmetry constraints to atoms using the current G98
package, unphysical mixing of the s and d orbitals can occur,
leading to noninteger occupancies. For example, both B3LYP
and mPW1PW91 give d1xz d0z2.01 d0x2.0-3y2s1.96 as the lowest energy
atomic configuration, and the resulting determinant shows 7%
contamination of the nominally doublet wave function. Lying
0.6 kcal/mol above this state, we also find the configuration
d0z2.25 d0x2.7-5y2s2. Although neither state has integral orbital oc-
cupancies, the latter configuration exhibits no spin contamination
and we use it to represent the ground-state term Y (4d15s2, 2D).
We use d0xz.97 dz02.04 dx02.9-9y2s1 to represent the excited-state term Y
2
Finally, the basis set superposition error (BSSE)46 at TSent is
small and increases the energy of the transition state by only
0.03 kcal/mol.
Thus at the current level of theory we find that the Y +
ethylene entrance channel PES is very flat with essentially no
barrier to the approach of the ground-state metal and alkene.
As discussed in an earlier theoretical study,11 Y ultimately binds
to ethylene in two distinct modes. The geometry and energy of
“long-range” YsC2H4 (2B2) (1a) and “short-range” YC2H4 (2A1)
(1b) are both described in Figure 6 and Table 2. In the long-
range complex 1a, R ) 2.45 Å and there is minimal distortion
of ethylene (RCC ) 1.39 Å). In the short-range metallacyclo-
propane complex 1b, R ) 2.14 Å and the ethylene π bond has
been essentially broken (RCC ) 1.53 Å). The geometries of these
species are nearly identical to the ones calculated at the HF
level.11 The binding energy of the metallacyclopropane 1b
calculated at the level of B3LYP/Stuttgart+6-311++G(d,p)
compares favorably with the energy computed using MCPF+PCI-
80 (Table 2).3 The binding energy of 1a was not reported in
the latter study, but here we find the long-range complex to be
substantially bound, located 16 kcal/mol below ground-state
reactants. For Y-C2H4 (2B2) (1a), the computed value of Sˆ2
shows a deviation of 8% from a pure doublet spin description.
Using the smaller LANL2DZ basis set, we performed a potential
energy surface scan decreasing R from the long-range complex
minimum, but found no energy barrier in the region of the curve-
crossing between 1a and 1b.
2
(4d25s1, F). Again, d orbital mixing is unavoidable and the
computed value of Sˆ2 is 1.75p2, quite different from the pure
doublet value of 0.75p2. The calculated excitation energies to
this state are 35 and 33 kcal/mol using B3LYP and mPW1PW91,
respectively; it is 44 kcal/mol experimentally.43 Both the spin
contamination and computed excitation energy to this excited
state indicate a contribution of the low-lying term Y (4d25s1,
4F), lying 31 kcal/mol above ground-state reactants experimen-
tally.43 The difficulty of computing accurate atomic state
splittings in transition metals using density functional theory is
well documented.42,44
B. Results. With the electronic wave function constrained
to doublet spin multiplicity, we calculate those features of the
Y + ethylene PES illustrated in Figure 5. Below the energy
level diagram, we include the key structures that illustrate the
major atomic motions along the computed reaction paths, while
the detailed structural parameters of all reaction intermediates
and transition states are given in Figure 6. Table 2 summarizes
the energetics of these species, using the Stuttgart+6-311++G-
(d,p) basis set with two different density functional methods.
Also in Table 2, we collect results from prior theoretical studies.
The individual s and d gross populations and the natural charge
of the Y metal center, as determined by a natural bond orbital
(NBO) analysis,45 are given in Table 3. Finally, rotational
constants and vibrational frequencies of all stationary points
calculated at the level of B3LYP/Stuttgart+6-311++G(d,p) are
collected in the Supporting Information to this paper.
The bonding mechanism in the two structurally different
complexes has been previously outlined by Blomberg et al.11
and is remarkably similar to the bonding in the analogous long
and short-range Zr(ethylene) complexes described in a more
recent study.10 Table 3 gives the electron configuration and
natural charge of Y in both 1a and 1b. Essentially, the s2-like
long-range complex (1a) binds to ethylene using two singly
occupied metal-carbon σ-bonds. Two nonbonding electrons
occupy an sd hybrid that effectively reduces repulsion with the
alkene. In the s1-like metallacyclopropane (1b), the metal-
carbon bonds are doubly occupied, leaving only one nonbonding
electron in the sd hybrid.
Next, we confirm the mechanism of primary CH bond
insertion by the metallacyclopropane complex (1b) implied by
earlier work.12,3 The geometry and energy of the CH insertion
transition state 2TSins that connects the metallacyclopropane (1b)
to an insertion intermediate (2b) are described in Figure 6 and
An important mechanistic question is whether there exists
an energy barrier to the association of the closed-shell s2
metal and the closed-shell ethylene molecule. Scanning the
distance R between Y and the midpoint of the CC bond, the
B3LYP/Stuttgart+6-311++G(d,p) calculations do locate a
small approach barrier to the formation of the long-range Ys
C2H4 (2B2) complex (1a, described below). The asymmetric
2
Table 2. In TSins the short Y-C (2.19 and 2.42 Å) and Y-H
(1.95 Å) bond distances and small C2-C1-Y (82°) bond angle
indicate the simultaneous interaction of the metal center with
the ethylene π-bond and the departing H atom. The CC bond
has twisted by nearly 180°, swinging the C2-H1 bond toward
the metal center. The substantial lengthening of this bond (1.11
Å) also reflects the presence of stabilizing agostic bonding
interactions. The transition vector for the imaginary frequency,
2
entrance channel transition state TSent is shown in Figure 6.
The transition state is located at very long range, R ) 4.42 Å,
and there is essentially no distortion of the ethylene molecule,
especially as compared to structures 1a and 1b. The transition
vector, also labeled, shows the relative motion of Y and ethylene

6660 J. Phys. Chem. A, Vol. 105, No. 27, 2001
Porembski and Weisshaar
Figure 5. B3LYP/Stuttgart+6-311++G(d,p) reaction path energetics along the doublet PES. All energies measured relative to ground-state Y
2
2
(4d15s2, D) + C2H4 reactants and corrected for zero-point energy. The location of the 4d25s1, F excited-state term is given by the experimental
excitation energy of 44 kcal/mol, not the B3LYP computed value (see text for details).
also labeled in Figure 6, clearly indicates the breaking of the
CH bond. The nonplanar structure of TSins is different from
in Table 2. Both 2a and 2b appear to be structurally distinct
from the insertion intermediate optimized at the HF level,
characterized by a C2-C1-Y bond angle of 115.1°.12 Here,
both DFT methods find the nonplanar (2a) and planar (2b)
structures to be nearly isoenergetic, separated by a small barrier
of 1 kcal/mol as revealed by the dihedral angle scan. Under all
reaction conditions with some 20 kcal/mol of internal energy,
rapid conversion between planar and nonplanar HYC2H3 will
occur. Nonetheless, at the level of B3LYP/Stuttgart+6-311++G-
(d,p), IRC scans from TSins, TSâH, and MCTS all led to
planar geometries with structural parameters consistent with the
insertion intermediate 2b.
To date, there has been no theoretical investigation of the
mechanism of H2 elimination from the insertion intermediate
HYC2H3 (2b). In our most recent theoretical study of the Zr
2
the original CH insertion transition state calculated at the HF
level, characterized by a bond angle of C2-C1-Y ) 128.2°.12
Carroll et al.3 do report the energy of a “new and lower” CH
insertion transition state than the one previously found12 but do
not describe its geometry.
The resulting planar insertion intermediate HYC2H3 (2b) is
described in Figure 6 and Table 2. Again, agostic interactions
account for the small C2-C1-Y bond angle (97°), relatively
short Y-C2 distance (2.80 Å), and long C2-H1 bond length
(1.11 Å). A potential energy surface scan of the H3-C1-Y-
H4 dihedral angle located a nonplanar insertion intermediate
2a (not shown here) at ΘH3-C1-Y-H4 ) -148°, with other
structural parameters similar to 2b. Its energy is also included
2
2
2

2
Reactions of Ground-State Y (4d15s2, D)
J. Phys. Chem. A, Vol. 105, No. 27, 2001 6661
Figure 6. Details of B3LYP/Stuttgart+6-311++G(d,p) optimized geometries of potential minima and transition states for Y + C2H4 along the
doublet PES. Distances in angstroms; angles in degrees. For each transition state, the transition vector corresponding to the imaginary frequencies
is shown.
(4d25s2, 3F) + ethylene reaction, we identified two reaction paths
on the triplet spin surface.10 The lower energy path involves
concerted H2 elimination from the planar insertion intermediate
HZrC2H3, while the higher energy path involves stepwise H2
elimination from HZrC2H3. Similarly, here we find that the Y
+ ethylene PES also branches from the planar insertion
intermediate 2b. The large-dashed path in Figure 5 follows the
the stepwise and concerted paths are very similar to those for
the stationary points identified along the analogous paths of the
Zr + ethylene triplet PES.
The âH transfer transition state 2TSâH that carries the planar
insertion intermediate (2b) to a dihydrido intermediate (3a) is
described in Figure 6 and Table 2. The labeled transition vector
clearly corresponds to the expected â-CH bond stretch. The
resulting dihydrido intermediate H2Y-C2H2 (3a), also described
in Figure 6 and Table 2, forms strong bonds to hydrogen and
weak bonds to acetylene. The geometry and bonding mechanism
of the YC2H2 unit of the complex are virtually identical to the
long-range Y-C2H2 (2B2) complex (4a) described below.
2
stepwise elimination of H2 via TSâH, while the small-dashed
2
path follows the concerted elimination of H2 via MCTS. The
latter is found to be the lowest energy path to exothermic YC2H2
(2A1) + H2 products. We find that the key structural and
electronic features of intermediates and transition states along

6662 J. Phys. Chem. A, Vol. 105, No. 27, 2001
Porembski and Weisshaar
TABLE 3: Natural Populations of Y for Doublet Stationary
agreement with the binding energy of 48.6 kcal/mol calculated
at the MCPF level.3 Finally, the bonding mechanisms in the
acetylene complexes parallel those already described for the
ethylene complexes. The additional interaction of the out-of-
plane π orbital with the metal d orbitals accounts for the very
strong metal-acetylene binding as discussed earlier.15,10
Next, we directly explore the connection of 3a and 3b to
products following either the stepwise H2 elimination reaction
path or the concerted H2 elimination reaction path. Along the
former path, we locate a large exit barrier 2TSexit of +23 kcal/
mol that connects the dihydrido intermediate H2Y-C2H2 (3a)
to products (Figure 6 and Table 2). The transition vectors, also
labeled in Figure 6, indicate the H1-Y-H4 bond-angle stretch
and reorientation of the acetylene group that accompanies the
conversion of 3a to products. In contrast, we find no exit channel
barrier along the concerted H2 elimination reaction path. Starting
from the H2-YC2H2 (3b) minimum, we compute the energy
while scanning the distance between Y and H4, optimizing the
geometry at each fixed value of Y-H4. At the level of B3LYP/
Stuttgart+6-311++G(d,p), we find that 3b dissociates smoothly
to YC2H2 (2A1) + H2 products with no barrier.
Finally, we note that the two DFT methods differ substantially
in their predictions of stationary point energetics along the Y
+ ethylene PES (Table 2). Compared to B3LYP, mPW1PW91
increases binding energies and lowers transition states relative
to ground-state reactants by about 5 kcal/mol on average. This
is quite similar to the performance of the same functionals on
the Zr + ethylene system.10 Next we use a statistical rate model
to assess the relative performance of the different theories in
explaining the behavior of the Y + ethylene reaction.
Points Using B3LYP/Stuttgart+6-311++G(d,p) Theory
species
natural chargea
4d
5s
2TSent
+0.007
+0.54
+1.11
+1.01
+1.23
+1.27
+1.54
+1.08
+1.84
+1.08
+1.65
+0.59
+1.14
1.04
0.63
0.89
1.02
0.81
0.78
0.96
0.99
0.79
0.99
1.13
0.58
0.85
1.96
1.80
0.99
0.95
0.92
0.90
0.50
0.91
0.37
0.90
0.23
1.81
1.00
Y-C2H4 (2B2) (1a)
YC2H4 (2A1) (1b)
2TSins
nonplanar HYC2H3 (2a)
planar HYC2H3 (2b)
2TSâH
2MCTS
H2Y-C2H2 (3a)
H2-YC2H2 (3b)
2TSexit
Y-C2H2 (2B2) (4a)
YC2H2 (2A1) (4b)
a The natural charge on Y in units of electrons, as found from an
NBO population analysis. In all cases, the 5p population was less than
0.06.
Furthermore, the geometry of the YH2 unit of the complex is
very similar to that calculated for YH2 (2A1) products (YH )
1.96 Å and HYH ) 114°; structure not shown).
On the lower energy reaction path, a multicenter transition
2
state MCTS (Table 2) carries the insertion intermediate (2b)
directly to a product-like complex (3b). The “late” transition
2
state MCTS is itself very product-like (Figure 6). The planar
geometry and bonding mechanism of the YC2H2 unit are similar
to the strongly bound YC2H2 (2A1) metallacyclopropene (4b)
described below. In addition, the Y-H distances are long, 2.11
and 2.02 Å, and the H4-H1 distance is small, 0.95 Å. The
NBO analysis45 reveals substantial donor-acceptor interactions
among four center (the two hydrogens, Y, and C1) involving
σHH donation to the metal center and σYC1 back-donation to
IV. Statistical Rate Model: Y + Ethylene
σ
HH*. The transition vector depicted in Figure 6 reveals the
concerted motion, the simultaneous â-CH stretch and formation
of molecular H2, that connects the insertion intermediate directly
to the precursor complex for H2 loss (3b). The resulting H2-
YC2H2 complex (3b) exhibits strong bonds to acetylene and a
weak interaction with molecular H2 (Figure 6 and Table 2). The
Y-H distances have increased to 2.36 and 2.26 Å and the H4-
H1 distance has decreased to 0.78 Å, nearly that of the
equilibrium bond length in H2, 0.744 Å. Since 2MCTS and H2-
YC2H2 (3b) have similar product-like geometries and bonding
mechanisms, it is not surprising that both stationary points lie
nearly isoenergetic to the YC2H2 (2A1) + H2 exit channel. We
note that although B3LYP/Stuttgart+6-311++G(d,p) finds H2-
YC2H2 (3b) to be a true minimum, ZPE effects place the
complex slightly above YC2H2 (2A1) + H2 products in energy.
Before discussing the Y + ethylene exit channel PES along
the stepwise and concerted reaction paths, we describe the
geometry and energetics of the YC2H2 + H2 products them-
selves. As for ethylene, we find that Y binds to acetylene in
two distinct modes. The geometry and energy of the Y-C2H2
(2B2) complex (4a) are described in Figure 6 and Table 2. Like
the Y-C2H4 (2B2) complex (1a), 4a binds at a long range. No
similar long-range Y-C2H2 complex has been reported in the
literature. The geometry and energy of the YC2H2 (2A1) (4b)
complex are also described in Figure 6 and Table 2. This short-
range complex is a true metallacyclopropene, since the acetylene
triple bond has been effectively broken. The geometry of 4b is
similar to the structure reported earlier, optimized at the HF
level.15 We calculate the binding energy of 4a to be only 20.9
kcal/mol, and as for the long-range complex 1a, there is some
spin contamination (6%) of the B3LYP determinant. We
compute a binding energy of 43.4 kcal/mol for 4b, in good
The electronic structure calculations presented in Section III-B
provide the necessary input for RRKM28,29 rate calculations
along the Y + ethylene PES. Here, we build a statistical model
of the decay of long-lived metallacyclopropane (1b) complexes.
The energies of key intermediates and transition states will be
adjusted to fit the collection of experimental data (Table 1),
providing both a quantitative mechanistic picture of primary
CH bond insertion and a critical test of the different levels of
theory (Table 2). With only the doublet surface important, Y
+ ethylene provides a cleaner test of DFT than Zr + ethylene,
for which both a singlet and a triplet surface are probably
involved in the dynamics.
The statistical rate model employed here differs from previous
work6 and will be described in detail below. In the model, long-
lived collision complexes are formed at a rate determined by
the Lennard-Jones rate constant kLJ and details of the long-
range interactions. Complexes (1b) then decay into three parallel
channels, dissociation back to reactants (kdiss), stabilization by
a third-body collision into the strongly bound metallacyclopro-
pane well (kQ[He]) and insertion into a CH bond (kins). We note
that the PIMS data determine the mass, not the structure, of
detected products. However, from RRKM theory we expect that
stabilized complexes will be dominated by the species 1b, due
to the large density of rovibrational states. Although theory
identifies two reaction paths emanating from HYC2H3 (2b), the
low-energy 2MCTS ensures that the concerted mechanism
dominates and that all insertion intermediates will go on to
eliminate H2, rather than become trapped in 2b (Figure 5).
As before, our model calculations are somewhat oversimpli-
fied; they are intended to demonstrate that a range of plausible
parameters can explain all of the data. The connection between

2
Reactions of Ground-State Y (4d15s2, D)
J. Phys. Chem. A, Vol. 105, No. 27, 2001 6663
the model and the experimental data is made via the microca-
nonical rate constant:
because lack of vibrational or rotational degrees of freedom
make He a rather inefficient quencher.3 In our rate model, we
allow kQ to vary by no more than a factor of 2 about this value,
placing kQ[He] within the range 1.5-6.0 × 106 s-1
.
Jmax
k(Eh) )
P(J)k(Eh,J)
In the absence of an approach barrier (model 1), we compute
∑
the total rate of bimolecular collisions as kLJ ) πb2maxVj ) πl2
/
J)0
max
µ2Vj. Here, lmax is the largest value of orbital angular momentum
that can penetrate the maximum of the effective potential, which
includes an estimated long-range attraction -C6/R6;47 µ is the
reduced mass of the collision pair; and Vj is the relative velocity
at Eht, the mean collision energy at 300 K. In such a model, Eht
) 0.9 kcal/mol and lmax ) 123, giving kLJ ) 7.74 × 10-10 cm3
s-1 for Y + C2H4. By this measure, the Y + C2H4 reaction
efficiency at room temperature, estimated as kH/kLJ, is only 1%.
Jmax
kins(Eh,J) + kQ[He]
) AkLJ
P(J)
(5)
∑
kins (Eh,J) + kdiss(Eh,J) + kQ[He]
J)0
Here Eh is the mean total energy including Eht, the average
kinetic energy of those thermal collisions that can surmount
the assumed entrance channel barrier for zero-impact parameter
collisions, and the mean ethylene vibrational energy. Notably,
we do not explicitly average eq 5 over a 300 K Boltzmann
distribution of energies, but instead carry out the microcanonical
rate calculations using the average energies, as defined above.
The total angular momentum is given by J with a distribution
P(J) ) 2J/Jmax2, where Jmax is the largest value of J for the
metallacyclopropane (1b) complexes. We assume there is no
ethylene rotational energy (bj ) 0) contributed to the reaction
rate so thatbj )Bl, whereBl is the orbital angular momentum of
the collision. The total rate of bimolecular collisions at 300 K
is given by kLJ, further described below. However, not all
collisions are “successful.” We define A as the reaction
inefficiency factor, i.e., the fraction of Lennard-Jones collisions
that actually gain access to the deep metallacyclopropane (1b)
well. Possible origins of reaction inefficiency will be discussed
in detail below.
The absence or presence of an approach barrier to the
formation of metallacyclopropane (1b) complexes distinguishes
two rate models, one for which E(2TSent) ) 0 kcal/mol (model
1) and one for which E(2TSent) > 0 kcal/mol (model 2).
Accordingly, each model ascribes very different meanings to
the inefficiency factor A. In model 1, we attribute a portion of
the overall reaction inefficiency to nonadiabatic or steric effects;
neither effect is easy to estimate a priori. Thus, A simply
represents the factor required to force the calculated Y +
ethylene rate constant into accord with the measured bulk
kinetics value. In model 2, A includes two factors, the fraction
of the 300 K Boltzmann distribution with kinetic energy above
the approach barrier energy threshold for l ) 0, including
differential zero-point energy effects; and the fraction of those
Model 2 assumes the presence of an approach barrier of
variable magnitude. We then carry out microcanonical RRKM
rate calculations for which Eh′t is now defined as the mean
kinetic energy of that fraction of collisions with energy aboVe
threshold, given a particular energy of 2TSent. The inclusion of
an approach barrier further constrains the range of orbital angular
2
momenta that can surmount the centrifugal barrier atop TSent
to reach the deep metallacyclopropane (1b) well. We calculate
the rate of association over 2TSent from kf πl′2 /µ2Vj′ where Vj′ is
max
the relative velocity corresponding to Eh′t and l′ ) µVj′bm′ ax. For
t
example, when E(2TSent) ) 2 kcal/mol, Eh′ ) 2.66 kcal/mol
t
and l′max ) 77, yielding kf ) 1.46 × 10-10 cm3 s-1 for Y +
C2H4. In this scenario, only a small fraction of collisions (0.079)
have sufficient energy to surmount the approach barrier and
angular momentum effects further reduce efficiency. Thus, the
rate of formation of long-lived complexes becomes 0.079kf ≡
AkLJ, yielding an effective A ) 0.015.
The Y + ethylene experimental data that constrain the
possible output of the statistical rate model (Table 1) include
the bimolecular rate constant k ) 7.4 ( 2.2 × 10-12 cm3 s-1
and the kinetic isotope effect kH/kD ) 1.75 ( 0.12. In addition,
we equate the experimental PIMS intensity ratios to the
branching ratio of stabilized complexes to elimination products,
i.e., to kQ[He]/kins in the rate model. Thus, kQ[He]/kins e 0.10
( 0.03 for Y + C2H4 and kQ[He]/kins ) 0.33 ( 0.09 for Y +
C2D4. Possible experimental complications were discussed in
Section II-B.
A. Model 1: No Entrance Channel Barrier. The electronic
structure calculations described above find no energy barrier
to the approach of Y and ethylene along the adiabatic entrance
channel. Thus, our first model assumes that metallacyclopropane
complexes (1b) dissociate over a loose, orbiting transition state,
whose placement is defined as the maximum in the effective
potential for Y + ethylene collisions, as usual.28,29 The results
using the mPW1PW91 binding energy for 1b (-31 kcal/mol),
the estimate kQ[He] ) 3 × 106 s-1, and a variety of assumptions
about the energy of 2TSins are collected in Table 4. Superscripts
H or D on rate constants refer to the reaction Y + C2H4 or Y
+ C2D4, respectively. Vibrational frequencies, rotational con-
stants and other input parameters are suitably modified for
calculations of Y + C2D4. In Table 4, we note that the values
of kH and kD calculated from eq 5 use A ) 1, so that the
predictions of the model are made readily apparent. The factor
A will later be adjusted to fit the bulk kinetics rate constant
(Table 5), given the model parameters found to best fit the
isotope effect and product branching ratios.
sufficiently energetic collisions that can surmount the centrifugal
2
barrier atop TSent
.
Each unimolecular rate constant in eq 5 is computed from
the usual RRKM expression by a program based on the work
of Yi et al.24 We explore how the barrier heights for 2TSent and
2TSins and the binding energy of YC2H4 (2A1) (1b) impact the
microcanonical rate constants kdiss and kins and what combination
of stationary point energies best explains the experimental data.
Although both B3LYP and mPW1PW91 compute the energy
2
of TSent to lie slightly below ground-state reactants, we test
approach barrier heights that range from 0 to 3 kcal/mol.
Correspondingly, the treatment of the entrance channel transition
state varies from loose to tight. In all calculations, the transition
state 2TSins is modeled as tight. We use the B3LYP/Stuttgart+6-
311++G(d,p) rotational constants and vibrational frequencies
for all three stationary points, collected in the Supporting
Information to this paper.
In the flow tube at 0.8 Torr total pressure, quenching occurs
via collisions with the buffer gas (mostly He) at a rate kQ[He]
Equation 5 reveals that the experimentally observed rate
constant and kinetic isotope effect are determined by the
competition between the rates of dissociation, stabilization and
insertion. When kdiss dominates both kQ[He] and kins, the model
≈ 3 × 106 s-1. This estimate uses kQ ) 1 × 10-10 cm3 s-1
,
about 5 times smaller than the hard-spheres collision rate,

6664 J. Phys. Chem. A, Vol. 105, No. 27, 2001
Porembski and Weisshaar
TABLE 4: RRKM Rate Constants for Model 1: No
Entrance Channel Barriera
TABLE 6: RRKM Rate Constants for Model 2: Non-Zero
Entrance Channel Barriera
E(2TSins) (kcal/mol)b
E(2TSins) (kcal/mol)b
rate or ratio
-1
-2
-3
-4
-5
rate or ratio
0
-1
-2
-3
-4
kHins(108 s-1
)
)
kHins(108 s-1
)
)
0.019
0.071
0.003
0.216
0.014
0.575
0.047
1.400
0.140
0.333
0.013
0.987
0.062
2.56
5.99
0.645
9
13.0
1.69
8
kDins(108 s-1
kHins/kDins
kDins(108 s-1
kHins/kDins
0.0005
0.218
38
24
15
12
143
85.4
1.68
0.05
0.64
10
26
16
12
kH(10-12 cm3 s-1
kD(10-12 cm3 s-1
kH/kD
)
)
18.3
35.8
0.51
1.56
61.9
36.0
39.5
0.91
0.42
9.58
75.1
52.4
1.43
0.14
2.22
242
148
1.64
0.02
0.21
kH(10-12 cm3 s-1
kD(10-12 cm3 s-1
kH/kD
)
)
7.2
3.4
2.1
0.04
1.14
9.7
6.0
1.62
0.015
0.241
10.8
9.1
1.19
0.006
0.069
11.3
10.9
1.03
0.003
0.023
11.5
11.7
0.98
0.001
0.009
kH[He]kH
kQ[He]/kH
ins
ins
kQ[He]kD
kQ[He]/kD
ins
ins
a
a Calculations use E(1b) ) -31 kcal/mol and kQ[He] ) 3 × 106
s-1. The factor A ) 1 and will be adjusted later (Table 5). In the absence
of an approach barrier, Eht ) 0.9 kcal/mol, and the estimated association
Calculations use E(2TSent) ) 2 kcal/mol, E(1b) ) -26 kcal/mol,
and kQ[He] ) 1.5 × 106 s-1. Here, Eht ) 2.66 kcal/mol, and A becomes
0.015 and 0.016 for Y + C2H4 and Y + C2D4, respectively (see text
for details). Assuming metallacyclopropane complexes dissociate over
the tight transition state 2TSent, the computed complex dissociation rates
rates for Y + C2H4 and Y + C2D4 are: kH ) 7.74 × 10-10 cm3 s-1
LJ
and kLDJ ) 7.36 × 10-10 cm3 s-1. Assuming metallacyclopropane
complexes dissociate over a loose, orbiting transition state, the computed
complex dissociation rates are kHdiss ) 2.24 × 108 s-1 and kDdiss ) 6.33
are kHdiss ) 2.06 × 107 s-1 and kDdiss ) 7.21 × 106 s-1
.
b Placement of
2TSins for each column of RRKM calculations. The mPW1PW91
functional gives 2TSins at -3 kcal/mol relative to Y + C2H4 reactants.
× 107 s-1
.
b Placement of 2TSins for each column of RRKM calculations.
The mPW1PW91 functional gives 2TSins at -3 kcal/mol relative to Y
+ C2H4 reactants.
if we decrease the binding energy of 1b by 2 kcal/mol, kH/kD
increases to 1.94, while the branching ratios kQ[He]/kins decrease
to 0.03 and 0.35 for Y + C2H4 and Y + C2D4, respectively.
Alternatively, if kQ[He] is decreased to 2.3 × 106 s-1, kH/kD
becomes 1.80 while the branching ratios are 0.04 and 0.49 for
Y + C2H4 and Y + C2D4, respectively.
TABLE 5: Rate Constants, Isotope Effects, and Branching
Ratios as Determined by Experiment and RRKM Models 1
and 2
rate or ratio
experimenta
model 1b
model 2c
kH(10-12 cm3 s-1
kD(10-12 cm3 s-1
kH/kD
)
)
7.4 ( 0.3
4.2 ( 0.12
1.75 ( 0.12
e0.10 ( 0.03
0.33 ( 0.09
7.15
4.26
1.68
0.05
0.64
9.7
6.0
1.62
0.015
0.241
2
Table 4 also shows that for the energies of 1b and TSins
that approximately reproduce the observed isotope effect and
product branching data, the overall reaction rate kH is about 19
times faster than experiment (Table 1). As mentioned earlier,
in this model, we must attribute the overall reaction inefficiency
to either nonadiabatic or steric effects. Both would limit the
fraction A of Y + ethylene collisions that gain access to the
deep metallacyclopropane (1b) well as discussed further below.
Specifically, to match the 300 K bulk rate constant when
E(2TSins) ) -4 kcal/mol, we require that A ≈ 0.05.
kD[He]kH
ins
kD[He]kD
ins
a Values as determined by the present work; see text and Table 1
for details. Error estimates refer to the precision of the experiments.
Absolute accuracy of kH and kD are (30%. b Values as determined by
model 1 which uses A ) 0.05, E(1b) ) -31 kcal/mol, E(2TSins) ) -4
kcal/mol and kQ[He] ) 3 × 106 s-1; see text and Table 4 for details.
c Values as determined by model 2 which uses E(2TSent) ) 2 kcal/mol
(A ) 0.015), E(1b) ) -26 kcal/mol, E(2TSins) ) -1 kcal/mol and
kQ[He] ) 1.5 × 106 s-1; see text and Table 6 for details.
Table 5 directly compares representative “good fit” results
from model 1 to the available experimental data. To summarize,
for A ≈ 0.05, E(1b) ) -31 ( 2 kcal/mol, E(2TSins) ) -4 (
1 kcal/mol, and for reasonable values of kQ[He], model 1 can
explain all of the Y + ethylene data. Furthermore, since
mPW1PW91 does not find an approach barrier and places 1b
and 2TSins at -31 and -3 kcal/mol, respectively, this functional
yields energies consistent with experiment within the limitations
predicts a relatively inefficient reaction, since a large fraction
of complexes dissociate; and an inverse isotope effect, since
2
YC2D4 complexes are longer lived. As TSins is lowered, the
rate of insertion and the overall reaction rate rapidly increase.
When 2TSins lies at least 11 kcal/mol below reactants (data not
shown), kins dominates kQ[He] and kdiss. The model then predicts
that the overall reaction rate kH approaches the rate of bimo-
lecular collisions kHLJ and the deuterium isotope effect disap-
pears, with kHLJ/kLDJ ) 1.05. In this limit, essentially all com-
plexes go on to insert into a CH bond of ethylene and a
negligible fraction become stabilized into the deep metalla-
cyclopropane (1b) well, since kins . kQ[He].
The experimental data rule out either extreme, dominant
dissociation or dominant insertion. Furthermore, the observation
of a small normal isotope effect, the absence of stabilized YC2H4
complexes within experimental sensitivity, and the small
measured fraction of stabilized YC2D4 complexes constrain the
magnitude of 2TSins. When E(2TSins) ) -4 kcal/mol the model
comfortably agrees with these data. The overall reaction rate is
much less sensitive to the binding energy of 1b and the precise
value of kQ[He]; instead, these parameters primarily impact the
isotope effect and branching ratios. Given E(2TSins) ) -4 kcal/
mol, minor adjustments to the other parameters can tune the
ratios toward even better agreement with the data. For example,
2
of our approximate rate model. In contrast, the 1b and TSins
energies of -22.3 and +4.8 kcal/mol from B3LYP are much
too high.
B. Model 2: Non-Zero Entrance Channel Barrier. Given
the modest isotope effect and the small fraction of stabilized
complexes, the statistical rate modeling thus far shows clearly
that the 1% efficiency of the Y + ethylene reaction is only partly
due to the failure of metallacyclopropane (1b) complexes to
insert in a CH bond of ethylene. Model 1 attributes the bulk of
the reaction efficiency to nonadiabatic or steric effects. Next
we explore model 2, for which a real energy barrier in the
entrance channel limits the fraction of collisions that gain access
to the deep metallacyclopropane (1b) well. Thus we assume
that the complexes dissociate over the tight transition state
2TSent, whose energy is allowed to vary from 0 to 3 kcal/mol.
Representative results for E(2TSent) ) 2 kcal/mol are collected
in Table 6, using E(1b) ) -26 kcal/mol, the estimate kQ[He]
) 1.5 × 106 s-1, and a variety of assumptions about the energy
of 2TSins. Aside from reducing the number of successful

2
Reactions of Ground-State Y (4d15s2, D)
J. Phys. Chem. A, Vol. 105, No. 27, 2001 6665
Y-ethylene collisions, the primary effect of the model with an
approach barrier is to dramatically decrease kdiss at a given
collision energy. Consequently, for relatively large values of
2TSins, kins dominates kdiss and kQ[He], resulting in the absence
of an isotope effect and a small fraction of stabilized complexes,
as explained above. Although almost all complexes insert when
2TSins lies at -3 kcal/mol, only a small fraction (0.015) of
collisions are able to surmount the association barrier. When
5s2, 2D) atom has only three valence electrons, the lowest energy
pathway must again involve the concerted elimination of H2,
now via MCTS on the doublet PES.
2
Although the B3LYP and mPW1PW91 functionals predict
very different stationary point energies along the Y + ethylene
PES, the statistical rate models described above help to
independently constrain the energetics of the key reaction
intermediates and transition states involved in primary CH bond
insertion. The small normal isotope effect, the absence of
stabilized YC2H4 complexes, and the small fraction of stabilized
YC2D4 complexes indicate that insertion competes with dis-
sociation, i.e., neither process can dominate. The comparison
of well-tuned models 1 and 2 (details above) with experiment
(Table 5) shows that the parameters preferred by model 1 are
very similar to the mPW1PW91 results. Larger, upward adjust-
ments to the mPW1PW91 energies are demanded by model 2.
Regardless of which model is chosen, it is clear that 2TSins must
lie 1-4 kcal/mol below ground-state reactants. This is 6-9 kcal/
mol lower than the B3LYP value, but within (2 kcal/mol of
the mPW1PW91 value, depending on the model. There is by
now abundant evidence that B3LYP consistently overestimates
the energy of key transition states by a similar amount in many
related transition metal-hydrocarbon systems.24-27
The remaining subtle issue involves the origin of the room-
temperature reaction efficiency, which is about 1% for Y +
C2H4 and increases to about 16% for Y + C3H6. For Y + C2H4,
the reaction efficiency is a product of two probabilities: the
fraction of collisions that reach the deep metallacyclopropane
well (1b) times the fraction of those complexes that are
collisionally stabilized or eliminate H2 rather than dissociating
back to reactants. The former fraction is given by the inef-
ficiency factor A in eq 5, attributed to nonadiabatic effects or
steric effects in model 1 and to a small entrance channel barrier
and its associated angular momentum effects in model 2. The
latter fraction is determined by the J-averaged model ratio (kins
+ kQ[He])/(kins + kQ[He] + kdiss), which is quite sensitive to
isotope effect and product branching ratios. For this discussion,
call the latter fraction B. From the calculations in Table 5, for
model 1, A ) 0.05 and B ) 0.185. For model 2, A ) 0.015 and
B ) 0.841. The calculated reaction rates kH are then given by
ABkLJ and are listed in Table 5; both models are in reasonable
agreement with the experimentally determined kH. However, the
two models differ substantially in their partitioning of the
reaction inefficiency. Additional experimental data would further
constrain the RRKM calculations, possibly ruling out one model
or the other. For example, at high pressures the fraction B will
E(2TSent) ) 2 kcal/mol, E(1b) ) -26 kcal/mol and E(2TSins
)
) -1 kcal/mol, the model approximately matches the experi-
mental reaction rate, isotope effect and branching ratios. Within
the assumptions of model 2, the energy of 2TSent is well
constrained by the 300 K bulk rate constant. Additional RRKM
2
calculations considered other combinations of 1b, TSins, and
kQ[He]. Such changes either produce less acceptable agreement
with the experimental data, deviate further from the electronic
structure results, or conflict with our expectations about kQ[He].
Predictions of model 2 are compared directly with experiment
in Table 5. Model 2 places the best-fit values of 2TSent, 1b, and
2TSins at energies 3.4, 5.0, and 2.0 kcal/mol higher than those
predicted by mPW1PW91. These adjustments are larger than
those called for by model 1. The required adjustments remain
2
substantial for the B3LYP functional, which computes TSent
,
1b, and 2TSins to be 2.3, 3.7, and 5.8 kcal/mol too high relative
to the values favored by model 2. As discussed below, we favor
model 1 since it more readily fits the experimental data and
requires only minor adjustments to the mPW1PW91 electronic
structure results, which required only small adjustments for the
Zr + ethylene reaction as well.10
V. Discussion
Before discussing the impact of the statistical rate modeling
on our understanding of the Y (4d15s2, 2D) + ethylene reaction,
we compare the new theoretical results to our recent findings
3
in the related Zr (4d25s2, F) + ethylene system.10 In neither
case does DFT in its B3LYP or mPW1PW91 form find any
substantial barrier to the formation of a long-range metal-
ethylene complex along the adiabatic entrance channel, despite
the fact that the reactions exhibit 300 K efficiencies of only
1% and 7%, respectively.2,5 Along both the doublet Y +
ethylene PES and the triplet Zr + ethylene PES, two distinct
H2 elimination paths have been found, a stepwise and a
concerted path. For both reactions, regardless of the DFT method
2
employed, a high-energy exit barrier (here, TSexit) precludes
H2 elimination via the stepwise path at low collision energies.
Along the low-energy path, the strongly bound insertion
intermediate (here, 2b) rearranges to H2 elimination products
in a concerted fashion via a multicenter transition state (here,
2MCTS).
be driven toward unity as kQ[He] dominates both kins and kdiss
.
In this regime, because all collision complexes are stabilized,
the bulk kinetics measures the fraction of complexes that have
gained access to the metallacyclopropane (1b) well. With B )
1, model 1 and model 2 predict overall reaction rates that differ
by more than a factor of 3.
Theory has also identified key MCTSs involved in the H2
elimination mechanisms of Fe+ (3d64s1, 6D), Co+ (3d8, 3F) and
2
3
Ni+ (3d9, D) with small alkanes, and of Pt (5d96s1, D) with
methane.24,27,48-51 Apparently, the MCTSs are important when
the metal center lacks sufficient bonding capacity to form
covalent bonds to both hydrogen atoms and to the remaining
alkene or alkyne. For example, in Zr + ethylene, the Zr (4d2-
We can suggest several effects that may contribute to reaction
inefficiency. The first is a small entrance channel barrier, which
provides both an energy cutoff and an angular momentum cutoff
to limit the rate of formation of metallacyclopropane complexes,
as described above. We might expect the entrance channel
barrier height to be similar for ethylene and propylene, so it is
not easy to understand the far greater reactivity of Y +
propylene in this context. The second is a steric effect, meaning
that not all Y + alkene approach angles are energetically
favorable. Presumably the extra methyl group on propylene
presents greater steric difficulty to an incoming Y atom, in
disagreement with the greater reaction efficiency for propylene.
5s2, F) atom has four valence electrons, but they are triplet
3
coupled. Only for Zr in its singlet spin state are the reaction
intermediates and transition states along the stepwise H2
elimination path favorable in energy.3,10 In fact, the excited
singlet spin surface is expected to play an important role in the
behavior of the Zr + ethylene system, competing with primary
CH insertion and concerted H2 elimination on the triplet surface,
as argued recently.10 In the present study, because the Y (4d1-

6666 J. Phys. Chem. A, Vol. 105, No. 27, 2001
Porembski and Weisshaar
The third effect involves the fraction of collisions that find the
attractive potential surface rather than one of the several
repulsive surfaces emanating from Y + alkene reactants. With
VI. Conclusion
The identification of reaction products and determination of
kinetic isotope effects for Y (4d15s2, D) with ethylene and
2
2
one unpaired d electron, the Y (4d15s2, D) reactant atom has
propylene provides much needed experimental data from which
to infer reaction mechanisms. New density functional theory
calculations sharpen the theoretical view of the Y + ethylene
system, including explicit characterization of all reaction
intermediates and transition states along the doublet PES. A
new lowest-energy path involving the concerted elimination of
H2 was identified, consistent with the observation of YC2H2
products.
5-fold spatial degeneracy corresponding to occupancy of the
five different d orbitals. Naively, only one in five collisions
will find the attractive potential for which the good acceptor
orbital dyz is occupied. Nonadiabatic transitions at long range
in the entrance channel may alter this fraction, and, in fact, Y
+ propylene may be more successful at finding the attractive
potential than Y + ethylene. The reason is that the nonadiabatic
process at long range is analogous to a nonradiative transition
between nearly degenerate electronic states. The larger density
of rovibrational states for Y-propylene may enhance transitions
to the more stable potential as the collision proceeds. Finally,
Y + propylene could be more efficient than Y + ethylene
because additional reaction channels involving CC insertion may
contribute, or the presence of the methyl group may significantly
In the absence of an approach barrier, as suggested by theory,
a statistical rate model is able to explain all of the experimental
data with essentially no adjustments of the mPW1PW91
energetics calculated for primary CH bond insertion. A second
model that includes the effects of a rate-determining approach
barrier demands somewhat larger adjustments (+2 to +5 kcal/
mol). Once again, both models indicate that B3LYP overesti-
mates 2TSins by at least 6-9 kcal/mol, comparable to its
performance on other related systems.24-27,10 Therefore, this
study provides additional evidence that mPW1PW91 more
accurately describes the energetics of electronically complex
transition metal-hydrocarbon species. The results are satisfy-
ingly consistent with recent experimental and theoretical studies
2
stabilize TSins, allowing a greater fraction of complexes to
eliminate rather than dissociate.
The two rate models are extreme cases that attribute the
reaction inefficiency either entirely to nonadiabatic and/or steric
effects (model 1) or entirely to a small barrier at 2TSent (model
2). Of course a combination of factors could come into play.
However, we tend to favor model 1 (no approach barrier) over
model 2 in part because it more comfortably fits the experi-
mental data and in part because theory consistently fails to find
such an entrance channel barrier. Although the prevailing view
has been that repulsive long-range forces arise from the
interaction of closed-shell s2 metal atoms with close-shell
alkenes, density functional theory does not find a barrier along
3
of Zr (4d25s2, F) + ethylene.10
Despite the more detailed understanding of transition metal-
alkene chemistry afforded by such work, the issue of room-
temperature reaction inefficiency remains unresolved. Nonadi-
abatic and steric effects may both contribute, and we cannot
rule out the possibility that some or all of the reaction
inefficiency is due to a small approach barrier along the adiabatic
2
the adiabatic entrance channel for Y (4d15s2, D) + ethylene.
2
3
entrance channel for both Y(4d15s2, D) and Zr(4d25s2, F)
reactions with ethylene. If so, density functional theory fails to
capture the long-range energetics. More sophisticated multi-
configurational methods may be required to accurately describe
the subtle transition metal-hydrocarbon long-range potential.
The definitive answer will come from experimental studies of
the Y + ethylene and Zr + ethylene reactions using a variable-
temperature flow reactor or a merged-beam apparatus at very
low collision energies (<1 kcal/mol).
From a theoretical point of view, mPW1PW91 may provide a
better model of such long-range interactions than B3LYP. The
mPW1PW91 functional was specifically designed to correct
deficiencies in the B3LYP functional, which has been shown
to underestimate the attractive interactions of weakly bound
systems.22,23,17 If theory is correct, then we must attribute the
overall Y + ethylene reaction inefficiency primarily to nona-
diabatic and/or steric effects, as described above.
The close agreement between the mPW1PW91 energetics and
the experimental data, as revealed by model 1, further supports
the viability of this hybrid density functional. Increased
confidence in the mPW1PW91 functional in turn strengthens
Finally, additional studies in progress52,53 will help provide
a unified picture of the interactions of neutral transition metals
with small alkenes. Y + propylene has four unique bond
insertion possibilities; perhaps CC bond activation is important
or some of the CH insertion barriers are significantly lower than
in Y + ethylene. In contrast to Y and Zr, ground-state Nb (4d4-
3
our recent analysis of the more complicated Zr(4d25s2, F) +
ethylene reaction.10 In that system, both triplet and singlet spin
surfaces are probably important. RRKM calculations based on
the mPW1PW91 energies (3TSins and 1TSins at -6.2 and -15.3
kcal/mol, respectively) indicate that facile CH bond insertion
on either the triplet or singlet surface is in accord with the
measured isotope effect kH/kD ) 1.04 ( 0.05 and the absence
of stabilized ZrC2H4 or ZrC2D4 complexes at low collision
energies. The competition between dissociation and insertion
on the triplet surface with the unknown triplet-to-singlet
intersystem crossing rate kISC then determines the branching
between the two surfaces, as discussed earlier.10 In contrast to
Y + ethylene, essentially all Zr(ethylene) collision complexes
react since kins dominates, i.e., B ∼ 1. If the mPW1PW91
functional is correct in finding no barrier to the approach of
ground-state Zr with ethylene, then the measured 7% reaction
inefficiency must be attributed entirely to nonadiabatic/steric
effects. Consequently, the factor A becomes 0.07, similar to the
factor necessary to explain the reduced efficiency of Y +
ethylene.
6
5s1, D) has an s1 configuration which is expected to be more
strongly attractive to alkenes. In Nb, the interplay of the low-
lying sextet and quartet spin surfaces will likely be important
in determining the reaction efficiency, analogous to the two
surfaces in play for Zr.
Acknowledgment. Generous support of this research has
come from the National Science Foundation (Grants CHE-
9616724 and NSF-0071458) and the donors of the Petroleum
Research Fund (Grant 33441-AC6). M.P. thanks the UW-
Madison Department of Chemistry for a Martha Weeks graduate
fellowship.
Supporting Information Available: Rotational constants
and vibrational frequencies of all stationary points calculated
at the B3LYP/Stuttgart+6-311++G(d,p) level. This material
is available free of charge via the Internet at http://pubs.acs.org.

2
Reactions of Ground-State Y (4d15s2, D)
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