Effects of Molecular Dynamics on Electrical Conductance of Single Molecular Junction in Aqueous Solution: First Principles Calculations

e-Journal of Surface Science and Nanotechnology 23 January 2010 e-J. Surf. Sci. Nanotech. Vol. 8 (2010) 38-43 Conference - ACSIN-10 -

Eﬀects of Molecular Dynamics on Electrical Conductance of Single Molecular Junction in Aqueous Solution: First Principles Calculations∗

Arihiro Tawara,† Tomofumi Tada, and Satoshi Watanabe Department of Materials Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan (Received 9 October 2009; Accepted 4 December 2009; Published 23 January 2010)

The electronic transport in benzene-1,4-dithiolate molecule in aqueous solution sandwiched between gold electrodes have been investigated by the ab initio nonequilibrium Green’s function method combined with Car- Parrinello molecular dynamics. We have found that the C–S bond length shows clear negative correlation with the conductance both in aqueous solution and vacuum, whereas the Au–S and C–C bond lengths have little correlation. This originates from large local density of states around C–S bonds at the Fermi level. [DOI: 10.1380/ejssnt.2010.38] Keywords: Electrical transport; Benzene dithiolate (BDT); Water; Density functional calculations; Nonequilibrium Green’s function method; Molecular dynamics

I. INTRODUCTION neither been understood enough nor explored in theoretical fashion so far except for our previous calculation [32]. Recently, electronic transport properties of single In our previous study [32], we calculated a large number molecular junctions have been increasingly important and of conductances of a system consisting of a BDT molecule actively investigated in experimental and theoretical stud- and water molecules sandwiched between Au(100) sur- ies to deepen our understanding of nanoscale devices on faces in order to clarify bare effects of aqueous solution single molecule level. Benzene-1,4-dithiolate (BDT) be- on conductance at room temperature, using the ab ini- tween gold electrodes has been studied extensively as a tio NEGF-DFT method and Car-Parrinello molecular dy- benchmark molecule both in experimental [1–4] and the- namics (CPMD). Analyses of conductance histograms re- oretical [5–13] studies. In particular, theoretical inves- vealed the effects of the aqueous solution on the conduc- tigations on various effects such as adsorption configura- tance of the BDT: the peak of the conductance histogram tion [14–18], inelastic current [19–23] and limitation of the shifts downward by 0.01-0.02 G0 (5-10 %), which is at- density functional theory (DFT) [24–27] deepened our un- tributed to the electrostatic effects of the aqueous solu- derstanding on the transport properties of molecular junction. We also found that subsidiary peaks in histograms tions. In experimental studies, conductance is mostly ob- emerge or disappear due to dynamical effects of the aque- served at room temperature and in solution, and the fluc- ous solution, which is correlated with the C–S stretching tuation of conductance is usually analyzed through his- mode of the BDT molecule. However, the analyses were tograms. Thus theoretical analyses based on conductance focused only on the C–S stretching, and the dynamical ef- histograms considering the fluctuations of the molecular fects induced by other modes have not been investigated junction are expected to provide a better understanding in the previous work [32]. In this paper, we assess the of the electronic transport through a single molecule. A correlations of conductance with Au–S and C–C lengths few such studies have already been reported for single as well as C–S length to understand the effects on con- molecular junctions in vacuum [28–30]. ductance in terms of the dynamical behavior of the BDT Several interesting studies on solution effects have also molecular junction further. been reported. In a mechanically controllable break junction experiment, Taniguchi et al. observed that the − Au−[Ni(dmit)2] −Au molecular junction shows almost II. COMPUTATIONAL METHOD the same conductance but different stability between the cases in a trichlorobenzene solution and in vacuum at Figure 1 shows the schematic of our computational room temperature [31]. Cao et al. studied the calculated model for the molecular junction adopted in our study, conductance histograms of a perylene tetracarboxylic di- where a BDT molecule is sandwiched between Au(100) imides molecule between gold electrodes in aqueous solu- surfaces. Fig. 1(a) and 1(b) show the unit cells of the sys- tion at different temperatures and showed a temperature tems with and without the aqueous solution (1.0 g/cm3, dependence of conductance due to water molecules [30]. 22 H2O), respectively. We abbreviate the systems in the However, much room for discussion is left and further in- aqueous solution and in vacuum as “Aq” and “Vac”. The vestigations are needed in the electronic transport proper- BDT is assumed to be adsorbed at the hollow site of ties affected by solution. Even for the simple BDT molec- Au(100) surface shown in Fig. 1(c) with the distance be- ular junction, the solution effects on conductance have tween electrodes of 9.8 A.˚ The computational model and procedures are identical to those given in our previous study [32]. Here, the computational procedures are described briefly. The procedure consists of three steps as ∗This paper was presented at 10th International Conference on follows. Atomically Controlled Surfaces, Interfaces and Nanostructures (ACSIN-10), Granada Conference Centre, Spain, 21-25 September, Step I: We perform CPMD calculations using the 2009. CPMD code [33, 34] to obtain a large number of configu- †Corresponding author: [email protected] rations in the Aq/Vac systems. Note that water and BDT

ISSN 1348-0391 °c 2010 The Surface Science Society of Japan (http://www.sssj.org/ejssnt) 38 e-Journal of Surface Science and Nanotechnology Volume 8 (2010)

(a) Aq (b) Vac 10 0.22 dcane( (G nce nducta o C

9 z 0.20 8

0.18

0.16

0.14

5 (Å) 0.12 z 4

0.10

0.08

2 0 )

0.06 1

0 0.04

0.0 0.5 1.0 1.5

Time (ps)

(c) FIG. 2: Calculated time evolution of oxygen positions (left vertical axis, black lines) and conductance (right vertical axis, 1st layer red points) of BDT in the aqueous solution. Here, the z-axis on top bridge is perpendicular to the Au(100) surface, and z = 0 and 9.8 A˚ + + j 2nd layer z correspond to the topmost layers of the Au(100) surfaces. + hollow i

(a) (b)

60 60

FIG. 1: (a) Snapshot of the system with the aqueous solu- Aq Vac tion (Aq), (b) the system without the aqueous solution (Vac) 50 50 and (c) the top view of a Au(100) surface together with the 40 40 adsorption sites (hollow, bridge, and on-top). Au, S, C, O, 30 30 Counts Counts and H atoms are depicted with gold, yellow, gray, red, and 20 20

white, respectively, in the snapshots. The z direction is per- 10 10

pendicular to the Au(100) surface. The white squares deﬁne 0 0 the scattering region in conductance calculations with ATK. 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3

Conductance (G ) Conductance (G )

0 0

(c) (d) molecules are relaxed while Au atoms are ﬁxed during the 60 60 Aq Vac dynamics. 50 50 Step II: We perform NEGF-DFT calculations using the 40 40 code Atomistix ToolKit (ATK) [35–38] to obtain zero-bias 30 30 Counts Counts conductances of BDT in Aq/Vac for hundreds of conﬁgu- 20 20 rations selected from the CPMD calculations in Step I. 10 10

Step III: We make conductance histograms using the 0 0 calculated conductance and analyze the Aq/Vac his- 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3

Conductance (G ) Conductance (G ) tograms to characterize the effects of the aqueous solution 0 0 on conductance. Note that in the present study we use the same compu- FIG. 3: Calculated conductance histograms for the cases (a) tational data obtained in our previous work [32] to analyze in the aqueous solution (Aq) and (b) in vacuum (Vac). Fig- histogram in more detail. ure 3(a) is obtained from the calculated conductance shown as red points in Fig 2. The total number of samples is 400 in the Aq histogram (Fig. 3(a)) and 448 in the Vac one ( Fig. 3(b)). Figures 3(c) and 3(d) show conductance histograms obtained III. RESULTS AND DISCUSSION from 300 samples randomly selected from the histograms of Figs. 3(a) and 3(b), respectively. The bin sizes are 0.006 G0 in Figure 2 shows the time evolution of oxygen z-positions all the histograms. The light blue and black lines denote the of water molecules and conductance of BDT in water. Wa- respective Gaussian functions obtained by the fitting and the ter molecules confined in the nano-gapped space between sum of the Gaussians, respectively. the electrodes form three layers with each interlayer distance of about 2.5 A,˚ and the uppermost and lowermost layers appear at about 2.5 A˚ apart from the gold surfaces. lines, we found two peaks in the histogram of Fig. 3(a) In Step II of our method, we picked up the CPMD con- and three in Fig. 3(b) as listed in Table I. The main figurations every 20 time-steps (1.94 fs) and calculated peaks are the 1st peak in the Aq histogram and the 3rd the zero-bias conductances for the selected configurations peak in the Vac, respectively. The downward shift of the using ATK to make conductance histograms. Figures 3(a) main peak in Aq is caused by the electrostatic effect of and (b) show the conductance histograms of the Aq and water molecules [32], i.e. the electrostatic potential due Vac systems, respectively. As described in the previous to the water dipoles. The peak at 0.102 G0 obtained in work, the important feature found in the histograms of the histogram of Fig. 3(b) disappears in the histogram of Figs. 3(a) and 3(b) is the appearance of several peaks and Fig. 3(a), while the peak at 0.168 G0 in the histogram of the shift of main peak. Fitting the histograms with sev- Fig. 3(a) is not confirmed in the histogram of Fig. 3(b). eral Gaussian functions as shown with light blue-colored Here, the total number of samples is 400 in the Aq his- http://www.sssj.org/ejssnt (J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) 39 Volume 8 (2010) Tawara, et al.

TABLE I: Peak positions and standard deviations in the respective conductance histograms shown in Fig. 3. N denotes the number of samplings of conductance histograms. PP1, PP2 and PP3 denote the 1st, 2nd and 3rd peak position, and SD1, SD2 and SD3 the corresponding 1st, 2nd and 3rd standard deviation. 1st peak 2nd peak 3rd peak System Figure N PP1 (/G0) SD1 (/G0) PP2 (/G0) SD2 (/G0) PP3 (/G0) SD3 (/G0) Aq Fig. 3(a) 400 0.127 ± 0.020 0.168 ± 0.004 Aq Fig. 3(c) 300 0.128 ± 0.040 0.169 ± 0.011 Vac Fig. 3(b) 448 0.102 ± 0.008 0.142 ± 0.005 0.159 ± 0.027 Vac Fig. 3(d) 300 0.100 ± 0.012 0.141 ± 0.007 0.157 ± 0.054

(a) (b) (a) (b)

25 25

0.25 0.25

G 0.160 G G 0.110 G G 0.180 G G 0.110 G ) )

0 0 0 0 0 0

20 20

0.20 0.20

Aq Vac

15 15

0.15 0.15

10 10 0.10 0.10 Counts Counts

5 5 0.05 0.05 Aq Vac Conductance (G Conductance (G

0 0.00 0.00 0

2.40 2.45 2.50 2.55 2.60 2.65 2.40 2.45 2.50 2.55 2.60 2.65

-0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15

Au-S Bond Length (Å) Au-S Bond Length (Å)

d (Å) d (Å)

0.25 0.25 ) ) 0 0 FIG. 5: (a) The Aq C–S bond length histogram and (b) the 0.20 0.20 Aq Vac Vac C–S bond length histogram. In Fig. 5(a), the total his-

0.15 0.15

togram is divided into two groups depending on the conduc-

0.10 0.10 tance value G: G ≥ 0.160 G0 (Group 1, red) and G < 0.110

0.05 0.05

Conductance (G Conductance (G G0 (Group 2, blue). In ﬁg. 5(b), G ≥ 0.180 G0 (Group 3, red)

0.00 0.00 and G < 0.110 G0 (Group 4, blue). 1.70 1.75 1.80 1.85 1.90 1.95 1.70 1.75 1.80 1.85 1.90 1.95

C-S Bond Length (Å) C-S Bond Length (Å)

(e) (f)

0.25 0.25 ) ) bond lengths (Au–S, C–S, C–C) and conductance. Fig- 0 0

0.20 0.20 ure 4 shows correlations between the bond lengths and

0.15 0.15

conductance ((a, b) for Au–S, (c, d) for C–S and (e, f)

0.10 0.10 for C–C). Since there are four Au–S bonds for the ad-

0.05 0.05 Aq Vac sorption of hollow site and two C–S bonds of BDT in our Conductance (G Conductance (G

0.00 0.00

2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 system, we estimated the average of the shortest Au–S

C-C Length (Å) C-C Length (Å) length in the upper and that in the lower surface, and the average of the two C–S lengths in our system at FIG. 4: The correlations with conductance of the Au–S bond each time step in drawing Fig. 4. The C–C length in length in (a) Aq and (b) Vac, those of the C–S bond length in Figs. 4(e) and 4(f) is the distance between the two car- (c) Aq and (d) Vac, and those of the C–C length and conduc- bon atoms adjacent to the two sulfur atoms of BDT. tance in (e) Aq and (f) Vac. The straight lines in Figs. 4 were all obtained by lin- ear regression and the correlation coefficients are estimated as 0.279/0.131 for Figs. 4(a)/(b), −0.748/−0.848 togram and 448 in the Vac one, and the bin sizes are 0.006 for 4(c)/(d) and 0.216/0.260 for 4(e)/(f), respectively. G0 in the both histograms. To comfirm the reliability of Therefore, we can say that the Au–S bond length and our peak analyses, we constructed the histograms using the C–C length have little correlation with the conduc- other sets of conductance data. Figures 3(c) and 3(d) tance whereas the C–S bond length shows clear negative show conductance histograms obtained from 300 samples correlation both in the aqueous solution and vacuum. randomly selected from the ones of Figs. 3(a) and 3(b), Next, we confirmed the correlation between C–S bond respectively. As listed in Table I, the numbers of peaks, length and conductance further with additional analysis the positions of peaks and the standard deviations in focusing on the peaks at 0.168 G0 in Fig. 3(a) and 0.102 Figs. 3(c) and 3(d) are regarded as the same as Figs. 3(a) G0 in Fig. 3(b) as follows. From the conductance his- and 3(b). Therefore the numbers of samples in Figs. 3(a) tograms of Figs. 3(a) and (b), we extract two groups and 3(b) are enough in the analyses of the conductance in each case in terms of the conductance value G, con- peaks. More detailed discussion on the number of samples sidering the peak positions, the standard deviations and and bin size in the histogram is described in Appendix. downward peak shift of 0.02 G0 due to the electrostatic Comparing the standard deviations of the main peaks lo- effect of water molecules [32], as follows; ‘Group 1: G > cated at 0.127 G0 in Aq (Fig. 3(a)) and at 0.159 G0 in Vac 0.160 G0’ and ‘Group 2: G < 0.110 G0’ in Fig. 3(a), and (Fig. 3(b)), the conductance fluctuation is found to be re- ‘Group 3: G > 0.180 G0’ and ‘Group 4: G < 0.110 G0’ duced by the presence of the solution. A similar tendency in Fig. 3(b). Figures 5(a) and 5(b) show the C–S bond is seen in the experiment by Taniguchi et al. [31]. length histograms in Aq and Vac systems, and ∆d cor- To elucidate dynamical effects on conductance in terms responds to the difference of the bond length d from the of the BDT motions, we examined the correlation between optimized value dopt (= 1.81 A)˚ in CPMD (∆d = d−dopt).

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Parrinello molecular dynamics to obtain better understanding on the effects of the aqueous solution on the conductance of benzene-1,4-dithiolate molecular junction between gold electrodes. It is found that the C–S bond length shows clear negative correlation both in aqueous solution and vacuum whereas the Au–S bond length and C–C length has little correlation with the conductance. Further, it is shown that this difference can be understood from the distribution of LDOS in the molecular junction in our system. The present analyses reveals that 0.0000.015 0.030 0.045 0.060 the dynamical changes of the C–S stretching mode in the molecular junction induced by the aqueous solution play an important role in the considerable changes in the conductance histograms. FIG. 6: Local density of states (LDOS) at the Fermi level in the Vac system. The LDOS is depicted on the plane including the gold atoms “i” and “j” in Fig. 1(c) and C–S bonds. The Acknowledgments color bar shows the intensity of LDOS, in the unit of eV−1A˚−3. The present work was partially supported by the Grant- in-Aid for Scientific Research on Priority Area “Linked Here, like in Figs. 4(c) and 4(d), the averages of the two molecule in nano-scale (448)”(20027003) and for Global C–S bonds in the BDT molecule were used in the bond COE Program “Global Center of Excellence for Mechan- length histograms. The values of counts (vertical axes) ical Systems Innovation” by the Ministry of Education, in Figs. 5 mean the sum of the numbers of two groups: Culture, Sports, Science and Technology of Japan. A part Group 1 + Group 2 in Fig. 5(a) and Group 3 + Group 4 of the calculations was carried out on the SR11000 system in Fig. 5(b). Group 1 (red) in Fig. 5(a) in Aq is clearly of the Super Computer Center at the Institute for Solid constituted of the BDT configurations with a relatively State Physics and HA8000 Cluster System (T2K Open shortened C–S bond length of 1.75-1.81 A.˚ Thus, the con- Supercomputer) at the Information Technology Center, ductance peak at 0.168 G0 in Fig. 3(a) can be assigned to The University of Tokyo. the BDT configurations with shortened C–S bonds. On the other hand, Group 4 (blue) in Fig. 5(b) in the Vac is almost perfectly constituted of the BDT configurations Appendix with a lengthened C–S bond length of 1.85-1.91 A˚ and the conductance peak at 0.102 G0 can be assigned to the In making conductance histograms (Step III), we BDT configurations with lengthened C–S bonds. These checked the binning and number of samples in the his- results also reflect the clear negative correlation between tograms so as to analyze the histograms properly. In gen- the C–S bond length and conductance in aqueous solution eral, there is no unified rule to decide the best bin size in and vacuum as shown in Figs. 4(c) and 4(d). Although a histogram. We referred to Sturges’ formula [39], Scott’s distributions of Group 2 (blue) in Fig. 5(a) and Group 3 choice [40] and Freedman-Diaconis’ rule [41] as formulas (red) in Fig. 5(b) show roughly negative correlations be- to derive a reasonable bin size. According to Sturges’ for- tween the C–S bond length and conductance, they do not mula [39], the number of columns in the histograms k can have so clear assignments as Group 1 and Group 4. Thus, be calculated with the number of samples n as follows: we can say that the emergence of sub peaks linked to the shortened/lengthened C–S bond length depends on the k = 1 + 3.322 log n. (1) environment of aqueous solution/vacuum, and the peak Using Scotts’ rule [40], the data-based optimal choice for positions are drastically altered by the existence of the the bin sizes in histograms h is calculated as follows: surrounding water molecules. − 1 Finally, we investigated the local density of states h = 3.49sn 3 , (2) (LDOS) at the Fermi level in order to clarify the reason why only the C–S bonds in BDT molecular junction show where s is the estimated standard deviations for the his- clear correlation with the conductance. Figure 6 shows tograms. The bin sizes h is calculated with Freedman- the LDOS at the Fermi level in the Vac system. This fig- Diaconis’ rule [41] with the interquartile range of sample ure reveals that the considerable amplitude of LDOS is standard deviations (IQR): seen on the C–S bonds, whereas such bonding states are − 1 h = 2 · IQR · n 3 . (3) not seen on Au–S and C–C bonds. Therefore, it is reasonable that the conductance of BDT is sensitive to only From Eq. (1) according to Sturges’ formula, the num- the C–S bonds. ber of columns in the Aq and Vac conductance histograms of Fig. 3(a) and Fig. 3(b), kAq and kVac, can be calculated with the number of samples n as follows: IV. SUMMARY kAq = 1 + 3.322 × log 400 ' 10, We extended our previous study using the ab ini- kVac = 1 + 3.322 × log 448 ' 10. tio nonequilibrium Green’s function method and Car- http://www.sssj.org/ejssnt (J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) 41 Volume 8 (2010) Tawara, et al.

TABLE II: Dependence of peak positions and standard deviations on bin sizes for the conductance histograms of Figs. 3(a) and 3(b). PP1, PP2 and PP3 denote the 1st, 2nd and 3rd peak position, and SD1, SD2 and SD3 the corresponding 1st, 2nd and 3rd standard deviation. 1st peak 2nd peak 3rd peak System Figure Bin size PP1 (/G0) SD1 (/G0) PP2 (/G0) SD2 (/G0) PP3 (/G0) SD3 (/G0) Aq 0.010 0.125 ± 0.019 0.168 ± 0.010 Aq Fig. 3(a) 0.006 0.127 ± 0.020 0.168 ± 0.004 Aq 0.004 0.126 ± 0.019 0.168 ± 0.008 Vac 0.010 0.103 ± 0.005 0.142 ± 0.006 0.159 ± 0.029 Vac Fig. 3(b) 0.006 0.102 ± 0.008 0.142 ± 0.005 0.159 ± 0.027 Vac 0.004 0.103 ± 0.008 0.142 ± 0.005 0.159 ± 0.026

TABLE III: Dependence of peak positions and standard deviations on the number of samples N for the conductance histograms of Figs. 3(a) and 3(b). PP1, PP2 and PP3 denote the 1st, 2nd and 3rd peak position, and SD1, SD2 and SD3 the corresponding 1st, 2nd and 3rd standard deviation. 1st peak 2nd peak 3rd peak System Figure N Samples PP1 (/G0) SD1 (/G0) PP2 (/G0) SD2 (/G0) PP3 (/G0) SD3 (/G0) Aq Fig. 3(a) 400 (total) 0.127 ± 0.020 0.168 ± 0.004 Aq Fig. 3(c) 300 (set 1) 0.128 ± 0.040 0.169 ± 0.011 Aq 300 (set 2) 0.123 ± 0.036 0.164 ± 0.029 Aq 300 (set 3) 0.124 ± 0.033 0.162 ± 0.027 Aq 300 (set 4) 0.123 ± 0.034 0.154 ± 0.037 Vac Fig. 3(b) 448 (total) 0.102 ± 0.008 0.142 ± 0.005 0.159 ± 0.027 Vac Fig. 3(d) 300 (set 1) 0.100 ± 0.012 0.141 ± 0.007 0.157 ± 0.054 Vac 300 (set 2) 0.102 ± 0.012 0.141 ± 0.007 0.158 ± 0.056 Vac 300 (set 3) 0.100 ± 0.018 0.142 ± 0.011 0.164 ± 0.061 Vac 300 (set 4) 0.102 ± 0.017 0.141 ± 0.008 0.158 ± 0.049

The calculated conductance values in Fig. 3(a) range from size in the Aq and Vac histograms. The peak positions 0.05 to 0.20 G0 and Fig. 3(b) from 0.09 to 0.24 G0, and and standard deviations in Table II are nearly constant the both of bin sizes are thereby 0.015 G0. Using Eq. (2) regardless of the bin sizes in the region from 0.004 to of Scotts’ rule, the data-based optimal choice for the bin 0.01 G0, and the number of the histogram peaks does not sizes in the Aq and Vac conductance histograms, hAq and change. hVac, are calculated as follows: We also checked the number of samples. Figures 3(c) and 3(d) show conductance histograms obtained from 3.49 × 0.020 hAq = √ ' 0.009, 300 samples randomly selected from the configurations 3 400 of Figs. 3(a) and 3(b), respectively. We obtained two and 3.49 × 0.027 three peaks in the conductance histograms of Figs. 3(c) hVac = √ ' 0.012, 3 448 and 3(d). The respective peak positions and standard deviations in Figs. 3(c) and 3(d), which are shown as set 1 in Table III, are nearly same as the corresponding ones where we used s of 0.020 G0 (SD1 in Fig. 3(a)) and 0.027 in Figs. 3(a) and 3(b). Three sets of samples, set 2, set G0 (SD3 in Fig. 3(b)) as listed in Table I. The bin sizes 3 and set 4, in Table III show the peak positions and calculated with Eq. (3) of Freedman-Diaconis’ rule are as standard deviations in other three histograms obtained follows: from 300 samples randomly selected from the histograms 2 × 1.349 × 0.020 of Figs. 3(a) and 3(b). The peak positions and standard hAq = √ ' 0.007, 3 400 deviations in Table III are almost constant regardless of 2 × 1.349 × 0.027 the number and set of samples. In all the sets of sam- √ hVac = 3 ' 0.010. ples, we obtained two and three peaks in the Aq and Vac 448 histograms. On the basis of results shown in Table II and III, we can Considering these estimations of bin sizes, we carried say that the sample numbers and bin sizes adopted in our out histogram analysis using bin sizes h from 0.004 to study are sufficient to draw reliable values of the number 0.01 G0. Table II shows the peak positions and standard of peaks, the peak positions and the standard deviations deviations of fitted Gaussians as a function of the bin of the fitted Gaussians.

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