Molecular Distances Determined with Resonant ... - Rice University
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Molecular Distances Determined with Resonant Vibrational Energy Transfers Hailong Chen,† Xiewen Wen,† Jiebo Li, and Junrong Zheng* Department of Chemistry, Rice University, 6100 Main Street, Houston, Texas 77005-1892, United States S Supporting Information *
ABSTRACT: In general, intermolecular distances in condensed phases at the angstrom scale are difficult to measure. We were able to do so by using the vibrational energy transfer method, an ultrafast vibrational analogue of Förster resonance energy transfer. The distances among SCN− anions in KSCN crystals and ion clusters of KSCN aqueous solutions were determined with the method. In the crystalline samples, the closest anion distance was determined to be 3.9 ± 0.3 Å, consistent with the XRD result. In the 1.8 and 1 M KSCN aqueous solutions, the anion distances in the ion clusters were determined to be 4.4 ± 0.4 Å. The clustered anion distances in aqueous solutions are very similar to the closest anion distance in the KSCN crystal but significantly shorter than the average anion distance (0.94−1.17 nm) in the aqueous solutions if ion clustering did not occur. The result suggests that ions in the strong electrolyte aqueous solutions can form clusters inside of which they have direct contact with each other.
1. INTRODUCTION Short-ranged transient intermolecular interactions, for example, guest/host interactions, ion/ion interactions, and ion/molecule bindings, play significant roles in chemistry and biology. Electronic energy transfer methods, for example, Förster resonance energy transfer (FRET), are routinely applied to measure the distance between two molecules in many of these interactions.1,2 In these methods, chromophores that can transfer and accept energy in the visible and near-IR frequency range are required to attach to molecules. The chromophores are typically at the size of 1−2 nm or larger, which is not only much larger than many of the intermolecular distances but also can perturb the molecular interactions under investigation. To probe molecular distances at the angstrom scale, what is needed is an ultrafast vibrational analogue of FRET of which the chromophores are simple chemical bonds (1−2 Å). Here, by measuring ultrafast vibrational energy transfers in model systems of KSCN (potassium thiocyanate) crystals and aqueous solutions, we demonstrate that angstrom molecular distances in condensed phases can be determined. In this paper, we first present anisotropy decay and vibrational energy exchange data to show that vibrational energy can transfer resonantly among SCN− anions and nonresonantly from SCN− to S13C15N− with much slower rates in KSCN/KS13C15N mixed © 2014 American Chemical Society
crystals. A kinetic model is then developed to quantitatively analyze the resonant energy transfer data to obtain the resonant energy transfer time between two adjacent anions in the crystals. To convert the energy transfer time into distance, an equation based on the dephasing mechanism and the timedependent Schrodinger equation is derived to correlate the energy transfer time with the energy donor/acceptor coupling strength, which is quantitatively correlated to the donor/ acceptor distance under the dipole/dipole interaction. On the basis of the energy transfer equation and the dipole/dipole interaction equation and the measured energy transfer time, the distance between the two adjacent anions is calculated. The calculated distance is then compared to the distance measured by XRD. Within experimental uncertainty, the two values are the same. The uncertainty introduced by the point dipole assumption used in the method is then calculated based on the monopole theory. It is found that the uncertainty is very small, only ∼1.3%, mainly because the donor/acceptor distance is more than three times larger than the sizes of the donor/ acceptor. After benchmarking the vibrational energy transfer Received: January 17, 2014 Revised: March 17, 2014 Published: March 18, 2014 2463
dx.doi.org/10.1021/jp500586h | J. Phys. Chem. A 2014, 118, 2463−2469
The Journal of Physical Chemistry A
Article
measurements, the nitrile stretch of one anion, for example, SCN−, is excited to its first excited state. The decay of the anisotropy value of this vibrational excitation signal is then monitored in real time. Simultaneous analyses on the anisotropy decays in samples with different mixed KSCN/ KS13C15N ratios quantitatively yield both the resonant energy transfer rate constant and the rotational time constant of the anion. 2.3. Samples. Unless specified, chemicals were purchased from Sigma−Aldrich and used without further purification. KS13C15N was purchased from Cambridge Isotope Laboratory. The samples were thin films of polycrystalline KSCN/ KS13C15N mixed crystals with different molar ratios blended with ∼50 wt % PMMA. The thickness of the sample is estimated to be a few hundred nm based on the CN stretch optical density. An optical image of a sample is provided in Figure S1 in the Supporting Information (SI). The function of PMMA was to suppress scattered light. The samples were placed in a vacuum chamber during measurements. There are four reasons to use KSCN and KS13C15N and KS13CN as model systems: (1) the vibrational lifetimes of the nitrile stretches are relatively long (longer than 30 ps in D2O solutions and even longer in the crystals as included in the SI);4 (2) the transition dipole moments of the nitrile stretches are relatively large (0.3−0.4 D);4 (3) the distance between any two anions in the KSCN crystal is well-characterized with XRD and neutron scattering, which can be used to benchmark the distance determined by the vibrational energy transfer method;7 and (4) the phonon dispersion in the KSCN crystal was wellcharacterized before.8,9 The purpose of using different-isotope-labeled thiocyanate anion mixed samples, for example, a KSCN/KS13C15N mixed crystal with different molar ratios, is 2-fold, (1) to shift the vibrational frequency of the nitrile stretch as the vibrational frequency of the CN stretch is different from that of the 13C15N stretch and (2) to change the number of resonant energy transfer acceptors in the samples without changing the crystalline structure or ion cluster structure because the isotope labeling has negligible effects on the intermolecular interaction between two thiocyanate anions that determines their distance and relative orientation.10
method with the crystalline sample, the method is used to determine the anion distance in the ion clusters of KSCN aqueous solutions. It is found that within experimental uncertainty, the anion distance in the clusters is the same as that in the crystal.
2. EXPERIMENTS AND METHODS 2.1. Laser System. A ps amplifier and a fs amplifier are synchronized with the same seed pulse. The ps amplifier pumps an OPA to produce ∼0.8 ps (vary from 0.7−0.9 ps in different frequencies) mid-IR pulses with a bandwidth of 10−35 cm−1 in a tunable frequency range from 400 to 4000 cm−1 with an energy of 1−40 μJ/pulse (1−10 μJ/pulse for 400−900 cm−1 and >10 μJ/pulse for higher frequencies) at 1 kHz. Light from the fs amplifier is used to generate a high-intensity mid-IR and terahertz supercontinuum pulse with a duration of > 2 ), eqs.S30~32 can be
numerically solved with
p0 (t ) ≈ ∏ j
−2 k0 j t
1+ e 2
−2 k0 j 't
1+ e ∏j ' 2
−2 k0 j t
1 1 1+ e p|| (t ) ≈ + e −2 k⊥t − ∏ 2 2 2 j
p⊥ (t )=
1 1 −2 k⊥t − e , 2 2
eq.S33 −2 k0 j 't
1+ e ∏j ' 2
eq.S34
eq.S35
where kij and kij ' are correlated to each other through eqs.13&14 in the main text.
14
Calculations of clustered anion distances in 1M and 1.8M KSCN aqueous solutions
µ= 0.33D , We have the parameters: n = 1.5 , µ= D A
κ = 2 / 3 (randomized,
because the rotation of anion is faster than the energy transfer) and the dephasing time ( τ=
τ = 0.29 ps
(τ =
1 100 ps ) 2p 18 × 3
for
the
1.8M
solution
and
τ = 0.27 ps
1 100 ps ) for the 1M solution from the energy mismatch dependent 2p 20 × 3
experiments (fig.S6~7) 1,10. For the 1.8 M solution 1 1 2 τ = k β= 2 −2 (2 β ) + τ 18 ps
β=
1 µD µ A κ 3 n 2 4πε 0 rDA
⇒β = 2.3 cm−1
⇒ rDA = 4.4 A
For the 1M solution 1 1 2 τ k β= = 2 −2 (2 β ) + τ 15 ps
β=
1 µD µ A κ 3 n 2 4πε 0 rDA
⇒β = 2.5 cm−1
⇒ rDA = 4.3 A
If n = 1.42 is used (averaging the refractive indexes of D 2 O and KSCN), rDA = 4.5 A are for the 1.8M and 1M solutions respectively. and rDA = 4.4 A
15
0.014
1.0
Normalized Population
Normalized Population
0.012
KSCN KS13C15N
0.8
0.6
(A)
0.4
0.2
0.0
Flowing down Pumping up
0.010 0.008 0.006
(B)
0.004 0.002 0.000 -0.002
0
50
100
150
200
250
0
Waiting Time (ps)
50
100
150
200
250
Waiting Time (ps)
1.2
Normalized Anisotropy
1.0
2:98 50:50 100:0
0.8 0.6 0.4
(C)
0.2 0.0 -0.2 0
10
20
30
40
50
Waiting Time (ps)
Figure S6. Data and calculations of nonresonant [(A) and (B)] energy transfer between SCN- and S13C15N- (SCN-/S13C15N-=1/1) and resonant [(C)] energy transfers among SCN- anions for a 1.8M KSCN aqueous solution from our previous publication.1 Dots are data, and lines are calculations. Calculations for (A) and (B) are with input parameters: (ps −1 ); kSCN − slow 1/ 22 (ps −1 ); kS13C15 N − fast 1/ 2.7 (ps −1 ); kS13C15 N − slow 1/ 29 (ps −1 ); kSCN − fast 1/1.4 = = = = kclu →iso
1/10 (ps −1 ); K=0.55; kSCN − → S13C15 N − 1/160 (ps −1 ); D=0.70 =
with pre-factors of the subgroups and offset of the bi-exponential = ASCN − fast 0.16; = ASCN − slow 0.84;= AS13C15 N − fast 0.22; = AS13C15 N − slow 0.78; = offset 0 . (C) τ or = 4.5 ps is experimentally determined, which is the rotation time of the clustered ions. ntot = 4 , τ = 18 ps . Therefore, for the same number of acceptors, the resonant transfer time is 18/2=9ps, and that for ∆ω = 75 cm −1 is 160ps. Based on eq.9, the dephasing width is determined to be 15.5 cm-1.
16
0.008
KSCN KS13C15N
0.8
Normalized Population
Normalized Population
1.0
0.6
(A)
0.4
0.2
Flowing down Pumping up
0.006
0.004
(B)
0.002
0.000
0.0 -0.002 0
50
100
150
200
250
0
Waiting Time (ps)
50
100
150
200
250
Waiting Time (ps)
1.0
10:90 50:50 100:0
Normalized Anisotropy
0.8
0.6
0.4
(C)
0.2
0.0
-0.2 0
10
20
30
40
50
Waiting Time (ps)
Figure S7. Data and calculations of nonresonant [(A) and (B)] energy transfer between SCN- and S13C15N- (SCN-/S13C15N-=1/1) and resonant [(C)] energy transfers among SCN- anions for a 1.0M KSCN aqueous solution from our previous publication.1 Dots are data, and lines are calculations. Calculations for (A) and (B) are with input parameters: = = = = k SCN − fast 1/1.7 (ps −1 ); k SCN − slow 1/ 21 (ps −1 ); k S13C15 N − fast 1/1.6 (ps −1 ); kS13C15 N − slow 1/ 28 (ps −1 ); kclu →iso
= 1/10 (ps −1 ); K=0.38; kSCN − → S13C15 N − 1/180 (ps −1 ); D=0.70
with pre-factors of the subgroups and offset of the bi-exponential = = = ASCN − fast 0.25; ASCN − slow 0.75;= AS13C15 N − fast 0.21; = AS13C15 N − slow 0.79; offset 0 . (C) τ or = 4.3 ps is experimentally determined, which is the rotation time of the clustered ions. ntot = 3 , τ = 15 ps . Therefore, for the same number of acceptors, the resonant transfer time is 15/1.5=10ps, and that for ∆ω = 75 cm −1 is 180ps. Based on eq.9, the dephasing width is determined to be 15.5 cm-1.
17
0.5
Model
Lorentz
Equation
y = y0 + (2*A/PI)*(w/(4*(x-xc)^ 2 + w^2))
Value
D
Standard Error
y0
-0.01231
4.53192E-4
xc
2063.54727
0.05117
Lorentz
Equation
y = y0 + (2*A/PI)*(w/(4*(x-xc)^ 2 + w^2))
w
32.36957
0.1768
A
23.27113
0.10815
H
0.45768
(A)
0.2
1.58653E-4
Reduced Chi-Sqr
0.4
0.99093
Adj. R-Square
Value
0.3 OD
OD
Model
0.99164
Adj. R-Square
0.3
0.5
1.10401E-4
Reduced Chi-Sqr
0.4
D
(B)
Standard Error
y0
-0.02054
5.38998E-4
xc
2063.76032
0.05248
w
31.73273
0.18054
A
26.40325
0.12735
H
0.5297
0.2
0.1
0.1
0.0
0.0
1900 1950 2000 2050 2100 2150 2200 -1
Wavenumber (cm )
1900 1950 2000 2050 2100 2150 2200 Wavenumber (cm-1)
Figure S8. FTIR spectra of 1M and 1.8M KSCN in D 2 O solutions. The Lorentzian line width for both samples is 32 cm-1, giving the dephasing width 16 cm-1.
18
Calculations of the transition dipole moment of the CN stretch in KSCN crystal The transition dipole moment of the CN stretch was calculated by comparing the FTIR spectra and the 2D-IR signals for both of the CO stretch in the molecule (CH 3 C 5 H 4 )Mn(CO) 3 and the CN stretch in KSCN crystal. Under the same experimental conditions, e.g. the pump intensity, the signal intensities can be expressed as: 2 I FTIR −CO = C1µCO
eq.S36
2 I FTIR −CN = C1µCN
eq.S37
4 I 2 DIR −CO = C2 µCO
eq.S38
4 . I 2 DIR −CN = C2 µCN
eq.S39
It is based the fact that, FTIR measurement is a linear spectroscopy, while 2D-IR method measures the 3rd order nonlinear response. From eq.S36-S39, we have
/I I µCN = 2 DIR −CN 2 DIR −CO . I FTIR −CN / I FTIR −CO µCO
eq.S40
The transition dipole moment of the CO stretch was obtained by measuring the FTIR spectra of (CH 3 C 5 H 4 )Mn(CO) 3 in CCl 4 solution with the concentration 0.029 M and the path length 25 m m (fig.S9A), and using the expression = µ 2 9.186 ×10−3 n ∫ [ε (n ) / n ]dn ,
eq.S41
where µ is in the unit of Debye, ν is the wavenumber in cm −1 , εν ( ) is the molar decadic extinction coefficient in L/(mol cm) at wavenumberν , and n is the refractive index of the media
11
. For CCl 4 solution at around 2022 cm −1 , n = 1.44 .
From the FTIR data and these parameters, we got µCO = 0.45 D . The intensity ratios in eq. S40 were obtained from the FTIR and 2D-IR measurements for two samples. We got I FTIR −CN / I FTIR −CO = 0.48 / 0.44 (fig.S9A&B), and I 2 DIR −CN / I 2 DIR −CO = 0.032 / 0.067 for the intensity of red peak (fig.S10A&B)
I 2 DIR −CN / I 2 DIR −CO = 0.029 / 0.052 for the intensity of blue peak (fig.S10C&D). By inserting all these ratios into eq.S40, we finally got the transition dipole moment of the CN stretch µCN = 0.30 D (using red peak), or µCN = 0.32 D (using blue peak). As a result, we chose µCN = 0.31D for the calculations in this work. 19
0.5 0.4
0.5
(A)
0.4
0.3
(B)
OD
OD
0.3
0.2
0.2
0.1
0.1
0.0
0.0
1960 1980 2000 2020 2040 2060 2080
1900 1950 2000 2050 2100 2150 2200
Wavenumber (cm-1)
Wavenumber (cm-1)
Figure S9. FTIR spectra of (A) (CH 3 C 5 H 4 )Mn(CO) 3 in CCl 4 solution with the concentration 0.029 M and the path length 25 m m , and (B) the thin film of polycrystalline KSCN crystals blended with ~50 wt% PMMA.
Population
0.06
(A)
0.03
0.04
0.02
0.02
0.01
0.00
0.00 -20
0
20
40
60
0.02
-20
0
20
40
60
0
20
40
60
0.01
(C)
(D)
0.00
0.00
-0.02
-0.01
-0.02
-0.04
-0.06
(B)
-0.03 -20
0
20
40
60
-20
Time Delay (ps)
Figure S10. Waiting time dependent intensities of the (A) red peak for (CH 3 C 5 H 4 )Mn(CO) 3 solution, (B) red peak for KSCN crystal, (C) blue peak for (CH 3 C 5 H 4 )Mn(CO) 3 solution, and (D) blue peak for KSCN crystal, all of which were measured by 2D-IR method, with the pump at 2022 cm-1 for the (CH 3 C 5 H 4 )Mn(CO) 3 solution and 2050 cm-1 for the KSCN crystal.
20
The derivation of the relation between the one-way energy transfer rate and the total energy transfer rate For the energy transfer between a pair of donor and acceptor, without considering the vibrational relaxations, we have the rate equations
d D(t ) = −k DA D(t ) + k AD A(t ) dt
eq.S42
d A(t ) = −k AD A(t ) + k DA D(t ) , dt
eq.S43
where D(t ) and A(t ) are the populations of the exited state for the donor and acceptor, respectively, and k DA (or k AD ) is the energy transfer rate from the donor (or acceptor) to the acceptor (or the donor). For both of the donor and the acceptor are identical SCN- anions, we chose = k k DA + k AD to denote the one-way energy transfer rate. Considering the initial conditions, D(0) = 1 and A(0) = 0 . We get the solutions
= D(t )
1 − kt 1 e + 2 2
1 1 A(t ) = − e − kt + . 2 2 Therefore, the total energy transfer rate is = k k DA + k AD , and k DA = e
eq.S44
eq.S45 DωDA RT
k AD .
21
Energy transfer dephasing time of the KSCN crystalline samples The energy transfer dephasing time in eq.9 is a situation dependent parameter. If the dephasings of the donor and acceptor are uncorrelated, the energy transfer dephasing time (in terms of line width) is the convoluted line width of the donor and acceptor. For Lorentzian lineshapes, it is the sum of both donor and acceptor line widths, which determines the fastest value of possible energy transfer dephasing time. In many electronic energy transfers, such a dephasing time is assumed as the donor and the acceptor are several nm away so that it is reasonable to assume that the donor/acceptor dephasings are uncorrelated. For vibrational energy transfers, such an assumption may not be valid as the majority of experimentally measurable intermolecular vibrational energy transfers occur within distances smaller than 1 nm. Within such short distances, a molecular event which causes the dephasing of one molecule will inevitably affect the vibration of another molecule nearby. Therefore, the dephasings of the donor and the acceptor are correlated. The result of such a correlation is that the energy transfer dephasing time must be longer than that determined by the donor/acceptor convoluted line width. How long the energy transfer dephasing time can be is determined by how the dephasings of donor and acceptor are correlated. Experimentally, the energy transfer dephasing time can be obtained from the energy mismatch ∆ω dependent energy transfers based on eq.9. In principle, if we know the energy transfer rate constants for two energy mismatches, mathematically we can derive the coupling strength and the dephasing time from eq.9. The condition for such a treatment is that the nonresonant energy transfer through the dephasing mechanism (eq.9) is much faster than that of the phonon compensation mechanism 12,13. We expect that the condition can be fulfilled for most liquid solutions with relatively small energy mismatches ( ∆ω < RT ) where the dephasing is fast and well defined phonon motions are scarce. The condition can be invalid for some solid samples where the dephasing is relatively slow and the density of phonons with energy the same as the donor/acceptor mismatch is high.
22
w3 (cm-1)
(A) 50 ps
0 ps
2060
100 ps a
d 2020
b
c
1980 2000
2040
2080 -1
ω1 (cm )
1.0
1.0
(B)
(C)
0.8
Intensity (a.u.)
Intensity (a.u.)
0.8
0.6
0.4
0.2
0.6
0.4
0.2
0.0
0.0 0
200
400
600
Delay (ps)
800
1000
0
200
400
600
800
1000
Delay (ps)
Figure S11. (A):Waiting time dependent vibrational energy exchange 2D IR spectra of a KSCN/KS13CN=1/1 mixed crystal at room temperature. The growth of cross peaks indicates how fast the vibrational energy exchange proceeds between SCN- and S13CN-. (B)-(C) Waiting time dependent normalized intensities of peaks a, b, c, and d. Dots are experimental data, and curves are calculations based on the energy exchange kinetic model and experimentally measured vibrational lifetimes.Calculation parameters are pfa=0.081; pfb=0.022; kfa=1/1.05; kfb=1/0.73; ka=1/589; kb=1/688; kab=1/96; and kba=1/123. The KSCN crystalline sample with a mismatch 75 cm-1 is such an example. As analyzed in the main text, in the KSCN/KS13C15N=1/1 sample, the energy transfer
1 time constant ( ) from SCN to S13C15N ( ∆ω = 75 cm −1 ) is 99ps, and that from SCN k to SCN ( ∆ω = 0 cm −1 ) is 3.6ps. As shown in fig.S11, in the KSCN/KS13CN=1/1
1 sample, the energy transfer time constant ( ) from SCN to S13CN ( ∆ω = 50 cm −1 ) is k 96ps. The results with three different energy mismatches cannot be described by eq.9 as it predicts that the energy transfer with ∆ω = 50 cm −1 should be 100% faster than that with ∆ω = 75 cm −1 . The essential reason for the energy transfer from SCN to S13C15N is much faster than the prediction from eq.9 is that the measured energy transfer is from the phonon compensation mechanism. As we can see from both 23
Raman and Neutron scattering data, the phonon densities at ~ 75 cm-1 are much higher than those at around 50 cm-1. Therefore, the energy transfer dephasing time can only be calculated from the results of samples with ∆ω = 50 cm −1 and ∆ω = 0 cm −1 . Calculations show that the dephasing time is 8 cm-1 (in terms of line width), narrower than the convoluated line width (10 cm-1). However, 8 cm-1 is only the fast limit of the dephasing time, as we don’t know how much portion of the measured ∆ω = 50 cm −1 energy transfer is from the energy compensation mechanism. To test the uncertainty range of determined distance caused by this dephasing time uncertainty, we varied the dephasing time from 3 cm-1 to 8 cm-1, and found that the calculated distance between two closest anions varies from 4.4 to 3.9 angstroms. The determined values are within experimental uncertainty (~10% of the actual distance 4 angstroms). 1.0
KSCN/KS13C15N=1/1 RT KSCN/KS13CN=1/1 RT
200000 150000
0.8 G(E) (Normalized)
Intensity (counts)
250000
(A)
100000 50000 0
0.6
(B)
0.4 0.2 0.0
25
50
75
100
Frequency (cm-1)
125
150
50
100 E (cm-1)
150
Figure S12. (A) Raman spectra of KSCN/KS13C15N=1/1 and KSCN/KS13CN=1/1 samples at room temperature; (B) Neutron scattering data at 10K from literature 14.
24
Table of parameters of SCN- in KSCN crystal Table S2. Calculated parameters of every SCN- in KSCN crystal (3*3*3, 500 anions). The orientation factors were calculated based on eq.S9, with A∞ = 0.7 .The coupling
µ= 0.31D . Since the constants were calculated based on eq.10, with n = 1.5 , µ= D A energy transfer rate for the donor to each of the acceptors is proportional to
β 2 / [(2β ) 2 + τ −2 ] (see eq.9), the 1/k ET for every SCN- can be calculated by distributing the total energy transfer rate 1/1.8ps, which was obtained from experimental result, with τ −1 = 8 cm −1 . No . 0 2 4 6 8 10 12 14 16 18 20 22 24
Distanc e Orientatio (Å n factor ) 0.000 0.000 4.017 0.378 4.026 0.706 4.802 0.477 5.576 0.597 6.160 0.740 6.451 0.728 6.532 0.950 6.673 0.626 6.715 0.672 7.517 1.182 7.543 0.899 7.789 0.991
Coupling constant (cm-1)
1/k ET
No .
Distanc e (Å)
Orientatio n factor
Coupling constant (cm-1)
1/k ET
0.000 1.252 2.324 0.926 0.740 0.681 0.583 0.732 0.452 0.477 0.598 0.450 0.451
0 29 8 53 82 97 133 84 220 198 126 222 222
1 3 5 7 9 11 13 15 17 19 21 23 25
4.017 4.026 4.802 5.576 6.160 6.451 6.532 6.673 6.715 7.517 7.543 7.789 7.789
0.378 0.706 0.477 0.597 0.740 0.728 0.950 0.626 0.672 1.182 0.899 0.991 0.991
1.252 2.324 0.926 0.740 0.681 0.583 0.732 0.452 0.477 0.598 0.450 0.451 0.451
222
27
8.550
0.385
0.132
29
8.550
0.385
0.132
31
8.932
1.139
0.343
33
9.338
0.523
0.138
35
9.338
0.523
0.138
37
9.380
0.417
0.109
39
9.380
0.417
0.109
26
7.789
0.991
0.451
28
8.550
0.385
0.132
30
8.550
0.385
0.132
32
8.932
1.139
0.343
34
9.338
0.523
0.138
36
9.338
0.523
0.138
38
9.380
0.417
0.109
2.57E+0 3 2.57E+0 3 3.82E+0 2 2.37E+0 3 2.37E+0 3 3.82E+0
25
29 8 53 82 97 133 84 220 198 126 222 222 222 2.57E+0 3 2.57E+0 3 3.82E+0 2 2.37E+0 3 2.37E+0 3 3.82E+0 3 3.82E+0
40
9.380
0.417
0.109
42
9.467
1.784
0.452
44
9.467
1.784
0.452
46
9.507
1.054
0.263
48
9.731
0.932
0.217
50
9.861
0.844
0.189
52
10.071
0.507
0.107
54
10.071
0.507
0.107
56
10.099
0.492
0.103
58
10.099
0.492
0.103
60
10.315
1.253
0.245
62
10.467
0.715
0.134
64
10.761
0.755
0.130
66
11.012
0.990
0.159
68
11.150
0.528
0.082
70
11.150
0.528
0.082
72
11.399
0.333
0.048
74
11.620
1.146
0.157
76
11.699
0.763
0.102
78
11.911
0.449
0.057
80
12.104
0.874
0.106
3 3.82E+0 3 2.21E+0 2 2.21E+0 2 6.49E+0 2 9.54E+0 2 1.26E+0 3 3.96E+0 3 3.96E+0 3 4.27E+0 3 4.27E+0 3 7.49E+0 2 2.52E+0 3 2.66E+0 3 1.78E+0 3 6.74E+0 3 6.74E+0 3 1.93E+0 4 1.83E+0 3 4.30E+0 3 1.38E+0 4 4.02E+0 3
41
9.467
1.784
0.452
43
9.467
1.784
0.452
45
9.507
1.054
0.263
47
9.731
0.932
0.217
49
9.861
0.844
0.189
51
10.071
0.507
0.107
53
10.071
0.507
0.107
55
10.099
0.492
0.103
57
10.099
0.492
0.103
59
10.315
1.253
0.245
61
10.467
0.715
0.134
63
10.761
0.755
0.130
65
11.012
0.990
0.159
67
11.150
0.528
0.082
69
11.150
0.528
0.082
71
11.399
0.333
0.048
73
11.620
1.146
0.157
75
11.699
0.763
0.102
77
11.911
0.449
0.057
79
12.104
0.874
0.106
81
12.104
0.874
0.106
26
3 2.21E+0 2 2.21E+0 2 6.49E+0 2 9.54E+0 2 1.26E+0 3 3.96E+0 3 3.96E+0 3 4.27E+0 3 4.27E+0 3 7.49E+0 2 2.52E+0 3 2.66E+0 3 1.78E+0 3 6.74E+0 3 6.74E+0 3 1.93E+0 4 1.83E+0 3 4.30E+0 3 1.38E+0 4 4.02E+0 3 4.02E+0 3
82
12.104
0.874
0.106
84
12.104
0.874
0.106
86
12.104
0.874
0.106
88
12.136
0.726
0.087
90
12.136
0.726
0.087
92
12.262
0.446
0.052
94
12.318
0.858
0.099
96
12.467
0.852
0.095
98
12.508
0.424
0.047
12.625
1.249
0.133
12.667
1.248
0.132
12.778
0.866
0.089
12.778
0.866
0.089
12.855
1.242
0.126
12.995
1.579
0.155
13.050
0.472
0.046
13.208
0.496
0.046
13.208
0.496
0.046
13.346
0.626
0.057
13.348
0.731
0.066
13.348
0.731
0.066
13.430
0.672
0.060
10 0 10 2 10 4 10 6 10 8 11 0 11 2 11 4 11 6 11 8 12 0 12 2 12 4
4.02E+0 3 4.02E+0 3 4.02E+0 3 5.92E+0 3 5.92E+0 3 1.67E+0 4 4.63E+0 3 5.05E+0 3 2.08E+0 4 2.53E+0 3 2.59E+0 3 5.67E+0 3 5.67E+0 3 2.85E+0 3 1.89E+0 3 2.17E+0 4 2.10E+0 4 2.10E+0 4 1.41E+0 4 1.03E+0 4 1.03E+0 4 1.27E+0 4
83
12.104
0.874
0.106
85
12.104
0.874
0.106
87
12.136
0.726
0.087
89
12.136
0.726
0.087
91
12.262
0.446
0.052
93
12.318
0.858
0.099
95
12.467
0.852
0.095
97
12.508
0.424
0.047
99
12.625
1.249
0.133
12.667
1.248
0.132
12.778
0.866
0.089
12.778
0.866
0.089
12.855
1.242
0.126
12.995
1.579
0.155
13.050
0.472
0.046
13.208
0.496
0.046
13.208
0.496
0.046
13.346
0.626
0.057
13.348
0.731
0.066
13.348
0.731
0.066
13.430
0.672
0.060
13.438
0.523
0.046
10 1 10 3 10 5 10 7 10 9 11 1 11 3 11 5 11 7 11 9 12 1 12 3 12 5
27
4.02E+0 3 4.02E+0 3 5.92E+0 3 5.92E+0 3 1.67E+0 4 4.63E+0 3 5.05E+0 3 2.08E+0 4 2.53E+0 3 2.59E+0 3 5.67E+0 3 5.67E+0 3 2.85E+0 3 1.89E+0 3 2.17E+0 4 2.10E+0 4 2.10E+0 4 1.41E+0 4 1.03E+0 4 1.03E+0 4 1.27E+0 4 2.10E+0 4
12 6 12 8 13 0 13 2 13 4 13 6 13 8 14 0 14 2 14 4 14 6 14 8 15 0 15 2 15 4 15 6 15 8 16 0 16 2 16 4 16 6 16 8
13.438
0.523
0.046
13.854
0.862
0.070
13.854
0.862
0.070
13.913
0.573
0.046
13.937
1.229
0.098
13.937
1.229
0.098
14.177
0.413
0.031
14.177
0.413
0.031
14.280
0.795
0.059
14.280
0.795
0.059
14.439
0.447
0.032
14.527
0.453
0.032
14.612
0.875
0.060
14.859
0.971
0.064
14.859
0.971
0.064
14.940
1.527
0.098
14.940
1.527
0.098
14.945
0.806
0.052
14.985
0.861
0.055
14.996
1.556
0.099
14.996
1.556
0.099
15.007
0.508
0.032
2.10E+0 4 9.30E+0 3 9.30E+0 3 2.16E+0 4 4.74E+0 3 4.74E+0 3 4.65E+0 4 4.65E+0 4 1.31E+0 4 1.31E+0 4 4.42E+0 4 4.47E+0 4 1.24E+0 4 1.11E+0 4 1.11E+0 4 4.66E+0 3 4.66E+0 3 1.67E+0 4 1.49E+0 4 4.59E+0 3 4.59E+0 3 4.32E+0 4
12 7 12 9 13 1 13 3 13 5 13 7 13 9 14 1 14 3 14 5 14 7 14 9 15 1 15 3 15 5 15 7 15 9 16 1 16 3 16 5 16 7 16 9
13.854
0.862
0.070
13.854
0.862
0.070
13.913
0.573
0.046
13.937
1.229
0.098
13.937
1.229
0.098
14.177
0.413
0.031
14.177
0.413
0.031
14.280
0.795
0.059
14.280
0.795
0.059
14.439
0.447
0.032
14.527
0.453
0.032
14.612
0.875
0.060
14.859
0.971
0.064
14.859
0.971
0.064
14.940
1.527
0.098
14.940
1.527
0.098
14.945
0.806
0.052
14.985
0.861
0.055
14.996
1.556
0.099
14.996
1.556
0.099
15.007
0.508
0.032
15.086
0.899
0.056 28
9.30E+0 3 9.30E+0 3 2.16E+0 4 4.74E+0 3 4.74E+0 3 4.65E+0 4 4.65E+0 4 1.31E+0 4 1.31E+0 4 4.42E+0 4 4.47E+0 4 1.24E+0 4 1.11E+0 4 1.11E+0 4 4.66E+0 3 4.66E+0 3 1.67E+0 4 1.49E+0 4 4.59E+0 3 4.59E+0 3 4.32E+0 4 1.42E+0 4
17 0 17 2 17 4 17 6 17 8 18 0 18 2 18 4 18 6 18 8 19 0 19 2 19 4 19 6 19 8 20 0 20 2 20 4 20 6 20 8 21 0 21 2
15.086
0.899
0.056
15.152
0.825
0.051
15.285
1.285
0.077
15.330
0.467
0.028
15.330
0.467
0.028
15.403
0.485
0.029
15.403
0.485
0.029
15.410
0.477
0.028
15.410
0.477
0.028
15.431
0.428
0.025
15.431
0.428
0.025
15.614
0.344
0.019
15.614
0.344
0.019
15.990
0.588
0.031
16.015
0.886
0.046
16.059
0.726
0.038
16.059
0.726
0.038
16.083
0.359
0.019
16.083
0.359
0.019
16.253
0.573
0.029
16.407
1.656
0.081
16.473
0.488
0.023
1.42E+0 4 1.74E+0 4 7.55E+0 3 5.81E+0 4 5.81E+0 4 5.54E+0 4 5.54E+0 4 5.74E+0 4 5.74E+0 4 7.19E+0 4 7.19E+0 4 1.20E+0 5 1.20E+0 5 4.72E+0 4 2.10E+0 4 3.18E+0 4 3.18E+0 4 1.31E+0 5 1.31E+0 5 5.49E+0 4 6.95E+0 3 8.21E+0 4
17 1 17 3 17 5 17 7 17 9 18 1 18 3 18 5 18 7 18 9 19 1 19 3 19 5 19 7 19 9 20 1 20 3 20 5 20 7 20 9 21 1 21 3
15.152
0.825
0.051
15.285
1.285
0.077
15.330
0.467
0.028
15.330
0.467
0.028
15.403
0.485
0.029
15.403
0.485
0.029
15.410
0.477
0.028
15.410
0.477
0.028
15.431
0.428
0.025
15.431
0.428
0.025
15.614
0.344
0.019
15.614
0.344
0.019
15.990
0.588
0.031
16.015
0.886
0.046
16.059
0.726
0.038
16.059
0.726
0.038
16.083
0.359
0.019
16.083
0.359
0.019
16.253
0.573
0.029
16.407
1.656
0.081
16.473
0.488
0.023
16.496
0.732
0.035 29
1.74E+0 4 7.55E+0 3 5.81E+0 4 5.81E+0 4 5.54E+0 4 5.54E+0 4 5.74E+0 4 5.74E+0 4 7.19E+0 4 7.19E+0 4 1.20E+0 5 1.20E+0 5 4.72E+0 4 2.10E+0 4 3.18E+0 4 3.18E+0 4 1.31E+0 5 1.31E+0 5 5.49E+0 4 6.95E+0 3 8.21E+0 4 3.67E+0 4
21 4 21 6 21 8 22 0 22 2 22 4 22 6 22 8 23 0 23 2 23 4 23 6 23 8 24 0 24 2 24 4 24 6 24 8 25 0 25 2 25 4 25 6
16.496
0.732
0.035
16.496
0.732
0.035
16.513
0.722
0.034
16.513
0.722
0.034
16.528
0.789
0.038
16.560
0.570
0.027
16.678
0.937
0.043
16.678
0.937
0.043
16.704
0.577
0.027
16.736
1.099
0.050
16.736
1.099
0.050
16.736
1.099
0.050
16.736
1.099
0.050
16.787
1.124
0.051
16.787
1.124
0.051
16.787
1.124
0.051
16.787
1.124
0.051
16.796
0.476
0.022
16.796
0.476
0.022
16.812
0.738
0.033
17.368
1.208
0.050
17.551
0.840
0.033
3.67E+0 4 3.67E+0 4 3.80E+0 4 3.80E+0 4 3.20E+0 4 6.20E+0 4 2.40E+0 4 2.40E+0 4 6.38E+0 4 1.78E+0 4 1.78E+0 4 1.78E+0 4 1.78E+0 4 1.73E+0 4 1.73E+0 4 1.73E+0 4 1.73E+0 4 9.67E+0 4 9.67E+0 4 4.04E+0 4 1.84E+0 4 4.04E+0 4
21 5 21 7 21 9 22 1 22 3 22 5 22 7 22 9 23 1 23 3 23 5 23 7 23 9 24 1 24 3 24 5 24 7 24 9 25 1 25 3 25 5 25 7
16.496
0.732
0.035
16.513
0.722
0.034
16.513
0.722
0.034
16.528
0.789
0.038
16.560
0.570
0.027
16.678
0.937
0.043
16.678
0.937
0.043
16.704
0.577
0.027
16.736
1.099
0.050
16.736
1.099
0.050
16.736
1.099
0.050
16.736
1.099
0.050
16.787
1.124
0.051
16.787
1.124
0.051
16.787
1.124
0.051
16.787
1.124
0.051
16.796
0.476
0.022
16.796
0.476
0.022
16.812
0.738
0.033
16.934
1.320
0.058
17.551
0.840
0.033
17.551
0.840
0.033 30
3.67E+0 4 3.80E+0 4 3.80E+0 4 3.20E+0 4 6.20E+0 4 2.40E+0 4 2.40E+0 4 6.38E+0 4 1.78E+0 4 1.78E+0 4 1.78E+0 4 1.78E+0 4 1.73E+0 4 1.73E+0 4 1.73E+0 4 1.73E+0 4 9.67E+0 4 9.67E+0 4 4.04E+0 4 1.32E+0 4 4.04E+0 4 4.04E+0 4
25 8 26 0 26 2 26 4 26 6 26 8 27 0 27 2 27 4 27 6 27 8 28 0 28 2 28 4 28 6 28 8 29 0 29 2 29 4 29 6 29 8 30 0
17.551
0.840
0.033
17.810
0.450
0.017
17.810
0.450
0.017
17.810
0.450
0.017
17.810
0.450
0.017
17.824
0.590
0.022
17.824
0.590
0.022
17.832
0.475
0.018
17.832
0.475
0.018
18.091
0.884
0.032
18.111
1.258
0.046
18.184
1.539
0.055
18.275
0.526
0.019
18.275
0.526
0.019
18.291
0.713
0.025
18.291
0.713
0.025
18.300
0.462
0.016
18.371
0.502
0.017
18.371
0.502
0.017
18.394
0.874
0.030
18.538
1.122
0.038
18.639
0.908
0.030
4.04E+0 4 1.54E+0 5 1.54E+0 5 1.54E+0 5 1.54E+0 5 9.00E+0 4 9.00E+0 4 1.39E+0 5 1.39E+0 5 4.39E+0 4 2.18E+0 4 1.49E+0 4 1.31E+0 5 1.31E+0 5 7.19E+0 4 7.19E+0 4 1.72E+0 5 1.49E+0 5 1.49E+0 5 4.95E+0 4 3.15E+0 4 4.97E+0 4
25 9 26 1 26 3 26 5 26 7 26 9 27 1 27 3 27 5 27 7 27 9 28 1 28 3 28 5 28 7 28 9 29 1 29 3 29 5 29 7 29 9 30 1
17.810
0.450
0.017
17.810
0.450
0.017
17.810
0.450
0.017
17.810
0.450
0.017
17.824
0.590
0.022
17.824
0.590
0.022
17.832
0.475
0.018
17.832
0.475
0.018
18.091
0.884
0.032
18.111
1.258
0.046
18.184
1.539
0.055
18.275
0.526
0.019
18.275
0.526
0.019
18.291
0.713
0.025
18.291
0.713
0.025
18.300
0.462
0.016
18.371
0.502
0.017
18.371
0.502
0.017
18.394
0.874
0.030
18.538
1.122
0.038
18.639
0.908
0.030
18.678
0.487
0.016 31
1.54E+0 5 1.54E+0 5 1.54E+0 5 1.54E+0 5 9.00E+0 4 9.00E+0 4 1.39E+0 5 1.39E+0 5 4.39E+0 4 2.18E+0 4 1.49E+0 4 1.31E+0 5 1.31E+0 5 7.19E+0 4 7.19E+0 4 1.72E+0 5 1.49E+0 5 1.49E+0 5 4.95E+0 4 3.15E+0 4 4.97E+0 4 1.75E+0 5
30 2 30 4 30 6 30 8 31 0 31 2 31 4 31 6 31 8 32 0 32 2 32 4 32 6 32 8 33 0 33 2 33 4 33 6 33 8 34 0 34 2 34 4
18.678
0.487
0.016
18.678
0.487
0.016
18.800
1.089
0.035
18.934
1.784
0.056
18.934
1.784
0.056
18.935
1.038
0.033
18.998
0.871
0.027
19.042
0.564
0.018
19.042
0.564
0.018
19.243
0.531
0.016
19.279
0.585
0.018
19.279
0.585
0.018
19.373
0.838
0.025
19.476
0.883
0.026
19.570
0.993
0.028
19.597
0.353
0.010
19.619
1.097
0.031
19.619
1.097
0.031
19.626
0.580
0.016
19.797
0.718
0.020
19.797
0.718
0.020
20.051
0.383
0.010
1.75E+0 5 1.75E+0 5 3.63E+0 4 1.41E+0 4 1.41E+0 4 4.18E+0 4 6.05E+0 4 1.47E+0 5 1.47E+0 5 1.76E+0 5 1.47E+0 5 1.47E+0 5 7.35E+0 4 6.84E+0 4 5.56E+0 4 4.43E+0 5 4.63E+0 4 4.63E+0 4 1.66E+0 5 1.14E+0 5 1.14E+0 5 4.32E+0 5
30 3 30 5 30 7 30 9 31 1 31 3 31 5 31 7 31 9 32 1 32 3 32 5 32 7 32 9 33 1 33 3 33 5 33 7 33 9 34 1 34 3 34 5
18.678
0.487
0.016
18.800
1.089
0.035
18.908
0.331
0.011
18.934
1.784
0.056
18.934
1.784
0.056
18.935
1.038
0.033
19.042
0.564
0.018
19.042
0.564
0.018
19.090
0.847
0.026
19.243
0.531
0.016
19.279
0.585
0.018
19.279
0.585
0.018
19.373
0.838
0.025
19.570
0.993
0.028
19.570
0.880
0.025
19.619
1.097
0.031
19.619
1.097
0.031
19.626
0.580
0.016
19.797
0.718
0.020
19.797
0.718
0.020
19.947
0.680
0.018
20.051
0.383
0.010 32
1.75E+0 5 3.63E+0 4 4.06E+0 5 1.41E+0 4 1.41E+0 4 4.18E+0 4 1.47E+0 5 1.47E+0 5 6.59E+0 4 1.76E+0 5 1.47E+0 5 1.47E+0 5 7.35E+0 4 5.56E+0 4 7.08E+0 4 4.63E+0 4 4.63E+0 4 1.66E+0 5 1.14E+0 5 1.14E+0 5 1.33E+0 5 4.32E+0 5
34 6 34 8 35 0 35 2 35 4 35 6 35 8 36 0 36 2 36 4 36 6 36 8 37 0 37 2 37 4 37 6 37 8 38 0 38 2 38 4 38 6 38 8
20.139
0.705
0.019
20.142
0.507
0.013
20.142
0.507
0.013
20.162
0.703
0.018
20.198
0.492
0.013
20.198
0.492
0.013
20.203
0.416
0.011
20.203
0.416
0.011
20.291
0.509
0.013
20.291
0.509
0.013
20.381
0.899
0.023
20.381
0.899
0.023
20.381
1.453
0.037
20.381
1.453
0.037
20.441
0.779
0.020
20.450
0.805
0.020
20.702
0.369
0.009
20.882
0.621
0.015
21.082
1.000
0.023
21.182
0.456
0.010
21.232
0.552
0.012
21.232
0.552
0.012
1.31E+0 5 2.54E+0 5 2.54E+0 5 1.33E+0 5 2.73E+0 5 2.73E+0 5 3.83E+0 5 3.83E+0 5 2.63E+0 5 2.63E+0 5 8.66E+0 4 8.66E+0 4 3.31E+0 4 3.31E+0 4 1.17E+0 5 1.10E+0 5 5.65E+0 5 2.10E+0 5 8.57E+0 4 4.24E+0 5 2.93E+0 5 2.93E+0 5
34 7 34 9 35 1 35 3 35 5 35 7 35 9 36 1 36 3 36 5 36 7 36 9 37 1 37 3 37 5 37 7 37 9 38 1 38 3 38 5 38 7 38 9
20.142
0.507
0.013
20.142
0.507
0.013
20.162
0.703
0.018
20.198
0.492
0.013
20.198
0.492
0.013
20.203
0.416
0.011
20.203
0.416
0.011
20.291
0.509
0.013
20.291
0.509
0.013
20.381
0.899
0.023
20.381
0.899
0.023
20.381
1.453
0.037
20.381
1.453
0.037
20.441
0.779
0.020
20.450
0.805
0.020
20.522
0.403
0.010
20.702
0.369
0.009
20.943
0.632
0.015
21.082
1.000
0.023
21.182
0.456
0.010
21.232
0.552
0.012
21.232
0.552
0.012 33
2.54E+0 5 2.54E+0 5 1.33E+0 5 2.73E+0 5 2.73E+0 5 3.83E+0 5 3.83E+0 5 2.63E+0 5 2.63E+0 5 8.66E+0 4 8.66E+0 4 3.31E+0 4 3.31E+0 4 1.17E+0 5 1.10E+0 5 4.48E+0 5 5.65E+0 5 2.06E+0 5 8.57E+0 4 4.24E+0 5 2.93E+0 5 2.93E+0 5
39 0 39 2 39 4 39 6 39 8 40 0 40 2 40 4 40 6 40 8 41 0 41 2 41 4 41 6 41 8 42 0 42 2 42 4 42 6 42 8 43 0 43 2
21.232
0.552
0.012
21.232
0.552
0.012
21.236
0.853
0.019
21.244
0.463
0.010
21.244
0.463
0.010
21.272
0.564
0.013
21.272
0.564
0.013
21.272
0.564
0.013
21.272
0.564
0.013
21.279
0.426
0.009
21.279
0.426
0.009
21.498
1.591
0.034
21.505
0.648
0.014
21.580
0.378
0.008
21.616
0.716
0.015
21.797
0.737
0.015
21.797
0.737
0.015
21.983
0.725
0.015
22.116
0.852
0.017
22.378
0.523
0.010
22.508
0.468
0.009
22.679
0.799
0.015
2.93E+0 5 2.93E+0 5 1.23E+0 5 4.18E+0 5 4.18E+0 5 2.84E+0 5 2.84E+0 5 2.84E+0 5 2.84E+0 5 5.00E+0 5 5.00E+0 5 3.81E+0 4 2.30E+0 5 6.89E+0 5 1.94E+0 5 1.93E+0 5 1.93E+0 5 2.10E+0 5 1.57E+0 5 4.49E+0 5 5.80E+0 5 2.08E+0 5
39 1 39 3 39 5 39 7 39 9 40 1 40 3 40 5 40 7 40 9 41 1 41 3 41 5 41 7 41 9 42 1 42 3 42 5 42 7 42 9 43 1 43 3
21.232
0.552
0.012
21.232
0.552
0.012
21.244
0.463
0.010
21.244
0.463
0.010
21.272
0.564
0.013
21.272
0.564
0.013
21.272
0.564
0.013
21.272
0.564
0.013
21.279
0.426
0.009
21.279
0.426
0.009
21.382
0.862
0.019
21.498
1.591
0.034
21.505
0.648
0.014
21.580
0.378
0.008
21.616
0.716
0.015
21.797
0.737
0.015
21.797
0.737
0.015
22.116
0.852
0.017
22.175
0.712
0.014
22.401
0.472
0.009
22.589
0.444
0.008
22.679
0.799
0.015 34
2.93E+0 5 2.93E+0 5 4.18E+0 5 4.18E+0 5 2.84E+0 5 2.84E+0 5 2.84E+0 5 2.84E+0 5 5.00E+0 5 5.00E+0 5 1.26E+0 5 3.81E+0 4 2.30E+0 5 6.89E+0 5 1.94E+0 5 1.93E+0 5 1.93E+0 5 1.57E+0 5 2.29E+0 5 5.55E+0 5 6.57E+0 5 2.08E+0 5
43 4 43 6 43 8 44 0 44 2 44 4 44 6 44 8 45 0 45 2 45 4 45 6 45 8 46 0 46 2 46 4 46 6 46 8 47 0 47 2 47 4 47 6
22.810
0.744
0.013
22.924
0.472
0.008
22.924
0.472
0.008
23.005
0.755
0.013
23.144
0.577
0.010
23.351
0.454
0.008
23.556
0.890
0.015
23.571
0.817
0.013
23.571
0.817
0.013
23.710
0.526
0.008
23.839
0.892
0.014
23.979
0.619
0.010
23.999
1.160
0.018
24.209
0.899
0.014
24.209
0.899
0.014
24.209
0.874
0.013
24.209
0.874
0.013
24.259
0.616
0.009
24.466
0.499
0.007
24.480
0.473
0.007
24.754
0.688
0.010
24.890
0.537
0.007
2.49E+0 5 6.36E+0 5 6.36E+0 5 2.54E+0 5 4.51E+0 5 7.68E+0 5 2.11E+0 5 2.51E+0 5 2.51E+0 5 6.28E+0 5 2.25E+0 5 4.85E+0 5 1.39E+0 5 2.43E+0 5 2.43E+0 5 2.57E+0 5 2.57E+0 5 5.25E+0 5 8.40E+0 5 9.41E+0 5 4.75E+0 5 8.06E+0 5
43 5 43 7 43 9 44 1 44 3 44 5 44 7 44 9 45 1 45 3 45 5 45 7 45 9 46 1 46 3 46 5 46 7 46 9 47 1 47 3 47 5 47 7
22.810
0.744
0.013
22.924
0.472
0.008
22.924
0.472
0.008
23.005
0.755
0.013
23.144
0.577
0.010
23.351
0.454
0.008
23.571
0.817
0.013
23.571
0.817
0.013
23.710
0.526
0.008
23.789
0.884
0.014
23.839
0.892
0.014
23.999
1.160
0.018
24.209
0.899
0.014
24.209
0.899
0.014
24.209
0.874
0.013
24.209
0.874
0.013
24.259
0.616
0.009
24.451
0.459
0.007
24.480
0.473
0.007
24.710
0.517
0.007
24.890
0.537
0.007
25.163
0.543
0.007 35
2.49E+0 5 6.36E+0 5 6.36E+0 5 2.54E+0 5 4.51E+0 5 7.68E+0 5 2.51E+0 5 2.51E+0 5 6.28E+0 5 2.26E+0 5 2.25E+0 5 1.39E+0 5 2.43E+0 5 2.43E+0 5 2.57E+0 5 2.57E+0 5 5.25E+0 5 9.90E+0 5 9.41E+0 5 8.30E+0 5 8.06E+0 5 8.41E+0 5
47 8 48 0 48 2 48 4 48 6 48 8 49 0 49 2 49 4 49 6 49 8
25.163
0.543
0.007
25.413
0.610
0.008
25.413
0.610
0.008
25.602
0.487
0.006
26.005
0.554
0.007
26.055
0.422
0.005
26.117
0.893
0.011
26.771
0.867
0.010
27.443
0.431
0.004
28.347
0.704
0.007
28.743
0.459
0.004
8.41E+0 5 7.07E+0 5 7.07E+0 5 1.16E+0 6 9.82E+0 5 1.71E+0 6 3.89E+0 5 4.78E+0 5 2.25E+0 6 1.02E+0 6 2.62E+0 6
47 9 48 1 48 3 48 5 48 7 48 9 49 1 49 3 49 5 49 7 49 9
25.413
0.610
0.008
25.413
0.610
0.008
25.602
0.487
0.006
25.924
0.552
0.007
26.055
0.422
0.005
26.117
0.893
0.011
26.359
0.460
0.005
27.347
0.790
0.008
27.443
0.431
0.004
28.743
0.459
0.004
30.161
0.895
0.007
Reference and Notes: (1) Bian, H. T.; Wen, X. W.; Li, J. B.; Chen, H. L.; Han, S. Z.; Sun, X. Q.; Song, J. A.; Zhuang, W.; Zheng, J. R. Ion Clustering in Aqueous Solutions Probed with Vibrational Energy Transfer. Proc. Natl. Acad. Sci. U. S. A. 2011, 108, 4737-4742. (2) Lipari, G.; Szabo, A. Effect of Librational Motion on Fluorescence Depolarization and Nuclear Magnietic Resonance Relaxation in Macromolecules and Membranes. Biophys. J. 1980, 30, 489-506. (3) Kinosita Jr, K.; Kawato, S.; Ikegami, A. A Theory of Fluorescence Polarization Decay in Membranes. Biophysical journal 1977, 20, 289-305. (4) Soleillet, P. Sur Les Parametres Caracterisant La Polarisation Partielle De La Lumiere Dans Les Phenomenes De Fluorescence. Ann. Phys. 1929, 12, 23-97. (5) Dale, R. E.; Eisinger, J.; Blumberg, W. The Orientational Freedom of Molecular Probes. The Orientation Factor in Intramolecular Energy Transfer. Biophys. J. 1979, 26, 161-193. (6) Li, J. B.; Bian, H. T.; Chen, H. L.; Hoang, B.; Zheng, J. R. Ion Association in Aqueous Solutions Probed through Vibrational Energy Transfers among Cation, Anion and Water Molecules. J. Phys. Chem. B. 2012, 117, 4274-4283. (7) Chang, J. C. Monopole Effects on Electronic Excitation Interactions between Large Molecules. I. Application to Energy Transfer in Chlorophylls. The Journal of 36
7.07E+0 5 7.07E+0 5 1.16E+0 6 9.74E+0 5 1.71E+0 6 3.89E+0 5 1.55E+0 6 6.55E+0 5 2.25E+0 6 2.62E+0 6 9.18E+0 5
chemical physics 1977, 67, 3901. (8) Baumann, J.; Fayer, M. Excitation Transfer in Disordered Two-Dimensional and Anisotropic Three-Dimensional Systems: Effects of Spatial Geometry on Time-Resolved Observables. The Journal of Chemical Physics 1986, 85, 4087. (9) Yang, M.; Li, F.; Skinner, J. Vibrational Energy Transfer and Anisotropy Decay in Liquid Water: Is the Förster Model Valid? The Journal of Chemical Physics 2011, 135, 164505. (10) Bian, H. T.; Chen, H. L.; Li, J. B.; Wen, X. W.; Zheng, J. R. Nonresonant and Resonant Mode-Specific Intermolecular Vibrational Energy Transfers in Electrolyte Aqueous Solutions. J. Phys. Chem. A. 2011, 115, 11657-11664. (11) Knox, R. S.; van Amerongen, H. Refractive Index Dependence of the Forster Resonance Excitation Transfer Rate. J. Phys. Chem. B 2002, 106, 5289-5293. (12) Holstein, T.; Lyo, S.; Orbach, R. Phonon-Assisted Radiative Transfer. Physical Review B 1977, 16, 934. (13) Holstein, T.; Lyo, S.; Orbach, R. Excitation Transfer in Disordered Systems. In Laser Spectroscopy of Solids; Springer: New York, 1986; pp 39-82. (14) Natkaniec, I.; Smirnov, L.; Solov'ev, A. Ammonium Dynamics in the Ordered and Disordered Phases of K 1-x (NH 4 ) x SCN Solid Solutions. Physica B: Condensed Matter 1995, 213, 667-668.
37
Molecular Distances Determined with Resonant Vibrational Energy Transfers Hailong Chen,† Xiewen Wen,† Jiebo Li, and Junrong Zheng* Department of Chemistry, Rice University, 6100 Main Street, Houston, Texas 77005-1892, United States S Supporting Information *
ABSTRACT: In general, intermolecular distances in condensed phases at the angstrom scale are difficult to measure. We were able to do so by using the vibrational energy transfer method, an ultrafast vibrational analogue of Förster resonance energy transfer. The distances among SCN− anions in KSCN crystals and ion clusters of KSCN aqueous solutions were determined with the method. In the crystalline samples, the closest anion distance was determined to be 3.9 ± 0.3 Å, consistent with the XRD result. In the 1.8 and 1 M KSCN aqueous solutions, the anion distances in the ion clusters were determined to be 4.4 ± 0.4 Å. The clustered anion distances in aqueous solutions are very similar to the closest anion distance in the KSCN crystal but significantly shorter than the average anion distance (0.94−1.17 nm) in the aqueous solutions if ion clustering did not occur. The result suggests that ions in the strong electrolyte aqueous solutions can form clusters inside of which they have direct contact with each other.
1. INTRODUCTION Short-ranged transient intermolecular interactions, for example, guest/host interactions, ion/ion interactions, and ion/molecule bindings, play significant roles in chemistry and biology. Electronic energy transfer methods, for example, Förster resonance energy transfer (FRET), are routinely applied to measure the distance between two molecules in many of these interactions.1,2 In these methods, chromophores that can transfer and accept energy in the visible and near-IR frequency range are required to attach to molecules. The chromophores are typically at the size of 1−2 nm or larger, which is not only much larger than many of the intermolecular distances but also can perturb the molecular interactions under investigation. To probe molecular distances at the angstrom scale, what is needed is an ultrafast vibrational analogue of FRET of which the chromophores are simple chemical bonds (1−2 Å). Here, by measuring ultrafast vibrational energy transfers in model systems of KSCN (potassium thiocyanate) crystals and aqueous solutions, we demonstrate that angstrom molecular distances in condensed phases can be determined. In this paper, we first present anisotropy decay and vibrational energy exchange data to show that vibrational energy can transfer resonantly among SCN− anions and nonresonantly from SCN− to S13C15N− with much slower rates in KSCN/KS13C15N mixed © 2014 American Chemical Society
crystals. A kinetic model is then developed to quantitatively analyze the resonant energy transfer data to obtain the resonant energy transfer time between two adjacent anions in the crystals. To convert the energy transfer time into distance, an equation based on the dephasing mechanism and the timedependent Schrodinger equation is derived to correlate the energy transfer time with the energy donor/acceptor coupling strength, which is quantitatively correlated to the donor/ acceptor distance under the dipole/dipole interaction. On the basis of the energy transfer equation and the dipole/dipole interaction equation and the measured energy transfer time, the distance between the two adjacent anions is calculated. The calculated distance is then compared to the distance measured by XRD. Within experimental uncertainty, the two values are the same. The uncertainty introduced by the point dipole assumption used in the method is then calculated based on the monopole theory. It is found that the uncertainty is very small, only ∼1.3%, mainly because the donor/acceptor distance is more than three times larger than the sizes of the donor/ acceptor. After benchmarking the vibrational energy transfer Received: January 17, 2014 Revised: March 17, 2014 Published: March 18, 2014 2463
dx.doi.org/10.1021/jp500586h | J. Phys. Chem. A 2014, 118, 2463−2469
The Journal of Physical Chemistry A
Article
measurements, the nitrile stretch of one anion, for example, SCN−, is excited to its first excited state. The decay of the anisotropy value of this vibrational excitation signal is then monitored in real time. Simultaneous analyses on the anisotropy decays in samples with different mixed KSCN/ KS13C15N ratios quantitatively yield both the resonant energy transfer rate constant and the rotational time constant of the anion. 2.3. Samples. Unless specified, chemicals were purchased from Sigma−Aldrich and used without further purification. KS13C15N was purchased from Cambridge Isotope Laboratory. The samples were thin films of polycrystalline KSCN/ KS13C15N mixed crystals with different molar ratios blended with ∼50 wt % PMMA. The thickness of the sample is estimated to be a few hundred nm based on the CN stretch optical density. An optical image of a sample is provided in Figure S1 in the Supporting Information (SI). The function of PMMA was to suppress scattered light. The samples were placed in a vacuum chamber during measurements. There are four reasons to use KSCN and KS13C15N and KS13CN as model systems: (1) the vibrational lifetimes of the nitrile stretches are relatively long (longer than 30 ps in D2O solutions and even longer in the crystals as included in the SI);4 (2) the transition dipole moments of the nitrile stretches are relatively large (0.3−0.4 D);4 (3) the distance between any two anions in the KSCN crystal is well-characterized with XRD and neutron scattering, which can be used to benchmark the distance determined by the vibrational energy transfer method;7 and (4) the phonon dispersion in the KSCN crystal was wellcharacterized before.8,9 The purpose of using different-isotope-labeled thiocyanate anion mixed samples, for example, a KSCN/KS13C15N mixed crystal with different molar ratios, is 2-fold, (1) to shift the vibrational frequency of the nitrile stretch as the vibrational frequency of the CN stretch is different from that of the 13C15N stretch and (2) to change the number of resonant energy transfer acceptors in the samples without changing the crystalline structure or ion cluster structure because the isotope labeling has negligible effects on the intermolecular interaction between two thiocyanate anions that determines their distance and relative orientation.10
method with the crystalline sample, the method is used to determine the anion distance in the ion clusters of KSCN aqueous solutions. It is found that within experimental uncertainty, the anion distance in the clusters is the same as that in the crystal.
2. EXPERIMENTS AND METHODS 2.1. Laser System. A ps amplifier and a fs amplifier are synchronized with the same seed pulse. The ps amplifier pumps an OPA to produce ∼0.8 ps (vary from 0.7−0.9 ps in different frequencies) mid-IR pulses with a bandwidth of 10−35 cm−1 in a tunable frequency range from 400 to 4000 cm−1 with an energy of 1−40 μJ/pulse (1−10 μJ/pulse for 400−900 cm−1 and >10 μJ/pulse for higher frequencies) at 1 kHz. Light from the fs amplifier is used to generate a high-intensity mid-IR and terahertz supercontinuum pulse with a duration of > 2 ), eqs.S30~32 can be
numerically solved with
p0 (t ) ≈ ∏ j
−2 k0 j t
1+ e 2
−2 k0 j 't
1+ e ∏j ' 2
−2 k0 j t
1 1 1+ e p|| (t ) ≈ + e −2 k⊥t − ∏ 2 2 2 j
p⊥ (t )=
1 1 −2 k⊥t − e , 2 2
eq.S33 −2 k0 j 't
1+ e ∏j ' 2
eq.S34
eq.S35
where kij and kij ' are correlated to each other through eqs.13&14 in the main text.
14
Calculations of clustered anion distances in 1M and 1.8M KSCN aqueous solutions
µ= 0.33D , We have the parameters: n = 1.5 , µ= D A
κ = 2 / 3 (randomized,
because the rotation of anion is faster than the energy transfer) and the dephasing time ( τ=
τ = 0.29 ps
(τ =
1 100 ps ) 2p 18 × 3
for
the
1.8M
solution
and
τ = 0.27 ps
1 100 ps ) for the 1M solution from the energy mismatch dependent 2p 20 × 3
experiments (fig.S6~7) 1,10. For the 1.8 M solution 1 1 2 τ = k β= 2 −2 (2 β ) + τ 18 ps
β=
1 µD µ A κ 3 n 2 4πε 0 rDA
⇒β = 2.3 cm−1
⇒ rDA = 4.4 A
For the 1M solution 1 1 2 τ k β= = 2 −2 (2 β ) + τ 15 ps
β=
1 µD µ A κ 3 n 2 4πε 0 rDA
⇒β = 2.5 cm−1
⇒ rDA = 4.3 A
If n = 1.42 is used (averaging the refractive indexes of D 2 O and KSCN), rDA = 4.5 A are for the 1.8M and 1M solutions respectively. and rDA = 4.4 A
15
0.014
1.0
Normalized Population
Normalized Population
0.012
KSCN KS13C15N
0.8
0.6
(A)
0.4
0.2
0.0
Flowing down Pumping up
0.010 0.008 0.006
(B)
0.004 0.002 0.000 -0.002
0
50
100
150
200
250
0
Waiting Time (ps)
50
100
150
200
250
Waiting Time (ps)
1.2
Normalized Anisotropy
1.0
2:98 50:50 100:0
0.8 0.6 0.4
(C)
0.2 0.0 -0.2 0
10
20
30
40
50
Waiting Time (ps)
Figure S6. Data and calculations of nonresonant [(A) and (B)] energy transfer between SCN- and S13C15N- (SCN-/S13C15N-=1/1) and resonant [(C)] energy transfers among SCN- anions for a 1.8M KSCN aqueous solution from our previous publication.1 Dots are data, and lines are calculations. Calculations for (A) and (B) are with input parameters: (ps −1 ); kSCN − slow 1/ 22 (ps −1 ); kS13C15 N − fast 1/ 2.7 (ps −1 ); kS13C15 N − slow 1/ 29 (ps −1 ); kSCN − fast 1/1.4 = = = = kclu →iso
1/10 (ps −1 ); K=0.55; kSCN − → S13C15 N − 1/160 (ps −1 ); D=0.70 =
with pre-factors of the subgroups and offset of the bi-exponential = ASCN − fast 0.16; = ASCN − slow 0.84;= AS13C15 N − fast 0.22; = AS13C15 N − slow 0.78; = offset 0 . (C) τ or = 4.5 ps is experimentally determined, which is the rotation time of the clustered ions. ntot = 4 , τ = 18 ps . Therefore, for the same number of acceptors, the resonant transfer time is 18/2=9ps, and that for ∆ω = 75 cm −1 is 160ps. Based on eq.9, the dephasing width is determined to be 15.5 cm-1.
16
0.008
KSCN KS13C15N
0.8
Normalized Population
Normalized Population
1.0
0.6
(A)
0.4
0.2
Flowing down Pumping up
0.006
0.004
(B)
0.002
0.000
0.0 -0.002 0
50
100
150
200
250
0
Waiting Time (ps)
50
100
150
200
250
Waiting Time (ps)
1.0
10:90 50:50 100:0
Normalized Anisotropy
0.8
0.6
0.4
(C)
0.2
0.0
-0.2 0
10
20
30
40
50
Waiting Time (ps)
Figure S7. Data and calculations of nonresonant [(A) and (B)] energy transfer between SCN- and S13C15N- (SCN-/S13C15N-=1/1) and resonant [(C)] energy transfers among SCN- anions for a 1.0M KSCN aqueous solution from our previous publication.1 Dots are data, and lines are calculations. Calculations for (A) and (B) are with input parameters: = = = = k SCN − fast 1/1.7 (ps −1 ); k SCN − slow 1/ 21 (ps −1 ); k S13C15 N − fast 1/1.6 (ps −1 ); kS13C15 N − slow 1/ 28 (ps −1 ); kclu →iso
= 1/10 (ps −1 ); K=0.38; kSCN − → S13C15 N − 1/180 (ps −1 ); D=0.70
with pre-factors of the subgroups and offset of the bi-exponential = = = ASCN − fast 0.25; ASCN − slow 0.75;= AS13C15 N − fast 0.21; = AS13C15 N − slow 0.79; offset 0 . (C) τ or = 4.3 ps is experimentally determined, which is the rotation time of the clustered ions. ntot = 3 , τ = 15 ps . Therefore, for the same number of acceptors, the resonant transfer time is 15/1.5=10ps, and that for ∆ω = 75 cm −1 is 180ps. Based on eq.9, the dephasing width is determined to be 15.5 cm-1.
17
0.5
Model
Lorentz
Equation
y = y0 + (2*A/PI)*(w/(4*(x-xc)^ 2 + w^2))
Value
D
Standard Error
y0
-0.01231
4.53192E-4
xc
2063.54727
0.05117
Lorentz
Equation
y = y0 + (2*A/PI)*(w/(4*(x-xc)^ 2 + w^2))
w
32.36957
0.1768
A
23.27113
0.10815
H
0.45768
(A)
0.2
1.58653E-4
Reduced Chi-Sqr
0.4
0.99093
Adj. R-Square
Value
0.3 OD
OD
Model
0.99164
Adj. R-Square
0.3
0.5
1.10401E-4
Reduced Chi-Sqr
0.4
D
(B)
Standard Error
y0
-0.02054
5.38998E-4
xc
2063.76032
0.05248
w
31.73273
0.18054
A
26.40325
0.12735
H
0.5297
0.2
0.1
0.1
0.0
0.0
1900 1950 2000 2050 2100 2150 2200 -1
Wavenumber (cm )
1900 1950 2000 2050 2100 2150 2200 Wavenumber (cm-1)
Figure S8. FTIR spectra of 1M and 1.8M KSCN in D 2 O solutions. The Lorentzian line width for both samples is 32 cm-1, giving the dephasing width 16 cm-1.
18
Calculations of the transition dipole moment of the CN stretch in KSCN crystal The transition dipole moment of the CN stretch was calculated by comparing the FTIR spectra and the 2D-IR signals for both of the CO stretch in the molecule (CH 3 C 5 H 4 )Mn(CO) 3 and the CN stretch in KSCN crystal. Under the same experimental conditions, e.g. the pump intensity, the signal intensities can be expressed as: 2 I FTIR −CO = C1µCO
eq.S36
2 I FTIR −CN = C1µCN
eq.S37
4 I 2 DIR −CO = C2 µCO
eq.S38
4 . I 2 DIR −CN = C2 µCN
eq.S39
It is based the fact that, FTIR measurement is a linear spectroscopy, while 2D-IR method measures the 3rd order nonlinear response. From eq.S36-S39, we have
/I I µCN = 2 DIR −CN 2 DIR −CO . I FTIR −CN / I FTIR −CO µCO
eq.S40
The transition dipole moment of the CO stretch was obtained by measuring the FTIR spectra of (CH 3 C 5 H 4 )Mn(CO) 3 in CCl 4 solution with the concentration 0.029 M and the path length 25 m m (fig.S9A), and using the expression = µ 2 9.186 ×10−3 n ∫ [ε (n ) / n ]dn ,
eq.S41
where µ is in the unit of Debye, ν is the wavenumber in cm −1 , εν ( ) is the molar decadic extinction coefficient in L/(mol cm) at wavenumberν , and n is the refractive index of the media
11
. For CCl 4 solution at around 2022 cm −1 , n = 1.44 .
From the FTIR data and these parameters, we got µCO = 0.45 D . The intensity ratios in eq. S40 were obtained from the FTIR and 2D-IR measurements for two samples. We got I FTIR −CN / I FTIR −CO = 0.48 / 0.44 (fig.S9A&B), and I 2 DIR −CN / I 2 DIR −CO = 0.032 / 0.067 for the intensity of red peak (fig.S10A&B)
I 2 DIR −CN / I 2 DIR −CO = 0.029 / 0.052 for the intensity of blue peak (fig.S10C&D). By inserting all these ratios into eq.S40, we finally got the transition dipole moment of the CN stretch µCN = 0.30 D (using red peak), or µCN = 0.32 D (using blue peak). As a result, we chose µCN = 0.31D for the calculations in this work. 19
0.5 0.4
0.5
(A)
0.4
0.3
(B)
OD
OD
0.3
0.2
0.2
0.1
0.1
0.0
0.0
1960 1980 2000 2020 2040 2060 2080
1900 1950 2000 2050 2100 2150 2200
Wavenumber (cm-1)
Wavenumber (cm-1)
Figure S9. FTIR spectra of (A) (CH 3 C 5 H 4 )Mn(CO) 3 in CCl 4 solution with the concentration 0.029 M and the path length 25 m m , and (B) the thin film of polycrystalline KSCN crystals blended with ~50 wt% PMMA.
Population
0.06
(A)
0.03
0.04
0.02
0.02
0.01
0.00
0.00 -20
0
20
40
60
0.02
-20
0
20
40
60
0
20
40
60
0.01
(C)
(D)
0.00
0.00
-0.02
-0.01
-0.02
-0.04
-0.06
(B)
-0.03 -20
0
20
40
60
-20
Time Delay (ps)
Figure S10. Waiting time dependent intensities of the (A) red peak for (CH 3 C 5 H 4 )Mn(CO) 3 solution, (B) red peak for KSCN crystal, (C) blue peak for (CH 3 C 5 H 4 )Mn(CO) 3 solution, and (D) blue peak for KSCN crystal, all of which were measured by 2D-IR method, with the pump at 2022 cm-1 for the (CH 3 C 5 H 4 )Mn(CO) 3 solution and 2050 cm-1 for the KSCN crystal.
20
The derivation of the relation between the one-way energy transfer rate and the total energy transfer rate For the energy transfer between a pair of donor and acceptor, without considering the vibrational relaxations, we have the rate equations
d D(t ) = −k DA D(t ) + k AD A(t ) dt
eq.S42
d A(t ) = −k AD A(t ) + k DA D(t ) , dt
eq.S43
where D(t ) and A(t ) are the populations of the exited state for the donor and acceptor, respectively, and k DA (or k AD ) is the energy transfer rate from the donor (or acceptor) to the acceptor (or the donor). For both of the donor and the acceptor are identical SCN- anions, we chose = k k DA + k AD to denote the one-way energy transfer rate. Considering the initial conditions, D(0) = 1 and A(0) = 0 . We get the solutions
= D(t )
1 − kt 1 e + 2 2
1 1 A(t ) = − e − kt + . 2 2 Therefore, the total energy transfer rate is = k k DA + k AD , and k DA = e
eq.S44
eq.S45 DωDA RT
k AD .
21
Energy transfer dephasing time of the KSCN crystalline samples The energy transfer dephasing time in eq.9 is a situation dependent parameter. If the dephasings of the donor and acceptor are uncorrelated, the energy transfer dephasing time (in terms of line width) is the convoluted line width of the donor and acceptor. For Lorentzian lineshapes, it is the sum of both donor and acceptor line widths, which determines the fastest value of possible energy transfer dephasing time. In many electronic energy transfers, such a dephasing time is assumed as the donor and the acceptor are several nm away so that it is reasonable to assume that the donor/acceptor dephasings are uncorrelated. For vibrational energy transfers, such an assumption may not be valid as the majority of experimentally measurable intermolecular vibrational energy transfers occur within distances smaller than 1 nm. Within such short distances, a molecular event which causes the dephasing of one molecule will inevitably affect the vibration of another molecule nearby. Therefore, the dephasings of the donor and the acceptor are correlated. The result of such a correlation is that the energy transfer dephasing time must be longer than that determined by the donor/acceptor convoluted line width. How long the energy transfer dephasing time can be is determined by how the dephasings of donor and acceptor are correlated. Experimentally, the energy transfer dephasing time can be obtained from the energy mismatch ∆ω dependent energy transfers based on eq.9. In principle, if we know the energy transfer rate constants for two energy mismatches, mathematically we can derive the coupling strength and the dephasing time from eq.9. The condition for such a treatment is that the nonresonant energy transfer through the dephasing mechanism (eq.9) is much faster than that of the phonon compensation mechanism 12,13. We expect that the condition can be fulfilled for most liquid solutions with relatively small energy mismatches ( ∆ω < RT ) where the dephasing is fast and well defined phonon motions are scarce. The condition can be invalid for some solid samples where the dephasing is relatively slow and the density of phonons with energy the same as the donor/acceptor mismatch is high.
22
w3 (cm-1)
(A) 50 ps
0 ps
2060
100 ps a
d 2020
b
c
1980 2000
2040
2080 -1
ω1 (cm )
1.0
1.0
(B)
(C)
0.8
Intensity (a.u.)
Intensity (a.u.)
0.8
0.6
0.4
0.2
0.6
0.4
0.2
0.0
0.0 0
200
400
600
Delay (ps)
800
1000
0
200
400
600
800
1000
Delay (ps)
Figure S11. (A):Waiting time dependent vibrational energy exchange 2D IR spectra of a KSCN/KS13CN=1/1 mixed crystal at room temperature. The growth of cross peaks indicates how fast the vibrational energy exchange proceeds between SCN- and S13CN-. (B)-(C) Waiting time dependent normalized intensities of peaks a, b, c, and d. Dots are experimental data, and curves are calculations based on the energy exchange kinetic model and experimentally measured vibrational lifetimes.Calculation parameters are pfa=0.081; pfb=0.022; kfa=1/1.05; kfb=1/0.73; ka=1/589; kb=1/688; kab=1/96; and kba=1/123. The KSCN crystalline sample with a mismatch 75 cm-1 is such an example. As analyzed in the main text, in the KSCN/KS13C15N=1/1 sample, the energy transfer
1 time constant ( ) from SCN to S13C15N ( ∆ω = 75 cm −1 ) is 99ps, and that from SCN k to SCN ( ∆ω = 0 cm −1 ) is 3.6ps. As shown in fig.S11, in the KSCN/KS13CN=1/1
1 sample, the energy transfer time constant ( ) from SCN to S13CN ( ∆ω = 50 cm −1 ) is k 96ps. The results with three different energy mismatches cannot be described by eq.9 as it predicts that the energy transfer with ∆ω = 50 cm −1 should be 100% faster than that with ∆ω = 75 cm −1 . The essential reason for the energy transfer from SCN to S13C15N is much faster than the prediction from eq.9 is that the measured energy transfer is from the phonon compensation mechanism. As we can see from both 23
Raman and Neutron scattering data, the phonon densities at ~ 75 cm-1 are much higher than those at around 50 cm-1. Therefore, the energy transfer dephasing time can only be calculated from the results of samples with ∆ω = 50 cm −1 and ∆ω = 0 cm −1 . Calculations show that the dephasing time is 8 cm-1 (in terms of line width), narrower than the convoluated line width (10 cm-1). However, 8 cm-1 is only the fast limit of the dephasing time, as we don’t know how much portion of the measured ∆ω = 50 cm −1 energy transfer is from the energy compensation mechanism. To test the uncertainty range of determined distance caused by this dephasing time uncertainty, we varied the dephasing time from 3 cm-1 to 8 cm-1, and found that the calculated distance between two closest anions varies from 4.4 to 3.9 angstroms. The determined values are within experimental uncertainty (~10% of the actual distance 4 angstroms). 1.0
KSCN/KS13C15N=1/1 RT KSCN/KS13CN=1/1 RT
200000 150000
0.8 G(E) (Normalized)
Intensity (counts)
250000
(A)
100000 50000 0
0.6
(B)
0.4 0.2 0.0
25
50
75
100
Frequency (cm-1)
125
150
50
100 E (cm-1)
150
Figure S12. (A) Raman spectra of KSCN/KS13C15N=1/1 and KSCN/KS13CN=1/1 samples at room temperature; (B) Neutron scattering data at 10K from literature 14.
24
Table of parameters of SCN- in KSCN crystal Table S2. Calculated parameters of every SCN- in KSCN crystal (3*3*3, 500 anions). The orientation factors were calculated based on eq.S9, with A∞ = 0.7 .The coupling
µ= 0.31D . Since the constants were calculated based on eq.10, with n = 1.5 , µ= D A energy transfer rate for the donor to each of the acceptors is proportional to
β 2 / [(2β ) 2 + τ −2 ] (see eq.9), the 1/k ET for every SCN- can be calculated by distributing the total energy transfer rate 1/1.8ps, which was obtained from experimental result, with τ −1 = 8 cm −1 . No . 0 2 4 6 8 10 12 14 16 18 20 22 24
Distanc e Orientatio (Å n factor ) 0.000 0.000 4.017 0.378 4.026 0.706 4.802 0.477 5.576 0.597 6.160 0.740 6.451 0.728 6.532 0.950 6.673 0.626 6.715 0.672 7.517 1.182 7.543 0.899 7.789 0.991
Coupling constant (cm-1)
1/k ET
No .
Distanc e (Å)
Orientatio n factor
Coupling constant (cm-1)
1/k ET
0.000 1.252 2.324 0.926 0.740 0.681 0.583 0.732 0.452 0.477 0.598 0.450 0.451
0 29 8 53 82 97 133 84 220 198 126 222 222
1 3 5 7 9 11 13 15 17 19 21 23 25
4.017 4.026 4.802 5.576 6.160 6.451 6.532 6.673 6.715 7.517 7.543 7.789 7.789
0.378 0.706 0.477 0.597 0.740 0.728 0.950 0.626 0.672 1.182 0.899 0.991 0.991
1.252 2.324 0.926 0.740 0.681 0.583 0.732 0.452 0.477 0.598 0.450 0.451 0.451
222
27
8.550
0.385
0.132
29
8.550
0.385
0.132
31
8.932
1.139
0.343
33
9.338
0.523
0.138
35
9.338
0.523
0.138
37
9.380
0.417
0.109
39
9.380
0.417
0.109
26
7.789
0.991
0.451
28
8.550
0.385
0.132
30
8.550
0.385
0.132
32
8.932
1.139
0.343
34
9.338
0.523
0.138
36
9.338
0.523
0.138
38
9.380
0.417
0.109
2.57E+0 3 2.57E+0 3 3.82E+0 2 2.37E+0 3 2.37E+0 3 3.82E+0
25
29 8 53 82 97 133 84 220 198 126 222 222 222 2.57E+0 3 2.57E+0 3 3.82E+0 2 2.37E+0 3 2.37E+0 3 3.82E+0 3 3.82E+0
40
9.380
0.417
0.109
42
9.467
1.784
0.452
44
9.467
1.784
0.452
46
9.507
1.054
0.263
48
9.731
0.932
0.217
50
9.861
0.844
0.189
52
10.071
0.507
0.107
54
10.071
0.507
0.107
56
10.099
0.492
0.103
58
10.099
0.492
0.103
60
10.315
1.253
0.245
62
10.467
0.715
0.134
64
10.761
0.755
0.130
66
11.012
0.990
0.159
68
11.150
0.528
0.082
70
11.150
0.528
0.082
72
11.399
0.333
0.048
74
11.620
1.146
0.157
76
11.699
0.763
0.102
78
11.911
0.449
0.057
80
12.104
0.874
0.106
3 3.82E+0 3 2.21E+0 2 2.21E+0 2 6.49E+0 2 9.54E+0 2 1.26E+0 3 3.96E+0 3 3.96E+0 3 4.27E+0 3 4.27E+0 3 7.49E+0 2 2.52E+0 3 2.66E+0 3 1.78E+0 3 6.74E+0 3 6.74E+0 3 1.93E+0 4 1.83E+0 3 4.30E+0 3 1.38E+0 4 4.02E+0 3
41
9.467
1.784
0.452
43
9.467
1.784
0.452
45
9.507
1.054
0.263
47
9.731
0.932
0.217
49
9.861
0.844
0.189
51
10.071
0.507
0.107
53
10.071
0.507
0.107
55
10.099
0.492
0.103
57
10.099
0.492
0.103
59
10.315
1.253
0.245
61
10.467
0.715
0.134
63
10.761
0.755
0.130
65
11.012
0.990
0.159
67
11.150
0.528
0.082
69
11.150
0.528
0.082
71
11.399
0.333
0.048
73
11.620
1.146
0.157
75
11.699
0.763
0.102
77
11.911
0.449
0.057
79
12.104
0.874
0.106
81
12.104
0.874
0.106
26
3 2.21E+0 2 2.21E+0 2 6.49E+0 2 9.54E+0 2 1.26E+0 3 3.96E+0 3 3.96E+0 3 4.27E+0 3 4.27E+0 3 7.49E+0 2 2.52E+0 3 2.66E+0 3 1.78E+0 3 6.74E+0 3 6.74E+0 3 1.93E+0 4 1.83E+0 3 4.30E+0 3 1.38E+0 4 4.02E+0 3 4.02E+0 3
82
12.104
0.874
0.106
84
12.104
0.874
0.106
86
12.104
0.874
0.106
88
12.136
0.726
0.087
90
12.136
0.726
0.087
92
12.262
0.446
0.052
94
12.318
0.858
0.099
96
12.467
0.852
0.095
98
12.508
0.424
0.047
12.625
1.249
0.133
12.667
1.248
0.132
12.778
0.866
0.089
12.778
0.866
0.089
12.855
1.242
0.126
12.995
1.579
0.155
13.050
0.472
0.046
13.208
0.496
0.046
13.208
0.496
0.046
13.346
0.626
0.057
13.348
0.731
0.066
13.348
0.731
0.066
13.430
0.672
0.060
10 0 10 2 10 4 10 6 10 8 11 0 11 2 11 4 11 6 11 8 12 0 12 2 12 4
4.02E+0 3 4.02E+0 3 4.02E+0 3 5.92E+0 3 5.92E+0 3 1.67E+0 4 4.63E+0 3 5.05E+0 3 2.08E+0 4 2.53E+0 3 2.59E+0 3 5.67E+0 3 5.67E+0 3 2.85E+0 3 1.89E+0 3 2.17E+0 4 2.10E+0 4 2.10E+0 4 1.41E+0 4 1.03E+0 4 1.03E+0 4 1.27E+0 4
83
12.104
0.874
0.106
85
12.104
0.874
0.106
87
12.136
0.726
0.087
89
12.136
0.726
0.087
91
12.262
0.446
0.052
93
12.318
0.858
0.099
95
12.467
0.852
0.095
97
12.508
0.424
0.047
99
12.625
1.249
0.133
12.667
1.248
0.132
12.778
0.866
0.089
12.778
0.866
0.089
12.855
1.242
0.126
12.995
1.579
0.155
13.050
0.472
0.046
13.208
0.496
0.046
13.208
0.496
0.046
13.346
0.626
0.057
13.348
0.731
0.066
13.348
0.731
0.066
13.430
0.672
0.060
13.438
0.523
0.046
10 1 10 3 10 5 10 7 10 9 11 1 11 3 11 5 11 7 11 9 12 1 12 3 12 5
27
4.02E+0 3 4.02E+0 3 5.92E+0 3 5.92E+0 3 1.67E+0 4 4.63E+0 3 5.05E+0 3 2.08E+0 4 2.53E+0 3 2.59E+0 3 5.67E+0 3 5.67E+0 3 2.85E+0 3 1.89E+0 3 2.17E+0 4 2.10E+0 4 2.10E+0 4 1.41E+0 4 1.03E+0 4 1.03E+0 4 1.27E+0 4 2.10E+0 4
12 6 12 8 13 0 13 2 13 4 13 6 13 8 14 0 14 2 14 4 14 6 14 8 15 0 15 2 15 4 15 6 15 8 16 0 16 2 16 4 16 6 16 8
13.438
0.523
0.046
13.854
0.862
0.070
13.854
0.862
0.070
13.913
0.573
0.046
13.937
1.229
0.098
13.937
1.229
0.098
14.177
0.413
0.031
14.177
0.413
0.031
14.280
0.795
0.059
14.280
0.795
0.059
14.439
0.447
0.032
14.527
0.453
0.032
14.612
0.875
0.060
14.859
0.971
0.064
14.859
0.971
0.064
14.940
1.527
0.098
14.940
1.527
0.098
14.945
0.806
0.052
14.985
0.861
0.055
14.996
1.556
0.099
14.996
1.556
0.099
15.007
0.508
0.032
2.10E+0 4 9.30E+0 3 9.30E+0 3 2.16E+0 4 4.74E+0 3 4.74E+0 3 4.65E+0 4 4.65E+0 4 1.31E+0 4 1.31E+0 4 4.42E+0 4 4.47E+0 4 1.24E+0 4 1.11E+0 4 1.11E+0 4 4.66E+0 3 4.66E+0 3 1.67E+0 4 1.49E+0 4 4.59E+0 3 4.59E+0 3 4.32E+0 4
12 7 12 9 13 1 13 3 13 5 13 7 13 9 14 1 14 3 14 5 14 7 14 9 15 1 15 3 15 5 15 7 15 9 16 1 16 3 16 5 16 7 16 9
13.854
0.862
0.070
13.854
0.862
0.070
13.913
0.573
0.046
13.937
1.229
0.098
13.937
1.229
0.098
14.177
0.413
0.031
14.177
0.413
0.031
14.280
0.795
0.059
14.280
0.795
0.059
14.439
0.447
0.032
14.527
0.453
0.032
14.612
0.875
0.060
14.859
0.971
0.064
14.859
0.971
0.064
14.940
1.527
0.098
14.940
1.527
0.098
14.945
0.806
0.052
14.985
0.861
0.055
14.996
1.556
0.099
14.996
1.556
0.099
15.007
0.508
0.032
15.086
0.899
0.056 28
9.30E+0 3 9.30E+0 3 2.16E+0 4 4.74E+0 3 4.74E+0 3 4.65E+0 4 4.65E+0 4 1.31E+0 4 1.31E+0 4 4.42E+0 4 4.47E+0 4 1.24E+0 4 1.11E+0 4 1.11E+0 4 4.66E+0 3 4.66E+0 3 1.67E+0 4 1.49E+0 4 4.59E+0 3 4.59E+0 3 4.32E+0 4 1.42E+0 4
17 0 17 2 17 4 17 6 17 8 18 0 18 2 18 4 18 6 18 8 19 0 19 2 19 4 19 6 19 8 20 0 20 2 20 4 20 6 20 8 21 0 21 2
15.086
0.899
0.056
15.152
0.825
0.051
15.285
1.285
0.077
15.330
0.467
0.028
15.330
0.467
0.028
15.403
0.485
0.029
15.403
0.485
0.029
15.410
0.477
0.028
15.410
0.477
0.028
15.431
0.428
0.025
15.431
0.428
0.025
15.614
0.344
0.019
15.614
0.344
0.019
15.990
0.588
0.031
16.015
0.886
0.046
16.059
0.726
0.038
16.059
0.726
0.038
16.083
0.359
0.019
16.083
0.359
0.019
16.253
0.573
0.029
16.407
1.656
0.081
16.473
0.488
0.023
1.42E+0 4 1.74E+0 4 7.55E+0 3 5.81E+0 4 5.81E+0 4 5.54E+0 4 5.54E+0 4 5.74E+0 4 5.74E+0 4 7.19E+0 4 7.19E+0 4 1.20E+0 5 1.20E+0 5 4.72E+0 4 2.10E+0 4 3.18E+0 4 3.18E+0 4 1.31E+0 5 1.31E+0 5 5.49E+0 4 6.95E+0 3 8.21E+0 4
17 1 17 3 17 5 17 7 17 9 18 1 18 3 18 5 18 7 18 9 19 1 19 3 19 5 19 7 19 9 20 1 20 3 20 5 20 7 20 9 21 1 21 3
15.152
0.825
0.051
15.285
1.285
0.077
15.330
0.467
0.028
15.330
0.467
0.028
15.403
0.485
0.029
15.403
0.485
0.029
15.410
0.477
0.028
15.410
0.477
0.028
15.431
0.428
0.025
15.431
0.428
0.025
15.614
0.344
0.019
15.614
0.344
0.019
15.990
0.588
0.031
16.015
0.886
0.046
16.059
0.726
0.038
16.059
0.726
0.038
16.083
0.359
0.019
16.083
0.359
0.019
16.253
0.573
0.029
16.407
1.656
0.081
16.473
0.488
0.023
16.496
0.732
0.035 29
1.74E+0 4 7.55E+0 3 5.81E+0 4 5.81E+0 4 5.54E+0 4 5.54E+0 4 5.74E+0 4 5.74E+0 4 7.19E+0 4 7.19E+0 4 1.20E+0 5 1.20E+0 5 4.72E+0 4 2.10E+0 4 3.18E+0 4 3.18E+0 4 1.31E+0 5 1.31E+0 5 5.49E+0 4 6.95E+0 3 8.21E+0 4 3.67E+0 4
21 4 21 6 21 8 22 0 22 2 22 4 22 6 22 8 23 0 23 2 23 4 23 6 23 8 24 0 24 2 24 4 24 6 24 8 25 0 25 2 25 4 25 6
16.496
0.732
0.035
16.496
0.732
0.035
16.513
0.722
0.034
16.513
0.722
0.034
16.528
0.789
0.038
16.560
0.570
0.027
16.678
0.937
0.043
16.678
0.937
0.043
16.704
0.577
0.027
16.736
1.099
0.050
16.736
1.099
0.050
16.736
1.099
0.050
16.736
1.099
0.050
16.787
1.124
0.051
16.787
1.124
0.051
16.787
1.124
0.051
16.787
1.124
0.051
16.796
0.476
0.022
16.796
0.476
0.022
16.812
0.738
0.033
17.368
1.208
0.050
17.551
0.840
0.033
3.67E+0 4 3.67E+0 4 3.80E+0 4 3.80E+0 4 3.20E+0 4 6.20E+0 4 2.40E+0 4 2.40E+0 4 6.38E+0 4 1.78E+0 4 1.78E+0 4 1.78E+0 4 1.78E+0 4 1.73E+0 4 1.73E+0 4 1.73E+0 4 1.73E+0 4 9.67E+0 4 9.67E+0 4 4.04E+0 4 1.84E+0 4 4.04E+0 4
21 5 21 7 21 9 22 1 22 3 22 5 22 7 22 9 23 1 23 3 23 5 23 7 23 9 24 1 24 3 24 5 24 7 24 9 25 1 25 3 25 5 25 7
16.496
0.732
0.035
16.513
0.722
0.034
16.513
0.722
0.034
16.528
0.789
0.038
16.560
0.570
0.027
16.678
0.937
0.043
16.678
0.937
0.043
16.704
0.577
0.027
16.736
1.099
0.050
16.736
1.099
0.050
16.736
1.099
0.050
16.736
1.099
0.050
16.787
1.124
0.051
16.787
1.124
0.051
16.787
1.124
0.051
16.787
1.124
0.051
16.796
0.476
0.022
16.796
0.476
0.022
16.812
0.738
0.033
16.934
1.320
0.058
17.551
0.840
0.033
17.551
0.840
0.033 30
3.67E+0 4 3.80E+0 4 3.80E+0 4 3.20E+0 4 6.20E+0 4 2.40E+0 4 2.40E+0 4 6.38E+0 4 1.78E+0 4 1.78E+0 4 1.78E+0 4 1.78E+0 4 1.73E+0 4 1.73E+0 4 1.73E+0 4 1.73E+0 4 9.67E+0 4 9.67E+0 4 4.04E+0 4 1.32E+0 4 4.04E+0 4 4.04E+0 4
25 8 26 0 26 2 26 4 26 6 26 8 27 0 27 2 27 4 27 6 27 8 28 0 28 2 28 4 28 6 28 8 29 0 29 2 29 4 29 6 29 8 30 0
17.551
0.840
0.033
17.810
0.450
0.017
17.810
0.450
0.017
17.810
0.450
0.017
17.810
0.450
0.017
17.824
0.590
0.022
17.824
0.590
0.022
17.832
0.475
0.018
17.832
0.475
0.018
18.091
0.884
0.032
18.111
1.258
0.046
18.184
1.539
0.055
18.275
0.526
0.019
18.275
0.526
0.019
18.291
0.713
0.025
18.291
0.713
0.025
18.300
0.462
0.016
18.371
0.502
0.017
18.371
0.502
0.017
18.394
0.874
0.030
18.538
1.122
0.038
18.639
0.908
0.030
4.04E+0 4 1.54E+0 5 1.54E+0 5 1.54E+0 5 1.54E+0 5 9.00E+0 4 9.00E+0 4 1.39E+0 5 1.39E+0 5 4.39E+0 4 2.18E+0 4 1.49E+0 4 1.31E+0 5 1.31E+0 5 7.19E+0 4 7.19E+0 4 1.72E+0 5 1.49E+0 5 1.49E+0 5 4.95E+0 4 3.15E+0 4 4.97E+0 4
25 9 26 1 26 3 26 5 26 7 26 9 27 1 27 3 27 5 27 7 27 9 28 1 28 3 28 5 28 7 28 9 29 1 29 3 29 5 29 7 29 9 30 1
17.810
0.450
0.017
17.810
0.450
0.017
17.810
0.450
0.017
17.810
0.450
0.017
17.824
0.590
0.022
17.824
0.590
0.022
17.832
0.475
0.018
17.832
0.475
0.018
18.091
0.884
0.032
18.111
1.258
0.046
18.184
1.539
0.055
18.275
0.526
0.019
18.275
0.526
0.019
18.291
0.713
0.025
18.291
0.713
0.025
18.300
0.462
0.016
18.371
0.502
0.017
18.371
0.502
0.017
18.394
0.874
0.030
18.538
1.122
0.038
18.639
0.908
0.030
18.678
0.487
0.016 31
1.54E+0 5 1.54E+0 5 1.54E+0 5 1.54E+0 5 9.00E+0 4 9.00E+0 4 1.39E+0 5 1.39E+0 5 4.39E+0 4 2.18E+0 4 1.49E+0 4 1.31E+0 5 1.31E+0 5 7.19E+0 4 7.19E+0 4 1.72E+0 5 1.49E+0 5 1.49E+0 5 4.95E+0 4 3.15E+0 4 4.97E+0 4 1.75E+0 5
30 2 30 4 30 6 30 8 31 0 31 2 31 4 31 6 31 8 32 0 32 2 32 4 32 6 32 8 33 0 33 2 33 4 33 6 33 8 34 0 34 2 34 4
18.678
0.487
0.016
18.678
0.487
0.016
18.800
1.089
0.035
18.934
1.784
0.056
18.934
1.784
0.056
18.935
1.038
0.033
18.998
0.871
0.027
19.042
0.564
0.018
19.042
0.564
0.018
19.243
0.531
0.016
19.279
0.585
0.018
19.279
0.585
0.018
19.373
0.838
0.025
19.476
0.883
0.026
19.570
0.993
0.028
19.597
0.353
0.010
19.619
1.097
0.031
19.619
1.097
0.031
19.626
0.580
0.016
19.797
0.718
0.020
19.797
0.718
0.020
20.051
0.383
0.010
1.75E+0 5 1.75E+0 5 3.63E+0 4 1.41E+0 4 1.41E+0 4 4.18E+0 4 6.05E+0 4 1.47E+0 5 1.47E+0 5 1.76E+0 5 1.47E+0 5 1.47E+0 5 7.35E+0 4 6.84E+0 4 5.56E+0 4 4.43E+0 5 4.63E+0 4 4.63E+0 4 1.66E+0 5 1.14E+0 5 1.14E+0 5 4.32E+0 5
30 3 30 5 30 7 30 9 31 1 31 3 31 5 31 7 31 9 32 1 32 3 32 5 32 7 32 9 33 1 33 3 33 5 33 7 33 9 34 1 34 3 34 5
18.678
0.487
0.016
18.800
1.089
0.035
18.908
0.331
0.011
18.934
1.784
0.056
18.934
1.784
0.056
18.935
1.038
0.033
19.042
0.564
0.018
19.042
0.564
0.018
19.090
0.847
0.026
19.243
0.531
0.016
19.279
0.585
0.018
19.279
0.585
0.018
19.373
0.838
0.025
19.570
0.993
0.028
19.570
0.880
0.025
19.619
1.097
0.031
19.619
1.097
0.031
19.626
0.580
0.016
19.797
0.718
0.020
19.797
0.718
0.020
19.947
0.680
0.018
20.051
0.383
0.010 32
1.75E+0 5 3.63E+0 4 4.06E+0 5 1.41E+0 4 1.41E+0 4 4.18E+0 4 1.47E+0 5 1.47E+0 5 6.59E+0 4 1.76E+0 5 1.47E+0 5 1.47E+0 5 7.35E+0 4 5.56E+0 4 7.08E+0 4 4.63E+0 4 4.63E+0 4 1.66E+0 5 1.14E+0 5 1.14E+0 5 1.33E+0 5 4.32E+0 5
34 6 34 8 35 0 35 2 35 4 35 6 35 8 36 0 36 2 36 4 36 6 36 8 37 0 37 2 37 4 37 6 37 8 38 0 38 2 38 4 38 6 38 8
20.139
0.705
0.019
20.142
0.507
0.013
20.142
0.507
0.013
20.162
0.703
0.018
20.198
0.492
0.013
20.198
0.492
0.013
20.203
0.416
0.011
20.203
0.416
0.011
20.291
0.509
0.013
20.291
0.509
0.013
20.381
0.899
0.023
20.381
0.899
0.023
20.381
1.453
0.037
20.381
1.453
0.037
20.441
0.779
0.020
20.450
0.805
0.020
20.702
0.369
0.009
20.882
0.621
0.015
21.082
1.000
0.023
21.182
0.456
0.010
21.232
0.552
0.012
21.232
0.552
0.012
1.31E+0 5 2.54E+0 5 2.54E+0 5 1.33E+0 5 2.73E+0 5 2.73E+0 5 3.83E+0 5 3.83E+0 5 2.63E+0 5 2.63E+0 5 8.66E+0 4 8.66E+0 4 3.31E+0 4 3.31E+0 4 1.17E+0 5 1.10E+0 5 5.65E+0 5 2.10E+0 5 8.57E+0 4 4.24E+0 5 2.93E+0 5 2.93E+0 5
34 7 34 9 35 1 35 3 35 5 35 7 35 9 36 1 36 3 36 5 36 7 36 9 37 1 37 3 37 5 37 7 37 9 38 1 38 3 38 5 38 7 38 9
20.142
0.507
0.013
20.142
0.507
0.013
20.162
0.703
0.018
20.198
0.492
0.013
20.198
0.492
0.013
20.203
0.416
0.011
20.203
0.416
0.011
20.291
0.509
0.013
20.291
0.509
0.013
20.381
0.899
0.023
20.381
0.899
0.023
20.381
1.453
0.037
20.381
1.453
0.037
20.441
0.779
0.020
20.450
0.805
0.020
20.522
0.403
0.010
20.702
0.369
0.009
20.943
0.632
0.015
21.082
1.000
0.023
21.182
0.456
0.010
21.232
0.552
0.012
21.232
0.552
0.012 33
2.54E+0 5 2.54E+0 5 1.33E+0 5 2.73E+0 5 2.73E+0 5 3.83E+0 5 3.83E+0 5 2.63E+0 5 2.63E+0 5 8.66E+0 4 8.66E+0 4 3.31E+0 4 3.31E+0 4 1.17E+0 5 1.10E+0 5 4.48E+0 5 5.65E+0 5 2.06E+0 5 8.57E+0 4 4.24E+0 5 2.93E+0 5 2.93E+0 5
39 0 39 2 39 4 39 6 39 8 40 0 40 2 40 4 40 6 40 8 41 0 41 2 41 4 41 6 41 8 42 0 42 2 42 4 42 6 42 8 43 0 43 2
21.232
0.552
0.012
21.232
0.552
0.012
21.236
0.853
0.019
21.244
0.463
0.010
21.244
0.463
0.010
21.272
0.564
0.013
21.272
0.564
0.013
21.272
0.564
0.013
21.272
0.564
0.013
21.279
0.426
0.009
21.279
0.426
0.009
21.498
1.591
0.034
21.505
0.648
0.014
21.580
0.378
0.008
21.616
0.716
0.015
21.797
0.737
0.015
21.797
0.737
0.015
21.983
0.725
0.015
22.116
0.852
0.017
22.378
0.523
0.010
22.508
0.468
0.009
22.679
0.799
0.015
2.93E+0 5 2.93E+0 5 1.23E+0 5 4.18E+0 5 4.18E+0 5 2.84E+0 5 2.84E+0 5 2.84E+0 5 2.84E+0 5 5.00E+0 5 5.00E+0 5 3.81E+0 4 2.30E+0 5 6.89E+0 5 1.94E+0 5 1.93E+0 5 1.93E+0 5 2.10E+0 5 1.57E+0 5 4.49E+0 5 5.80E+0 5 2.08E+0 5
39 1 39 3 39 5 39 7 39 9 40 1 40 3 40 5 40 7 40 9 41 1 41 3 41 5 41 7 41 9 42 1 42 3 42 5 42 7 42 9 43 1 43 3
21.232
0.552
0.012
21.232
0.552
0.012
21.244
0.463
0.010
21.244
0.463
0.010
21.272
0.564
0.013
21.272
0.564
0.013
21.272
0.564
0.013
21.272
0.564
0.013
21.279
0.426
0.009
21.279
0.426
0.009
21.382
0.862
0.019
21.498
1.591
0.034
21.505
0.648
0.014
21.580
0.378
0.008
21.616
0.716
0.015
21.797
0.737
0.015
21.797
0.737
0.015
22.116
0.852
0.017
22.175
0.712
0.014
22.401
0.472
0.009
22.589
0.444
0.008
22.679
0.799
0.015 34
2.93E+0 5 2.93E+0 5 4.18E+0 5 4.18E+0 5 2.84E+0 5 2.84E+0 5 2.84E+0 5 2.84E+0 5 5.00E+0 5 5.00E+0 5 1.26E+0 5 3.81E+0 4 2.30E+0 5 6.89E+0 5 1.94E+0 5 1.93E+0 5 1.93E+0 5 1.57E+0 5 2.29E+0 5 5.55E+0 5 6.57E+0 5 2.08E+0 5
43 4 43 6 43 8 44 0 44 2 44 4 44 6 44 8 45 0 45 2 45 4 45 6 45 8 46 0 46 2 46 4 46 6 46 8 47 0 47 2 47 4 47 6
22.810
0.744
0.013
22.924
0.472
0.008
22.924
0.472
0.008
23.005
0.755
0.013
23.144
0.577
0.010
23.351
0.454
0.008
23.556
0.890
0.015
23.571
0.817
0.013
23.571
0.817
0.013
23.710
0.526
0.008
23.839
0.892
0.014
23.979
0.619
0.010
23.999
1.160
0.018
24.209
0.899
0.014
24.209
0.899
0.014
24.209
0.874
0.013
24.209
0.874
0.013
24.259
0.616
0.009
24.466
0.499
0.007
24.480
0.473
0.007
24.754
0.688
0.010
24.890
0.537
0.007
2.49E+0 5 6.36E+0 5 6.36E+0 5 2.54E+0 5 4.51E+0 5 7.68E+0 5 2.11E+0 5 2.51E+0 5 2.51E+0 5 6.28E+0 5 2.25E+0 5 4.85E+0 5 1.39E+0 5 2.43E+0 5 2.43E+0 5 2.57E+0 5 2.57E+0 5 5.25E+0 5 8.40E+0 5 9.41E+0 5 4.75E+0 5 8.06E+0 5
43 5 43 7 43 9 44 1 44 3 44 5 44 7 44 9 45 1 45 3 45 5 45 7 45 9 46 1 46 3 46 5 46 7 46 9 47 1 47 3 47 5 47 7
22.810
0.744
0.013
22.924
0.472
0.008
22.924
0.472
0.008
23.005
0.755
0.013
23.144
0.577
0.010
23.351
0.454
0.008
23.571
0.817
0.013
23.571
0.817
0.013
23.710
0.526
0.008
23.789
0.884
0.014
23.839
0.892
0.014
23.999
1.160
0.018
24.209
0.899
0.014
24.209
0.899
0.014
24.209
0.874
0.013
24.209
0.874
0.013
24.259
0.616
0.009
24.451
0.459
0.007
24.480
0.473
0.007
24.710
0.517
0.007
24.890
0.537
0.007
25.163
0.543
0.007 35
2.49E+0 5 6.36E+0 5 6.36E+0 5 2.54E+0 5 4.51E+0 5 7.68E+0 5 2.51E+0 5 2.51E+0 5 6.28E+0 5 2.26E+0 5 2.25E+0 5 1.39E+0 5 2.43E+0 5 2.43E+0 5 2.57E+0 5 2.57E+0 5 5.25E+0 5 9.90E+0 5 9.41E+0 5 8.30E+0 5 8.06E+0 5 8.41E+0 5
47 8 48 0 48 2 48 4 48 6 48 8 49 0 49 2 49 4 49 6 49 8
25.163
0.543
0.007
25.413
0.610
0.008
25.413
0.610
0.008
25.602
0.487
0.006
26.005
0.554
0.007
26.055
0.422
0.005
26.117
0.893
0.011
26.771
0.867
0.010
27.443
0.431
0.004
28.347
0.704
0.007
28.743
0.459
0.004
8.41E+0 5 7.07E+0 5 7.07E+0 5 1.16E+0 6 9.82E+0 5 1.71E+0 6 3.89E+0 5 4.78E+0 5 2.25E+0 6 1.02E+0 6 2.62E+0 6
47 9 48 1 48 3 48 5 48 7 48 9 49 1 49 3 49 5 49 7 49 9
25.413
0.610
0.008
25.413
0.610
0.008
25.602
0.487
0.006
25.924
0.552
0.007
26.055
0.422
0.005
26.117
0.893
0.011
26.359
0.460
0.005
27.347
0.790
0.008
27.443
0.431
0.004
28.743
0.459
0.004
30.161
0.895
0.007
Reference and Notes: (1) Bian, H. T.; Wen, X. W.; Li, J. B.; Chen, H. L.; Han, S. Z.; Sun, X. Q.; Song, J. A.; Zhuang, W.; Zheng, J. R. Ion Clustering in Aqueous Solutions Probed with Vibrational Energy Transfer. Proc. Natl. Acad. Sci. U. S. A. 2011, 108, 4737-4742. (2) Lipari, G.; Szabo, A. Effect of Librational Motion on Fluorescence Depolarization and Nuclear Magnietic Resonance Relaxation in Macromolecules and Membranes. Biophys. J. 1980, 30, 489-506. (3) Kinosita Jr, K.; Kawato, S.; Ikegami, A. A Theory of Fluorescence Polarization Decay in Membranes. Biophysical journal 1977, 20, 289-305. (4) Soleillet, P. Sur Les Parametres Caracterisant La Polarisation Partielle De La Lumiere Dans Les Phenomenes De Fluorescence. Ann. Phys. 1929, 12, 23-97. (5) Dale, R. E.; Eisinger, J.; Blumberg, W. The Orientational Freedom of Molecular Probes. The Orientation Factor in Intramolecular Energy Transfer. Biophys. J. 1979, 26, 161-193. (6) Li, J. B.; Bian, H. T.; Chen, H. L.; Hoang, B.; Zheng, J. R. Ion Association in Aqueous Solutions Probed through Vibrational Energy Transfers among Cation, Anion and Water Molecules. J. Phys. Chem. B. 2012, 117, 4274-4283. (7) Chang, J. C. Monopole Effects on Electronic Excitation Interactions between Large Molecules. I. Application to Energy Transfer in Chlorophylls. The Journal of 36
7.07E+0 5 7.07E+0 5 1.16E+0 6 9.74E+0 5 1.71E+0 6 3.89E+0 5 1.55E+0 6 6.55E+0 5 2.25E+0 6 2.62E+0 6 9.18E+0 5
chemical physics 1977, 67, 3901. (8) Baumann, J.; Fayer, M. Excitation Transfer in Disordered Two-Dimensional and Anisotropic Three-Dimensional Systems: Effects of Spatial Geometry on Time-Resolved Observables. The Journal of Chemical Physics 1986, 85, 4087. (9) Yang, M.; Li, F.; Skinner, J. Vibrational Energy Transfer and Anisotropy Decay in Liquid Water: Is the Förster Model Valid? The Journal of Chemical Physics 2011, 135, 164505. (10) Bian, H. T.; Chen, H. L.; Li, J. B.; Wen, X. W.; Zheng, J. R. Nonresonant and Resonant Mode-Specific Intermolecular Vibrational Energy Transfers in Electrolyte Aqueous Solutions. J. Phys. Chem. A. 2011, 115, 11657-11664. (11) Knox, R. S.; van Amerongen, H. Refractive Index Dependence of the Forster Resonance Excitation Transfer Rate. J. Phys. Chem. B 2002, 106, 5289-5293. (12) Holstein, T.; Lyo, S.; Orbach, R. Phonon-Assisted Radiative Transfer. Physical Review B 1977, 16, 934. (13) Holstein, T.; Lyo, S.; Orbach, R. Excitation Transfer in Disordered Systems. In Laser Spectroscopy of Solids; Springer: New York, 1986; pp 39-82. (14) Natkaniec, I.; Smirnov, L.; Solov'ev, A. Ammonium Dynamics in the Ordered and Disordered Phases of K 1-x (NH 4 ) x SCN Solid Solutions. Physica B: Condensed Matter 1995, 213, 667-668.
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