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Annu. Rev. Phys. Chern. 1992.43:497-523 Copyright © 1992 by Annual Reviews Inc. All rights reserved
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ANALYSIS OF FEMTOSECOND Annu. Rev. Phys. Chem. 1992.43:497-523. Downloaded from www.annualreviews.org by Duke University on 07/09/12. For personal use only.
DYNAMIC ABSORPTION SPEC�TRA OF NONSTATIONARY STAT'ES W.
Thomas Pollard and Richard
A.
Mathies
Department of Chemistry, University of California, Berkeley, California 94720 KEY WORDS:
femtosecond pump-probe spectroscopy, time-dependent wave packets, transition state spectroscopy, hole-burning, reaction dy namics, resonant impulsive Raman
INTRODUCTION
Over the last decade, dramatic advances in optical pulse technology have ushered in a new era in the study of photochemical reaction dynamics. Today, pulses shorter than 1 00 fs are routinely available in many lab oratories, and optical pulse-compression techniques allow the production of pulses as short as 6 fs ( 1 , 2). Because femtosecol).d pulses are shorter than the hme scale for most intramolecular motions, they provide tools for directly studying the dynamics and relaxation of molecular vibrations (3-1 6). More importantly, this technology now allows the "real-time" observation of photochemical reactions as they proceed on the initi ally populated excited-state potential surfaces, which are the "transition states" of photoinitiated reactions (1 7-22). The state of this already large and rapidly expanding field has been addressed in recent reviews (22-26). This review foc:uses on one particular type of femtosecond experiment that we believe will become i ncreasingly important: pump-probe absorption spectroscopy in the dispersed continuum-probe configuration, or "femto second dynamic absorption spectroscopy" (see Figure 1 ) (27-29). This method is uniquely powerful in that it offers simultaneously high resolution 497 0066--426Xj92jl lOl-0497$02.00
Annu. Rev. Phys. Chem. 1992.43:497-523. Downloaded from www.annualreviews.org by Duke University on 07/09/12. For personal use only.
498
POLLARD
&
MATHIES
in both the frequency and time domains when ultrashort femtosecond pulses are used, thus sidestepping the limitations that the time-energy uncertainty principle places on most other experimental uses of femto second pulses. The traditional method in pump-probe absorption spectroscopy treats the probe pulse as being effectively monochromatic. The absorbance of the sample at the probe wavelength is found as a function of the pump-probe delay time by measuring the loss of energy of the pulse to the sample, either directly using a photodetector (30), or indirectly using such methods as laser-induced fluorescence (31). The repetition of the experiment for probe pulses of different wavelengths then yields a picture of the time dependent absorption spectrum of the photo-excited system (Figure 1 bottom). In this scheme, however, the absorbance of the sample is averaged over the actual spectrum of the probe pulse; the spectra are thus broadened by the spectral width of the pulses, which varies as the inverse of the pulse Spectrograph
MCD
(3)
P (k,m) sample
14 wavelength
Photodetector
(3)
P (k,m) sample
Figure 1 Comparison of time-resolved pump-probe absorption spectroscopy in the dis persed-continuum-probe (top) and integrated-probe (bottom) configurations. MCD multi channel detector. =
Annu. Rev. Phys. Chem. 1992.43:497-523. Downloaded from www.annualreviews.org by Duke University on 07/09/12. For personal use only.
FEMTOSECOND ABSORPTION SPECTROSCOPY
499
length. To observe a particular process, there is generally an optimal pulse length, which balances the time resolution required to monitor the dynamical process, with the frequency resolution required to distinguish the spectral changes in the system (32). In thc continuum-probe or dynamic absorption technique, the transient absorption spectrum of a sample is obtained by measuring how the spec trum of an optical pulse is changed by transmission through the sample (Figure 1, top). This approach is most useful when ultrashort, spectrally broad pulses are available; the useful spectral width of a transform-limited 6-fs pulse, for instance, is on thc order of 3500 cm I, sufficient to measure the complete absorption spectrum for individual electronic transitions in many molecules (1 4). The femtosecond dynamic absorption technique has become the basis for an increasing number of studies ( 1 3, 1 4, 1 6-- 1 8, 20, -
21, 33-43}.
Because: the pulse spectrum can be measured with essentially arbitrary precision, dynamic absorption spectroscopy allows one to monitor the frequency-resolved absorption spectrum of an evolving system with the time resolution of ultrafast femtosecond pUlses. What is often perceived to be paradoxical is that the time resolution and frequency resolution involved in the dynamic absorption method are completely independent, i.e. they are not related by the Heisenberg uncertainty principle. This is possible because the actual measurement of the spectrum for a given pump probe delay is not time resolved. Rather, the brief interaction of the probe pulse with the material evokes a polarization response, i.e. it stimulates a transient oscillating dipole moment, whose subsequent emission has the same wavevector as the probe pulse. Because the measurement of the probe pulse spectrum is not time resolved, both the spectrum of the transmitted pulse and any other emission in thc probe direction are mea sured together. When the interference between the original pulse and the subsequent emission is destructive, the result is the net absorption of energy from the pulse. The sharpness of the spectral features measured in this experiment is determined only by the dephasing rate of the induced polar ization, i.e. by the natural optical linewidths of the vibronic transitions in the material, and has nothing to do with the pulse width. The dynamic absorption experiment gives time-resolved information about nonstationary states, because the induced polarization, which gives rise to the measured signals through its long-time emission, is completely created in the brief instant that the femtosecond probe pulse interacts with the system and, thus, records the transient state of the system. This information, stored in the electronic coherence created by the probe pulse, is revealed over time in the emitted radiation. The continued evolution of the initial system after the probe-pulse interaction modulates the induced
500
POLLARD & MATHIES
polarization, but the essential characteristics of the dynamic absorption spectrum, e.g. its strength, position, and breadth, are determined purely by the state of the system during the initial probe-pulse interaction (26, 27). THEORY OF DYNAMIC ABSORPTION
Annu. Rev. Phys. Chem. 1992.43:497-523. Downloaded from www.annualreviews.org by Duke University on 07/09/12. For personal use only.
SPECTROSCOPY
As noted above, it is important to focus on how the final transmitted probe pulse actually interacts with the sample to understand the properties of dynamic absorption spectra. In ordinary linear absorption spectroscopy, by contrast, the details of the interaction of the probe light with the sample are less important. Whether the absorption spectrum is measured with white light or with a monochromatic laser beam, linear response theory says that the interaction can be considered independently for each fre quency component of the probe radiation. The absorption cross-section, defined through the rate of loss of energy by each frequency component of the probe light, is proportional to the imaginary part of the linear susceptibility, O'(w) ex w 1m [x(1)(w)]. The lowest-order description of pump-probe absorption spectroscopy is third-order in the electric fields of the pump and probe pulses (second order in the pump and first-order in the probe). It is possible to define a linear susceptibility that describes an effective first-order interaction of the probe pulse and the pump-excited sample. However, the physics of this interaction is quite different than in ordinary linear absorption spec troscopy. Likewise, the interpretation of this effective linear susceptibility and of the observed dynamic absorption spectra are in many ways quali tatively new. To clarify how the signal arises in dynamic absorption spectroscopy, we first define the differential transmittance and absorbance spectra in terms of the classical electric fields of the laser pulses and discuss the relationship between the transmitted pulses and the polarization induced in the sample by the incident pulses. Under the slowly-varying-envelope approximation, the output field spectrum is simply proportional to the spectrum of the source polarization; thus, we can concentrate on the latter. The quantum mechanical system assumed throughout is then introduced, and its polar ization response under a semiclassical interaction Hamiltonian defin ed. We present the results of the third-order perturbation expansion of the polarization response in the incident electric fields and label the different contributions to the differential absorbance that arise at this level. Next, the third-order response is expressed in terms of an effective first-order linear susceptibility that captures the interaction of the probe pulse with
FEMTOSECOND ABSORPTION SPECTROSCOPY
50 I
the pump-perturbed sample. This first-order theory is then specialized to the case of a well-defined initial nonstationary state of the quantum system, and the features of the dynamic absorption spectra are discussed in terms of the properties of this initial state.
Annu. Rev. Phys. Chem. 1992.43:497-523. Downloaded from www.annualreviews.org by Duke University on 07/09/12. For personal use only.
Definition of Differential Transmittance Spectrum In dynamic pump-probe absorption spectroscopy, the photochemical pro cesses initiated by an ultrashort pump pulse are monitored by observing the differential transmittance spectrum (pump-on minus pump-off) of a delayed probe pulse. Because all radiative transitions are directly stimu lated by the laser pulses, the radiation field can be described in terms of the classical electric fields of the pump and probe pulses. The total time dependent electric field at location r is written l. Here, the pump (probe) pulse i s characterized by the wavevector, kpu (kpr), which describes its wavelength and direction of propagation, a central frequency, mpu (mpr), and a slowly varying electric field envelope, Epu(r, t) [Epr(r, t)] (c.c. complex conjugate). The transmittance spectrum of the sample is defined as the ratio of the power spedra of the transmitted and incident pulses. For a probe pulse traveling along the z-axis, the transmittance spectrum is defined as =
2. where Epr(m,z) S dt e-iwtEpr(t,z) is the spectral amplitude of the probe pulse after propagating a distance z into the sample. The probe pulse field can usually be separated into a linear component that reflects the normal propagation of a weak pulse through the sample and a smaller nonlinear component caused by the action of the pump pulse on the system, i.e. Epr(m,z) E��)(m,z)+E�;)(m,z), where E�;)(m, O) O. Assuming that the effect of the pump is weak, the intensity of the transmitted light is IEpr(m,zW IE��)(m,z)12+E��)(m,z)*E�;)(m,z)+c.c. The differential transmittance spectrum (pump-on minus pump-off) then contains the spectral response only of the photo-excited systems: =
=
=
=
3.
Dividing by T(m) gives the normalized differential transmittance, which is closely related to the differential absorption spectrum, ACT(m). Under our assumption that the effect of the pump is weak,
502
POLLARD & MATHIES
�(J(W)
oc
�T(w) - T(w)
=
E�;)(w, z) -2ReE�!)(w,z)"
4.
Annu. Rev. Phys. Chem. 1992.43:497-523. Downloaded from www.annualreviews.org by Duke University on 07/09/12. For personal use only.
Signal Generation under the Slowly-Varying Envelope Approximation The excitation of the sample by the incident pulses results in the creation of a polarization (dipole moment), Per, t), which acts back on the initial electric fields. When the spatial extent of the pulses is large compared with the wavelength, one can invoke the slowly varying envelope approximation (SVEA) to reduce the wave equation to a simple first-order differential equation, which describes the evolution of the electric field spectrum of the pulse (44, 45). Under the SVEA, a pulse propagating along the z-axis develops according to (44)
oE(w,z) _--=-_ =
oz
2nw ·k i--P(w, z) e�l +· nc
Z
,
5.
lW .
In particular, the case of a spatially uniform source polarization yields E(w) oc iwP(w). For optically thin samples, the depletion of the initial pulse is neglected, and the sample polarization response is described in terms of the linear susceptibility, X(l)(w), and the nonlinear �;usceptibility, X(3)(w), that reflects the effects of the pump pulse, i.e. 6. The differential absorption spectrum in this limit is the imaginary part of the third-order susceptibility, _
�(J(w) - -
�T(w) (3) P�;l(w) oc -wImx (w) - -wIm Epr(w)· T(w) _
7.
Definition of Quantum-Mechanical Polarization Response The polarization response of the quantum-mechanical system is obtained by considering the evolution of the initially relaxed system in the presence of the incident pulses. The state of the system is represented most usefully by the density matrix, p(t), which evolves according to the quantum Liouville equation, dp(t)/dt = -U/h)[H(t), pet)] - rpet). The relaxation (damping) term, rp(t), accoullts for the effects of the environment on the system (27, 46, 47). Our main interest, however, lies in the nuclear dynamics of the system under H(t), and its manifestation in the time-dependent absorption spectra. Treating the matter-radiation interaction in the electric-dipole approxi mation (48), the total Hamiltonian, H(t), is
FEMTOSECOND ABSORPTION SPECTROSCOPY
H(Q, t; r) = Ho( Q) - p.( Q) ' E(r, t),
503
8.
where E(r, t) is the time-dependent electric field for a molecule located at po sitio n r (Equation 1). Ho( Q) is the Hamiltonian of an individual molecule with internal dis placement coordinates Q. The molecular Hamiltonian for a system of two electronic states in the Born-Oppenheimer approximation is
Annu. Rev. Phys. Chem. 1992.43:497-523. Downloaded from www.annualreviews.org by Duke University on 07/09/12. For personal use only.
9a. where hWn is the energy of electronic state I n) at the ground state ori gin
,
Q=O; 9b. is the multidimensional vibrational Hamiltonian associated with that elec tronic statt:; and mj is the reduced mass of internal vibrational coordinate
Qj. p.(Q) is the coordinate-dependent transition dipole operator, p.( Q) = I l)p.IO( Q) (01+10)p.1o( Q)
Annu. Rev. Phys. Chern. 1992.43:497-523 Copyright © 1992 by Annual Reviews Inc. All rights reserved
Further
Quick links to online content
ANALYSIS OF FEMTOSECOND Annu. Rev. Phys. Chem. 1992.43:497-523. Downloaded from www.annualreviews.org by Duke University on 07/09/12. For personal use only.
DYNAMIC ABSORPTION SPEC�TRA OF NONSTATIONARY STAT'ES W.
Thomas Pollard and Richard
A.
Mathies
Department of Chemistry, University of California, Berkeley, California 94720 KEY WORDS:
femtosecond pump-probe spectroscopy, time-dependent wave packets, transition state spectroscopy, hole-burning, reaction dy namics, resonant impulsive Raman
INTRODUCTION
Over the last decade, dramatic advances in optical pulse technology have ushered in a new era in the study of photochemical reaction dynamics. Today, pulses shorter than 1 00 fs are routinely available in many lab oratories, and optical pulse-compression techniques allow the production of pulses as short as 6 fs ( 1 , 2). Because femtosecol).d pulses are shorter than the hme scale for most intramolecular motions, they provide tools for directly studying the dynamics and relaxation of molecular vibrations (3-1 6). More importantly, this technology now allows the "real-time" observation of photochemical reactions as they proceed on the initi ally populated excited-state potential surfaces, which are the "transition states" of photoinitiated reactions (1 7-22). The state of this already large and rapidly expanding field has been addressed in recent reviews (22-26). This review foc:uses on one particular type of femtosecond experiment that we believe will become i ncreasingly important: pump-probe absorption spectroscopy in the dispersed continuum-probe configuration, or "femto second dynamic absorption spectroscopy" (see Figure 1 ) (27-29). This method is uniquely powerful in that it offers simultaneously high resolution 497 0066--426Xj92jl lOl-0497$02.00
Annu. Rev. Phys. Chem. 1992.43:497-523. Downloaded from www.annualreviews.org by Duke University on 07/09/12. For personal use only.
498
POLLARD
&
MATHIES
in both the frequency and time domains when ultrashort femtosecond pulses are used, thus sidestepping the limitations that the time-energy uncertainty principle places on most other experimental uses of femto second pulses. The traditional method in pump-probe absorption spectroscopy treats the probe pulse as being effectively monochromatic. The absorbance of the sample at the probe wavelength is found as a function of the pump-probe delay time by measuring the loss of energy of the pulse to the sample, either directly using a photodetector (30), or indirectly using such methods as laser-induced fluorescence (31). The repetition of the experiment for probe pulses of different wavelengths then yields a picture of the time dependent absorption spectrum of the photo-excited system (Figure 1 bottom). In this scheme, however, the absorbance of the sample is averaged over the actual spectrum of the probe pulse; the spectra are thus broadened by the spectral width of the pulses, which varies as the inverse of the pulse Spectrograph
MCD
(3)
P (k,m) sample
14 wavelength
Photodetector
(3)
P (k,m) sample
Figure 1 Comparison of time-resolved pump-probe absorption spectroscopy in the dis persed-continuum-probe (top) and integrated-probe (bottom) configurations. MCD multi channel detector. =
Annu. Rev. Phys. Chem. 1992.43:497-523. Downloaded from www.annualreviews.org by Duke University on 07/09/12. For personal use only.
FEMTOSECOND ABSORPTION SPECTROSCOPY
499
length. To observe a particular process, there is generally an optimal pulse length, which balances the time resolution required to monitor the dynamical process, with the frequency resolution required to distinguish the spectral changes in the system (32). In thc continuum-probe or dynamic absorption technique, the transient absorption spectrum of a sample is obtained by measuring how the spec trum of an optical pulse is changed by transmission through the sample (Figure 1, top). This approach is most useful when ultrashort, spectrally broad pulses are available; the useful spectral width of a transform-limited 6-fs pulse, for instance, is on thc order of 3500 cm I, sufficient to measure the complete absorption spectrum for individual electronic transitions in many molecules (1 4). The femtosecond dynamic absorption technique has become the basis for an increasing number of studies ( 1 3, 1 4, 1 6-- 1 8, 20, -
21, 33-43}.
Because: the pulse spectrum can be measured with essentially arbitrary precision, dynamic absorption spectroscopy allows one to monitor the frequency-resolved absorption spectrum of an evolving system with the time resolution of ultrafast femtosecond pUlses. What is often perceived to be paradoxical is that the time resolution and frequency resolution involved in the dynamic absorption method are completely independent, i.e. they are not related by the Heisenberg uncertainty principle. This is possible because the actual measurement of the spectrum for a given pump probe delay is not time resolved. Rather, the brief interaction of the probe pulse with the material evokes a polarization response, i.e. it stimulates a transient oscillating dipole moment, whose subsequent emission has the same wavevector as the probe pulse. Because the measurement of the probe pulse spectrum is not time resolved, both the spectrum of the transmitted pulse and any other emission in thc probe direction are mea sured together. When the interference between the original pulse and the subsequent emission is destructive, the result is the net absorption of energy from the pulse. The sharpness of the spectral features measured in this experiment is determined only by the dephasing rate of the induced polar ization, i.e. by the natural optical linewidths of the vibronic transitions in the material, and has nothing to do with the pulse width. The dynamic absorption experiment gives time-resolved information about nonstationary states, because the induced polarization, which gives rise to the measured signals through its long-time emission, is completely created in the brief instant that the femtosecond probe pulse interacts with the system and, thus, records the transient state of the system. This information, stored in the electronic coherence created by the probe pulse, is revealed over time in the emitted radiation. The continued evolution of the initial system after the probe-pulse interaction modulates the induced
500
POLLARD & MATHIES
polarization, but the essential characteristics of the dynamic absorption spectrum, e.g. its strength, position, and breadth, are determined purely by the state of the system during the initial probe-pulse interaction (26, 27). THEORY OF DYNAMIC ABSORPTION
Annu. Rev. Phys. Chem. 1992.43:497-523. Downloaded from www.annualreviews.org by Duke University on 07/09/12. For personal use only.
SPECTROSCOPY
As noted above, it is important to focus on how the final transmitted probe pulse actually interacts with the sample to understand the properties of dynamic absorption spectra. In ordinary linear absorption spectroscopy, by contrast, the details of the interaction of the probe light with the sample are less important. Whether the absorption spectrum is measured with white light or with a monochromatic laser beam, linear response theory says that the interaction can be considered independently for each fre quency component of the probe radiation. The absorption cross-section, defined through the rate of loss of energy by each frequency component of the probe light, is proportional to the imaginary part of the linear susceptibility, O'(w) ex w 1m [x(1)(w)]. The lowest-order description of pump-probe absorption spectroscopy is third-order in the electric fields of the pump and probe pulses (second order in the pump and first-order in the probe). It is possible to define a linear susceptibility that describes an effective first-order interaction of the probe pulse and the pump-excited sample. However, the physics of this interaction is quite different than in ordinary linear absorption spec troscopy. Likewise, the interpretation of this effective linear susceptibility and of the observed dynamic absorption spectra are in many ways quali tatively new. To clarify how the signal arises in dynamic absorption spectroscopy, we first define the differential transmittance and absorbance spectra in terms of the classical electric fields of the laser pulses and discuss the relationship between the transmitted pulses and the polarization induced in the sample by the incident pulses. Under the slowly-varying-envelope approximation, the output field spectrum is simply proportional to the spectrum of the source polarization; thus, we can concentrate on the latter. The quantum mechanical system assumed throughout is then introduced, and its polar ization response under a semiclassical interaction Hamiltonian defin ed. We present the results of the third-order perturbation expansion of the polarization response in the incident electric fields and label the different contributions to the differential absorbance that arise at this level. Next, the third-order response is expressed in terms of an effective first-order linear susceptibility that captures the interaction of the probe pulse with
FEMTOSECOND ABSORPTION SPECTROSCOPY
50 I
the pump-perturbed sample. This first-order theory is then specialized to the case of a well-defined initial nonstationary state of the quantum system, and the features of the dynamic absorption spectra are discussed in terms of the properties of this initial state.
Annu. Rev. Phys. Chem. 1992.43:497-523. Downloaded from www.annualreviews.org by Duke University on 07/09/12. For personal use only.
Definition of Differential Transmittance Spectrum In dynamic pump-probe absorption spectroscopy, the photochemical pro cesses initiated by an ultrashort pump pulse are monitored by observing the differential transmittance spectrum (pump-on minus pump-off) of a delayed probe pulse. Because all radiative transitions are directly stimu lated by the laser pulses, the radiation field can be described in terms of the classical electric fields of the pump and probe pulses. The total time dependent electric field at location r is written l. Here, the pump (probe) pulse i s characterized by the wavevector, kpu (kpr), which describes its wavelength and direction of propagation, a central frequency, mpu (mpr), and a slowly varying electric field envelope, Epu(r, t) [Epr(r, t)] (c.c. complex conjugate). The transmittance spectrum of the sample is defined as the ratio of the power spedra of the transmitted and incident pulses. For a probe pulse traveling along the z-axis, the transmittance spectrum is defined as =
2. where Epr(m,z) S dt e-iwtEpr(t,z) is the spectral amplitude of the probe pulse after propagating a distance z into the sample. The probe pulse field can usually be separated into a linear component that reflects the normal propagation of a weak pulse through the sample and a smaller nonlinear component caused by the action of the pump pulse on the system, i.e. Epr(m,z) E��)(m,z)+E�;)(m,z), where E�;)(m, O) O. Assuming that the effect of the pump is weak, the intensity of the transmitted light is IEpr(m,zW IE��)(m,z)12+E��)(m,z)*E�;)(m,z)+c.c. The differential transmittance spectrum (pump-on minus pump-off) then contains the spectral response only of the photo-excited systems: =
=
=
=
3.
Dividing by T(m) gives the normalized differential transmittance, which is closely related to the differential absorption spectrum, ACT(m). Under our assumption that the effect of the pump is weak,
502
POLLARD & MATHIES
�(J(W)
oc
�T(w) - T(w)
=
E�;)(w, z) -2ReE�!)(w,z)"
4.
Annu. Rev. Phys. Chem. 1992.43:497-523. Downloaded from www.annualreviews.org by Duke University on 07/09/12. For personal use only.
Signal Generation under the Slowly-Varying Envelope Approximation The excitation of the sample by the incident pulses results in the creation of a polarization (dipole moment), Per, t), which acts back on the initial electric fields. When the spatial extent of the pulses is large compared with the wavelength, one can invoke the slowly varying envelope approximation (SVEA) to reduce the wave equation to a simple first-order differential equation, which describes the evolution of the electric field spectrum of the pulse (44, 45). Under the SVEA, a pulse propagating along the z-axis develops according to (44)
oE(w,z) _--=-_ =
oz
2nw ·k i--P(w, z) e�l +· nc
Z
,
5.
lW .
In particular, the case of a spatially uniform source polarization yields E(w) oc iwP(w). For optically thin samples, the depletion of the initial pulse is neglected, and the sample polarization response is described in terms of the linear susceptibility, X(l)(w), and the nonlinear �;usceptibility, X(3)(w), that reflects the effects of the pump pulse, i.e. 6. The differential absorption spectrum in this limit is the imaginary part of the third-order susceptibility, _
�(J(w) - -
�T(w) (3) P�;l(w) oc -wImx (w) - -wIm Epr(w)· T(w) _
7.
Definition of Quantum-Mechanical Polarization Response The polarization response of the quantum-mechanical system is obtained by considering the evolution of the initially relaxed system in the presence of the incident pulses. The state of the system is represented most usefully by the density matrix, p(t), which evolves according to the quantum Liouville equation, dp(t)/dt = -U/h)[H(t), pet)] - rpet). The relaxation (damping) term, rp(t), accoullts for the effects of the environment on the system (27, 46, 47). Our main interest, however, lies in the nuclear dynamics of the system under H(t), and its manifestation in the time-dependent absorption spectra. Treating the matter-radiation interaction in the electric-dipole approxi mation (48), the total Hamiltonian, H(t), is
FEMTOSECOND ABSORPTION SPECTROSCOPY
H(Q, t; r) = Ho( Q) - p.( Q) ' E(r, t),
503
8.
where E(r, t) is the time-dependent electric field for a molecule located at po sitio n r (Equation 1). Ho( Q) is the Hamiltonian of an individual molecule with internal dis placement coordinates Q. The molecular Hamiltonian for a system of two electronic states in the Born-Oppenheimer approximation is
Annu. Rev. Phys. Chem. 1992.43:497-523. Downloaded from www.annualreviews.org by Duke University on 07/09/12. For personal use only.
9a. where hWn is the energy of electronic state I n) at the ground state ori gin
,
Q=O; 9b. is the multidimensional vibrational Hamiltonian associated with that elec tronic statt:; and mj is the reduced mass of internal vibrational coordinate
Qj. p.(Q) is the coordinate-dependent transition dipole operator, p.( Q) = I l)p.IO( Q) (01+10)p.1o( Q)
