Potential energy surfaces for electron dynamics modeled by floating and breathing Gaussian wave packets with valence-bond spin-coupling: An analysis of high-harmonic generation spectrum
PPotential energy surfaces for electron dynamics modeled by floating and breathing Gaussian wavepackets with valence-bond spin-coupling: An analysis of high-harmonic generation spectrum
Koji Ando ∗ Department of Information and Sciences, Tokyo Woman’s Christian University,2-6-1 Zenpukuji, Suginami-ku, Tokyo 167-8585, Japan (Dated: November 10, 2018)A model of localized electron wave packets (EWPs), floating and breathing Gaussians with non-orthogonalvalence-bond spin-coupling, is applied to compute the high-harmonic generation (HHG) spectrum from a LiHmolecule induced by an intense laser pulse. The characteristic features of the spectrum, a plateau up to 50harmonic-order and a cut-off, agreed well with those from the previous time-dependent complete active-spaceself-consistent-field calculation [T. Sato and K. L. Ishikawa, Phys. Rev. A , 023417 (2015)]. In contrastwith the conventional molecular orbital picture in which the Li 2s and H 1s atomic orbitals are strongly mixed,the present calculation indicates that an incoherent sum of responses of single electrons reproduces the HHGspectrum, in which the contribution from H 1s electron dominates the plateau and cut-off, whereas the Li 2selectron contributes to the lower frequency response. The results are comprehensive in terms of the shapes ofsingle-electron potential energy curves constructed from the localized EWP model. INTRODUCTION
Electron dynamics in molecules is an emerging field of re-search driven by recent advances of attosecond time-resolvedlaser spectroscopies [1–4]. In particular, the high-harmonicgeneration (HHG) spectra induced by intense laser field hasbeen a major subject [5–9] vitalized by a proposed possibilityof self-probing molecules by their own electrons, which aimsas far as to probe electronic wave functions (more preciselythe Dyson orbitals) via the so-called molecular orbital (MO)tomography [10–13].Theoretical studies of HHG spectra with quantum dynam-ical calculations of realistic systems have been rather limitedto small atoms and molecules such as H, He, H +2 , H , HeH + ,D +3 , and LiH [14–20]. Treating more electrons in larger andmore complex molecules seems too demanding at present un-less invoking the time-dependent mean-field approximationsof various levels [21–27] or the density functional theory(DFT) [28–30]. The conventional MO and DFT calculationsare based on atomic orbitals (AOs) that are clamped at nuclearcenters, with the time-dependence carried by the coefficientsof MO or the configuration-interaction. They essentially relyon the concept of one-electron orbitals in the mean-field thatare normally delocalized over the molecule according to itsspatial symmetry. To describe the dynamics of delocalizedwave functions by spatially fixed basis functions, those oflarge wave numbers or high angular momenta are needed.To obtain an alternative perspective, we have been study-ing a model of localized electron wave packets (EWPs) withnon-orthogonal valence-bond (VB) spin-coupling [31–34]. Itwas originally developed for a polarizable and reactive force-field model in condensed phase simulations to be combinedwith nuclear wave packets for light atoms [35–38]. For smallmolecules such as H , LiH, BeH , CH , H O, and NH in ∗ E-mail: ando [email protected] the ground electronic state, the model gives reasonably ac-curate potential energy surfaces with the minimal number ofEWPs [32]. The accuracy is considered to come from theflexibility to describe the static correlation by the VB cou-pling, the dynamic correlation by the EWP breathing, andthe polarization by the EWP floating. The electronic excitedstates of LiH have been also examined by quantizing the po-tential energy curves constructed in the same way as in thiswork [33]. The present work is an extension to study thereal-time quantum electron dynamics. Along this line, wehave reported recently the semiquantal WP dynamics of Li 2sEWP in LiH induced by an intense laser pulse [34]. The com-puted HHG spectrum exhibited the intensity up to a hundredof harmonic order, but not the characteristic plateau and cut-off. We considered this would be due to the lack of quantumcoherence in the simple semiquantal dynamics of the local-ized EWP, and could be remedied by introducing the MonteCarlo integration of coherent-state path-integral (CSPI) prop-agator [39, 40]. Nonetheless, for a small diatomic moleculesuch as LiH, the full quantum dynamics in one-dimension isstraightforward on the effective one-electron potential energycurves constructed from the EWPs. This will be a test for theadequacy of the model before proceeding to the CSPI calcu-lations.After an outline of the theory and computation in Sec. II,potential energy curves for electron motion in a LiH moleculeare presented. Single-electron quantum dynamics on these po-tential curves induced by an intense laser pulse are computedand the HHG spectra are examined. Section IV concludes.
THEORY AND COMPUTATION
As in our previous reports [31–34], The electronic wavefunction is assumed to be an antisymmetrized product of spa-tial and spin functions,
Ψ(1 , · · · , N ) = A [Φ( r , · · · , r N )Θ(1 , · · · , N )] , (1) a r X i v : . [ phy s i c s . c h e m - ph ] F e b with the spatial part modeled by a product of one-electronfunctions, Φ( r , · · · , r N ) = φ ( r ) · · · φ N ( r N ) . (2)In contrast with the conventional VB methods that use theAOs clumped at nuclear centers, we employ ‘floating andbreathing’ spherical Gaussian WPs of variable position q i andwidth ρ i , φ i ( r ) = (2 πρ i ) − exp[ −| r − q i | / ρ i ] . (3)In our previous report [34], q i and ρ i were time-dependent,but in this work, the EWPs are used just to construct the ef-fective potential energy curves along displacements of q i onwhich the full quantum dynamics are evolved. The spin part Θ(1 , · · · , N ) consists of the spin eigenfunctions. In this work,we employ a single configuration of the perfect-pairing form, Θ = θ (1 , θ (3 , · · · θ ( N − , N ) , (4)with θ ( i, j ) = ( α ( i ) β ( j ) − β ( i ) α ( j )) / √ . The electronicenergy, E = (cid:104) Ψ | ˆ H | Ψ (cid:105) / (cid:104) Ψ | Ψ (cid:105) , is thus a function of thevariables { q i } and { ρ i } at a given nuclear geometry { R I } .We first optimized { q i } and { ρ i } to minimize the energy E ( { q i } , { ρ i } ; { R I } ) , to determine the optimal values { q (0) i } and { ρ (0) i } . The effective potential function for the j -th elec-tron V j ( q ) was then constructed by fixing all the variablesother than q j at the optimal values, V j ( q ) = E ( q (0)1 , · · · , q (0) j − , q , q (0) j +1 , · · · , q (0) N ,ρ (0)1 , · · · , ρ (0) N ; { R I } ) , (5)on which the time-dependent Schr¨odinger equation was nu-merically solved with the effective one-electron Hamiltonianfor the j -th electron, ˆ H j = − (cid:126) m ∇ q + V j ( q ) . (6)It might appear that the electronic kinetic energies weredouble-counted since they have been computed in E = (cid:104) Ψ | ˆ H | Ψ (cid:105) / (cid:104) Ψ | Ψ (cid:105) . However, those in V j ( q ) are constants withthe fixed { ρ (0) i } as the kinetic energy expectation for φ i ( r ) ofEq. (3) is (cid:126) / (8 mρ i ) .Although the procedure is essentially a one-electron ap-proximation under the field of other EWPs, it is distinct fromthose of MO and Kohn-Sham models. The use of such EWPpotentials is related to the coherent-state path-integral theoryin which the Gaussian WPs are identified as the coordinaterepresentation of the coherent-state basis and the action inte-gral is determined by the energy expectation with respect tothe WPs [39, 40]. It also has a technical advantage of remov-ing the singularity of Coulomb potential that can cause prob-lems with the numerical grid methods of quantum dynamicalcalculation. (This problem would be a reason for the use ofsoft Coulomb potential of a form / √ r + c in many of the previous studies, including Ref. [27] with which our resultswill be compared.)The scheme was applied to a LiH molecule under an in-tense laser pulse. The parameters were taken from Ref. [27]that employed the time-dependent complete-active-space self-consistent-field (TD-CASSCF) calculation. The internucleardistance was fixed at 2.3 bohr. The EWP centers q j weredisplaced along the bond direction and the electronic ener-gies were calculated to construct the potential curves V j ( q ) in one-dimension. The electron dynamics were induced by alaser pulse with time-dependent electric field E ( t ) = E sin( ω t ) sin ( πt/τ ) , ≤ t ≤ τ, (7)parallel to the bond direction. The frequency ω correspondsto the wavelength of 750 nm, the duration τ is of three op-tical cycles, τ = 3(2 π/ω ) (cid:39) . fs, and the field inten-sity E is . × V/cm with the laser intensity . × W/cm . The length of the simulation box was taken to be1200 bohr, with the transmission-free absorbing potential [41]of 120 bohr length at both ends. The initial condition of theelectronic wave function at t = 0 was a Gaussian function ofthe center q (0) j and width ρ (0) j , i.e., those optimized withoutthe external field. The wave functions were propagated withthe Cayley’s hybrid scheme [42] with the spatial grid lengthof 0.2 bohr and the time-step of 0.01 au ( ∼ − throughout the simulation. The HHG spectra were com-puted from the Fourier transform of the dipole accelerationdynamics.We note that the quantum dynamical calculations in thiswork are one-dimensional: the EWPs are spherical in three-dimension, but they are used just to construct the effec-tive potential curves V j ( q ) along the bond axis. The one-dimensional treatment is in accord with the TD-CASSCF cal-culation of Ref. [27] which we take as the reference for com-parison. RESULTS AND DISCUSSION
Figure 1 displays the potential energy curves V j ( q ) for theelectrons in LiH. Two of them are deeply bound to the Li nu-clear center and correspond to the Li 1s core electrons. Theircontributions to the HHG spectrum have been analyzed pre-viously with the ordinary frozen-core treatment [27]. There-fore, we focus on the more labile Li 2s and H 1s electronswith much shallower potential wells in Fig. 1. The poten-tials for EWPs are modulated under the external laser field viathe field-dipole interaction. The modulations at the maximumand minimum of the field E ( t ) in Eq. (7) (see also the upperpanel of Fig. 3) are displayed in Fig. 2. The potentials indi-cate that the dynamics of Li 2s electron will be directly drivenby the laser field without energy barriers, whereas the H 1selectron will be basically bound near the proton but with pos-sibilities of tunneling out in both directions. These picturesare confirmed in Fig. 3 that plots the position expectation and −1−0.5 0−15 −10 −5 0 5 10 15 Li H ene r g y ( au ) coordinate (bohr) Li 2sH 1sLi 1sLi 1s
FIG. 1: Potential energy curves for displacements of the wave packetcenters in the singlet X Σ + state of LiH. The inset shows the wavepackets represented by circles with the radius of wave packet width ρ i −15 −10 −5 0 5 10 15 H 1s ene r g y ( au ) displacement (bohr) 2 3 4 Li 2s ene r g y ( au ) FIG. 2: Potential energy curves for Li 2s and H 1s electron wavepacket centers modulated by the laser field of Eq. (7) via the field-dipole interaction. its root-mean-squares (rms) deviation of the time-dependentwave function.Figure 4 displays the HHG spectra from the dipole accel-eration dynamics of Li 2s and H 1s electrons, and their in-coherent sum with an equal weight, as the dipole moment isadditive, neglecting the cross-correlation in the power spec-trum. For comparison, data from the TD-CASSCF calcula-tion [27] was included. It is seen that the incoherent sum ofthe Li 2s and H 1s spectra agrees well with the TD-CASSCFspectrum, particularly the plateau up to ∼
50 harmonic order(HO) and the cut-off. The spectra from individual electronsindicate that the plateau and cut-off come almost solely fromthe H 1s dynamics. The spectrum of Li 2s electron dominates −40−20 0 20 40 0 2 4 6 8 d i s p l a c e m en t ( boh r) time (fs) Li 2s rmsH 1s rms
Laser field
FIG. 3: Time evolution of the position expectation value and its root-mean-squares (rms) deviation. The upper panel displays the laserfield of Eq. (7). −12−10−8−6−4−2 0 20 40 60 80 100 l og [I n t en s i t y ] ( a r b . un i t s ) harmonic order Li 2sH 1sLi2s+H1sTD−CASSCF
FIG. 4: Fourier transforms of the dipole acceleration that give thehigh-harmonic generation spectra. The abscissa is the harmonic or-der ω/ω . The TD-CASSCF data is from Ref. [27]. the low-frequency peak at 1 HO, but decays by ∼
20 HO. Thisis comprehended in terms of the potential shape and its modu-lation in the upper panel of Fig. 2: the potential well for Li 2sEWP is shallow such that the dynamics will be rather similarto that of a free electron directly driven by the external field,which will result in the dominant contribution of the first HOin the spectrum. The agreement of the incoherent sum withthe TD-CASSCF spectrum also implies that the correlationbetween the H 1s and Li 2s electrons is minor. This is againcomprehensive with the results in Fig. 3: the small amplitudeoscillation of the position expectation of H 1s electron indi-cates that the mean-field treatment for the calculation of Li 2selectronic potential was adequate, and the dynamics of thesetwo electrons with different amplitudes of spatial oscillationare mostly decoupled.In the conventional MO picture, the Li 2s and H 1s AOsare strongly mixed in the valence MOs, from which the time-dependent wave functions are described by mixing many con-figuration state functions. The picture is thus fundamentallydifferent from the present VB WP, but by applying some lo-calization or projection analysis to the complex MO-CI wavefunction, a connection might be found. Similar argumentwould apply to the picture of population transfers amongMOs, in particular, the natural orbitals [24, 26, 43]: in thiscase, a projection of the present wave functions to the conven-tional natural orbitals would complement the picture. Anotherintriguing issue is on the different treatments of Coulomb in-teraction, the soft Coulomb potential in Ref. [27] and the or-dinary Coulomb potential in the present work. The presentscheme removes the singularity of /r by averaging with theEWPs φ i ( r ) , to give an effectively softened potential V j ( q ) without empirical adjustments. CONCLUSION
The single-electron quantum dynamics on the electronicpotential energy curves constructed with use of the local-ized floating and breathing electron wave packets with theVB spin-coupling produced the HHG spectrum of a LiHmolecule in good agreement with the previous TD-CASSCFcalculation. The electronic potential energy curves providesa unique picture for understanding the dynamics. Althoughmore case studies and improvements for numerical efficien-cies are needed, we envisage that its applications will extendnot only to other molecules but also to the electron conductionand optical processes in condensed matters.
Acknowledgment
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