Intermode Coupling Drives the Irreversible ... - ACS Publications

Letter pubs.acs.org/JPCL

Intermode Coupling Drives the Irreversible Tautomerization in Porphycene on Copper(111) Induced by Scanning Tunnelling Microscopy Dino Novko,† María Blanco-Rey,†,‡ and Jean Christophe Tremblay*,¶ †

Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, 20018 Donostia−San Sebastián, Spain Departamento de Fı ́sica de Materiales, Facultad de Quı ́micas UPV/EHU, Apartado 1072, 20018 Donostia−San Sebastián, Spain ¶ Institut für Chemie und Biochemie, Freie Universität Berlin, Takustrasse 3, 14195 Berlin, Germany ‡

S Supporting Information *

ABSTRACT: In this contribution, we develop a nonadiabatic theory that explains, from first-principles, the recently reported irreversible trans → cis tautomerization of porphycene on Cu(111) induced by a scanning tunnelling microscope at finite bias. The inelastic contribution to the STM current is found to excite a large number of skeletal vibrational modes of the molecule, thereby inducing a deformation of the potential energy landscape along the hydrogen transfer coordinate. Above a threshold bias, the stability of the tautomers is reversed, which indirectly drives the reaction via intermode coupling. The proposed potential deformation term accounts effectively for the excitation of all internal vibrational modes without increasing the dimensionality of the problem. The model yields information about reaction rates, explains the reaction irreversibility at low temperatures, and accounts for the presence of resonant processes.

I

low-dimensional space, where only a few relevant degrees of freedom are active. In the porphycene/Cu(111) system, the tautomerization reaction coordinate can be identified as H atom migration between the inequivalent imine and amine groups inside of the porphycene cavity (see the inset of Figure 1). The potential energy profile involves a low trans → cis activation barrier (∼150 meV) and a small energy difference (∼85 meV) between trans and cis.33 An important distortion of the molecular frame is observed along the adiabatic tautomerization path leading to the cis configuration, which binds closer to the surface and buckles to produce characteristic STM images.33 This implies that many of the porphycene internal vibrational modes are involved in the H transfer coordinate. A fully quantum dynamical description of a problem of such dimensions is beyond any conventional method. In this Letter, we put forward a microscopic dynamical model based on density functional theory (DFT) calculations that circumvents the need for high-dimensional quantum dynamics and rationalizes the experimental details of the STM-induced trans → cis tautomerization reaction in porphycene/Cu(111) in simple physical terms. The concept unifying energy relaxation and STM is the coupling of the adsorbate vibrations to electron− hole pairs in the metal substrate.27 Our model consists of treating intermode coupling as a deformation of the vibrationally

ndirect adsorbate transformation by hot electrons has gained prominence as a tool to control the reactivity of species in the condensed phase. In this context, tautomerization reactions are particularly attractive because large changes in chemical reactivity are associated with simple intramolecular transformations.1,2 In recent years, much effort has been devoted to harness their potential as molecular switches,3−12 for which porphycene on Cu surfaces appears as a promising candidate.13−17 In particular, the trans → cis tautomerization of porphycene on Cu(111) was accomplished by injection of electrons via scanning tunnelling microscopy (STM).15 The process was observed only above a specific threshold voltage, below the resonant frequency of the NH stretching mode, and in the vibrational band of the skeletal modes. The reaction takes place up to tens of nanometers away from the STM tip position, and the backward cis → trans tautomerization can be induced only by thermal activation. To our knowledge, no theory is able to explain all of these observations simultaneously. The driving force for STM-induced reactions on metallic surfaces is the nonadiabatic coupling (NAC) between the excited electrons of the substrate and the adsorbate internal vibrations.18−22 Reactions involving large-amplitude motions require proper treatment of intermode couplings beyond the harmonic approximation.23,24 A few theories have been proposed to account for anharmonic coupling in the NAC terms, either along the reaction path,25−28 by introducing coupling terms with specific soft modes,29−32 or by increasing dimensionality.25−27 Although promising, these methods remain only tractable in a © 2017 American Chemical Society

Received: January 19, 2017 Accepted: February 15, 2017 Published: February 15, 2017 1053

DOI: 10.1021/acs.jpclett.7b00141 J. Phys. Chem. Lett. 2017, 8, 1053−1059

Letter

The Journal of Physical Chemistry Letters

In physical terms, the STM creates hot electrons in the metal that are propagated as an S-wave through the substrate to reach the adsorbate.37 This accounts for the fact that tautomerization can be triggered in molecules far away from the injection point of the tunnelling electrons.15 The NAC between hot surface electrons and the molecular vibrations indirectly distorts the potential nonuniformly along the reaction path via intermode coupling, thereby reversing the thermodynamical stability of the trans and cis configurations (see the cartoon in Figure 1). The H atom tunnels through the tautomerization barrier (with time τ0 ≫ τrelax) and relaxes rapidly (τrelax → 0) through electron−hole pair coupling to the thermodynamically more stable cis configuration. At higher bias voltages, resonant excitation of the tautomerization mode increases the tunnelling rate (τ1 < τ0) along with the global reaction rate. In formal terms, the deformation is modeled as a sum of contributions coming from vibrational excitation of all modes perpendicular to the reaction path q via NAC

Figure 1. Schematic representation of the STM-induced tautomerization of porphycene on Cu(111). After switching on the bias voltage, the molecule is found in the trans configuration (upper curve, thick red line). The molecule tunnels to the cis configuration within a time τ0 and relaxes almost instantaneously (τrelax → 0) to the local ground state (thick blue line). At a bias above the resonance, the trans molecule becomes partially excited to a state with shorter tunnelling time τ1 < τ0, accelerating the tautomerization dynamics. After switching off the bias voltage (lower curve), the molecule remains trapped in the cis configuration, τc → ∞.

ΔV (q; U , I ) =

α

(1)

A large part of the intermode coupling is included in the zerothorder potential, V0(q), by adiabatic relaxation of the atomic positions along the constrained reaction path (see the Supporting Information (SI)). The potential deformation ΔV(q; U, I), where (U, I) are respectively the STM bias voltage and current intensity, originates from the excitation of all modes orthogonal to the reaction path via NAC to hot electrons in the metal. Note that a field-induced potential distortion and intermode coupling could also in principle arise from the proximity of the STM tip. In the case of porphycene/Cu(111), the tautomerization potential is largely determined by the intramolecular H bonds that are parallel to the surface and, consequently, should remain insensitive to the electrostatic field perpendicular to the surface. Complementary DFT calculations including dipole correction (see the SI) confirm that the fieldinduced interactions are marginal for the studied system and can therefore be neglected. The even weaker polarizability effects were found to be insufficient to explain the switching behavior in similar systems.34−36

α) Γ(i → j

(2)

As shown in the SI, this expression is obtained by statistical averaging of the total Hamiltonian over the remaining vibrational degrees of freedom, in which these so-called “phonon bath modes” depend parametrically on the system reaction coordinate. Intermode coupling is thus included in the reduced-dimensional dynamics as a mean field, quasi-thermal potential deformation via the associated vibrational frequencies ω(α) 01 (q). These are evaluated in the local harmonic approximation at each ab initio point along the reaction path (see the SI). The term in parentheses describes the probability of excitation of the locally harmonic mode Qα as the ratio of upward to downward rates. As demonstrated elsewhere,27,28,37 these rates are a function of the STM parameters (bias voltage U and current intensity I), the temperature, and the intermode coupling. Note that eq 2 is the low-bias, low-temperature limit of a more general equation (see the SI), such that only the first excited state of each mode contributes to the potential deformation. The associated populations can thus simply be obtained from the ratio of the downward to upward rates, that is, (α) and Γ(α) Γ1→0 0→1, respectively. Their parametric dependence on the tautomerization coordinate q accounts for intermode coupling. The rates required in eq 2 originate from the creation/ annihilation of electron−hole pairs in the metal, which can be computed perturbatively using Fermi’s Golden Rule. In recent work,28 it was shown how the anharmonic coupling rates for all vibrational transitions i → j can be calculated as

adiabatic potential energy curve (PEC) along the H transfer coordinate q V (q; U , I ) = V0(q) + ΔV (q; U , I )

⎛ Γ(α) ⎞ 0→1 ⎟ (α) ℏω01 (q) (α) ⎟ ⎝ Γ1 → 0 ⎠(q)

∑ ⎜⎜

⎧ ⎡ ⎤ ⎛ ⎞ ⎪ ⎢B(|ℏω |) + w (U , I )⎜ e|U | − |ℏωij| ⎟Θ(e|U | − |ℏω |)⎥γ (α) if Ej > Ei ij α ij ⎜ ⎟ ⎪ ⎢ ⎥⎦ ij |ℏ | ω ⎝ ⎠ ij ⎣ ⎪ =⎨ ⎤ ⎪⎡ ⎛ e|U | − |ℏωij| ⎞ ⎪ ⎢1 + B(|ℏωij|) + wα(U , I )⎜⎜ ⎟⎟Θ(e|U | − |ℏωij|)⎥γij(α) otherwise ⎥⎦ ⎪ ⎢⎣ |ℏωij| ⎝ ⎠ ⎩

which is expressed here in a form also valid in the low-bias limit.38,39 The Bose−Einstein distribution B(|ℏωij|) ensures that the static NAC rates obey the detailed balance. The last term in

(3)

the brackets describes the efficiency of the STM perturbation, where the step function Θ ensures that only vibrational states at energies below e|U| can be excited. Note that throughout this 1054


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Figure 2. (a) Vibrational DOS of the porphycene for three points along the tautomerization coordinate, cis, TS, and trans configurations, broadened using a Lorentzian with fwhm = 1 meV. The vertical dashed lines represent the frequency of the H transfer coordinate mode. (b) Harmonic lifetime of all vibrational modes due to electron−hole pair coupling at the metal. The associated rates are computed using the independent atom model in the harmonic limit; see eq 4.

Figure 3. (a) Potential deformation induced by indirect coupling of the skeletal vibrational modes to electron−hole pairs of the metal for increasing STM biases (from 0 mV in blue to 400 mV in red) and fixed STM current intensity I = 100 pA. The temperature is T = 5 K. The trans configuration (left structure) is seen to be more strongly affected by the presence of the STM than the cis configuration (right structure). The field-induced potential deformation is neglected. The inset shows the variation of the projected electronic density of states around the Fermi energy, which affects the efficiency of the NAC. (b) Vibrational ground-state density for different bias voltages, shifted at the zero-point energy. Despite the small potential deformation, the higher biases are seen to favor the cis configuration over the thermodynamically more stable trans configuration.

Letter the wording “STM perturbation” refers to an inelastic electron tunnelling process and the resulting NAC. As already mentioned, other possible perturbations, such as chemical interaction between the STM tip and the molecule and dipole−tip electrostatic interaction, are not included here. The Iτ STM enhancement factor, wα(U , I ) = eρ (αU ) , accounts for the

numerical resolution of the transition moment, eq 4, accounts for all anharmonic effects on the relaxation rates.25−28 The state- and position-independent electron−phonon coupling constant γ*α can be determined from first-principles by taking the harmonic 2/3

neq Mα limit of eq 4 at the equilibrium geometry, Γ1(α→) 0 = γα* 2ℏ2 . A unique expression is then obtained by relating these fundamental harmonic relaxation rates to the friction tensor.43 Although current efforts aim at developing more accurate models for the friction tensor,47,48 we make use of the established independent atom approximation.44,46 The amplitude of the potential deformation induced by STM excitation depends on two main factors: the variation of the NAC strength of the skeletal modes along the reaction path and the intermode coupling strength. To understand the effect of the latter, Figure 2a shows the vibrational DOS for the cis configuration, the transition state (TS), and the trans configuration (top to bottom panels, respectively). The dashed lines representing the H transfer mode are seen to have the same transition frequency for cis and trans configurations, ∼350 meV. The higher frequencies at >370 meV correspond to CH stretch modes of the outer rings, which are decoupled from the reaction path. This behavior is also observed in the gas phase.49 In the bias voltage range investigated in the experiment, only the skeletal

0

current dependence and the residence time τα of an electron on mode Qα. The number of one-electron states per atom at a given bias, ρ0(U) = |ρ(eU) − ρ(ϵF)|, can be estimated using the projected density of states (DOS) on the copper atoms.28,37 The static NAC rate between vibrational states |i⟩ and |j⟩ for mode Qα takes the form γij(α)

=

γα* |ℏωij|

1/3

⟨i|n

∂ | j⟩ ∂Q α

2

(4)

A similar equation can be obtained for the reaction coordinate q. The embedding density, n1/3, is related to the phase shift of oneelectron wave functions upon scattering in a nonuniform electron gas via the local density approximation.40−46 It describes the electron density surrounding a spherical impurity at the position of the transferred H atom, and it effectively modulates the strength of the NAC elements along coordinate Qα. The 1055


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Figure 4. (a) Below-threshold dynamical evolution of the nuclear density upon application of a U = 50 mV bias. The density remains preferentially in the thermodynamically stable trans configuration. (b) Above-threshold dynamical evolution of the nuclear density upon application of a U = 200 mV bias. The forward tautomerization reaction trans → cis under the influence of the NAC to the metal is favored. (c) Static NAC rate (eq 4) for the trans → cis ground-state-to-ground-state transition (blue circles) and difference between the zero-point energy levels of the trans and cis wells (red diamonds) as a function of the bias voltage. (d) Tautomerization time (blue circles) and efficiency (red diamonds) for different bias values obtained from dynamical simulations. An acceleration is observed for biases above the resonant frequency of the tautomerization mode. At U = 0 and at the temperature considered, the tautomerization time goes to infinity while the efficiency is strictly zero.

surface and the charge exchange between the unsaturated N atoms of the cavity and the Cu atoms.33 The adiabatic PEC obtained from periodic DFT calculations is depicted as a dark blue curve in Figure 3a. The inset shows ρ0(U), which exhibits a linear dependence at low bias voltages. More information about the fitting procedure for the distorted PECs can be found in the SI. Distorted PECs at various biases are shown in panel (a) of Figure 3 for a fixed STM current intensity of I = 100 pA and temperature of T = 5 K. It can be seen that the STM perturbation generally destabilizes the molecule at the surface. This originates from the quasi-thermal fluctuations induced by excitation of the skeletal modes according to eq 2. The cis configuration is less destabilized with increasing U compared to the trans configuration. This can be rationalized from the NAC strengths and intermode coupling, as presented in Figure 2a,b. Because the cis configuration is closer to the surface, the projected electronic density of states is larger and the relaxation rates are greater. Consequently, the excitation probability according to eq 3 is lower and the potential deformation is smaller in this region of the PEC. The trans configuration is longer-lived and more affected by the bias voltage. Intermode coupling enters the potential deformation via the position dependence of the harmonic modes in the resolution of the NAC rates used in eq 2. Importantly, the nonuniformity of the potential deformation along the tautomerization coordinate is not sufficient to reverse the energetic ordering of the two potential wells. The vibrational bound states sustained by each PEC were calculated by diagonalizing the matrix representation of the

vibrations can be excited. The vibrational DOS of these skeletal modes below 200 meV remains largely unaffected along the reaction path. We can conclude that the potential deformation does not originate from the variation of the vibrational DOS along the tautomerization coordinate. The electronic DOS varies significantly along the reaction coordinate. This is reflected in the behavior of the nonadiabatic lifetimes for selected snapshots along the reaction path (see Figure 2b). The mode lifetimes are calculated as the inverse rate for the fundamental transition, eq 4. Comparison of the lower (trans) and upper (cis) panels reveals an important change in the lifetimes of the trans (∼5−10 ps) and cis (∼2−5 ps) configurations. The origin of this change can be understood from eq 4; the embedding density n1/3 gives a measure of the electronic density of states projected on the adsorbate, which modulates the intensity of the NAC. In the trans configuration, the porphycene is planar, at about 3.5 Å above the surface. The buckled cis configuration binds to the surface via the unsaturated N atoms, at about 2.5 Å. The variation of the adsorbate height along the reaction path implies that n1/3 increases significantly along the tautomerization coordinate. This reduces the static lifetime of all vibrational modes and the associated STM excitation probabilities. Thus, an impinging electron from the STM source resides more briefly on the adsorbate in the cis configuration due to its proximity to the surface. The potential deformation, eq 2, is reduced accordingly. The molecule−surface interaction is a compromise between dispersion interactions of the outer ring of the molecule with the 1056


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in Figure 4c. Acceleration of the reaction is observed at biases above the resonance energy of the tautomerization mode. As it becomes directly activated, the first vibrationally excited state of the trans species is populated (see Figure 1), which tunnels faster than the local ground state (τ1 < τ0). In addition to the simultaneous tunnelling/relaxation mechanism, an indirect channel via the excited state becomes available. This latter mechanism eventually dominates at higher bias voltages. On the other hand, the tunneling rates become vanishingly small at zero bias, which explains the observed irreversibility of the STMinduced tautomerization reaction at low temperatures; once the transformation occurs, the molecule remains trapped in the cis configuration due to inefficient coupling to electron−hole pairs of the metal. In Figure 4d, the tautomerization time is found to vary with U. On the contrary, the efficiency indicates a quantitative reaction for any above-threshold bias (dPcis ≃ 1). That is, if a hot electron with sufficiently high energy reaches the molecule, the tautomerization driven by the potential deformation is quantitative. This accounts for single-molecule tautomerization, but the observed yield is subject to the statistical availability of hot electrons. The yield is determined experimentally by counting the number of molecules found in the cis configuration after application of the STM pulse in a limited area of a few nm2. Recalling that the hot electrons propagate as an S-wave through the metal,37 the number of electrons impinging on each molecule will thus decrease as a function of the distance to the STM injection point. Consequently, the observed effective rates will decrease. In summary, we demonstrate that intermode coupling is the driving mechanism for the STM-induced trans → cis tautomerization of porphycene on Cu(111). Quasi-thermal excitation of skeletal vibrations by NAC to the metal electrons distorts the potential along the reaction coordinate. All parameters entering the potential deformation can be estimated from first principles. The model explains the experimentally observed stability reversal of the two configurations at a given threshold bias. The tautomerization rates are dominated by simultaneous tunnelling and relaxation to the cis ground state and dictated by the energetic alignment of the local vibrational ground states. The observed dependency of the tautomerization yield on the applied voltage can be rationalized from geometrical arguments based on the availability of electrons traveling as Swaves in the metallic substrate. The tautomerization channel is opened by skeletal vibrations excited by hot electrons impinging from the STM. Because quasi-thermal fluctuations are suppressed at zero bias, the molecule remains trapped in the nearest local minimum of the potential energy landscape. The reverse cis → trans reaction can only be triggered by thermal activation, where it is mediated by hopping diffusion. The model presented here has a general character and could be applied to other similar systems that have already been investigated experimentally, for example, azobenzene derivatives on Au(111)53 or diarylethene on Ag(111).54

reaction path Hamiltonian (see the SI). In Figure 3b, the zeropoint energy is found close to the top of the tautomerization barrier in all PECs, with some degree of delocalization in both wells. At low biases, the trans configuration is favored, in line with the experiment. With increasing bias voltage, the vibrational ground-state density (shaded areas) becomes more localized in the cis well, with a tipping point at ∼150 mV. This threshold value is sensitive to the barrier height, the skeletal mode vibrational frequencies, and the NAC rates. To recover the exact experimental tipping point, the DFT value of ρ0(U) was adjusted to (0.003 V−1) × |U|. The slight asymmetry of ρ0(U) in the lowbias regime is not sufficient to produce a significant bias polarity dependence of the potential deformation. On the other hand, ρ0(U) is found to be more asymmetric for potential biases larger than the ones considered in this Letter and is expected to play an important role in that case. In the presence of thermal and STM-induced fluctuations, the system will react and relax to the most stable configuration due to NAC. The relaxation process in each potential well is in the picosecond regime, but tunnelling is in the microseconds; see Figure 2b. Consequently, coherences between states localized within one well can be neglected on the time scale of the STMinduced tautomerization. The system dynamics is thus determined by the Pauli master equation for the nuclear probability density. To account for molecular frame reorganization during the reaction,50 the vibrational functions used to represent the density and calculate the rates entering the Pauli equation are localized prior to the dynamics using a formalism similar to the one used in other reduced-dimensional dynamical simulations51,52 (see the SI for details). Figure 4a,b shows the evolution of the nuclear probability density for below- (50 mV) and above-threshold (200 mV) conditions, respectively. The PECs contain the instantaneous reaction of the skeletal vibrational modes to the STM perturbation. The H atom, initially found in the trans configuration, evolves from this initial condition in the perturbed energy landscape. Below threshold, the density remains localized in the trans well. The quasi-thermal fluctuations induced by the STM are insufficient to trigger the tautomerization reaction. Above threshold, a smooth transfer from the trans to the cis configuration is seen, without probability density in the tunnelling region or nodal structure building up in either well. For such low biases, the tautomerization dynamics is dominated by the direct trans → cis transfer. This corresponds to the first scenario of Figure 1. The tautomerization rate is determined by the static NAC rate, γt→c, as described by eq 4. The latter correlates with the alignment of the fundamental level in the trans and cis wells, as shown in Figure 4c. The lifetime broadening (blue circles) reaches a maximum at U = 100 mV, where the zeropoint energies of the trans and cis configurations are quasidegenerate (red diamonds). At higher voltages, the levels drift apart and the rate decreases. Although the NAC rates are larger at U = 100 mV, the tautomerization efficiency is smaller at below-threshold biases. Figure 4d reports the tautomerization time (τ, blue circles) and efficiency (dPcis, red diamonds) at various STM biases. They are obtained by fitting the population of the vibrational ground state of the cis isomer to an exponential form, Pcis(t) = dPcis(1 − e−(t−t0)/τ). All τ values are in the microsecond regime, well below the upper limit inferred from experimental observations. For biases below the resonance energy of 350 mV (i.e., the N−H stretch mode), the tautomerization times correlate with the energy level alignment and the behavior of the NAC rates shown

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b00141. Derivation of the potential deformation equation and details about the electronic structure calculations, the 1057


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potential energy curves fitting, the localization procedure, and the projected vibrational density of states (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Jean Christophe Tremblay: 0000-0001-8021-7063 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank T. Kumagai, T. Frederiksen, and P. Saalfrank for stimulating discussions. J.C.T. acknowledges financial support from the Deutsche Forschungsgemeinschaft (Project: TR 1109/ 2-1). D.N. acknowledges funding from the DIPC. This work was supported by the Basque Departamento de Educación, Universidades e Investigación, the University of the Basque Country UPV/EHU (Grant No. IT-756-13) and the Spanish Ministerio de Economiá y Competitividad (Grant No. FIS201348286-C2-2-P). Computing resources were provided by the DIPC computing center.

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