Chemical Physics 120 (1988) 327-333 North-Holland, Amsterdam

327

DYNAMIC ASPECTS OF ELECTRONIC EXCITATION

M.V. RAMA KRISHNA’ and R.D. COALSON

Department of Chemistg., University of Pittsburgh, Pittsburgh, PA 15260, USA

Received 27 July 1987

Electronic excitation with a continuous wave light source is shown to result in the creation of a “Raman wavefunction” on the excited surface. In this connection a misconception regarding the regime of validity of Heller’s localized wavepacket approach to spectroscopy is clarified.

1. Introduction

A common experimental situation in optical spectroscopy is this: a light pulse of finite dura- tion and strength is shone upon a molecule which subsequently undergoes electronic excitation if the spectral bandwidth of the light contains the angu- lar frequency w equal to the energy difference of the initial and final rovibrational energy levels in the ground and the excited electronic states di- vided by A. However, there is some confusion in the literature as to the exact nature of the wave- function that is created on the excited electronic surface by this process, especially in the limit that the duration of the light pulse is very short or very long. Such confusion has arisen partly because of a misreading of Heller’s formulation of spec- troscopy using localized wavepackets, which pre- scribes that the initial localized Franck-Condon wavepacket be vertically displaced to the excited surface and propagated there [1,2]. Since the tem- poral character of the light pulse has no explicit role to play in this formulation, it seems to have generated a belief that the optical excitation pro- cess creates a localized wavepacket [3]. In ad- dition, because Heller’s computational scheme starts with a localized wavepacket on the excited surface [1,2], this time-dependent approach has been unfairly criticized as belonging to the broad-

’ Present address: Department of Chemistry, University of California, Berkeley, CA 94720, USA.

band ultrashort pulse excitation limit [4]. For these reasons, a closer look at the optical excitation in the short and long pulse limits appears in order. A primary objective of this paper is to show that a continuous wave (cw) light source creates a de- localized wavepacket on the excited surface which is essentially an object introduced in a different context by Heller and co-workers, and termed by them a “Raman wavefunction” [5]. While doing so it will be clarified #’ that Heller’s time-depen- dent approach to spectroscopy falls within cw light excitation limit and hence is applicable to

almost all experimental situations that are possible with currently available light sources.

Some of the results derived in this paper have

been obtained by Heller and co-workers in differ- ent ways [1,7]. But in our opinion the electronic excitation aspect of these papers has not received as much attention as it deserves (which is perhaps the source of the abovementioned confusion) be- cause that was not the primary focus of these papers. Also, to our knowledge it has never been demonstrated in the literature that a cw light source creates a Raman wavefunction on the ex- cited surface.

The approach adopted in this paper is as fol- lows. The wavefunction created on the excited

In fact, no such clarification is really necessary since Heller’s method is based on transcription of golden rule rate expres-

sions from frequency domain to time domain. See refs.

11961.

0301-0104/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

328 M. V. Rama Krishna, R. D. Coalson / Dynamic aspects ojelectronic excitation

surface by an ultrashort light pulse is discussed first. This enables us to make contact with the long pulse limit in a simple way. It will then be shown that the cw light source creates a Raman wavefunction which is obtainable by following Heller’s prescription of half-Fourier transforming the propagating wavepacket [5]. All these results are obtained from first principles using time-de- pendent perturbation theory [8].

2. Results and discussion

Consider a ground and an excited potential surface of arbitrary shape and dimensionality. When a light source of finite duration and strength is turned on, the two surfaces are coupled by a transition dipole operator c l E, where p and E are the transition dipole moment vector and the electric field vector of the light, respectively. This coupling mechanism enables transfer of amplitude between the ground and excited surfaces. Treating the radiation field classically, at any time t the amplitudes #g and #, on the ground and excited surfaces are given by the solution to the time-de- pendent Schrijdinger equation [8,9], i.e.

ihaG@= H&,(t) + ~,E(+J,(t),

=l+,/at = K&0) + P,E(+&(t)Y (1)

where pe is the transition dipole moment in the direction of the electric field vector and E(t) = A(t)& cos(wt) is the electric field amplitude of the light pulse whose temporal profile is defined by A(t). In addition, G,(O) is the rovibrational eigenstate of the ground surface in which the system is initially prepared, and q,(O) = 0; H, and H, are the zeroth-order Hamiltonians for the ground and excited surfaces, respectively. Trans- forming these equations into integral equation form one obtains

G,(t) =exp(-iE,t/A)+,(O)

- ipt’exp[ -iH,(t- t’)/h]

%Wbk&‘)~

G,(t)= -gd t’ exp[ -iH,(t - t’)/A]

xcl,E(t’)G,(t’). (2)

For arbitrary strengths and functional forms of E(t) (more specifically, A(t)), the above equa- tions need to be solved numerically. However, in the weak-field limit

q,(t) = exp( -iEst/h)#,(O) - @PE)2Y

J/,(t)= -i/fdt’exp[-iH,(t-t’)/h] 0

xE(t’)d&‘)

z - +/*dt’ A(t') cos(wt’) 0

xexp[ -iH,(t- t’)/A]

xexp( -iEgt’/h)+g(0), (3)

where c+(t) = pe+,(t) is a Franck-Condon wave- packet m Heller’s terminology [2]. Now, imposing the rotating wave approximation [lo] cos( at’) = sexp( - iwt ‘) which is valid near resonance, one

obtains

t),(t) = - 3 (dt' A( t’) exp( -iEt’/A)

xexp[ -iH,(t - t’>/h] +g(O), (4) where E = Aw + Eg is the total energy of the sys- tem subsequent to optical excitation.

Two limiting forms are possible for the tem- poral profile of the light pulse. One such limit is the S-function limit and the other the cw limit. Although our objective is to obtain the functional form for the wavefunction created on the excited surface by a cw light source, a digression on short pulse excitation will be useful.

2. I. a-function limit

In this limit the temporal profile of the light pulse is taken to be a Dirac delta function 8(t). Then, 4,(t) and #,(t) become

&.(t) = exp(-iEsVA)&(O),

4,(t) = (-iE,/2h) exp(-iH,t/ti)$s(O). (5)

(Eq. (5) is valid in the strong pulse (large E,) limit as well.) Thus, in this limit a wavepacket is sud- denly promoted from the ground to the excited surface where it is not an eigenstate. Subsequent

M. V. Rnma Krishna, R. D. Coalson / Dynamic aspects of electronic excitation 329

to its creation, this wavepacket evolves for time t on the excited surface according to the exponenti- ated Hamiltonian operator exp( - iH,t/A) ap- propriate for that surface.

In actual experimental practice such a S-func- tion radiation field is impossible to achieve. Con- sequently, the creation and evolution of wave- packets on the excited surface is best illustrated with a specific numerical example.

In all the calculations presented here we have adopted natural units in which A, the particle mass m, and the ground state oscillator frequency wp are all equal to one. In addition, the frequency w of the light source (see our definition of the electric field amplitude above) is chosen to be equal to the resonance frequency, defined accord- ing to hw = V,(x,) - +Aw,, where V, is the excited state potential and xeq the equilibrium position of the ground surface. Further details regarding the calculations are presented at ap- propriate places below.

Consider a harmonic ground and an exponen- tial repulsive excited surface, as shown in fig. 1. Assume that at time t = 0 the molecule is in the vibrationless level of the ground surface and that the transition moment ~1, for excitation from the ground state is constant (taken to be 1) over the spatial extent spanned by the zero-point vibra- tional motion of the molecule. Consequently, the initial wavepacket, also shown in fig. 1, is a node- less real valued Gaussian. Although this is a one- dimensional example with specific initial condi- tions, the essential results concerning the spatial-temporal dynamics of the photoexcitation process are expected to be valid in the general case.

Fig. 2 shows a series of plots of the wavepacket on the excited surface at various times t. These wavefunctions are obtained by solving the coupled differential equations (1) numerically exactly using standard “grid” techniques [9,11]. The temporal profile of the radiation field is a Gaussian with a full width at half maximum intensity of about 6.5 fs, and it requires about 16 fs to reach full power. In addition, the integrated intensity of the light pulse is chosen to be equal to 71. Such an ultra- short light pulse is achievable experimentally [12]. At t = 0 there is no wavepacket on the excited

T 25

;; 5

0

-25

-50-1 1 -5 5 15 25 35 45

X-

Fig. 1. Schematic depiction of the ground and excited surfaces used to illustrate the creation of wavepackets on the excited

surface by the radiation field and their subsequent evolution.

The ground surface is a harmonic well, whereas the excited

surface is exponentially repulsive. The wavefunction shown in

the harmonic well is the initial state corresponding to the

vibrationless level of the ground surface.

surface since at this time the radiation field is barely turned on and the magnitude of the electric field is essentially zero. However, soon afterwards the amplitude starts to build on the excited surface. For example at t = 11 fs there is a small imaginary valued localized Gaussian wavepacket on the ex- cited surface, the real component being almost absent. (The fact that the imaginary component is created first may be seen from eq. (2).) This wavepacket is essentially the wavefunction on the ground surface. As this wavepacket starts to move on the excited surface, the real part starts to grow. In addition, the net wavepacket on the ground surface (not shown in the figure) also starts to move and develop both real and imaginary com- ponents. During this same period, the radiation field, which is still on, brings in additional pieces of the ground state wavepacket upstairs as both real and imaginary wavefunctions [l]. These new wavepackets also start to evolve on the excited surface and develop nodes in the wavefunction. After the radiation field is off, both the real and imaginary components of the wavepacket start to evolve on the excited surface as shown in the last four panels of fig. 2. By the time t = 220 fs, the wavepacket has reached the asymptotic region of the excited surface and from there on it evolves like a free particle.

330 h4. V. Rama Krishna, R.D. Coalson / Dynamic uperrs of electronic excitation

0.8

0.8

-0.8- -5 15

v 35 -5 35 -5 15 35 -5 15 35

x-

Fig. 2. This series of plots depicts the wavefunction on the exponential repulsive excited surface at times shown on each plot in femtoseconds for the Gaussian pulse excitation process discussed in section 2.1. The solid and dashed lines are the real and imaginary components of the wavefunction, respectively. The radiation field is a gaussian with full width at half maximum of about 5.5 fs and it

reaches full power in about 16 fs.

2.2. cw limil

This limit is defined by t + co, A(t) = 1, and E, = 1. Eq. (4) then reduces to

&= lim -jij--” dt’ exp( -iEt’/A)$(t - t’) I--r00 0

(64

= lim - & exp( -i&/A) t-00

x oWdU exp(iEu/A)$( u) I (6b)

= lim - & exp(-iEt/tt)#n, 1’00

(6~)

where the substitution u = t - t’ has been utilized in order to get from (6a) to (6b), and $(u) is the wavefunction obtained by propagating +a on the excited surface for time U. Finally, $a is the half-Fourier transform integral in (6b) which has been previously designated as the “Raman wave- function” [5]. According to eq. (6), at each instant t’ a wavepacket is created on the excited surface with a phase factor of exp( -iEt’/A). Each of these wavepackets then evolve independently un- der the influence of the propagator exp( - i H,( t - t’)/h). Thus within a phase and a multiplicative factor, a Raman wavefunction is created on the excited surface by a cw monochromatic light source.

Xe = J m dt exp(iEt/h)cpi(t) (74

= G(#,). (7b)

(eq. (7b) follows from (7a) because $$(O) = pcl,$,(0) is real.) This eigenfunction enters into the compu- tation of the total absorption cross sections via the well-known Franck-Condon formula [14], whose transcription into the time domain yields [I].

e(E) = S_m_df exp(iEt/ft)(~s(O)l~~(t))

= 2 Re(&(O) I 4,)). On the other hand, the Raman cross sections are given by the squared modulus of the Kramers- Heisenberg-Dirac (KHD) amplitude [15], whose transcription into the time domain yields [l]

where x< is the rovibronic eigenfunction of the ground surface. Thus, only the real part of the Raman wavefunction is needed for the computa- tion of the total absorption cross sections, whereas both real and imaginary parts are essential for computing Raman cross sections.

It is interesting to note some nice properties of The computational procedure for obtaining the

the Raman wavefunction which makes it a con- Raman wavefunction is clear from its definition as

venient tool for computing various transition rates. The real part of #a is related to an eigenfunction of the excited surface, xe, by the relationship [13],

M. V. Rama Krishna, R.D. Coalson / Dynamic aspects of electronic excitation 331

x

3

-54 I -5 4 13 22 3( 40

N 30

z F.----T - I

OI -5 4 13 22 31 40

x-

Fig. 3. Top panel: A Raman wavefunction created on the exponential repulsive surface by the cw light source of about

340 fs. Bottom panel: The square of the magnitude of the wavefunction shown in top panel.

a half-Fourier transform integral of a time-depen- dent wavepacket. Following Heller’s prescription

[1,2], place the initial wavepacket (ground state wavefunction multiplied by the transition dipole

moment) on the excited surface and propagate it there. Subsequently, half-Fourier transform the propagating wavepacket to obtain the Raman wavefunction. Thus Heller’s wavepacket method in its application to spectroscopy is clearly a

lowest-order correction (first-order for absorption and second-order for Raman) valid in the weak- field and cw limit [6]. However, since Heller’s scheme starts with a vertical displacement of the localized Franck-Condon wavepacket to the ex- cited surface [2], false impressions persist that that is what is created by optical excitation. In fact, the wavepackets are projected upwards at various times t by a weak cw light source [l] and the half-Fourier transformation ensures that these are added with correct phase relationships.

Fig. 3 shows the wavefunction obtained with a cw light source. This wavefunction is obtained for the excited potential surface depicted in fig. 1 and initial conditions already mentioned above, except that the radiation field is now a square pulse of about 340 fs. The integrated intensity of the light pulse is again equal to IT. The real and imaginary parts of the Raman wavefunction dove-tail per- fectly in the asymptotic regions of the exponential repulsive excited surface and the probability den- sity is non-zero in these regions. Thus it is seen that a delocalized wavepacket is created by the cw light source. This wavepacket is essentially a coherent superposition of several wavepackets,

each of which is created on the excited surface at different times by the oscillating radiation field. In

fact the wavefront of the wavepacket that is created near the earliest times is seen clearly at the asymptotic regions of the potential surface (cf. figs. 2 and 3). Since both real and imaginary parts of the Raman wavefunction are about equal in

-2.5 1.9 6.3 -2.5 I .9 6.3 -2.5 1.9 6.3 -2.5 1.9 6.3

x-

Fig. 4. This series of plots depicts the wavefunction on the harmonic excited surface. The solid and dashed lines represent the real and imaginary components of the wavefunction, respectively. The radiation field is a square pulse of integrated intensity equal to n. The temporal duration of the light pulses is shown in each plot in units of one vibrational period of the oscillator, which is about 278

fs.

332 M. V. Rama Krishna, R. D. Coalson / Dynamic aspects of electronic excitation

magnitude, contribution to the Raman cross sec- tion arising from the imaginary part cannot be neglected. In addition, since it takes only a few femtoseconds for the light source to approach cw limit, most spectroscopic experiments fall within this limit and Heller’s wavepacket approach is suitable to all these situations.

Finally, we consider a case in which both the ground and excited state potentials are harmonic wells and light pulses of varying temporal dura- tion are applied to excite the molecule, which is initially present in the vibrationless level of the ground state. The oscillator frequencies of the ground and excited surfaces are taken to be about 120 and 96 wavenumbers, respectively, and the mass of the system is chosen to be 1 atomic mass unit.

-0.21 I I 1 i J - 2.5 -0.3 1.9 4.1 6.3 a.5

X-

Fig. 5. A Raman wavefunction created on a harmonic oscilla- tor potential by a cw light source of about 278 fs, which is one vibrational period of the oscillator. The solid and dashed lines represent real and imaginary components of the wavefunction,

respectively.

The excited potential surface is displaced by 2.5 atomic units with respect to the ground surface, whose effect is to yield maximum Franck-Condon overlap of the initial wavefunction with the second eigenstate of the excited potential. The light pulses are all constant amplitude pulses of integrated intensity equal to IT. The frequency w of the light pulses is equal to the frequency difference of the second eigenstate of the excited surface and the vibrationless level of the ground surface.

fig. 5. The real part (solid line) of this wavefunc- tion is clearly the second eigenstate of the harmonic well, which illustrates the claim made above that the real part of the Raman wavefunc- tion is an eigenfunction.

3. Conclusions

Fig. 4 shows the wavefunctions created on the It has been shown in this paper that a cw light excited surface when the duration of the light source creates essentially a delocalized Raman pulses is varied. These wavefunctions are obtained wavefunction. This Raman wavefunction is a by solving the coupled differential equations (1) coherent superposition of the wavepackets (initial exactly using the standard grid techniques [9,11]. ground state wavefunction multiplied by the tran- When the temporal duration of the light pulse is one thousandth of one vibrational period of the

sition dipole moment) that are promoted by the resonant light from the ground to the excited

excited state oscillator (about 0.278 fs), an almost exact replica of the ground state wavefunction is

surface at various times t with a relative phase exp(iEt/fi). The real and imaginary parts of this

created on the excited surface. As the duration of the light pulses is increased, the wavefunction on

Raman wavefunction are about equal in magni- tude (at least on resonance for unbound excited

the excited surface starts to acquire both real and imaginary components, as well as nodes. As the

surfaces; see below), although only the real part is an eigenfunction of the excited surface. In ad-

successive panels of fig. 4 show, the wavefunction slowly evolves towards the second eigenstate of

dition, for the computation of total absorption cross sections only the real part of the Raman

the excited surface. When the duration of the light wavefunction is needed, whereas both real and pulse is exactly one vibrational period of the ex- imaginary parts in general are needed for the cited state oscillator (about 278 fs), one compo- computation of Raman cross sections. It has also nent of the wavefunction is very much the second eigenstate of the harmonic oscillator potential.

been clarified that Heller’s wavepacket dynamics

The Raman wavefunction for this case is shown in approach to spectroscopy is within cw light excita- tion limit.

,U. V. Rama Krishna, R. D. Coalson / Dynamic aspecls of electronic excitation 333

In concluding, we wish to point to some very recent and interesting work by Williams and Imre [16]. Their immediate focus is on the conceptual and computational utility of the Raman wave- function for understanding the effect upon the KHD formula of various input parameters (poten- tial surfaces, incident and scattered laser frequen- cies, etc.). However, several of their observations are pertinent to the present discussion.

In particular, they have examined the effect of detuning the incident frequency away from reso- nance. They find a rich variety of behaviors. For example, their results indicate that our conclusion concerning the topology of the Raman wavefunc- tion, namely that it is a delocalized object consist- ing of dove-tailing real and imaginary parts which contribute with equal magnitude, is valid only on resonance for dissociative excited state surfaces. Off resonance, the situation is different, as are the localization/dominance properties of real and imaginary parts in the bound excited surface case. The interested reader should consult their paper for further details.

Acknowledgement

It is a pleasure to thank Professors J.A. Bes- wick, Jack Simons and D. Waldeck for fruitful discussions. One of us (RDC) has profited from numerous conversations on excited state spec- troscopy with Professor D.G. Imre. MVRK is grateful to Professors K.D. Jordan and D.W. Pratt for fostering in him a healthy scientific tempera-

ment. This work was supported in part by a grant from the National Science Foundation.

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