18282
J. Phys. Chem. 1996, 100, 18282-18288
MRCI Calculations of the Ground and Excited State Potential Energy Surfaces of the 2,4-Pentadien-1-iminium Cation J. A. Dobado and M. Nonella* Physikalisch-Chemisches Institut der UniVersita¨ t Zu¨ rich, Winterthurerstr. 190, CH-8057 Zu¨ rich, Switzerland ReceiVed: June 5, 1996; In Final Form: August 23, 1996X
Quantum chemical calculations of the electronic ground and first excited singlet states S0 and S1 of the protonated Schiff base 2,4-pentadien-1-iminium cation (CH2dCHsCHdCHsCHdNH2)+ are presented. To compare different multireference CI approaches that differ in their choice of reference configurations and basis sets, potential energy surfaces with respect to two dihedral angles have been calculated for the states S0 and S1. The study reveals that the characteristic features of the two potential energy surfaces, i.e., the appearance of two minima and four maxima in the case of the S0 surface and of three maxima and two minima in the case of the S1 surface, are correctly predicted at all applied levels of theory. The energies of torsional barriers and of higher energy maxima, however, depend considerably on the applied quantum chemical procedure. Ground state calculations are also compared to the results of Møller-Plesset perturbation theory calculations up to the MP4 level.
1. Introduction Bacteriorhodopsin (BR) is the light-driven proton pump of the purple membrane of Halobacterium salinarium.1-3 The light-absorbing chromophore of BR is an all-trans retinal4,5 that is bound to lysine 216, forming a protonated Schiff base (PSB). After light excitation, a catalytic cycle is induced, which reveals several intermediates with different electronic absorption spectra.6 During this photocycle the chromophore changes its conformation and its protonation state such that one proton per absorbed photon is pumped across the membrane. Two possible mechanisms have been postulated for the primary photochemical process of BR: the first mechanism proposes a single all-trans f 13-cis isomerization7 and a second model proposes an alltrans f 13,14-dicis double isomerization.8-10 The primary step after light absorption is a very fast process, which within the first 200 fs takes place on the electronically first excited state.11-15 This photocycle has been subject to numerous theoretical investigations. Simulations of the photocycle of the retinal Schiff base in the protein have been carried out by means of classical molecular dynamics simulation techniques. Owing to the lack of an appropriate potential energy surface (PES) of the S1 state, isomerizations of the chromophore were either enforced by adjusting torsional barriers and energy minima of the involved dihedral angles16-18 or enforced by using hypothetical PES’s.19 Quantum chemical methods have been applied to the chromophore in order to get energies of scattered points of the ground and electronically first excited states.20-22 Owing to the size of the system, only semiempirical methods could be applied in these calculations. All these studies considered the two dihedral angles τ13 and τ14 of the retinal Schiff base, corresponding to dihedrals τ3 and τ4 in the model molecule depicted in Figure 1, since those coordinates most likely govern the isomerization reactions of the chromophore. Furthermore, rotational barriers of CdN and CdC double bonds of the 2,4pentadien-1-iminium cation have been investigated by using a MNDO/CI method.23 More detailed knowledge of the PES of the electronic states S0 and S1 is certainly essential for more * To whom correspondence should be sent. X Abstract published in AdVance ACS Abstracts, October 1, 1996.
S0022-3654(96)01639-5 CCC: $12.00
Figure 1. Structure of the 2,4-pentadien-1-iminium cation C5H8N+: (a) atom labeling and definition of the dihedrals τ3 and τ4; (b) SCF optimized geometrical parameters (bond lengths in angstroms and bond angles in degrees).
realistic simulations of the primary step and, thus, for a better understanding of the photocycle in general. Single point calculations of the electronic spectrum of a model molecule of 11-cis-retinal containing six conjugated double bonds have been carried out with ab initio multireference configuration interaction (MRCI) methods by Du and Davidson.24 After freezing of core and virtual orbitals and after selection of important configurations by means of second-order perturbation theory, their CI contained ∼158 000 configurations, which could account for about half the correlation energy. The full correlation energy was then extrapolated by means of perturbation theory. PES’s of electronically excited states have been calculated for several molecules in the past. Among them are H2O,25 HONO,26-28 FNO,28,29 ClNO,30 and CH3ONO.31-33 Although in HONO a basis set of ANO34 quality and quantum chemical methods that account for static and dynamic correlation effects35,36 have been applied, in the case of methyl nitrite, a very limited MCSCF expansion and a small basis set of not even double-ζ quality with polarization functions only on carbon and oxygen atoms had been used.31 Nevertheless, this latter PES © 1996 American Chemical Society
2,4-Pentadien-1-iminium Cation allowed prediction of molecular properties such as the shape of the electronic absorption spectrum or the energy distribution of the fragments after photodissociation, which both were in good agreement with experimental data.32,33 The retinal Schiff base is considerably larger than the systems previously investigated. Since the investigation of a multidimensional PES often requires the calculation of the energies of up to several hundred conformations, the determination of an excited state PES for this molecule is not tractable with today’s methods and computer equipments. Calculations on smaller model molecules, however, are believed to reveal valuable details of such a PES, which could help to model PES’s of good quality for the real chromophore. The longer the conjugated system of such a model molecule can be chosen the better the corresponding PES should compare to the PES of the real system. Even in the case of such model molecules, however, restrictions concerning the quality of the applied quantum chemical method and the size of the basis set are unavoidable. In this contribution we present results of a series of different MRCI calculations on the small model molecule 2,4-pentadien1-iminium cation (CH2dCHsCHdCHsCHdNH2)+ depicted in Figure 1. We compare the quality of the different calculations by comparing calculated energies of selected points on a twodimensional PES defined by the two dihedrals τ3 and τ4 to the results obtained with a “reference calculation”, i.e., a calculation that we consider at the limit concerning disk space and CPU time requirements for the purpose of the calculation of a multidimensional PES. The presented results enable us (i) to define the minimal requirements necessary for a sufficiently correct description of the electronic states of interest and (ii) to estimate errors in cases where additional approximations are necessary. 2. Methods of Calculation MRCI calculations with single and double excitations have been carried out with the COLUMBUS37,38 package of programs, using the standard STO-3G, 6-31G, and 6-31G* basis sets39-41 as they are implemented in GAUSSIAN92/DFT.42 The complete geometry optimization of the (CH2dCHsCHd CHsCHdNH2)+ ion in the all-trans conformation was carried out with the GAUSSIAN92/DFT package of programs at the SCF/6-31G* level of theory. This optimized geometry was then used for all following calculations. All geometrical parameters were kept fix except the two torsional angles τ3 and τ4 (see Figure 1), which have been changed stepwise between 0 and 180° in order to generate the grid for the calculation of the PES. Two different grid sizes were used. In a first grid, a step size of 10° was applied, leading to 361 points (levels I and II; see later). In a second grid, the step size was 20°, but the points at 90° were also included in the grid, leading to a total of 121 points (levels III-V). In previous semiempirical calculations of the retinal PSB, it has been demonstrated that in order to correctly predict the electronic absorption spectrum as well as a blue shift of correct size upon addition of a negative charge near the chromophore, at least two reference configurations and single and double excitations from these reference configurations have to be taken into account.43 Important contributions of several configurations have also been found in an ab initio MRCI study of the electronic spectrum of the protonated Schiff base of a model molecule of 11-cis-retinal.24 The selection of appropriate reference configurations is more difficult in the case of the calculation of a PES than it is for a single point calculation. To find the most important configurations, we have carried out single reference configuration
J. Phys. Chem., Vol. 100, No. 46, 1996 18283 interaction calculations with single and double excitations (SDCI) on points distributed over the whole PES. Configurations that contributed 0.1 or more to the CI expansion vector of the S1 state in one of these points were considered as reference configurations. This selection resulted in 10 important reference configurations. This setup resulted in a large number of configuration state functions (CSF’s) between 1.0 and 3.7 million. Thus, such a CI problem is already computationally very demanding. In the case of a longer conjugated system like the retinal PSB, even more configurations might be important. To check how sensitive the shape of the PES depends on the chosen set of reference configurations, we have reduced the number of reference configurations by considering only those configurations that show a coefficient g0.3 at least in one point of the surface. Three such configurations have been found. Their electronic configurations are [...(22)2], [...(22)1(23)1], and [...(22)0(23)2] and, thus, correspond to the determinant of the SCF wave function and to determinants formed after single and double HOMO f LUMO excitations. Five different MRCI levels of theory were employed in our calculations. (a) Level I: MRCI/STO-3G single point energy calculation with 3 reference states and 6 frozen core and 6 frozen virtual molecular orbitals (MO’s) of a total of 38 basis functions. The active space yielded 47 601 CSF’s. (b) Level II: MRCI/6-31G single point energy calculation with 3 reference states and 6 frozen core and 6 frozen virtual MO’s of a total of 70 basis functions. The active space yielded 915 601 CSF’s. (c) Level III: MRCI/6-31G single point energy calculation with 10 reference states and 12 frozen core and 12 frozen virtual MO’s of a total of 70 basis functions. The active space yielded 1 083 929 CSF’s. (d) Level IV: MRCI/6-31G* single point energy calculation with 3 reference states and 6 frozen core and 6 frozen virtual MO’s of a total of 100 basis functions. The active space yielded 2 720 731 CSF’s. (e) Level V: MRCI/6-31G* single point energy calculation with 10 reference states and 12 frozen core and 12 frozen virtual MO’s of a total of 100 basis functions. The active space yielded 3 766 784 CSF’s. Level V is the best level of theory we can presently afford, considering the size of the system and the required number of single point energy calculations necessary to characterize the PES. For the ground state we have also carried out SCF and Møller-Plesset perturbation theory calculations up to the MP4 level. In all these calculations the 6-31G* basis set was applied. Furthermore, to compare our ab initio results to results obtained with a method similar to that used by Orlandi and Schulten20 in calculations of a larger PSB system, we have also carried out semiempirical configuration interaction calculations with the program package MOPAC744 applying the AM1 Hamiltonian. Within an active space containing 4 electrons and 6 MO’s, single and double excitations (AM1/CISD) yielded 255 CSF’s. 3. Results and Discussion 3.1. Potential Energy Surface of the Ground State S0. The geometrical parameters and the dihedral angles τ3 and τ4 for the optimized structure of the 2,4-pentadien-1-iminium cation (C5H8N)+ at the SCF/6-31G* level are depicted in Figure 1. The geometrical parameters agree with previously reported45,46 optimized geometries at the Hartree-Fock level applying the STO-3G, 3-21G, and 4-31G basis sets.
18284 J. Phys. Chem., Vol. 100, No. 46, 1996
Figure 2. (a) Contour plot of the electronic ground state S0; (b) contour plot of the first excited state S1; (c) 3D plot of the electronic ground and first excited states; (d) contour plot of the energy difference between states S1 and S0. Contour levels are between 0 and 90 kcal mol-1 in parts a and b and between 0 and 125 kcal mol-1 in part d, all with a step size of 5 kcal mol-1.
The PES of the ground state S0 as a function of the two dihedral angles τ3 and τ4 calculated at level V of theory is shown in parts a and c of Figure 2 as a 2D contour plot and a 3D plot, respectively. In Table 1 relative energies of characteristic points on the ground state surface calculated at different levels of theory are listed. This table also contains results from semiempirical calculations, from Hartree-Fock calculations, and from MøllerPlesset perturbation theory calculations up to the MP4 level. All levels of theory predict a similar shape of the ground state PES, which exhibits the following characteristic stationary points. Four saddle points are found on the PES that all correspond to transition states of isomerization reactions. The points τ3 ) 90°, τ4 ) 180° and τ3 ) 90°, τ4 ) 0°, denoted in the following as (90°, 180°) and (90°, 0°), correspond to the transition states for the isomerization reactions of all-trans f 3-cis and (3-trans, 4-cis) f 3,4-dicis conformations, respectively. Points (180°, 90°) and (0°, 90°) correspond to the barriers of the all-trans f 4-cis and (3-cis, 4-trans) f 3,4-dicis isomerizations, respectively. Furthermore, two additional energy maxima are found at points (90°, 90°), which corresponds to the global energy maximum, and (0°, 0°). The latter conformation corresponds to the 3,4-dicis geometry, which, due to sterical interactions, does not correspond to an energy minimum conformation. Four energy minima were found on the ground state PES: three of them correspond to the planar (3-cis, 4-trans), (3-trans, 4-cis), and all-trans conformations, and the fourth minimum corresponds to a twisted structure in the proximity of the 3,4dicis conformation. Our calculation at level V (see Table 1) reveals torsional barriers for all-trans f 3-cis and all-trans f 4-cis isomerizations of 59.6 and 19.9 kcal mol-1, respectively. These barriers are between those expected for real double and single bonds in the corresponding unprotonated Schiff base model molecule that we have determined at the MP2 level to be 100-110 and 8-10
Dobado and Nonella kcal mol-1, respectively. Similar trends are also shown by the previous semiempirical calculations of torsional barriers of the protonated retinal Schiff base.22 It is apparent from Table 1 that the semiempirical calculations predict significantly lower barriers than those from all ab initio calculations. Inclusion of a very limited CI within MOPAC occasionally tends to worsen the result. Among all ab initio calculations a remarkable agreement of relative energies is found. Compared to the calculation at level V, the SCF calculation overestimates the energies of points (90°, 90°) and (0°, 0°) by approximately 13 and 6 kcal mol-1, respectively. All other deviations are smaller than 3 kcal mol-1. Taking correlation effects into account by means of perturbation theory methods decreases the energy of point (0°, 0°) by ∼8 kcal mol-1 and brings it closer to the corresponding energy predicted at level V. Perturbation theory methods, however, do not lower the energy of point (90°, 90°). The agreement among all perturbation theory calculations is very good. Also, the relative energies of all planar conformations and of all barrier heights for single rotations agree within less than 3 kcal mol-1 with those calculated at level V. A considerable deviation between perturbation theory calculations and the calculation at level V is only found in the case of point (90°, 90°). The energies predicted by perturbation theory methods are more than 10 kcal mol-1 higher than that predicted at level V. To test the quality of the chosen 6-31G* basis set, we have additionally calculated a ground state PES with a larger 6-311G(2d,p) basis set at the Hartree-Fock level (see Table 1). No rotational barrier is affected by more than 0.6 kcal mol-1. Thus, the 6-31G* basis set is most likely sufficient for the description of the electronic ground state. We now discuss the results of the various MRCI calculations. The smallest MRCI calculation that makes use of the STO-3G basis set and takes into account only three reference configurations (level I) can qualitatively correctly predict the shape of the S0 PES. This level, however, cannot accurately predict the torsional barriers for a rotation around double bond C3dC4, i.e., for the isomerization reactions all-trans f 3-cis and (3trans, 4-cis) f 3,4-dicis, which are both overestimated by ∼14 kcal mol-1. Both barriers are already better described at the SCF level. The energy of point (90°, 90°) is underestimated by ∼5.5 kcal mol-1 at level I and overestimated by ∼13 kcal mol-1 at the SCF level. Replacing the STO-3G basis set by the larger 6-31G basis and keeping the reduced set of three reference configurations (level II) slightly increases the energy of point (90°, 90°) by ∼3 kcal mol-1 and significantly lowers both torsional barriers for rotation around the C3dC4 double bond by ∼10 kcal mol-1. The inaccurate description of the torsional barriers for rotations around C3dC4 at level I, therefore, is most likely caused by the small STO-3G basis set. Besides these correcting effects, we also find a deterioration in the case of point (0°, 0°) whose energy rises by 7 kcal mol-1. At level III the same 6-31G basis set as at level II is applied, but 10 reference configurations are now taken into account. Four points are predominantly affected by this larger CI. The energy of the 3,4-dicis conformation (0°, 0°) decreases dramatically by more than 11 kcal mol-1 and agrees well with the corresponding value determined at level V. Furthermore, all transition states and higher energy maxima involved in rotations around the C4-C5 single bond (points (90°, 90°), (180°, 90°), and (0°, 90°)) increase by ∼3 kcal mol-1 in energy. Addition of polarization functions to carbon and nitrogen at level II reveals level IV. Similar trends in energy changes as they have been found for the transition from STO-3G to 6-31G
2,4-Pentadien-1-iminium Cation
J. Phys. Chem., Vol. 100, No. 46, 1996 18285
TABLE 1: Relative Energy of C5H8N+ (kcal mol-1) for Characteristic Points of the Ground State S0 PES at Different Levels of Theory multireference (τ3, τ4) (180°, 180°)a (90°, 90°) (0°, 0°) (180°, 0°) (0°, 180°) (90°, 0°) (90°, 180°) (180°, 90°) (0°, 90°) a
semiempirical AM1 AM1/CISD 0 68.4 48.0 4.3 2.1 41.3 38.2 9.4 9.6
0 48.3 48.8 2.8 3.0 35.5 33.5 10.2 10.1
6-31G*
SCF 6-311G(2d,p)
0 109.5 54.8 7.5 5.4 59.5 57.1 20.1 21.6
0 109.4 57.0 7.8 5.6 60.1 57.5 19.8 21.5
Møller-Plesset MP2 MP3 MP4
level I
level II
level III
level IV
level V
0 110.7 47.1 6.6 4.9 62.2 59.8 18.4 19.6
0 91.2 50.1 4.6 3.4 76.5 74.5 19.9 20.9
0 94.2 57.1 7.2 5.0 66.7 65.1 21.6 22.8
0 97.5 45.6 5.2 2.3 68.3 65.4 25.2 25.3
0 96.4 52.1 7.0 5.4 64.4 61.9 21.3 22.7
0 96.7 48.9 6.6 4.0 62.2 59.6 19.9 21.4
0 110.0 47.7 6.6 4.7 62.1 59.6 17.3 18.6
0 108.9 47.0 6.5 4.7 62.0 59.4 17.5 18.8
Reference energy.
TABLE 2: Relative Energy of C5H8N+ (kcal mol-1) for Characteristic Points of the Excited State S1 PES at Different Levels of Theory (τ3, τ4) 180°)a
(180°, (90°, 90°) (0°, 0°) (180°, 0°) (0°, 180°) (90°, 0°) (90°, 180°) (180°, 90°) (0°, 90°)
AM1/ CISD
level I
level II
level III
level IV
level V
0 (591) 0.0 50.1 1.3 3.2 -3.4 -1.6 18.0 19.0
0 (224) 29.1 51.7 0.9 1.9 -47.3 -44.1 6.2 5.2
0 (242) 4.8 54.7 -0.4 1.6 -40.5 -37.9 0.6 -2.1
0 (253) 11.4 42.6 -2.3 0.0 -32.3 -30.6 7.4 6.6
0 (235) -4.5 49.5 -0.5 2.0 -40.7 -38.7 0.8 -1.9
0 (248) 2.0 45.9 -0.5 -1.4 -38.3 -36.3 -2.3 -1.8
a
Reference energy. Excitation energies in nm are given in parentheses.
are also found in this case. The energy of point (90°, 90°) rises by ∼2 kcal mol-1, while point (0°, 0°) and the barriers for rotation around double bond C3dC4 decrease in energy. A generally good agreement between the results of levels IV and V is found. The largest difference is found in the case of point (0°, 0°) whose energy decreases by 3.2 kcal mol-1 to 48.9 kcal mol-1 upon increase of the number of reference configurations. A generally good agreement is also found between level V and perturbation theory calculations, confirming that level V allows an accurate prediction of the PES of the electronic ground state of the 2,4-pentadien-1-iminium cation even though the reference configurations have been solely selected according to their contribution to the wave function of the S1 state. We can compare our results with recent ab initio HartreeFock calculations of rotational barriers of a protonated Schiff base containing two conjugated double bonds.47 After constrained minimization of all structures along the isomerization reaction, a barrier of ∼10 kcal mol-1 was calculated for a rotation around the C-C single bond. We have applied our approach for the determination of the corresponding barrier in the molecule used in ref 47 and have calculated barriers of 13.0 kcal mol-1 (nonoptimized transition state) and 11.4 kcal mol-1 (optimized transition state) at the SCF/6-31G* level and, thus, find similar results such as in this previous study. It has previously been noted that with the applied semiempirical method no accurate torsional barriers could be calculated. Scattered points on the S0 and S1 surfaces of a protonated Schiff base model molecule containing five conjugated double bonds have been calculated previously using a semiempirical MINDO3/ CI method.20 By use of this method, torsional barriers for the points (90°, 180°), (180°, 90°), and (90°, 90°) were predicted to be 21.1, 11.4, and 36.6 kcal mol-1, respectively (Table 3). To check for the effect of a longer conjugated system on these torsional barriers, we have carried out MP2 calculations on systems with four and five conjugated double bonds. The results
TABLE 3: Torsional Barriers (kcal mol-1) for Ground State S0 Isomerizations All-Trans f 3-Cis, All-Trans f 4-Cis, and All-Trans f 3,4-Dicis as a Function of the Number of Conjugated Double Bonds MINDO3/CIa
MP2/6-31G*
number of double bonds (τ3, τ4)
3
4
5
5
(90°, 180°) (180°, 90°) (90°, 90°)
59.8 18.4 110.7
48.0 22.8 107.4
41.2 25.9 103.3
21.1 11.4 36.6
a
Reference 20.
of these preliminary calculations are presented in Table 3. The following trends are obvious from these data. The barrier for an all-trans f 13-cis isomerization is significantly reduced when the conjugated system becomes longer, while that for an alltrans f 14-cis isomerization is slightly raised. Only a slight decrease in energy is found for point (90°, 90°). The semiempirical calculation in ref 20 clearly underestimates all these torsional barriers. A similar result is found in a semiempirical multireference MNDO/CI study of the 2,4-pentadien-1-iminium cation,23 which predicts a barrier of approximately 30 kcal mol-1 for a rotation around τ3 in the ground state. All geometries had been optimized in this study. At the SCF/6-31G* level we calculate a barrier of 45 kcal mol-1 for this rotation after minimization of the ground state and saddle point. Since correlation effects do not seem to considerably affect torsional barriers in the ground state, we conclude that this MNDO/CI calculation also tends to underestimate the corresponding barrier. 3.2. Potential Energy Surface of the Excited State S1. A contour plot of the PES of the first electronically excited state S1 calculated at level V of theory is presented in Figure 2 b. The topology of this surface is completely different and considerably more complex than that of the ground state PES. Two minima were found in the regions around the points (90°, 0°) and (90°, 180°), i.e., at conformations that are planar with respect to the C4-C5 dihedral and nonplanar with respect to the C3dC4 dihedral. Furthermore, three maxima were found on this PES: one for the 3,4-dicis conformation, which as in the case of the electronic ground state is caused by steric interactions, and two in the neighborhood of the points (45°, 90°) and (135°, 90°). The PES is almost flat in the regions corresponding to the planar all-trans, (3-cis, 4-trans), and (3trans, 4-cis) conformations. Relative energies of the electronically excited state are listed in Table 2, together with calculated excitation energies. Comparison of excitation energies calculated at levels II-V reveals the following trends. Addition of polarization functions tends to slightly raise the excitation energy, while a larger number of reference configurations tend to decrease it. Our MRCI
18286 J. Phys. Chem., Vol. 100, No. 46, 1996 calculations reveal an excitation energy of ∼250 nm for the S1 r S0 transition. The difficulty of calculating accurate electronic transition energies of a protonated Schiff base has been pointed out by Du and Davidson.24 By use of MRCI methods combined with additional corrections, an excitation energy of 326 nm had been calculated for a protonated Schiff base with six conjugated double bonds,24 which considerably deviates from the experimental absorption maxima of 445 and 477 nm of the protonated Schiff bases of all-trans48 and 11-cis-retinal,49,50 respectively. An improvement was achieved upon using different sets of orbitals in ground and excited states, which revealed an excitation energy of 460 nm,24 which at a first glance is in good agreement with experimental data. The calculation, however, did not take into account the negatively charged counterion of the protonated chromophore, which is usually modeled by adding a negative charge within a distance of ∼3 Å to the proton of the Schiff base51 and which is known to induce a blue shift of the electronic absorption spectrum. For a protonated Schiff base with six conjugated double bonds and with a counterion at infinite distance, PPP-MRCI calculations have predicted an excitation energy of 650 nm.52 This result agrees very well with the electronic absorption spectrum of BR in the O640 state6 of the photocycle, i.e., a state where the direct counterion of the protonated chromophore is most likely neutralized. By use of a semiempirical INDO/CI method,53 we have calculated an excitation energy of 640 nm for a system with six conjugated double bonds that is in good agreement with the results of ref 52. The same method reveals an excitation energy of 380 nm for a system with three conjugated double bonds. A somewhat lower excitation energy of 450 nm for the 2,4-pentadien-1-iminium cation has been calculated with a multireference MNDO/CI method.23 Thus, in agreement with the MRCI calculations of Du and Davidson,24 our MRCI calculations most likely overestimate the S1 r S0 excitation energy. Since the goal of this study consists of the description of the shape of the PES, we consider the errors in the prediction of the excitation energies to be of minor significance. The excitation energy predicted by the semiempirical AM1/CISD method, however, is most likely considerably underestimated. As apparent from Figure 2, torsional barriers in the excited state are very different from those in the ground state. In agreement with the previous semiempirical calculations20 the relative energy of point (90°, 90°) is considerably lower than in the ground state and the barrier for a rotation around bond C3dC4 becomes negative. All MRCI calculations, however, predict a much larger negative barrier than the semiempirical calculation. In contradiction to the previous semiempirical results as well as to our AM1/CISD calculations, barriers for all-trans f 4-cis and (3-cis, 4-trans) f 3,4-dicis isomerizations are smaller in the excited state than in the ground state. From Table 2 it is obvious that the semiempirical method most likely fails in the prediction of all torsional barriers. The energies of point (90°, 90°) and of all transition states for rotation around either C3dC4 or C4sC5 considerably depend on the level of theory. The smallest calculation (level I) clearly overestimates the energy of point (90°, 90°). Furthermore, the barriers for rotation around C4-C5 and the negative barriers for rotation around C3dC4 are both overestimated by up to 9 kcal mol-1 compared to the results of the calculation at level V. The comparison above makes clear that the semiempirical AM1/ CISD method cannot be applied in order to determine a PES of the S1 state with respect to an energetically acceptable description of torsional barriers. A somewhat better description is achieved with the cheapest MRCI calculation (level I), which,
Dobado and Nonella
Figure 3. Dominant CI coefficients in state S1 during rotation around τ3 (a) and during simultaneous rotation around τ3 and τ4 (b).
however, for particular conformations predicts energies that deviate considerably from those calculated at level V. The results of the calculation at level II agree well with those of level V except in the case of the 3,4-dicis conformation, which is predicted to be ∼9 kcal mol-1 higher in energy at level II than at level V. This deviation is probably not very important, since it is very unlikely that the dicis conformation will be reached in the S1 state. All torsional barriers agree well with the results of level V. Expanding the number of reference configurations at level II to 10 leads to level III and increases the energies of all nonplanar conformations. Addition of polarization functions to level II reveals level IV and affects mainly the energies of the points (90°, 90°) and (0°, 0°), which decrease by 9.3 and 5.2 kcal mol-1, respectively. Level IV generally agrees satisfactorily with the reference calculation at level V. All applied methods agree in that they predict the highest energy for the dicis conformation and that they reveal negative torsional barriers for rotations around bond C3dC4. Qualitatively, the predicted shape of the S1 PES agrees among all applied quantum chemical methods. Although the relative energies of the planar conformations (180°, 0°) and (0°, 180°) do not depend considerably on the applied level of calculation, all rotational barriers are shown to be very sensitive to the quality of the basis set and to the definition of the MRCI expansion. On the basis of semiempirical calculations of excited state PES’s of protonated Schiff bases,23,52 the second excited state (denoted as 1A-g) has been found to play an important role in photochemical processes on the S1 surface. The CdC antibonding character of this state causes an avoided crossing of the S1 and S2 surfaces upon rotation around a CdC double bond and, thus, a lowering of the energy of the S1 state. We have analyzed our CI wave function with respect to the most important configurations in the S1 state for the isomerization coordinate τ3 (Figure 3a) and for a simultaneous rotation around τ3 and τ4 (Figure 3b). From this analysis, it becomes clear that no higher excitations contribute significantly to the S1 wave function. Nevertheless, we find an energy profile that parallels that found in ref 23 and a negative barrier that agrees well with this previous calculation. On the other hand, although an avoided crossing between the states S0 and S1 has been found along τ3 in the MNDO/CI study, our calculation predicts this behavior only in regions around point (90°, 90°). Furthermore, it is interesting to compare our S1 energy surface to the S1 surface calculated for cis-hexatriene.54 The sequence of the lowest excited states has been shown to be different in
2,4-Pentadien-1-iminium Cation polyenes and in protonated Schiff base polyenes.52,55 Different properties of these two S1 surfaces can therefore be expected. The observation that the energy rises upon rotation either around a single or a double bond actually indicates that the S1 state of cis-hexatriene more likely corresponds to the S2 state of the 2,4-pentadien-1-iminium cation, which we have calculated at level III but not included in Figure 2. Transition states for cis f trans isomerizations of cis-hexatriene in ground and excited states have been found before a torsional angle of 90° was reached. Close to such transition states, regions of conical intersections have been found that allow a fast decay into the electronic ground state.56 Our calculation of the electronic ground state, however, clearly predicts transition states at 90° of either τ3 or τ4. In agreement with findings of ref 54 possible regions for conical intersections are not found at stationary points but, in our case, more likely around points (90°, 50°) and (90°, 130°). 3.3. Possible Implications of the Photocycle of Bacteriorhodopsin. A characteristic feature of the excited state PES found at all levels of calculation is the very flat surface around the points (180°, 0°), (0°, 180°), and (180°, 180°). Relatively small gradients in the Franck-Condon region are also shown by the model PES’s used in successful molecular dynamics simulations of the primary step of the photocycle of BR19 and thus are not in contradiction with the time scale of this very fast process. The two excited state model PES’s used in this simulation study differ in that the one applied for the all-trans f 13-cis isomerization has a saddle point around (90°, 90°) while the potential for the all-trans f 13,14-dicis reaction has the same energy at τ13 ) 90° for all values of τ14 . Our calculated PES agrees better with the model potential for an all-trans f 13-cis isomerization in that it also exhibits the characteristic saddle point at (90°, 90°), which should most likely prevent a torsion around the C14-C15 bond in the retinal chromophore. After light excitation and relaxation along the C13dC14 dihedral the molecule contains a considerable amount of internal energy. In the simulation, this energy allows in 6% of the cases a crossing of the saddle point, and thus, the molecule ends up in the 13,14-dicis conformation.19 The barrier height chosen in the simulations was 10 kcal mol-1. Calculated barrier heights are very unstable among the different MRCI approaches and are between 34 kcal mol-1 (level IV) and 73 kcal mol-1 (level I). Such barriers would make a transition to the 13,14-dicis conformation most likely impossible. However, it has to be expected that the negative barrier of τ3 as well as the relative energy of the saddle point (90°, 90°) might depend on the size of the conjugated system. In comparison with the effects found in the ground state, we can roughly assume that the negative barrier of τ3 is reduced by not more than 50% and that the saddle point approximately keeps its energy. We would then end up with a barrier of ∼15 kcal mol-1 at the level, which predicts the smallest barrier (level IV). A transition to a dicis conformation can, therefore, be considered very unlikely. Thus, our calculation tends to favor an all-trans f 13-cis isomerization. A decay from the S1 PES to the ground state is most efficient in regions where the energy difference between the two potential surfaces is small.57 Since our excitation energies are not properly described, we cannot discuss this topic quantitatively but only in a qualitative way. The calculated energy difference between the two PES’s calculated at level V are presented in Figure 2d. Due to the saddle point in the excited state PES, the minimal energy difference is not found at (90°, 90°) but closer to the points (90°, 50°) and (90°, 130°). In these regions, conical intersections between the states S0 and S1 are most likely
J. Phys. Chem., Vol. 100, No. 46, 1996 18287 possible54,57 and would allow a very fast decay into the electronic ground state. These findings suggest that the decision about the final conformation with respect to τ4 is most likely made in the excited state and not in the ground state. 4. Conclusions We have characterized the PES’s of the electronic ground and first excited states S0 and S1 as a function of the two dihedrals τ3 and τ4 of the 2,4-pentadien-1-iminium cation using different multireference configuration interaction (MRCI) approaches. The ground state PES shows positive torsional barriers for the isomerization reactions all-trans f 3-cis and all-trans f 4-cis and a global maximum at point (90°, 90°). All calculations predict the all-trans conformation to have the lowest energy. The PES of the first excited state S1 is more complicated. All calculations predict the 3,4-dicis conformation to have the highest energy. Additional energy maxima are found around points (45°, 90°) and (135°, 90°). Two energy minima are found at the points (90°, 0°) and (90°, 180°), i.e., the torsional barriers for rotations around bond C3dC4 are predicted to be negative and, thus, isomerizations around bond C3dC4 are predicted to be activationless by all calculations. Point (90°, 90°) corresponds to a saddle point in the excited state. Its energy is most likely close to that of the all-trans conformation. The comparison of the different methods makes clear that semiempirical methods give inaccurate energies for all rotational barriers in the electronic ground and excited states. In the case of state S0, all ab initio calculations agree relatively well. The largest variations in energy are found in points that involve a rotation around the C3dC4 double bond. Level I underestimates the energy of point (90°, 90°) and clearly overestimates the torsional barriers for rotation around bond C3dC4. Thus, for a quantitative correct description of the PES this level is not appropriate. A quite accurate prediction of the ground state PES, however, is already achieved with the SCF method. The largest uncertainty is found for the energy of point (90°, 90°) where we find an energy difference between MP4 and level V of ∼12 kcal mol-1. In the excited state S1, level I cannot accurately enough predict energies of torsional barriers of rotations around bond C3dC4. The energy of point (90°, 90°) is considerably higher than in all other calculations, and the negative barriers for single rotations around bond C3dC4 are up to 9 kcal mol-1 larger than at level V. A better description compared to the results of level V is found at level II, which only fails in the prediction of the energy of the dicis conformation, a conformation that most likely will never be reached in the excited state. As in the case of the ground state PES, the largest variations among the different methods are found in the energies of points (90°, 90°), (90°, 0°), and (90°, 180°). In contrast to the electronic ground state, the MRCI results cannot be compared presently to results of other methods. We can conclude that although the qualitative features of the two-dimensional PES’s agree well within all applied quantum chemical methods, relative energies can considerably depend on this method. In particular, semiempirical methods have been shown to be generally unsuitable for quantitative predictions of rotational barriers. A somewhat better agreement with level V is already achieved with the calculation at level I. Even though this calculation has been shown to inaccurately predict the energies of particular points on the surface, it could already be helpful in the first steps of the construction of a more realistic model PES that then can be employed in molecular dynamics simulations.
18288 J. Phys. Chem., Vol. 100, No. 46, 1996 The presented calculations are restricted to a two-dimensional PES as a function of the two dihedrals τ3 and τ4. We have chosen this restricted set of variables to compare the results of different levels of theory and also to compare our calculated PES to the model PES applied in molecular dynamics simulations.19 The comparison of those model PES’s with our calculated PES shows good agreement in the case of the PES applied for the simulation of the all-trans f 13-cis isomerization reaction. Therefore, our calculation in combination with the dynamics simulations of Humphrey et al.19 tend to support the 13-cis model, which postulates an all-trans f 13-cis isomerization as the primary step after light excitation.7 It has, however, also been suggested that additional degrees of freedom, in particular the out of plane coordinate of the proton at C4 (which corresponds to C14 in the retinal PSB), could play an important role in the isomerization mechanism.20,58 The possible effects of such additional degrees of freedom have to be investigated in future work. Acknowledgment. M.N. is grateful to Paul Tavan and J. Robert Huber for helpful discussions on effects of higher excited states, excited state dynamics, and avoided crossings. Support by the Swiss National Foundation (Project 31-39376.93) is gratefully acknowledged. Computing time has been provided by the Rechenzentrum der Universita¨t Zu¨rich. References and Notes (1) Oesterhelt, D.; Stoeckenius, W. Proc. Natl. Acad. Sci. U.S.A. 1973, 70, 2853. (2) Oesterhelt, D. Angew. Chem., Int. Ed. Engl. 1976, 15, 16. (3) Henderson, R. Annu. ReV. Biophys. Bioeng. 1977, 6, 87. (4) Harbison, G. S.; Smith, S. O.; Pardoen, J. A.; Courtin, J. M. L.; Lugtenburg, J.; Herzfeld, J.; Mathies, R. A.; Griffin, R. G. Biochemistry 1985, 24, 6955. (5) Lin, S. W.; Mathies, R. A. Biophys. J. 1989, 56, 653. (6) Lozier, R. H.; Bogomolni, R. A.; Stoeckenius, W. Biophys. J. 1975, 15, 955. (7) Fodor, S. P. A.; Ames, J. B.; Gebhard, R.; van den Berg, E. M. M.; Stoeckenius, W.; Lugtenburg, J.; Mathies, R. A. Biochemistry 1988, 27, 7097. (8) Schulten, K.; Tavan, P. Nature 1978, 272, 85. (9) Schulten, K. In Energetics and Structure of Halophilic Microorganism; Caplan, S. R., Ginzburg, M., Eds.; Elsevier: Amsterdam, 1978; p 331. (10) Gerwert, K.; Siebert, F. EMBO J. 1986, 4, 805. (11) Dobler, J.; Zinth, W.; Kaiser, W.; Oesterhelt, D. Chem. Phys. Lett. 1988, 144, 215. (12) Mathies, R. A.; Cruz, C. H. B.; Pollard, W. T.; Shank, C. V. Science 1988, 240, 777. (13) Zinth, W.; Oesterhelt, D. In Photobiology; Ricklis, E., Ed.; Plenum Press: New York, 1991; p 531. (14) Arlt, T.; Schmidt, S.; Zinth, W.; Haupts, U.; Oesterhelt, D. Chem. Phys. Lett. 1995, 241, 559. (15) Xu, D.; Martin, C.; Schulten, K. Biophys. J. 1996, 70, 453. (16) Nonella, M.; Windemuth, A.; Schulten, K. Photochem. Photobiol. 1991, 54, 937. (17) Zhou, F.; Windemuth, A.; Schulten, K. Biochemistry 1993, 32, 2291. (18) Xu, D.; Sheves, M.; Schulten, K. Biophys. J. 1995, 69, 2745. (19) Humphrey, W.; Xu, D.; Sheves, M.; Schulten, K. J. Phys. Chem. 1995, 99, 14549. (20) Orlandi, G.; Schulten, K. Chem. Phys. Lett. 1979, 64, 370. (21) Tavan, P.; Schulten, K.; Ga¨rtner, W.; Oesterhelt, D. Biophys. J. 1985, 47, 349. (22) Tavan, P.; Schulten, K.; Oesterhelt, D. Biophys. J. 1985, 47, 415.
Dobado and Nonella (23) Dormans, G. J. M.; Groenenboom, G. C.; van Dorst, W. C. A.; Buck, H. M. J. Am. Chem. Soc. 1988, 110, 1406. (24) Du, P.; Davidson, E. R. J. Phys. Chem. 1990, 94, 7013. (25) Brown, F. B; Shavitt, I.; Shepard, R. Chem. Phys. Lett. 1984, 105, 363. (26) Hennig, S.; Untch, A.; Schinke, R.; Nonella, M.; Huber, J. R. Chem. Phys. 1989, 119, 93. (27) Suter, H. U.; Huber, J. R. Chem. Phys. Lett. 1989, 155, 203. (28) Cotting, R. PhD Thesis, Universita¨t Zu¨rich, Zu¨rich, Switzerland, 1995. (29) Suter, H. U.; Huber, J. R.; von Dirke, M.; Untch, A.; Schinke, R. J. Chem. Phys. 1992, 96, 6727. (30) Schinke, R.; Nonella, M.; Suter, H. U.; Huber, J. R. J. Chem. Phys. 1990, 93, 1098. (31) Nonella, M.; Huber, J. R. Chem. Phys. Lett. 1986, 131, 376. (32) Schinke, R.; Hennig, S.; Untch, A.; Nonella, M.; Huber, J. R. J. Chem. Phys. 1989, 91, 2016. (33) Nonella, M.; Huber, J. R.; Untch, A.; Schinke, R. J. Chem. Phys. 1989, 91, 194. (34) Almlo¨f, J.; Taylor, P. R. J. Chem. Phys. 1987, 86, 4070. (35) Anderson, K.; Malmqvist, P.-Å.; Roos, B. O.; Sadlej, A. J.; Wolinski, K. J. Phys. Chem. 1990, 94, 5483. (36) Andersson, K.; Malmqvist, P.- Å.; Roos, B. O. J. Chem. Phys. 1992, 96, 1218. (37) Lischka, H.; Shepard, R.; Brown, F. B.; Shavitt, I. Int. J. Quantum Chem. Symp. 1981, 15, 91. (38) Shepard, R.; Shavitt, I.; Pitzer, R. M. Int. J. Quantum Chem. Symp. 1988, 22, 149. (39) Hehre, W. J.; Stewart, R. F.; Pople, J. A. J. Chem. Phys. 1969, 51, 2657. (40) Hehre, W. J.; Ditchfield, R.; Stewart, R. F.; Pople, J. A. J. Chem. Phys. 1970, 52, 2769. (41) Francl, M. M.; Pietro, W. J.; Hehre, W. J.; Binkley, J. S.; Gordon, M. S.; de Frees, D. J.; Pople, J. A. J. Chem. Phys. 1982, 77, 3654. (42) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Wong, M. W.; Foresman, J. B.; Robb, M. A.; Head-Gordon, M.; Replogle, E. S.; Gomperts, R.; Andres, J. L.; Raghavachari, K.; Binkley, J. S.; Gonzalez, C.; Martin, R. L.; Fox, D. J.; Defrees, D. J.; Baker, J.; Stewart, J. J. P.; Pople, J. A. Gaussian 92, Revision F.2.; Gaussian, Inc.: Pittsburgh, PA, 1993. (43) Nonella, M.; DeDecker, B.; Schulten, K. Unpublished results. (44) Stewart, James J. P. MOPAC 7: Molecular Orbital Package, Version 7; QCPE-455; Stewart Computational Chemistry: Colorado Springs, CO. (45) Foucrault, M.; Hobza, P.; Sandorfy, C. J. Mol. Struct.: THEOCHEM 1987, 152, 231. (46) Sreerama, N.; Vishveshwara, S. J. Mol. Struct.: THEOCHEM 1989, 194, 61. (47) Nina, M.; Roux, B.; Smith, J. C. Biophys. J. 1995, 68, 25. (48) Spudich, J. L.; McCain, D. A.; Nakanishi, K.; Okabe, M.; Shimizu, N.; Rodman, H.; Honig, B.; Bogomolni, R. Biophys. J. 1986, 49, 479. (49) Birge, R. R.; Murray, L. P.; Pierce, B. M.; Akita, H.; Balog-Nair, V.; Findsen, L. A.; Nakanishi, K. Proc. Natl. Acad. Sci. U.S.A. 1985, 82, 4117. (50) Birge, R. R. Acc. Chem. Res. 1986, 19, 138. (51) Nakanishi, K.; Balog-Nair, V.; Arnaboldi, M.; Tsujimoto, K.; Honig, B. J. Am. Chem. Soc. 1980, 102, 7945. (52) Grossjean, M. F.; Tavan, P. J. Chem. Phys. 1988, 88, 4884. (53) Ridley, J.; Zerner, M. Theor. Chim. Acta 1973, 32, 111. (54) Olivucci, M.; Bernardi, F.; Celani, P.; Ragazos, I.; Robb, M. A. J. Am. Chem. Soc. 1994, 116, 1077. (55) Schulten, K.; Dinur, U.; Honig, B. J. Chem. Phys. 1980, 73, 3927. (56) Manthe, U.; Ko¨ppel, H. J. Chem. Phys. 1980, 73, 3927. (57) Olivucci, M.; Ragazos, I.; Bernardi, F.; Robb, M. A. J. Am. Chem. Soc. 1993, 115, 3710. (58) Bonacic-Koutecky´, V.; Ishimara, S. J. Am. Chem. Soc. 1977, 99, 8134.
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