A. Theory of Vibrational Third-Order Nonlinear Spectroscopy
Tải bản đầy đủ - 0trang
Proteins and Peptides
283
obtained (46,76,77):
h
01
00 i
0
10
Dj
10 j
with the line shape function, g
2
exp
iω10
g
(4a)
, given by:
0
g
D
d
0
d 00 hυω10
00
υω10 0 i
(4b)
0
0
The corresponding double-sided Feynman diagram is shown in Fig. 2a.
The double-sided diagrams, which have been used for many years, (78)
have a very simple interpretation for spectroscopic or resonant processes
because of the applicability of the rotating wave approximation in that
case. The vertical lines are time lines with the earliest time at the bottom.
The left and right time lines display properties of the light ﬁelds as they
would appear on the ket and bra sides of the initial density operator 00
in a conventional expansion of the density matrix as a power series in the
ﬁelds. The indices of the density matrix element created by the interaction
are displayed horizontally. By convention the arrow pointing upward from
left to right signiﬁes a positive wavevector, whereas upward from right
to left is a negative wave vector. Since the complex ﬁeld is assumed to
have the form exp[iωt kr], a negative wave vector ﬁeld implies a positive
frequency, which causes a transition from a lower to a higher energy state
on the bra time line or from a higher to lower energy state on the ket time
line. The last arrow refers to the generated ﬁeld, which we will always
take as being emitted as the system undergoes a downward transition on
the ket side. Thus in the correlation function containing 01 t 10 0 00
the ﬁrst interaction is on the left (ket) side of the initial density operator
and the wave vectors of the outgoing and input ﬁelds cancel. After the ﬁrst
interaction the system is in a 1-0 coherence. The sign of the term is 1 n ,
where n is the number of interactions on the right (bra) side.
Within the same formalism, the response functions of the thirdorder Feynman diagrams in Fig. 2b can be written down immediately. For
example, the response function of the ﬁrst (rephasing) diagram in Fig. 2b is:
R1 D h
Dj
01
10 j
4
3
10
e
iω10
1
3
00
1
2
01
0
10
Ð exp
2
i
3
i
υω10
0
d
0
0
1
Ci
υω10
0
2
d 0Ci
0
Copyright © 2001 by Taylor & Francis Group, LLC
0
υω10
0
d
0
5
284
Hamm and Hochstrasser
Figure 2 The double-sided Feynman diagrams, which have to be considered in
(a) a linear absorption experiment and (b) a nonlinear third-order experiment such
as photon echo, pump-probe, transient grating. The diagrams are arranged according
to the possible time orderings, as discussed in the text and illustrated in Fig. 4.
Copyright © 2001 by Taylor & Francis Group, LLC
Proteins and Peptides
285
Again with the help of the cumulant expansion (46) we obtain:
R1 D j
10 j
4
g
3
e
iω10
1
3
1
2
g
3
Ð exp Cg
2 Cg
3
g
2
1
1
g
2
6
with the same line shape function g as the one used for the linear absorption spectrum [Equation (4)]. By changing the absolute times 1 , 2 , and 3
to the time intervals t1 D 1 , t2 D 2
1 , and t3 D 3
2 corresponding
to the delay times that are under experimental control, we obtain the ﬁnal
result for the response function:
R1 D R2 D j
10 j
e iω10 t3 t1 exp g t1 C g t2
g t3 C t2 C g t1 C t2 C t3
4
g t3
g t2 C t1
7
In order to calculate the response function of the Feynman diagram R3 , it is
further assumed that the transition frequency ω12 is anharmonically shifted
with respect to the ground states transition frequency so that, ω12 D ω01
. Another assumption that can be made (see later for a discussion of these
assumptions) is that the ﬂuctuations between both level pairs are strictly
correlated υω12 D υω01 . This implies that only the harmonic part of the
potential surface is perturbed by the bath ﬂuctuations and the anharmonicity
of the vibrator is unaffected. We then obtain for R3 :
j 10 j2 j 21 j2 e i ω10 t3 ω10 t1 Ð exp g t1 C g t2
g t2 C t1
g t3 C t2 C g t1 C t2 C t3
R3 D
g t3
8
The response functions for the nonrephasing Feynman diagram (see Fig. 2b)
are calculated accordingly:
R4 D R5 D j 10 j4 e iω10
C g t3 C t2
R6 D
t3 Ct1
Ð exp g t1
g t1 C t2 C t3
g t2
g t3 C g t2 C t1
9
j 10 j2 j 21 j2 e i ω10 t3 Cω10 t1 Ð exp g t1
g t2
g t1 C t2 C t3
g t3 C g t2 C t1 C g t3 C t2
10
The remaining time orderings, for which there is no corresponding
process in a photon echo experiment on an electronic two-level system, are:
R7 D j
R8 D
j 21 j2 e iω10 t3 C2t2 Ct1 eit2 Ð exp Cg t1
g t2 C t1
g t3 C t2
g t1 C t2 C t3
10 j
2
j 10 j2 j 21 j2 e iω10
C g t3
g t2 C t1
t3 C2t2 Ct1
g t2 C g t3
ei t3 Ct2 Ð exp Cg t1
g t3 C t2
g t1 C t2 C t3
Copyright © 2001 by Taylor & Francis Group, LLC
11
g t2
12
286
Hamm and Hochstrasser
As expected, we ﬁnd that the total response function 3lD1 Rl D 6lD4 Rl D
8
lD7 Rl D 0 (i.e., for each possible time ordering) vanishes exactly in the
harmonic case, deﬁned by D 0 and j 21 j2 D 2j 10 j2 . Furthermore, it can
be easily seen that in the case of a strict separation of time scales of homogeneous and inhomogeneous broadening, the line shape function becomes
g t D t/T2 C 2 t2 /2 and the total response function reduces exactly to
the result obtained within a Bloch picture (see, for example Refs. 52 and
75), e.g.,
3
Rl D 2j
10 j
4
e
iω10 t3 t1
Ð 1
eit3 Ð exp
t1 C t3 /T2
lD1
ð exp
2
t1
t3 2 /2
13
In particular, the time coordinate t2 disappears in Equation (13), which
emphasizes the fact that three-pulse photon echo experiments are necessary
to characterize effects that are beyond the Bloch picture.
Vibrational energy relaxation T1 is taken into account phenomenologically through the multiplicative factors:
exp
t1 /2 C t2 C t3 /2
T1
exp
t1 /2 C t2 C 3t3 /2
T1
for R1 , R2 , R4 , and R5
(14a)
and
for R3 and R6
(14b)
which assumes that vibrational relaxation from the D 2 to the D 1 level
is twice as fast as from the D 1 to the D 0 level and that it is constant
throughout the inhomogeneous distribution. Orientational relaxation for R1
to R6 is taken into account through the factor:
h
z
0
z t1
z t1
C t2
z t1
C t2 C t3 i D
2 1
2
C e
3 6 15
6Dt2
e
2D t1 Ct3
15
where h. . .i is an orientational average, z t is the projection of the transition dipole at t onto the laboratory z-axis, and D is the rotational diffusion coefﬁcient. Thus, during the period t1 and t3 , where the system is
in a coherence, the four-wave mixing signal decays like the ﬁrst-order
Legendre polynomial P1 cos Â just as is found in a two-pulse echo (74).
However, during the period t2 , where the system is in a population state, it
Copyright © 2001 by Taylor & Francis Group, LLC
Proteins and Peptides
287
decays fractionally like P2 cos Â . The formulas for other polarization conditions and for nonspherical rotors are easily obtained by standard methods
(79,80). Generally such simple diffusional models are not correct for small
molecules in solutions. Their orientational correlation functions are not
usually simple exponentials with parameters that depend on the molecular
size, but they also involve varying amounts of inertial motion, and there is
no general form for the intermediate region. Furthermore, there is in general
no relation between the decays of P1 and P2 such as occurs when there is
diffusive motion. However, if the orientational correlation functions could
be measured independently it would be straightforward to introduce them
in place of Equation (15) and then to solve the problem numerically.
The third-order polarization is obtained by convolution of the response
functions with the electric ﬁelds of the three laser pulses:
1
P3 t D
1
dt3
0
ð E2 t
0
t3
1
dt2
t2 e
Rl t1 , t2 , t3 Ð E3 t
dt1
0
iω t t3 t2
t3 e
iω t t3
l
E1 Ł t
t3
t2
t1 eiω t
t3 t2 t1
16
where the E1,2,3 are the ﬁeld envelopes of the laser pulses with wave vectors
vector k1 , k2 , and k3 , respectively, and ω is the carrier frequency of the light
pulses. When the laser pulses are not overlapping in time, the time ordering
is the same in the whole integration area and the sum in Equation (16) runs
over those Feynman diagrams relevant for that particular time ordering
(l D 1, 2, 3, l D 4, 5, 6, or l D 8, 9, respectively). In the case of overlapping
laser pulses, the set of response functions is interchanged each time the
integration in Equation (16) switches the time ordering.
In a homodyne detection scheme, such as in the stimulated photon
echo experiments described in the next paragraph, the detector measures
the t-integrated intensity of the square of the third-order polarization
S T,
1
D
jP 3 T, , t j2 dt
(17)
0
where and T are the time delays between the peaks of pulse k1 and k2
and pulse k2 and k3 , respectively (see Section III.D).
B. Limitations of the Stochastic Model
The response function R2 , which develops on the vibrational ground state
after the second light interaction, is the same as R1 [see Equation (7)]. This
is a consequence of the stochastic ansatz in Equation (3), which implies that
the bath inﬂuences the vibrational frequency of the solute but excitation of
Copyright © 2001 by Taylor & Francis Group, LLC
288
Hamm and Hochstrasser
the solute does not perturb the bath. In optical spectroscopy there is a
Stokes shift which is due to the response of the bath molecules to the
changed charge distribution of the solute after electronic excitation (81).
This effect causes a nonzero imaginary part in the line shape function g t
and the response functions R1 and R2 to become different (46). Vibrational transitions might undergo dynamical spectral shifts analogous to the
Stokes shifts of electronic transitions. The nuclear motion dependence of
the dipole or higher electric multipoles could induce a dynamical vibrational shift that will inﬂuence the interpretations of vibrational dynamics
and cooling in general. Such dynamics is not included in, for example,
Redﬁeld theory (82–86) of multiple vibrational level systems coupled to
bath. The usual electronic Stokes shift is caused by the response of the bath
molecules to the changed charge distribution of the solute after excitation
(87,88). For high-frequency vibrations the change in the static, averaged
electric moment would be sensed by the solvent. However, when the solvent
responds comparably or faster than the solute motion, both the curvature and
the absolute energy of the potential surface may be changed, and any shift
of a vibrational frequency will not simply record the change in free energy
computed for a change in electric moment. Clearly the dipole moment as a
function of the internuclear distance is already partly incorporated into the
potential of mean force, which determines the mean frequencies in the BornOppenheimer approximation. Nevertheless, the vibrational state dependence
of the dipole moment is well known (89–92) and changes of 0.03–0.2 D
between D 0 and D 1 have been reported. Although the dipole (or electric quadrupole for nearest neighbor effects) changes are much smaller than
for electronic transitions, the spectral resolution is improved in the infrared
by a similar factor; therefore, when such dynamical shifts are present they
might be observable. In summary, excitation of D 1 by a short pulse
should create a nonequilibrium distribution of solvent/solute conﬁgurations
because of the mixed mode anharmonicity. Both the negative stimulated
emission peak and the new (1 ! 2 transition) absorption in a IR-pump–IR
probe experiment could shift and change width in time, while the bleaching
peak should remain constant. It should be possible to measure shifts of less
than one twentieth of the bandwidth. In some cases, speciﬁc modes (93)
may dominate the anharmonicity while in others more complex solvent
motions may be involved. In any event, further experimental work is needed
to evaluate the assumption that R1 D R2 .
Another simplifying assumption included at this point in our model
so far is the strict correlation of the ﬂuctuations between the 0-1 and the
1-2 level pairs υω12 D υω01 . This assumption is based on the notion that
Copyright © 2001 by Taylor & Francis Group, LLC