Direct signature of light-induced conical intersections in diatomics
DDirect signature of light-induced conicalintersections in diatomics
G. J. Halász, † Á. Vibók, ∗ , ‡ and L. S. Cederbaum ¶ Department of Information Technology, University of Debrecen, H-4010 Debrecen, PO Box12, Hungary, Department of Theoretical Physics, University of Debrecen, H-4010 Debrecen,PO Box 5, Hungary, and Theoretische Chemie, Physikalish-Chemisches Institut, UniversitätHeidelberg, H-69120, Germany
E-mail: [email protected]
Abstract
Nonadiabatic effects are ubiquitous in physics, chemistry and biology. They are stronglyamplified by conical intersections (CIs) which are degeneracies between electronic statesof triatomic or larger molecules. A few years ago it has been revealed that CIs in molec-ular systems can be formed by laser light even in diatomics. Due to the prevailing strongnonadiabatic couplings, the existence of such laser-induced conical intersections (LICIs)may considerably change the dynamical behavior of molecular systems. By analyzing thephotodissociation process of the D + molecule carefully, we found a robust effect in theangular distribution of the photofragments which serves as a direct signature of the LICIproviding undoubted evidence for its existence. ∗ To whom correspondence should be addressed † Department of Information Technology, University of Debrecen, H-4010 Debrecen, PO Box 12, Hungary ‡ Department of Theoretical Physics, University of Debrecen, H-4010 Debrecen, PO Box 5, Hungary ¶ Theoretische Chemie, Physikalish-Chemisches Institut, Universität Heidelberg, H-69120, Germany a r X i v : . [ phy s i c s . a t m - c l u s ] N ov or Table of Contents Only.It is well known that conical intersections (CIs) are widely recognized to be ubiquitous inpolyatomics, and to play an important role in several different fields like spectroscopy, chem-ical reaction dynamics, photophysics, photochemistry . At CIs the nuclear and electronicmotions couple strongly, giving rise to often unexpected — so-called nonadiabatic — phe-nomena. The nonadiabatic coupling matrix elements between electronic states are the largestpossible at CIs where they become singular . Therefore, one expects that such intersectionsgive rise to the strongest possible nonadiabatic phenomena. It was demonstrated for a varietyof examples that conical intersections provide the mechanism for ultrafast chemical processesas, e.g. photodissociation, photoisomerization and internal conversion to the electronic groundstate . Clearly, the CIs must be considered as photochemically relevant decay channels. Thepresence of CIs in biomolecules, like the building blocks of DNA and proteins contribute to thephotostability of these important molecules. CIs can be formed between different electronicstates starting from triatomic systems to truly large polyatomic molecules. Several importantbooks, review articles and publications have documented the existence and relevance of suchintersections .CIs can exist only if the molecular system possesses at least two independent nuclear de-grees of freedom. This is why diatomics which have only one nuclear vibrational coordinatecannot exhibit a CI. Importantly, this statement only holds in field free space, a fact which hasbeen generally overlooked. When a laser field is present, then due to the interaction of thediatomic with this field, the rotational degree of freedom comes into play and serves as an addi-2ional degree of freedom. It was revealed in earlier studies that CIs can be induced both byrunning or standing laser waves even in diatomics. Interestingly, in the standing wave case alsothe position of the center of mass becomes a new degree of freedom in addition to the nuclearvibrational and rotational motions. In order to demonstrate how the rotation supplies the neces-sary degree of freedom to facilitate the shaping of a CI in the general case of propagating laserwaves, one may adopt the dressed state representation . In this picture the molecule-lightinteraction is explicitly included into the Hamiltonian, and the changes of nuclear dynamicsdue to the light field can be considered as originating from the appearance of a “light-inducedconical intersection” (LICI) . The laser frequency determines the position of the LICI, whilethe laser intensity determines the strength of the nonadiabatic couplings. Our results in the lastfew years undoubtedly demonstrated that LICIs exert strong effects on the quantum dynamicseven for weak laser fields .The present study goes beyond previous investigations and makes an attempt to provideand analyze a physical event which may serve as an undoubted evidence of the laser- or light-induced conical intersection (LICI), giving a “direct signature” of the presence of this inter-section. It is known in the field of nonadiabatic molecular dynamics that due to the extremebreakdown of the Born-Oppenheimer approximation, conical intersections are responsible forultrafast radiationless processes, typically on the femtosecond time scale. They provide path-ways for extremely fast population transfer between electronic states. This latter effect is proba-bly the most important inherent feature of the CIs. Nevertheless, until now one could not find anunambiguous experimentally measurably quantity which reflects directly this population trans-fer between electronic states for a LICI. The present work discusses a physical process, wherean ultrafast population transfer takes place between the electronic states of a diatomic molecule,providing direct evidence for the existence of the LICI. The photodissociation process of theD + molecule serves as a show case physical example. This molecule has already been studiedin vast amount of works , mainly because of its simplicity. It has the advantage that we cancompute it accurately and easily study the light-induced nonadiabatic phenomena separatelyfrom other phenomena.It is important to note that recently, motivated by theoretical predictions on the LICIs in3iatomics, experiments on the laser-induced isomerization and photodissociation of polyatomicmolecules were qualitatively interpreted using the concept of LICIs . Furthermore, theorydevoted to generalize the LICI phenomenon for polyatomics has been derived showing the largepotential of LICIs in controlling reactions . Description of the system.
The two relevant electronic states of the D + ion (see Fig. 1),which will be considered in the calculations are the ground (V = s σ g ) and the first excited(V = p σ u ) eigenstates of the field-free Hamiltonian. For describing the dissociation mech-anism we assume that initially the D + ion is in its ground electronic (1 s σ g ) as well as in itsground rotational state and in one of its vibrational eigenstates (see Fig. 1). Exciting the elec-tronic ground state by a resonant laser pulse to the repulsive 2 p σ u state, the two electronicstates are resonantly coupled. The nonvanishing dipole matrix elements are responsible for thelight-induced electronic transitions. Within these two electronic states representation the totaltime-dependent Hamiltonian for the rovibronic nuclear motion reads H = − µ ∂ ∂ R + L θϕ µ R − µ ∂ ∂ R + L θϕ µ R + (1) V ( R ) − ε f ( t ) d ( R ) cos θ cos ω L t − ε f ( t ) d ( R ) cos θ cos ω L t V ( R ) . Here, R and ( θ , ϕ ) are the molecular vibrational and rotational coordinates, respectively, µ isthe reduced mass, and L θ ϕ denotes the angular momentum operator of the nuclei. Here θ isthe angle between the polarization direction and the direction of the transition dipole and thusone of the angles of rotation of the molecule. V ( R ) (1 s σ g ) and V ( R ) (2 p σ u ) are the potentialenergies of the two electronic states coupled by the laser (whose frequency is ω L and amplitudeis ε ), f ( t ) is the envelop function and d ( R ) (cid:0) = − (cid:10) ψ e (cid:12)(cid:12) ∑ j r j (cid:12)(cid:12) ψ e (cid:11)(cid:1) is the transition dipole matrixelement ( e = m e = ¯ h =
1; atomic units are used throughout the article). We used the quantities V ( R ) and V ( R ) and d ( R ) published in . Light-induced conical intersection (LICI).
In the dressed representation the laser lightshifts the energy of the 2 p σ u repulsive excited potential curve by ¯ h ω L and a crossing between4he ground ( V ) and the shifted excited ( V − ¯ h ω L ) potential energy curves is created. Bydiagonalizing the potential energy matrix one obtains the adiabatic potential surfaces V lower and V upper (Fig. 1). These two surfaces cross each other at a single point R , θ , giving rise toa conical intersection whenever the conditions cos θ = ( θ = π / ) and V ( R ) = V ( R ) − ¯ h ω L are simultaneously fulfilled .The characteristic properties of the LICI, i.e., the location of the intersections and thestrengths of the nonadiabatic couplings, can be directly controlled by the laser frequency andintensity. This opens the possibility to control the nonadiabatic effects emerging from the LI-CIs. To demonstrate the impact of the LICI on the photodissociation dynamics of D + we haveto solve the time-dependent nuclear Schrödinger equation (TDSE) using the Hamiltonian ˆ H given by Eq. ( ?? ). The angular distribution of the photofragments P ( θ ) were calculated withthe solution of the TDSE equation. Methods.
The MCTDH (multi configuration time-dependent Hartree) method was usedto solve the TDSE . It is one of the most efficient approaches for solving the TDSE.For describing the vibrational degree of freedom we have applied FFT-DVR (Fast FourierTransformation-Discrete Variable Representation) with N R basis elements distributed withinthe range from 0.1 a.u. to 80 a.u. for the internuclear separation. The rotational degree offreedom was represented by Legendre polynomials { P J ( cos θ ) } j = , , , ··· , N θ . These so calledprimitive basis sets ( χ ) were used to build up the single particle functions ( φ ), which in turnwere applied to represent the wave function: φ ( q ) j q ( q , t ) = N q ∑ l = c ( q ) j q l ( t ) χ ( q ) l ( q ) q = R , θ (2) ψ ( R , θ , t ) = n R ∑ j R = n θ ∑ j θ = A j R , j θ ( t ) φ ( R ) j R ( R , t ) φ ( θ ) j θ ( θ , t ) . In the actual simulations N R = N θ =
199 were used. On both diabatic surfaces andfor both degrees of freedom a set of n R = n θ =
20 single particle functions were applied toconstruct the nuclear wave packet of the system. The calculations converged properly by usingthese chosen parameters. Using the solution of equation (2) one can calculate the angular5 E n e r gy ( e V ) -0.2-0.10.0 Interatomic distance (A) -6-5-4-3-2-101 E n e r gy ( e V ) -0.2-0.10.0 Interatomic distance (A) L = 200 nm= 6.19921 eVI = 3.0 10 W/cm (a.u.) ¯ hω L sσ g pσ u ¯ hω L pσ u − ¯ hω L Figure 1: Potential energies of the D + molecule and the light-induced conical intersection(LICI). Left panel:
The dressed adiabatic surfaces as a function of the interatomic distance R and the angle θ between the molecular axis and the laser polarization exhibiting the LICIfor a field intensity of 3 × Wcm . Right panel:
The diabatic energies of the ground ( s σ g ) and the first excited ( p σ u ) states of the D + molecule are displayed by solid green and redlines, respectively. The field dressed excited state (2 p σ u − ¯ h ω L ; dashed red line) forms a LICIwith the ground state. A cut through the adiabatic surfaces at θ = R LICI = .
53 Å = . a . u . and E LICI = − . eV ). 6istribution of the photofragments : P ( θ j ) = w j ∞ ˆ dt < ψ ( t ) | W θ j | ψ ( t ) > (3)where − iW θ j is the projection of the complex absorbing potential (CAP) on a specific point ofthe angular grid ( j = , .. N θ ) , and w j is the weight related to this grid point according to theapplied DVR. The simulations . Linearly polarized Gaussian laser pulses centered around t = f s wereapplied in the numerical calculations with a carrier wavelength of λ L = nm . The pulseduration at full width of half maximum (FWHM) is t pulse = f s . The initial nuclear wavepacket (at t = − f s ) was chosen to be in its rotational ground state ( J =
0) and in one ofits vibrational eigenstates ( ν = , , , ) . Under such conditions, the angular distribution ofthe photofragments can provide accurate details of the photodissociation of single vibrationallevels .Full two dimensional calculations (2d) in R − θ space have been performed. In order tohave a better understanding of the obtained results, we have also performed calculations witha restricted one dimensional (1d) approach where we have eliminated the rotational motionfrom the system by putting L θ ϕ to zero in the Hamiltonian Eq. ( ?? ). This way the molecule’sinitial orientation can not change during the dissociation process, and the TDSE can be solvedwith independent 1d calculations for each value of θ using the “effective field strength” ε e f f = ε · cos θ ( intensity I e f f = I · cos θ ) at that value of θ . In these 1d calculations the Hamiltoniandepends only parametrically on the rotational degree of freedom ( θ ) and none of the individualcalculations are able to take into account the effects of the LICI. The respective potential energycurves exhibit avoided crossings as can be seen for θ = × Wcm are displayed in Figs. 2a-2d for the fourdifferent initial vibrational levels ν = −
7. The scale on Fig. 2 was chosen such that the disso-ciation rate of 1 implies that the dissociation in a given direction is complete. Consequently, in1d the dissociation rate can not be larger than 1. Larger values in the full 2d calculation mean7 .00.51.01.5 D i ss o c i a ti on r a t e ( a r b . un it s ) Angle, (rad) (A) =4 D i ss o c i a ti on r a t e ( a r b . un it s ) Angle, (rad) (B) =5 D i ss o c i a ti on r a t e ( a r b . un it s ) Angle, (rad) (C) =6 D i ss o c i a ti on r a t e ( a r b . un it s ) Angle, (rad) (D) =7
Figure 2: Fragment angular distributions of the dissociating D + molecule for four differentinitial vibrational states ( ν = , , , × W / cm .8 /2 O r i e n t a ti on , (r a d )
10 20 30 40 50
Interatomic distance, R (a.u.) t = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fst = -30 fs
I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm (A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A)(A) O r i e n t a ti on , (r a d )
10 20 30 40 50
Interatomic distance, R (a.u.) t = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fst = -10 fs
I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm I(t) = 7.3 10 W/cm (B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B)(B) O r i e n t a ti on , (r a d )
10 20 30 40 50
Interatomic distance, R (a.u.) a t = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fst = 0 fs I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm I(t) = 1.0 10 W/cm (C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C) O r i e n t a ti on , (r a d )
10 20 30 40 50
Interatomic distance, R (a.u.) a t = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fst = 5 fs I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm I(t) = 9.3 10 W/cm (D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D)(D) O r i e n t a ti on , (r a d )
10 20 30 40 50
Interatomic distance, R (a.u.) ab t = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fst = 15 fs I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm I(t) = 5.0 10 W/cm (E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E)(E) O r i e n t a ti on , (r a d )
10 20 30 40 50
Interatomic distance, R (a.u.) ab t = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fst = 20 fs I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm I(t) = 2.9 10 W/cm (F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F)(F) O r i e n t a ti on , (r a d )
10 20 30 40 50
Interatomic distance, R (a.u.) abc t = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fst = 30 fs
I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm I(t) = 6.2 10 W/cm (G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G)(G) O r i e n t a ti on , (r a d )
10 20 30 40 50
Interatomic distance, R (a.u.) abc t = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fs
I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm (H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H)(H) O r i e n t a ti on , (r a d )
10 20 30 40 50
Interatomic distance, R (a.u.) t = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fst = 52 fs
I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm I(t) = 2.4 10 W/cm (I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I)(I) Figure 3: Snapshots from the real-time evolution of the nuclear density of the ν = + due to a Gaussian laser pulse of peak intensity 1 × W / cm and 30 f s duration.The nuclear density exhibits severe interference effects and splits at larger distances around θ = π /
2. The instantaneous intensity is shown in the individual snapshots. The yellow crossdenotes the position of the LICI. The points a, b and c are explained in the text. All panelsshow the results of the full 2d calculation expect of the last panel (I) which is computed in 1dfor comparison. Note the jump in the interatomic scale.9hat some parts of the dissociating particles were rotated by the field to this direction from somedifferent initial directions.Let us study first the 1d curves in Fig. 2. For the case of ν = θ and is very close to the value of1 up to θ = π /
12. This is related to the fact that the energy of this vibrational level is justabove the energy of the LICI and, therefore, most of the initial wavepacket belongs to the lowerdissociative adiabatic surface and can easily dissociate. For larger ν the 1d dissociation ratecurves display one or two ( ν =
7) local maxima as a function of the angle θ . This behavioris related to the accidental increase of the so - called bond hardening effect at particularwavelengths and intensities of the laser field . The 1d dissociation rate predicts a significantbond hardening effect for ν = ν =
7, even at the parallel orientation of the molecules. Ina Gaussian laser pulse the field rises smoothly and independently of the peak intensity starts tointeract with the molecular system at weak intensities, and therefore the initial populations onthe lower and on the upper adiabatic surfaces are determined by the behavior of the system atlow intensities. In our earlier work , it was shown for the ν = ν = λ L = nm wavelength leads to a close to an “ideal” bond hardening case atweak intensities. This is the reason why we find here an significant bond hardening effect upto 1 × Wcm field intensities at this wavelength.Let us now turn to the 2d results. For the initial ν = . The differences arise due to our finding that the laser light not just drivesthe dissociation process, but also starts to rotate the molecules. As a result the dissociationrate decreases at large θ angles and at the same time increases at the small values of θ . Asthe induced rotation towards the polarization direction of the electric field creates rotationalnodes, the angular distribution of the dissociation rate is further affected by the appearance ofadditional structures .For the ν = θ = π / ν = ν = θ (cid:38) π /
24, whoseheights are one ( ν =
7) or even two ( ν =
5) orders of magnitude larger than the correspondingdissociation rates in the 1d calculations. This characteristic structure of the 2d curves in thisangle region can not be explained without the strong nonadiabatic effect due to the existenceof the LICI. The peaks displayed on the 2d curves are direct fingerprints of the existence of thelaser-induced conical intersection in the studied system. We note here that such peaks can alsobe seen at smaller intensities (1 × Wcm , 3 × Wcm ), but in less pronounced form.To understand the underlying dynamical process more deeply, we analyzed the nucleardensity function | ψ ( R , θ , t ) | (cid:16) = (cid:12)(cid:12) ψ s σ g ( R , θ , t ) (cid:12)(cid:12) + (cid:12)(cid:12) ψ p σ u ( R , θ , t ) (cid:12)(cid:12) (cid:17) . Snapshots of the nu-clear wave packet density functions are shown in Fig. 3. After the pulse is over ( t = f s , Fig.3h) we can identify three peaks – on both sides of the symmetry axis θ = π / π / (cid:46) θ (cid:46) π /
24 of the outgoing wavepacket. These peaks – especially themost right one – are clearly responsible for the characteristic structure of the huge excess in thedissociation probability close to θ = π /
2. Their “leading edge” are labeled by “a”, “b” and “c”on the subfigures whenever they are recognizable.In what follows we study the time evaluation of the nuclear density in order to see the originof these peaks. It can be seen that at t = − f s (Fig. 3a) the molecules start to dissociate onthe lower adiabatic surface. This process evolves in time, but at t = − f s one can clearly rec-ognize also the fingerprint of the rotational motion induced on the upper surface (see the arrowin Fig. 3b). At t = f s (Fig. 3c, labeled by “a”) tracks show up that an additional dissociationprocess starts close to θ = π /
2, presumably from the upper adiabatic surface. From t = f s (Fig. 3e) on it is clearly seen that additional dissociation takes place continuously, and from t = f s (Fig. 3f) on the dissociated fragments explicitly appear in the dissociation region( R > a . u . ). The much less populated “b” and “c” peaks appear for the first time on subfigures11ig. 3e and Fig. 3g, respectively.The last snapshot displays the results of the 1d simulation at t = f s , the same time asin Fig. 3h for the full 2d calculation. The dynamics during the initial time period, when theinstantaneous intensity is low, are very similar in both the 1d and 2d models. After reachingthe maximum intensity, however, two significant differences can be observed. The first one is,as expected, the unequivocal fingerprint of the presently discussed nonadiabatic effect leadingto a strong enhancement of fragment density near θ = π / + molecule. The structure and magnitude ofthe 2d ( ν = ,
7) dissociation rates close to θ = π / Acknowledgements
The authors acknowledge financial support by the Deutsche Forschungsgemeinschaft (ProjectID CE10/50-2). Á. V. also acknowledges the TÁMOP-4.2.4.A/ 2-11/1-2012-0001 ‘NationalExcellence Program’ and the OTKA (NN103251) project. This research was supported in partby the National Science Foundation under Grant No. NSF PHY11-25915.12 eferences (1) Zener, C. Non-adiabatic crossing of energy level. Proc. R. Soc. Lond. A , 137,696-702.(2) Teller, E. The crossing of potential energy surfaces. J. Phys. Chem. , 41, 109-116.(3) Köppel, H.; Domcke, W.; Cederbaum L. S. Multimode molecular dynamics beyond theBorn-Oppenheimer approximation. Adv. Chem. Phys . , 57, 59-246.(4) Truhlar, D. G.; Mead, A. The relative likelihood of encountering conical intersections onthe potential energy surfaces of polyatomic molecules. Phys. Rev. A , 68, 032501.(5) Baer, M. Introduction to the theory of electronic non-adiabatic coupling terms in molecu-lar system. Phys. Rep . , 358, 75-142.(6) Worth, G. 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