Chin. J. Chem. Eng., 15(1) 44—50 (2007)
Theoretical Study of Reaction Paths and Transition States on Conversion Methane into C2 Hydrocarbons Through Plasma*
WANG Baowei(王保伟)a,**, YANG Encui(杨恩翠)b, XU Genhui(许根慧)a and HAO Jinku(郝金库)b
a
Key Laboratory for Green Chemical Technology of State Education Ministry, School of Chemical Engineering &
Technology, Tianjin University, Tianjin 300072, China
b
College of Chemistry and Life Sciences, Tianjin Normal University, Tianjin 300074, China
Abstract The direct synthesis of C2 hydrocarbons (ethylene, acetylene and ethane) from methane is one of the
most important task in C1 chemistry. Higher conversion of methane and selectivity to C2 hydrocarbons can be realized through plasma reaction. In order to explore the reaction process and mechanism, the possible reaction paths
(1)—(4) were proposed on coupling reaction of methane through plasma and studied theoretically using semi-PM3
method [PM3 is parameterization method of modified neglect of diatomic overlap (MNDO)] including determining
the transition state, calculating the activation energy and thermodynamic state functions and analyzing the bond order and intrinsic reaction coordinate. The reaction heat results indicate that the reactions (2) and (4) are exothermic,
while reactions of (1) and (3) are endothermic. The activation energy results show that activation energy for reactions (1) and (2) was much lower than that of reaction paths (3) and (4). Therefore, paths (1) and (2) is the favorable
reaction path energetically. More interestingly by comparing the intrinsic reaction coordinated (IRC) of the reaction
paths (1) and (2), it is found that the variations of bond lengths in reaction path (1) has a crucial effect on the potential energy, while in reaction path (2), the adjustment of the system geometry also contributes to the whole potential
energy of the system.
Keywords reaction path, transition state, methane, plasma, PM3
1
INTRODUCTION
With large increase in natural gas reserves proven
worldwide, it can be expected that natural gas will
play an increasingly important role in energy and
chemicals supplies in the 21st century. Methane is
mainly being used as a fuel for industrial and residential heating at present. Since much more methane is
produced than that is required for that purpose, it may
be commercially advantageous to convert methane
into industrial chemicals. Selective activation of saturated hydrocarbons like methane is difficult due to the
similar high dissociation energy of carbon-hydrogen
bond. Plasma chemical processing is a promising
route for synthesis of chemicals that have high activation energies, because very high energy can be gained
from plasma and many kinds of reactive particles,
electrons, free radicals, ions, metastable species and
photons are produced in a plasma chemical processing
system. Recently, the synthesis of C2 hydrocarbons by
coupling reaction of methane through plasma has
made rapid progress[1—6].
The reaction was mainly carried out between
radicals, instead between ion-molecules after methane
is activated in nonequilibrium plasma[7]. Only when
heavy ions were accelerated to under specific condition,
reaction was mainly carried out between ions and
molecules. Hiraoka et al. used radical scavenger I2 and
ion scavenger NH3 to investigate the reaction mechanism under the condition of PCH 4 =133.3Pa in order to
confirm contribution of radicals and ions during the
course of reaction[8]. The chemical species of the
neutral radicals in methane microwave discharge
plasma was studied with Li+ ion attachment technology in combination with quadrupole mass spectrometry[9]. Main radicals such as CnH2n+1 (n=2—11) and
CnH2n－1 (n=2—11) were detected. Summing up many
references[3] and the experimental facts, we think the
C2 hydrocarbons are produced through radicals interaction with methane molecules. The probable reaction
mechanism is that methane is excited to become radicals: CH3 ·, CH 2 ·, CH·, H·, C in the plasma field. In
succession all kinds of radicals react with the first
available species-methane to produce the main product-ethane. The final distribution of products is relied
on the rate of C2 hydrocarbon formation and decomposition. Due to the complexity of the active species
in the gaseous phase and the similarity of chemical
bonds, the characteristic feature of plasma reaction is
that fast reaction and complexity of products of reaction, which offers a formidable challenge for investigating the mechanisms solely by experimental methods. In order to improve the conversion of methane
and selectivity to C2 hydrocarbons, it is widely
thought as a fairly wise idea to divert the research interest from the experimental methods to the theoretical
methods. According to the conservation of mass and
spin multiplicity, the probable reaction mechanism is
put forward as follows:
CH 4 + CH3 ⋅ ⎯⎯
→ C2 H 6 + H ⋅
(1)
CH 4 + CH 2 ⋅ (singlet) ⎯⎯
→ C2 H6
(2)
CH 4 + CH 2 ⋅ (triplet) ⎯⎯
→[C2 H5 − H] ⋅ ⎯⎯
→
[C2 H 4 − H] ⋅ + H ⋅ ⎯⎯
→ C2 H 4 + 2H ⋅
Received 2005-12-26, accepted 2006-07-24.
* Supported by the National Natural Science Foundation of China (No.20606023).
** To whom correspondence should be addressed. E-mail: [email protected]
(3)
Theoretical Study of Reaction Paths and Transition States on Conversion Methane into C2 Hydrocarbons Through Plasma
CH 4 + CH ⋅ ⎯⎯
→ TS1 → intermediate(IM) ⎯⎯
→
→ C2 H 4 + H ⋅
TS2 ⎯⎯
(4)
In this paper, the probable reaction paths were
studied with semi-PM3 method [PM3 is parameterization method of modified neglect of diatomic overlap
(MNDO)] including determination the transition state,
calculation the activation energy and thermodynamic
state functions and analysis the bond order and the
intrinsic reaction coordinate in order to offer rather
reliable and useful information to experimental chemists.
2
COMPUTATIONAL METHODS
The geometry configurations involved in the reaction system including reactants, transition states and
resultants of reaction were fully optimized with
Baker’s eigenvector following (EF) method at precise
level and without any symmetry limiting factor[10,11].
And then, the force analyses were also carried out in
order to confirm the stability of the geometry configurations. The transition state (TS) method was used
to search for the transition states of all reaction
paths[10,12]. The vibration analysis was also conducted in order to testify that their Hessian matrixes
have only one negative eigenvalue and their positive
and negative imaginary frequencies corresponded to
reactants and resultant of reaction, respectively. The
potential energy and the geometry configurations of
Table 1
Reaction path
reaction path 1
reaction path 2
reaction path 3
[C2H5-H]·
45
all stagnation point over the potential energy surfaces
were obtained through the intrinsic reaction coordinate
(IRC) analyses from positive and negative direction of
imaginary frequency of their transition states. All calculations were carried out using the PM3 method in
the MOPAC 6.0 package program[13—17] on a PentiumIII/500 computer.
2.1
Geometry configuration of transition states
The geometry configuration of transition states of
four reaction paths was completely optimized with the
TS method. Their atom codes and optimization geometry configurations of transition state are shown in
Fig.1. The parameters of optimization geometry configurations of transition state of four reactions path are
listed in Table 1.
Figure 1
Atom code of transition state
Geometric parameters of transition states
Bond length r, nm
C6C1
0.1761
H2C1
H3C1
Bond angles θ, (o)
H7C1H2
89.8
0.1219
H4C1H3
0.1103
H5C1H4
H4C1
0.1103
H5C1
0.1103
H7C1
H8C1
H9C1
0.1090
C2C1
Dihedral angles θ, (o)
H4C1H3H2
89.7
120.0
H5C1H4H3
179.4
120.0
H6C1H2H5
87.6
H6C1H2
180.0
H7C6C1H5
－180.0
H7C6C1
109.3
H8C6H7C1
119.8
0.1090
H8C6H7
109.6
H9C6H8H7
－120.4
0.1090
H9C6H8
109.6
0.1987
H3C1C2
45.0
H4C1H3C2
－123.8
H3C1
0.1248
H4C1C2
94.1
H5C1H4H3
－0.03
H4C1
0.1087
H5C1C2
136.2
H6C1C2H3
123.7
H5C1
0.1098
H6C1C2
94.1
H7C2C1H3
62.3
H6C1
0.1807
H7C2C1
111.0
H8C2C1H3
－62.3
H7C1
0.1807
H8C2C1
111.0
H8C1
0.1807
C2C1
H3C1
H4C1
H5C2
H6C2
H7C2
H8C2
0.1686
0.1077
0.1077
0.1147
0.1147
0.1099
0.1100
H3C1C2
H4C1C2
H5C2C1
H6C2C1
H7C2C1
H8C2C1
120.05
120.4
72.7
72.7
124.2
125.4
H4C1H3C2
H5C2C1H3
H6C2C1H3
H7C2C1H3
H8C2C1H3
180.0
89.5
－89.5
180.0
0
Chin. J. Ch. E. 15(1) 44 (2007)
Chin. J. Ch. E. (Vol. 15, No.1)
46
Table 1 (continued)
Reaction path
Bond angles θ, (o)
Bond length r, nm
reaction path 3
[C2H4-H]·
reaction path 4
C2C1
0.1312
H3C1C2
123.3
H4C1H3C2
H3C1
0.1087
H4C1C2
123.3
H5C2C1H3
179.2
H4C1
0.1087
H5C2C1
122.8
H6C2C1H4
－179.2
H5C2
0.1095
H6C2C1
122.9
H7C2C1H3
90.2
H6C2
0.1095
H7C2C1
128.7
H7C2
0.1710
C2C1
0.1822
H3C2C1
45.9
H4C1C2H3
－123.7
0.1398
H4C1C2
96.2
H5C1C2H3
0
H4C1
0.1091
H5C1C2
135.8
H6C1C2H3
123.7
H5C1
0.1102
H6C1C2
96.2
H7C2C1H3
0
H6C1
0.1091
H7C2C1
128.1
H7C2
0.1095
C2C1
0.1822
H3C2C1
25.5
H4C1C2H3
－1.18
H3C2
0.2114
H4C1H3
148.3
H5C1H4H3
177.6
H4C1
0.1095
H5C1H4
114.3
H6C1C2H4
－107.6
H5C1
0.1105
H6C1H5
124.5
H7C2C1H5
179.2
H6C1
0.1710
H7C2C1
123.3
H7C2
0.1087
Heat of formation and activation energy
The steady geometric configurations of all radiTable 2
Diagnostic parameters of transition states
Reaction path
Imaginary fre- Negative force
quency, cm－1 constant, N·cm－1
reaction path 1
－1431.2
－4.84
reaction path 2
－1334.1
－0.99
reaction path 3 [C2H5-H]·
－1062.4
－1.73
[C2H4-H]·
－1049.3
－0.48
TS1
－1246.0
－1.96
TS2
－1246.0
－1.96
reaction path 4
February, 2007
179.6
H3C2
The results of optimization geometry configurations indicate that there is only one transition state in
reaction paths (1) and (2). It also indicates that the
reaction path (3) is a consecutive reaction, and it is
composed of two elementary reactions, whose have
one transition state, respectively. There are two transition states in reaction path (4).
The optimized transition states have been further
confirmed by vibration analyses: they all have only one
imaginary vibration frequency, which are － 1431.2,
－
－ 1334.1, － 1062.4, － 1049.3, － 1246.0cm 1 and
－1
－1246.0cm , respectively. The optimized transition
states have also been further validated by force constant analyses: they all have only one negative eigenvalue, which are － 4.84, － 0.99, － 1.73, － 0.48,
－
－
－1.96cm 1 and －1.96cm 1, respectively. The results
are shown in Table 2.
2.2
Dihedral angles θ, (o)
cals and molecules involved in this reaction system
were obtained through geometric configuration optimization[18—21]. The values of heat of formation are
shown in Table 3. The data of activation energy and
the heat of formation of different reaction paths are
presented in Table 4.
Species
Table 3 The heat of formation
Heat of formaHeat of formaSpecies
－
tion, kJ·mol－1
tion, kJ·mol 1
CH4
－54.40
C2H6
－75.81
CH3·
124.44
C2H5·
72.43
CH2· (singlet)
473.29
C2H4
69.51
CH2· (triplet)
316.19
H·
217.79
CH·
613.75
The data in Table 4 indicate that the reaction
paths (2) and (4) are exothermic and the reaction path
(1) and reaction path (3) is endothermic.
The values in Table 4 also indicate that the activation energy of reaction path (2) is the lowest among
－
the four reactions, and it is 48.75kJ·mol 1. Reaction
path (3) is a consecutive reaction, which is controlled
by the second elementary reaction whose activation
energy is higher than that of the first one. Therefore,
the second elementary is the control step of reaction
path (3). There are two transition states during the
course of reaction path (4). However, the second transition state is the main transition state. Moreover, its
activation energy is higher than that of the first one.
Theoretical Study of Reaction Paths and Transition States on Conversion Methane into C2 Hydrocarbons Through Plasma
Table 4
47
Activation energy and heat of reaction (kJ⋅mol－1)
Reaction path 1
Reaction path 2
Reaction path 3-1
Reaction path 3-2
Reaction path 4
reactant
70.03
418.88
261.79
290.21
559.34
reaction complex
66.36
392.39
260.78
72.43
533.14
transition state
260.46
441.14
389.81
291.82
608.84(TS1)
291.82(TS2)
72.47(IM)
－69.78
264.40
262.95
141.98
－75.81
290.21
505.08
287.30
51.04
－494.69
28.42
214.87
－272.05
194.10
48.75
129.39
219.39
resultant of reaction complex
117.12
resultant of reaction
heat of reaction ΔH
activation energy E
275.38
75.70 (TS1)
219.35 (TS2)
Note: ΔH=∑Hf, resultant of reaction－∑Hf, reactant; E=∑Hf, transition state－∑Hf, reactant.
Thus, the reaction path (4) is not a favorable elementary reaction.
According to the energy analysis, reaction paths
(1) and (2) appear preferable to reaction paths (3) and
(4). This is consistent with the experimental fact that
the main resultant of reaction is ethane. The activation
energy of reaction path (2) is lower than that of reaction path (1). Moreover, reaction path (2) is an exothermic one. The theoretic calculation results suggest
that the resultant of reaction ethane of reaction come
mainly from reaction path (2).
Bond order analyses
The bond orders of transition states of the four
reaction paths are presented in Table 5. It indicates
that the carbon-carbon and C6 H4 bond order of the
transition state of reaction path (1) is 0.56 and 0.66,
respectively. However, the bond order of other three
bonds in methane and three bonds in methyl is yet
0.91and 0.93. This suggests that the carbon atom of
methane is closer to the carbon atom of methyl inorder
to form resultant ethane. Meanwhile, one hydrogen
atom breaks away from methane when reaction (1)
happens.
Table 5 also suggests that when reaction path (2)
occurs, the bond order of the formation of carbon-carbon bond, the transfer of hydrogen atom (H3)
from methane molecule with carbon atom (C1) of
methane molecule and with carbon atom (C2) of
singlet methylene is 0.36, 0.60 and 0.38, respectively.
These indicate that C1 C2 and C2 H3 bond is formed;
C2 H3 bond breaks. At this time, the transition state is
near to reactant, which belongs to the earlier transition
state.
It can be seen from Table 5, there are two elementary reactions in reaction path (3). The bond order
of the carbon-carbon is 0.56 and the bond orders of
C2 H5 and C2 H6 in methane molecules reduce from
1.0 to 0.74 in the first elementary reaction. This fact
suggests that the carbon-carbon bond has just formed
and the two hydrogen atom H5 and H6 will break away
from methane. The carbon-carbon bond order has
reached 1.75 in the second elementary reaction, which
Table 5
Bond order of transition states
Reaction path
Bond
Bond order
reaction path 1
C1 C6
0.56
0.66
reaction path 2
C6 H4
C1 H2, C1 H3
C1 H5
C6 H7, C6 H8
C6 H9
C1 C2
0.36
2.3
reaction path 3 [C2H5-H]·
[C2H4-H]·
reaction path 4
TS1
TS2
0.93
0.91
C1 H3
0.60
C2 H3
0.38
C2 H7, C2 H8
0.97
C1 H4, C1 H6
0.99
C1 H5
0.96
C1 C2
0.56
C2 H5
0.74
C2 H6
0.74
C1 C2
1.75
C2 H5
0.29
C1 C2
0.45
C1 H3
0.53
C1 H6
0.98
C2 H3
0.43
C1 C2
1.74
C1 H3
0
C1 H6
0.29
C2 H3
0.97
C2 H6
0
is very close to 2.0 of the carbon-carbon double bond
of ethylene molecules. Meanwhile, the bond order of
C2 H5 is 0.29, suggesting that the carbon-carbon
double bond has formed and one hydrogen atom has
Chin. J. Ch. E. 15(1) 44 (2007)
Chin. J. Ch. E. (Vol. 15, No.1)
48
been eliminated from the methane molecule.
Table 5 also shows that when reaction path (4) is
happening, the bond order of the carbon-carbon is
0.45 and the bond order of C1 H3 in methane molecule reduces to 0.53 in the first transition state. At the
same time, the bond order of C2 H3 reaches to 0.47 in
the first transition state. These suggest that the carbon-carbon bond has almost formed and the hydrogen
atom H3 that will break away from methane is transferring to methyl. The carbon-carbon bond order has
reached 1.74 in the second transition state, which is
very close to 2.0 of the carbon-carbon double bond of
ethylene molecules. Meanwhile, the bond order of
C1 — H3, H6 — C1 and C2 — H3 is 0, 0.29 and 0.97,
respectively, suggesting that carbon-carbon double
bond and C2—H3 have been formed. At the same time,
hydrogen atom H6 has dropped off from methane
molecule.
2.4
IRC analyses
On the basis of energy analysis, the reaction
paths (1) and (2) appear more preferable to reaction
paths (3) and (4). The experimental facts showed that
the main resultant of reaction is ethane. The intrinsic
reaction coordinate of reaction paths (1) and (2) was
analyzed. Potential energy curves, bond lengths curves
and bond angles of reaction paths (1) and (2) are
shown in Fig.2 to 7.
Figure 2
Figure 3
Potential energy versus No. of stagnation point
Bond length of C1—C6 and C1—H2 versus No. of
stagnation point
■ C1—H2; ● C1—C6
February, 2007
The relationship of potential energy with number
of stagnation point in the course of reaction path (1) in
Fig.2 indicates that the place of transition state is near
to resultant of reaction and the energy difference of
resultant of reaction to reactant is not large.
The variation of bond lengths of C1 C6 and
C1 H2 are shown in Fig.3 in the course of reaction of
methane with methyl before and after the formation of
transition state. The distance of C1 and C6 decreases
and the bond length of C1 and H2 has no obvious
change before the formation of transition state. The
bond length of C1 and C6 quickly decreases near the
formation of transition state. After that, the bond
length of C1 and C6 slowly decreases and the distance
of C1 H2 is elongated fast. This suggests that H2
completely breaks away from C1. It seems that the
breakage and formation of chemical bond is stepwise
and continuous, but not synchronously when reaction
process is carried through elementary reaction path (1).
Figure 4 shows that H2C1H3 bond angle gradually
diminishes when the reaction is going on according to
reaction path (1). There are two inflexions that are
corresponding to the inflexion of Fig.2 and Fig.3 at
stagnation point －10000 (towards reactant) and stagnation point 10000 (towards resultant of reaction),
respectively. This further reveals that the configuration
adjustment of transition state is obvious at stagnation
points －10000 and 10000.
Figure 4
Bond angle of H2C1H3 versus No. of
stagnation point
Figure 5 shows that the energy of reactant is
much higher than that of resultant of reaction in reaction path (2). The change of energy is obviously remarkable when the reaction is going on according to
reaction path (2). The energy slowly increases from
reactant to transition state. The energy reaches the
maximum while the transition state come into being.
And then, the energy of system drops fast. The energy
is the lowest after the product ethane is produced.
However, the energy barrier is low.
Figure 6 shows that the formation of carbon-carbon single bond is the main process through
the intrinsic coordinate analysis of reaction path (2)
before the formation of transition state. That is to say,
the carbon-carbon bond length decreases before the
formation of transition state. However, the distance
between carbon-carbon temporarily increases at the
Theoretical Study of Reaction Paths and Transition States on Conversion Methane into C2 Hydrocarbons Through Plasma
Figure 5
49
Potential energy versus No. of stagnation point
Figure 7
Bond angle of H3C1C2 versus No. of
stagnation point
3
Figure 6
Bond length of C1—C2, C1—H3 and C2—H3
versus No. of stagnation point
■ C1—C2; ● C1—H3; ▲ C2—H3
beginning of the formation of carbon-carbon bond.
The bond length of H3 C1 and H3 C2 has no obvious
change before the formation of transition state. The
reason is that hydrogen atom in tetrahedron methane
molecule transfers with the adjustment of bond angles
and dihedral angles in order to afford space to carbon-carbon bond. The carbon-carbon bond length is
always decreasing with the formation of transition
state. This indicates that the strength of carbon-carbon
bonding is also ceaselessly strengthened. At the same
time, H3 begins to transfer. It appears that the C1 H3
bond length increases and C2 H3 bond length decreases. The transfer of hydrogen lags behind the formation of carbon-carbon bond during the wholly reaction process.
It be can seen from Fig.7 that H3C1C2 bond angle
increases with the formation of transition states. The
bond angle comes to the maximum when the transition state has formed. And then, H3C1C2 bond angle
decreases during the course of transformation transition state into resultant of reaction. This indicates that
the bond angle is always changing when the reaction
process is taking place in order to meet the conformation and energy expectations.
CONCLUSIONS
① Transition states analysis indicates that there
is only one transition state in reaction paths (1) and (2),
but there are two transition states in reaction path (4).
It also indicates that reaction path (3) is a consecutive
reaction composed of two elementary reactions, with
one transition state for each, respectively.
② Reaction paths (2) and (4) are exothermic and
reaction paths (1) and (3) are endothermic reaction.
③ The activation energy of reaction path (2) is
the lowest among the four reaction processes, which is
－
48.75kJ·mol 1. Reaction path (3) is a consecutive reaction and the activation energy of the second elementary reaction is the highest among the four reactions. Therefore, the second elementary reaction is the
control step of reaction path (3). There are two transition states in reaction path (4). However, the second
transition state is the main transition state. Moreover,
its activation energy is higher. Thus, reaction paths (3)
and (4) are not favorable ones.
④ On the basis of energy analysis, reactions
paths (1) and (2) appear more preferable to reaction
paths (3) and (4). The main resultant of reaction is
ethane. Meanwhile, the activation energy of reaction
path (2) is lower than that of reaction path (1). The
theoretic calculation suggests that the product ethane
mainly comes from reaction path (2). This is consistent with the experiment.
⑤ The breakage and formation of chemical
bond is stepwise and continuous when reaction process is carried through elementary reaction path (1).
The transfer of hydrogen is later than the formation of
carbon-carbon bond when reaction process is carried
through reaction path (2).
REFERENCES
1
2
3
Liu, C., Mallinson, R., Lobban, L., “Comparative investigations on plasma catalytic methane conversion to
higher hydrocarbons over zeolites”, Applied Catalysis,
178(1), 17—27(1999).
Wang, B., Xu, G., “Plasma technology application in
natural gas chemical engineering”, J. Chem. Ind. Eng.,
51(12), 207—210(2000).
Wang, B., Xu, G., “Conversion natural gas to C2 hydrocarbons
Chin. J. Ch. E. 15(1) 44 (2007)
Chin. J. Ch. E. (Vol. 15, No.1)
50
4
5
6
7
8
9
10
11
12
through dielectric-barrier discharge plasma catalysis”,
Science in China (B), 45(3), 299—310(2002).
Istadi, Amin NAS, “Co-generation of synthesis gas and
C2+ hydrocarbons from methane and carbon dioxide in a
hybrid catalytic-plasma reactor: A review”, Fuel. 85(5/6),
577—592(2006).
Tarverdi, H.S., Mortazavi, Y., Khodadadi, A.A.,
Mohajerzadeh, S., “Synergetic effects of plasma, temperature and diluant on nonoxidative conversion of
methane to C2+ hydrocarbons in a dielectric barrier discharge reactor”, Iranian. Chem. & Chem. Eng. (Inter.
Eng. Ed.), 24(4), 63—71(2005).
Eliasson, B., Liu, C., Kogelschatz, U.,“Direct conversion
of methane and carbon dioxide to higher hydrocarbons
using catalytic dielectric-barrier discharges with zeolites”,
Ind. & Eng. Chem. Research, 39(5), 1221—1227(2000).
Lebedev, Y. A., Epshtein, I. L., “Ion composition of no
equilibrium hydrogen- methane plasma”, High Temp.,
36(4), 510—516(1998).
Hiraoka, K., Aoyama, K., Morise, K.A., “Study of reaction mechanisms of methane in a radio-frequency glow
discharge plasma using radical and ion scavengers”, Can.
J. Chem., 63(11), 2899—2905(1985).
Toshihiro, F., Ken-ichi, S., “Mass spectro-metric detection of neutral radicals in CH4 microwave discharge by
use of Li+ ion attachment techniques”, J. Appl. Phys.,
74(5), 3009—3015(1993).
Baker, J., “An algorithm for the location of transition
states”, J. Comp. Chem., 7, 385—389(1986).
Fletcher, R., “A new approach to variable-metric algorithms”, Computer Journal, 13, 317—322(1970).
Doubleday, C., Mclver, J., Page, M., Zielinski, T., “Tem-
February, 2007
13
14
15
16
17
18
19
20
21
perature dependence of the transition-state structure of
the disproportionation of hydrogen atom of the ethyl
radical”, J. American Chem. Society, 107, 5800 —
5801(1985).
Stewar, J. J. P., “Optimization of parameters for semi empirical methods”. J. Comp. Chem., 10, 209—220(1989).
Andrew, R. L., Molecular Modelling Principles and Applications, (pp90-105). London: Addison Wesley Longman (1996).
David, C. Y., Computational Chemistry: A Practical
Guide for Applying Technique to Real-world Problems,
(pp32-41). New York: John Wiley & Sons (2001).
Houk, K.N, Gonzalze, J., Li, Y., “Pericyclic reaction
transition states: passions and punctilios1993—1995 ”,
Accounts of Chemical Research, 28, 81—90(1995).
Vinter, J.G., “Extended electron distributions applied to
molecular mechanics of some intermolecular interactional”, J. Computer-Aided Molecular Design, 8, 653—
668(1994).
Judson, R.S., Jaeger, E.P., Treasurywala, A.M., Peterson,
M.L., “Conformational searching methods for small
molecules (Ⅱ) Genetic algorithm approach”, J. Comp.
Chem., 14(11), 1407—1414(1993).
Broyden, C.G., “The convergence of a class of double-rank minimization algorithms (2) The new algorithm”, J. Inst. Math. & Appl., 6, 222—231(1970).
Shanno, D.F., “Conditioning of quasi-Newton methods
for function minimization”, Mathematics of Computation,
24, 647—656(1970).
Goldfarb, D., “A family of variable-metric algorithms
derived by variational means”, Mathematics of Computation, 24, 22—26(1970).