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High pressure effect on the intramolecular electron transfer process
Chemical Physics Letters 378 (2003) 263–268 www.elsevier.com/locate/cplett
High pressure effect on the intramolecular electron transfer process Liming He a, Hong Li a, Jiufeng Fan a, Shayu Li a, Quan Gan a, Guoqi Zhang a, Baowen Zhang b, Yi Li b,*, Guoqiang Yang a,* a
Key Laboratory of Photochemistry, Institute of Chemistry, Chinese Academy of Sciences, Beijing 100080, China b Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Beijing, 100101, China Received 15 May 2003; in final form 4 August 2003 Published online:
Abstract An intramolecular electron transfer compound (4-(10-cyano-9-anthracenylmethyl)-N,N-dimethylaniline) has been studied under different pressure up to about 6 GPa. The electron transfer (ET) rate constants (ket ) with pressure have been obtained from its fluorescence emission and decay processes at different pressures. By analogy with the theory of optical transition, the free energy change (DG) and solvent reorganization energy (k) during the ET process can be expressed by the functions including the variable pressure. The experimental results can be understood well by analyzing the influence of DG and k on the ket with pressure. Ó 2003 Elsevier B.V. All rights reserved.
1. Introduction The process of electron transfer (ET) plays a major role in chemistry and biology, notably in photosynthesis, and has been studied in recent years [1–6]. Theories have predicted the normal region, where increasing the driving force of the reaction will increase its rate [7,8]. Both for cognitive reasons, the underlying physical mechanisms should be understood in detail. To gain a proper understanding of these processes, a wide range of equilibrium and dynamical effects need to be characterized. *
Corresponding authors. Fax: +861082617315. E-mail addresses: [email protected] (Y. Li), gqyang@iccas. ac.cn (G. Yang).
There are two commonly used experimental approaches to study various aspects of ET: by varying the temperature [9,10] and by changing the chemical composition of the solvent [11–13]. As an universal thermodynamic variable factor, pressure may offer an attractive experimental approach on the electron transfer process. Continuous tuning of the properties of the molecules over a wide range without changing its structure and chemical identity can be achieved in a controlled manner [14]. In this Letter, the pressure dependent fluorescence emission and lifetime of an intramolecular electron transfer compound, 4-(10-cyano-9-anthracenylmethyl)-N,N-dimethylaniline (CADMA) and its model compound 9-cyanoanthracence (the stereo-structures are shown in Scheme 1) blending
0009-2614/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0009-2614(03)01335-6
264
L. He et al. / Chemical Physics Letters 378 (2003) 263–268 1 at m 1. 95 G Pa 2. 8 G Pa 3. 62 GPa 4. 55 G Pa
200000
N 150000 100000
N N
C
C
Intensity(a.u.)
50000 0
1 at m 1. 31 G P a 2. 60 G P a 3. 70 G P a 4. 71 G P a 5. 95 G P a 6. 66 G P a
40000 30000
CADMA
Th e m o d e l c o m p o u n d
Scheme 1. The stereo-structure of CADMA and the model compound.
20000 10000 0
in PMMA are measured. The pressure dependence for the photo-induced electron transfer rate is investigated.
2. Experiment CADMA was synthesized in our lab and characterized by NMR, MS, IR, and elemental analysis. Polymethylmethacrylate (PMMA medium molecular weight) was used after reprecipitation. Spectral grade chloroform was used as the solvent to dissolve CADMA, 9-cyanoanthracence, and PMMA. The blend PMMA films of the samples were obtained according to the process described in another paper [15]. The films were transparent and with low enough concentration (for CADMA 0.067%, and the model 0.034%) without any phase separation. The instruments used in fluorescence detecting are described in another Letter [16]. For the lifetime measurements, LifeSpec-ps fluorescence lifetime analytical spectrometer (Edinburgh Instruments) was used and it was modified for the DAC. A pulsed diode laser with 403 nm (PDL 800B, Pico Quant) was used to excite the samples.
3. Results The typical pressure dependent fluorescence spectra of CADMA and the model compound are shown in Fig. 1.
350
400
450
500
550
600
Wavelength/nm
Fig. 1. Typic pressure dependent fluorescence spectra of CADMA in PMMA (up) and the model compound in PMMA (low).
At 1 atm, the spectrum of CADMA has three peaks like that of the model compound, but because of the substitute effect the peak location has 10 nm difference. With the increase of pressure the peak position has a red shift, about 20 nm when the pressure is around 5.0 GPa. The small red shift of the emission spectra with pressure implies the pressure will tune the energy parameter of the electronics state CADMA in a relative small range. The lifetimes of CADMA at different pressure are detected and presented in Table 1. At all the pressures the fluorescence decay of CADMA and the model compound in PMMA can be fitted with
Table 1 The lifetime of CADMA and the model in PMMA with pressures (averaged by two runs) Pressure (GPa) 0 1.0 2.0 3.0 4.0 5.0
s of CADMA (ns)
s of the model (ns)
10.66 10.48 10.33 10.16 10.00 9.80
13.24 12.50 12.03 11.93 11.92 11.81
L. He et al. / Chemical Physics Letters 378 (2003) 263–268
single exponential. Both of them decrease slightly with of pressure. The lifetime of CADMA is smaller than the model compound, which is the result of intramolecular electron transfer process as one of the channels for radiationless energy relaxation. According to the lifetimes and the relationship between the electron transfer rate constant (ket ) and the lifetimes, ket ¼ 1=s 1=smodel , the electron transfer rate constants of CADMA at different pressure are calculated and plotted against pressure, as shown in Fig. 2. It is observed that the rate constants are not beyond 108 . It indicates that the electron transfer is not a very fast process for the relaxation of the excited state. In general the favor interaction between the donor and acceptor through space requires parallel facing of the two groups. In the CADMA, however, the donor and the acceptor are not parallel but make an obtuse angle. So it is understandable that the conformation of this molecule will result in an unfavorable coupling and lead to the relatively slow charge separation (electron transfer) process. In Fig. 2, it is clearly illustrated that before 2.0 GPa the ET rate constant drops from about 1.83 107 to 1.37 107 s1 and then rise to about 1.74 107 s1 at the higher pressure. This implies a complicated influence of pressure on the ET related properties, such as free energy change, solvent reorganization energy, etc.
1.9
4. Discussion As we have known, the existing ET theories do not treat pressure effects in details. A general discussion by Jortner [17] and based on the theory of optical transitions under pressure by Drickamer and Frank [14] should be acknowledged. And some others have discussed the pressure dependence ET rates at kilo bar range [18–20]. By analogy with chemical reactions, the electron transfer rate constant in the high temperature limit (kB T > hm), for all relevant vibrations m) can be expressed by the Arrhenius law [7] ket ¼ ð4p3 =h2 kkB T Þ
1=2
jV j
1=2
expðGþ =kB T Þ;
ð1Þ
þ
where G , the free energy of activation, is given by [8] 2
Gþ ¼ ðk þ DGÞ =4k:
ð2Þ
In Eqs. (1) and (2), h is the PlankÕs constant, kB is the BoltzmanÕs constant, T is the temperature, DG is the free energy gap, i.e., the reaction driving force, k is the reorganization free energy of the solvent in the presence of reactants and products, V is the electronic coupling matrix element between initial and final states. Eq. (1) represents the physical situation in the nonadiabatic weak-coupling limit, which is valid in most cases. In this case, the ET (and not the nuclear relaxation) is a rate-limiting process, proceeding slowly at the crossing point of the reactant and product potential energy surfaces. The commonly used exponential form of the distance dependence of the electronic coupling matrix element has been accepted [8] V ¼ V0 exp½bðR R0 Þ=2;
1.7
ð3Þ
where V0 and R0 are, respectively, the electronic coupling strength and donor–acceptor distance at 1 atm, b is the scaling constant, if set a ¼ ðR R0 Þ=p, then
-7
ket x 10 / s
-1
1.8
265
1.6 1.5
V ¼ V0 exp½b a p=2;
1.4
ð4Þ
and
1.3 0
1
2
3
4
5
Pressure / GPa Fig. 2. The pressure dependent electron transfer rate constant of CADMA (averaged by two runs).
olnjV j=op / baR0 ;
ð5Þ
this implies that V increases with pressure because R is reduced with pressure. From the pressure
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L. He et al. / Chemical Physics Letters 378 (2003) 263–268
dependence ET rate constants of CADMA, it is clearly seen that the V does not influence ket efficiently for two reasons: (1) the value of ket is relative small (less than 108 ) in the pressure range applied, which means the relative weak coupling between donor and acceptor and pressure did not influence the interaction greatly. (2) The ket drops first and then rise but did not increase all along as V with pressures. In fact, the ket is controlled by thermodynamics factors like DG and Gþ . By analogy with the theory of optical transitions [14], we suppose that the effect of external pressure on the reactant (S1 state) and product (ET state) through free potential energy wells (UR ðxÞ and UP ðxÞ), respectively, a schematic diagram is presented in Fig. 3. We assume initially a single configuration coordinate x, and harmonic potential wells with different frequencies, KR and KP , for the S1 excited state and ET excited state, respectively. At 1 atm, the value of x corresponding to the minimum of the potential well of the ET state is displaced from that of the S1 state by x0 . The energy difference between the bottoms of the two wells is DG, which at 1 atm labeled DGð0Þ and at high pressure labeled DGðpÞ. The crossing point of the two potential wells is Gþ . Addition of an external pressure displaces the equilibrium values of x for both ground and excited states. This effect is like adding a gravitational force to the system of a mass on a spring. Therefore, the potential energy of the S1 state, UR to be
In the second term on the right of the equation, one assumes that the pressure works against an area A that is independent of pressure, and in fact it does not change in DAC. So it can be incorporated into p or x. Similarly, for the ET state one can write 2
UP ðxÞ ¼ 1=2KP ðx x0 Þ þ px þ DGð0Þ: At the potential minimum oUR =op ¼ 0 ¼ KR x þ p;
ð8Þ
so ð9Þ
x ¼ p=KR ; and oUP =op ¼ 0 ¼ KP ðx x0 Þ þ p;
ð10Þ
so x x0 ¼ p=KP :
ð11Þ
In this part of the analysis the KR and KP could be assumed independent of pressure. Then: ðUR Þmin ¼ p2 =2KR ;
ð12Þ
ðUP Þmin ¼ p2 =2KP þ px0 þ DGð0Þ: 0
ð6Þ
ð13Þ
0
If we let x ¼ x þ ðp=KR Þ, x ¼ 0 is then the bottom of the new S1 state (S01 ). Substituting in (12) and (13), we get UR ¼ ðUR Þmin þ 1=ð2KR x02 Þ;
ð14Þ
and 2
UR ðxÞ ¼ ð1=2ÞKR x2 þ px:
ð7Þ
UP ¼ ðUP Þmin þ 1=½2KP ðx0 x00 Þ ;
ð15Þ
where x00 ¼ x0 þ pð1=KR 1=KP Þ:
ð16Þ
Eq. (15) shows that we can eliminate the explicit pressure dependence by going to a new coordinate x0 and considering harmonic wells of altered energies and relative displacements in coordinate, but of unaltered spring constants. Then DG, the difference between the potential wells is DGðpÞ ¼ DGð0Þ þ px0 þ p2 ð1=KR 1=KP Þ=2: ð17Þ At 1 atm Fig. 3. The single coordinate model for ET.
k ¼ x02 0 KP =2:
ð18Þ
L. He et al. / Chemical Physics Letters 378 (2003) 263–268
At another pressure p, from Eqs. (16) and (18) 2
kðpÞ ¼ KP ½x0 þ pð1=KR 1=KP Þ =2:
ð19Þ
As we have known, ET state has little vibration fine structure and more broad energy distribution along the single configuration coordinate, and that means KP < KR and 1=KR 1=KP < 0. So we get: 0 < jð1=KR 1=KP Þj < 1; 0 < ð1=KR 1=KP Þ2 < 1;
ð20Þ 2
jð1=KR 1=KP Þj > ð1=KR 1=KP Þ :
ð21Þ
Therefore we can relate and compare the two conics Eqs. (17) and (19) in the same coordinate system, and that is shown in Fig. 4. In Fig. 4, before k and DG reach the turning point according to (17), (19):
In this case, because the target compound is an intramolecular ET system and is blended in solid PMMA, the configuration of the molecule is supposed to be hard to reorganize when electron transfer happens and so k can be supposed to be constant. And because of the weak coupling between donor and acceptor of CADMA all along the pressure range applied and the relative low pressure range, DG can be supposed to be adjusted in the normal region in the Marcus ET theory. According to Eqs. (1), (2) 2
Ket / exp½ðDG þ kÞ k ¼ constant:
ð25Þ
ð22Þ ð23Þ
4.2. The k descending with pressure
So jok=opj=oDG=op ¼ ð1 KP =KR Þ < 1:
4.1. The simplest mode (k does not change with pressure)
So ket has opposite change trend to DG, which implies ket will drop first and then rise with pressure and our experimental results just reflect this trend.
ok=op ¼ ½x0 þ pð1=KR 1=KP ÞðKP =KR 1Þ < 0;
oDG=op ¼ x0 þ ðð1=KR 1=KP Þp > 0:
267
ð24Þ
It is clear from Eq. (24) that k changes slower than DG. As it has been referred in Eq. (2), Gþ is controlled by DG and k together. When compressed, Gþ and ket will be analyzed and determined through the relative change of DG and k.
In another case, where k decreases with pressure (k changes along the descending part of conic with pressure, k1 > k2 ) when DG is tuned in normal region. So we get jDGj < k and DG þ k > 0:
ð26Þ
At any pressure p1 and p2 , set p1 < p2 then according to Eq. (24), so DG2 DG1 > k1 k2 ;
ð27Þ
and DG2 þ k2 > DG1 þ k1 ;
ð28Þ
because k1 > k2 , so 2
2
ðDG2 þ k2 Þ =4k2 > ðDG1 þ k1 Þ =4k1 ;
ð29Þ
according to (2), so þ Gþ 2 > G1 :
ð30Þ
When DG reaches the tuning point in normal region and according to our assumptions k still decreases with pressure Fig. 4. The schematic relationship between k and DG under pressure.
DG2 < DG1 and k2 < k1 ;
ð31Þ
268
L. He et al. / Chemical Physics Letters 378 (2003) 263–268
so þ DG2 þ k2 < DG1 þ k1 and Gþ 2 < G1 :
References ð32Þ
When k and V is insensitive to pressure, in Eq. (1) the exponential item will contribute much more than pre-exponential factors, which include k and V , and it is reasonable to ignore the contribution of k and V to ket . So ket / expðGþ Þ:
ð33Þ
Therefore from Eq. (30) before DG reaches turning point with the increase of pressure, ket2 < ket1 , which means ket drops with pressure and after DG reaches turning point, from Eqs. (32) and (33), ket2 > ket1 , it means ket rises with pressure. 5. Conclusion The fluorescence spectra and the decay lifetime of CADMA have been detected under different pressures up to about 6 Gpa. With the single configuration coordinate and by analogy with the theory of optical transition, the changing trend for the pressure dependence of electron transfer rate constant in CADMA is obtained and can well explain our experimental result. The rate constant drops at beginning with pressure and then rises up at higher pressures. The tuning of the free energy level and other thermodynamic factors give us a better understanding of electron transfer process at different pressures.
[1] R.D. Cannon, Electron Transfer Reactions, Butterworths, London, 1980. [2] M.D. Newton, N Sutin, Ann. Rev. Phys. Chem. 35 (1984) 437. [3] R.A. Marcus, N. Sutin, Biochim. Biophys. 811 (1985) 265. [4] M.R. Wasielewski, Chem. Rev. 92 (1992) 435. [5] G.L. Closs, J.R. Miller, Science 240 (1988) 440. [6] L.R. Ground, S. Farid, Acc. Chem. Res. 129 (1996) 522. [7] R.A. Marcus, J. Chem. Phys. 24 (1956) 966. [8] C.C. Moser, J.M. Keske, K. Warncke, R.S. Farid, P. Leslie Dutton, Nature 355 (1992) 796. [9] P. Finckh, H. Heitele, M. Volk, M.E. Michel-Beyerle, J. Phys. Chem. 92 (1998) 6584. [10] Z. Wang, J. Tang, J.R. Norris, J. Chem. Phys. 97 (10) (1992) 7251. [11] B.D. MacQueen, K.S. Schanze, J. Am. Chem. Soc. 113 (1991) 7470. [12] I.H.M. Van Stokkum, T. Scherer, A.M. Brouwer, J.W. Verhoeven, J. Phys. Chem. 98 (1994) 852. kesson, G.C. Walker, P.F.J. Barbara, Chem. Phys. 95 [13] E. A (6) (1991) 4188. [14] H.G. Drickamer, C.W. Frank, Electronic Transitions and the High Pressure Chemistry and Physics of Solid, Chapman and Hall, London, 1973. [15] G. Yang, Y. Li, J.O. White, H.G. Drickmaner, J. Phys. Chem. B 103 (1999) 7853. [16] X.F. Lu, H. Zhang, Y. Li, J. Liu, G.Q. Yang, Chem. Phys. Lett. 342 (2001) 545. [17] J. Jortner, Biochim. Biophys. Acta (1980) 594. [18] A. Freiberg, A. Ellervee, M. Tars, K. Timpmann, A. Laisaar, Biophys. Chem. 68 (1997) 189. [19] Kimihiko Hara, Trends Chem. Phys. (1997) 557. [20] N.L. Redline, M.W. Windsor, Chem. Phys. Lett. 198 (3–4) (1992) 334.
High pressure effect on the intramolecular electron transfer process Liming He a, Hong Li a, Jiufeng Fan a, Shayu Li a, Quan Gan a, Guoqi Zhang a, Baowen Zhang b, Yi Li b,*, Guoqiang Yang a,* a
Key Laboratory of Photochemistry, Institute of Chemistry, Chinese Academy of Sciences, Beijing 100080, China b Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Beijing, 100101, China Received 15 May 2003; in final form 4 August 2003 Published online:
Abstract An intramolecular electron transfer compound (4-(10-cyano-9-anthracenylmethyl)-N,N-dimethylaniline) has been studied under different pressure up to about 6 GPa. The electron transfer (ET) rate constants (ket ) with pressure have been obtained from its fluorescence emission and decay processes at different pressures. By analogy with the theory of optical transition, the free energy change (DG) and solvent reorganization energy (k) during the ET process can be expressed by the functions including the variable pressure. The experimental results can be understood well by analyzing the influence of DG and k on the ket with pressure. Ó 2003 Elsevier B.V. All rights reserved.
1. Introduction The process of electron transfer (ET) plays a major role in chemistry and biology, notably in photosynthesis, and has been studied in recent years [1–6]. Theories have predicted the normal region, where increasing the driving force of the reaction will increase its rate [7,8]. Both for cognitive reasons, the underlying physical mechanisms should be understood in detail. To gain a proper understanding of these processes, a wide range of equilibrium and dynamical effects need to be characterized. *
Corresponding authors. Fax: +861082617315. E-mail addresses: [email protected] (Y. Li), gqyang@iccas. ac.cn (G. Yang).
There are two commonly used experimental approaches to study various aspects of ET: by varying the temperature [9,10] and by changing the chemical composition of the solvent [11–13]. As an universal thermodynamic variable factor, pressure may offer an attractive experimental approach on the electron transfer process. Continuous tuning of the properties of the molecules over a wide range without changing its structure and chemical identity can be achieved in a controlled manner [14]. In this Letter, the pressure dependent fluorescence emission and lifetime of an intramolecular electron transfer compound, 4-(10-cyano-9-anthracenylmethyl)-N,N-dimethylaniline (CADMA) and its model compound 9-cyanoanthracence (the stereo-structures are shown in Scheme 1) blending
0009-2614/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0009-2614(03)01335-6
264
L. He et al. / Chemical Physics Letters 378 (2003) 263–268 1 at m 1. 95 G Pa 2. 8 G Pa 3. 62 GPa 4. 55 G Pa
200000
N 150000 100000
N N
C
C
Intensity(a.u.)
50000 0
1 at m 1. 31 G P a 2. 60 G P a 3. 70 G P a 4. 71 G P a 5. 95 G P a 6. 66 G P a
40000 30000
CADMA
Th e m o d e l c o m p o u n d
Scheme 1. The stereo-structure of CADMA and the model compound.
20000 10000 0
in PMMA are measured. The pressure dependence for the photo-induced electron transfer rate is investigated.
2. Experiment CADMA was synthesized in our lab and characterized by NMR, MS, IR, and elemental analysis. Polymethylmethacrylate (PMMA medium molecular weight) was used after reprecipitation. Spectral grade chloroform was used as the solvent to dissolve CADMA, 9-cyanoanthracence, and PMMA. The blend PMMA films of the samples were obtained according to the process described in another paper [15]. The films were transparent and with low enough concentration (for CADMA 0.067%, and the model 0.034%) without any phase separation. The instruments used in fluorescence detecting are described in another Letter [16]. For the lifetime measurements, LifeSpec-ps fluorescence lifetime analytical spectrometer (Edinburgh Instruments) was used and it was modified for the DAC. A pulsed diode laser with 403 nm (PDL 800B, Pico Quant) was used to excite the samples.
3. Results The typical pressure dependent fluorescence spectra of CADMA and the model compound are shown in Fig. 1.
350
400
450
500
550
600
Wavelength/nm
Fig. 1. Typic pressure dependent fluorescence spectra of CADMA in PMMA (up) and the model compound in PMMA (low).
At 1 atm, the spectrum of CADMA has three peaks like that of the model compound, but because of the substitute effect the peak location has 10 nm difference. With the increase of pressure the peak position has a red shift, about 20 nm when the pressure is around 5.0 GPa. The small red shift of the emission spectra with pressure implies the pressure will tune the energy parameter of the electronics state CADMA in a relative small range. The lifetimes of CADMA at different pressure are detected and presented in Table 1. At all the pressures the fluorescence decay of CADMA and the model compound in PMMA can be fitted with
Table 1 The lifetime of CADMA and the model in PMMA with pressures (averaged by two runs) Pressure (GPa) 0 1.0 2.0 3.0 4.0 5.0
s of CADMA (ns)
s of the model (ns)
10.66 10.48 10.33 10.16 10.00 9.80
13.24 12.50 12.03 11.93 11.92 11.81
L. He et al. / Chemical Physics Letters 378 (2003) 263–268
single exponential. Both of them decrease slightly with of pressure. The lifetime of CADMA is smaller than the model compound, which is the result of intramolecular electron transfer process as one of the channels for radiationless energy relaxation. According to the lifetimes and the relationship between the electron transfer rate constant (ket ) and the lifetimes, ket ¼ 1=s 1=smodel , the electron transfer rate constants of CADMA at different pressure are calculated and plotted against pressure, as shown in Fig. 2. It is observed that the rate constants are not beyond 108 . It indicates that the electron transfer is not a very fast process for the relaxation of the excited state. In general the favor interaction between the donor and acceptor through space requires parallel facing of the two groups. In the CADMA, however, the donor and the acceptor are not parallel but make an obtuse angle. So it is understandable that the conformation of this molecule will result in an unfavorable coupling and lead to the relatively slow charge separation (electron transfer) process. In Fig. 2, it is clearly illustrated that before 2.0 GPa the ET rate constant drops from about 1.83 107 to 1.37 107 s1 and then rise to about 1.74 107 s1 at the higher pressure. This implies a complicated influence of pressure on the ET related properties, such as free energy change, solvent reorganization energy, etc.
1.9
4. Discussion As we have known, the existing ET theories do not treat pressure effects in details. A general discussion by Jortner [17] and based on the theory of optical transitions under pressure by Drickamer and Frank [14] should be acknowledged. And some others have discussed the pressure dependence ET rates at kilo bar range [18–20]. By analogy with chemical reactions, the electron transfer rate constant in the high temperature limit (kB T > hm), for all relevant vibrations m) can be expressed by the Arrhenius law [7] ket ¼ ð4p3 =h2 kkB T Þ
1=2
jV j
1=2
expðGþ =kB T Þ;
ð1Þ
þ
where G , the free energy of activation, is given by [8] 2
Gþ ¼ ðk þ DGÞ =4k:
ð2Þ
In Eqs. (1) and (2), h is the PlankÕs constant, kB is the BoltzmanÕs constant, T is the temperature, DG is the free energy gap, i.e., the reaction driving force, k is the reorganization free energy of the solvent in the presence of reactants and products, V is the electronic coupling matrix element between initial and final states. Eq. (1) represents the physical situation in the nonadiabatic weak-coupling limit, which is valid in most cases. In this case, the ET (and not the nuclear relaxation) is a rate-limiting process, proceeding slowly at the crossing point of the reactant and product potential energy surfaces. The commonly used exponential form of the distance dependence of the electronic coupling matrix element has been accepted [8] V ¼ V0 exp½bðR R0 Þ=2;
1.7
ð3Þ
where V0 and R0 are, respectively, the electronic coupling strength and donor–acceptor distance at 1 atm, b is the scaling constant, if set a ¼ ðR R0 Þ=p, then
-7
ket x 10 / s
-1
1.8
265
1.6 1.5
V ¼ V0 exp½b a p=2;
1.4
ð4Þ
and
1.3 0
1
2
3
4
5
Pressure / GPa Fig. 2. The pressure dependent electron transfer rate constant of CADMA (averaged by two runs).
olnjV j=op / baR0 ;
ð5Þ
this implies that V increases with pressure because R is reduced with pressure. From the pressure
266
L. He et al. / Chemical Physics Letters 378 (2003) 263–268
dependence ET rate constants of CADMA, it is clearly seen that the V does not influence ket efficiently for two reasons: (1) the value of ket is relative small (less than 108 ) in the pressure range applied, which means the relative weak coupling between donor and acceptor and pressure did not influence the interaction greatly. (2) The ket drops first and then rise but did not increase all along as V with pressures. In fact, the ket is controlled by thermodynamics factors like DG and Gþ . By analogy with the theory of optical transitions [14], we suppose that the effect of external pressure on the reactant (S1 state) and product (ET state) through free potential energy wells (UR ðxÞ and UP ðxÞ), respectively, a schematic diagram is presented in Fig. 3. We assume initially a single configuration coordinate x, and harmonic potential wells with different frequencies, KR and KP , for the S1 excited state and ET excited state, respectively. At 1 atm, the value of x corresponding to the minimum of the potential well of the ET state is displaced from that of the S1 state by x0 . The energy difference between the bottoms of the two wells is DG, which at 1 atm labeled DGð0Þ and at high pressure labeled DGðpÞ. The crossing point of the two potential wells is Gþ . Addition of an external pressure displaces the equilibrium values of x for both ground and excited states. This effect is like adding a gravitational force to the system of a mass on a spring. Therefore, the potential energy of the S1 state, UR to be
In the second term on the right of the equation, one assumes that the pressure works against an area A that is independent of pressure, and in fact it does not change in DAC. So it can be incorporated into p or x. Similarly, for the ET state one can write 2
UP ðxÞ ¼ 1=2KP ðx x0 Þ þ px þ DGð0Þ: At the potential minimum oUR =op ¼ 0 ¼ KR x þ p;
ð8Þ
so ð9Þ
x ¼ p=KR ; and oUP =op ¼ 0 ¼ KP ðx x0 Þ þ p;
ð10Þ
so x x0 ¼ p=KP :
ð11Þ
In this part of the analysis the KR and KP could be assumed independent of pressure. Then: ðUR Þmin ¼ p2 =2KR ;
ð12Þ
ðUP Þmin ¼ p2 =2KP þ px0 þ DGð0Þ: 0
ð6Þ
ð13Þ
0
If we let x ¼ x þ ðp=KR Þ, x ¼ 0 is then the bottom of the new S1 state (S01 ). Substituting in (12) and (13), we get UR ¼ ðUR Þmin þ 1=ð2KR x02 Þ;
ð14Þ
and 2
UR ðxÞ ¼ ð1=2ÞKR x2 þ px:
ð7Þ
UP ¼ ðUP Þmin þ 1=½2KP ðx0 x00 Þ ;
ð15Þ
where x00 ¼ x0 þ pð1=KR 1=KP Þ:
ð16Þ
Eq. (15) shows that we can eliminate the explicit pressure dependence by going to a new coordinate x0 and considering harmonic wells of altered energies and relative displacements in coordinate, but of unaltered spring constants. Then DG, the difference between the potential wells is DGðpÞ ¼ DGð0Þ þ px0 þ p2 ð1=KR 1=KP Þ=2: ð17Þ At 1 atm Fig. 3. The single coordinate model for ET.
k ¼ x02 0 KP =2:
ð18Þ
L. He et al. / Chemical Physics Letters 378 (2003) 263–268
At another pressure p, from Eqs. (16) and (18) 2
kðpÞ ¼ KP ½x0 þ pð1=KR 1=KP Þ =2:
ð19Þ
As we have known, ET state has little vibration fine structure and more broad energy distribution along the single configuration coordinate, and that means KP < KR and 1=KR 1=KP < 0. So we get: 0 < jð1=KR 1=KP Þj < 1; 0 < ð1=KR 1=KP Þ2 < 1;
ð20Þ 2
jð1=KR 1=KP Þj > ð1=KR 1=KP Þ :
ð21Þ
Therefore we can relate and compare the two conics Eqs. (17) and (19) in the same coordinate system, and that is shown in Fig. 4. In Fig. 4, before k and DG reach the turning point according to (17), (19):
In this case, because the target compound is an intramolecular ET system and is blended in solid PMMA, the configuration of the molecule is supposed to be hard to reorganize when electron transfer happens and so k can be supposed to be constant. And because of the weak coupling between donor and acceptor of CADMA all along the pressure range applied and the relative low pressure range, DG can be supposed to be adjusted in the normal region in the Marcus ET theory. According to Eqs. (1), (2) 2
Ket / exp½ðDG þ kÞ k ¼ constant:
ð25Þ
ð22Þ ð23Þ
4.2. The k descending with pressure
So jok=opj=oDG=op ¼ ð1 KP =KR Þ < 1:
4.1. The simplest mode (k does not change with pressure)
So ket has opposite change trend to DG, which implies ket will drop first and then rise with pressure and our experimental results just reflect this trend.
ok=op ¼ ½x0 þ pð1=KR 1=KP ÞðKP =KR 1Þ < 0;
oDG=op ¼ x0 þ ðð1=KR 1=KP Þp > 0:
267
ð24Þ
It is clear from Eq. (24) that k changes slower than DG. As it has been referred in Eq. (2), Gþ is controlled by DG and k together. When compressed, Gþ and ket will be analyzed and determined through the relative change of DG and k.
In another case, where k decreases with pressure (k changes along the descending part of conic with pressure, k1 > k2 ) when DG is tuned in normal region. So we get jDGj < k and DG þ k > 0:
ð26Þ
At any pressure p1 and p2 , set p1 < p2 then according to Eq. (24), so DG2 DG1 > k1 k2 ;
ð27Þ
and DG2 þ k2 > DG1 þ k1 ;
ð28Þ
because k1 > k2 , so 2
2
ðDG2 þ k2 Þ =4k2 > ðDG1 þ k1 Þ =4k1 ;
ð29Þ
according to (2), so þ Gþ 2 > G1 :
ð30Þ
When DG reaches the tuning point in normal region and according to our assumptions k still decreases with pressure Fig. 4. The schematic relationship between k and DG under pressure.
DG2 < DG1 and k2 < k1 ;
ð31Þ
268
L. He et al. / Chemical Physics Letters 378 (2003) 263–268
so þ DG2 þ k2 < DG1 þ k1 and Gþ 2 < G1 :
References ð32Þ
When k and V is insensitive to pressure, in Eq. (1) the exponential item will contribute much more than pre-exponential factors, which include k and V , and it is reasonable to ignore the contribution of k and V to ket . So ket / expðGþ Þ:
ð33Þ
Therefore from Eq. (30) before DG reaches turning point with the increase of pressure, ket2 < ket1 , which means ket drops with pressure and after DG reaches turning point, from Eqs. (32) and (33), ket2 > ket1 , it means ket rises with pressure. 5. Conclusion The fluorescence spectra and the decay lifetime of CADMA have been detected under different pressures up to about 6 Gpa. With the single configuration coordinate and by analogy with the theory of optical transition, the changing trend for the pressure dependence of electron transfer rate constant in CADMA is obtained and can well explain our experimental result. The rate constant drops at beginning with pressure and then rises up at higher pressures. The tuning of the free energy level and other thermodynamic factors give us a better understanding of electron transfer process at different pressures.
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