Article

Isomerization around a C=N double bond and a C=C double bond with a nitrogen atom attached: Thermal and photochemical routes

Department of Physical Chemistry, Farkas Center for Light Induced Processes, The Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel.
Photochemical and Photobiological Sciences (Impact Factor: 2.27). 01/2004; 2(12):1256-63. DOI: 10.1039/B306137J
Source: PubMed

ABSTRACT

The Longuet-Higgins phase change theorem is used to show that, in certain photochemical reactions, a single product is formed via a conical intersection. The cis-trans isomerization around the double bond in the formaldiminium cation and vinylamine are shown to be possible examples. This situation is expected to hold when the reactant can be converted to the product via two distinct elementary ground-state reactions that differ in their phase characteristics. In one, the total electronic wavefunction preserves its phase in the reaction; in the other, the phase is inverted. Under these conditions, a conical intersection necessarily connects the first electronic excited state to the ground state, leading to rapid photochemical isomerization following optical excitation. Detailed quantum chemical calculations support the proposed model. The possibility that a similar mechanism is operative in other systems, among them the rapid photo-induced cis-trans isomerization of longer protonated Schiff bases (the parent chromophores of rhodopsins), is discussed.

Full-text

Available from: Shmuel Zilberg
Isomerization around a C
N double bond and a C
C double bond
with a nitrogen atom attached: thermal and photochemical routes
Shmuel Zilberg and Yehuda Haas
Department of Physical Chemistry and the Farkas Center for Light Induced Processes,
The Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel.
E-mail: yehuda@chem.ch.huji.ac.il
Received 30th May 2003, Accepted 26th June 2003
First published as an Advance Article on the web 15th July 2003
The Longuet-Higgins phase change theorem is used to show that, in certain photochemical reactions, a single
product is formed via a conical intersection. The cistrans isomerization around the double bond in the
formaldiminium cation and vinylamine are shown to be possible examples. This situation is expected to hold when
the reactant can be converted to the product via two distinct elementary ground-state reactions that dier in their
phase characteristics. In one, the total electronic wavefunction preserves its phase in the reaction; in the other, the
phase is inverted. Under these conditions, a conical intersection necessarily connects the rst electronic excited state
to the ground state, leading to rapid photochemical isomerization following optical excitation. Detailed quantum
chemical calculations support the proposed model. The possibility that a similar mechanism is operative in other
systems, among them the rapid photo-induced cistrans isomerization of longer protonated Schi bases (the parent
chromophores of rhodopsins), is discussed.
I Introduction
The photo-induced cistrans isomerization of protonated
Schi bases (PSBs) is one of the fastest known reactions, at
least within the opsin matrix.
1
The isomer is formed in its
ground state within 200 fs (rhodopsin)
2,3
or perhaps 500 fs
(bacteriorhodopsin).
4
In rhodopsins, this isomerization is also
highly selective—a single product is observed, in spite of the
fact that many are potentially possible. Conical intersections
were suggested as funnels eciently connecting the electronic
excited state and the ground state,
5
but their exact nature has
not been explored for a while. More recently, some models
based on quantum chemical calculations have been put for-
ward, suggesting specic structures for the conical intersections
involved in the process.
6,7
In a series of papers, Olivucci and co-workers presented evi-
dence over the last few years for the existence of conical inter-
sections in PSBs of various sizes.
6,8
The model supports the
two-state model proposed many years earlier.
9–14
The compu-
tational search for the conical intersections was generally made
by following the minimum energy path (MEP) from the
Franck–Condon (FC) region on the rst excited state (S
1
) down
to the minimum point on the conical intersection hyper-surface,
and from there on the ground-state (S
0
) surface. The MEP is
determined by using the initial relaxation direction, which is the
local steepest descent direction.
15
The extraordinary selectivity is another intriguing unique
feature of these systems—of several possible reactions, only
isomerization around a single bond takes place in rhodopsins.
Energetic considerations of the PSB moiety appear to be incap-
able of accounting for such extreme selectivity, making it likely
that other factors must be involved. It has been proposed
recently that the intersection’s topography could explain this
phenomenon.
7
It was pointed out that the nature of the conical
intersections of PSBs is quite dierent from that of the struc-
turally similar polyenes. In the latter, several products are
expected, as the conical intersection can lead to several dierent
ground-state minima.
The physical basis for the high selectivity is still not com-
pletely understood. The unique environment created by the
protein has been put forward as the primary reason,
1,16
but no
specic model has been advanced. The possibility that the
properties of the ground-state potential surface of PSBs are of
central importance has not yet been explored in detail. We
hereby wish to propose a mechanism that specically takes into
account the greater electronegativity of the nitrogen atom with
respect to carbon and its tendency to acquire sp
3
hybridization.
The model is based on the Longuet-Higgins phase-change
theorem,
17
which allows the location of conical intersections
by considering properties of the ground-state potential surface
only. The basic idea of the model is demonstrated using two
simple nitrogen-containing molecules, the iminium ion (I) and
vinylamine (II), shown in Scheme 1. For I, a full computational
analysis is presented, while for II, only a partial numerical
characterization of the conical intersection was achieved. The
potential extension of the model to a longer PSB, such as III
(Scheme 1), is qualitatively discussed.
II Locating conical intersections
The use of the Longuet-Higgins phase change theorem
17
for nding conical intersections was recently discussed in
detail.
18–22
Conical intersections are located by looking for
points surrounded by phase-inverting Longuet-Higgins loops.
As a full description can be found in ref. 22, only a brief discus-
sion will be presented here, focussing on the concept of the
phase change in chemical reactions, which is central to this
paper.
Consider a system consisting of two species, R and P, that
dier only by their spin-pairing schemes (we use the term
anchors for their designation,
18,22
). Within the Born–
Oppenheimer approximation, the corresponding electronic
wavefunctions are |R> and |P>, respectively. |R> and |P> are
dierent, but not necessarily orthogonal to each other. At
certain nuclear congurations, Q
R
and Q
P
, respectively, they lie
Scheme 1
1256 Photochem. Photobiol. Sci., 2003, 2, 1256–1263 DOI: 10.1039/b306137j
This journal is © The Royal Society of Chemistry and Owner Societies 2003
Page 1
at local minima on the ground-state potential surface. If
motion along the coordinate connecting the two species (the
reaction coordinate) involves a single local maximum, the
reaction R
P is an elementary one.
The electronic wavefunction of the system along the reaction
coordinate, Q, may be written as the two linear combin-
ations:
2326
k
R
and k
P
are coecients such that k
R
= 1 and k
P
= 0 at Q
R
,
while k
R
= 0 and k
P
= 1 at Q
P
. As the system moves along the
reaction coordinate, k
R
varies smoothly from unity to zero and
k
P
from zero to unity. At a certain point, Q
RP
, along the co-
ordinate, k
R
= k
P
and the potential surfaces of R and P cross. If
the two states interact at this point, which is usually the case, the
degeneracy is lifted:
27,28
two adiabatic potential surfaces are
formed, a ground state and an excited state. This is a standard
quantum mechanical problem;
27,28
the electronic wavefunctions
of these adiabatic states are formed by linear combinations of
the original wavefunctions, one is the in-phase combination,
|R> |P>, and the other the out-of-phase one, |R> |P>. The
electronic energy function E
el
(Q), which is the ground-state
potential surface for nuclear motion, has a local maximum at
Q
RP
.
22,24
As shown elsewhere,
22,29
the in-phase combination is the
ground state if the number of exchanged electron pairs is odd
(3, 5, . . .), while in cases where that number is even (2, 4, . . .
pairs), the out-of-phase combination is the ground state.
(If the total number of the electrons is odd, one of the electron
pairs is occupied by a single electron, but the same rule
applies.) It follows that the electronic wavefunction of the tran-
sition state for the reaction |TS> may be expressed as either the
in-phase or the out-of-phase combinations of |R> and |P>.
Reactions for which the wavefunction |TS> is the in-phase
combination are denoted phase-preserving reactions, and those
for which it is the out-of-phase combination, phase-inverting
reactions.
According to the Longuet-Higgins theorem,
30
a conical inter-
section can be found within a loop that is phase inverting. Reac-
tion coordinates of elementary chemical reactions can be used
to form a loop:
22
The reaction of interest, from a reactant R to
a product P, is one reaction. A third molecule, S, is, in general,
required to form a closed loop by serving as a third anchor; the
complete loop is R
P S R. The loop is phase inverting
if, and only if, one or three of these reactions are phase invert-
ing. A corollary of this idea is that a photochemical reaction
usually involves two products. Starting at R, the return from the
excited state can lead to the formation of the desired product, P,
and another one, S. Indeed, as many photochemical reactions
are known to lead to two or more products, this method was
shown to properly portray them.
1822, 31
This is a correct model
for photochemical reactions going through a conical inter-
section, in the case of covalent molecules, where each of the
three molecules R, P and S is described by a dierent dominant
covalent spin-pairing structure (anchors).
It seems that the application of the three anchors model to
the rhodopsin systems leads us to a contradiction, because the
rhodopsin photo-reaction leads to a single product. We wish to
discuss here the circumstances by which a conical intersection
necessarily leads to a single product.
A possible way to form a phase-inverting loop with only one
product suggests itself if the reaction connecting R and P can
proceed along two dierent reaction coordinates with two dier-
ent transition statesone that is phase inverting and another
that is phase preserving. The electronic wavefunctions of the
two transition states may be written as the respective combin-
ations of the wavefunctions of the reactant and the product
(TS
IP
and TS
OOP
denote the in-phase and out-of-phase transi-
tion states, respectively):
(Q) = k
R
|R> ± k
P
|P> (1)
Recently, this situation was shown to hold in unimolecular
charge transfer reactions in alkane radical cations, in which the
charge moves across the molecular structure.
32
As explained
there, this state of aairs can arise when an even number of
electron pairs are spin re-pairedthe reaction proceeds via a
phase-inverting transition state. If two other electrons that are
not part of the bonds that undergo a change can be added to or
taken away from the spin exchange scheme, a phase-preserving
transition state becomes possible for the same reaction.
Another example can be found in isomerization reactions
around a heteropolar double bond. In these systems, the elec-
tronic wavefunctions of the reactant and the product are
expected to contain considerable contributions from both
covalent (A
1
or A
2
) and polar (B) valence bond (VB) structures
(Fig. 1):
In symmetric cases, c
A
1
= c
A
2
.
In VB language, a polar structure arises when two electrons
that were shared by two atoms are placed in an orbital belong-
ing to just one of them. Therefore, a proper description of the
bonding in this system requires three VB structures. In addition
to the two possible spin-pairing schemes of the four electrons
forming the covalent component of the double bond, a
polar one is added, in which two electrons are placed in a
non-bonding orbital of one of the atoms.
A specic example may serve to illustrate the general idea. In
the case of the formaldiminium cation, the three VB structures
shown in Fig. 1 can be constructed. Due to the higher electro-
negativity of the nitrogen atom, structure B is expected to con-
tribute to the stability of the two isomers, which are depicted in
Fig. 1(b). The VB wavefunctions of the two possible transition
states are:
Two distinct transition states connecting the reactant and
the product can, therefore, exist: one is of a biradical, purely
covalent in nature (TS
OOP
), the other has a predominantly polar
character (TS
IP
). A loop connecting the two isomers via the two
transition states is shown in Fig. 1(c). This phase-inverting loop
is actually constructed of three anchors (the three VB struc-
tures) and encircles a conical intersection. A photochemical
reaction that proceeds via this conical intersection will result in
a single product.
III Method of calculation and computational
details
IIIa Molecular orbitals
In this section, the practical implementation of the method is
described, using the formaldiminium cation as an example.
By analogy with olens, the phase-inverting transition state is
expected to have a perpendicular biradicaloid structure in
which the CN bond is stretched and the CH
2
moiety is rotated
by 90 with respect to the NH
2
(C
2v
symmetry). At rst sight,
therefore, the ZE isomerization reaction is a four-electron
problem,
33
so that a calculation based on these four electrons
|TS
IP
> = |R> |P> (2)
|TS
OOP
> = |R> |P> (3)
Reactant: |R> = c
A
1
|A
1
> c
B
|B> (4)
Product: |P> = c
A
2
|A
2
> c
B
|B> (5)
In-phase combination: |TS
IP
> = |R> |P> = (|A
1
>
|B>) (|A
2
> |B>) = |A
1
> |A
2
> 2|B> (6)
Out-of-phase combination: |TS
OOP
> = |R> |P> =
(|A
1
> |B>) (|A
2
> |B>) = |A
1
> |A
2
> (7)
Photochem. Photobiol. Sci., 2003, 2, 12561263 1257
Page 2
Fig. 1 (a) VB representation of the two covalent structures A
1
and A
2
, and the polar structure B used to construct the electronic wavefunctions for
the cistrans isomerization of the formaldiminium cation. The numbers in curved braces show the spin-pairing schemes. (b) VB representation of the
electronic wavefunctions of cis and trans isomers of the formaldiminium cation. (c) The phase-inverting loop formed by the two reaction routes for
isomerization of the formaldiminium cation. The electronic wavefunction of the phase-inverting transition state is |A
1
> |A
2
> and that of the
phase-preserving transition state is |2B> |A
1
> |A
2
>. The two main coordinates involved in these routes are torsion around the C
N bond CN
stretch and pyramidalization.
leads to acceptable results. A conguration interaction method
is required, as at least two congurations are required to
adequately represent the phase-inverting transition state. We
chose to use the CAS method,
34
implemented in the GAMESS
suite of programs.
35
A 4e/4o active space, however, is not
appropriate for the present problem, as one also needs to con-
sider the second transition state. All electrons and orbitals that
are involved in this reaction mode must be included.
The phase-preserving transition state is obtained when the
two π electrons of the double bond are placed in the third 2p
orbital of the nitrogen atom [Fig. 1(a)]. The carbon atom
becomes positively charged, leading to an inverted dipole
moment (with respect to the stable molecule). The resulting
charge-translocated species is stabilized by rehybridization of
the N atom from sp
2
to sp
3
, forcing it to pyramidalize. This
transition state formally involves two electrons that change
hybridization, and is therefore phase preserving, in contrast
with the biradical state, which is phase inverting. The former
will be referred to as the pyramidalized transition state.
Inclusion of the pyramidalized transition state requires that
all three p orbitals of the nitrogen atom have to be included in
the analysis, as well as the 2s orbital. These orbitals are sym-
metry coupled to the molecular orbital (MO) formed by the
symmetric 1s
H
1
1s
H
2
combination, namely a symmetric σ
NH
2
orbital. Due to the symmetry of the molecule, this MO is
coupled to the σ
CH
2
of the carbon atom. Inclusion of the
symmetric 1s
H
1
1s
H
2
combination also requires the inclusion
of the antisymmetric one (1s
H
1
1s
H
2
), which is coupled to the
in-plane p orbital. All in all, we have six bonding and six anti-
bonding orbitals, leading to a 12e/12o active space. This is in
fact a full conguration interaction (CI) treatment (considering
valence electrons only). One of the virtual σ
valence orbitals
was found to strongly couple to a Rydberg-type orbital. This
introduces complications, as one need to consider more
Rydberg orbitals, and the calculation becomes tedious and
erratic. Thus, this virtual orbital was deleted, and the nal
active space used in the actual calculation was 12e/11o.
It transpires that a proper MOCI solution of the problem
and calculation of the properties of the two transition states
and the associated conical intersection therefore requires an
almost full CI computation.
IIIb Calculation of critical points
The upshot of the analysis in the previous section is that while a
VB description of the reactant and product include substantial
contributions from both the covalent and the polar spin-paired
structures, those of the transition states are purely covalent or
principally polar. The computer analysis starts by looking for
the stationary points in the system, for which the rst derivative
of the potential energy vanishes along a certain coordinate. In
the present case, it is convenient to start by searching for transi-
tion states rather than for the minima, as they turn out to be
simpler VB constructs (they consist of one major VB compon-
ent, while the reactant and product are combinations of two).
The covalent transition state TS
OOP
is found by guessing a
perpendicular structure and optimizing it. A vibrational analy-
sis performed at the critical point, conrms the existence of a
single mode having an imaginary frequency. The system is
allowed to move in the direction of this mode (both ways) to the
next stationary point, leading presumably to the reactant and to
the product. A vibrational analysis indeed conrmed that they
are minima on the potential surface. The optimized out-of-
phase transition state of the formaldiminium cation is denoted
TS
1
. The structure of the second transition state (TS
IP
) is esti-
mated based on its VB propertiesthe nitrogen atom in this
case has sp
3
hybridization. The search for a stationary point
begins by transforming the system so that the nitrogen atom
1258 Photochem. Photobiol. Sci., 2003, 2, 12561263
Page 3
assumes a pyramidal conguration and nding the geometry at
which the rst derivative vanishes. The optimized in-phase tran-
sition state of the formaldiminium cation found in this way is
denoted TS
2
. Once a critical point is found, the search for the
stable structures connected by it in an elementary reaction is
conducted using the same procedure: the Hessian matrix is cal-
culated, a single imaginary frequency is found and the system is
allowed to move in the vector direction of this mode. This leads
to the same two minima as for the other transition state, con-
rming the existence of two dierent, independent transition
states for the same reaction.
IIIc Finding the conical intersection
The method used here is similar to that described in ref. 21. In
the case of the formaldiminium cation, the two transition states
are expected to have one common symmetry elementthe
CH
1
H
2
plane (common C
s
symmetry). The search is helped
by the fact that the electronic wavefunction of the charge-
translocated pyramidal transition state transforms as the totally
symmetric irreducible representation (irrep.) A and that of the
biradical TS as the non-symmetric one, A. The rst vertical
excited states of these species have the opposite symmetry,
respectively, as they can be represented (at the given nuclear
geometry) by the inverse combination (cf. eqn. 1).
The search for the conical intersection starts by placing the
system in the rst excited state of one of the transition states.
Starting, say, with the charge-translocated pyramidal transition
state, the excited state [which is of A symmetry (see section II,
eqn. 1)] is calculated at the FranckCondon geometry. The
system at this point turns out to be non-stationary; it is then
allowed to freely evolve from this state, apart from maintaining
the A symmetry. This leads smoothly to the ground state at the
geometry of the biradical transition state, and the conical inter-
section lies somewhere along this trajectory. Its geometry is
found by calculating at each point along the trajectory the ener-
gies of both the A and the A states; the calculation is stopped
when they are less than 1 kcal mol
1
apart.
IV Results
IVa Isomerization around a C
N bond—the formaldiminium
cation
Guided by the VB model described in section II, the two transi-
tion states were sought at approximately a perpendicular struc-
ture. Both were conrmed to have a single imaginary frequency,
as discussed in section III; the vector displacement of the two
imaginary modes is shown schematically in Fig. 2.
The two transition states were found to lead to the same two
stable isomersboth relate to the Z E conversion. The calcu-
lation was carried out without any symmetry restrictions (C
1
symmetry), but the resulting structures were found to belong
to a common C
s
symmetry, as predicted (see above). TS
1
Fig. 2 The vector displacements of the two transition states calculated
for the formaldiminium cation: (a) TS
1
; (b) TS
2
. The CN bond
stretching, not shown explicitly in the gure, is also part of the vector
displacement.
transforms as the assymmetric A irrep., and is assigned as the
biradical transition state TS
OOP
(eqn. 3 and 7). TS
2
transforms
as the totally symmetric A irrep., and is assigned to the charge-
translocated pyramidal transition state, this is TS
IP
(eqn. 2 and
6). The calculated structures are shown at the top of Fig. 3.
The conical intersection was found as described in section
IIIc. By symmetry, both transition states exist in two proto-
chiral forms (they would be chiral if the substituents were dis-
similar). Therefore, there are in fact two phase-inverting loops
of the type shown in Fig. 1 and, thus, two conical intersections.
For the purposes of this paper, they are equivalent, and we shall
continue to refer to a single representative of each species. The
structure of the conical intersection is shown schematically at
the bottom of Fig. 3, and some of its properties are listed in
Table 1.
As shown in Fig. 1 and 2, the biradical-type reaction co-
ordinate is a combination of the CN bond stretching and the
torsion around it. For the other route, the pyramidalization of
the nitrogen atom must be added to these motions. The CN
bond in TS
2
(the charge-translocated pyramidal transition
state) is longer than in the ground state, but shorter than in TS
1
(the biradical-type transition state). The structure of the con-
ical intersection is seen to be intermediate between those of the
two transition states. The calculated energies of the two transi-
tion states are nearly equal, and the conical intersection lies
slightly above them. The energies of the biradicaloid and
pyramidal transition states are 84.3 and 87.5 kcal mol
1
[CAS
(12/11)/DZV] above the ground state, respectively, while the
conical intersection is found to lie at 89.1 kcal mol
1
.
The dipole moment is pointing from the carbon to the
nitrogen atom in the ground-state molecule, as well as in the
biradical transition state. Its sign is reversed in the pyramidal
transition state. It is noted that while the length of the CN bond
in the conical intersection is comparable to the biradical-like
TS, the dihedral and pyramidal angles are similar to that of the
carbon charge-carrying one. The geometry of the conical
intersection is thus shown to be intermediate between the two
transition states.
IVb Isomerization around a C
C bond in vinylamine
Vinylamine (II) is the simplest amine-substituted ethylene. Its
isomer, acetaldehyde imine, is much more stable and so special
precautions are required to store vinylamine.
36
For our pur-
poses, it serves as a prototype for aminated polyenes, in which
the nitrogen atom plays an important role. This molecule is
used to discuss the properties of two transition states that are
involved in the cistrans isomerization around a C
C double
bond with an adjacent nitrogen atom. In this case, a zwit-
terionic transition state can be formed if the two electrons of
the nitrogen lone pair are allowed to participate in the spin re-
pairing scheme. In this larger molecule, a full CI treatment
(even for valence electrons only) is not feasible. The following
eleven orbitals (ve occupied and six virtual) were chosen and
are described approximately as follows. (The C
s
group sym-
metry elements are added to help visualize the orbitals, as the
molecule in the ground state is nearly of C
s
symmetry.) Occu-
pied orbitals: a pseudo π
CH
2
(a), in the C
NH
2
plane in the
molecule and perpendicular to it in the transition states; a
σ orbital, σ
CC
σ
CH
2
(a), a non-bonding nitrogen orbital
(carrying the two lone-pair electrons in the molecule) (a),
another σ orbital σ
CC
σ
CH
(a) and the π*
C
C
orbital (a).
Virtual orbitals: the π*
C
C
orbital (a), four σ* orbitals, σ*
CH
2
(a), σ
CC
σ
CN
σ
CH
(a) and two σ*
CC
σ*
CH
(a) orbitals,
and a pseudo π*
CN
(a) that mixes with π*
C
C
in the charge-
separated transition state.
Fig. 4 shows the postulated VB structures and the results of
the calculation. Some further details are listed in Table 2. In the
ground state, the four electrons forming the C
C double bond
may be considered as two carbene moieties (Fig. 1).
33
The
Photochem. Photobiol. Sci., 2003, 2, 12561263 1259
Page 4
Table 1 Calculated properties of the formaldimium cation species (CASSF/DZV 12/11)
Species Energy/E
h
(E/kcal mol
1
) Dipole moment/D CN bond length/Å Dihedral angle/
Ground state 94.50258 0.59 1.311 0.0
TS
1
(biradical) 94.36827 (84.3) 2.11 1.455 90.0
TS
2
(ionic) 94.36307 (87.5) 2.47 1.392 70.2
Conical intersection 94.36055 (89.1) 1.445 73.3
Table 2 Calculated properties of the vinylamine species [CASSF/(10/11)/DZV]
Species Energy/E
h
(E/kcal mol
1
) Dipole moment/D CC bond length/Å CN bond length/Å
Ground state 133.14233 1.5 1.353 1.394
TS
3
(biradical) 133.0382 (65.3) 1.6 1.468 1.475
TS
4
(zwitterionic) 133.0441 (61.6) 6.2 1.488 1.314
Fig. 3 The calculated [CAS(12,11)DZV] structures of the iminium cation, the two transition states and the conical intersection. Note that the
nitrogen atom is tetrahedral in TS
2
and the conical intersection.
Fig. 4 The calculated [CAS(10,11)DZV] structures of vinylamine ground state and the two transition states. The top row shows the VB structures
of these species schematically.
biradical transition state (TS
3
), shown on the left, is intermedi-
ate between the cis and trans isomers, as the CH
2
moiety is
twisted by 90. This is thus a four-electron transition state and
is phase inverting (TS
OOP
). On the right, the other transition
state (TS
4
) is shown: one of the lone-pair electrons moves to
the central carbon atom, forming a C
N double bond. Two
electrons are located on the terminal carbon atom, making this
structure a zwitterion. This is a phase-preserving (TS
IP
) six-
electron transition state. The calculated structures of the
ground state and of the two transition states are shown in the
second row of Fig. 4. Both are calculated to be approximately
the same energy above the ground state [65.3 and 61.6 kcal
1260 Photochem. Photobiol. Sci., 2003, 2, 12561263
Page 5
mol
1
for TS
3
and TS
4
, respectively, CAS(10,11)/DZV] level. In
this case, both transition states belong to the C
s
point group,
and transform as two dierent irreps: the electronic wave-
function of TS
3
transforms as the A irrep., and that of TS
4
as
the totally symmetric A irrep. The dipole moment of the zwit-
terionic TS
4
(6.2 D) is much larger than that of the biradicaloid
TS
3
(1.6 D), as expected. The calculated geometries are as pre-
dicted from the VB model: in TS
3
the CN bond length is 1.475
Å, almost as long as a CN single bond, and the CC bond length
(1.468 Å) is close to that of a CC single bond. The terminal
carbon is bonded to two hydrogen atoms. In TS
4
, the CN bond
length (1.314 Å) is as expected for a CN double bond and the
terminal carbon atom is tetrahedral, bonded to two hydrogen
atoms and to the other carbon atom, and carrying an electron
lone pair.
The search for the conical intersection was begun as detailed
in section III. It was found that the two electronic states
approach each other, but since no symmetry element exists in
the system, they are of the same symmetry (point group C
1
).
The two transition states have C
s
symmetry, but the symmetry
planes are dierent, so that the conical intersection is asym-
metric. This frustrates the attempts to optimize the energies of
the two states. The situation arises from the fact that the two
states are too close together. By the Longuet-Higgins theorem,
a degeneracy must exist within the stipulated loop.
V Discussion
Va The formaldiminium iona numerical check of the
Longuet-Higgins theorem
In agreement with the qualitative model, two transition states
are found for the EZ isomerization reaction of the form-
aldiminum cation. The two transition states, which are calcu-
lated to be essentially isoenergetic, have some features in
common, but dier considerably in others. Both have a nearly
perpendicular geometry, and a larger dipole moment than the
stable molecules. The calculated dipole moment is 0.59 D at
the minimum, and 2.11 and 2.47 D at the biradical and
ionic transition states, respectively. The sign reversal of the di-
pole moment found for the ionic transition state indicates the
transfer of the positive charge from the nitrogen to the carbon
atom. The CN bond length increases from 1.311 Å at the mini-
mum to 1.392 and 1.455 Å for the pyramidal and biradical
transition states, respectively. The latter is a value which might
be expected for a single bond.
A conical intersection was calculated to lie within a loop
formed by the two transition states and the two stable isomers.
The data provide a means for directly monitoring the sign of
the electronic wavefunction when transported adiabatically
around a complete loop, discussed in section II using the
VB representation. The molecular orbitals used for the active
space (12e/11o) of CH
2
NH
2
in the geometry of the biradical
transition state were chosen for this purpose. The leading con-
guration of this transition state is an open shell congur-
ation (OSC), {(a)
2
(a)
2
(a)
2
(a)
2
(a)
2
(a)
1
(a)
1
(a)
0
(a)
0
(a)
0
(a)
0
}
(excluding the two core orbitals), and was found to have a co-
ecient of 0.97 (the next largest is 0.07). The singly occupied
orbitals of this conguration are 2p
x
of N (2p orbital per-
pendicular to NH
2
fragment) and 2p
z
of C (2p orbital per-
pendicular to CH
2
fragment), respectively. The calculation
showed that the leading conguration of the optimized struc-
ture of the pyramidalized charge-translocated transition state is
a closed shell conguration in which the six lowest molecular
orbitals are lled. We denote this as CSC, (a)
2
(a)
2
(a)
2
(a)
2
(a)
2
(a)
2
(a)
0
(a)
0
(a)
0
(a)
0
(a)
0
. The coecient of this conguration
is 0.96, the next largest coecient is 0.11.
The two congurations (OSC and CSC) also dominate in the
optimized wavefunctions of the reactant and product, with
almost equal weights. The CSC and OSC coecients in the case
of the reactant are 0.57 and 0.57, respectively, and in the
product, 0.57 and 0.57, respectively (compare with eqn. 2
and 3). We note that an out-of-phase combination of CSC and
OSC congurations represents the reactant and an in-phase
combination represents the product.
Fig. 5 shows a schematic view of the loop, and two represen-
tations of the electronic wavefunction: VB and MOCI
The expected phase change was checked directly using the
results of the MOCI calculation. All wavefunctions are real, so
that the phase change in this case appears as a sign change of
the function. The two leading congurations are used to moni-
tor the signs (all other congurations follow suit). In the wave-
function representing the E isomer, the signs of the two
congurations (OSC and CSC) are equal; we take both to be
positive. The system is taken through TS
1
to the Z isomer, in
which the CSC carries the opposite sign to the OSC. Moving
now through TS
2
, which preserves the sign of the CSC, the
system returns to the E isomer. In the E isomer, the sign of the
OSC must equal that of the CSC, which is negative. Thus,
the signs of both are negative, inverting the sign of the wave-
function upon traversing the complete loop. It may be argued
that the argument fails, as the MOs constituting the congur-
ations may change their signs in the process. A careful check
allowed this possibility to be ruled out. Consequently, in these
systems, irradiation is expected to produce a single product,
unless other conical intersections with a lower energy exist.
Vb Extension to larger systems
The data for vinylamine are less dependable than those for the
formaldiminium cation. Its larger size precluded the inclus-
ion of all required molecular orbitals, and the fact that the
molecule has no symmetry element in the two transition states
frustrated attempts to exactly locate the conical intersection.
Nonetheless, it was shown that two dierent transition states,
one phase preserving, the other phase inverting, exist in this
system. From the general arguments, a conical intersection
necessarily should be located in their vicinity. We chose to
investigate this molecule, in spite of these shortcomings, as it is
demonstrated that the principles outlined for the iminium
cation apply also to a neutral molecule.
Vc Possible application of the model to the photochemistry of
Schi bases and rhodopsins
Taijkhorshid et al.
37
calculated the barriers to the rotation of
dierent dihedrals in a model protonated Schi base using
density functional theory. The calculated barrier to isomeriz-
ation around the C
N double bond is reduced from about
80 kcal mol
1
for the iminium cation to about 25 kcal mol
1
for
the protonated Schi base III. We repeated the calculation,
obtaining similar results. The likely origin of this dramatic
change, which was not specied in ref. 37, is the resonance sta-
bilization of the conjugated bonds in the transition state of III.
According to their calculation, the barrier for rotation around
the C
2
C
3
bond is only slightly higher. The NMR experimental
studies of Sheves and Basov
38
on another protonated Schi
base showed that the isomerization around the C
N bond is
rapid on the NMR time scale at room temperature, and the
isomerization around the C
2
C
3
bond is slightly slower. These
results are consistent with our calculations, as well as those of
ref. 37 for the free PSB.
In photo-excitation of rhodopsins, isomerization around the
C
N bond is not observed, while isomerization around one of
the C
C bonds is strongly dominant. If isomerization around
the C
N bond is made energetically expensive because of inter-
action with the protein, the C
C isomerization channel
becomes the lowest energy route, and with it, the associated
conical intersection. By analogy with the iminium ion and the
vinylamine case, we propose that an ionic transition state also
exists for C
C bond isomerization. Fig. 6 shows a schematic
Photochem. Photobiol. Sci., 2003, 2, 12561263 1261
Page 6
Fig. 5 A schematic view of the Longuet-Higgins loop for the iminium cation, and two representations of the electronic wavefunction: VB and
MOCI. The loop encloses a conical intersection whose geometry is related to the geometry of the transition states (see Fig. 1 and 3).
representation of the two transition states for a larger Schi
base.
In the isolated olens, the polar transition state is likely to be
higher in energy than the biradical one. However, in the pres-
ence of a strong electronegative atom (such as nitrogen) and in
ionic systems, like PSBs, the two routes may be of comparable
energy. Therefore, we suggest that in these systems, a conical
intersection exists that leads from the excited state to a single
ground-state product.
Recently, Zadok et al.
39
showed that light-induced charge
redistribution is required to initiate the bacteriorhodopsin
photo-cycle. This result underscores the importance of ionic
species in the system. The possible role of ionic species was
pointed out many years ago,
5
and its connection to a conical
intersection was suggested. In this work it has been shown that
an ionic transition state is likely to be an important species in
the ground state of PSBs. This might explain the role of some
charged amino acids (such as tryptophane) in making certain
photoisomerization routes preferable over others. For instance,
in bacteriorhodopsin, a tryptophane residue lies very near
(3.44 Å) to the C
C bond that is involved in the isomerization
reaction.
40
It may be part of the ionic environment required for
stabilizing this particular pathway by lowering the energy of the
corresponding conical intersection.
VI Comparison with previous work
Bonacic-Koutecký and co-workers
5
highlighted the importance
of ionic and covalent (dot-dot) structures in the photo-
chemistry of PSBs. They concluded that a conical intersection
is likely to be responsible for the ultrafast rst step in the
vision process. At that time, no specic suggestion was made as
to the structure of the conical intersection. Olivucci and co-
workers
8,15
calculated the structure of conical intersections for
some model systems. They limited the basis set to p electrons
only and found that it was dierent from the conical inter-
section in the analogous polyene systems: no tetra-radicaloid
structure was found. The main motion along the reaction co-
ordinate was torsion around the C
C double bond, accom-
panied by CC skeletal stretching. Martinez et al.
7
utilized a
similar approach.
Fig. 6 The bonding structure of the biradicaloid (left) and C-cationic
(right) transition states proposed for isomerization around a C
C bond
(marked with a curved arrow) in III. In both, the system is stabilized by
charge delocalization. In the C-cationic transition state, the positive
charge is spread over three C
C units. In the biradicaloid transition
state, a cationic allyl-type resonance stabilizes the NCC group and the
second electron is delocalized over three C
C units.
The conical intersection discussed in this work is dened by
two coordinates: one is a combination of torsion and CC
stretch and the second is pyramidalization of the amino group,
which was not considered in the other models. A fundamental
aspect is the use of the Longuet-Higgins phase change theorem
for the study of the topology of the ground-state PES around a
conical intersection. The charge translocation is an essential
component in the characterization of the conical intersection in
the present modelit imparts an ionic character to the carbon
chain in the conical intersection. It is likely that both types of
transition states are present in the larger PSBs. Future experi-
mental work may elucidate which is operative under the various
experimental conditions met in practice.
VII Summary
The main premise of this paper is that in the protonated Schi
base system, the cistrans isomerization can occur along two
distinct reaction coordinates, one of which is phase preserving,
the other phase inverting. Consequently, a conical intersection
connecting the ground state and the S
1
potential surfaces exists,
whose structure is determined by the transition states of these
reaction routes. The coordinates leading to this conical inter-
section include changes in the coordinates of all heavy atoms, in
particular of the nitrogen atom. Model calculations performed
on the formaldiminium cation and vinylamine indicate that the
barrier to cistrans conversion is similar for the two routes,
whose transition states are biradical (the phase-inverting one)
and charge translocated (the phase-preserving one). The
charge-translocated route is important in polar media, such as
the environment created by the protein around the chromo-
phore.
40
In contrast with previous computational studies,
σ electrons (and not just p electrons) must be included in the
active space while searching for the conical intersection. The
geometry change of the nitrogen atom from planar to pyram-
idal is a crucial coordinate. The situation discussed in this
paper, of a conical intersection connected to only two minima
(rather than the usual three) on the ground-state surface, may
turn out to be a frequent occurrence when ionic transition
states are of comparable energy to covalent ones.
Acknowledgements
This research was supported by the Israel Science Foundation
founded by the Israel Academy of Sciences and Humanities
and by the VolkswagenStiftung. The Farkas Center for Light
Induced Processes is supported by the Minerva Gesellschaft
mbH. We thank Mr S. Cogan for many helpful discussions and
two referees for their valuable comments.
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Page 8
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