Download Fast hydrogen elimination from the †Ru„PH3…3„CO…„H…2‡ and

JOURNAL OF CHEMICAL PHYSICS
VOLUME 121, NUMBER 13
1 OCTOBER 2004
Fast hydrogen elimination from the †Ru„PH3 … 3 „CO…„H… 2 ‡ and †Ru„PH3 … 4 „H… 2 ‡
complexes in the first singlet excited states: A diabatic quantum
dynamics study
Oriol Vendrell, Miquel Moreno, and José M. Lluch
Departament de Quı́mica, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
共Received 10 May 2004; accepted 24 June 2004兲
The photodissociation dynamics of 关 Ru共PH3 ) 3 (CO兲共H) 2 ] and cis- 关 Ru共PH3 ) 4 (H) 2 ] is theoretically
analyzed in the lowest two excited singlet states. Energies obtained through electronic density
functional theory calculations that use the time-dependent formalism are fitted to analytical reduced
two-dimensional potential energy surfaces 共2D-PES兲. The metal-H2 (R) and H-H (r) distances are
the variables of these 2D-PES, the rest of the parameters being kept frozen at the values of the
minimum energy structure in the ground electronic state. The time evolution in these 2D-PES is
exactly followed by means of a fast Fourier transform algorithm applied to solve the time-dependent
Schrödinger equation. A simple diabatization scheme is devised to take into account the probability
of transitions between both excited states. The quantum dynamics results point out that
photoelimination is almost inexistent if the H2 fragment is to be expelled without further
rearrangement of the rest of the complex. Conversely, when the geometries of the complex are
optimized by keeping r and R frozen at the hydrogen elimination barrier coordinates, the new
2D-PES so obtained are highly dissociative, the H2 fragment being expelled in less than 100 fs.
Finally the picture of the whole reaction that emerges from our theoretical results is described
and the main differences between both complexes are examined. © 2004 American Institute of
Physics. 关DOI: 10.1063/1.1783171兴
I. INTRODUCTION
lowered to the picosecond range. However, even then they
were not able to measure the rate of the process, concluding
that the photolysis of the metal-hydrogen bond and the formation of the H-H bond require less than 6 ps 共the instrumental response time兲. A more recent work by the same authors on the related cis- 关 Ru共PMe3 ) 4 (H) 2 ] complex24 also
reveals an ultrafast loss of dihydrogen. However, in this case
the reductive elimination competes with loss of the PMe3
ligand, a parallel process that has a quantum yield ca. 4.5
times that for loss of H2 . This suggests that the ligands attached to the metal may largely affect the dynamics of the
dihydrogen elimination.
From a theoretical point of view, the dynamics of the
photoelimination of molecular hydrogen have been analyzed
by Daniel and co-workers taking the Fe共CO) 4 (H) 2 complex
as a model.25–27 The diabatic potential energy curves of the
ground and the lowest ten singlet and triplet excited states
were modeled by fitting analytical functions to complete active space SCF calculations with multireference contracted
configuration interaction 共CASSCF/CCI兲. Only one dimension, corresponding to the synchronic increase of the two
Fe-H distances, was considered. A rigorous quantum dynamics on such a one-dimensional 共1-D兲 potential energy study
disclosed that dihydrogen elimination was completed in a
time scale of around 40 fs. More recently, some of us28
have studied the photodissociation dynamics of
关 Ru共PH3 ) 3 (CO兲共H) 2 ], a very realistic model of one of the
systems
experimentally
studied
by
Perutz
and
co-workers.22,23 The lowest two singlet excited state energies
were calculated through time-dependent density functional
Transition metal polyhydride complexes represent a fascinating class of compounds both because of their importance in homogeneous catalysis and because of their unusual
properties related to the peculiar chemical properties of the
metal-hydride bond.1 Most of these phenomena cannot be
explained through ‘‘conventional’’ chemical concepts but require the use of quantum mechanics to deal with the motion
of the light hydrogen nuclei.2
Electronic excitation of transition metal organometallic
complexes may lead to ultrafast metal-ligand bond splitting.3
Photochemical bond splitting of many metal dihydride complexes with mutually cis hydride ligands usually leads to
reductive elimination of molecular hydrogen and produces
coordinatively unsaturated Lewis acidic intermediates which
can act as active catalysts or photoinitiators.4 Given the great
stability of the dihydride complex in the ground electronic
state, the easiness to eliminate dihydrogen upon irradiation
also opens a way to safely store and transport molecular
hydrogen.5,6 Even if many experimental works devoted to
such processes can be found in the literature,7–24 only few of
them carry on time-resolved data of these processes. This is
due to the rapidness of such a processes that are usually
faster than the instrumental risetime. Up to now the best
estimates of the time scale of the reductive dihydrogen elimination processes have been reported by Perutz and coworkers in a series of works4,15–24 using an ultrafast laser
equipped with IR detection. In the study of the
关 Ru共PPh3 ) 3 (CO兲共H) 2 ] complex22,23 the detection time was
0021-9606/2004/121(13)/6258/10/$22.00
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© 2004 American Institute of Physics
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J. Chem. Phys., Vol. 121, No. 13, 1 October 2004
FIG. 1. Schematic illustration of the structure of both complexes.
theory 共TDDFT兲 electronic calculations and a bidimensional
potential energy surface was obtained for each state. The two
dimensions were the Ru-H2 and the H-H distances. A quantum dynamics study of each state separately 共i.e., not allowing the jump between one state and the other兲 indicated that
the reaction does not take place in the first singlet excited
state 共which has a high transition probability兲, at least not at
the picosecond time scale as experimentally found. Conversely, the photodissociation is very fast 共less than 60 fs兲 in
the other excited state which has a very small transition
probability.
In this work we undertake a more challenging study of
the photodissociation dynamics of dihydride complexes. In
particular we will reconsider the 关 Ru共PH3 ) 3 (CO兲共H) 2 ] complex and we will compare it with the cis- 关 Ru共PH3 ) 4 (H) 2 ]
system 共see Fig. 1兲.
The latter is also a slight modelization 共the methyl
groups on the phosphines are substituted by hydrogen atoms兲
of another of the systems where Perutz and co-workers have
reported time-resolved data.24 In both cases we will obtain
two-dimensional potential energy surfaces by fitting analytical functions to electronic calculations and will analyze the
nuclear dynamics on them through the use of rigorous quantum mechanical procedures. In order to explain the experimental time scale we will perform simulations that take into
account the probability of transitions between the lowest two
singlet excited states. To this aim we have developed a practical diabatization scheme in terms of only the adiabatic
energies.
This paper is organized as follows: In Sec. II A the quantum chemistry methods used in the present study are described. In Sec. II B the connection between adiabatic and
diabatic representations is outlined and the fast Fourier wave
packet propagation method used to solve the nuclear quantum dynamics is briefly reviewed. In Sec. II C the diabatization scheme used is described. In Sec. III the results are
presented and discussed. Section IV summarizes and gives
some conclusions.
II. THEORY AND CALCULATIONAL DETAILS
A. Electronic calculations
In order to solve the electronic Schrödinger equation the
density functional theory was used. In particular the functional used was the Becke’s three-parameter hybrid method
with the Lee–Yang–Parr correlation functional, a method
widely known as B3LYP.29 An effective core potential was
used to replace the 36 innermost electrons of ruthenium
Fast hydrogen elimination
6259
atom.30,31 The basis set for the metal was that associated with
the potential with a standard double-␨ LANL2DZ contraction. The double-␨ quality 6-31G was used for the rest of the
system.32 Finally, p polarization functions were added to the
two hydride ligands.33 The electronic excited states were calculated within the time-dependent formalism34 –36 using the
same B3LYP functional. Such an approach has been previously used in related studies, where the TDDFT results
proved to be similar to CASSCF/CASPT2 excitation
energies.28,37 In a more recent study the TDDFT method
proved to be comparable to CASPT2 calculations in describing a modelization of the retinal molecule, giving a qualitatively correct description of the intersection region.38 The
method of optimization of Schlegel using redundant internal
coordinates was used to locate the minimum energy structures in the ground electronic state by full geometry
optimization.39 The same method was used when restricted
optimizations were required. All the electronic structure results were obtained through the use of the GAUSSIAN 98
program.40
B. Nuclear quantum dynamics
The time evolution of the system can be followed in the
two low lying singlet excited states in the adiabatic representation. To account for the coupled dynamics in both electronic states the time-dependent Schrödinger equation
共TDSE兲 has to be solved:
iប
d
兩 ␺ 典 ⫽ 共 Tad⫹Vad兲 兩 ␺ 典 ,
dt
共1兲
where the nuclear wave packet is represented as a column
vector:
兩␺典⫽
冉 典典 冊
兩␹1
兩␹2
共2兲
.
兩 ␹ 1 典 and 兩 ␹ 2 典 are the parts of the total nuclear wave packet
on the first and second singlet electronic adiabatic states,
respectively. Tad is a 2⫻2 matrix containing the nuclear kinetic energy operators T̂ and the nonadiabatic coupling elements T̂ i j :
Tad⫽
冉
T̂
T̂ 12
T̂ 21
T̂
冊
,
共3兲
where
n
T̂⫽⫺
兺
k⫽1
1 ⳵2
,
2m k ⳵ q 2k
T̂ i j ⫽ 具 ␰ i 兩 “ ␰ j 典 “⫹ 具 ␰ i 兩 ⵜ 2 ␰ j 典 ,
共4a兲
共4b兲
and n is the number of nuclear degrees of freedom q k . Here
兩 ␰ i 典 is the eigenvector corresponding to the adiabatic electronic state i for the electronic Schrödinger equation:
i
Ĥ el兩 ␰ i 典 ⫽V ad
兩 ␰ i典 .
共5兲
Vad is a 2⫻2 diagonal matrix containing the adiabatic potential energy terms, which are the eigenvalues of the electronic
Schrödinger equation for a given set of the two coordinates:
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6260
J. Chem. Phys., Vol. 121, No. 13, 1 October 2004
Vendrell, Moreno, and Lluch
R 共the M -H2 distance where M refers to the metal complex
except the two hydride atoms兲 and r 共the H-H distance兲:
冉
Vad⫽
1
V ad
0
0
2
V ad
冊
.
共6兲
The adiabatic representation of the coupled nuclearelectronic problem has two main drawbacks. First, the T̂ i j
terms must be known, which implies the tedious differenciation of the electronic wave functions with respect to the
nuclear coordinates. Second, in case the T̂ i j terms were
known, the integration of Eq. 共1兲 may result in numerical
instabilities due to the oscillating character of T̂ i j , which
may become singular at the vicinities of conical intersections. To circumvent these problems it is possible to work in
the diabatic representation where the T̂ i j are forced to vanish.
The adiabatic representation and the diabatic one are connected by a unitary transformation matrix U which must fulfill the following relation:41,42
“U⫹⌳U⫽0,
共7兲
where ⌳ is a matrix of vectors, each one of the dimensionality of the nuclear problem, defined as
⌳⫽
冉
0
⌳ 12
⌳ 21
0
冊
共8兲
with
⌳ i j ⫽ 具 ␰ i兩 “ ␰ j 典 .
共9兲
The nuclear wave packets represented as column vectors are
thus transformed according to
兩 ␺ 典 ⫽U兩 ␾ 典 ,
where
兩␾典⫽
冉 典典 冊
兩␸1
兩␸2
共10兲
共11兲
.
兩 ␸ 1 典 and 兩 ␸ 2 典 are the parts of the total nuclear wave packet
on the first and second singlet electronic diabatic states, respectively. A new potential matrix containing the diabatic
nondiagonal elements (V idij ) and the diabatic potentials (V iidi)
is obtained by unitary transformation of the adiabatic potential energy matrix Vad :
Vdi⫽U⫹ VadU.
共12兲
The Schrödinger equation in the diabatic representation reads
iប
d
兩 ␾ 典 ⫽ 共 Tdi⫹Vdi兲 兩 ␾ 典 ,
dt
共13兲
Tdi⫽
冉 冊
0
0
T̂
共14兲
and the diabatic potential matrix is given by
Vdi⫽
冉
V 11
di
V 12
di
V 12
di
V 22
di
冊
.
具 q 兩 ␾ 典 t⫹⌬t ⫽U
冉
1
e ⫺0.5iV ad⌬t
0
0
e ⫺0.5iV ad⌬t
⫻U⫹ e ⫺iTdi⌬t U
⫻U⫹ 具 q 兩 ␾ 典 t .
共15兲
2
冉
1
冊
e ⫺0.5iV ad⌬t
0
0
e ⫺0.5iV ad⌬t
2
冊
共16兲
The potential and kinetic operators are applied locally, so a
forward FFT is performed on the wave packet after the first
half potential operator has been applied, transforming the
wave packet in the position representation, 具 q 兩 ␾ 典 , to the
momentum representation, 具 p 兩 ␾ 典 . After the momentum operator has been applied locally the wave packet is transformed back to the position representation by a reverse FFT.
In order to follow the evolution of the system upon photoexcitation the two-dimensional potential energy surfaces
共PES兲 were constructed for both S 1 and S 2 adiabatic states.
The S 0 PES was also constructed in order to obtain its lowest
vibrational state. The reduced dimensional PES were constructed performing single-point calculations at different values of the R and r coordinates and keeping the rest of the
coordinates frozen. The R coordinate was scanned for both
metal complexes from 0.5 to 2.5 Å while the r coordinate
was scanned from 0.5 to 3.5 Å. Steps of 0.1 Å were used.
The obtained PES were interpolated by cubic splines and
diabatized following the procedure outlined below. The reduced masses are those associated with each coordinate:
1
1
1
⫽
⫹
,
␮ r m H2 m M
共17a兲
1
1
1
⫽
⫹
.
␮R mH mH
共17b兲
The propagation was performed on a grid of 2 N ⫻2 N points
with N⫽6. A time step ⌬t⫽0.05 atomic time units was used
(1 atu⫽2.419⫻10⫺17 s). A complex linear absorbing potential was used to avoid back reflections of the wave packet at
the edge of the defined grid. The absorbing potential is activated when R is larger than R max and is given by
V⫽⫺b 共 R⫺R max兲 i,
where the kinetic matrix is now given by
T̂
The time-dependent Schrödinger equation in the diabatic
representation 关Eq. 共13兲兴 is solved numerically using a Fourier representation of the Hamiltonian. The Fourier transforms are evaluated using the fast Fourier transform 共FFT兲
algorithm.43,44 The propagation of the wave packet is performed using the split-operator technique.45 The propagation
program basically consists in using the recursive formula:
共18兲
where the slope b was adjusted after a series of trial and error
runs until neither reflections nor side effects caused by the
imaginary potential were observed. The vibrational levels on
S 0 were calculated using the discrete variable representation
sinc-DVR method.46 The initial wave packet was in each
case the corresponding to the lowest vibrational level on the
S 0 potential energy surface. Initially the wave packet was set
on the electronic state having the greater oscillator strength.
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J. Chem. Phys., Vol. 121, No. 13, 1 October 2004
Fast hydrogen elimination
6261
C. Diabatization procedure
An approximation that may be used47,48 when dealing
with avoided crossings in one-dimensional nuclear dynamics
cases is to represent the ⌳ i j terms as Lorentzian functions of
the nuclear coordinate, where the maximum of the Lorentzian kinetic coupling is centered in the avoided crossing region. The matrix Eq. 共7兲 then turns to
⳵
U⫹⌳U⫽0.
⳵q
共19兲
The system of differential equations resulting from Eq. 共19兲
can be easily solved numerically. For instance one may follow the numerical recipe outlined in Ref. 48. That approximation becomes nonpractical when more than one nuclear
dimension are considered. In order to have a single-valued
solution to Eq. 共7兲 the curl condition must hold for each pair
of nuclear coordinates:42
⳵
⳵
⌳p ⫺
⌳ ⫽ 关 ⌳p ,⌳q 兴 .
⳵q
⳵p q
冉冓 冏
The ⌳p matrix is defined as
⌳p ⫽
⳵
␰j
␰
⳵p i
冓冏 冔
␰i
0
冔
⳵
␰
⳵p j
0
共20兲
冊
.
共21兲
Due to the fact that we are dealing with a twodimensional problem, here we have adopted a different diabatization strategy. The approximate procedure we are following is based on the facts that the diabatic coupling is
exactly known at the avoided crossing region and that the
diabatic coupling quickly vanishes as the distance to the
avoided crossing region increases. It is also reasonably assumed that the diabatic potential crossing occurs when the
energy difference between adiabatic potentials is on a minimum. Only the adiabatic potentials are needed in such
scheme. We define a parametric curve c„R p (t),r p (t)… which
follows along the avoided crossing region. R p and r p are
functions of the parameter t defining the curve. The avoided
crossing region follows the one-dimensional seam of the gap
function obtained as the difference between the two adiabatic
potentials:
2
1
G 共 R,r 兲 ⫽V ad
共 R,r 兲 ⫺V ad
共 R,r 兲 .
共22兲
We also define a distance function d c (R,r) as the minimum
distance from any coordinates to the c curve, and the
R c (R,r) and r c (R,r) functions, which return the coordinates
of the point in c closest to a given pair of coordinates R
and r.
The diabatization procedure is easily derived if we work
back from the diabatic representation to the adiabatic one.
The eigenvalues of Vdi , namely, the adiabatic potential values, read
1
V ad
⫽S⫺⌬,
共23a兲
2
⫽S⫹⌬,
V ad
共23b兲
where
FIG. 2. For the Ru共PH3 ) 3 (CO兲共H) 2 complex: Gap function G(R,r) 共a兲,
parametric curve along the avoided crossing seam c„R p (t),r p (t)… 共b兲, and
diabatic potential coupling V di
12 共c兲.
22
S⫽ 12 共 V 11
di ⫹V di 兲 ,
共24a兲
22 2
12 2
⌬⫽ 21 冑共 V 11
di ⫺V di 兲 ⫹4 共 V di 兲 .
共24b兲
At the region where the diabatic potentials cross, the region
22
described by c, it holds by definition that V 11
di ⫽V di , thus
having from Eq. 共24b兲 that
共25兲
⌬⫽V 12
di .
Equations 共22兲, 共23兲, and 共25兲 can be combined to yield the
coupling value at the crossing region
1
V 12
di „R p 共 t 兲 ,r p 共 t 兲 …⫽ 2 G„R p 共 t 兲 ,r p 共 t 兲 ….
共26兲
V 12
di (R,r)
is then apThe nonadiabatic coupling function
proximated for the rest of the coordinates as a Gaussian function of the distance to the crossing region:
2
12
⫺ ␣ 关 d c 共 R,r 兲兴
.
V 12
di 共 R,r 兲 ⫽V di „R c 共 R,r 兲 ,r c 共 R,r 兲 …e
共27兲
12
V di
(R,r)
are depicted in Fig. 2
G(R,r), c„R p (t),r p (t)…, and
for the Ru共PH3 ) 3 (CO兲共H) 2 case. The results are not very
sensitive to the ␣ parameter, the width of the Gaussian function, which has been finally set to a value of 50 Å⫺2. The
following equations may be derived for the diabatic potentials as functions of the adiabatic potentials already known
and the V 12
di calculated from Eq. 共27兲:
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6262
J. Chem. Phys., Vol. 121, No. 13, 1 October 2004
Vendrell, Moreno, and Lluch
TABLE I. Vertical transition energies and oscillator strenghts for the two
metal complexes calculated at the TDDFT level. Transitions are given in
nanometers, and oscillator strenghts are in parenthesis.
Ru共PH3 ) 3 (CO兲共H) 2
Ru共PH3 ) 4 (H) 2
279.65 共0.0217兲
268.79 共0.0027兲
288.02 共0.0002兲
286.36 共0.0261兲
S0 → S1
S0 → S2
1
12
2
2
V 11
di 共 R,r 兲 ⫽S 共 R,r 兲 ⫾ 2 冑G 共 R,r 兲 ⫺4V di 共 R,r 兲 ,
共28a兲
1
12
2
2
V 22
di 共 R,r 兲 ⫽S 共 R,r 兲 ⫿ 2 冑G 共 R,r 兲 ⫺4V di 共 R,r 兲 .
共28b兲
It should be noted that the procedure outlined here is completely local, thus providing the two values for the diagonal
elements of Vdi from Eq. 共28兲 but not their unique assigna22
tion to V 11
di (R,r) and V di (R,r). This assigment is done on
the basis of the regions in the 2D space that are delimited by
the curve c. The ⫾ signs in Eq. 共28兲 will be taken as either
28a⫹, 28b⫺, or 28a⫺, 28b⫹ depending on the region at
which some particular R and r values belong. The U matrix
needed for the TDSE propagation in Eq. 共16兲 is obtained
after the diagonalization of Vdi .
Finally, the theoretical simulation of the absorbing electronic spectra can be obtained through calculation of the autocorrelation function at regular periods of time:
A 共 t 兲 ⫽ 具 ␾ 共 r,R,0兲 兩 ␾ 共 r,R,t 兲 典 ,
共29兲
where the integral extends over the whole range of R and r
coordinates and it is assumed that 兩 ␾ (r,R,0) 典 and
兩 ␾ (r,R,t) 典 are normalized. It is worth noting that Eq. 共29兲 is
only an approximation to the exact procedure that would
instead need the obtention of the time correlation function of
the dipolar moment. It can be demonstrated that the use of
the autocorrelation function instead leads to approximately
correct results provided that the Franck–Condon principle is
obeyed and the temperature is low enough so that the vibrationally excited states are not appreciably populated in the
ground electronic state. The absorption electronic spectrum
in its frequency-dependent form ␴共␯兲 is finally obtained by
numerically calculating the Fourier transform of A(t): 49
冋冕
1
␴ 共 ␯ 兲 ⫽ Re
h
⬁
⫺⬁
册
A 共 t 兲 e 2 ␲ i ␯ t dt .
共30兲
III. RESULTS AND DISCUSSION
A. Electronic calculations
A full geometry optimization leads to octaedral dihydride complexes in both Ru共PH3 ) 3 (CO兲共H) 2 and
Ru共PH3 ) 4 (H) 2 cases. In order to describe the orbitals and
electronic excitations the xy plane will be taken as the plane
containing the metal atom and both hydrides. The y axis is
equidistant to the two hydride ligands. The z axis is then
defined as perperdicular to the xy plane 共see Fig. 1兲. The
ground state minimum energy geometries belong to the C s
and C 2 v point groups for Ru共PH3 ) 3 (CO兲共H) 2 and
Ru共PH3 ) 4 (H) 2 , respectively.
For the Ru共PH3 ) 3 (CO兲共H) 2 complex the vertical transitions from S 0 to S 1 and S 2 and their corresponding oscillator
TABLE II. Dissociation energies for both complexes at the ground and
excited electronic states.
S0
S1
S2
Ru共PH3 ) 3 (CO兲共H) 2
Ru共PH3 ) 4 (H) 2
25.67
⫺26.09
⫺16.42
32.65
⫺25.44
⫺7.94
strengths are given in Table I. As the oscillator strength is
much larger for the S 0 to S 1 transition, it is clear that the S 1
state will be the most accessible upon irradiation. The lowest
energy transition between singlet states arises basically from
a HOMO-1–LUMO contribution. HOMO and LUMO represent highest occupied and lowest unoccupied molecular orbitals. The HOMO-1 orbital is basically the 4d xz orbital of
the Ru atom while the LUMO orbital is basically a combination of 4d x 2 ⫺z 2 and 4d z 2 of Ru atom with a non-negligible
contribution of the 1 ␴ g orbital of the H2 fragment interacting
in an antibonding manner with the 4d x 2 ⫺z 2 of the Ru atom.
The S 2 excited state arises basically from a HOMO-LUMO
transition. The HOMO orbital is fundamentally the 4d xy orbital.
Analogous results for Ru共PH3 ) 4 (H) 2 complex are also
shown in Table I. Now the transition from S 0 to S 1 has an
almost null oscillator strength while the oscillator strength of
the S 0 to S 2 transition is quite high. So, for this complex the
most accessible state upon excitation will be S 2 instead of
S 1 . The excitation from S 0 to S 1 is basically described by a
HOMO-LUMO transition while the S 0 to S 2 excitation is
basically described by a HOMO-1–LUMO transition. The
HOMO-1 orbital is basically the 4d xz orbital of the Ru atom,
the HOMO orbital is basically contributed by the 4d xy orbital, and the LUMO is, as for the other metal complex, a
combination of 4d x 2 ⫺z 2 and 4d z 2 of Ru atom with a contribution of the 1 ␴ g orbital of the H2 fragment interacting in an
antibonding manner.
The main difference between the two metal complexes is
the ordering of the two electronic excitations. In the
Ru共PH3 ) 3 (CO兲共H) 2 the HOMO-1→LUMO bright transition
is the less energetic one, so initially the molecule is excited
to the S 1 state upon irradiation. Conversely in the
Ru共PH3 ) 4 (H) 2 case the HOMO-1→LUMO transition is
more energetic than the dark HOMO→LUMO transition, so
the system will be initially excited to the S 2 state. In terms of
orbitals, the p orbital of the CO ligand lying on the xy plane
stabilizes the HOMO 4d xy orbital of the Ru atom, thus making more energetically favorable the transition from the
HOMO-1 orbital, the one with the major oscillator strength.
Finally, the dissociation energies for both complexes at
the considered electronic states are given in Table II. These
energies correspond to differences between the relaxed products 共dihydrogen ⫹ the 16-electron complex兲 and the original dihydride complex at the geometry corresponding to the
minimum in the ground electronic state. These geometries
are kept frozen in the excited state TDDFT calculations so
that the Franck-Condon dissociation energies are reported for
both S 1 and S 2 . It is to be noted that H2 elimination is an
endothermic process in S 0 but it is clear exothermic in both
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J. Chem. Phys., Vol. 121, No. 13, 1 October 2004
Fast hydrogen elimination
FIG. 3. Diabatic states populations along the first 500 fs of wave packet
propagation. 共a兲 Ru共PH3 ) 3 (CO兲共H) 2 , 兩 ␸ 1 典 is the initially populated state,
共b兲 Ru共PH3 ) 4 (H) 2 , 兩 ␸ 2 典 is the initially populated state.
excited states. Given that some relaxation of the nuclear geometry can follow the electronic excitation the actual dissociation energies in the excited states will be even more negative so that from a static 共i.e., thermodynamic兲 point of view
the photoelimination of H2 is feasible in both excited states
and in both complexes. Themodynamics also tells us that
excitation to S 1 results in the more favorable 共i.e., exothermic兲 H2 elimination. In this state there is no difference between both complexes. Conversely, upon excitation to S 2 the
elimination in complex 1 is more favorable than in 2.
B. Nuclear dynamics
The diabatic TDSE 关Eq. 共13兲兴 was propagated according
to Eq. 共16兲 for both complexes. A first set of propagations for
the two coordinates R and r was performed by keeping the
rest of the metal complexes coordinates at the values of the
equilibrium ground electronic state. The initial conditions for
the propagations were in both cases set to
具 ␸ i 兩 ␸ i 典 ⫽1,
共31a兲
具 ␸ j 兩 ␸ j 典 ⫽0,
共31b兲
where i denotes the electronic state with the higher oscillator
strength. 兩 ␸ i 典 was chosen to be the lowest vibrational state
on the S 0 surface, thus using the Franck–Condon approximation. We adopted the convention of numbering the two
diabatic states in ascending order according to their energy at
the S 0 minimum energy geometry. The quantum dynamics
was propagated in both cases for 500 fs. Calibration propagations were performed in conditions where the wave packet
was not able to reach the absorbing potential zone in order to
estimate the numerical error introduced by the loss of wave
packet norm. This error was estimated, for the chosen propagation conditions, to be of (2.5⫻10⫺2 )% every 1000 propa-
6263
FIG. 4. Diabatic state energies along the minimum energy path leading to
the H2 elimination on the 2D-PES obtained after scanning both R and r
coordinates with the rest of the complex frozen at the values of the S 0
minimum. The energies are represented as a function of the R coordinate at
each point on the path. The vertical arrows indicate the point above the S 0
minimum corresponding to the Franck–Condon vertical excitation and the
initially populated state according to the calculated oscillator strengths. The
vertical dashed lines indicate the R coordinate value for the constrained
geometry optimizations. r is also frozen to its corresponding value during
the energy minimizations. 共a兲 Ru共PH3 ) 3 (CO兲共H) 2 , 共b兲 Ru共PH3 ) 4 (H) 2 .
gation steps, which is equivalent to (2.0⫻10⫺2 )% per fs.
The dependence of electronic state populations on time is
depicted in Fig. 3. The population oscillates in both cases
between the two diabatic states within a period of 400–500
fs. No hydrogen molecule elimination is observed for the
Ru共PH3 ) 3 (CO兲共H) 2 case in accordance to Fig. 4共a兲 where
energy barriers in the two diabatic PES are encountered
along the H2 elimination path.
For the Ru共PH3 ) 4 (H) 2 metal-complex case a small fraction of H2 elimination is observed arising from state V 22
di as
depicted in Fig. 3共b兲. This small portion of the wave packet
has enough energy to go across the energy barrier encoun11
tered on the V 22
di surface after the crossing with the V di surface, again in accordance with the energy profile along the
minimum energy path for this case shown in Fig. 4共b兲. The
already discussed propagations also show how in both cases
the population density may transfer between the two diabatic
states in less than half a picosecond. Then, our results seem
to lead to the conclusion that the ground state equilibrium
geometry is not dissociative towards the formation of molecular hydrogen upon electronic excitation for the first analyzed metal complex, and for the second case the reaction
occurs just in a small fraction within the whole time of
propagation. This is not in accordance to the experimental
observations22,23 suggesting that the hydrogen elimination is
completed in a period of less than 6 ps, the time resolution of
the cited experiments.
At this point it has to be noted that obviously, motions of
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6264
J. Chem. Phys., Vol. 121, No. 13, 1 October 2004
TABLE III. Energy difference between the absolute minimum and the constrained optimization geometry, obtained with the R and r coordinates fixed
at the values corresponding to the H2 elimination barrier on the low lying
excited states 共see Fig. 4兲. Energies are in kcal/mol.
S0
S1
S2
Ru共PH3 ) 3 (CO兲共H) 2
Ru共PH3 ) 4 (H) 2
18.8
⫺7.3
⫺0.3
22.4
⫺4.5
⫺1.5
the complex are not necessarily restricted to the R and r
coordinates. In fact, rearrangements of other degrees of freedom may lead to a decrease in the reaction barrier and so
they can pave the way for the hydrogen elimination. In order
to investigate this point, the geometries of both metal complexes were minimized on the S 0 electronic state by keeping
R and r frozen at the hydrogen elimination barrier coordinates for the two diabatic states. As an approximation, it is
assumed here that the metal-complex rearrangements produced during the H2 abstraction in S 0 will be similar in the
low lying singlet excited states.
The energy difference between the constrained optimization geometry and the geometry corresponding to the S 0
minimum was computed for the S 1 , S 2 , and S 0 states. Results are reported in Table III. The energy difference is positive in S 0 indicating that the H2 elimination is not energetically favorable in such electronic state. The elimination
would imply at least a barrier of around 20 kcal/mol. Conversely, in S 1 the energy differences are negative, so a dissociative channel towards H2 elimination exists for both
complexes. The S 2 state has no appreciable energy difference
between both geometries, indicating that coordinate deformations leading to geometries where the H2 release is favorable can also take place when the system is photoexcited to
S 2 , as it is found for the Ru共PH3 ) 4 (H) 2 complex. The dissociative channel implies several heavy atom coordinates
that are not included in the two-dimensional model based on
R and r coordinates. The main geometry changes occuring in
both complexes imply ligand-metal-ligand angle variations.
The most relevants are given in Table IV.
The vector connecting both initial and final structures in
the minimization process was projected against all the normal mode vectors of the minimum energy structure for all
the atoms except hydrogens. The projected values are found
in Table V.
The modes with small frequencies are mainly breathing
oscillations of the metal complex implying basically ligandmetal-ligand angles. Mode v 42 is the exception to this and
TABLE IV. Most representative geometry changes when the metal-complex
geometries are optimized with coordinates R and r frozen to their values for
the H2 elimination energy barrier in the low lying singlet excited states 共see
Fig. 4兲. Axial refers to the ligand-metal-ligand angle for the axial ligands.
Equatorial refers to the angle formed between the metal and the ligands
lying on the same plane than H2 . Values are given in degrees.
Axial
Equatorial
Ru共PH3 ) 3 (CO兲共H) 2
Ru共PH3 ) 4 (H) 2
153→170
101→112
154→169
100→109
Vendrell, Moreno, and Lluch
TABLE V. Projection of the vector representing the geometry change between the two geometries, the constrained minimization geometry, and the
absolute minimum geometry, on the normal modes at the absolute minimum
energy geometry on S 0 . Only the modes with a projection value bigger than
0.1 are shown.
Ru共PH3 ) 3 (CO兲共H) 2
Ru共PH3 ) 4 (H) 2
Mode
␯ 共cm⫺1兲
vជ • vជ i
Mode
␯ 共cm⫺1兲
vជ • vជ i
v2
v6
v 11
v 42
80.98
200.84
392.44
3443.57
0.28
0.11
0.19
0.21
v2
v7
v 10
83.28
142.70
307.91
0.24
0.15
0.19
corresponds to the C-O stretching mode of the CO ligand,
indicating some degree of change in the equilibrium C-O
distance as the H2 is released.
The R and r coordinates were scanned again, now keeping the geometry of the metal complexes fixed at the structures obtained after the constrained minimization process. R
and r were explored for the same range as before. The potential energy surfaces for the S 1 and S 2 states were diabatized and different sets of propagations were performed on
the obtained diabatic potentials. The new energy profiles are
depicted in Fig. 5. These simulations are not providing the
whole process time scale since the heavy atom motions are
not included in the dynamical model. Anyway they will provide a lower limit to the H2 elimination time and information
on how the H2 molecule is released in each case when the
adequate geometry is reached by the metal complex. At this
point it could be argued about the most important vibrational
FIG. 5. Diabatic state energies along the minimum energy path leading to
the H2 elimination on the 2D-PES obtained after scanning both R and r
coordinates with the rest of the complex frozen at the values obtained after
a constrained minimization process in S 0 共see text兲. The energies are represented as a function of the R coordinate at each point on the path. The
arrows indicate the point above the constrained S 0 minimum for these
metal-complex coordinates. 共a兲 Ru共PH3 ) 3 (CO兲共H) 2 , 共b兲 Ru共PH3 ) 4 (H) 2 .
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J. Chem. Phys., Vol. 121, No. 13, 1 October 2004
FIG. 6. Diabatic state populations along the first 80 fs of wave packet
propagation. The propagations depicted here correspond to the excitation to
V 11
di 共see Fig. 5兲. The densities are integrated for R values smaller than 2.5 Å,
so the total density will become smaller than 1 as part of the wave packet
reaches the PES elimination channel. 共a兲 Ru共PH3 ) 3 (CO兲共H) 2 , 共b兲
Ru共PH3 ) 4 (H) 2 .
modes in the deformation towards H2 elimination being obtained for S 0 where the process is not spontaneous. However,
it should be noted that the vibrational period for an oscillation of 100 cm⫺1 共see Table V兲 is around 330 fs, so that the
rate limiting process in both cases will probably correspond
to metal-complex deformations occurring after the system
has been electronically excited. As discussed above these
deformations are barrierless upon photoexcitation.
For the Ru共PH3 ) 3 (CO兲共H) 2 case on the geometry obtained after constrained optimization, the avoided crossing
regions were found to be away from the regions accessible to
the wave packet, so there is no density transfer between the
two diabatic states. The density as a function of time is depicted in Fig. 6共a兲 with initial conditions set to 具 ␸ 1 兩 ␸ 1 典 ⫽1
and 具 ␸ 2 兩 ␸ 2 典 ⫽0. The reverse case is not depicted since no
reaction is found due to the noticeable energy barrier still
11
found in V 22
di . In these simulations V di maps to S 1 for all
the regions accessible to the wave packet. The H2 elimination is found to proceed extremely fast once the complex
has reached the adequate geometry. Conversely, the
Ru共PH3 ) 4 (H) 2 complex presents an avoided crossing at the
elimination region 关Fig. 5共b兲兴. The H2 elimination proceeds
in an amount of 40% in around 60 fs in this case, while the
same amount of H2 has been eliminated in around 30 fs in
the Ru共PH3 ) 3 (CO兲共H) 2 complex. The simulation results for
the initial conditions 具 ␸ 1 兩 ␸ 1 典 ⫽1 and 具 ␸ 2 兩 ␸ 2 典 ⫽0 are depicted in Fig. 6共b兲. The diabatic state V 11
di maps to S 1 for
values of R smaller than 1.7 Å. Afterwards such diabatic
state maps to S 2 . When the simulation is started on V 22
di no
reaction is found as it was previously obtained for the
Fast hydrogen elimination
6265
Ru共PH3 ) 3 (CO兲共H) 2 case due to the barrier energy found
on V 22
di .
At this point the main differences in the two metal complexes may be discussed. In the Ru共PH3 ) 3 (CO兲共H) 2 case for
the coordinates of the minimum on S 0 , the electronic excitation populates the V 11
di diabatic state, as depicted in Fig.
4共a兲. It is possible a partial density exchange between both
diabatic states in a period of less than half a picosecond 关see
Fig. 3共a兲兴. The dynamics performed on the PES obtained
after the constrained minimization process indicate that the
V 11
di state is highly dissociative when the correct geometry is
reached by the metal complex as seen in Fig. 6共a兲. On the
other hand, the electronic excitation takes the Ru共PH3 ) 4 (H) 2
complex to the V 22
di electronic state, which in this case is
slightly dissociative from the geometry corresponding to the
minimum of S 0 , although hundreds of femtoseconds would
be needed for an appreciable amount of H2 to be released.
There is also density exchange between both diabatic states,
trough to a less extent. However, the initially populated state
V 22
di is not dissociative for the metal-complex geometry obtained after the constrained minimization proces. The H2
elimination process can take place arising only from the V 11
di
state as shown in Fig. 6共b兲, but the yield is clearly lower than
for the other metal complex.
Two facts have been identified that make the H2 elimination from the Ru共PH3 ) 4 (H) 2 the slowest one: First, the
electronic excitation takes the molecule to an electronic state
V 22
di dissociative only to a very few extent, so the wave
packet has to go through an avoided crossing before elimination can take place in V 11
di . This is not true for the other
complex where the initially populated state V 11
di will lead to
the elimination after the proper deformations of the metal
complex. Second, the elimination from the reactive electronic state V 11
di is slower in the Ru共PH3 ) 4 (H) 2 complex than
for the Ru共PH3 ) 3 (CO兲共H) 2 case.
Finally we have obtained a theoretical simulation of the
electronic absorption spectra for both complexes. As explained in the methodological section, for this we only need
to calculate the autocorrelation function that directly comes
out from the time evolution of the nuclear wave packet 关see
Eqs. 共29兲 and 共30兲 in the preceding section兴. For this calculation we have used the propagation in r and R without previous relaxation of the complex as in our calculations the
time scale for the motions of the complex other than R and r
is not explicitly considered. Given that, from the frequency
analysis, the motions of the rest of the complex relevant for
the H2 dissociation are expected to be quite slow 共no larger
than 350 cm⫺1兲 we cannot follow the wave packet evolution
for a long time. This implies a lost in the precision of the
spectra in the frequency domain. The final results of the absorption spectra in the 共usual兲 frequency domain for both
complexes are given in Fig. 7. The total time for the propagation was 320 fs which gives a precision of about 100
cm⫺1. A shorter propagation time did not modify the shape
of the spectra except for the fact that the observed bands
were wider. For the Ru共PH3 ) 3 (CO兲共H) 2 complex 关Fig. 7共a兲兴
there is a major single peak at 29 600 cm⫺1 which probably
corresponds to the 0-0 adiabatic vibrational transition. Another band can be observed at higher energy 共31 300 cm⫺1兲
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6266
J. Chem. Phys., Vol. 121, No. 13, 1 October 2004
FIG. 7. Electronic absorption spectra for Ru共PH3 ) 3 (CO兲共H) 2 共a兲 and
Ru共PH3 ) 3 (H) 2 共b兲.
but it is of very minor intensity. This result reflects the fact
that the initial excitation in this case puts the system near the
minimum of the V 11 excited state which is quite bonded at
the vertical transition point 关see Fig. 4共a兲兴. The spectrum for
the Ru共PH3 ) 4 (H) 2 complex is somehow different as this
mainly consist of two overlapped peaks at 29 400–29 500
cm⫺1. To resolve these two 共or more兲 peaks a higher time
resolution would be needed. In addition, other minor bands
are found in the range 29 800–31 300 cm⫺1. This somewhat
more complex pattern arises from the fact that in this case
the electronic vertical excitation puts the system in the V 22
state which is not bonded 关Fig. 4共b兲兴.
As far as we know these spectra have not been yet reported so that our data may be in the future of great help for
the experimentalists in order to understand the behavior of
the molecular system in the subpicosecond time scale. The
absorption electronic spectra at higher resolution time will
probably have certain differences mainly due to the slow
oscillation modes that start to play its role after the few first
hundreds of femtoseconds. As explained above, these modes
are not dynamically taken into account in our modelization,
where we have explicitly considered only the r and R coordinates. The inclusion of some of that modes in future modelizations will require the parallel devolopment of practical
strategies to obtain diabatic potentials for complex reactive
systems.
IV. CONCLUSIONS
The
photodissociation
dynamics
of
关 Ru共PPh3 ) 3 (CO兲共H) 2 ] 共1兲 and cis- 关 Ru共PH3 ) 4 (H) 2 ] 共2兲
complexes has been theoretically analyzed in the lowest two
excited singlet states by means of electronic calculations for
the excited states performed at a time-dependent DFT level.
Results were fitted to analytical PES that only consider two
Vendrell, Moreno, and Lluch
coordinates: r 共H-H distance兲 and R 共metal-H2 distance兲. The
exact time evolution of the wave packet has been calculated
within a procedure based on the FFT algorithm. The probability of transition between the lowest two singlet excited
states has been taken into account through a simple diabatization procedure that only needs a previous calculation of the
adiabatic potentials. Analysis of the FFT results has revealed
that photoelimination of molecular hydrogen does not take
place noticeably when the PES is constructed in such a way
that, at any given pair of (r,R) values, the geometry of the
rest of the complex is kept fixed at the values of the minimum energy geometry in the ground state 共Franck–Condon
excitation兲. On the other hand, the FFT calculations have
disclosed that in both complexes the electronic state populations oscillate between the two diabatic states within a period
of 400–500 fs.
Conversely, the H2 elimination occurs in less than 100 fs
if the wave packet propagations are performed on the PES
obtained after relaxation of the metal complexes at r and R
values corresponding to the H2 elimination energy barrier.
This value provides a lower limit for the whole H2 elimination process as it previously requires a geometry reorganization of the complex. Analysis of the differences between the
initial and the optimum geometries has revealed that this
relaxation mainly involves low-frequency deformations of
the metal-ligand angles. These modes have periods much
longer than the time needed for the H2 elimination so that the
most likely picture for the elimination process involves slow
motions of the molecular skeleton that eventually lead to a
geometry where the H2 fragment is quickly expelled. Comparing the obtained results between both complexes it is seen
that there are two factors that make the process less likely in
2 as compared with 1: First, the electronic excitation in 2
takes the molecule to an electronic state that, irrespective of
the heavy-atom relaxation, does not noticeably dissociate.
Second, even if the more favorable diabatic state is accessed
upon diabatic crossing, the H2 elimination in 2 is clearly
slower. These results are in accordance with previous experimental evidence collected for complexes fairly similar to 1
and 2.
ACKNOWLEDGMENTS
We are grateful for financial support from the Spanish
‘‘Ministerio de Ciencia y Tecnologı́a’’ and the ‘‘Fondo Europeo de Desarrollo Regional’’ through Project No. BQU200200301, and the use of the computational facilities of the
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