Effects of Molecular Symmetry on Quantum Reaction Dynamics: Novel

Article pubs.acs.org/JPCA

Effects of Molecular Symmetry on Quantum Reaction Dynamics: Novel Aspects of Photoinduced Nonadiabatic Dynamics Salih Al-Jabour† and Monika Leibscher*,†,‡ †

Institut für Chemie und Biochemie, Freie Universität Berlin, Takustr. 3, 14195 Berlin, Germany Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstr. 2, 30167 Hannover, Germany

‡

S Supporting Information *

ABSTRACT: Nonadiabatic coupling terms (NACTs) between different electronic states lead to fast radiationless decay in photoexcited molecules. Using molecular symmetry, i.e., symmetry with respect to permutation of identical nuclei and inversion of the molecule in space, the irreducible representations of the NACTs can be determined with a combination of molecular symmetry arguments and quantization rules. Here, we extend these symmetry rules for electronic states and coupling elements and demonstrate the importance of molecular symmetry for nonadiabatic nuclear dynamics. As an example, we consider the NACTs related to the torsion around the CN bond in C5H4NH. We present the results of quantum dynamical simulations of the photoinduced large amplitude torsion on three coupled electronic states and show how the interference between wavepackets leads to radiationless decay, which depends on the symmetry of the NACTs. Moreover, we show that the nuclear spin of the system determines the symmetry of the initial nuclear wave function and thus influences the torsional dynamics. This may open new possibilities for nuclear spin selective laser control of nuclear dynamics.

I. INTRODUCTION Photoisomerization processes play an important role in many chemical and biochemical systems. Activated by light, the molecules may undergo large amplitude intramolecular torsion in excited electronic states. Recently, it has been demonstrated that rotational1,2 and torsional3−5 dynamics of molecules is influenced by their nuclear spin. Controlling intramolecular torsion thus opens new possibilities for the separation of nuclear spin isomers. An adequate description of the nuclear dynamics of photoinduced large amplitude torsion is thus crucial for the theoretical investigation of the laser control of the above-mentioned processes. In general, large amplitude nuclear motion destroys the point group symmetry of a molecule. A global representation of the electronic states of molecules with identical nuclei has therefore to take into account the molecular symmetry of the system, as it has been demonstrated in ref 6 for single electronic states in the Born− Oppenheimer approximation. Molecular symmetry considers all feasible permutations of identical nuclei and the inversion of all nuclear and electronic coordinates7−9 and thus determines all relevant equivalent configurations of a molecule. Photoinduced torsion may guide the system to regions where strongly coupled electronic states, for example, at conical intersections, lead to fast radiationless deactivation of the system. This type of decay has first been quantified by Landau10 and Zener11 in a semiclassical framework for one-dimensional models. More accurate ways to calculate the radiationless decay are based on quantum mechanical simulations of nuclear dynamics on the coupled electronic states. This requires the determination of the coupling strength between different electronic states. A common approach is the multimode © 2014 American Chemical Society

vibronic coupling method, where the diabatic or potential couplings are modeled by linear or higher order functions of the nuclear coordinates.12−15 It has also been suggested to obtain the diabatic couplings by a quasi-diabatization procedure from the adiabatic potential energies.16−18 Today, algorithms for ab initio calculation of the nonadiabatic coupling terms (NACTs), which appear in the adiabatic representation of the molecular Hamiltonian, are implemented in quantum chemical program packages like Molpro19 or Columbus.20 Molecular symmetry adapted potential energy surfaces have also been constructed for electronic states coupled by conical intersections.21,22 In these studies, diabatic potential energy surfaces are constructed such that they are invariant under the operations of the molecular symmetry (MS) group. Our approach is based on the adiabatic representations of the electronic states, which allows a direct quantum chemical calculation of the NACTs. In ref 23, it was demonstrated that the symmetry of the NACTs can be determined by a combination of molecular symmetry arguments, quantization rules for the NACTs24−26 and quantum chemical considerations. As an example, the equivalent NACTs related to the torsion around the CN bond in model cyclopenta-2,4dienimine (C5H4NH) have been considered. Very recently, symmetry adapted NACTs have also been determined for Na3 and K3 cluster.27,28 The present study consists of two parts: in the first one, we present some extensions to the symmetry rules derived in ref 23 Received: September 23, 2014 Revised: December 16, 2014 Published: December 16, 2014 271

DOI: 10.1021/jp509604e J. Phys. Chem. A 2015, 119, 271−280

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The Journal of Physical Chemistry A

Table 1. Character Table of C2v(M) with Transformation Properties of Cartesian Coordinates and of Derivatives with Respect to the Nuclear Coordinates; the Last Row Shows the Effect of the Symmetry Operations on the Torsion Angle 0 ≤ φ0 ≤ π/2

and propose an alternative approach for the construction of molecular symmetry adapted adiabatic electronic states and coupling terms, which makes use of the correlation between point groups and the MS group of the system. In the second part, we demonstrate the importance of molecular symmetry adapted wave functions and coupling terms. We present the results of quantum dynamical simulations of the photoinduced torsion on three coupled electronic states and show that the symmetry of the NACTs and transition dipole moments influences the results of simulations of intramolecular torsion and radiationless decay. The one-dimensional simulations serve as a proof of principle for the importance of symmetry adapted NACTs and transition dipole moments. Furthermore, we demonstrate how the symmetry of torsional wave functions of C5H4NH is determined by the nuclear spin of the molecule and investigate the resulting nuclear spin selective nuclear dynamics, which opens possibilities for the control and manipulation of nuclear spin isomers.

C2v(M)

E

(12)

E*

(12)*

coord

deriv

A1 A2 B1 B2

1 1 1 1 φ0

1 1 −1 −1 φ0 − π

1 −1 −1 1 −φ0

1 −1 1 −1 π − φ0

z

∂/∂r ∂/∂φ ∂/∂y ∂/∂x

y x

written in the form of matrix elements on the basis of the electronic states, transform according to the irreducible representations as well. In ref 23, we showed that the irreducible representations of the NACTs can be determined in an indirect way by a combination of quantum chemistry with symmetry arguments and quantization rules for the NACTs.24−26 A. Symmetry Rules for NACTs and Transition Dipole Moments. The nonadiabatic coupling terms (NACTs) between the electronic states ψi and ψj are defined as

II. MOLECULAR SYMMETRY AND TRANSFORMATION PROPERTIES In this study, we investigate cyclopenta-2,4-dienimine (C5H4NH) (see Figure 1a) and the NACTs related to the

τkij =

ψi

∂ ψ ∂sk j

(2)

where sk is the symmetry adapted nuclear coordinate, which transform according to one of the irreducible representations of the MS group. Here, we consider the torsion of the NH fragment relative to the C4H5 ring, which is equivalent to motion of the H atom in a plane perpendicular to the z-axis (see Figure 1). Its motion can thus be described either by the Cartesian coordinates x and y or by the polar coordinates r and ϕ. It has been shown that for a three-state system with an Abelian MS group, the irreducible representations of the NACTs between different states, eq 2, are connected via the relation23

Figure 1. (a) Torsion of the H atom around the CN bond in C5H4NH. (b) Potential energy curves V0 (dash-dotted line), V1 (dashed line), and V2 (solid line) as a function of the torsion angle φ (CASSCF with cc-pVDZ(10/9), using Molpro19), combined with the irreducible representations of the electronic states in the molecular symmetry group C2v(M), as determined in the present study. The remaining nuclear coordinates are fixed; for their values see ref 23. The cut presented here corresponds to r = 0.9 Å. (b) Adapted with permission from ref 23. Copyright 2010 American Chemical Society.

⎛ ∂ ⎞ Γ(τk01) × Γ(τk02) × Γ(τk12) = Γ⎜ ⎟ ⎝ ∂sk ⎠

(3)

Equation 3 follows directly from ⎛ ∂ ⎞ Γ(τkij) = Γ(ψi ) × Γ⎜ ⎟ × Γ(ψj) ⎝ ∂sk ⎠

torsion of the H atom around the CN bond. Figure 1b shows a cut through the potential energy surfaces of the three lowest electronic states S0, S1 and S2 as a function of the torsion angle φ.23 The potential curves point to several equivalent conical intersections, which can be found if one of them is known, by applying the operations of the molecular symmetry group.7 The molecular symmetry (MS) group of C5H4NH is C2v(M ) = {E , (12), E*, (12)*}

(4)

since Γ(ψi) × Γ(ψi) = A1 for one-dimensional irreducible representations. As a consequence of eq 3, the symmetry of the third NACT is known once the irreducible representations of the other two NACTs are determined. Note that this theorem can be generalized to N coupled electronic states, which have N(N − 1)/2 coupling elements. As we show as Supporting Information, the irreducible representations of (N − 1)(N − 2)/2 coupling elements can be determined by relations similar to eq 3 if the transformation properties of N − 1 elements are known. Similar theorems can be derived to relate the irreducible representations of different molecular properties to each other. For example, for any MS group with only one-dimensional representations, the dipole moments μiiρ always transform as ρ, where ρ = X, Y, Z are the Cartesian components of the electronic coordinates. In the process described in this study, the system is activated by a laser pulse,

(1)

where E is the identity, (12) denotes the permutation of the C2H2 fragments 1 and 2 (Figure 1a), the operation E* is the inversion of all nuclear and electronic coordinates, and (12)* = (12)E* = E*(12). Table 1 shows the character table of C2v(M) and the effect of its operations on the torsion angle φ. Since the electronic Hamiltonian is invariant under the transformations of the molecular symmetry group, the electronic wave functions have to transform according to its irreducible representations. As a consequence, molecular properties like nonadiabatic coupling terms and transition dipole moments, which can be 272


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Table 2. Character Table of Cs and Cs(M) (Left) and of C̃ s and C̃ s(M) (Right); the Irreducible Representations of C̃ s and C̃ s(M) Are Labeled with a Tilde to Distinguish Them from the Irreducible Representations of Cs and Cs(M)

which interacts with the molecule via its electric dipole moment. The transition dipole moments are defined as μρij = ⟨ψi|μρ |ψj⟩

(5)

Their transformation properties are also shown in Table 1. The definitions 2 and 5 lead to ⎛ ∂ ⎞ Γ(μρij ) = Γ(τkij) × Γ⎜ ⎟ × Γ(ρ) ⎝ ∂sk ⎠

(6)

E

σxz

E 1 1

E* 1 −1

C̃ s C̃ s(M) Ã ′ Ã ″

E

σyz

E 1 1

(12)* 1 −1

representations of the MS groups of the nonrigid system, as shown in Figure 2. If the irreducible representations of an

for i ≠ j since Γ(μρij ) = Γ(ψi ) × Γ(ρ) × Γ(ψj)

Cs Cs(M) A′ A″

(7)

With the irreducible representations of the NACTs, we therefore also determine the irreducible representations of the transition dipole moments. For a three state system, it is therefore sufficient to know the irreducible representations of two coupling elements in order to determine the transformation properties of the remaining ones. In ref 23 the irreducible representations of the remaining two NACTs of C5H4NH have been assigned with the help of properties of the conical intersection, in particular the quantization rules for the NACTs, which result from their singularity at a conical intersection.24−26 By this indirect approach, it has been circumvented to assign the irreducible representation of the molecular symmetry group to the electronic states themselves. In the following, we present a direct approach, where we determine the molecular symmetry of the adiabatic electronic states in order to obtain the irreducible representations of the coupling terms. B. Molecular Symmetry Properties of the Electronic States. An alternative procedure to obtain the irreducible representations of the NACTs and transition dipole moments according to the molecular symmetry group is to directly determine the irreducible representations of the adiabatic electronic states by correlating the irreducible representations of the point groups with those of the MS group, similar to the approach presented in ref 29. Therefore, we first establish the effect of the symmetry operations of the MS group on an arbitrary electronic coordinate, following the procedure developed in ref 7. Accordingly, (12) has the same effect as a C2 rotation, E* corresponds to a σxz reflection, and (12)* corresponds to σyz. Here, the electronic coordinates are given in a molecule fixed coordinate system. Note that, in a space-fixed coordinate system, the nuclear permutation (12) does not affect the electronic coordinates since it describes the exchange of identical nuclei only. However, the operation (12) transforms the molecule fixed coordinate system with respect to the spacefixed one (see ref 7). Therefore, in the molecule fixed coordinate system, the electronic coordinates are affected by (12). Next, we consider two symmetric configurations of the molecule, the equilibrium configuration with torsion angle φ = 0 with the point group Cs = {E, σxz} and a twisted configuration with φ = π/2 (with all other nuclear coordinates fixed), where the point group is C̃ s = (E, σyz). The MS group of a rigid molecule is always isomorphic to the point group at the same nuclear configuration.7 The MS group corresponding to Cs is Cs(M) = {E,E*}, and the MS group of a rigid system with point group C̃ s is C̃ s(M) = {E, (12)*}. The character tables are shown in Table 2. The irreducible representations of Cs(M) and C̃ s(M) can be correlated with those of C2v(M), i.e., with the irreducible

Figure 2. Correlations between the irreducible representations of the MS groups Cs(M), C̃ s(M), and C2v(M).

electronic state at φ = 0 and φ = π/2 in Cs and C̃ s can be identified, one can also determine its irreducible representations according to the MS group C2υ(M). For example, an electronic state with A′ symmetry at φ = 0 and Ã ′ symmetry at φ = π/2 transforms according to A1 in C2v(M). For C5H4NH, the electronic states S0, S1, and S2 all transform as A′ at φ = 0. A sketch of the potential energy for the two lowest electronic states at φ = π/2 as a function of the radius r is shown in Figure 3. Accordingly, a potential curve of an electronic state with Ã ′

Figure 3. Sketch of the potential energy curves for the S0 and S1 electronic states of C5H4NH at φ = π/2 as a function of the coordinate r close to the intersection with radius r = rX = 0.96 Å. The figure also shows the irreducible representations of the corresponding electronic states according to the point group C̃ s.

symmetry crosses one with Ã ″ symmetry. The crossing is at the radius is r = rX = 0.96 Å. For r < rX, the lowest state (S0) has Ã ′ symmetry and the higher one (S2) has Ã ″ symmetry. For r > rX, the order of the states changes. The third state, S2 transforms as Ã ′ at φ = π/2. The symmetry assignment at φ = 0 and φ = π/2 is based on CAS(10/9)/cc-pVDZ calculations of the electronic states, see also Figures 3 and 4 of ref 23. This information combined with the correlation diagram in Figure 2 determines the transformation properties of the electronic states according to the molecular symmetry group C2v(M). As a consequence, the irreducible representations of the NACTs and transition dipole moments can be calculated as direct products of the irreducible representations of the respective electronic states and coordinates. The results are summarized in Table 3. In Figure 1, we present a cut through the potential energy surfaces of C5H4NH with r = 0.9 Å. Since r < rX, the NACTs 273


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assume that the C5H4N ring is planar. The remaining coordinates are chosen such that the potential curves are very close to the conical intersections between the S1 and S2 states.23 It can be seen that four equivalent crossings between S1 and S2 exist at φ = ±0.33π and φ = π ±0.33π. At φ = ±π/2 the potential gap between the states S0 and S1 is small, indicating two equivalent conical intersections between S0 and S1 at a slightly different configuration of C5H4NH. The nuclear Hamiltonian in adiabatic representation can be written as24

Table 3. Irreducible Representations of the Adiabatic Electronic States, Nonadiabatic Coupling Terms (NACTs), and Transition Dipole Moments (TDMs) Calculated along a Ring with r < rX (Case I) and r > rX (Case II) r < rX (case I) C2v(M)

el. states

A1 A2 B1 B2

S0, S2

NACTs

TDMs μ02 Z

τ02 φ 12 τ01 φ , τφ S1

r > rX (case II)

12 μ01 Z ,μZ

el. states

NACTs τ12 φ 02 τ01 φ , τφ

S0

TDMs μ12 Z

S1, S2

02 μ01 Z , μZ

H=−

and transition dipole moments transform according to case I in Table 3. Using this correlation approach, it becomes obvious that, in the presence of a conical intersection, it is not possible to assign a single irreducible representation to an electronic state in the adiabatic representation, which is valid for all nuclear configurations. In ref 23, we have determined the transformation properties for a specific range of nuclear configurations, i.e., for a ring φ = 0 to φ = 2π with a radius r = 1.0 Å. At φ = π/2, this configuration is very close to the S0/S1 conical intersection. We have obtained the same results as with the present correlation method for r > rX = 0.96 Å (case II in Table 3). However, we have further assumed that the same assignment is valid for all rings with different values of r. The correlation method clearly contradicts this assumption. In particular, we have to correct the results for the ring with r = 0.9 Å < rX. Here, the NACTs and transition dipole moments according to the irreducible representations presented in case I in Table 3 and not according to case II as predicted in ref 23. Moreover, one should note that the correlation method for the determination of the irreducible representations can only be applied if the electronic states are single-valued. It is known that electronic states change their sign if they are transported around a single conical intersection or around an odd number of conical intersections.30,31 In such case, the present approach has to be generalized to extended MS groups.32 The sign change of the adiabatic electronic states in view of globally symmetry adapted electronic states has also be addressed in ref 33. In the case considered here, the closed path along the torsion angle always encloses an even number of conical intersections. The sign of the electronic wave function is therefore unchanged along a path with constant r, and the correlation approach presented above can thus be applied.

⎞2 ℏ2 ⎛ ∂ ⎜ 1 + τφ⎟ + V 2Ired ⎝ ∂φ ⎠

(8)

where Ired = 0.75u Å2 is the reduced moment of inertia, 1 denotes the unity matrix, and ⎛ V2(φ) 0 0 ⎞ ⎟ ⎜ V1(φ) 0 ⎟ V=⎜ 0 ⎟⎟ ⎜⎜ 0 V0(φ)⎠ ⎝ 0

(9)

The matrix ⎛ 0 τφ12 τφ02 ⎞ ⎜ ⎟ ⎜ 12 01 ⎟ 0 τφ ⎟ τφ = ⎜ −τφ ⎜ 02 ⎟ 01 ⎝−τφ −τφ 0 ⎠

(10)

represents the NACTs τφij(φ) = ⟨ψi(φ)|

∂ ψ (φ )⟩ ∂φ j

(11)

where |ψi(φ)⟩ are the electronic wave functions that depend parametrically on the torsion angle φ. Since we are considering real electronic wave functions, the diagonal elements τiiφ are zero and the matrix eq 10 is antisymmetric.24 The NACTs τ01 φ 02 (dashed lines), τ12 φ (solid lines), and τφ (dash-dotted lines) are shown in Figure 4a. They have been calculated ab initio using CASSCF with cc-pVDZ(10/9).23 Close to a crossing between two electronic states, the corresponding NACT is very large, and it decreases rapidly to approximately zero where the gap between the potentials increases. Since there is no neardegeneracy between the states S0 and S2, the term τ02 φ is almost zero for all angles φ. As we have shown above, the NACTs 12 02 τ01 φ (φ) and τφ (φ) transform as B1 in C2v(M) and τφ (φ) transforms as A2. In order to simulate the nuclear dynamics on the coupled potentials, one usually transforms Hamiltonian eq 8 to a diabatic representation where

III. PHOTOINDUCED NUCLEAR DYNAMICS In the previous section, we have determined the irreducible representations of the adiabatic electronic states, NACTs, and transition dipole moments. In the following, we show that a correct symmetry assignment is important for simulating the photoinduced nuclear dynamics. As a proof of principle, we simulate the one-dimensional photoinduced torsional dynamics of C5H4NH and calculate the resulting radiationless decay. In order to emphasize the role of molecular symmetry in quantum dynamics, we compare simulations with symmetry adapted coupling terms with simulation performed with coupling elements with deliberately wrong symmetry. In the following, we present results of quantum dynamical simulations of the movement of the H atom. A one-dimensional cut through the potential energy surface belonging to the three lowest electronic states S0, S1, and S2 of C5H4NH can be seen in Figure 1b (adapted from ref 23). It shows the potentials Vi, with i = 0, 1, 2 as a function of the torsion angle φ. Here, we

Hdia = A†HA = −

ℏ2 ∂ 2 1+W 2Ired ∂φ 2

(12)

with the diabatic potential matrix ⎛W22 W12 W02 ⎞ ⎜ ⎟ W = ⎜ W12 W11 W01 ⎟ ⎜ ⎟ ⎝W02 W01 W00 ⎠

(13)

Here the coupling between the electronic states has been transformed to potential couplings Wij, which are smooth functions of the nuclear coordinates. The unitary transformation matrix A, called the A-matrix, is determined from the NACTs by the expression24 274


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12 02 NACTs τ01 φ (φ), τφ (φ), and τφ (φ) transform according to B1, A2, and B1, respectively (see case II in Table 3). Figure 4e shows the diagonal elements of the A-matrix. It can be seen that eq 16 is fulfilled here as well. We conclude that a set of NACTs can result in an A-matrix, which leads to single valued diabatic potentials even if the symmetry assignment of the NACTs is incorrect. This emphasizes the importance of the symmetry analysis presented in ref 23 and in section II. The diabatic potentials Wii obtained by eq 12 for both cases are shown in Figure 4c,f. Note that the adiabatic-to-diabatic transformation as it is carried out here does not conserve the molecular symmetry of the electronic states. This can be seen in Figure 4c,f where, for example, W22(φ = π) ≠ W00(φ = 0). Likewise, the elements of the A-matrix, as shown in Figure 4b,d do not transform according to the irreducible representations of the molecular symmetry group. This is, however, not a contradiction to the fact that the conformations with φ = 0 and φ = π are indistinguishable. It is the total molecular wave function, i.e., the product of electronic wave function and nuclear wavepacket that has to reflect the symmetry of the system. In our approach, this is guaranteed by using molecular symmetry adapted potentials, coupling terms, and nuclear wavepackets in the adiabatic representation. We simulate the movement of the H atom after excitation with an ultra short laser pulse by solving the time-dependent Schrödinger equation

12 02 Figure 4. Panel (a) shows the NACTs τ01 φ (φ), τφ (φ), and τφ (φ) depicted in dashed, solid, and dash-dotted lines, respectively. For comparison, the adiabatic potentials are also shown (gray lines). The diagonal elements of the A-matrix can be seen in panel (b) in dashdotted (A00), dashed (A11), and solid (A22) lines. Panel (c) shows the diabatic potentials W00 (dash-dotted line), W11 (dashed line), and W22 (solid line). Panels (d−f) show the same properties, but for NACTs that transform according to different irreducible representations. Panels (a) and (b) are reprinted with permission from ref 23. Copyright 2010 American Chemical Society.

A(φ) = ζexp[−

∫φ

iℏ

dφτφ(φ)]A(φ0)

⎛ χ (φ , t ) ⎞ ⎜ 2 ⎟ χ (φ , t ) = ⎜ χ1 (φ , t ) ⎟ ⎜ ⎟ ⎜ χ (φ , t )⎟ ⎝ 0 ⎠

(14)

∫φ

φj

j−1

dφτφ(φ)]A(φj − 1)

H int(t ) = −μZ E(t )

(15)

if the interval Δφ = φj − φj−1 is chosen such that τφ(φ) is approximately constant inside this interval. To ensure that the diabatic potentials are single-valued, it is required that Aii (φ0 + 2π ) = ±1

(20)

Here, we assume that the molecule is oriented along the polarization direction of the laser pulse (see, e.g., refs 34 and 35). The components of the dipole matrix μz = (μijZ) are defined in eq 5. We assume that the initial state, χ0(φ, 0), is the torsional ground state, i.e., the lowest eigenstate of V0. If the laser pulse is short enough to induce a vertical transition to the excited electronic state S2, the wave function immediately after the interaction can be written as

(16)

and Aij (φ0 + 2π ) = 0

(19)

is the nuclear wave function in the adiabatic basis. The molecular Hamiltonian H is given in eq 8, and the interaction of the molecule with a z-polarized laser pulse with the electric field E(t) is described by

where ζ is the ordering operator. As initial value we choose A(φ0 = 0) = 1. The A-matrix can be calculated iteratively since A(φj) = exp[−

(18)

where

φ

0

∂ χ (φ , t ) = [H + H int(t )]χ (φ , t ) ∂t

(17)

⎛ χ (φ , 0)⎞ ⎟ ⎜ 2 χ (φ , 0) = ⎜ 0 ⎟ ⎟ ⎜ ⎝ 0 ⎠

for i ≠ j. The diagonal matrix elements Aii(φ) are depicted in Figure 4b, and the corresponding diabatic potentials Wii(φ) for i = 0, 1, 2 are shown in Figure 4c. Figure 4b shows that condition eq 16 is fulfilled for i = 0,1,2 with Aii(φ0 + 2π) = +1. The plus sign indicates that an even number of conical intersections is enclosed by the loop from φ = 0 to φ = 2π. The nondiagonal terms of the A-matrix are consistent with eq 17 (not shown here). The fact that eqs 16 and 17 are fulfilled confirms the symmetry assignment of the previous section. Different assignments for the symmetry of the NACTs can, however, also lead to an A-matrix that is consistent with conditions 16 and 17, as it is shown in Figure 4d,e. Here the 24−26

(21)

with χ2 (φ , 0) = 5μZ02 χ0 (φ , 0)

(22)

where 5 is the normalization constant. Further, we assume that the transition dipole moment is constant in the Franck− Condon region (Condon approximation), i.e., at φ ≈ 0 and φ ≈ π and simulate the time-evolution in the diabatic 275


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The Journal of Physical Chemistry A representation with Hamiltonian eq 12 using the split operator method implemented in the program package Wavepacket.36 Afterward, the results are transformed back to the adiabatic representation. We will first discuss how the transformation properties of the NACTs and transition dipole moments affects the nonadiabatic nuclear dynamics. Afterwords, we show how the torsional wavepackets are connected with the nuclear spin of the molecules and demonstrate that different nuclear spin isomers show different torsional dynamics and radiationless decay. A. Does Symmetry of the NACTs and Transition Dipole Moments Matter? We first assume that the initial state χ0(φ, 0) is the lowest eigenstate of V0, i.e., the symmetric torsional wave function shown in Figure 5 in dash-dotted red

Figure 5. Sketch of the symmetric initial wave functions χ0(φ, 0) (dash-dotted red line) and of the wave functions after excitation with a short laser pulse, χ2(φ, 0) (solid red line). Figure 6. Population of the (adiabatic) electronic states S2 (solid lines), S1 (dashed lines), and S0 (dash-dotted lines). Panel (a) shows the result of a simulation where the irreducible representations of the NACTs and transition dipole moments determined in Section II have been considered. The simulations in panel (b) are carried out using the NACTs shown in Figure 4d and transition dipole moments that are in accordance with those NACTs. Finally, panel (c) shows the adiabatic populations resulting from simulations with the same NACTs as in panel (b) but with transition dipole moments with different irreducible representations.

lines, which transforms as A1 in C2v(M). Since the transition dipole moment μ02 Z also transforms as A1 (see Table 3), the wave function after the interaction, χ2(φ, 0) (solid red line in Figure 5) has again A1 symmetry. After the interaction with the laser pulse, the wavepacket starts moving on the three coupled potentials. Figure 6a shows the population of the three electronic states S2 (solid), S1 (dashed), and S0 (dash-dotted) as a function of time. Here, the NACTs and transition dipole moments transform according to the irreducible representations determined in Section II. After 10 fs, the wavepacket decays from S2 to S1 through the S2/S1 crossing located at φ = ± 0.33π and φ = π ± 0.33π. Another 5 fs later, a small part of the population is transferred to the ground state, due to the NACTs between S1 and S0 at φ = ±π/2 and φ = 3π/2, but as it can be seen Figure 6, during the first 100 fs after photoexcitation, the population mainly oscillates between the states S2 and S1 and the ground state is barley populated. In Figure 6 we also present the results of simulations where we deliberately use NACTs and transition dipole moments with wrong symmetry. The blue curves in Figure 6b show the results of a simulation where we used the NACTs shown in Figure 4d, i.e., NACTs that do not transform according to the irreducible representation determined in Section II for r = 0.9 Å, but nevertheless lead to single valued diabatic potentials. Here, the irreducible representations of the NACTs are Γ(τ01 φ ) = B1, 02 Γ(τ12 ) = A , and Γ(τ ) = B . According to eq 6, the transition φ 2 φ 1 dipole moment μZ02 then transforms as B2. The nuclear dynamics after photoexcitation is shown in Figure 6b. During the first 20 fs after excitation, the adiabatic populations are the same as in Figure 6a. However, after t ≈ 22 fs, differences between the red and blue curves can be observed, especially in the excited electronic states S2 (solid lines) and S1 (dashed

lines). In Figure 6a the transfer of the wave function to the intermediate state S1 is weaker than in panel b. After t ≈ 65 fs after excitation, 34% of the population is in state S1 in panel a (dashed red line), compared to 49% in panel b (dashed blue line). We can further analyze the differences in the wavepacket evolution by calculating the angular expectation value ⟨cos(2φ)⟩. If a wavepacket is located at φ ≈ 0 and φ ≈ π, the angular expectation value is ⟨cos(2φ)⟩ ≈ 1. A wave function localized at φ ≈ ± π/2 or 3π/2 has an expectation value ⟨cos(2φ)⟩ ≈ −1, and a uniform distribution in the interval −π/ 2 ≤ φ ≤ 3π/2 corresponds to ⟨cos 2φ⟩ ≈ 0. In Figure 7a,b, we show the expectation values for the electronic states S2 and S1 ⟨cos(2φ)⟩i =

⟨χi |cos(2φ)|χi ⟩ ⟨χi |χi ⟩

(23)

for i = 2 (a) and i = 1 (b). Again, the red and blue curves correspond to the deliberately wrong NACTs shown in Figure 4a,d, respectively. At t = 0, the wavepackets are located for both cases in S2 around φ = 0 and φ = π, and therefore, initially ⟨cos(2φ)⟩2 ≈ 1. As the wavepackets disperse and become 276


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Figure 7. Expectation values ⟨cos(2φ)⟩i for i = 2 (a) and i = 1 (b) as a function of time. Panels (c) and (d) show |χ2(φ, t)|2 and |χ1(φ, t)|2 at t = 44 fs. The red and blue curves correspond to the NACTs shown in Figure 4a,d, respectively. Since a numerically meaningful expectation value can be calculated only if there is enough population in the electronic state to evaluate the corresponding integrals, therefore, ⟨cos(2φ)⟩1 is shown only for t ≥ 20 fs.

Figure 8. Sketch of the excitation of para-C5H4NH: antisymmetric initial wave functions χ0(φ, 0) (dash-dotted green line) and wave function after excitation with a short laser pulse, χ2(φ, 0) (solid green line).

localized again with time ⟨cos(2φ)⟩2 decreases and increases again. The difference in the torsional dynamics between the two cases is most pronounced in state S1 (Figure 7b). In the case that is in accordance with the molecular symmetry (red line), the angular expectation value varies between −0.8 and −0.5. The wave function is trapped in the potential minima of V1 at φ = π/2 and φ = 3π/2. In the hypothetical case (blue line in Figure 7b), however, ⟨cos 2φ⟩1 increases up to ∼0.37. At t = 44 fs, the wavepacket is spread over all torsion angles. This proof of principle shows that NACTs, which have the wrong transformation properties, result in misleading wavepacket dynamics. In Figure 6c we finally present simulations with the same NACTs as in Figure 6b but with a transition dipole moment that is not consistent with eq 6, i.e., has A1-symmetry. The wave function χ2(φ, 0) is therefore generated as a symmetric wavepacket as shown in solid red lines in Figure 5. As it can be seen in Figure 6, the population of the three states is the same as in the previous simulations for the first 20 fs. However, afterward, considerable differences are observed. In the simulations depicted in Figure 6c, almost 50% of the population decays to the ground state, while only approximately 20% of the population remains in the intermediate state S1. This is qualitatively different from the simulations (a) and (b) where the ground state is barely populated. We therefore conclude that the irreducible representations and therefore the relative phase of the NACTs and of the transition dipole moments affect significantly the radiationless decay and the dispersion of the torsional wavepackets in the excited electronic states. In simulations of nuclear dynamics where large amplitude torsion is expected, the irreducible representations of the NACTs and transition dipole moments have to be determined correctly, e.g., by a symmetry analysis as presented in Section II. Otherwise, the results of the simulations can significantly deviate, especially if the irreducible representations of the NACTs are not consistent with those of the transition dipole moments, as the differences in the adiabatic population in Figure 6c compared to a and b demonstrate. B. Nuclear Spin Selective Nuclear Dynamics. So far, we have assumed that the initial state is the lowest eigenstate of the potential V0, which is the symmetric torsional wave function as shown in Figure 5 in dashed green lines. However, the potential barrier of V0 is so high that the antisymmetric torsional eigenstate, shown in Figure 8, is practically degenerate with the symmetric one. As is has been shown in refs 3−5, 37, and 38, the symmetry of torsional states is determined by the nuclear

spin of the molecule. C5H4NH occurs in two different nuclear spin modifications: ortho-C5H4NH has a nuclear spin state that is symmetric with respect to the permutation (12). Its statistical weight is g = 10. Para-C5H4NH, which has an nuclear spin state that is antisymmetric with respect to (12), occurs with a statistical weight of g = 6. The number of nuclear spin isomers and their statistical weights have been determined as described in the Appendix of ref 5. Note that we consider C5H4NH with the isotope 12 C, which has zero nuclear spin. The symmetrization postulate states that the total wave function ψmol of the molecule remains the same if two bosons are exchanged and changes its sign upon the exchange of two Fermions. The symmetry operation (12) corresponds to an exchange of two pairs of Fermions, e.g., an effective exchange of two bosons and thus (12)ψmol(φ) = ψmol(φ − π ) = ψmol(φ)

(24)

If the molecule is in its electronic, vibrational and rotational ground state, which is totally symmetric, its molecular symmetry is determined by the irreducible representations of the torsional and nuclear spin states, i.e. Γ(ψmol) = Γ(ψtor) × Γ(ψnuc.spin)

(25)

where ψtor = χ0(φ, 0). Thus, Pauli’s principle (eq 24) requires that ortho-C5H4NH can only have a symmetric torsional ground state, while para-C5H4NH has an antisymmetric torsional ground state. Beside the ortho-isomer with symmetric initial state (Figure 5) that leads to the nonradiative decay shown in Figure 6a, a sample of C5H4NH molecules also contains para-isomers with the antisymmetric initial state shown in Figure 8. Since the transition dipole moment μ02 Z transforms as A1, the wave function after photoexcitation, χ2(φ, 0), is antisymmetric for para-C5H4NH. In Figure 9a we show the radiationless decay of a sample of C5H4NH molecules with their nuclear spin isomers distributed according to their statistical weights. In average 40% of the S2 populations decays after photoexcitation, and the population of the states S1 and S0 varies in time between 20% and 40%. The torsional dynamics and radiationless decay is thus considerably different from the radiationless decay expected from a purely symmetric initial state as shown in Figure 6a. The role of the nuclear spin becomes even more obvious if the radiationless decay for the two nuclear spin isomers is compared, as it can be seen in Figure 9b. The green 277


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themselves if different electronic states are coupled, either internally via nonadiabatic coupling elements or due to an external field via transition dipole moments. Determining the global symmetry of an electronic state, i.e., its relative phase at equivalent nuclear configurations by quantum chemistry alone can be very tedious and difficult.39 Therefore, an indirect way has been proposed23 in order to obtain the irreducible representations of the NACTs that employs symmetry rules and general properties of conical intersections. Here, we generalize some of the symmetry rules derived in ref 23 to other molecular properties and to N-state systems. We also propose an alternative method for direct determination of the molecular symmetry of the adiabatic electronic states based on correlations between the molecular point groups and the MS group of the system.7 The correlation method reveals that in the presence of a conical intersection, the electronic states and properties like NACTs and transition dipole moments cannot be assigned with a single irreducible representation of the MS group for all nuclear configurations. The reason is that the adiabatic electronic states are not single valued in the presence of a conical intersection but can gain a geometrical phase if they are transported around a conical intersection. A more general molecular symmetry analysis of the adiabatic electronic states requires extended molecular symmetry groups, which takes into account multivalued molecular states32 In the second part of this article, we have demonstrated the importance of a correct symmetry assignment of molecular properties for the simulation of the nuclear dynamics. The photoexcitation and subsequent nuclear dynamics and radiationless decay in three coupled electronic states of C5H4NH have been considered. The results for NACTs with correct irreducible representations have been compared with simulations where we deliberately choose coupling elements with wrong symmetry. In all simulations, the absolute values of the NACTs and transition dipole moments are indistinguishable, and the only differences are their irreducible representations, i.e., their different relative phases. The photoinduced nuclear dynamics shows significant differences, which can be observed in the time-dependent populations of the electronic states and in the angular expectation value, which describes the dispersion of the torsional wave functions. This proof of principle demonstrates the importance of the (relative) sign, i.e., the irreducible representation of the NACTs and transition dipole moments for a correct description of the nuclear dynamics. We also showed that the radiationless decay to the electronic ground state is strongly influenced by the symmetry of the initial nuclear state. Since the symmetry of the initial torsional wave function is determined via the symmetrization principle by the nuclear spin of the molecule, C5H4NH is also a promising candidate for investigating the possibilities of nuclear spin selective photoinduced nuclear dynamics and laser control.1,3,40,41 In this study, we considered a model with one nuclear degree of freedom for a proof of principle that molecular symmetry of the electronic states affects nonadiabatic nuclear dynamics. It should be noted, however, that the description of the nuclear dynamics through conical intersections requires at least two nuclear coordinates. Thus, models including several nuclear degrees of freedom will have to be employed in future research for a more realistic prediction of the photochemistry of C5H4NH. What the one-dimensional simulations do show is the importance of molecular symmetry for the photoinduced nuclear dynamics with large amplitude torsion.

Figure 9. Population of the (adiabatic) electronic states S2 (solid lines), S1 (dashed lines), and S0 (dash-dotted lines). Panel (a) shows the radiationless decay of para-C5H4NH and panel (b) represents the statistical average of ortho- and para-C5H4NH, which have the statistical weights g = 10 and g = 6, respectively.

curves display the adiabatic populations for an initially antisymmetric torsional wavepacket (para-C5H4NH). Following the decay from S2 to S1 after 10 fs, the electronic ground state S0 becomes populated. The ground state population increases to approximately 50% at t ≈ 45 fs. This is considerably different compared to the nonadiabatic dynamics of ortho-C5H4NH with almost no population in the ground state, i.e., 2% at t ≈ 45 fs as it can be seen from the red curves in Figure 9b or Figure 6a. These simulations show that, whenever a molecule occurs in the form of nuclear spin isomers, its torsional dynamics depends on the symmetry of its nuclear spin state. Similar observations have been reported for other molecules that occur in the form of different nuclear isomers and undergo large amplitude torsions after photoexcitation.3−5 We therefore emphasize that the statistical weight of each nuclear spin state has to be included to predict the nuclear dynamics of a sample of molecules if large amplitude torsion is expected. The observation that torsional dynamics depends on the nuclear spin of the molecules also opens possibilities for selective manipulation of nuclear spin isomers. As we suggest in refs 3 and 5, sequences of laser pulses could be applied to photoexcite only a single isomer. This can be a starting point for separation of nuclear spin isomers, e.g., by selective ionization.

IV. CONCLUSIONS In this study, we have investigated the influence of the molecular symmetry for electronic wave functions on nuclear dynamics. Whenever large amplitude motion proceeds via several equivalent configurations, nuclear dynamics depends strongly on the global symmetry of the electronic wave functions. A prominent example for large amplitude motion is intramolecular torsion. Molecular symmetry effects reveal 278


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ASSOCIATED CONTENT

S Supporting Information *

Generalization of the symmetry rules for NACTs from a threestate system (see Section IIA) to a N-state system is provided. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +49 511 7624833. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank PD Dr. D. Andrae (Freie Universität Berlin), Prof. M. Baer (The Hebrew University of Jerusalem), Prof. O. Deeb (Al-Quds University), Prof. L. González (Universität Wien), Dr. T. Grohmann (Freie Universität Berlin), Prof. J. Manz (Freie Universität Berlin) and Prof. S. Zilberg (The Hebrew University of Jerusalem) for stimulating discussions and advice and PD Dr. B. Schmidt for access to his program wavepacket.36 Financial support by the Deutsche Forschungsgemeinschaft (project MA 515/22-3) is gratefully acknowledged.

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