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: [email protected] 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 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.

I

Introduction

The photo-induced cis–trans isomerization of protonated Schiﬀ 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 eﬃciently 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 forward, suggesting speciﬁc structures for the conical intersections involved in the process.6,7 In a series of papers, Olivucci and co-workers presented evidence over the last few years for the existence of conical intersections in PSBs of various sizes.6,8 The model supports the two-state model proposed many years earlier.9–14 The computational search for the conical intersections was generally made by following the minimum energy path (MEP) from the Franck–Condon (FC) region on the ﬁrst excited state (S1) down to the minimum point on the conical intersection hyper-surface, and from there on the ground-state (S0) 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 incapable 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 diﬀerent from that of the structurally similar polyenes. In the latter, several products are expected, as the conical intersection can lead to several diﬀerent ground-state minima. The physical basis for the high selectivity is still not completely understood. The unique environment created by the protein has been put forward as the primary reason,1,16 but no 1256

speciﬁc 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 speciﬁcally takes into account the greater electronegativity of the nitrogen atom with respect to carbon and its tendency to acquire sp3 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.

Scheme 1

II

Locating conical intersections

The use of the Longuet-Higgins phase change theorem 17 for ﬁnding 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 discussion 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 diﬀer 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 diﬀerent, but not necessarily orthogonal to each other. At certain nuclear conﬁgurations, QR and QP, respectively, they lie

Photochem. Photobiol. Sci., 2003, 2, 1256–1263 This journal is © The Royal Society of Chemistry and Owner Societies 2003

DOI: 10.1039/b306137j

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 P is an elementary one. reaction R The electronic wavefunction of the system along the reaction coordinate, Q, may be written as the two linear combinations: 23–26 (Q) = kR|R> ± kP|P>

(1)

kR and kP are coeﬃcients such that kR = 1 and kP = 0 at QR, while kR = 0 and kP = 1 at QP. As the system moves along the reaction coordinate, kR varies smoothly from unity to zero and kP from zero to unity. At a certain point, QRP, along the coordinate, kR = kP 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 Eel(Q), which is the ground-state potential surface for nuclear motion, has a local maximum at QRP.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 transition 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 intersection can be found within a loop that is phase inverting. Reaction 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 P S R. The loop is phase inverting complete loop is R if, and only if, one or three of these reactions are phase inverting. 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.18–22, 31 This is a correct model for photochemical reactions going through a conical intersection, in the case of covalent molecules, where each of the three molecules R, P and S is described by a diﬀerent 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 diﬀerent reaction coordinates with two diﬀerent transition states—one that is phase inverting and another that is phase preserving. The electronic wavefunctions of the two transition states may be written as the respective combinations of the wavefunctions of the reactant and the product (TSIP and TSOOP denote the in-phase and out-of-phase transition states, respectively):

|TSIP> = |R> ⫹ |P>

(2)

|TSOOP> = |R> ⫺ |P>

(3)

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 aﬀairs can arise when an even number of electron pairs are spin re-paired—the 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 electronic wavefunctions of the reactant and the product are expected to contain considerable contributions from both covalent (A1 or A2) and polar (B) valence bond (VB) structures (Fig. 1): Reactant: |R> = cA |A1> ⫹ cB|B>

(4)

Product: |P> = cA |A2> ⫹ cB|B>

(5)

1

2

In symmetric cases, cA = cA . In VB language, a polar structure arises when two electrons that were shared by two atoms are placed in an orbital belonging 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 speciﬁc 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 electronegativity of the nitrogen atom, structure B is expected to contribute to the stability of the two isomers, which are depicted in Fig. 1(b). The VB wavefunctions of the two possible transition states are: 1

2

In-phase combination: |TSIP> = |R> ⫹ |P> = (|A1> ⫹ |B>) ⫹ (|A2> ⫹ |B>) = |A1> ⫹ |A2> ⫹ 2|B>

(6)

Out-of-phase combination: |TSOOP> = |R> ⫺ |P> = (|A1> ⫹ |B>) ⫺ (|A2> ⫹ |B>) = |A1> ⫺ |A2>

(7)

Two distinct transition states connecting the reactant and the product can, therefore, exist: one is of a biradical, purely covalent in nature (TSOOP), the other has a predominantly polar character (TSIP). 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 structures) 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 oleﬁns, the phase-inverting transition state is expected to have a perpendicular biradicaloid structure in which the CN bond is stretched and the CH2 moiety is rotated by 90⬚ with respect to the NH2 (C2v symmetry). At ﬁrst sight, therefore, the Z–E isomerization reaction is a four-electron problem,33 so that a calculation based on these four electrons Photochem. Photobiol. Sci., 2003, 2, 1256–1263

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Fig. 1 (a) VB representation of the two covalent structures A1 and A2, and the polar structure B used to construct the electronic wavefunctions for the cis–trans 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 |A1> ⫺ |A2> and that of the phase-preserving transition state is |2B> ⫹ |A1> ⫹ |A2>. The two main coordinates involved in these routes are torsion around the C᎐᎐N bond ⫹ CN stretch and pyramidalization.

leads to acceptable results. A conﬁguration interaction method is required, as at least two conﬁgurations 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 consider 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 sp2 to sp3, 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 symmetry coupled to the molecular orbital (MO) formed by the symmetric 1sH ⫹ 1sH combination, namely a symmetric σ⫹NH orbital. Due to the symmetry of the molecule, this MO is coupled to the σ⫹CH of the carbon atom. Inclusion of the symmetric 1sH ⫹ 1sH combination also requires the inclusion of the antisymmetric one (1sH ⫺ 1sH ), which is coupled to the in-plane p orbital. All in all, we have six bonding and six antibonding orbitals, leading to a 12e/12o active space. This is in fact a full conﬁguration 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 1

2

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2

1

2

1

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Rydberg orbitals, and the calculation becomes tedious and erratic. Thus, this virtual orbital was deleted, and the ﬁnal active space used in the actual calculation was 12e/11o. It transpires that a proper MO–CI 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 ﬁrst derivative of the potential energy vanishes along a certain coordinate. In the present case, it is convenient to start by searching for transition states rather than for the minima, as they turn out to be simpler VB constructs (they consist of one major VB component, while the reactant and product are combinations of two). The covalent transition state TSOOP is found by guessing a perpendicular structure and optimizing it. A vibrational analysis performed at the critical point, conﬁrms 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 conﬁrmed that they are minima on the potential surface. The optimized out-ofphase transition state of the formaldiminium cation is denoted TS1. The structure of the second transition state (TSIP) is estimated based on its VB properties—the nitrogen atom in this case has sp3 hybridization. The search for a stationary point begins by transforming the system so that the nitrogen atom

assumes a pyramidal conﬁguration and ﬁnding the geometry at which the ﬁrst derivative vanishes. The optimized in-phase transition state of the formaldiminium cation found in this way is denoted TS2. 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 calculated, 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, conﬁrming the existence of two diﬀerent, independent transition states for the same reaction.

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 element—the CH1H2 plane (common Cs symmetry). The search is helped by the fact that the electronic wavefunction of the chargetranslocated 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 ﬁrst 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 ﬁrst 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 Franck–Condon 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 intersection lies somewhere along this trajectory. Its geometry is found by calculating at each point along the trajectory the energies of both the A⬘ and the A⬙ states; the calculation is stopped when they are less than 1 kcal mol⫺1 apart.

transforms as the assymmetric A⬙ irrep., and is assigned as the biradical transition state TSOOP (eqn. 3 and 7). TS2 transforms as the totally symmetric A⬘ irrep., and is assigned to the chargetranslocated pyramidal transition state, this is TSIP (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 protochiral forms (they would be chiral if the substituents were dissimilar). 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 coordinate 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 TS2 (the charge-translocated pyramidal transition state) is longer than in the ground state, but shorter than in TS1 (the biradical-type transition state). The structure of the conical intersection is seen to be intermediate between those of the two transition states. The calculated energies of the two transition 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.

IV

IVb

IIIc

Finding the conical intersection

Results

IVa Isomerization around a C᎐᎐N bond—the formaldiminium cation Guided by the VB model described in section II, the two transition states were sought at approximately a perpendicular structure. Both were conﬁrmed 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.

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 purposes, 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 cis–trans isomerization around a C᎐᎐C double bond with an adjacent nitrogen atom. In this case, a zwitterionic transition state can be formed if the two electrons of the nitrogen lone pair are allowed to participate in the spin repairing scheme. In this larger molecule, a full CI treatment (even for valence electrons only) is not feasible. The following eleven orbitals (ﬁve occupied and six virtual) were chosen and are described approximately as follows. (The Cs group symmetry elements are added to help visualize the orbitals, as the molecule in the ground state is nearly of Cs symmetry.) Occupied orbitals: a pseudo πCH (a⬘), in the C᎐᎐NH2 plane in the molecule and perpendicular to it in the transition states; a σ orbital, σCC ⫹ σCH (a⬘), a non-bonding nitrogen orbital (carrying the two lone-pair electrons in the molecule) (a⬙), another σ orbital σCC ⫹ σCH (a⬘) and the π*C᎐᎐C orbital (a⬙). Virtual orbitals: the π*C᎐᎐C orbital (a⬙), four σ* orbitals, σ*CH (a⬘), σCC ⫹ σCN ⫹ σCH (a⬘) and two σ*CC ⫹ σ*CH (a⬘) orbitals, and a pseudo π*CN (a⬙) that mixes with π*C᎐᎐C in the chargeseparated 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 2

2

Fig. 2 The vector displacements of the two transition states calculated for the formaldiminium cation: (a) TS1; (b) TS2. The CN bond stretching, not shown explicitly in the ﬁgure, is also part of the vector displacement.

The two transition states were found to lead to the same two stable isomers—both relate to the Z ↔ E conversion. The calculation was carried out without any symmetry restrictions (C1 symmetry), but the resulting structures were found to belong to a common Cs symmetry, as predicted (see above). TS1

2

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Table 1

Table 2

Calculated properties of the formaldimium cation species (CASSF/DZV 12/11) Species

Energy/Eh (∆E/kcal mol⫺1)

Dipole moment/D

CN bond length/Å

Dihedral angle/⬚

Ground state TS1 (biradical) TS2 (ionic) Conical intersection

⫺94.50258 ⫺94.36827 (84.3) ⫺94.36307 (87.5) ⫺94.36055 (89.1)

⫺0.59 ⫺2.11 2.47 —

1.311 1.455 1.392 1.445

0.0 90.0 70.2 73.3

Calculated properties of the vinylamine species [CASSF/(10/11)/DZV] Species

Energy/Eh (∆E/kcal mol⫺1)

Dipole moment/D

CC bond length/Å

CN bond length/Å

Ground state TS3 (biradical) TS4 (zwitterionic)

⫺133.14233 ⫺133.0382 (65.3) ⫺133.0441 (61.6)

1.5 1.6 6.2

1.353 1.468 1.488

1.394 1.475 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 TS2 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 (TS3), shown on the left, is intermediate between the cis and trans isomers, as the CH2 moiety is twisted by 90⬚. This is thus a four-electron transition state and is phase inverting (TSOOP). On the right, the other transition state (TS4) is shown: one of the lone-pair electrons moves to the central carbon atom, forming a C᎐᎐N double bond. Two 1260

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electrons are located on the terminal carbon atom, making this structure a zwitterion. This is a phase-preserving (TSIP) sixelectron 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

mol⫺1 for TS3 and TS4, respectively, CAS(10,11)/DZV] level. In this case, both transition states belong to the Cs point group, and transform as two diﬀerent irreps: the electronic wavefunction of TS3 transforms as the A⬙ irrep., and that of TS4 as the totally symmetric A⬘ irrep. The dipole moment of the zwitterionic TS4 (6.2 D) is much larger than that of the biradicaloid TS3 (1.6 D), as expected. The calculated geometries are as predicted from the VB model: in TS3 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 TS4, 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 C1). The two transition states have Cs symmetry, but the symmetry planes are diﬀerent, so that the conical intersection is asymmetric. 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.

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 conﬁgurations represents the reactant and an in-phase combination represents the product. Fig. 5 shows a schematic view of the loop, and two representations of the electronic wavefunction: VB and MO–CI The expected phase change was checked directly using the results of the MO–CI calculation. All wavefunctions are real, so that the phase change in this case appears as a sign change of the function. The two leading conﬁgurations are used to monitor the signs (all other conﬁgurations follow suit). In the wavefunction representing the E isomer, the signs of the two conﬁgurations (OSC and CSC) are equal; we take both to be positive. The system is taken through TS1 to the Z isomer, in which the CSC carries the opposite sign to the OSC. Moving now through TS2, 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 wavefunction upon traversing the complete loop. It may be argued that the argument fails, as the MOs constituting the conﬁgurations 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.

V

Vb

Discussion

Va The formaldiminium ion—a numerical check of the Longuet-Higgins theorem In agreement with the qualitative model, two transition states are found for the E–Z isomerization reaction of the formaldiminum cation. The two transition states, which are calculated to be essentially isoenergetic, have some features in common, but diﬀer 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 dipole 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 minimum 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 CH2NH2⫹ in the geometry of the biradical transition state were chosen for this purpose. The leading conﬁguration of this transition state is an open shell conﬁguration (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 coeﬃcient of 0.97 (the next largest is 0.07). The singly occupied orbitals of this conﬁguration are 2px of N (2p orbital perpendicular to NH2 fragment) and 2pz of C (2p orbital perpendicular to CH2 fragment), respectively. The calculation showed that the leading conﬁguration of the optimized structure of the pyramidalized charge-translocated transition state is a closed shell conﬁguration in which the six lowest molecular orbitals are ﬁlled. 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 coeﬃcient of this conﬁguration is 0.96, the next largest coeﬃcient is 0.11. The two conﬁgurations (OSC and CSC) also dominate in the optimized wavefunctions of the reactant and product, with almost equal weights. The CSC and OSC coeﬃcients in the case

Extension to larger systems

The data for vinylamine are less dependable than those for the formaldiminium cation. Its larger size precluded the inclusion 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 diﬀerent 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 Schiﬀ bases and rhodopsins Taijkhorshid et al.37 calculated the barriers to the rotation of diﬀerent dihedrals in a model protonated Schiﬀ base using density functional theory. The calculated barrier to isomerization 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 Schiﬀ base III. We repeated the calculation, obtaining similar results. The likely origin of this dramatic change, which was not speciﬁed in ref. 37, is the resonance stabilization of the conjugated bonds in the transition state of III. According to their calculation, the barrier for rotation around the C2–C3 bond is only slightly higher. The NMR experimental studies of Sheves and Basov 38 on another protonated Schiﬀ 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 C2–C3 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 interaction 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, 1256–1263

1261

Fig. 5 A schematic view of the Longuet-Higgins loop for the iminium cation, and two representations of the electronic wavefunction: VB and MO–CI. The loop encloses a conical intersection whose geometry is related to the geometry of the transition states (see Fig. 1 and 3).

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.

representation of the two transition states for a larger Schiﬀ base. In the isolated oleﬁns, the polar transition state is likely to be higher in energy than the biradical one. However, in the presence 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 photochemistry of PSBs. They concluded that a conical intersection is likely to be responsible for the ultrafast ﬁrst step in the vision process. At that time, no speciﬁc suggestion was made as to the structure of the conical intersection. Olivucci and coworkers 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 diﬀerent from the conical intersection in the analogous polyene systems: no tetra-radicaloid structure was found. The main motion along the reaction coordinate was torsion around the C᎐᎐C double bond, accompanied by CC skeletal stretching. Martinez et al.7 utilized a similar approach. 1262

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The conical intersection discussed in this work is deﬁned 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 model—it 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 experimental 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 Schiﬀ base system, the cis–trans 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 S1 potential surfaces exists, whose structure is determined by the transition states of these reaction routes. The coordinates leading to this conical intersection 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 cis–trans 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 chromophore.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 pyramidal 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.

References and notes 1 For reviews on rhodopsins, see: Isr. J. Chem., 1995, 35(3, 4). 2 G. Haran, K. Wynne, X. Aihua, Q. He, M. Chance and R. M. Hochstrasser, Chem. Phys. Lett., 1996, 261, 389.

3 J. E. Kim, D. W. McCamant, L. Zhu and R. A. Mathies, J. Phys. Chem. B., 2001, 105, 1240. 4 J. Herbst, K. Heyne and R. Diller, Science, 2002, 297, 822. 5 (a) J. Michl and V. Bonacic-Koutecky, Electronic Aspects of Organic Photochemistry, Wiley, New York, 1990, ch. 6; (b) V. BonacicKoutecky, J. Koutecky and J. Michl, Angew. Chem.,Int. Ed. Engl., 1987, 26, 170. 6 For a recent summary, see: (a) R. Gonzales Luque, M. Garavelli, F. Bernardi, M. Merchán, M. A. Robb and M. Olivucci, PNAS, 2000, 97, 9379; (b) M. Garavelli, P. Celani, F. Bernardi, M. A. Robb and M. Olivucci, J. Am. Chem. Soc., 1997, 119, 6891; (c) M. Garavelli, T. Vreven, P. Celani, F. Bernardi, M. A. Robb and M. Olivucci, J. Am. Chem. Soc., 1998, 120, 1285; (d ) M. Garavelli, F. Bernardi, M. A. Robb and M. Olivucci, J. Mol. Struct. (THEOCHEM), 1999, 463, 59; (e) L. De Vico, C. S. Page, M. Garavelli, F. Bernardi, R. Basosi and M. Olivucci, J. Am. Chem. Soc., 2002, 124, 4124. 7 M. Ben-Nun, F. Molnar, K. Schulten and T. J. Martinez, Proc. Natl. Acad. Sci. U. S. A., 2002, 99, 1769. 8 D. Sampedro Ruiz, A. Cembran, M. Garavelli, M. Olivucci and W. Fuss, Photochem. Photobiol., 2002, 76, 622. 9 T. Rosenfeld, B. Honig and M. Ottolenghi, Pure Appl. Chem., 1977, 49, 341. 10 J. B. Hurley, T. G. Ebrey, B. Honig and M. Ottolenghi, Nature, 1977, 270, 540. 11 R. M. Weiss and A. Warshel, J. Am. Chem. Soc., 1979, 101, 6131. 12 R. W. Schoenlein, L. A. Peteanu, R. A. Mathies and C. V. Shank, Science, 1991, 254, 412. 13 R. A. Mathies, C.-H. Brito-Cruz, T. Pollard and C. V. Shank, Science, 1988, 240, 777. 14 J. Dobler, W. Zinth, K. Kaiser and D. Oesterhelt, Chem. Phys. Lett., 1988, 144, 215. 15 M. Garavelli, F. Bernardi, P. Celani, M. A. Robb and M. Olivucci, J. Photochem. Photobiol., A, 1998, 114, 109. 16 L. Song, M. A. El-Sayed and J. K. Lanyi, Science, 1993, 261, 891. 17 (a) H. C. Longuet-Higgins, Proc. R. Soc London, Ser. A., 1975, 344, 147; (b) G. Herzberg and H. C. Longuet-Higgins, Discuss. Faraday Soc., 1963, 35, 77. 18 S. Zilberg and Y. Haas, Eur. J. Chem., 1999, 5, 1755. 19 S. Zilberg and Y. Haas, Chem. Phys., 2000, 259, 249.

20 21 22 23 24 25 26 27 28 29 30

31 32 33 34 35

36 37 38 39 40

S. Zilberg and Y. Haas, J. Photochem. Photobiol., 2001, 144, 221. S. Zilberg and Y. Haas, Phys. Chem. Chem. Phys., 2002, 4, 34. Y. Haas and S. Zilberg, Adv. Chem. Phys., 2002, 124, 433. M. Polanyi, Nature, 1943, 151, 96. S. S. Shaik, J. Am. Chem. Soc., 1981, 103, 3692. S. Shaik and P. C. Hiberty, Adv. Quantum Chem., 1995, 26, 99. D. M. Silver, J. Am. Chem. Soc., 1974, 96, 5959. L. D. Landau and E. M. Lifshitz, Quantum Mechanics (NonRelativistic Theory), Pergamon Press, Oxford, 3rd edn., 1977, p. 139. C. Cohen-Tanoudji, B. Diu and F. Laloë, Quantum Mechanics, Wiley, New York, 1977, ch. 4. S. Zilberg and Y. Haas, Int. J. Quant. Chem., 1999, 71, 133. The theorem may be stated as follows. Let S be any simply connected surface in nuclear conﬁguration space, bounded by a closed loop L. Let (q,Q) be the total electronic wavefunction deﬁned by the Born– Oppenheimer approximation, with q and Q the electron and nuclear coordinates, respectively. Then, if (q,Q) changes sign when transported adiabatically round L, there must be at least one point on S at which (q,Q) is discontinuous, implying that its potential energy surface intersects that of another electronic state.’ We deﬁne a loop as phase inverting if (q,Q) changes sign upon being transported around it, and phase preserving if (q,Q) maintains its sign. C. Bornemann and M. Klessinger, Chem. Phys., 2000, 259, 263. (a) S. Zilberg and Y. Haas, J. Phys. Chem., A., 2003, 107, 1222; (b) S. Zilberg and Y. Haas, J. Am. Chem. Soc., 2003, 125, 1810. J. J. C. Mulder, Nouv. J. Chim., 1980, 4, 283. B. O. Roos, Adv. Chem. Phys., 1987, 69, 399. M. W. Schmidt, K. K. Baldrige, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis and J. A. Montgomery, GAMESS version 5, Iowa State University, USA, 1998. O. Mo, M. Yáñez, M. DeCouzon, J.-F. Gal, P.-C. Maria and J.-C. Guillemin, J. Am. Chem. Soc., 1999, 121, 4653. E. Taijkhorshid, B. Paizs and S. Suhai, J. Phys. Chem., 2002, 124, 11 844. M. Sheves and T. Baasov, J. Am. Chem. Soc., 1984, 106, 6840. U. Zadok, A. Khatchatouriants, A. Lewis, M. Ottolenghi and M. Sheves, J. Am. Chem. Soc., 2002, 124, 11 844. J. K. Lanyi, J. Mol. Biol., 2002, 321, 715 and references therein.

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1263

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.

I

Introduction

The photo-induced cis–trans isomerization of protonated Schiﬀ 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 eﬃciently 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 forward, suggesting speciﬁc structures for the conical intersections involved in the process.6,7 In a series of papers, Olivucci and co-workers presented evidence over the last few years for the existence of conical intersections in PSBs of various sizes.6,8 The model supports the two-state model proposed many years earlier.9–14 The computational search for the conical intersections was generally made by following the minimum energy path (MEP) from the Franck–Condon (FC) region on the ﬁrst excited state (S1) down to the minimum point on the conical intersection hyper-surface, and from there on the ground-state (S0) 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 incapable 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 diﬀerent from that of the structurally similar polyenes. In the latter, several products are expected, as the conical intersection can lead to several diﬀerent ground-state minima. The physical basis for the high selectivity is still not completely understood. The unique environment created by the protein has been put forward as the primary reason,1,16 but no 1256

speciﬁc 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 speciﬁcally takes into account the greater electronegativity of the nitrogen atom with respect to carbon and its tendency to acquire sp3 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.

Scheme 1

II

Locating conical intersections

The use of the Longuet-Higgins phase change theorem 17 for ﬁnding 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 discussion 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 diﬀer 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 diﬀerent, but not necessarily orthogonal to each other. At certain nuclear conﬁgurations, QR and QP, respectively, they lie

Photochem. Photobiol. Sci., 2003, 2, 1256–1263 This journal is © The Royal Society of Chemistry and Owner Societies 2003

DOI: 10.1039/b306137j

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 P is an elementary one. reaction R The electronic wavefunction of the system along the reaction coordinate, Q, may be written as the two linear combinations: 23–26 (Q) = kR|R> ± kP|P>

(1)

kR and kP are coeﬃcients such that kR = 1 and kP = 0 at QR, while kR = 0 and kP = 1 at QP. As the system moves along the reaction coordinate, kR varies smoothly from unity to zero and kP from zero to unity. At a certain point, QRP, along the coordinate, kR = kP 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 Eel(Q), which is the ground-state potential surface for nuclear motion, has a local maximum at QRP.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 transition 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 intersection can be found within a loop that is phase inverting. Reaction 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 P S R. The loop is phase inverting complete loop is R if, and only if, one or three of these reactions are phase inverting. 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.18–22, 31 This is a correct model for photochemical reactions going through a conical intersection, in the case of covalent molecules, where each of the three molecules R, P and S is described by a diﬀerent 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 diﬀerent reaction coordinates with two diﬀerent transition states—one that is phase inverting and another that is phase preserving. The electronic wavefunctions of the two transition states may be written as the respective combinations of the wavefunctions of the reactant and the product (TSIP and TSOOP denote the in-phase and out-of-phase transition states, respectively):

|TSIP> = |R> ⫹ |P>

(2)

|TSOOP> = |R> ⫺ |P>

(3)

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 aﬀairs can arise when an even number of electron pairs are spin re-paired—the 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 electronic wavefunctions of the reactant and the product are expected to contain considerable contributions from both covalent (A1 or A2) and polar (B) valence bond (VB) structures (Fig. 1): Reactant: |R> = cA |A1> ⫹ cB|B>

(4)

Product: |P> = cA |A2> ⫹ cB|B>

(5)

1

2

In symmetric cases, cA = cA . In VB language, a polar structure arises when two electrons that were shared by two atoms are placed in an orbital belonging 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 speciﬁc 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 electronegativity of the nitrogen atom, structure B is expected to contribute to the stability of the two isomers, which are depicted in Fig. 1(b). The VB wavefunctions of the two possible transition states are: 1

2

In-phase combination: |TSIP> = |R> ⫹ |P> = (|A1> ⫹ |B>) ⫹ (|A2> ⫹ |B>) = |A1> ⫹ |A2> ⫹ 2|B>

(6)

Out-of-phase combination: |TSOOP> = |R> ⫺ |P> = (|A1> ⫹ |B>) ⫺ (|A2> ⫹ |B>) = |A1> ⫺ |A2>

(7)

Two distinct transition states connecting the reactant and the product can, therefore, exist: one is of a biradical, purely covalent in nature (TSOOP), the other has a predominantly polar character (TSIP). 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 structures) 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 oleﬁns, the phase-inverting transition state is expected to have a perpendicular biradicaloid structure in which the CN bond is stretched and the CH2 moiety is rotated by 90⬚ with respect to the NH2 (C2v symmetry). At ﬁrst sight, therefore, the Z–E isomerization reaction is a four-electron problem,33 so that a calculation based on these four electrons Photochem. Photobiol. Sci., 2003, 2, 1256–1263

1257

Fig. 1 (a) VB representation of the two covalent structures A1 and A2, and the polar structure B used to construct the electronic wavefunctions for the cis–trans 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 |A1> ⫺ |A2> and that of the phase-preserving transition state is |2B> ⫹ |A1> ⫹ |A2>. The two main coordinates involved in these routes are torsion around the C᎐᎐N bond ⫹ CN stretch and pyramidalization.

leads to acceptable results. A conﬁguration interaction method is required, as at least two conﬁgurations 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 consider 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 sp2 to sp3, 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 symmetry coupled to the molecular orbital (MO) formed by the symmetric 1sH ⫹ 1sH combination, namely a symmetric σ⫹NH orbital. Due to the symmetry of the molecule, this MO is coupled to the σ⫹CH of the carbon atom. Inclusion of the symmetric 1sH ⫹ 1sH combination also requires the inclusion of the antisymmetric one (1sH ⫺ 1sH ), which is coupled to the in-plane p orbital. All in all, we have six bonding and six antibonding orbitals, leading to a 12e/12o active space. This is in fact a full conﬁguration 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 1

2

2

2

1

2

1

1258

2

Photochem. Photobiol. Sci., 2003, 2, 1256–1263

Rydberg orbitals, and the calculation becomes tedious and erratic. Thus, this virtual orbital was deleted, and the ﬁnal active space used in the actual calculation was 12e/11o. It transpires that a proper MO–CI 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 ﬁrst derivative of the potential energy vanishes along a certain coordinate. In the present case, it is convenient to start by searching for transition states rather than for the minima, as they turn out to be simpler VB constructs (they consist of one major VB component, while the reactant and product are combinations of two). The covalent transition state TSOOP is found by guessing a perpendicular structure and optimizing it. A vibrational analysis performed at the critical point, conﬁrms 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 conﬁrmed that they are minima on the potential surface. The optimized out-ofphase transition state of the formaldiminium cation is denoted TS1. The structure of the second transition state (TSIP) is estimated based on its VB properties—the nitrogen atom in this case has sp3 hybridization. The search for a stationary point begins by transforming the system so that the nitrogen atom

assumes a pyramidal conﬁguration and ﬁnding the geometry at which the ﬁrst derivative vanishes. The optimized in-phase transition state of the formaldiminium cation found in this way is denoted TS2. 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 calculated, 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, conﬁrming the existence of two diﬀerent, independent transition states for the same reaction.

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 element—the CH1H2 plane (common Cs symmetry). The search is helped by the fact that the electronic wavefunction of the chargetranslocated 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 ﬁrst 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 ﬁrst 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 Franck–Condon 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 intersection lies somewhere along this trajectory. Its geometry is found by calculating at each point along the trajectory the energies of both the A⬘ and the A⬙ states; the calculation is stopped when they are less than 1 kcal mol⫺1 apart.

transforms as the assymmetric A⬙ irrep., and is assigned as the biradical transition state TSOOP (eqn. 3 and 7). TS2 transforms as the totally symmetric A⬘ irrep., and is assigned to the chargetranslocated pyramidal transition state, this is TSIP (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 protochiral forms (they would be chiral if the substituents were dissimilar). 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 coordinate 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 TS2 (the charge-translocated pyramidal transition state) is longer than in the ground state, but shorter than in TS1 (the biradical-type transition state). The structure of the conical intersection is seen to be intermediate between those of the two transition states. The calculated energies of the two transition 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.

IV

IVb

IIIc

Finding the conical intersection

Results

IVa Isomerization around a C᎐᎐N bond—the formaldiminium cation Guided by the VB model described in section II, the two transition states were sought at approximately a perpendicular structure. Both were conﬁrmed 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.

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 purposes, 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 cis–trans isomerization around a C᎐᎐C double bond with an adjacent nitrogen atom. In this case, a zwitterionic transition state can be formed if the two electrons of the nitrogen lone pair are allowed to participate in the spin repairing scheme. In this larger molecule, a full CI treatment (even for valence electrons only) is not feasible. The following eleven orbitals (ﬁve occupied and six virtual) were chosen and are described approximately as follows. (The Cs group symmetry elements are added to help visualize the orbitals, as the molecule in the ground state is nearly of Cs symmetry.) Occupied orbitals: a pseudo πCH (a⬘), in the C᎐᎐NH2 plane in the molecule and perpendicular to it in the transition states; a σ orbital, σCC ⫹ σCH (a⬘), a non-bonding nitrogen orbital (carrying the two lone-pair electrons in the molecule) (a⬙), another σ orbital σCC ⫹ σCH (a⬘) and the π*C᎐᎐C orbital (a⬙). Virtual orbitals: the π*C᎐᎐C orbital (a⬙), four σ* orbitals, σ*CH (a⬘), σCC ⫹ σCN ⫹ σCH (a⬘) and two σ*CC ⫹ σ*CH (a⬘) orbitals, and a pseudo π*CN (a⬙) that mixes with π*C᎐᎐C in the chargeseparated 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 2

2

Fig. 2 The vector displacements of the two transition states calculated for the formaldiminium cation: (a) TS1; (b) TS2. The CN bond stretching, not shown explicitly in the ﬁgure, is also part of the vector displacement.

The two transition states were found to lead to the same two stable isomers—both relate to the Z ↔ E conversion. The calculation was carried out without any symmetry restrictions (C1 symmetry), but the resulting structures were found to belong to a common Cs symmetry, as predicted (see above). TS1

2

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Table 1

Table 2

Calculated properties of the formaldimium cation species (CASSF/DZV 12/11) Species

Energy/Eh (∆E/kcal mol⫺1)

Dipole moment/D

CN bond length/Å

Dihedral angle/⬚

Ground state TS1 (biradical) TS2 (ionic) Conical intersection

⫺94.50258 ⫺94.36827 (84.3) ⫺94.36307 (87.5) ⫺94.36055 (89.1)

⫺0.59 ⫺2.11 2.47 —

1.311 1.455 1.392 1.445

0.0 90.0 70.2 73.3

Calculated properties of the vinylamine species [CASSF/(10/11)/DZV] Species

Energy/Eh (∆E/kcal mol⫺1)

Dipole moment/D

CC bond length/Å

CN bond length/Å

Ground state TS3 (biradical) TS4 (zwitterionic)

⫺133.14233 ⫺133.0382 (65.3) ⫺133.0441 (61.6)

1.5 1.6 6.2

1.353 1.468 1.488

1.394 1.475 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 TS2 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 (TS3), shown on the left, is intermediate between the cis and trans isomers, as the CH2 moiety is twisted by 90⬚. This is thus a four-electron transition state and is phase inverting (TSOOP). On the right, the other transition state (TS4) is shown: one of the lone-pair electrons moves to the central carbon atom, forming a C᎐᎐N double bond. Two 1260

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electrons are located on the terminal carbon atom, making this structure a zwitterion. This is a phase-preserving (TSIP) sixelectron 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

mol⫺1 for TS3 and TS4, respectively, CAS(10,11)/DZV] level. In this case, both transition states belong to the Cs point group, and transform as two diﬀerent irreps: the electronic wavefunction of TS3 transforms as the A⬙ irrep., and that of TS4 as the totally symmetric A⬘ irrep. The dipole moment of the zwitterionic TS4 (6.2 D) is much larger than that of the biradicaloid TS3 (1.6 D), as expected. The calculated geometries are as predicted from the VB model: in TS3 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 TS4, 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 C1). The two transition states have Cs symmetry, but the symmetry planes are diﬀerent, so that the conical intersection is asymmetric. 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.

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 conﬁgurations represents the reactant and an in-phase combination represents the product. Fig. 5 shows a schematic view of the loop, and two representations of the electronic wavefunction: VB and MO–CI The expected phase change was checked directly using the results of the MO–CI calculation. All wavefunctions are real, so that the phase change in this case appears as a sign change of the function. The two leading conﬁgurations are used to monitor the signs (all other conﬁgurations follow suit). In the wavefunction representing the E isomer, the signs of the two conﬁgurations (OSC and CSC) are equal; we take both to be positive. The system is taken through TS1 to the Z isomer, in which the CSC carries the opposite sign to the OSC. Moving now through TS2, 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 wavefunction upon traversing the complete loop. It may be argued that the argument fails, as the MOs constituting the conﬁgurations 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.

V

Vb

Discussion

Va The formaldiminium ion—a numerical check of the Longuet-Higgins theorem In agreement with the qualitative model, two transition states are found for the E–Z isomerization reaction of the formaldiminum cation. The two transition states, which are calculated to be essentially isoenergetic, have some features in common, but diﬀer 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 dipole 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 minimum 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 CH2NH2⫹ in the geometry of the biradical transition state were chosen for this purpose. The leading conﬁguration of this transition state is an open shell conﬁguration (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 coeﬃcient of 0.97 (the next largest is 0.07). The singly occupied orbitals of this conﬁguration are 2px of N (2p orbital perpendicular to NH2 fragment) and 2pz of C (2p orbital perpendicular to CH2 fragment), respectively. The calculation showed that the leading conﬁguration of the optimized structure of the pyramidalized charge-translocated transition state is a closed shell conﬁguration in which the six lowest molecular orbitals are ﬁlled. 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 coeﬃcient of this conﬁguration is 0.96, the next largest coeﬃcient is 0.11. The two conﬁgurations (OSC and CSC) also dominate in the optimized wavefunctions of the reactant and product, with almost equal weights. The CSC and OSC coeﬃcients in the case

Extension to larger systems

The data for vinylamine are less dependable than those for the formaldiminium cation. Its larger size precluded the inclusion 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 diﬀerent 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 Schiﬀ bases and rhodopsins Taijkhorshid et al.37 calculated the barriers to the rotation of diﬀerent dihedrals in a model protonated Schiﬀ base using density functional theory. The calculated barrier to isomerization 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 Schiﬀ base III. We repeated the calculation, obtaining similar results. The likely origin of this dramatic change, which was not speciﬁed in ref. 37, is the resonance stabilization of the conjugated bonds in the transition state of III. According to their calculation, the barrier for rotation around the C2–C3 bond is only slightly higher. The NMR experimental studies of Sheves and Basov 38 on another protonated Schiﬀ 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 C2–C3 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 interaction 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, 1256–1263

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Fig. 5 A schematic view of the Longuet-Higgins loop for the iminium cation, and two representations of the electronic wavefunction: VB and MO–CI. The loop encloses a conical intersection whose geometry is related to the geometry of the transition states (see Fig. 1 and 3).

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.

representation of the two transition states for a larger Schiﬀ base. In the isolated oleﬁns, the polar transition state is likely to be higher in energy than the biradical one. However, in the presence 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 photochemistry of PSBs. They concluded that a conical intersection is likely to be responsible for the ultrafast ﬁrst step in the vision process. At that time, no speciﬁc suggestion was made as to the structure of the conical intersection. Olivucci and coworkers 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 diﬀerent from the conical intersection in the analogous polyene systems: no tetra-radicaloid structure was found. The main motion along the reaction coordinate was torsion around the C᎐᎐C double bond, accompanied by CC skeletal stretching. Martinez et al.7 utilized a similar approach. 1262

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The conical intersection discussed in this work is deﬁned 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 model—it 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 experimental 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 Schiﬀ base system, the cis–trans 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 S1 potential surfaces exists, whose structure is determined by the transition states of these reaction routes. The coordinates leading to this conical intersection 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 cis–trans 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 chromophore.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 pyramidal 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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