J. Phys. Chem. 1994, 98, 11428-11438
11428
Wavepacket Calculations of Femtosecond Pump-Probe Experiments on the Sodium Trimer A. J. Dobbynt and Jeremy M. HutSon* Department of Chemistry, University of Durham, Durham DHI 3LE, England Received: April 11, 1994; In Final Form: August 17, 1994@
Wavepacket calculations are carried out to simulate femtosecond pump-probe experiments involving pseudorotational motion on the B state of the sodium trimer. The calculations are carried out in two dimensions, corresponding to the radical and angular pseudorotation modes. Simple potential energy surfaces appropriate for second-order Jahn-Teller and pseudo-Jahn-Teller effects are used. The initial wavepacket on the B state is created by explicit modeling of the short-pulse excitation process, taking proper account of interference effects that occur on the B-state surface. The resulting wavepacket is propagated for up to 20 ps by using the Chebyshev propagator. The ionization signal is then simulated in three different ways: (i) explicit modeling of the probe laser; (ii) overlap between the wavepacket on the B state and the ground-state ion wavefunction; (iii) the zero-time autocorrelation function on the B state. The ionic overlap method is found to be in fair agreement with the exact method, but the zero-time autocorrelation function is not. The results show reasonable agreement with the frequencies obtained from the experimental work, but the intensities do not agree well.
to the approach or intersection of several Bom-Oppenheimer surfaces. In the electronic states of interest here, there is only There has recently been a great deal of interest in timea small potential barrier between the various acute and obtuse resolved experiments which use femtosecond laser pulses. angled isosceles mangle geometries,and the molecule can move These laser pulses are shorter (10-100 fs) than even the fastest from one geometry to another with relative ease. This motion vibrational motions or dissociation reactions, allowing the atomic is generally referred to as pseudorotation and is of particular motions to be resolved and the dynamics of a chemical system interest as a molecular manifestation of Berry's geometrical to be followed in real time.' phase."J* In systems with bound potential energy surfaces, femtosecond A. Background. There has been a considerable amount of time-resolved experiments can be used to view wavepacket spectroscopic work on the sodium trimer. Experiments have motion directly. For example, in experiments on 12, Bowman been carried out to investigate both the ground state and various et aL2 used a pump laser pulse to prepare a coherent superposielectronically excited states. The excitation spectrum has been tion of a few vibrational states in the B state of 12. This systematically investigated from 700 to 330 nm, both by twoproduced a wavepacket moving on the B-state potential curve. photon ionization experiments and by depletion experiment^.'^,'^ A probe laser pulse, delayed by up to 3000 fs from the pump, Five band systems originating in the X state are observed, was then used to excite the molecule to a higher excited state, corresponding to the A, B, B', C, and D excited states. The B which decays by fluorescence. For a probe laser at a particular state, which is bound, is the excited state that is involved in the frequency, the transition is resonant only when the molecule is femtosecond experiments of interest here. The B X band in a certain configuration or the wavepacket is in a particular system shows a long progression of nearly equally spaced bands position on the potential curve. Bowman et al. observed a (spacing about 128 cm-l), which are split into doublets. To fluorescence signal that oscillated as a function of the time delay high frequency of each doublet is a set of closely spaced bands between the pump and probe pulses, with a period of about of steadily increasing separation. The spectrum has been 300 fs, characteristic of the wavepacket motion on the B state. interpreted in terms of the pseudorotational motion of the sodium Time-dependent experiments have also been carried out to trimer. investigate wavepacket motion in the Naz s y ~ t e m , and ~.~ The vibronic structure of the Na3 ground state has been calculations simulating the experiments have been p e r f ~ r m e d . ~ . ~ investigated both by the analysis of hot bands in the two-photon The experiments have recently been extended to the sodium ionization spectral5 and by stimulated emission pumping (SEP) where the multidimensional nature of the problem s p e c t r o ~ c o p y . ~The ~ * *B~ X system exhibits a clear hot-band allows much more complicated dynamics. The aim of the structure, which has been analyzed to give X-state vibrational present paper is to simulate the femtosecond pump-probe frequencies. The wavenumbers are 139 cm-' for the symmetric experiments on the sodium trimer and thus to investigate the stretch and 49 and 87 cm-' for the asymmetric stretch and bend. additional features that arise for wavepacket motion in more Similar results are obtained from the hot-band analysis of the than one dimension. spectra involving the A and C states. In the SEP experiment, The ground electronic state and many of the excited states numerous resonances are observed in the spectrum in the range of Na3 are orbitally degenerate at an equilateral configuration. from 150 to 1000 cm-'. A strong and simple pattem found in This leads to a Jahn-Teller (or pseudo-Jahn-Teller) distorthe low-resolution spectrum can again be interpreted in terms tion, lo producing equilibrium geometries that are isosceles of ground-state normal-mode wavenumbers of 139, 49, and 87 triangles. Thus, spectroscopic and dynamical studies of the cm-l. sodium trimer offer the opportunity to investigate effects due Time-Resolved Experiments. In the femtosecond pump-probe experiment^,^-^ Na3 is prepared in its ground electronic state Present address: Max-Planck-Institut ftir Stromungsforschung, Bunin a supersonic beam. The same laser is used for both the pump senstrasse 10, 37073 Gottingen, Germany. and the probe. The time profile of the pulse is roughly Gaussian, Abstract published in Advance ACS Abstracts, October 1, 1994.
I. Introduction
-
-
@
0022-3654/94/2098-11428$04.50/0
0 1994 American Chemical Society
Wavepacket Calculations on Na3 with a temporal full-width at half-maximum (FWHM) of about 70 fs.18 The spectral distribution is non-Gaussian, with significant intensity from about 608 to 626 nm and a peak around 623 nm; the distribution is somewhat wider than implied by the temporal width alone. Molecules in the beam are excited by the pump pulse to the B state. The pump laser has only sufficient energy to populate about the lowest 300 cm-I of the B state. After a given time delay, a probe pulse is used to ionize the Na3: there is only just enough energy in the probe pulse for ionization to occur. The ions that are produced are detected by time-of-flight mass spectroscopy, allowing the masses of the ionic fragments and the corresponding kinetic energy release to be determined. The experiment is repeated for many different pump-probe delay times, up to a maximum of approximately 12 ps. The ion signal takes a few femtoseconds to appear and then oscillates, though not back to zero. The principal period of oscillation is 320 fs, corresponding to a wavenumber of 105 cm-I. The Fourier transform of the ion signal shows peaks at 12 (m), 19 (m), 34 (w), 50 (m), 73 (m), 105 (vs), 90 (s), 123 (w), and 140 (w) cm-' and higher frequencies (w = weak, m = medium, s = strong, vs = very strong). Pump-probe experiments have also been performed in which the zero kinetic energy (ZEKE) electrons that are ejected on ionization are d e t e ~ t e d .These ~ ~ ~ experiments used a slightly shorter wavelength laser pulse (peak 618 nm) than those in which the ion signal was detected. The time-dependent ZEKE signal is similar to the ion signal, and the same frequencies are obtained in the Fourier transform, though the intensities are somewhat different. The oscillations in the Na3+ signal are due to the motion of the time-dependent wavefunction, created by the pump laser pulse, on the potential surfaces of the B and X electronic states of Na3. Each frequency observed in the ionization signal corresponds to the difference between two energy levels (but not necessarily to a spectroscopically allowed transition). The X-state frequencies occur because the pump laser can produce a wavepacket on the ground state by a two-photon excitation via the B state, and the probe laser can ionize it by another two-photon process. The contribution of the X-state frequencies to the total ion signal becomes less significant at lower powers.'* This in principle allows the frequencies to be assigned to the two different states, though the procedure is experimentally difficult. The frequencies have tentatively been assigned as foll o w ~ : ~12, - ~19, and 35 cm-' as beat frequencies between the pseudorotation modes on the B state; 50 cm-I as the asymmetric stretch frequency on the X state; 73 cm-' as the bending frequency on the B state; 90 cm-' as the bending frequency on the X state; 105 cm-l as the symmetric stretch frequency on the B state; and 140 cm-' as the symmetric stretch frequency on the X state. Transient two-photon ionization experiments are also being carried out on the sodium trimer, using picosecond pump-probe techniques followed by mass-selective dete~tion.l~-*~ 11. The Coordinates
The complete nuclear permutation-inversion group of the sodium trimer is D3h(kf). Use of this group ensures that the symmetry of the wavefunctions with respect to exchange of atoms is properly accounted for.** However, in the present work, rotations of the molecule are neglected, and it is adequate to work with the point group D3h (which is isomorphic to D34M)). The normal vibrational coordinates of Na3 belong to the irreducible representations a'l and e' of D3h and may be written
J. Phys. Chem., Vol. 98, No. 44, 1994 11429
['
Qs = 31/2 -xl
+ (;x2 - 7 y 2 ) 4-(;x3 + q y 3 ) ]
(3)
where xi and yi are the Cartesian coordinates of atom i. Qsis the symmetric stretch, and Qx and Qy are the two components of the normal e'-type displacements. Qx and Qy may be described as bend and asymmetric stretch coordinates, but different authors do not necessarily agree about which description applies to which component. Qswill not be considered until later, since it is only indirectly involved in the Jahn-Teller effect. Consider now a Cartesian space in Qx and Qy. At the origin of this space, the sodium trimer has an equilateral geometry. For a displacement in the positive Qx direction the molecule has an acute isosceles geometry, while for a displacement in the negative Qx direction the molecule has an obtuse geometry. Because of the Jahn-Teller effect, the equilibrium geometry is an obtuse isosceles triangle, which lies some distance from the origin; there are three equivalent acute geometries and three equivalent obtuse geometries arranged symmetrically around the origin. The molecule can move from one obtuse isosceles geometry to another without going through the equilateral configuration, passing instead through an acute isosceles geometry. This corresponds to motion along a near-circular path around the origin; the change in configuration gives the appearance of a rotation of the molecule, and is usually referred to as a pseudorotation. Our wavepacket calculations are initially carried out in two dimensions, corresponding to Qx and Qy. However, it is more convenient to carry out the actual calculations in a polar coordinate system, using coordinates r and 4 defined by (4) The angle 4 will be referred to as the angular pseudorotation coordinate, and the distance r as the radial coordinate; r should not be confused with the symmetric stretch coordinate Qs. Introducing a reduced wavefunction W ( r , 4) = r-1'2Y(r,
4)
(5)
the two-dimensional kinetic energy operator appropriate for Y' is
where m is the mass of a sodium atom. It is convenient to work with Y' rather than Y when carrying out propagations.
111. Potentials The theory needed to describe the effect of Jahn-Teller distortions on the potential energy surface of the ground and excited states of Na3 has been discussed e ~ t e n s i v e l y . * ~Only -~~ a brief summary will be given here, in order to establish the notation that we have used and to provide a foundation for the discussion of the dynamics. The notation used elsewhere varies in both the symbols and the definitions used for the various vibronic coupling constants.
Dobbyn and Hutson
11430 J. Phys. Chem., Vol. 98, No. 44, 1994 A. The Ground State. The vibrational levels of the ground electronic state of Na3 can be understood in terms of E €3 e mixing between the components of a degenerate electronic state of E symmetry induced by a vibration of e symmetry. Consider an E’ electronic state in a molecule of D3h symmetry. The two normalized electronic wavefunctions of the E’ term, These at the equilateral geometry, are denoted ly; and.:$I functions can be used to expand the wavefunction at nonequilateral geometries. The potential V(re, Q), which includes the dependence on the electronic coordinate re, is expanded as a Taylor series about the equilateral geometry. Using complex combinations of the degenerate electronic functions, ly+ = (lyf f i~+!f)/2~/~, and of the normal coordinates, Q+ = Qx & iQy = re*I@, linear and quadratic vibronic coupling constants FE and & are defined,
(7) In addition, a force constant KE is defined,
It is convenient to define two characteristic frequencies for the radial motion,
The two electronic wavefunctions +’+, which are the eigenvectors of the secular equations, are
where tan 52 =
2FE sin 4 - IG,lr sin 24 2FE COS 4 -k (GElrCOS 2 4
An interesting point to notice in this equation is that Q acts qualitatively like 4 when F E > lGlr but acts qualitatively like -24 when FE < I&lr. Thus, for the portion of space given the pseudorotation coordinate must make two by r < FE/JC&I, circuits of parameter space (4 = 0 4n) around the conical intersection before the electronic wavefunction retums to its initial value; for a single circuit, the electronic wavefunction changes sign. Thus, in order for the total wavefunction to be single-valued in parameter space, the nuclear wavefunction must also change its sign for a single circuit. This gives rise to halfinteger quantization of the nuclear motion. This is sometimes referred to as the adiabatic sign-change theorem and is a manifestation of Berry’s geometrical phase,” which has been discussed extensively in connection with E €3 e Jahn-Teller systems.29 The half-integer quantization has been used to explain the spectrum of the B state of the sodium trimer (though probably mistakenly, as discussed below).28 There have been several ab initio studies of the sodium trimer. Cocchini et a1.26 have carried out generalized valence bond (GVB) and configuration interaction (CI) calculations to characterize the ground and excited electronic states. In addition, Martin and Davidson30have carried out CI calculations on the ground-state potential. However, for our purposes it is very important to use a potential surface that accurately reproduces the experimental spectroscopic results. We have therefore used a ground-state potential surface with the functional form described above, based on the potential developed by Meiswinkel and Koppe128to reproduce the experimental line positions and intensities. The parameters of the potential surface are expressed in terms of dimensionless coupling constants. First, C& and KE are defined through eq 14 by GEIKE= -0.076 and wdc = 87 cm-’. Second, FE is defined through the dimensionless quantity FE/ (hW&)lI2 = 5.456. The resulting potential is shown in Figure 1: it has a stabilization energy of 1401hc cm-’ and a localization energy of 198hc cm-’. The value of ro is 0.737 8, (at 4 = 180”, corresponding to an obtuse equilibrium geometry). There are some quantitative differences between the potential obtained from the experimental spectrum and those calculated ab initio, but they are all very flat, with a conical intersection at the equilateral geometry. The frequency for the radial motion (or bending frequency at small distortions) compares well in all the potentials. The barrier to pseudorotation (198 cm-l) agrees reasonably well with the CI potential of Martin and Davidson (175.4 cm-’) but is in poorer agreement with the GVB potential (131 cm-I). Both the ab initio potentials give lower values of the Jahn-Teller stabilization energy than the present potential. +
If cubic and higher-order terms in the expansion are neglected, the secular determinant (in the basis set formed by the electronic wavefunctions, ly+ and ly-) can be written as a function of r and 4,
so that the adiabatic potential surfaces are
(10) The lower surface is sometimes described as a “warped Mexican hat”: along the bottom of the trough of the hat there are three wells, alternating regularly with three humps. At the equilateral geometry there is a conical intersection since the surfaces e+ separate linearly. The extremal points of the lower surface are at geometries (ro, q50) given by
where the upper and lower signs correspond to the cases FE > 0 and FE < 0, respectively. If FE and GE are of the same sign, the points with n = 1, 3, and 5 are saddle points and the points with n = 0, 2, and 4 are minima, whereas for FE and GE of opposite sign the two types of extremal points are interchanged. The depth of the trough, relative to the point at r = 0 where the two electronic components are degenerate, is
This is commonly referred to as the Jahn-Teller stabilization energy. The (minimum) barrier height between the minima is
This barrier height is often referred to as the Jahn-Teller localization energy.
(16)
Wavepacket Calculations on Na3
J. Phys. Chem., Vol. 98, No. 44, 1994 11431
Q, I A
2
i
X state wavefunction
1
T -2
-2
B state
1
Q,IA
ion X state
2
1
-2 -2 Figure 1. Two-dimensional potentials used for the ground and B excited states of Na3 and the ground state of Na3+. Contours are in millihartrees (1 m E h = 219.5hc cm-'). Also shown is the ground-state vibrational wavefunction for Na3 used as the initial state in the wavepacket calculations.
It should be emphasized that the functional form used here is actually very primitive. The potential is anharmonic if is not zero but is nevertheless based on quadratic expansions of the potentials and coupling terms about the equilateral geometry. It would be highly desirable to use a more sophisticated potential that includes higher-order terms. However, not enough information is yet available to allow the construction of such a potential. B. The B Excited State. The GVB and CI calculations for the excited electronic states of Na3 indicate that the B state involves a complicated mixing between one electronic state of E' symmetry and another of A'1 symmetry. The coupling between these two states dominates the usual coupling between the components of the E' state.26 This conclusion has been supported both by an analysis of the vibrational spectrumz8and by recent rotationally resolved experiment^.^^^^^ The theory needed to handle the three interacting states follows the same pattern as that for the two-state case, described in the previous section. The electronic wavefunctions for the two components of the E' state, at the equilateral geometry, are
denoted q,"'and q: and have energy EE'. The corresponding wavefunction for the nearby A'1 state is denoted and has energy EA',. These electronic wavefunctions are again used to expand the wavefunction at nonequilateral geometries. The secular determinant that must be solved in this case, again using complex combinations of the components of the E' state and of the normal coordinates, is26
= O (17) where FE and are the linear and quadratic vibronic coupling constants within the E' state, and f and g are the linear and quadratic vibronic coupling constants between the A'1 and E' states. These coupling constants are again the matrix elements of the coefficients in the potential expansion, and terms beyond quadratic have been neglected.
Dobbyn and Hutson
11432 J. Phys. Chem., Vol. 98, No. 44, 1994 A number of approximations are now made in order to obtain a reasonably simple solution.26 First, FEand & are set equal to zero, implying that there is no interaction within the E‘ state. In this case, for small displacements, the two surfaces with E’ symmetry separate quadratically, so there is not a conical intersection at the equilateral geometry. Second, the force constants in the A’1 and E’ states are assumed to be the same, K = KE = KA. Finally, If.lKI is set equal to 3, which is assumed to be much larger than - EA,,^; this is valid for large distortions. The three adiabatic potentials obtained are then
e* = 1/2Kr2f r [2f
+ 2fgr cos 3 4 4- 1/2g2r2]112
(19)
The B state corresponds to the lowest of these three surfaces, e-. An interesting point to notice here is the similarity in the expressions obtained above for two-state and three-state interactions, eqs 10 and 19; with 21/2gtaking the place of GE and 2’/2f taking the place of FE, the expressions are identical. Thus the position of the extremal points of the surface and the JahnTeller stabilization and localization energies can all be described in the same way as before by using eqs 11-13. The fit to the absorption spectrum2*again yields parameters that are quite different from those obtained from the GVB calculations. Once again, we have chosen to use a potential based mainly on the static spectroscopy. The dimensionless coupling constants are 21i2g/K = -0.0063 and 21/2f(h~oK)1/2 = 4.34 and wdc is 127 cm-’. The resulting potential is shown in Figure 1; it has a stabilization energy of 1204hc cm-I and a localization energy of 15hc cm-’. The barrier to pseudorotation is thus much lower for the B state than for the X state. The value of ro for the B state is 0.467 A (again at 4 = 180°, corresponding to an obtuse isosceles triangle). The two-photon ionization experiment^'^,^^ showed the origin of the 0-0 band of the B X system to be very close to 16 000 cm-’. We have therefore taken the ground vibronic state of the B-state potential to be 16 000 cm-l above that of the X-state potential. C. Pseudorotation Eigenstates and Eigenvalues. Before considering the dynamics of Na3, it is useful to have an understanding of the eigenstates and corresponding eigenvalues. Consider first the linear Jahn-Teller effect, where the quadratic coupling constant GE (or g) is zero, so that there is no barrier to pseudorotation. The potential for the lower surface is of the form
-
V(r) = ‘I2KEr2- rFE
(20)
which can be expressed as V(r) = 1/2KE(r- ro)2 - E,
(21) where r = FE/KEand Es = F E ~ / ~ KThis E . potential is harmonic about ro and independent of 4. Thus, the motion can be described by free “rotations” around the trough together with radial vibrations across the The eigenfunctions are products of angular and radial functions, and may be approximated \y
.=
UJ
( 2 ~ r ) - ”exp(ij$)NJiu[a(r ~ - ro)] exp[-’/,a(r
- r0)’]
(22) where j is a “pseudorotational” quantum number for the angular motion and u is a vibrational quantum number for radial motion.
The constant a is (mKE)ll2/h,and Hu[a(r - ro)] is a Hermite polynomial of order u, with normalization constant Nu. For the X state of Na3, there is a conical intersection at the origin and the quantum number j is a half-integer because of the adiabatic phase change described above. However, for the B state in the approximation described above there is not a conical intersection at the origin so that the quantum number j is an integer. Thus, provided the interaction between the A’1 and E’ states dominates that between the components of the E‘ state, it is not appropriate to use half-integral pseudorotational quantum numbers for the B state.28 The approximate description of the eigenfunctions (22) is valid in the absence of quadratic coupling terms, when the radial potential is harmonic and centrifugal distortion effects are neglected. The corresponding energy levels are
Euj= (U
+ ‘I2)huE+ Aj2
where A is a pseudorotational constant,
A = - =)i22mr02
4E,
This is still a good description of the energy levels for small nonzero GE or g, as in the B state of Na3, but breaks down when the quadratic coupling constant gets larger and the wells in the trough become deeper, so that the angular motion becomes localized. It is not a good description of the lower pseudorotation levels for the X state of Na3. Coupling between the angular and radial pseudorotation modes can in principle occur in two ways. First, the well depth can change as a function of angle, as reflected in the JahnTeller localization energy. Second, the position of the minimum can move in and out as a function of angle; this will be called the “circularity” of the potential. For the functional form of the potential described here, the localization energy and the circularity are both determined by the quadratic coupling constant and cannot be varied independently. A more sophisticated functional form would relax this constraint. D. The Cation Na+3. The ground state of Na3+ is not subject to Jahn-Teller distortions. Carter and M e ~ e have r~~ calculated an ab initio potential surface for the ground state and described an analytic form. Their potential was used in the present work and is shown in Figure 1. The normal-mode wavenumbers obtained by fitting to the calculated anharmonic spectrum were 142 and 101 cm-’ for the symmetric stretch and degenerate bend, respectively. These are close to recent experimental values obtained from the ZEKE photoelectron spectroscopy of Na3.9 In the present work, the value of the 0-0 ionization energy of Na3 was taken to be 32240hc ~ m - ’ ) .This ~ ~ is rather larger than the value of 31 363 f 5 cm-’, recently obtained from the ZEKE ~ p e c t r a . ~
IV. Vibrational Wavefunctions The ground-state vibrational wavefunction for the X state of Na3, which is used as the initial state in the wavepacket calculation, was calculated from the adiabatic potential surfaces by using the TRIATOM p r ~ g r a m , which ~ ~ , ~implements ~ the variational method of Sutcliffe and T e n n y ~ o n . Jacobi ~~ coordinates in three dimensions were used. However, the potential used for the X state has no coupling between the pseudorotation coordinates and Qs,so that the Schrodinger equation is separable. The calculation was canied out using basis functions centered in only one of the three equivalent wells (at 4 = 180”), so that the resulting wavefunction is localized in that well as shown in
Wavepacket Calculations on Na3
J. Phys. Chem., Vol. 98, No. 44, 1994 11433
Figure 1. This approximation is not expected to have any substantial effect on the wavepacket dynamics and avoids complications arising from the half-integer quantization of the pseudorotational motion for the X state. The two-dimensional zero-point energy is 69 cm-’. The ground-state wavefunction of Na3+, which is needed in one of the approximate methods for calculating the ion signal described in section V.C, was calculated similarly. However, in this case the Schrodinger equation is not separable, so that the wavefunction is not a simple product. The ionic overlap calculations described below require a two-dimensional ion wavefunction that neglects the stretching coordinate, and this was obtained by taking a cut through the three-dimensional wavefunction at the equilibrium value of Qs, 3.443 A. The twodimensional zero-point energy, needed to define the relative positions of the potential curves of the ion and the neutral in terms of the 0-0 ionization energy, was estimated as 101 cm-’.
V. Simulation of the Experiment In the present work, time-dependent wavepacket calculation^^^ are used to model the femtosecond pump-probe experiments on the sodium trimer. The procedure can be divided into three distinct parts. First, the initial state is prepared. Second, the wavepacket is propagated. Third, the observables are extracted. Each of these stages will be described separately. A. Preparation of the Initial State. Before the pump laser is tumed on, the Na3 molecule is in its ground vibrational and electronic state. During the pump laser pulse, some wavefunction amplitude is excited to vibrational levels of the B state, and a nonstationary state (wavepacket) is formed. The way in which the wavepacket moves on the B-state surface is quite sensitive to the details of how it is prepared. In order to describe the preparation as accurately as possible, we need to model the effect of the pump laser explicitly. To describe the short pulse excitation p r o c e ~ s ,time-dependent ~~,~~ first-order perturbation theory is used. The wavefunction is written as
where Y t is the initial (time-independent) wavefunction on the X state and UB and UX are the time evolution operators for the B and X states, defined by the equation Y(t At) = U(At)W(t). Since Y; is an eigenfunction of the ground-state Hamiltonian, Ux(At) is simply exp(-iExAt/h). The zero of energy is chosen as the energy of the lowest vibrational level of the B state, as described below, so that EX= - 16000hc cm-’. Note that eq 25 is written with the center of the pump pulse at t = 0, so that the propagation actually starts at t = -tfid, which is chosen to be before the start of the pulse (tmid = 120 fs in the present work). The transition dipole function, p ~ xis, taken to be independent of the coordinates in the present work. The electric field due to the laser is
+
where EOis the magnitude of the electric field vector and o = c/I is the central frequency of the pulse, with I = 620 nm (w/c = 16 129 cm-’) in the present calculations. The functionfit) describes the temporal shape of the laser pulse, which in the present case is taken to have a Gaussian profile,
with aT = 3.364 x fs-l. In physical terms, eq 25 can be interpreted as follows. The laser pulse, although short, is not a &function of time. Thus,
the initial wavefunction is not promoted to the excited state instantaneously, and the wavepacket created on the excited state does not correspond to vertical excitation of the ground-state wavefunction. The wavefunction evolves on the ground-state potentid surface until time t‘ (for time t‘ tmid) and then (after promotion) evolves on the excited-state surface for time t - t‘. All values of t’ within the laser pulse are possible, so the expression is integrated over t‘. The consequences of the shortpulse excitation process for the wavepacket in the excited state have been considered in detail by Williams and Imre.39540Since the phase of the wavefunction evolves differently on the ground and excited states, interference effects are possible between the parts of the wavefunction that are promoted at the beginning of the pulse and those that arrive later on. Williams and Imre showed that for certain laser pulses the norm on the excited state could initially increase, but then show a decrease, because of destructive interference occurring on the excited state. During the pump laser pulse, the wavefunction at time t At is calculated from that at time t as described by Engel,4l
+
+
The wavepacket propagation on the B state uses the Chebyshev propagator as discussed below. The quantity At in eq 28 must be chosen to be small enough for the excitation process to be properly described. In the present work, we used At = 5 fs, with 70 terms in the Chebyshev polynomial expansion. Sample calculations were repeated with shorter time steps in order to check that the calculations were converged in this respect. The Initial Wavepacket on the B State. The time-evolution of the wavepacket in the ( I , 4) plane was explored by viewing animations on a graphics workstation. Some “snapshots” showing the formation of the wavepacket on the B state during the pump laser pulse are shown in Figure 2. The femtosecond laser pulse has a fairly large energy spread (approximately 300hc cm-l), so that the wavepacket produced is a superposition of many different eigenfunctions of the B state. The wavepacket produced does not have as large an energy spread as the one that would be produced by vertical excitation but is nevertheless considerably radially excited, because the values of ro are quite different on the X and B surfaces (by approximately 0.27 A). At the beginning of the pump laser pulse, the wavepacket that begins to develop on the B state is very similar to the ground-state wavefunction on the X state. Initially, the molecule sees the laser pulse as “white light”, so that the wavefunction on the ground state is moved vertically to the B state. At the start of the pulse, the wavepacket is centered around the X-state equilibrium geometry, r = 0.737 A. However, as the laser pulse stays on, the wavepacket produced begins to show characteristics that depend on the frequency and true width of the laser pulse and on the B-state potential energy surface. This comes about as the different parts of the wavepacket, which were excited to the B state at different times, interfere with one another. It can be seen in Figure 2 that, halfway through the laser pulse, the wavepacket has moved substantially toward the value of ro for the B state, r = 0.467 A. By the end of the pulse, the wavepacket has moved on from its equilibrium radial value toward the equilateral geometry. In doing so it has started to reflect from the curved potential wall near the origin of the coordinate space and has spread out angularly as well as moving back toward slightly larger values of r. The overall effect of the short-pulse excitation process is to produce a wavepacket that reflects the energy spread of the laser
11434 J. Phys. Chem., Vol. 98, No. 44, 1994
Dobbyn and Hutson Q, 1 A 1.o
Q, t A 1.o t = -60 f S
-
t=O 0.5
0.5
1 I
1
I
I
I
l
I
I
(
1
I
I
-0'51
Q, I
"1
-1.o
-1 .o
A i.o
Q, 1 A 1.o t = +60 fs
0.51
7
i1
l
I
1.o
0.5
Q.IA
t=+120fs
0.5
n-!
1
1
I
-
l
l
l
0.5
(
o
r
I
~
1.o
-1.o
Q,IA
-0'51 -1 .o
-1 -0.51 .o
Figure 2. Evolution of the wavepacket on the B state during the femtosecond pump laser pulse. The pulse has a temporal width of 70 fs, centered at t = 0, with a central wavelength 1 = 620 nm. The wavepackets at different times are independently normalized.
pulse. For the femtosecond experiment, the wavepacket that is produced in this way is significantly different from that produced by vertical excitation. The restricted energy of the real laser pulse limits the extent to which higher-lying B-state levels are excited. B. Propagation of the Wavepacket. Several methods of propagating a wavepacket in time are available?* In the present work, the time-dependent Schrodinger equation was solved by using the Chebyshev propagator of Tal-Ezer and K o ~ l o f f In .~~ this approach, the time evolution optrator U(At) is expanded for a time-independent Hamiltonian H as N
U ( A t ) M &$,(-&Atln)
(29)
k=O
where the functions Pk are complex Chebyshev polynomials and the coefficients pk may be written in terms of Bessel functions. Calculating the effect ofthe time evolution operator requires the evaluation of functions HW, which involve repeated application of the Hamiltonian to the initial wavefunction. As usual, the effect of the kinetic energy operator was evaluated by transforming the function to momentum space, using fast Fourier transforms in the polar coordinates. In the present work, for a propagation time step At = 20 fs, it was found necessary to include 230 terms in the polynomial expansion. The propagation was done for times up to 20 ps. A grid with 64 points in each of the radial and angular coordinates was used. The grid extended from 0.1 to 2.1 8, in r and from 0 to 2x in
4.
The potential used in the propagation is that for the B excited state. In order to minimize irrelevant phase oscillations, the zero of energy was taken from an estimate of the ground-state energy on the B state, 69 cm-I above the bottom of the trough. The Propagating Wavefunction. After the wavepacket has been created by the pump laser pulse, it starts to vibrate along the radial coordinate, because the equilibrium position ro for the B state differs significantly from that for the X state. After a very short time (less than 100 fs) the wavepacket starts to spread out. The angular motion then quickly becomes disorderly, with the wavefunction spreading out over the whole of the angular space, though still vibrating backward and forward across the trough. An example of the wavepacket after 1000 fs is shown in Figure 3. C. Extracting the Observables. There are various ways in which information about the dynamics can be extracted from wavepacket calculations. This section will describe some of the possibilities and the ways in which the results have been analyzed in the present work. The Autocorrelation Function. One of the standard methods for interpreting wavepacket calculations is to consider the autocorrelation function. In the present work, the autoconelation function used is that between the wavefunction on the B state at time t and that at the end of the pump laser pulse
The Fourier transform of the autocorrelation function contains information on the energies of the eigenstates that are present in the wavepacket. Since the autocorrelation function is not
Wavepacket Calculations on Na3
J. Phys. Chem., Vol. 98, No. 44, 1994 11435
A
Vertical excitation
l'O-l
t = 1000 fs
0.2
-m-1
T
0.0
1.o
-1.o
5000
0
10000
IA
15000
20000
t lfs
-1 .o
Figure 3. Wavepacket after 1000 fs, under the conditions of the
femtosecond experiment. femtosecond pulse 0.7
15250
15500
15750
16000
16250
16500
16750
Wavenumber I cm'
Figure 5. Modulus and Fourier transform of the autocorelation function for a wavepacket on the B state produced by vertical excitation.
1OOOO
5000
0
15000
20000
t ns
s 10.0 7
'E
a
7.5
4
I
15250
15500
15750
16000
l
l
16250
605
16500
610
615
620
625
nm
16750
Wavenumber I cm" Figure 4. Modulus and Fourier transform of the autocorrelation function for a wavepacket on the B state produced by explicit modeling of the femtosecond laser pulse. wavsisnglrlnm
zero at the end of the propagation period, t = T, it is desirable to use a window function to avoid spurious oscillations in the Fourier transform. In the present work, we used the window function exp( - 1.4t/Tj; the appearance of the Fourier transform is not substantially affected by the window function used. The modulus of the autocorrelation function calculated for the conditions of the femtosecond experiment is shown in Figure 4, together with its Fourier transform. The pump laser pulse has an eriergy range of about 300 cm-', so that the wavepacket produced is made up of many different eigenstates. The energy levels correspond well with the static spectroscopy, but as expected the intensities of the bands are different. In particular, since the photon energy is restricted, there is little excitation of levels above 16 400 cm-'. We have also calculated the autocorrelation function corresponding to vertical excitation, with the initial wavepacket on the B state created by transferring the ground-state wavefunction of the X state to the B state. Figure 5 shows this autocorrelation function and its Fourier transform. The vertical excitation function contains considerably more high-frequency
Figure 6. Comparison of the experimental two-photon ionization spectrum for the B statez4with the Fourier transform of the autocor-
relation function for vertical excitation. oscillations than the exact function, because there is no restriction on the photon energy available in the vertical excitation case. The Fourier transform in this case should be identical to the experimental spectrum from the static spectroscopy and is compared with it in Figure 6; it may be seen that there is quite reasonable agreement in both line positions and intensities. The N u f Signal. The quantity that is actually measured in the femtosecond experiments is the Na3' signal. We therefore need to model the effect on the probe laser, which creates a wavepacket on the ground electronic state of Na3+ from the time-evolving wavepacket on the B state of Na3. The effect of the probe laser can be modeled in much the same way as that of the pump laser. There is an added complication in this case, since the state formed is ionic: the electron that is ejected can have a range of energies and a separate calculation must in principle be done for each of these energies.44 However, it is
Dobbyn and Hutson
11436 J. Phys. Chem., Vol. 98, No. 44, 1994 known experimentally that the signal due to electrons with zero kinetic energy (ZEKE) is similar to the ion accordingly, we have assumed that it is adequate to consider only electrons with zero kinetic energy. The wavefunction on the ionic potential is a function of two different times: the time delay t D between the pump and probe pulses and the total time t (measured from the center of the pump pulse). During the probe pulse, for a given value of fD, the ionic wavefunction is propagated according to
10.01
I1
Exact signal
I
5000
0
10000
15000
20000
t, ns
(ip,+,Edh)Atf( t
+ At)e-2niw(r-tD+At) UB(t -
10.01
tfid)yB(tfid)
,
Ion overlap
(31) where f ( t ) is the temporal profile of the probe laser pulse and in the present work is equal tofft - t ~ ) .Thus for each time step At within the profile of the probe laser, some wavefunction amplitude is created on the ionic surface. At the same time, the wavefunctions on both the B state of the neutral and the X state of the ion continue to evolve under the influence of their own potentials. This procedure takes proper account of the interference effects that occur during the probe pulse. The pump and probe laser are not permitted to overlap temporally, so that no calculation is done for values of t~ of less than 2tfid. Equation 3 1 assumes implicitly that the transition dipole function , independent of the spatial coordinates. for ionization, ~ x + B is The Na3+ signal s(fD) is taken to be the norm of the wavefunction on the ground electronic state of the ion at the end of the probe pulse,
S(~D> =~
JI~X+(~D&
tfid)I2
dQx dQy
0
10000
5000
15000
20000
t, Its
Figure 7. Time-dependent ion signal calculated by using (i, top)
explicit modeling of the probe laser pulse and (ii, bottom) the overlap of the wavepacket on the B state with the ground-state ion wavefunction.
(32)
This function is Fourier transformed to give a spectrum which contains frequencies characteristic of the motion on the B-state potential energy surface,
The last equality arises because the ion signal is real and symmetric about fD = 0. The magnitude of this function is used to represent the power density spectrum. In the present work, the lower limit of the integral is taken to be 2tmid rather than zero, because the ion signal is not calculated for shorter times. The Fourier transform used the same window function as for the autocorrelation function. The ion signal calculated in this way is shown in the upper panel of Figure 7 and its Fourier transform in the top panel of Figure 8. The largest peaks in the Fourier transform are the radial pseudorotation frequency (127 cm-l) and its overtone. However, there are also subsidiary peaks that can be attributed to angular pseudorotation motion. The oscillations in the ion signal do not take it back to zero on the time scale of the propagation carried out here. An Approximate Method. The full treatment of the probe laser described above is computationally expensive, because it involves propagating in two time dimensions. We have therefore investigated ways of approximating the Na3+ signal. To do this, we assume that the only state of the ion that is populated by the probe laser pulse is the ground vibrational state. The justification of this assumption is that the energy available in the probe laser is restricted, so that (for our potential surfaces) only the ground state of the ion is energetically accessible. The Na3+ signal is approximated by the square modulus A(tD) of the overlap of the wavepacket on the B state of Na3 with the
0
100
200
300
400
Wavenumber /cm"
0
100
200
300
400
Wavenumber Icm-'
0
100
200
300
400
Wavenumber Icm" Figure 8. Fourier transform of the ion signal calculated by using (i, bottom) explicit modeling of the probe laser pulse, (ii, middle) the overlap of the wavepacket on the B state with the ground-state ion wavefunction, and (iii, top) the modulus of the zero-time correlation function.
wavefunction of the ground state of Na-,+,
where Vx+ is the ground-state wavefunction of the ion and the delay time t~ is measured from the center of the pump laser pulse. The Fourier transform of this overlap function (over a
J. Phys. Chem., Vol. 98, No. 44, 1994 11437
Wavepacket Calculations on Na3 time from the end of the pump laser pulse to the end of the propagation) is used to give the power density spectrum. Another possibility that has been proposed is to use the modulus of the autocorrelation function on the B state as an approximate ion signal.21 This was suggested because, for the picosecond experiments, the largest signal is obtained at zero time delay; Gaus et aLZ1argued that this implies that the best conditions for ionization are those described by the initial wavefunction. However, this conclusion is rather surprising in physical terms, because there is relatively poor overlap between the ground-state vibrational wavefunctions of Na3 and Na3+. The peak at zero time delay might also be due to the greater laser intensity that arises when the pump and probe pulses overlap. In any case, it is not quite clear how this approach should be applied in our calculation, since the wavepacket in the B state develops during the pump pulse, rather than being placed there instantaneously. The procedure used for this approximation in the present work is to take the normalized overlap of the wavefunction with that at the middle of the pump laser pulse,
(35) This will be referred to as the zero-time autocorrelation function. The modulus of this function is then Fourier transformed to obtain an approximate power density spectrum. The ion signal calculated from the ionic overlap function is shown in the lower panel of Figure 7, and its Fourier transform in the center panel of Figure 8. The ionic overlap function is dominated by the two principal frequencies due to the radial pseudorotation mode and its overtone. Since there is very little radial anharmonicity in our B-state potential (because of the small value of j),the higher frequency is very nearly twice the lower one and the oscillations return close to zero. The ground vibrational state of the ion is centered at the origin, so that the overlap is largest when the wavepacket moves in toward the equilateral geometry. The average value of r was also calculated and shows a simple relationship to the ionic overlap function: when the value of r is at its minimum, the overlap function is at its maximum and vice versa. There are few other frequencies present in the ionic overlap function with any intensity. This arises mostly because the ground-state vibrational wavefunction of the ion is only weakly dependent on 4, so that the overlap between the wavepacket and the ion function does not change much as the wavepacket moves around the pseudorotation trough. It may be noted that the most recent value of the Na3 ionization energy? which was published after the present work was completed, is about 900 cm-’ smaller than the value used in our simulations. This implies that more states of the ion are energetically accessible in the femtosecond experiment than in our calculations. Highly excited wavefunctions of the ion are more strongly dependent on 4 than the ground state, so it is possible that this difference accounts in part for the lack of pseudorotation frequencies in our calculations. However, we did repeat the ionic overlap calculations using the wavefunction for the first excited bending state of the ion and obtained similar results to those obtained with the ground-state wavefunction. Figure 8 compares the Fourier transforms of the exact signal, the ionic overlap signal, and the modulus of the zero-time autocorrelation function. The signal obtained from the zerotime autocorrelation function shows more structure than the other signals, because it corresponds to an overlap with a
function that is localized in 4; thus the pseudorotation frequencies appear much more strongly in the autocorrelation function than in the accurately calculated ion signal. However, the argument that the initial wavepacket on the B state represents the optimum geometry for ionization appears physically implausible, because the ionic vibrational wavefunctions are concentrated around r = 0, and this region is only weakly sampled by the initial wavepacket.
VI. Conclusions We have carried out wavepacket calculations to model the results of femtosecond pump-probe experiments on the sodium trimer. We have used potential energy surfaces for the X and B states of Na3 obtained by combining information from static spectroscopy with the results of ab initio calculations. The resulting potentials are very simple in form, and it would be desirable in future work to use more sophisticated potentials. Nevertheless, the present potentials should have the essential features necessary to model the femtosecond experiments. The sodium trimer is a Jahn-Teller system, which can undergo pseudorotation between three equivalent isosceles triangle geometries. Our wavepacket calculations were carried out in two dimensions, corresponding to the degenerate bending mode of an equilateral molecule (and neglecting the symmetric stretching coordinate). Wavepackets were propagated by using the Chebyshev propagator of Tal-Ezer and Kosloff. It is important to model the creation of the initial wavepacket on the B state as accurately as possible. In the present work, the pump laser was modeled explicitly by the time-dependent perturbation theory. This takes account of the finite width of the pulse and (unlike vertical excitation) allows for interference effects that occur on the excited state during the pulse. However, the perturbative treatment does not take account of two-photon processes that return parts of the wavepacket to the ground electronic state. It becomes increasingly important to use explicit modeling of the pump laser for longer and more monochromatic pulses. The initial wavepacket created on the B state by the femtosecond laser pulse is made up of many states, which are excited in both radial and angular pseudorotation modes. The wavepacket has a large-amplitude motion in the radial direction. The angular motion quickly becomes disorderly, and the packet fills the available angular space. The oscillation of the wavepacket in the radial direction gives rise to an ion signal which oscillates with the frequency of the radial motion. Several different methods of modeling the ion signal have been investigated. The most accurate (and most expensive) method models the probe laser explicitly in the same way as the pump laser. An approximate method which appears to give useful results in the present case (because the probe laser has only just enough energy to ionize the cluster) is to approximate the ion signal as the time-dependent overlap between the wavepacket on the B state and the ground vibrational state of Na3+. Using this method, it should be feasible to extend the calculations to three dimensions, including the symmetric stretch coordinate. In three-dimensional calculations, it would not be possible to calculate the exact ion signal with the present computational resources. The present calculations show relatively little ion signal that oscillates at the frequency of the angular pseudorotational motion. This arises mostly because the ion wavefunction is only weakly dependent on this coordinate, so that motion in it has little effect on the overlap between the B-state wavepacket and the ionic wavefunction. The lack of pseudorotational frequencies may be contrasted with calculations based on the
11438 J. Phys. Chem., Vol. 98, No. 44, 1994 autocorrelation function, where the overlap involved is with a function that is localized in the angular coordinate. We cannot see a physical justification for the autocorrelation function approach in the present case. The calculations show some of the features of the femtosecond experiments. The experimental spectrum shows a small peak at 123 cm-’, which may correspond to the radial wavenumber of 127 cm-’ in the calculations. However, the pseudorotation frequencies appear in the calculations with considerably less intensity than in the experiments. The symmetric stretching mode, which appears to be important in the experiments, is not included in the present calculations. In future work, we will extend the calculations to include this degree of freedom. However, the static spectroscopy appears to show very little, if any, excitation in the symmetric stretching mode, which makes it surprising that the frequency should be so conspicuous in the femtosecond experiments.
Acknowledgment. The calculations were carried out on the Meiko Computing Surface belonging to the Durham and Newcastle Atomic and Molecular Physics groups and on the Intel iPSC computer at SERC’s Daresbury Laboratory. We are grateful to Dr. Lydia Heck for assistance with Meiko. A.J.D. is grateful to SERC for a Research Studentship. References and Notes (1) Zewail, A. H.; Bemstein, R. B. The Chemical Bond: Structure and Dynamics; Academic Press: New York 1992; p 223. (2) Bowman, R. M.; Dantus, M.; &wail, A. H. Chem. Phys. Lett. 1989, 161, 297. (3) Baumert. T.: Grosser., M.:. Thalweiser.. R.:. Gerber. G. Phvs. Rev. Lett.’i991, 67, 3753: (4) Baumert. T.: Buhler. B.: Grosser, M.: Thalweiser, R.; Weiss, V.: Wiedenmann, E.; Gerber, G. J . Phys. Chem. 1991, 95, 8103. (5) Baumert, T.; Engel, V.; Rottgermann, C.; Strunz, W. T.; Gerber, G. Chem. Phys. Lett. 1992, 191, 639. (6) Engel, V.; Baumert, T.; Meier, Ch.; Gerber, G. Z. Phys. D 1993, 28, 37. (7) Baumert, T.; Thalweiser, R.; Weiss, V.; Gerber, G. 2.Phys. D 1993, 26, 131. (8) Baumert, T.; Thalweiser, R.; Gerber, G. Chem. Phys. Lett. 1993, 209, 29. (9) Thalweiser, R.; Vogler, S.; Gerber, G. In Laser Techniquesfor Stateselected and State-to-state Chemistry; SPIE Proceedings, Vol. 1858, 1993. (10) Jahn, H. A.; Teller, E. Proc. R. Soc. (London), Ser. A 1937, 161, 220.
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