Geometric phase of light-induced conical intersections: Adiabatic time-dependent approach
GGeometric phase of light-induced conicalintersections: Adiabatic time-dependentapproach
Gábor J. Halász(1), Péter Badankó(2) and Ágnes Vibók(2),(3) ∗ , † † (1)Department of Information Technology, University of Debrecen, H-4002 Debrecen,PO Box 400, Hungary ‡ (2)Department of Theoretical Physics, University of Debrecen, H-4002 Debrecen, POBox 400, Hungary ¶ (3) ELI-ALPS, ELI-HU Non-Profit Ltd, H-6720 Szeged, Dugonics tér 13, Hungary E-mail: [email protected]
Abstract
Conical intersections are degeneracies between electronic states and are verycommon in nature. It has been found that they can also be created both by standingor by running laser waves. The latter are called light-induced conical intersections.It is well known that conical intersections are the sources for numerous topologicaleffects which are manifested e.g. in the appearance of the geometric or Berry phase.In one of our former works by incorporating the diabatic-to-adiabatic transformationangle with the line-integral technique we have calculated the Berry-phase of thelight-induced conical intersections.Here we demonstrate that by using the time dependent adiabatic approach sug-gested by Berry the geometric phase of the light-induced conical intersections canalso be obtained and the results are very similar to those of the time-independentcalculations. a r X i v : . [ phy s i c s . c h e m - ph ] D ec eywords: Born-Oppenheimer approximation; Conical intersections; Light-induced con-ical intersections; Geometric phase;
Introduction
Conical intersections (CIs) are degeneracies between two or more electronic states andplay an important mechanistic role in the nonadiabatic dynamics of polyatomic molecules .At the close vicinity of the CIs the Born-Oppenheimer adiabatic approximation breaksdown due to the strong nonadiabatic coupling between the nuclear and electronic mo-tions. Several important photophysical and photochemical processes like dissociation,proton transfer, isomerization or radiationless deactivation of the excited states are as-sociated with the appearance of CIs . These degeneracies are not isolated, rather theyare connected points forming a seam and can exist already between low lying states oftriatomic molecule. In truly large molecular systems they are very abundant.It is well-know that under “natural” (field-free) conditions CIs cannot be formed be-tween different electronic states in diatomic molecules . The one degree of freedompresents in these object is generally not enough to span a branching space and thereforeonly an avoided crossing result. However, applying standing or running laser waves ,CIs can be created even in diatomics. In the first situation, the laser light induces CIs(“light-induced” conical intersections, LICIs) which couple the center of mass motion withthe internal rovibrational degrees of freedom . In the latter case, the rotational motionprovides the missing degree of freedom allowing the formation of a LICI .Recently, several theoretical and experimental studies have demonstrated that simi-larly to the natural CIs the light-induced conical intersections have also significant impacton the different dynamical properties of molecules . Among others it can stronglymodify e. g. the spectra, the alignment, the dissociation probability or fragment angulardistribution of molecules . However, there are features in which the natural and light-induced CIs differ significantly. As long as the position of a natural CI and the strength ofits nonadiabatic effects are inherent properties of the electronic states of a molecule and2re difficult to modify, the energetic and spatial positions of the LICI can be controlledby changing the parameter settings (intensity, frequency) of the laser light. This lattercan open up a new direction in the field of molecular quantum control processes.The nonadiabatic couplings can become extremely large at the close vicinity of the CIs.Numerous statical and dynamical nonadiabatic phenomena originate from the presenceof a CI. Longuet-Higgins and Herzberg have demonstrated that each real adiabaticelectronic state changes sign when transported continuously along a closed path encirclingthe point of CI. Mead and Truhlar conjuncted this geometrical phase effect with the singleelectronic state problem and Berry generalized the theory , hence the name Berryphase. Making sure that the electronic wave function remains single valued one has tomultiply it by a phase factor and, as a consequence of it, this new electronic eigenfunction,instead of being real, becomes complex. The fact that the electronic eigenfunctions arealtered has a direct impact on the nuclear dynamics even if it happens in a single potentialenergy surface. Therefore having a nontrivial Berry phase in a molecular system can beseen as a direct fingerprint of the CI.In the last three decades several works made remarkable contributions to the subject ofmolecular topological features . However, all of these works related to the geometricalor topological properties, like Berry phase etc... of natural conical intersections.In our earlier papers we have calculated the Berry phase around the light-inducedconical intersection formed in the Na molecule in the presence of external electric field.Due to non vanishing transition dipole moment the nm light can resonantly couplethe X Σ + g and A Σ + u electronic states giving rise to a light-induced conical intersection.During the calculations 2 × with the time-independent line integral technique , we could calculate the Berry phase for differentclose contours which either surrounded or not surrounded the LICI. Obtained resultswere similar to those calculated for CIs given in nature demonstrating that a “true” CIhas been found.In the present work we would like to go beyond the time-independent description.3ere we intend to provide calculations for the geometric phase of the light-induced conicalintersection applying the time-dependent adiabatic approximation proposed by Berry inhis famous work . Our showcase example is the D +2 molecule. We demonstrate that forcertain initial conditions by assuming again 2 × The Hamiltonian
Let us define the Hamiltonian which governs the dynamics of the D +2 molecule. Twoelectronic eigenstates V ( R ) (ground, sσ g ) and V ( R ) (excited, pσ u ) are included inthe Hamiltonian which are coupled by a running laser wave (see in Figure 1). The non-vanishing transition dipole matrix element d ( R ) (cid:16) = − (cid:68) Φ e (cid:12)(cid:12)(cid:12)(cid:80) j r j (cid:12)(cid:12)(cid:12) Φ e (cid:69)(cid:17) is responsible forthe light-induced electronic transition. The corresponding Born-Oppenheimer potentialsand the transition dipole were taken from . As the nuclear coordinate R and themolecular orientation θ are taken as parameters during the calculations our Hamiltonianis defined by the potential energies and the laser-molecule interaction. This interaction isgiven in the dipole approximation as the scalar product of the transition dipole moment −→ d and the electric field vector −→ ε : −→ d · −→ ε = (cid:15) d ( R ) cos θcos ( ω L t ) . (1)In Eq. (1) (cid:15) is the maximum laser field amplitude, I ( ∼ (cid:15) ) is the laser intensity, θ denotes the angle between the polarization direction and the direction of the transitiondipole d ( R ) and ω L is the laser frequency which couples the two electronic states at R = 5 a.u. nuclear distance ( ω L = 1 . eV ). 4et us represent the Hamiltonian in the Floquet picture. Therefore, the originalHamiltonian is transformed into an equivalent static problem by using the leading termin the Fourier series expansion of the solution of the time-dependent Schrödinger equation.Then the field-dressed form reads ˆH = V ( R ) ( (cid:15) / d ( R ) cos θ ( (cid:15) / d ( R ) cos θ V ( R ) − (cid:126) ω L (2)In this dressed state representation the interaction between the molecule and the electro-magnetic field is obtained by shifting the energy of the excited potential curve by (cid:126) ω L .This picture is often used to explain various phenomena in the area of strong field physicswhenever only net one-photon is absorbed by the molecule.As a results of the dressed state representation a crossing is formed between thediabatic ground and the dressed excited potential energy curves. After diagonalizingthe diabatic potential matrix Eq. 2, the resulting adiabatic or light-induced surfaces( V lower and V upper ) form a light-induced conical intersection (see in Figure 1) wheneverthe following conditions are fulfilled : cosθ = 0 ( θ = π V ( R ) = V ( R ) − (cid:126) ω L . (3)An important feature of the light-induced conical intersections as compared to thenatural CIs is that their fundamental characteristics can be modified by the externalfield. It has already been shown that the intensity of the field determines the strengthof the nonadiabatic coupling, namely the steepness of the cone, while the energy of thefield specifies the position of the LICI. The methodology and the numerical details
The main subject of this section is to obtain the appropriate expression so as to computethe geometric phase. 5et us consider again the working Hamiltonian Eq. 2 which is parametrized by R and θ . If the system starts in an eigenstate Φ( R, θ ) with an energy E ( R, θ ) , then it evolves intothe state exp[ − iE ( R, θ ) t ]Φ( R, θ ) . Now let the parameters vary slowly, R = R ( t ) and θ = θ ( t ) , then due to the adiabatic theorem, the eigenstates Φ( R, θ ) are replaced by one of theactual eigenstates Φ( R ( t ) , θ ( t )) . If both R ( t ) and θ ( t ) are periodic functions of time witha period of T they describe a closed path in the configuration space. That is, for the time t = T the initial state Φ( R ( t = 0) , θ ( t = 0)) evolves into the final state which is identicalwith the initial state except for a phase factor: Φ( R ( T ) , θ ( T )) = exp ( iχ ) · Φ ( R (0) , θ (0)) . It is easy to see that the phase factor is identical with the autocorrelation function ( C ( t ) = (cid:104) Φ( R (0) , θ (0)) | Φ( R ( t ) , θ ( t )) (cid:105) ) at time T . Berry showed that χ is the sum of δ = − ´ T E ( R ( t (cid:48) ) , θ ( t (cid:48) )) dt (cid:48) and a quantity γ , latter is called the adiabatic phase. Here χ and δ are the overall and the dynamical phases, respectively.Both the χ and the δ functions can be generalized for any arbitrary time t . To obtainthe actual expressions for the χ ( t ) and δ ( t ) phases we refer to the work of Mukunda andSimon . Among others they have pointed out that the overall phase is the argument ofthe autocorrelation function χ ( t ) = arg (cid:104) Φ( R (0) , θ (0)) | Φ( R ( t ) , θ ( t )) (cid:105) (4)and the dynamical phase is as follows δ ( t ) = i ˆ t (cid:68) Φ( R ( t (cid:48) ) , θ ( t (cid:48) )) | ˙Φ( R ( t (cid:48) ) , θ ( t (cid:48) )) (cid:69) dt (cid:48) . (5)Aharonov and Anandan pointed out that γ is a purely geometrical property of thepath which is parametrically defined by the functions R = R ( t ) and θ = θ ( t ) and can becalculated as the difference of the χ ( t ) and δ ( t ) at the end of the closed path. Therefore itsvalue depends only upon the contour followed by the system in the configuration space.Hence the name of γ is geometric phase . If Φ( R ( t ) , θ ( t )) is the solution of the dynamical Here we allow a very slow time dependence of the R and θ parameters, so as to assume an implicitadiabatic time dependence of the working Floquet Hamiltonian. From now on we will refer to γ as Berry, geometric or adiabatic phase. i (cid:126) (cid:12)(cid:12)(cid:12) ˙Φ( R ( t ) , θ ( t )) (cid:69) = ˆ H ( R ( t ) , θ ( t )) | Φ( R ( t ) , θ ( t )) (cid:105) ,then we obtain for Eq. 5 δ ( t ) = 1 (cid:126) ˆ t (cid:68) Φ( R ( t (cid:48) ) , θ ( t (cid:48) )) | ˆ H ( R ( t ) , θ ( t )) | Φ( R ( t (cid:48) ) , θ ( t (cid:48) )) (cid:69) dt (cid:48) . (6)Using Eqs. 4 and 6 one can calculate the geometric phase γ , as difference of the χ ( t ) and δ ( t ) expressions at the end of the closed path for adiabatically slow changes ofthe parameters R ( t ) and θ ( t ) over the whole path. As we do not know in advance howslow change can be considered as an adiabatic one during the numerical simulations weconsider the quantity (cid:101) γ = χ ( T ) − δ ( T ) (7)as an approximation for the Berry phase for the given contour.To get the Φ( R ( t ) , θ ( t )) wave function we have solved numerically the time-dependentSchrödinger equation by using implicit 4th order Runge-Kutta integrator with Gaussianpoints implemented in the GNU Scientific Library . Results and discussion
So as to understand the meaning of the numerical results to be presented in this paper,we first discuss the geometrical situation for which the above approach is applied andthen analyze the numerical results.The light-induced adiabatic states ( V lower and V upper ) as well as the position of thecorresponding light-induced conical intersection are displayed in Figure 1. The geometri-cal arrangement used as contours in the geometric phase calculations are shown in Figure2. Here, four different ellipses are presented but only one of them surrounds the light-induced conical intersection. The numerical calculations take place along these closedpaths characterized by their centers ( R c , θ c ) and radii ( ρ R , ρ θ ) . The actual position is7iven by an angle β ( t ) = β + t/T · π : R ( t ) = R c + ρ R cos β ( t ) (8) θ ( t ) = θ c − ρ θ sin β ( t ) . The applied parameters for the different contours are displayed in Table 1. In Table 2the obtained approximated values ( (cid:101) γ ) for the geometric phase γ are presented (in unit of π ) for the contour C which encircles the LICI with the initial wave function chosen tobe the lower lying eigenstate of the Hamiltonian Eq. 2 at point S . The approximationis based on the difference of the argument of the autocorrelation function (Eq. 4) andthe dynamical phase (Eq. 6) at the end of the path (see Eq. 7). The applied photonenergy is (cid:126) ω L = 1 . eV . In the rows of Table 2 the different field intensities are chosento be between I = 1 × Wcm and I = 1 × Wcm . In the columns, the periodic timesof the round transport of the ellipse are indicated as a unit of the periodic time of thelaser pulse ( πω L ). The larger the value of T indicated here, the more adiabatic the processof encircling the ellipse. As the contour C surrounds only a single conical intersectionthe value of the geometric phase γ is expected to be ± π and so for long enough T values (cid:101) γ should be in the close vicinity of (2 n + 1) π (where n in an integer). We can observein Table 2 that except for the lowest studied intensities the value of (cid:101) γ is really close to π , whenever T ≥ × πω L . The numerical problem at small intensities are related tothe fact that for the field free case (zero intensity) the value of γ should be zero. As aconsequence, in weak fields we need extremely slow surrounding of the contour to be ableto consider it as an adiabatic one. For the extremely large values of T question arisesabout the accuracy concerning the numerical integration of the Schrödinger equation. Asa simple check for this issue we have also performed numerical integration over the samecontour for the field free case. The obtained (cid:101) γ values are displayed in the first row ofTable 2. All of these values are close to the expected value of γ = 0 .For larger intensities the adiabatic region can be reached before T = 500 × πω L . Table2 shows that for intensities larger than I = 1 × Wcm the beginning of this adiabatic8egion is moving towards larger values of T with the increasing intensities. This effectis related to the fact that at higher intensities the derivative of the adiabatic potentials( V lower and V upper ) respect to the position on the contour (controlled by parameter β ( t ) )become significantly larger than the same derivatives of the diabatic ones. As a result,slower change is requested in the value of the β ( t ) parameter so as to consider the processbeing adiabatic.In Figure 3 the difference of the χ ( t ) and δ ( t ) functions are displayed as a function oftime with three different time resolutions. Results are presented of the set of simulationsfor which the (cid:101) γ values are displayed in Table 2 at I = 1 × Wcm field intensity. It can beseen that the different curves display different shapes at t/T = 0 . , but all of them possessrelatively sudden jumps ranging from near zero to close to the final value. The phasejumps always take place at that position of t/T , where the value of the autocorrelationfunction tends to zero. In the panels of Figure 3 the positions of the phase jumps arealways positioned at t/T = 0 . . But this happens due to symmetry reasons. In thecurrent geometrical arrangement (see on Figure 2), the center of the ellipse is also theposition of the LICI. In this situation, four symmetry points can be considered concerningthe starting position of the encircling of the ellipse. These symmetry points are theendpoints of the small and the large axes of the ellipse. If one starts to circle the ellipseat one of these points the phase jumps always occur at t/T = 0 . . If this process startsfrom a different point then the phase jump happens at another value of t/T . E.g. for thecontour C the phase jump occurs at t/T = 0 . if the starting points are S or S (cid:48) but forstarting point S (cid:48)(cid:48) the jump happens at t/T ∼ = 0 . . Nevertheless, the values of the phasejumps are always very close to an odd multiple of π . The longer the encircling time, thesteeper the phase jump, but its value is getting closer and closer to the final value of (cid:101) γ .This effect is clearly recognizable as long as the time resolution gets finer (see on panelsof Figure 3).We have also computed the value of the approximate geometric phase (cid:101) γ along thoseellipses which do not surround LICI. Obtained results are always very close to zero.For completeness we have performed similar calculations on the upper adiabatic sur-9aces as well. All of these calculations provide the same results as for the case of lowersurface but always with an opposite sing for (cid:101) γ . Table 3 displays some values of (cid:101) γ at I = 1 × Wcm field intensity. We notice that for contour C starting the simulation atpoint S (cid:48) the calculated values of (cid:101) γ are always around of ± π or ± π depending upon theactual speed of the surrounding. (All of these values are odd multiples of π so they arein agreement with the expected value of the geometric phase γ = π .) This uncertainty isclearly related to the fact that the correct value of the autocorrelation function is zero at t/T = 0 . and therefore it is extremely hard to follow its argument during the numericalsimulations. Conclusions
By applying adiabatic time-dependent framework and Floquet representation for theHamiltonian we have calculated the geometric phase of the light-induced conical inter-section formed in the D +2 molecule. It has been demonstrated that assuming certainconditions for the initial wave functions the adiabatic time-dependent results for the ge-ometric phase are similar to those obtained from the time-independent solutions .Obviously, obtained numerical results are also in full agreement with the values of theBerry phase that hold for the natural conical intersections.In the future, our aim is to compute the Berry phase for the exact time-dependentlight-matter Hamiltonian, too. However, this is not an easy task because of the explicittime-dependence of the Hamiltonian. The latter gives rise to additional difficulties andthe adiabatic transport round a close path is far from being trivial. Acknowledgements
The supercomputing service of NIIF has been used for this work. This research wassupported by the EU-funded Hungarian grant EFOP-3.6.2-16-2017-00005. The authorsthank Tamás Vértesi for many fruitful discussions.10able 1: Parameters of the applied contours in the configuration space corresponding toFigure 2 and Eq. 8.Contour R c [ a.u. ] R θ [ rad ] ρ c [ a.u. ] ρ θ [ rad ] β [ rad ] C π/ π/ − π/ for S ; 0 for S (cid:48) ; − π/ for S (cid:48)(cid:48) C π/ π/ − π/ for S C π/ π/ − π/ for S C π/ π/ − π/ for S -3-2-1012 E n e r gy / e V Interatomic distance / a.u. L = 912.3 nm= 1.359 eVI = 1.0 10 W/cm ¯ hω L sσ g pσ u ¯ hω L pσ u − ¯ hω L Figure 1: A cut through the potential energy surface of the D +2 molecule as a functionof interatomic separation. Diabatic energies of the ground V ( R ) (1 sσ g ) and the firstexcited V ( R ) (2 pσ u ) states are displayed with solid blue and red lines, respectively. Thefield dressed excited state ( pσ u − (cid:126) ω L ; dashed red line) forms a light induced conicalintersection (LICI) with the ground state. For the case of a laser frequency (cid:126) ω L = 1 . eV and field intensity of × Wcm a cut through the adiabatic surfaces at θ = 0 (parallelto the field) is also shown by solid black lines marked with circles ( V lower ) and triangles( V upper ). We denote with a cross the position of the LICI ( R LICI = 5 . a.u. ). O r i e n t a ti on , / r a d Interatomic distance / a.u. CI S S S S S S C C C C Figure 2: Geometrical arrangement of the contours used in the geometrical phase calcula-tions. Four different geometrical arrangements are applied and only one curve surroundsthe LICI. The black cross shows the position of the LICI. Dots denote the starting pointsof the different simulations in the configuration space.11able 2: The difference of the total and dynamical phases in the units of π at the endof the paths ( t = T ) C surrounding the LICI (see on Fig.2) The staring point of thesurrounding is S and the initial wave function is chosen to be on the lower adiabaticsurface.Intensity T [2 π/ω L ][ W/cm ]
10 20 50 100 200 500field free <1 · − <1 · − <1 · − <1 · − <1 · − <1 · − · · · -0.1732 -7.4972 -5.1357 4.5659 -24.4448 -96.5521 · -0.5458 -7.2619 -3.5758 7.7951 -51.2940 -17.0145 · -0.9243 2.2194 -9.2701 -1.6251 -4.4822 -0.4876 · -0.6192 1.9429 -0.6340 -3.7004 -0.4275 0.4730 · · · · · · · · · · Intensity T [2 π/ω L ][ W/cm ] · − · − · − · − · − · − · -305.7453 -82.5002 -192.7763 -60.5332 -29.1699 -10.6957 · -148.5720 -49.4776 -12.8597 -5.5952 -2.2615 -0.3007 · -39.6079 -13.1177 -3.8062 -1.3639 -0.1775 0.5295 · -5.7173 -2.0801 -0.1961 0.4041 0.7023 0.8809 · · · · · · · · · · · · t )-( t ) / r a d t / T T=10 2 / L T=20 2 / L T=50 2 / L T=100 2 / L T=200 2 / L T=500 2 / L (A) ( t )-( t ) / r a d t / T T=100 2 / L T=200 2 / L T=500 2 / L T=1000 2 / L T=2000 2 / L T=5000 2 / L (B) ( t )-( t ) / r a d t / T T=1000 2 / L T=2000 2 / L T=5000 2 / L T=10000 2 / L T=20000 2 / L T=50000 2 / L (C) Figure 3: The difference of the total and the dynamical phases as a function of time. Thisquantity provides the value of the geometric phase γ at t/T = 1 if the T is long enough. T is the periodic time encircling the ellipse. The computation starts at the point S ofthe ellipse which surrounds the LICI (see on Fig.2). Different time intervals are depicted.In panel (A) the whole time period t/T { , ... } , in panel (B) the t/T { . , ... . } andin panel (C) the t/T { . ... . } are figured. The applied intensity is I = 1 × Wcm .13 eferences (1) M. Born, Ann. Phys. , 457 (1927).(2) J. von Neumann and E. P. Wigner, Zeit. fur Phys. , 467 (1929).(3) Y. Aharonov, and D. Bohm, Phys. Rev. , 485 (1959).(4) H. Köppel, W. Domcke, and L. S. Cederbaum, Adv. Chem. Phys . , 59 (1984).(5) M. Baer, Phys. Rep. , 75 (2002).(6) G. A. Worth and L. S. Cederbaum, Ann. Rev. Phys. Chem. , 127 (2004).(7) W. Domcke, D. R. Yarkony and H. Köppel, Conical Intersections: Electronic Struc-ture, Dynamics and Spectroscopy;
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