Experimental and ab initio calculational studies on 2,3-diketo- benzopiperazine

Article abstract of DOI:10.1016/j.molstruc.2004.06.027

The title compound, 2,3-diketo-benzopiperazine, has been synthesized and characterized by elemental analysis, IR, electronic spectroscopy and single crystal X-ray diffraction. Ab initio calculations of the structure, natural bond orbital, topological analysis and thermodynamic functions of the title compound were performed at HF/6-311G** and B3LYP/6-311G** levels of theory. Vibrational frequencies were also predicted, assigned, compared with the experimental values, and they supported each other. Electronic absorption spectra were calculated by the time-dependent density functional theory, which indicates that the two absorption bands are mainly derived from the contribution of bands π→π*. The calculation of the second order optical non-linearity was carried out, and the molecular hyperpolarisability was 3.282×10^-30 esu.

Full text of DOI:10.1016/j.molstruc.2004.06.027

Journal of Molecular Structure 705 (2004) 133–139
www.elsevier.com/locate/molstruc
Experimental and ab initio calculational studies
on 2,3-diketo-benzopiperazine
*
Fang-Fang Jian , Pu-Su Zhao
New Materials and Function Coordination Chemistry Laboratory, Qingdao University of Science and Technology,
Zhengzhou Road 25, Qingdao Shandong 266042, China
Received 13 April 2004; revised 21 June 2004; accepted 22 June 2004
Available online 20 July 2004
Abstract
The title compound, 2,3-diketo-benzopiperazine, has been synthesized and characterized by elemental analysis, IR, electronic
spectroscopy and single crystal X-ray diffraction. Ab initio calculations of the structure, natural bond orbital, topological analysis and
thermodynamic functions of the title compound were performed at HF/6-311G** and B3LYP/6-311G** levels of theory. Vibrational
frequencies were also predicted, assigned, compared with the experimental values, and they supported each other. Electronic absorption
spectra were calculated by the time-dependent density functional theory, which indicates that the two absorption bands are mainly derived
from the contribution of bands p/p*. The calculation of the second order optical non-linearity was carried out, and the molecular
hyperpolarisability was 3.282!10K30 esu.
q 2004 Elsevier B.V. All rights reserved.
Keywords: Synthesis; Crystal structure; Ab initio calculation; Vibrational frequencies; Second order optical non-linearity
1. Introduction
[4–8] accuracy using classical self-consistent field
molecular orbital Hartree–Fock theory (SCF-MO HF)
method. The achieved accuracy is generally improved in
more advanced calculations which include electron corre-
lation (e.g. Moller–Plesset second-order approximation,
MP2). The importance of size, presence of polarization and
diffuse functions in the basis set used for a proper theoretical
description of the molecules, which include multiple bonds,
heteroatoms and lone electron pairs, is also well recognized
[4,5]. On the other hand, the density functional theory (DFT)
is less computationally demanding in predicting physical
properties of molecular systems [5,9–13].
The investigation of the structure stability of heterocycle
compounds by both experimental techniques and theoretical
methods has been of interest for many years. Piperazine and
its derivations were important members of heterocycle
compounds and they were widely applied in many fields,
such as curatorial intermediate, bacteriophage and
insectifuge. Many synthesis methods for the piperazine
have been reported before [1–3]. However, to the best of our
knowledge, neither the synthesis nor the theory studies on
2,3-diketo-benzopiperazine have been available. So after we
synthesized this compound using hydrothermal method and
determined its crystal structure, we performed a further
theoretical study about it.
In this work, we report the synthesis of 2,3-diketo-
benzopiperazine and its crystal structure. We also give the
calculational results of the structure geometry, natural bond
orbital (NBO), electron density at bond critical points and
thermodynamic properties for the title compound at the
B3LYP/6-311G** and HF/6-311G** levels. On the other
hand, vibrational frequencies have been predicted, assigned
and compared with the experimental values. Furthermore,
the second-order optical non-linearity has been studied using
MOPAC program package.
With the recent advances in computer hardware
and software, it is possible to correctly describe the
physico-chemical properties of relatively small molecules
* Corresponding author. Tel./fax: C86-532-402-3606.
E-mail address: ffj2003@163169.net (F.-F. Jian).
0022-2860/$ - see front matter q 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.molstruc.2004.06.027

134
F.-F. Jian, P.-S. Zhao / Journal of Molecular Structure 705 (2004) 133–139
Table 1
2. Experimental
Summary of crystallographic results for the title compound
2.1. Physical measurements
Empirical formula
Formula weight
Temperature (K)
C8H6N2O2
162.15
293(2)
Elemental analyses for carbon, hydrogen and nitrogen
were performed using a Perkin–Elmer 240C elemental
instrument. The IR spectra were recorded in the
4000–400 cmK1 region using KBr pellets on a Nicolet
170SX spectrophotometer. Electronic absorption spectra
were measured on a Shimadzu UV3100 spectrophotometer
in EtOH.
˚
Wavelength (A)
0.71073
Crystal system, space group
Unit cell dimensions
Monoclinic, P21/c
˚
˚
aZ8.0497(16) A, bZ4.2859(9) A,
˚
cZ20.995(6) A, bZ102.14(3)8
708.1(3)
3
˚
Volume (A )
Z, calculated density (mg/m3)
Absorption coefficient (mmK1
F(000)
4, 1.521
)
0.113
336
q Range for data collection
Limiting indices
1.98–27.608
0%h%10, K5%k%5, K26%l%24
1812/1128 [RintZ0.1000]
Full-matrix least-squares on F2
1128/0/110
2.2. Synthesis
Reflections collected/unique
Refinement method
All chemicals were obtained from a commercial source
and used without further purification.
Data/restraints/parameters
Goodness-of-fit on F2
Final R indices [IO2s(I)]
R indices (all data)
0.988
An ethanol solution (20 ml) of o-diaminobenzene
(0.54 g, 5.0 mmol) and oxalic acid (0.450 g, 5.0 mmol) are
mixed, and the aqueous (15 ml) CuCl2 (0.336 g, 0.25 mmol)
was added with stirring, then the mixture was sealed in a
50 ml stainless–steel reactor with Teflon-lined reactor at
110 8C for 48 h, resulting in the formation of the colorless
crystals of the title compound. m.p. 116.7 8C. Yield: 85%.
Calc. for C8H6N2O2: C, 59.24; H, 3.73; N, 17.28. Found: C,
59.01; H, 3.52; N, 17.06.
R1Z0.0623, wR2Z0.1535
R1Z0.1288 wR2Z0.1749
0.265 and K0.316
Largest diff. peak
K3
˚
and hole (e A
)
Crystallographic data has been deposited with the
Cambridge Crystallographic Data Center as supplementary
Publication No. CCDC 227509.
2.4. Calculational methods
The synthesis pathway of the title compound is shown
in Scheme 1.
Initial molecular geometry was optimized using MMC
molecular modeling and semi-empirical AM1 methods [17]
(HYPERCHEM 6.0, Hypercube, Ont., Canada). In the next
step, the DFT calculations with a hybrid Functional B3LYP
(Becke’s three parameter Hybrid Functional Using the LYP
Correlation Functional) at 6-311G** basis set and the
Hatree–Fock calculations at 6-311G** basis set by the
Berny method [18,19] were performed with the GAUSSIAN 98
software package [8]. Vibrational frequencies calculated
ascertain the structure was characterized to be the stable
structure (no imaginary frequencies). NBO analyses and the
time-dependent density functional theory (TD-DFT)
[20–23] calculations of electronic absorption spectra were
performed on the optimized structure. The topological
analysis of electron density is based on the description of its
critical points, which are located between the nuclei of two
bonded atoms and were derived from AIM 2000 program
[24]. On the basis of the MNDO Hamiltonian [25] and PM3
parametrization [26] with the MOPAC [27] program package,
the molecular hyperpolarisability value was also calculated.
All calculations were performed on a Pentium IV
computer using the default convergence criteria.
2.3. X-ray structure determination
The selected crystal of the title compound was mounted
on a Rigaku Raxis-IV diffractometer. Reflection data were
measured at 20 8C using graphite monochromated Mo Ka
˚
(lZ0.71073 A) radiation. The collected data were reduced
using the program SAINT [14]. The structure was solved by
direct methods and refined by full-matrix least-squares
method on Fo2bs by using the SHELXTL software package [15].
All non-H atoms were anisotropically refined. The hydrogen
atom positions were fixed geometrically at calculated
distances and allowed to ride on the parent carbon atoms.
The final least-square cycle gave RZ0.0623, RwZ0.1535
for 671 reflections with I>2s(I); the weighting
scheme, wZ1=½s2ðF02ÞCð0:0867PÞ ꢀ;where PZðF02 C2Fc2
2
Þ=3: Atomic scattering factors and anomalous dispersion
corrections were taken from International Table for X-ray
Crystallography [16]. A summary of the key crystal-
lographic information is given in Table 1.
3. Results and discussions
3.1. Description of the crystal structure
The displacement ellipsoid plot with the numbering
scheme for the title compound is shown in Fig. 1.
Scheme 1. The synthesis pathway of the title compound.

F.-F. Jian, P.-S. Zhao / Journal of Molecular Structure 705 (2004) 133–139
135
Table 2
Atomic coordinates (!104) and equivalent isotropicdisplacement par-
2
ameters (A !10 ) for the title compound
3
˚
Atom
x
y
z
Ueq
O(1)
K1156(3)
K1229(3)
1073(3)
1080(3)
3593(4)
4838(4)
4828(5)
3568(4)
2325(4)
2341(4)
K127(4)
K145(4)
1076
5340(7)
3195(7)
203(7)
634(1)
54(1)
55(1)
42(1)
43(1)
48(1)
54(1)
57(1)
50(1)
39(1)
39(1)
42(1)
42(1)
50
O(2)
1870(1)
1838(1)
615(1)
1847(2)
1527(2)
901(2)
586(2)
906(2)
1536(2)
898(2)
1583(2)
2223
N(1)
N(2)
2178(7)
C(1)
K2940(9)
K3898(10)
K2929(10)
K963(9)
78(9)
C(2)
C(3)
C(4)
C(5)
C(6)
K915(8)
3376(9)
2233(9)
K480
C(7)
C(8)
Fig. 1. Molecular structure with the atomic numbering scheme for the title
compound.
H(1A)
H(2B)
H(1B)
H(2A)
H(3B)
H(4A)
1082
2758
222
51
3597
K3648
K5222
K3603
K331
2266
58
5697
1737
65
Fig. 2 shows a perspective view of the crystal packing in the
unit cell. Atomic parameters and equivalent isotropic
thermal parameters of non-hydrogen and hydrogen atoms
for the title compound are all given in Table 2. Selected
bond lengths and bond angles by X-ray diffractions are
listed in Table 3.
5675
692
68
3547
160
60
Ueq is defined as one-third of the trace of the orthogonalized Uij tensor.
The crystal structure of the title compound consists of
four monomers 2,3-diketo-benzopiperazine. Bond lengths
and angles in the phenyl ring are generally normal.
Table 3
Selected structural parameters by X-ray and theoretical calculations
Experiment
B3LYP/6-
311G**
HF/6-311G**
˚
The bond lengths of N–C (1.333–1.394 A) are all shorter
˚
than the typical N–C single bond distance (1.4685 A),
˚
Bond distances (A)
indicating that N–C bonds are of some character of double
bond. All the non-H atoms except O(1) atom define a plane
˚
with the largest deviation 0.032 A. In the crystal lattice,
O(1)–C(7)
O(2)–C(8)
N(1)–C(8)
N(1)–C(6)
N(2)–C(7)
N(2)–C(5)
C(1)–C(2)
C(1)–C(6)
C(2)–C(3)
C(3)–C(4)
C(4)–C(5)
C(5)–C(6)
C(7)–C(8)
1.226(4)
1.230(4)
1.333(4)
1.394(4)
1.343(4)
1.387(4)
1.380(5)
1.382(4)
1.375(5)
1.374(5)
1.391(5)
1.386(4)
1.522(5)
1.2079
1.2079
1.3805
1.3947
1.3805
1.3947
1.39
1.1819
1.1819
1.3605
1.3957
1.3605
1.3957
1.3826
1.3841
1.3843
1.3826
1.3841
1.3875
1.5366
there exist some intermolecular hydrogen bonds, potentially
weak (C–H/Y hydrogen bonds, YaO) [28,29]
intermolecular interactions, and p–p stacking interactions
[30,31]. There are two intermolecular hydrogen
bonds N(1)–H(1A)/O(2) [symmetry code: Kx, K1/2C
y, 1/2Kz] and N(2)–H(2B)/O(1) [symmetry code: K
x, 3Ky, Kz] with the separation distances of N(1)/O(2)
1.3957
1.3949
1.39
1.3957
1.4036
1.5364
Bond angles (8)
C(8)–N(1)–C(6)
C(7)–N(2)–C(5)
C(2)–C(1)–C(6)
C(3)–C(2)–C(1)
C(4)–C(3)–C(2)
C(3)–C(4)–C(5)
C(6)–C(5)–N(2)
C(6)–C(5)–C(4)
N(2)–C(5)–C(4)
C(1)–C(6)–C(5)
C(1)–C(6)–N(1)
C(5)–C(6)–N(1)
O(1)–C(7)–N(2)
O(1)–C(7)–C(8)
N(2)–C(7)–C(8)
O(2)–C(8)–N(1)
O(2)–C(8)–C(7)
N(1)–C(8)–C(7)
125.9(3)
125.0(3)
119.1(3)
121.1(3)
120.0(3)
119.6(3)
118.7(3)
120.0(3)
121.2(3)
120.1(3)
122.0(3)
117.9(3)
122.9(3)
120.4(3)
116.6(3)
123.9(3)
120.3(3)
115.8(3)
126.58
126.58
120.18
120.08
120.08
120.18
118.10
119.74
122.16
119.74
122.16
118.10
122.80
121.88
115.32
122.80
121.88
115.32
125.98
125.98
120.17
119.97
119.97
120.17
118.38
119.86
121.76
119.87
121.76
118.38
123.33
121.02
115.64
123.34
121.02
115.64
Fig. 2. A view of the crystal packing down the a axis for the title compound.

136
F.-F. Jian, P.-S. Zhao / Journal of Molecular Structure 705 (2004) 133–139
Fig. 3. IR spectra of the title compound:(top) experimental spectrum; (middle)B3LYP/6-311G** predicted spectrum; (bottom) HF/6-311G** predicted spectrum.
˚
˚
2.8231 A, N(2)/O(1) 2.8413 A, respectively. The O(2)
atom with C(2) atom form potentially weak C–H/O
intramolecular interactions, the donor and acceptor distance
3.2. Optimized geometry
Calculated geometric parameters at B3LYP/6-311G**
and HF/6-311G** levels are also listed in Table 3.
From the theoretical values, we can find that the molecule
of 2,3-diketo-benzopiperazine possess C2 symmetry and
most of the optimized bond lengths are slightly larger than
the experimental values, due to that the theoretical
calculations belong to isolated molecules in gaseous
phase and the experimental results belong to molecules
in solid state. Comparing bond angles and lengths of
˚
is 3.3344 A. There are two types p–p stacking interactions:
piperazine ring (X, Y, Z)–phenyl ring (X, 1CY, Z) and
phenyl ring (X, Y, Z)–piperazine ring (X,K1CY, Z).
˚
The center-to-center distances are all 3.574 A. The shortest
˚
interplanar distances above are 3.308 and 3.324 A,
respectively. In the solid state, all above extensive hydrogen
bonds in this structure connected molecule forms hydrogen
bonds networks, which stabilize the crystal structure.

F.-F. Jian, P.-S. Zhao / Journal of Molecular Structure 705 (2004) 133–139
137
Table 4
Selected vibrational frequencies and the intensities (in parentheses)
Experimental
B3LYP/6-311G**
HF/6-311G**
Description
3414 (95)
3435 (0.29) 3436 (65)
3040–3072 (2–11)
1727–1730 (118–1020)
1580–1587 (0.01–17.5)
1476 (61)
3417.8 (117) 3418 (0.47)
2958–2990 (7–14)
1774–1793 (117–1257)
1602–1605 (0–22)
1487 (7)
nN–H
3000 (15)
nC–H in phenyl ring
nCaO
1637 (50), 1617 (68)
1497 (6)
nPhenyl ring skeleton
dC–H phenyl ring
C–HCdN–H
1456 (8)
1425 (2)
1415 (28)
1442 (149)
1293 (301)
1349 (624)
nC(7)–C(8)
1265 (2)
1225 (12)
1162 (8)
1096 (14)
948 (3)
1243 (2)
1242 (2)
dC–H phenyl ring
dC–H phenyl ring CdN–H
nC(8)–N(1)CnC(7)–N(2)
dC–H phenyl ring
gC–H phenyl ring
gC–H phenyl ring
gC(8)–N(1)CgC(7)–N(2)
gC–H phenyl ring
gAll skeleton
1205–1219 (17–13)
1171 (16)
1207–1218 (25–10)
1187 (35)
1016–1137 (1–5)
933 (0)
1006–1172 (1–29)
933 (5) 971 (0)
848 (0)
796 (3)
817 (0)
748 (20)
722 (16)
695 (2)
773 (0)
807 (0)
724 (112)
746 (107)
682–688 (1–12)
614–667 (0–146)
680–685 (2–15)
623 (230)
625 (22)
dN–H
Frequencies in cmK1, the intensities for the calculatedvalues in km/mol and for the experimental values in %transmittance. Atomic numbering as shownin Fig. 1.
B3LYP with those of HF, as a whole, the formers are
bigger than the later and the HF calculated values
correlates well compared with the experimental results.
In spite of the differences, calculated geometric
parameters represent a good approximation and they are
the bases for calculating other parameters, such as atomic
charge, the vibrational frequencies, NBO analysis,
electronic absorption spectra, as we described below.
Based on B3LYP/6-311G** and HF/6-311G** optimized
geometry, the total energy of the title compound has been
calculated by MP2 method, which are K567.0346505
andK567.0282699 a.u., respectively.
3.3. Vibrational frequencies
The experimental IR spectra and the simulated infrared
spectra are shown in Fig. 3, where the calculated intensity is
plotted against the harmonic vibrational frequencies.
Vibrational frequencies calculated at B3LYP/6-311G**
level were scaled by 0.96 [32], and those calculated at
HF/6-311G** level were scaled by 0.89 [33]. Some selected
calculated harmonic frequencies and infrared intensities are
listed in Table 4 and compared with the experimental data.
The descriptions concerning the assignment have also been
indicated in Table 4. Gauss-view program [34] was used to
Table 5
Natural bond orbital occupancies and electronic densities at bond critical points
K3
˚
Electronic densities r (A
Occupancies (a.u.)
)
Bond
B3LYP
HF
Bond
B3LYP
HF
O(1)–C(7)
O(1)–C(7)p
O(1)–C(7)p*
N(2)–C(7)
N(2)–C(7)*
N(2)–C(5)
N(2)–C(5)*
C(7)–C(8)
C(5)–C(6)
C(5)–C(6)p
C(5)–C(4)
C(4)–C(3)
C(4)–C(3)p
C(3)–C(2)
C(2)–C(1)
C(2)–C(1)p
C(1)–C(6)
1.99305
1.98028
0.27464
1.98957
0.08197
1.98576
0.02996
1.97790
1.96440
1.60945
1.96997
1.97623
1.69713
1.97851
1.97623
1.69713
1.96997
1.99382
1.98666
0.20043
1.99017
0.06173
1.98721
0.02239
1.98104
1.96795
1.61827
1.97235
1.97804
1.69135
1.98000
1.98000
1.69135
1.97235
O(1)–C(7)
N(2)–C(7)
N(2)–C(5)
C(7)–C(8)
C(5)–C(6)
C(5)–C(4)
C(4)–C(3)
C(3)–C(2)
C(2)–C(1)
C(1)–C(6)
0.415915
0.3038857
0.2917525
0.2502625
0.307589
0.307069
0.309407
0.307742
0.309407
0.307069
0.441632
0.319957
0.292756
0.268491
0.328435
0.324244
0.324354
0.324564
0.324354
0.324244
(*) Anti-bond.

138
F.-F. Jian, P.-S. Zhao / Journal of Molecular Structure 705 (2004) 133–139
Table 6
temperature ranging from 200.00 to 800.00 K, due to that
the intensities of molecular vibration increase while the
temperature increases. The correlation equations between
these thermodynamic properties and temperatures are as
follows:
The thermodynamic properties of the title compound at different
temperatures at B3LYP/6-311G**
Cp0;m
(J molK1 KK1
Sm0
Hm0
(kJ molK1
T(K)
(J molK1 KK1
)
)
)
200.00
298.15
400.00
500.00
600.00
700.00
800.00
111.06
162.13
208.45
245.22
274.23
297.20
315.68
337.93
391.92
446.24
496.86
544.24
588.30
629.23
13.03
26.47
45.41
68.17
94.20
122.81
153.49
Cp0;m ZK12:36375 C0:68997T K3:51448!10K4T2;
S0m Z 247:2351 C0:48702T;
Hm0 ZK43:66001 C0:23686T
They provide helpful information for the further study on
the title compound.
assign the calculated harmonic frequencies. On the whole,
the predicted vibrational frequencies are in agreement with
the experimental results. Comparing the B3LYP and HF
methods, above 3000 cmK1, the predicted frequencies by
B3LYP are larger than those by HF, whereas under
3000 cmK1, most of calculated frequencies by HF are
larger than those by B3LYP.
3.6. Non-linear optical properties calculations
On the basis of the MNDO Hamiltonian and PM3
parametrization with the MOPAC program package, the
molecular hyperpolarisability value, bm, the vector
components along the dipole moment direction, of the title
compound was calculated to be 3.282!10K30 esu, which is
much greater than that of urea (0.14!10K30 esu).
The electronic absorption spectra of the title compound in a
solution of EtOH exhibit two intense bands near 200 and
280 nm, respectively. The TD-DFT electronic spectrum
calculation for the title compound also gives two electronic
transitions ca. 198.68 nm with oscillator strength 0.7557 and
284.70 nm with oscillator strength 0.2264, res-
pectively. The optical transitions associated with above
two bands are mainly derived from the contribution of bands
p/p*, according to the analysis of molecular orbital
components of the title compound. Therefore, the title
compound is a good candidate as second-order non-linear
optical material.
3.4. Natural bond orbital and electron densities
at bond critical point
The NBO occupancies along with the electronic densities
at bond critical points, which were calculated based on the
B3LYP/6-311G** and HF/6-311G** optimized structures,
are listed in Table 5.
NBO analyses reveal that the C–O bonds are double bonds
and the C(7)–C(8) exist as single bond. In phenyl-ring, the
C–C bonds are typical single–double arrangement, which
form the conjugate structure. The NBO occupancies of
N(2)–C(7) bonds are larger than those of N(2)–C(5) bonds,
indicating that the strengths of bonds N(2)–C(7) are stronger
than those of N(2)–C(5) bonds. All above predicted results
are consistent with the information obtained from the
bond distances of the crystal structure. The electronic
densities at bond critical points by HF method are all larger
than those by B3LYP method, but these two kinds analyses
all reveal the similar information got from NBO analyses and
they are good agreement with the information from the
crystal structure, such as that the strengths of bonds
N(2)–C(7) are stronger than those of N(2)–C(5) bonds and
the strength of bond C(7)–C(8) is weaker than those of C–C
bonds in phenyl ring.
4. Conclusions
From the before described results, the following
conclusions can be drawn:
(1) The optimized geometry parameters calculated at
B3LYP/6-311G** are slightly larger than those
calculated at HF/6-311G** level and the HF calculated
values correlates well compared with the single crystal
structure data on the whole.
(2) Comparison between the calculated vibrational
frequencies and the experimental values indicates that
both the methods of B3LYP/6-311G** and
HF/6-311G** can predict the IR spectra of the title
compound well.
3.5. Thermodynamic properties
On the basis of vibrational analysis at B3LYP/6-
311G** level and statistical thermodynamic, the standard
thermodynamic functions: heat capacities ðCp0;mÞ;
entropies ðS0mÞ and enthalpies ðHm0 Þ; were obtained and
listed in Table 6. The scale factor for frequencies is
also 0.96.
(3) The predicted results by NBO and electron densities at
bond critical point analyses are consistent with the
information obtained from the bond distances of the
crystal structure, which shows that the strength of
bonds N(2)–C(7) are stronger than those of N(2)–C(5)
From Table 6, it can be observed that the standard heat
capacities, entropies and enthalpies increase at any

F.-F. Jian, P.-S. Zhao / Journal of Molecular Structure 705 (2004) 133–139
139
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bonds and the strength of bond C(7)–C(8) is weaker
than those of C–C bonds in phenyl ring.
(4) The second-order non-linear optical hyperpolarisability
values, bm for the title compound are larger than that
of urea, indicating that the title compound is a
good candidate as second-order non-linear optical
material.
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Acknowledgements
This work was supported by Natural Science Foundation
of Shandong Province (No.Y2002B06), P.R. China.
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