Pyridinylidene-Phenoxide in Strong Electric Fields: Controlling

Article pubs.acs.org/JPCA

Pyridinylidene-Phenoxide in Strong Electric Fields: Controlling Orientation, Conical Intersection, and Radiation-Less Decay S. Belz,† S. Zilberg,‡ M. Berg,† T. Grohmann,† and M. Leibscher*,† †

Institut für Chemie und Biochemie, Freie Universität Berlin, Takustr. 3, 14195 Berlin, Germany Institute of Chemistry, The Edmond Safra Campus, Givat Ram, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel

‡

ABSTRACT: Strong electric fields open new routes for the control of radiation-less decay in molecules with conical intersections. Here, we present quantum chemical and quantum dynamical simulations which demonstrate that the radiation-less decay and related photoisomerization of pyridinylidene-phenoxide can be effectively manipulated with strong electric fields by shifting the conical intersection. Moreover, we show the effects of the electric field on the orientation of the molecules and on the photoexcitation and discuss the conditions for which the field induced coupling between rotational and vibronic states can be neglected. moment in the polar state, an electric field is expected to have a strong influence on the structure of PPO and the related lightinduced dynamics. Here, we investigate the effect of a strong electric field on the conical intersection and the related photoinduced torsion of PPO. Intense static or time-dependent electric fields are also known to efficiently control the rotational states of molecules. In particular, they induce orientation or alignment of molecules. Since molecular orientation and alignment has many applications, ranging from the control of chemical reaction dynamics to high harmonic generation, it has been studied intensively during the past decade (for a review, see refs 17 and 18). One method to orient polar molecules is to use strong static electric fields. If the energy of the interaction with the molecular dipole is much larger than the rotational energy, the molecules become oriented.19 The angular distribution of polarizable molecules can also be squeezed along the field polarization direction with strong nonresonant laser pules by creating pendular molecular states36 during the interaction with the field. Molecular orientation and alignment is commonly studied for molecules in the electronic ground state, where the molecules can often be considered as rigid under the experimental conditions and vibrational nuclear dynamics can be neglected. On the other hand, if the nuclear dynamics of photoexcited molecules is simulated, the rotational degrees of freedom are rarely taken into account. Strong electric fields, however, influence the rotation of the molecules as well as the shape of the potential energy surfaces and the coupling between electronic states, which can, for example, lead to light-induced

I. INTRODUCTION Controlling photochemical reactions with ultrafast laser pulses has become an extremely popular topic. Control is usually achieved by designing femtosecond laser pulses such that they steer the nuclear wavepacket to the desired product. Strong (static or time-dependent) electric fields also alter the shape of the potential energy surfaces, and it has recently been suggested to employ the Stark effect for controlling molecular processes.1 Usually, the changes in the shape of potential energy surfaces due to an electric field are rather small, but if two electronic states are almost degenerate and mixed, e.g., through a conical intersection or by spin−orbit coupling, small modifications of the electronic states can show a considerable influence on the coupling between the states and on the resulting nonadiabatic transitions. This has been successfully demonstrated by controlling the photodissociation of IBr using the dynamic Stark effect.2,3 The underlying nonadiabatic nuclear dynamics has been simulated both quantum mechanically4 and semiclassically.5 The possibilities to control photoinduced nuclear dynamics on coupled electronic states with the help of the dynamic Stark effect has been investigated for other diatomic molecules as well.6−9 Furthermore, the Stark effect has been suggested for the control of the ultrafast radiation-less decay through conical intersections.10−12 An important class of reactions which proceed via a conical intersection are photoisomerizations; the control of the related radiation-less decay by strong optical fields has also been investigated.13 Polar solvents have a similar effect on the location of a conical intersection as static electric fields.14 For instance, the change of photoreactivity of push− pull merocyanine in static electric fields, which has been predicted in ref 15, could be experimentally verified by changing the polarity of the solvent.16 A related compound is 4-(oxocyclohexadienylidene)-1,4-dihydropyridine or pyridinylidene-phenoxide (PPO) for short. Due to its large dipole © 2012 American Chemical Society

Special Issue: Jörn Manz Festschrift Received: May 25, 2012 Revised: August 3, 2012 Published: August 6, 2012 11189

dx.doi.org/10.1021/jp305090b | J. Phys. Chem. A 2012, 116, 11189−11198

The Journal of Physical Chemistry A

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conical intersections in diatomic molecules.20,21 Nevertheless, the effects of rotational degrees of freedom are often neglected in theoretical investigations of the control of radiation-less decay by strong electric fields, as in basically all theoretical studies mentioned above. In this paper, we study the effects of strong electric fields on both rotational and vibronic states of PPO and discuss the conditions under which the coupling between rotational and vibronic states due to the interaction with a strong field can be neglected. Therefore, we first demonstrate that PPO has a conical intersection associated with torsion of the two rings and show the related radiation-less decay under field-free conditions. Then, we investigate the effects of static and time-dependent electric fields on the rotational states of PPO and on the photoexcitation. Finally, we show how static electric fields influence the conical intersection, the nonadiabatic coupling terms, and the related radiation-less decay.

Figure 2. Longuet-Higgins loop for the conical intersection related to the cis−trans isomerization of PPO.

II. CONICAL INTERSECTION AND RADIATION-LESS DECAY Pyridinylidene-phenoxide (Figure 1) has a conical intersection between its ground and first excited electronic states which is

Table 1. Energies of the Ground State (S0) Minimum, the Transition States, and the Conical Intersection as well as the Energies of the Excited Electronic State (S1) for the Same Nuclear Geometriesa

S0 S1

GS minimum (eV)

biradical TS (eV)

zwitterionic TS (eV)

conical intersection (eV)

0 3.77

1.8019 1.9845

1.8090 1.9097

1.8332 1.8332

a

The energy difference between the electronic states at the conical intersection is ΔE = 4 × 10−8 eV.

Table 1. They are calculated on state-average complete active space self-consistent field level (CASSCF) with an active space of 11 molecular orbitals for 14 electrons in an aug-cc-pVDZ basis using the program package Molpro.25 A cut through the potential energy surface of the two lowest electronic states S0 and S1 is shown in Figure 3a. The potentials V0 and V1 are shown as a function of torsion angle φ, with all other nuclear coordinates frozen at a configuration close to the conical intersection. The electronic states ψ0 and ψ1 are coupled by the nonadiabatic coupling term Figure 1. Reference structure of pyridinylidene-phenoxide (PPO). The arrow indicates the torsion angle φ. The atoms are labeled as carbon = yellow, hydrogen = blue, oxygen = red, and nitrogen = magenta.

τφ(01) = ⟨ψ0|

∂ ψ⟩ ∂φ 1

(1)

where the integration is performed over the electronic coordinates and τ(01) is a function of the nuclear coordinate φ φ. The coupling term is shown in Figure 3b. Note that τ(01) has φ opposite signs at φ = π/2 and φ = −π/2, 3π/2. For a discussion of the symmetry of nonadiabatic coupling terms, see ref 26. The Longuet-Higgins phase change theorem states that the electronic states accumulate a geometrical phase if a conical intersection is encircled. This phase also affects the properties of the nonadiabatic coupling terms. For a two-state conical intersection, the contour integral of the nonadiabatic coupling terms along a closed loop Γ becomes

connected with the torsion around the C−C bond that links the two rings of PPO. It can be localized with the help of the Longuet-Higgins phase change theorem22 as implemented in, e.g., ref 23. The Longuet-Higgins loop shown in Figure 2 contains two transition states with a torsion angle of φ = 90°: a biradical transition state (TSb) which transforms as A2 in C2v and a zwitterionic transition state transforming as A1 (TSz). The Longuet-Higgins loop (cis configuration - biradical TS (A2) - zwitterionic TS (A1) - trans configuration) which is spanned by two reaction coordinates is a sign-inverting one; the degeneracy is a result of the crossing between A1 and A2 states along the coordinate connecting the two transition states. As a consequence, a single conical intersection has to be located inside the loop.24 It can be found by variation of the coordinate QD = QTSb − QTSz, where QTSb and QTSz are the nuclear coordinates of the two transition states. The energies of the transition states and of the conical intersection are shown in

∮Γ τ (01) dQ = π

(2)

if Γ encircles a conical intersection and

∮Γ τ (01) dQ = 0

(3)

if no conical intersection is enclosed. Here, τ = ⟨ψ0|∇ψ1⟩, where ∇ denotes the gradient with respect to the 27−29

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Figure 5. Absolute value of the autocorrelation function (a) and absorption spectrum of photoexcited PPO (b). The absorption spectrum has been calculated as a Fourier transform of the autocorrelation function propagated for t = 1000 fs.

connects the two rings of PPO, as shown in Appendix A. As a result, we obtain ∮ Γτ(01) dQ = 3.178 ± 0.037 or ∮ Γτ(01) dQ = 3.163 ± 0.013, depending on the contour Γ, which is in very good agreement with eq 2. The calculations provide an additional proof of the existence of the conical intersection as well as an indication for accurate quantum chemical calculation of the electronic states. In the following, we investigate the radiation-less decay of photoexcited PPO due to the conical intersection by solving the time-dependent nuclear Schrödinger equation along the torsion angle φ on two coupled electronic states, i.e.,

Figure 3. Panel a shows the potential energy curves of the electronic states S0 and S1 as a function of the torsion angle φ. The nonadiabatic coupling term τ(01) φ (φ) is depicted in panel b. The transition dipole is shown in panel c, and panel d displays the absolute moment μ(01) z (1) values of the dipole moments μ(0) z (solid lines) and μz (dashed lines).

iℏ

⎡ ⎤ ⎞2 ∂ ℏ2 ⎛ ∂ ψnuc(t ) = ⎢ − + τφ⎟ + V⎥ψnuc(t ) ⎜ ⎢⎣ 2Ired ⎝ ∂φ ⎥⎦ ∂t ⎠

(4)

2

where Ired = 44.23 Å mp is the reduced moment of inertia, mp is the proton mass, ⎛0 τφ(01)⎞⎟ ⎜ τφ = ⎜ ⎜−τ (01) 0 ⎟⎟ ⎝ φ ⎠

(5)

is the nonadiabatic coupling matrix, and ⎛ V1 0 ⎞ ⎟⎟ V = ⎜⎜ ⎝ 0 V0 ⎠ (1)

(6)

denotes the potential matrix. The total nuclear wave function can be written as

(1)

Figure 4. Excited state population P (t)/P (t = 0) after photoexcitation.

⎛|ψ (1)(t )⟩⎞ ⎜ nuc ⎟ ψnuc(t ) = ⎜ ⎟ (0) ⎝|ψnuc(t )⟩⎠

nuclear coordinates Q. The contour integral is calculated along a path around the conical intersection that includes the torsion angle φ and the bond length Z of the C−C bond which 11191

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can be neglected. The time evolution of the nuclear wave function ψnuc(t) is calculated in the diabatic representation (see, e.g., refs 27 and 34). We numerically propagate the diabatic nuclear wave function on the coupled electronic states using the split operator method as implemented in the program package “wavepacket”.30 The results are presented in the adiabatic representation. Figure 4 shows the change of population in S1 after photoexcitation, i.e., P(1)(t)/P(1)(0). After t ≈ 200 fs, about 50% of the initial population is transferred to the electronic ground state due to the strong coupling of the electronic states close to φ = π/2. More details about the dynamics of the torsional wavepacket can be obtained from the autocorrelation function (Figure 5a) or the absorption spectrum (Figure 5b), which has been calculated as a Fourier transformation of the autocorrelation function.35 The first recurrence of the autocorrelation function occurs at t1 = 526 fs. In a classical picture, this time interval corresponds to half a torsional period, i.e., a torsion by φ = 180°. In the absorption spectrum, this period is reflected by the (average) spacing between the absorption peaks, Δν = 1/t1 ≈ 1.8 × 1012 Hz or 7.4 × 10−3 eV. The center of the absorption spectrum indicates the energy of the vertical transition, which is 3.117 eV. Figure 3d shows the strong electronic dipole moments of PPO in the ground and excited electronic states. The difference between the dipole moments is particularly large close to the conical intersection at φ = π/2, which is an indication that applying an electric field can influence the nuclear dynamics close to the conical intersection. In the following, we will discuss how a strong electric field alters the torsional dynamics and the related radiation-less decay of PPO.

We assume that the molecules are initially in their rovibronic ground state, i.e., (1) |ψnuc (0)⟩ = 0

(8)

and (0) |ψnuc (0)⟩ = |χ0(0) ⟩|0, 0, 0⟩

(9)

|χ(0) 0 ⟩

where denotes the torsional ground state, i.e., the lowest eigenstate of V0. The rotational ground state is |J, K, M⟩ = |0, 0, 0⟩, where J, K, and M are the quantum numbers of a symmetric top rotor.31 In section III, we show that, for our purposes, PPO, which is an asymmetric top rotor, can be approximated as a symmetric top. The interaction between the molecules and a short, relatively weak laser pulse can be described in dipole approximation by Ĥ int(t ) = −μ̂ ·Epulse(t )

(10)

where μ̂ is the electric dipole operator and Epulse(t) = ezEpulse(t) cos(ωlt) is the electric component of a linear polarized laser pulse. If PPO is approximated as a symmetric top rotor, the interaction Hamiltonian reads Ĥ int(t ) = −μ ̂ Epulse(t ) cos θ − μ⊥̂ Epulse(t ) sin θ[sin χ − cos χ ]

(11)

with the components of the electric dipole operator μ̂∥ = μ̂z and μ̂ ⊥ = μ̂ x = μ̂y as well as the Euler angles θ and χ (see, e.g., ref 33). A short laser pulse with the pulse duration Δt and a central laser frequency ωl resonant to a transition to the excited electronic state can excite a wavepacket in S1. We assume that the pulse is short enough, so that the rotational and internal nuclear motion is frozen during the interaction, i.e., Epulse(t) can be approximated by a δ-function as Epulse(t) = E0Δt(2π)1/2δ(t). The excited state wavepacket at the time immediately after the interaction can then be written as 1 (1) |ψnuc (Δt )⟩ = − E0Δt 2π μz(01)|χ0(0) ⟩ cos θ|0, 0, 0⟩ iℏ (01) 1 E0Δt 2π μz =− |χ0(0) ⟩|1, 0, 0⟩ (12) iℏ 3

III. PPO IN STRONG ELECTRIC FIELDS: ORIENTATION, ALIGNMENT, AND PHOTO-EXCITATION In the following two sections, we investigate how a strong electric field influences the dynamics of PPO molecules. We Table 2. Rotational Constants and Dipole Moment and Polarisability Anisotropy for PPOa A (10−6 eV)

B (10−6 eV)

C (10−6 eV)

μ(0) z (D)

Δα (Å3)

11.67

1.67

1.46

12.56

40.39

a

The values are calculated at the CASSCF(14,11) level with the augcc-pVDZ basis using the program package Molpro.25

where E0 is the maximal field strength and μ(01) = ⟨ψ1|μ̂z|ψ0⟩ is z the z-component of the transition dipole moment. The transition dipole moment is almost constant over a large range of torsion angles but changes its sign at φ = π/2 and 3π/ 2, as it can be seen in Figure 3c. The torsional ground state |χ(0) 0 ⟩ is a symmetric function of the torsion angle φ. Due to the sign change of the transition dipole moment μ(01) z , the torsional wavepacket in the excited electronic state is antisymmetric. The interaction depends only on cos θ because μ(01) = μ(01) = 0. As a x y consequence, a transition occurs from J = 0 to J = 1, with conserved rotational quantum numbers K and M. For simulations of photoinduced nuclear dynamics, vertical excitation is widely assumed in order to define the initial wavepacket on the excited electronic state. Equation 12 presents an extension of the vertical transition approximation, which also takes into account rotational states. Including rotational states in the description of the photoexcitation will become important when we investigate photoexited PPO in a strong electric field. For now, we assume that the coupling between the rotational state |1, 0, 0⟩ and the torsional motion

consider both static electric fields and moderately intense nonresonant optical fields. The electric fields considered here affect the PPO molecules in several ways. First of all, the field alters the rotational states and creates so-called pendular states.19,37 Molecules with a large permanent dipole moment, like PPO, become oriented in the presence of a static electric field. A nonresonant optical field can also squeeze the angular distribution of the molecules, causing alignment.18 Second, photoexcitation depends on the relative orientation of the molecules with respect to the polarization direction of the electric field. Oriented or aligned molecules in a strong electric field will thus respond in a different way to a laser pulse than randomly oriented molecules. Finally, the potential energy surfaces are deformed by the electric field. The deformation is usually small, but if the electronic states are nearly degenerate, e.g., at a conical intersection, the electric field can cause a considerable change in the related coupling elements. In the following, we will discuss all three aspects of the Stark effect for the example of PPO. 11192



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Figure 6. Panels a and b show the orientation factor ⟨cos θ⟩ in the ground (solid lines) and excited (dashed lines) electronic states. Panels c and d display the relative excited state population P(1)(E)/P(1)(E = 0) immediately after photoexcitation. The field intensities range from 10−2 to 107 W/ cm2 in panels a and c and from 107 to 1012 W/cm2 in panels b and d. The vertical lines in panels b and d indicate the field intensities for which the influence of the electric field on the radiation-less decay is investigated: I = 8.5 × 109 W/cm2 (red), I = 1.4 × 1011 W/cm2 (blue), and I = 8.5 × 1011 W/cm2 (green).

−0.959, which is very close to the prolate symmetric top limit κ = −1. We therefore approximate PPO as a prolate symmetric top rotor with the symmetric top eigenfunctions |J, K, M⟩ and the eigenvalues EJ,K = B̅ J(J + 1) + (A − B̅ )K2, where B̅ = (B + C)/2.32 Within this approximation, the field-dressed Hamiltonian can be written (up to a constant) as FD Ĥ rot = Ĥ rot − Eμz cos θ

(15)

(0) since μ(0) x = μy = 0. The z-component of the dipole moment (0) μz is given in Table 2. The eigenstates of the field dressed Hamiltonian HFD rot can be written as a superposition of the symmetric top eigenfunctions

|χrot ⟩FD =

∑ cJ|J , K , M⟩ J

where the quantum numbers K and M are conserved in a linear polarized electric field. The coefficients cJ are determined by 19,37 diagonalizing Ĥ FD Here, rot in the basis of the free rotor states. we assume that the molecules are initially in their rotational ground state. The orientation of the molecules in the electric field can be measured by the orientation factor

Figure 7. Panel a shows the alignment factor ⟨cos θ⟩ in the ground (solid lines) and excited (dashed lines) electronic states. Panel c shows P(1)(E)/P(1)(E = 0) immediately after photoexcitation. The vertical lines indicate the field intensities for which the influence of the electric field on the radiation-less decay is investigated: I = 8.5 × 109 W/cm2 (red), I = 1.4 × 1011 W/cm2 (blue), and I = 8.5 × 1011 W/cm2 (green). 2

⟨cos θ ⟩ =

∑ c*J′cJ⟨J′, 0, 0| cos θ|J , 0, 0⟩ J ,J′

For molecules in the vibronic ground state, the rigid rotor approximation is well suited to model the effect of an electric field on the rotational states. If the molecules interact with a static electric field E, the field dressed rotational Hamiltonian can be written as FD Ĥ rot = Ĥ rot − μ E

(17)

where ⟨cos θ⟩ = 1 denotes the complete orientation of the molecules in the direction of the field. If ⟨cos θ⟩ = −1, the molecules are oriented antiparallel to the field, and ⟨cos θ⟩ = 0 refers to a uniform angular distribution. The solid lines in Figure 6a and b show the orientation factor of PPO as a function of the field intensity I = 1/2cϵ0E2, with the speed of light c and the dielectric constant ϵ0. For I = 0, the angular distribution of the molecules is uniform and ⟨cos θ⟩ = 0. Due to its large dipole moment, the orientation of PPO increases rapidly with rising field intensity. For intensities of I = 107 W/ cm2 and higher, the molecules are completely oriented. We conclude that even relatively weak electric fields strongly affect the rotation of PPO and cause almost complete orientation of the molecules. As a result, the initial state of molecules in an electric field before photoexcitation is

(13)

PPO is an asymmetric top rotor; i.e., the field-free Hamiltonian reads 2 2 2 Ĥ rot = AJâ + BJb̂ + CJc ̂

(16)

(14)

with the rotational constants A ≠ B ≠ C . The electric dipole moment is μ = ⟨ψ0|μ̂|ψ0⟩, where |ψ0⟩ denotes the electronic ground state and μ̂ the dipole operator. The rotational constants of PPO are shown in Table 2. They correspond to an asymmetry parameter38 κ = (2B − A − C)/(A − C) = 11193



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where Δα = αaa − αbb is the polarizability anisotropy and , (t) is the envelope of the electric field. Due to the cos2 θ dependence of the interaction, a nonresonant optical field can create alignment of molecules but no orientation. If the pulse duration is long compared to the time of the rotational dynamics, adiabatic alignment is induced during the pulse. To determine the degree of adiabatic alignment, we approximate , (t) = E, where E denotes the maximal field strength, and determine the rotational state in the presence of the electric field with eq 22 for the field-dressed rotational Hamiltonian. The resulting alignment factor ⟨cos2 θ⟩ is shown in Figure 7a in solid lines. For I = 107 W/cm2, the angular distribution is still isotropic. Since the interaction is proportional to E2, the intensities which are required to achieve complete alignment are higher than those which are necessary to completely orient molecules in a static field. However, for intensities of I ≥ 1010 W/cm2, the molecules are almost completely aligned. PPO molecules can be aligned by relatively weak optical fields because they have a particular large polarizability anisotropy. The relative population of the excited electronic state S1 after photoexcitation, eq 20, is shown in Figure 7b. For I ≥ 1010 W/ cm2, it converges to a value which is 3 times larger than the field-free excited state population. Moreover, we show the alignment factor for PPO in the excited electronic state S1 immediately after photoexcitation, i.e.,

(0) |ψnuc (0)⟩ = |χ0(0) ⟩ ∑ cJ |J , 0, 0⟩

(18)

J

which becomes eq 9 if the field strength E goes to zero. Similar to eq 12, the wavepacket immediately after the photoexcitation can then be written as |ψ (1)(Δt )⟩ = −

1 E0Δt 2π μz(01)|χ0(0) ⟩ ∑ cJ cos θ|J , 0, 0⟩ iℏ J (19)

Equation 19 describes the effect of a strong electric field on the photoexcited wavepacket. As it is well-known, the ability to absorb light depends on the orientation of the molecules. In particular, we calculate the population of the excited electronic state for molecules in a static field with field strength E compared to the excited state population for molecules without external electric field P(1)(E) (1)

P (E = 0)

= 3 ∑ c*J ′cJ⟨J ′, 0, 0| cos2 θ|J , 0, 0⟩ J ,J′

= 3⟨cos2 θ ⟩

(20)

where ⟨cos θ⟩ is the alignment factor of the molecules before photoexcitation. The alignment factor is ⟨cos2 θ⟩ = 1/3 for an isotropic angular distribution and increases to ⟨cos2 θ⟩ = 1 for completely aligned molecules, i.e., for molecules which are oriented either parallel or antiparallel to the field. The absorption of light is thus maximal if the molecules are completely aligned with the polarization direction of the field. Although the molecules interact with the field via their dipole moment, it is irrelevant whether they are also oriented or not. The relative population of S1, eq 20, for PPO is shown in Figure 6c and d. It increases rapidly with the field intensity up to 3 times the field-free limit. We also determine the orientation of the molecules in the excited state. Immediately after the interaction, the orientation factor of the excited wavepacket is 2

⟨ψ (1)(Δt )| cos θ|ψ (1)(Δt )⟩ ⟨ψ (1)(Δt )|ψ (1)(Δt )⟩

=

⟨cos3 θ ⟩ ⟨cos2 θ ⟩

⟨ψ (1)(Δt )| cos2 θ|ψ (1)(Δt )⟩ ⟨ψ (1)(Δt )|ψ (1)(Δt )⟩

= Hrot

1 − ,2(t )Δα cos2 θ 4

⟨cos 4 θ ⟩ ⟨cos2 θ ⟩

(23)

as dashed lines in Figure 7a. An intensity of I = 107 W/cm2 is too small to induce alignment. Here, the alignment factors are the same as in the field-free case. Without an external field, the rotational state in S1 after photoexcitation is |1, 0, 0⟩. The rotational state |1, 0, 0⟩ is already aligned with the field polarization direction, while the ground rotational state |0, 0, 0⟩ has an isotropic angular distribution. For small intensities, the alignment factor for PPO in the excited electronic state is therefore considerably larger than for PPO in the electronic ground state. With increasing intensity, the excited state alignment factor approaches 1. For I ≥ 1010 W/cm2, the excited state PPO is completely aligned. We conclude that also nonresonant optical fields with the maximal intensities indicated by the vertical lines in Figure 7 almost completely align the PPO molecules both in the ground and excited electronic state and increase the excited state population after photoexcitation by a factor of 3 compared to the field-free case.

(21)

It is shown for PPO in Figure 6a and b in dashed lines. For a given field intensity, the orientation factor of the excited state wavepacket is higher than that for the molecules in the ground state. Due to the first order interaction with the laser pulse, rotational states with the quantum numbers J′ = J + 1 are excited while the quantum numbers K and M are preserved, and rotational states with higher J and conserved values of K and M are better oriented. The vertical lines in Figure 6b and d indicate the field intensities which we will use in section IV to investigate how the electric field affects photoinduced nuclear dynamics. For all three intensities, the molecules are almost perfectly oriented. Moreover, the excited state population after photoexcitation is 3 times larger than that for molecules under field-free conditions. The rotational states of molecules can also be influenced by optical fields; moderately intense nonresonant laser pulses align molecules along the polarization direction of the field.17,18 The molecules interact with an optical field via their induced dipole moments, and eq 13, averaged over the fast optical frequencies, becomes18 FD Ĥ rot

=

IV. CONICAL INTERSECTION AND RADIATION-LESS DECAY IN ELECTRIC FIELDS In the following, we discuss how static electric fields influence the nuclear dynamics of photoexcited molecules. Therefore, we simulate the torsional dynamics for exemplary field strengths, namely, E = 2.5 × 108 V/m (I = 8.5 × 109 W/cm2), E = 1.0 × 109 V/m (I = 1.4 × 1011 W/cm2), and E = 2.5 × 109 V/m (I = 8.5 × 1011 W/cm2), as indicated in red, blue, and green lines in Figures 6 and 7. The field-dressed molecular Hamiltonian can be written as FD FD Ĥ = Ĥ el + TN̂

(24)

where T̂ N is the kinetic energy operator for the nuclei and

(22) 11194



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Figure 8. Potential energy curves (a) and nonadiabatic coupling terms (NACTs) (b) for PPO without electric field (black lines) and in a static electric field with E = E1 = 2.5 × 108 V/m (I = 8.5 × 109 W/cm2, red lines), E = E2 = 1.0 × 109 V/m (I = 1.4 × 1011 W/cm2, blue lines), and E = E3 = 2.5 × 109 V/m (I = 8.5 × 1011 W/cm2, green lines). Panels c and d show a magnification of the potential energy curves and NACTs close to the conical intersection. See also ref 39.

Figure 10. Absolute value of the autocorrelation function (a) and absorption spectrum (b) of PPO in a static electric field with E = E1 = 2.5 × 108 V/m (I = 8.5 × 109 W/cm2, red lines), E = E2 = 1.0 × 109 V/ m (I = 1.4 × 1011 W/cm2, blue lines), and E = E3 = 2.5 × 109 V/m (I = 8.5 × 1011 W/cm2, green lines). The spectrum has been determined by a Fourier transform of the autocorrelation function propagated for 1000 fs (black and red lines) and 1500 fs (blue and green lines).

and NACTs, it is not necessary to estimate in advance how many electronic states mix with the two states considered here due to the interaction with the field. A cut through the potential energy surface for the same nuclear coordinates as in Figure 3 has been calculated for different field strengths. Figure 8a and c shows the field-dressed potentials. Note that the overall shift of the potential energies toward lower energies with increasing field strength is neglected here, since it does not affect the nuclear dynamics. The electric field causes a slight decrease in the Franck−Condon energies (at φ = 0, π) with growing field strength. The slope of the potentials is deformed mainly close to φ = π/2, where the fieldfree potentials are almost degenerate. For E = 2.6 × 108 V/m (red line), the gap between V1 and V0 is even smaller. For stronger fields, the energy gap becomes larger. Small changes of the potential gap close to a conical intersection can lead to considerable changes in the nonadiabatic coupling elements 8 τ(01) φ , which are shown in Figure 8b and d. For E = 2.5 × 10 V/ m (red lines), the coupling at φ = π/2 increases compared to the field-free case. For E = 1.0 × 109 V/m (blue lines) and E = 2.5 × 109 V/m (green lines), the maximum of τ(01) φ decreases to almost zero, and the peaks become broader. In order to decide if there is still a conical intersection for the configuration QCI, which has been located at field-free conditions, we calculate the contour integral ∮ Γ τ(01) dQ along the same contours as in section II with the field-dressed electronic states. For E = 1.0 × 109 V/m and E = 2.5 × 109 V/m, we find that

Figure 9. Excited state population P(1)(t)/P(1)(t = 0) after photoexcitation in a static electric field with the field strength E = E1 = 2.5 × 108 V/m (I = 8.5 × 109 W/cm2, red line), E = E2 = 1.0 × 109 V/m (I = 1.4 × 1011 W/cm2, blue line), and E = E3 = 2.5 × 109 V/m (I = 8.5 × 1011 W/cm2, green line). The field-free case is depicted in solid black lines. See also ref 39. FD Ĥ el = Ĥ el − μẑ E

= Ĥ el − μẑ E cos θ − E sin θ(μŷ sin χ − μx̂ cos χ ) (25)

is the field-dressed electronic Hamiltionian. The field-free electronic Hamiltonian is denoted by Ĥ el, and the operators μ̂x, μ̂ y, and μ̂z are the components of the electric dipole operator. Since the PPO molecules are completely oriented in a static electric field (and aligned in an optical field), θ ≈ 0 and eq 25 is reduced to FD Ĥ el = Ĥ el − μẑ E

(26)

Due to the complete orientation (or alignment) of the molecules, the coupling between vibronic and rotational degrees of freedom caused by the interaction between the molecules and the electric field vanishes. The field-dressed potentials and nonadiabatic coupling terms have been determined by adding a finite dipole field to the one-electron Hamiltonian and the core energy of the system as implemented in the Molpro program package25 (see also ref 15). By direct quantum chemical calculation of the field dressed potentials

∮Γ τ (01) dQ ≈ 0 11195

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intensity of the laser pulse increases to 3 times the field-free value. Due to the complete orientation of the molecules in the electric fields considered here, the coupling between rotational and vibronic states caused by the interaction with the field can indeed be neglected for the simulation of photoinduced nuclear dynamics. Electronic states accumulate a geometrical phase when they are transported around a conical intersection. We employ this phasein the form of the Longuet-Higgins phase change theorem and by calculating contour integrals of the nonadiabatic coupling termsto localize the conical intersection and to show that the location, i.e., the nuclear coordinates of a conical intersection, can be shifted by strong electric fields. Quantum dynamical simulations of the nuclear dynamics along the torsion angle φ show that a static electric fieldwith the field strength as a control parametercan be used to control the photoinduced torsional dynamics and the amount of the related radiation-less decay. Here, we consider only one nuclear degree of freedom in the quantum dynamical simulations, namely, the torsion angle φ. It is well-known that additional fast vibrations may suppress the torsional motion. However, as we have shown in ref 40 and 41, properly designed excitation pulses can prevent such fast vibration and promote torsional motion in the excited electronic state and the related radiationless decay. Nevertheless, it will be necessary to include also additional nuclear coordinates in further studies, in particular the C−C bond connecting the two rings of PPO. Due to their large dipole moments, PPO molecules are completely oriented in the presence of the static electric fields, which are applied in order to control the radiation-less decay. If an electric field does not completely orient the molecules, for example, if a sample of molecules at higher initial temperatures or molecules with smaller dipole moments are investigated, the coupling between rotational and vibronic states might lead to additional interesting effects. It can be difficult to generate strong static electric fields in the laboratory. The lowest field strength considered here is of an order that is available experimentally but not yet the higher ones. A possible way to generate much higher field strengths is to employ nonresonant laser pulses. Here, we demonstrated that a long, nonresonant laser pulse can align the PPO molecules. A high degree of alignment also allows the rotational−vibronic coupling due to the interaction with the field to be neglected. An optical field interacts with the molecules mainly due to the polarizability anisotropy of the molecules. Since PPO has also a large polarizability anisotropy, we expect that effects similar to those which have been demonstrated here for a static electric field will also be observed in a nonresonant laser field.

i.e., no conical intersection is located inside the contour Γ. Finally, we simulate the photoinduced nuclear dynamics along the torsion angle φ in the electric field. The radiation-less decay after photoexcitation is shown in Figure 9, where the change of excited state population, P(1)(t)/P(1)(t = 0), is plotted for different field strengths. The field-free case is indicated by the black line. As discussed in section II, about 50% of the initial population is transferred to the electronic ground state The radiation-less decay is increased to almost 80% for E = 2.5 × 108 V/m (red line). For stronger fields (E = 1 × 109 V/m, blue line), the population transfer decreases. Only a few percent of the initial population decay to the electronic ground state, and for E = 2.5 × 109 V/m (green line), the excited state population is constant and no significant radiation-less decay is observed. We also compare the autocorrelation function (Figure 10a) of photoexcited PPO and the absorption spectrum (Figure 10b) for the different field strengths. The most noticeable difference in the absorption spectra lies in the total shift toward smaller energies with increasing field strengths. The shift is caused by the decrease in the Franck−Condon energies (vertical transition at φ = 0, π). A similar effect has been observed experimentally for merocyanine in solvation with different polarities.16 Moreover, the first recurrence of the autocorrelation function (Figure 10a) varies between t1 = 468.6 fs (red line) and t1 = 985.6 fs (green line). By the electric fields considered here, the period of torsion can thus be altered by a factor of 2. For E = 2.5 × 108 V/m (red line), the period of torsion is decreased compared to the field-free case (black line); for E = 1.0 × 109 V/m (blue line) and E = 2.5 × 109 V/m (green line), it is increased. Comparison with Figure 9 illustrates that enhanced radiation-less decay leads to shorter periods of torsion, while the suppression of radiation-less decay is accompanied by an increase in the torsional period; the large amount of kinetic energy which is released during the radiationless decay leads to faster torsional motion of the wavepacket. The different recurrence times in the autocorrelation function are also reflected in the spacing between the peak intensities of the absorption spectra, as has been discussed for the field-free case (section II). For E = 2.5 × 108 V/m (red line in Figure 10b), the spacing is larger than in the field-free case (black line); for the two higher field strengths (blue and green lines), it is smaller. Our simulations thus show that the torsional dynamics and the related radiation-less decay of photoexcited PPO can be effectively controlled by a strong electric field.

V. CONCLUSIONS In this study, we have investigated the effect of a strong electric field on the conical intersection and the resulting radiation-less decay of pyridinylidene-phenoxide (PPO). It is a promising system for observing the effects of an electric field, since it has a particularly large permanent dipole moment and polarizability anisotropy. The interaction with strong electric fields affects the vibronic and rotational degrees of freedom and, in general, leads to their coupling. Nevertheless, the rotations of molecules are often neglected in theoretical investigations of the Stark effect on photoinduced nuclear dynamics. Here, we also study the effect of electric fields on the rotational states of PPO and include rotational states in the treatment of the photoexcitation. We find that static electric fields with intensities of I ≥ 107 W/ cm2 completely orient the PPO molecules in the ground and excited electronic states. Nonresonant laser pulses lead to complete alignment for intensities of I ≥ 1010 W/cm2. As a consequence, the amount of photoexcited PPO at a given

VII. APPENDIX A: CONTOUR INTEGRALS In order to prove the existence of the conical intersection between the electronic states S0 and S1 connected with the internal torsion between the two rings of PPO, we calculate contour integrals of the form ∮ Γτ(01) dQ, with the nonadiabatic coupling matrix τ (01) = ⟨ψ0|∇ψ1⟩

(28)

as defined in eq 2 and below. We choose a path as sketched in Figure 11, where φ is the torsion angle and Z is the length of the C−C bond which connects the two rings of PPO. The conical intersection is located at Z0 = 1.47 Å. The contour integral then becomes 11196



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∮Γ τ (01) dQ = π

(33)

27−29

is fullfilled with good numerical accuracy, which proves the existence of the conical intersection located with the help of the Longuet-Higgins phase change theorem, as shown in Section II.

■

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

■

Figure 11. The blue arrows indicate the contour Γ for which the contour integrals ∮ Γ τ(01) dQ are calculated. 90 °+Δφ

∮Γ τ (01) dQ = ∫90°−Δφ

ACKNOWLEDGMENTS We wish to thank Prof. L. González (Universität Wien), Dr. J. González Vázquez (Universidad Complutense de Madrid), Prof. Y. Haas (Hebrew University of Jerusalem), Prof. J. Manz (Freie Universität Berlin) for stimulating discussions, and PD Dr. B. Schmidt (Freie Universität Berlin) for access to the program wavepacket. The computing facilities (ZEDAT) of the Freie Universität Berlin are acknowledged for computer time and the German Research Foundation (projects MA 515/22-3 and LE 2138/2-1) for financial support.

[τφ(01)(φ , Z0 + ΔZ)

− τφ(01)(φ , Z0 − ΔZ)] dφ Z0 +ΔZ

+

∫Z −ΔZ

[τZ(01)(90° − Δφ , Z)

−

τZ(01)(90°

+ Δφ , Z)] dZ

0

■

(29)

with τ(01) defined in eq 1 and φ τZ(01) =

ψ0

∂ ψ ∂Z 1

(30)

Figure 12. Nonadiabatic coupling terms τ(01) as a function of Z (panel Z as a function of φ (panel b) for ΔZ = 0.05 Å (dashed a) and τ(01) φ lines) and ΔZ = 0.005 Å (solid lines).

The coupling terms τ(01) and τ(01) along those contours are φ Z shown in Figure 12 for Δφ = 1° and ΔZ = 0.05 Å (dashed lines) or ΔZ = 0.005 Å (solid lines). We obtain

∮Γ τ (01) dQ = 3.178 ± 0.037

(31)

for the small contour and

∮Γ τ (01) dQ = 3.163 ± 0.013

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Pyridinylidene-Phenoxide in Strong Electric Fields: Controlling

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