Molecular dynamics simulation of liquid

JOURNAL OF CHEMICAL PHYSICS

VOLUME 115, NUMBER 23

15 DECEMBER 2001

Molecular dynamics simulation of liquid N2 O4 r2NO2 by orientationsensitive pairwise potential. I. Chemical equilibrium Toshiko Kato¯ Seibo Jogakuin Jr. College, Fukakusa, Fushimi-ku, Kyoto, 612-0878, Japan

Soichi Hayashi 51-2 Toji-in Nakamachi, Kita-ku, Kyoto, 603-8347, Japan

Katsunosuke Machida 2-1-308 Tatugaoka, Otsu-shi, 520-0803, Japan

共Received 29 June 2001; accepted 19 September 2001兲 This paper, the first of a series of papers, examines equilibrium properties of N2 O4 NO2 in liquid state by classical molecular dynamics simulations of liquid NO2 . An ab initio MO calculation has been carried out to elucidate NO2 – NO2 potential, and an orientation-sensitive pairwise potential 共OSPP兲, which can reproduce highly anisotropic character of covalent bonding between N–N, has been formulated. The OSPP potential is parameterized by the well depth D e and by two anisotropy factors: A ␪ (0⭐A ␪ ⭐1) the anisotropy factor for the rocking angle between NN bond and ONO direction, and A ␶ (0⭐A ␶ ⭐1) for torsional angle of the two NO2 about NN bond. The reactive liquid N2 O4 is modeled as liquid NO2 which interacts with the OSPP potential between N–N atoms and Lennard-Jones potentials between N–O and O–O atoms. Equilibrium properties were found to be very sensitive to the well depth D e and anisotropy factors of OSPP. The population of more than the NO2 dimer 共3-mer, 4-mer, . . . 兲 is considerable when anisotropy factors of the NN bond are small. On the other hand, the equilibrium liquid N2 O4 2NO2 is formed, that is, most NO2 form monomer or dimer and the population of more than 3-mer is very small when A ␪ ⫹A ␶ ⭓0.4– 0.5. In simulated liquid NO2 /N2 O4 , concentration of N2 O4 is found to increase as D e increases, A ␪ increases, and A ␶ decreases. The equilibrium constant for the dissociation reaction has been derived by computing the potential of mean force as a function of the N–N distance r c 共the reaction coordinate兲. The OSPP potential for D e ⫽0.12⫻10⫺18 J, A ␪ ⫽0.5 and A ␶ ⫽0.1 is found to reproduce the observed liquid phase equilibrium properties fairly well. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1417507兴

I. INTRODUCTION

Gas and liquid N2 O4 are known to be in thermal equilibrium for dissociation/association reactions N2 O4 2NO2 at room temperatures. The association of two NO2 produces N2 O4 with a planer structure (D 2h ) and an unusually long NN bond 共1.782 Å兲.1 The degrees of dissociation in the gas phase at the freezing point 共262.0 K兲 and at the boiling point 共294.3 K兲 are 8.6% and 15.9%, respectively.2,3 However, the dissociation is reported to be very much less in liquid phase 共0.018% at the freezing point and 0.12% at the boiling point兲.3 The Raman line shapes of the totally symmetric modes of liquid N2 O4 were measured at 262–298 K, and the vibrational and rotational relaxation functions of N2 O4 mol¯ and Katoˆ.4 These molecular ecules were calculated by Kato relaxations were also studied from molecular dynamics simulations of pure liquid N2 O4 by the present authors.5 In our previous papers, the reaction dynamics of a reactant NO2 pair activated above the dissociation limit in the gas phase and in inert solvent N2 O4 were studied in detail by molecular dynamics simulations including vibrational degrees of freedom.6 – 8 A detailed ab initio potential has not been available, and the potential between NO2 molecules was taken to be the sum of a Morse potential for the NN 0021-9606/2001/115(23)/10852/11/$18.00

bond with the observed dissociation energy D 0e ⫽56.9 kJ/ mol⫽0.0945⫻10⫺18 J and equilibrium N–N distance r 0 ⫽1.782 Å, and Lennard-Jones 共LJ兲 potentials between N–O and O–O atoms of parameters given in Table I. We named this potential Morse⫹LJ. A phase space surface E T ⫽0 was identified as the transition state 共TS兲 in both phases, where E T is the sum of the potential and kinetic energies of intermolecular motion. The dissociation or association reaction of an activated reactant NO2 pair in the gas phase was found to occur when the transition state surface is crossed by energy redistribution among vibrational, rotational, and intermolecular translational 共T兲 modes in the gas phase, while energy transfer between reactant’s T mode and solvent mode also plays an important role in crossing the TS in inert solvent. Real liquid N2 O4 is composed of N2 O4 and NO2 molecules, and it is in thermal equilibrium for reactions N2 O4 2NO2 , where N2 O4 molecule is the major species.3,9–11 All N2 O4 molecules are equivalent and have the same probability to dissociate. To model this property, all molecules in reactive liquid N2 O4 are assumed to be able to dissociate on the ground state potential energy surface in the present simulation. Results of the preliminary simulation of liquid NO2 interacting with Morse⫹LJ suggested interesting characters of chemical reactions in liquid state. However, the

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© 2001 American Institute of Physics

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Molecular dynamics simulation of the N2O4 2NO2 equilibrium

J. Chem. Phys., Vol. 115, No. 23, 15 December 2001 TABLE I. Atom–atom potential parameters of Lennard-Jones type for the NO2 – NO2 intermolecular interaction. Interaction

Parameters A(10

NO2 – NO2 N–O O–O

⫺18

6

JÅ )

B(10⫺18 J Å12)

0.5978 1.4177

115.14 646.3

potential Morse⫹LJ was found to overestimate the attractive forces between N–N because it neglects highly anisotropic character of chemical bonding. As a result, 3-mers and 4-mers of NO2 were often formed. In the present paper, we carry out ab initio MO calculation between NO2 molecules, and improve the Morse potential between N–N by incorporating the dependence of the attractive term on intermolecular orientation. Our goal in this paper is fourfold: first, to formulate an analytical intermolecular potential that can reproduce highly anisotropic character of chemical bonding; second, to invest a method of MD simulation for the orientation-sensitive pairwise potential and to carry out MD simulations of liquid NO2 for various OSPP potentials; third, to study chemical equilibrium of N2 O4 2NO2 in the liquid as a function of the potential parameters; and fourth, to derive the equilibrium constant by computing the potential of mean force as a function of the N–N distance r c 共the reaction coordinate兲. These results will be compared with observed liquid phase equilibrium properties. Microscopic reaction dynamics will be the subject in a forthcoming paper.

II. MOLECULAR DYNAMICS SIMULATION

In the present simulation of reactive liquid N2 O4 , all N2 O4 molecules are assumed to be able to dissociate on the ground state potential energy surface. If we define a molecule as an atomic aggregate which cannot dissociate, the simulation of reactive liquid N2 O4 can be considered as, in other words, the one of liquid NO2 . All atoms interact through intra- and intermolecular potentials, V intra and V inter , and the force Fi on the ith atom is given by Fi⫽⫺ⵜ i V intra ⫺ⵜ i V inter . The intramolecular potential of an NO2 molecule is assumed to be harmonic as to internal coordinates of NO2 : S 1 ⫽⌬r 1 , S 2 ⫽⌬r 2 , S 3 ⫽⌬ ␾ 12 , where r 1 and r 2 are the bond lengths between N and O and ␾ 12 is the angle between bonds 1 and 2. The equilibrium geometry of NO2 is assumed to be r 1 ⫽r 2 ⫽1.190 Å and ␾ 12⫽135.4°, and the force constants of NO2 in the internal coordinate representation are given in Table I of Ref. 7. A. Calculation of intermolecular potential by ab initio MO calculation

Here we carried out quantum mechanical ab initio MO calculations to elucidate the potential between a pair of NO2 molecules. It is calculated by the potential energy surface scans as a function of the NN distance for various relative intermolecular orientations: out-of-plane rocking, in-plane

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rocking, and torsional orientations 共see insets in Fig. 1兲. Energy calculations are performed using MP2 共Møller–Plesset兲 perturbation theory and 6-31G共d兲 basis sets. Examples of the calculated ab initio potential energy surfaces are compared with Morse⫹LJ potentials in Fig. 1. The ⫺18 calculated well depth D ab J for equilibrium e ⫽0.154⫻10 configuration is found to be considerably deeper than the observed dissociation energy D 0e ⫽0.0945⫻10⫺18 J, which is an average value of dissociation energy for various intermolecular orientations. The ab initio potential shows interesting behavior of chemical bond between N–N. As out-of-plane rocking angle ␪ out increases, Morse⫹LJ potential always has bonding character with shallower well depth, while the well of ab initio MO potential becomes very shallow and changes into repulsive potential for ␪ out⭓50° 关Figs. 1共a兲 and 1共b兲 for cis and trans out of plane rocking orientations兴. A similar trend is observed in the calculated ab initio potential for approach with in-plane-rocking orientation; it becomes repulsive for ␪ in⭓20° – 30° 关Fig. 1共c兲兴. Calculated ab initio potential shows a torsional barrier at rectangular orientation 关Fig. 1共d兲兴 as observed in vibrational spectroscopy, although the calculated barrier of about 0.068⫻10⫺18 J 共44% of D ab e ) is considerably higher than the barrier of about 0.014 ⫺0.021⫻10⫺18 J 共15%–22% of D 0e ) obtained from vibrational spectrum,12 and 0.011⫻10⫺18 共12% of D 0e ) from thermodynamic calculation.13 On the contrary, Morse⫹LJ gives a low torsional barrier at planer structure ␶ ⫽0 关Fig. 1共d兲兴. These comparisons show that the calculated ab initio potential is more repulsive than Morse⫹LJ for mutual orientations deviating from equilibrium N2 O4 configuration. This type of orientational dependence does not seem to be particular to the NO2 – NO2 interaction, but is general in most chemical bonding. The chemical bond is intrinsically highly anisotropic. Classical molecular dynamics simulations are usually performed for the atom–atom pairwise intermolecular potential which only depends on the distance between atoms. In order to study the reaction 共bond forming and breaking兲 dynamics in liquid state, it seems very important to invest a general method of molecular dynamics simulation that includes orientational dependence of the atom–atom potentials. Here we present an analytical potential that can reproduce highly anisotropic character of covalent bonding, between N–N in the present case. B. Orientation-sensitive pairwise potential between nonbonded atoms

Two atoms j and k belonging to different molecules are assumed to interact with each other according to a modified Morse potential, in which the attractive term is sensitive to mutual orientation of the two molecules in the form V 共 r jk 兲 ⫽V rep共 r jk 兲 ⫹V attr共 r jk 兲 ,

共1兲

⫽D e 兵 exp关 ⫺2a 共 r jk ⫺r eq jk 兲兴 ⫺2 f OR exp关 ⫺a 共 r jk ⫺r eq jk 兲兴其 ,

共2兲

r eq jk

where is the equilibrium distance between atoms j and k, and f OR , which let us call the orientation factor, is a function


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J. Chem. Phys., Vol. 115, No. 23, 15 December 2001

Kato¯, Hayashi, and Machida

FIG. 1. Examples of intermolecular potential between a pair of NO2 molecules by ab initio MO calculation are plotted as a function of N–N distance R共Å兲. Morse⫹LJ and OSPP⫹LJ potentials are shown for comparison. The relative orientations are: 共a兲cis out-of-plane rocking ( ␪ out); 共b兲 trans out-of-plane rocking ( ␪ out); 共c兲 trans in-plane rocking ( ␪ in); 共d兲 torsion ( ␶ ) as shown in the insets in MORSE⫹LJ of the figure, and angles are given in degrees.

of certain angular variables describing the orientation of the two molecules around the straight line connecting the atoms j and k. By introducing an orientation factor in the attractive term of the Morse potential, the potential covers from fully attractive Morse potential for f OR⫽1 to completely repulsive potential for f OR⫽0 as shown in Fig. 2. The potential can reproduce highly anisotropic character of chemical bonding, and we name this potential orientation-sensitive pairwise potential 共OSPP兲 between nonbonded atoms. This kind of orientational selectivity has been taken into account by intro⬘ in the bond potential as ducing an angle dependent factor f OR ⬘ V(r jk ) in the programs for biological macroV ⬘ (r jk )⫽ f OR molecules, CHARMM,14 and for complexes of proteins with small molecules, YETI.15 In our program of molecular dynamics simulation, the

potential and its first derivatives with respect to all the relevant atomic Cartesian coordinates are computed. Since f OR is independent of r jk , Eq. 共2兲 is differentiated with respect to r jk to give

⳵ V 共 r jk 兲 ⫽⫺2aD e 兵 exp关 ⫺2a 共 r jk ⫺r eq jk 兲兴 ⳵ r jk ⫺ f OR exp关 ⫺a 共 r jk ⫺r eq jk 兲兴其 .

共3兲

Let e jk be the unit vector along the line connecting the atoms j and k, with the direction from j to k. Then the relative orientation of the molecule containing the atom j against e jk is described in terms of the rocking angle ␪ j , which is de-




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FIG. 2. OSPP potential as a function of orientation factor. The potential can represent from fully attractive Morse potential for f OR⫽1 to completely repulsive potential for f OR⫽0. FIG. 3. Orientation subfactors: 共a兲 f ␪ ( ␪ ) for A ␪ ⫽0.5 and ␪ lim⫽60°, and 共b兲 f ␶ ( ␶ ) for A ␶ ⫽0.4.

fined as the angle formed by the vectors -e jk and t j , the latter being the sum of the bond unit vectors e ji ’s taken along all the bonds formed by j, tj ⫽

兺i eji .

共4兲

The mutual orientation of the two molecules containing j and k is specified by all the angles of types ␾ i jk and ␶ i jkl . Here ␾ i jk is the angle spanned by vectors e jk and e ji , and ␶ i jkl is the dihedral angle between the plane defined by vectors e jk and e ji and that defined by e jk and ekl . In order that the most probable orientation of the two molecules corresponds to the equilibrium structure of the association product of the interaction, the orientation factor is chosen to satisfy the condition 0⭐ f OR共 ␪ j ⬘ s, ␾ i jk ⬘ s, ␶ i jkl ⬘ s 兲 eq eq ⭐ f OR共 ␪ eq j ⬘ s, ␾ i jk ⬘ s, ␶ i jkl ⬘ s 兲 ⫽1,

共5兲

eq eq where ␪ eq j , ␾ i jk , and ␶ i jkl are the equilibrium intermolecular angles. Furthermore, the orientation factor is assumed to be written as a product of three subfactors in the form

f OR⫽ f ␪ 共 ␪ j 兲 f ␪ 共 ␪ k 兲 f ␶ 共 ␾ i jk ⬘ s, ␶ i jkl ⬘ s 兲 .

共6兲

The first two subfactors are a single variable function of the rocking angle ␪ j or ␪ k (0⭐ ␪ j , ␪ k ⭐ ␲ ) in the form f ␪共 ␪ j 兲 ⫽

再

1⫺A ␪ ⫹A ␪ 1⫺A ␪ ,

冉

cos ␪ j ⫺cos ␪ lim 1⫺cos ␪

lim

冊

2

,

if 0⭐ ␪ j ⭐ ␪ lim ,

if ␪ j ⭓ ␪ lim 共7兲

where 0⭐A ␪ ⭐1 and 0⭐ ␪ ⭐ ␲ . f ␪ ( ␪ j ) changes in the range from 1 at ␪ j ⫽0 to 1⫺A ␪ at ␪ j ⭓ ␪ lim as shown in Fig. 3共a兲, and we call A ␪ rocking anisotropy, or an anisotropy factor for rocking angle, and ␪ lim the limiting rocking angle for suitable bonding. lim

The last subfactor in Eq. 共6兲 includes several variables as given by f ␶ 共 ␾ i jk ⬘ s, ␶ i jkl ⬘ s 兲 ⫽1⫺

mj

⫻

兩 f n兩

兺n 兩 f n␶ 兩 ⫹ 兺n 2m j␶m k mk

兺兺 F 共 cos ␾ i jk 兲 F 共 cos ␾ jkl 兲

i⫽1 l⫽1

⫻ 共 1⫹ p ␶n cos n ␶ i jkl 兲 ,

共8兲

where m j and m k are the numbers of bonds formed by the interacting atoms j and k, respectively, and p ␶n ⫽ f n␶

冒再兩 f n␶ 兩 ⫽

⫹1 if f n␶ ⬎0, ⫺1 if f n␶ ⬍0.

共9兲

In Eq.共8兲, n⫽2, m j ⫽m k ⫽2 for the case that both j and k are bivalents such as NO2 -NO2 , and n⫽3, m j ⫽m k ⫽3 for interactions between trivalents such as CH3 –CH3 . In the former case, f ␶2 must be positive if the association product is planer. The factor F(cos ␾ i jk ) is introduced in Eq. 共8兲 in order to avoid the blow-up of the atomic Cartesian derivatives of f ␶ in the limit that ␾ i jk approaches 0 or ␲ , and it is given in Appendix A. F(cos ␾ i jk ) takes the maximum value 1 at ␾ i jk ⫽ ␾ eq i jk , and the minimum value 0 at ␾ i jk ⫽0 or ␲ . The torsional anisotropy A ␶ ⫽ 兺 n 兩 f ␶n 兩 in Eq. 共8兲 shows an anisotropy factor about torsional angle; A ␶ ⫽ 兩 f 2␶ 兩 for NO2 – NO2 interaction, and f ␶ changes from 1 at ␶ i jkl ⫽0° to 1⫺A ␶ at ␶ i jkl ⫽90° for f 2␶ ⬎0 and ␾ i jk ⫽ ␾ eq i jk 关see Fig. 3共b兲兴. In the following molecular dynamics simulations using OSPP poeq eq tentials, it is assumed that ␪ eq j ⫽ ␶ i jkl ⫽0 and ␾ i jk ⫽112.3° from the equilibrium planer N2 O4 structure. The limiting rocking angle was set ␪ lim⫽60°. We have formulated the OSPP potential between nonbonded atoms, which is characterized by the dissociation energy D e and two anisotropy factors for rocking and torsional


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angles. The method of computing the force, the first derivatives of OSPP with respect to all the relevant atomic Cartesian coordinates is given in Appendix B.

C. Intermolecular potential and initial conditions

The potential between a pair of NO2 molecules is modeled as a sum of OSPP potential between N–N atoms with r 0 ⫽1.782 Å, a⫽2.4987 Å⫺1 , 6 and Lennard-Jones potentials between N–O and O–O atoms of parameters given in Table I. We name this potential OSPP⫹LJ. In this formulation, Morse⫹LJ in our preceding papers6 – 8 corresponds to OSPP⫹LJ for D e ⫽0.0945⫻10⫺18 J and A ␪ ⫽A ␶ ⫽0. OSPP⫹LJ potentials are compared with ab initio MO potentials and Morse⫹LJ potentials in Fig. 1 for approach with rocking and torsional deviations from equilibrium orientation. The range of suitable rocking anisotropy that can reproduce the orientation dependence of ab initio potentials is A ␪ ⫽0.3– 1.0 for approach with cis and trans out-of-plane rocking deformations as is shown in Figs. 1共a兲 and 1共b兲, respectively. The suitable range is A ␪ ⭓0.5 for approach with trans in-plane rocking deformation 关Fig. 1共c兲兴, and the characteristic orientation dependence is reproduced, irrespective of A ␪ values, for approach with cis in-plane rocking deformation. The above results are combined to give a suitable range of rocking anisotropy A ␪ ⫽0.5– 1.0. It is noted that OSPP⫹LJ reproduces the orientation dependent bonding character of ab initio potential far better than Morse⫹LJ, but still ab initio potential is more repulsive than OSPP⫹LJ for mutual orientations deviating from equilibrium N2 O4 structure. We have seen that the torsional barrier of the ab initio potential is about 44% of the calculated dissociation energy D ab e , and that the barrier height from vibrational spectroscopy and thermodynamic calculation are around 12%–22% of the observed dissociation energy D 0e . 12,13 Considering that the observed dissociation energy D 0e is less than the one for equilibrium N2 O4 orientation, the range of suitable torsional anisotropy is presumed to be around A ␶ ⫽0.05⫺0.15, which gives the torsional barrier of height 0.06– 0.22D e 关Fig. 1共d兲兴. The reactive liquid N2 O4 is modeled as liquid NO2 which interact with the OSPP⫹LJ potential. Most simulations were carried out for OSPP of the observed dissociation energy D 0e , and some simulations were for deeper well ⫺18 J, the latdepths D 1e ⫽0.12⫻10⫺18 J and D ab e ⫽0.154⫻10 ter is the one from ab initio calculation. A system of 125 N2 O4 共250 NO2 ) molecules is contained in a periodic cube of length 23.40 Å, which corresponds to the density 1.491 g cm⫺3 of liquid N2 O4 at 273 K. A modified Verlet integration algorithm16 is used with a periodic boundary condition. The time step (⌬t兲 is set to 1/2048 ps, and most simulations were run for 64 ps. Temperature is adjusted every 32 steps. Simulations for the shallowest well depth D 0e were initiated from an equilibrium configuration and momenta of all atoms in the molecular dynamics simulation of pure liquid N2 O4 共we name this initial condition IN1兲,5 and an intramolecular potential of N2 O4 was replaced at t⫽0 by an intermolecular OSPP⫹LJ potential and an intramolecular potential of NO2 . The initial condition of simulations for deeper


well depths D 1e and D ab e was prepared carefully as follows. Starting from IN1, intramolecular potential of N2 O4 was replaced by a deep OSPP⫹LJ of D e ⫽0.30⫻10⫺18 J, A ␪ ⫽0.5, A ␶ ⫽0.1 between NO2 molecules and intramolecular potential of NO2 . The simulation during 32 ps gave pure N2 O4 (NO2 dimer兲 liquid structure, and next simulation at D e ⫽D 1e , A ␪ ⫽0.5, A ␶ ⫽0.1 gave completely dimerized N2 O4 structures during 32 ps of simulation, where some bonds are excited with NN distance 2.1Å⭐r c ⭐2.2Å. The final data was taken as the second initial condition IN2. Some simulations for D 0e were carried out both from IN1 and IN2 to see the effect of initial condition. All NO2 molecules are equivalent and no molecule is activated in these simulations. Equilibrium properties are derived by studying the change of NO2 clustering structure as a function of the simulation time. In order to see the temperature dependence of equilibrium properties, (N,V,T) simulations have been carried out at 273 K and 373 K. III. RESULTS AND DISCUSSIONS A. Dependence of chemical equilibrium on OSPP potential

In the simulation of reactive liquid N2 O4 initiated from IN1, the clustering structure of NO2 changes as a function of simulation time. Considering that two NO2 molecules are bound if their NN distance is smaller than 2.1 Å , the distribution of the size of NO2 clusters connected by NN bonds has been derived for D e ⫽D 0e and for anisotropy factors in the range 0⭐A ␪ ⭐1 and 0⭐A ␶ ⭐0.5 at ␪ lim⫽60°. The fractions ␣ i (i⫽1,2,3,⭓4) of NO2 ’s to form monomers, dimers, 3-mers, more than 3-mers are plotted as a function of simulation time in Fig. 4. Cluster size distribution, which initiates from 125 NO2 dimers, changes immediately after the abrupt change of intramolecular N2 O4 potential to intermolecular NO2 – NO2 potential at t⫽0, but afterward it quickly tends to constant distribution. The lifetime of initially prepared NO2 dimer may help to see whether the simulated liquid is in near equilibrium or not. In each simulation initiated from IN1, the number of surviving N2 O4 molecules N was plotted as a function of the simulation time as shown in Fig. 5. We defined an N2 O4 as surviving if the NO2 pair keeps the dimer structure of the NN distance within 2.1 Å with the same partner, and as dissociated if the pair is separated or if the pair is bound but forms more than dimer with other NO2 ’s. The N-t plot shows a jump from 125 to N 0 at t⫽0 corresponding to the abrupt change of potential, but afterward it decays monotonically. The lifetime ␶ D of initially prepared NO2 dimer has been derived by fitting the simulation data to N⫽N 0 exp(⫺t/␶D). The solid line shows our best fit of the data. The lifetime ␶ D changes from 1 ps to 10 ns as A ␪ increases from 0 to 1 at A ␶ ⫽0, and ␶ D decreases at most ⬃10⫺3 times as A ␶ increases at constant A ␪ . The state generated from IN1 may be considered to be in near equilibrium for systems characterized by short lifetimes ␶ D ⱗ100 ps. However, there may be some deviation from equilibrium for liquids of long ␶ D . The average values of ␣ i during 10– 64 ps for D 0e have been derived, and their dependence on anisotropy factors of




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FIG. 4. The fractions ␣ i (i⫽1,2,3, ⭓4) of NO2 ’s to form clusters connected by NN bonds are plotted as a function of simulation time in reactive liquid N2 O4 at 273 K for D e ⫽D e0 , ␪ lim⫽60°, A ␪ ⫽0.3, 0.4, and A ␶ ⫽0.1, 0.3: 䊉 monomer; 䊏 dimer; - 3-mer; ⫻ ⭓4-mer NO2 . Cluster size distribution, which initiates from 125 NO2 -dimers, changes immediately after t⫽0, but afterward it quickly tends to equilibrium distribution.

OSPP at 273 K is plotted in Fig. 6. Equilibrium properties of the reactive liquid N2 O4 are found to be very sensitive to the anisotropy factors of OSPP between N–N atoms. The population of more than NO2 dimer 共3-mer, 4-mer, 5-mer, . . . 兲 are considerable when anisotropy factors of the NN bond are small 共e.g., when A ␪ ⭐0.3 at A ␶ ⫽0 and A ␪ ⭐0.2 at A ␶ ⫽0.2; see Fig. 6兲. On the contrary, the equilibrium liquid NO2 /N2 O4 is formed, that is, most NO2 form monomer or

dimer and the population of more than 3-mer becomes small when A ␪ ⫹A ␶ ⭓0.4– 0.5. Thus both rocking and torsional anisotropies are found to determine the equilibrium between monomer⫹dimer and more than dimer 共3-mer, 4-mer, ..兲, i.e., whether the liquid is just NO2 liquid with some clustering structure or equilibrium liquid NO2 /N2 O4 . Torsional anisotropy of the NN bond plays an important role in determining the equilibrium of liquid NO2 /N2 O4 . Concentration of monomer NO2 increases 共equilibrium shifts to dissociation兲 as the torsional anisotropy A ␶ increases at constant A ␪ 关Fig. 6共b兲兴. On the other hand, change of A ␪ gives less effect on the equilibrium. In equilibrium liquid N2 O4 2NO2 where the fractions of NO2 ’s to form monomers and dimers satisfy ␣ 1 ⫹ ␣ 2 ⫽1, the mole fractions of two species are given as x N2O4 ⫽(1⫺ ␣ 1 )/(1⫹ ␣ 1 ) and x NO2 ⫽2 ␣ 1 /(1⫹ ␣ 1 ), and the equilibrium constant K x in terms of mole fractions can be obtained as a function of the degree of dissociation ␣ 1 as K x⫽

4 ␣ 21 1⫺ ␣ 21

,

共10兲

where activity coefficients are assumed to be close to unity. The equilibrium constant in terms of concentration is obtained as K c 共mol/l兲⫽ K x ⫻ 16.206 共mol/l of liquid N2 O4 ), and K c is related to the free energy difference ⌬G * l between N2 O4 共reactant兲 and separated NO2 pair in solvent NO2 / N2 O4 共product兲 as3 K c ⫽e ⫺ 共 ⌬G l* /RT 兲 . FIG. 5. The number of surviving NO2 pair of molecules in a simulation initiated from IN1 at 273 K is plotted as a function of the simulation time for D e ⫽D e0 and ␪ lim⫽60°; 共a兲 A ␪ ⫽0.4, A ␶ ⫽0.2, 共b兲 A ␪ ⫽0.5, A ␶ ⫽0.1. Dots show the simulation data and solid lines show our best fits of the data to single exponential decay with ␶ D shown in the figure.

共11兲

The equilibrium constants for liquid N2 O4 2NO2 have been derived by using Eq. 共10兲, and some of them are listed in Table II. The present OSPP⫹LJ potential is found to give a variety of liquids from K x ⬎1 at high A ␶ to K x ⬍1 at low


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FIG. 6. The average fractions of the size of NO2 cluster in reactive liquid N2 O4 for D e ⫽D e0 and ␪ lim⫽60° at 273 K are plotted 共a兲 as a function of A ␪ at constant A ␶ , and 共b兲 as a function of A ␶ at constant A ␪ : 䊊 monomer; 䊐 dimer; 䉱 3-mer; ⫻ ⭓4-mer NO2 .

A ␶ . Simulations for deeper well depths D 1e and D ab e show similar dependence of ␣ i on anisotropy factors of OSPP. Dominant species are NO2 and N2 O4 for A ␪ ⫹A ␶ ⭓0.4⫺0.5 as for D 0e , and the degree of dissociation decreases as the well depth increases. The liquid becomes pure N2 O4 ( ␣ 1 ⫽0兲 at D e ⫽D ab e , A ␪ ⭓0.4 and A ␶ ⫽0. The free energy of dissociation in equilibrium liquid N2 O4 2NO2 has been calculated by using ⌬G * l ⫽ ⫺RT ln(Kc), and the results are compared with the experimental data in Table II. The simulated free energy of dissociation 20–24 kJ/mol for D e ⫽D 1e ⫺D ab e , A ␪ ⫽0.4⫺0.5, and A ␶ ⭐0.1, is a little smaller than the experimental value 27.9 kJ/mol in pure liquid N2 O4 , 9,10 and is nearly the same as the dissociation free energy of N2 O4 in solution17 共Table II兲. However, we cannot calculate precisely the equilibrium constant of completely dimerized liquid ( ␣ 1 ⫽0) from fractions of molecular clusters in the molecular dynamics simulation of NO2 /N2 O4 liquid. Calculation of free energy of dissociation from the potential of mean force will help to determine the equilibrium constant more accurately 关Eq. 共11兲兴. Increasing the well depth of OSPP from the shallowest D 0e up to D ab e slows down ␶ D from 1 ns to ⬃10 ns at A ␪ ⫽0.5, A ␶ ⫽0.1. The liquid becomes almost static 共reaction seldom occurs during the simulation time兲 for deeper well depths D 1e and D ab e . Simulations at 273 K and 373 K show that equilibrium shifts to dissociation as the temperature increases. This temperature dependence is in accordance with experimental observation. The degree of dissociation in real liquid N2 O4 is found to be very low, ␣ 1 ⫽0.000 27 at 273 K.9 The equilibrium constant has been obtained by measuring optical

absorption,9,11 magnetic susceptibility,9 electron spin resonance,10 and 1 H n.m.r. shift.11 It should be noted that simulated equilibrium constants are dependent on the limiting distance of the NN bond 共we adopted here 2.1 Å兲. In order to compare the simulated results with the experimental data in more detail, the definition of the NN bond should be examined so as to accord with the method of measurement of NO2 concentration. The observed dominance of N2 O4 in real liquid corresponds to completely dimerized liquid NO2 for OSPP⫹LJ of D e ⫽D 1e ⫺D ab e , A ␪ ⭓0.4, and A ␶ ⭐0.1. This range agrees with the range of suitable anisotropy factors, A ␪ ⫽0.5– 1.0, A ␶ ⫽0.05– 0.15, derived from ab initio calculation and experimental data. The molecular structure of N2 O4 has been determined to be planer (D 2h ) by x-ray18 and neutron diffraction19 in crystals and by electron diffraction in the gas phase.20 Timeaveraged structural distribution of stable NO2 pairs in these OSPP⫹LJ liquids (A ␪ ⫽0.4⫺0.5, A ␶ ⫽0⫺0.2) has been calculated. The rocking angle between NN bond and ONO direction has maximum distribution at ␪ ⫽0° 关 P( ␪ )/sin ␪ in Fig. 7共a兲兴, in accordance with planer N2 O4 structure in the gas and crystal phases. The calculated N–N distance of maximum distribution was 1.85 Å for D e ⫽D 1e , A ␪ ⫽0.5 and A ␶ ⫽0.1 关Fig. 7共c兲兴, a little longer than the equilibrium N–N distance r 0 ⫽1.782 Å for OSPP potential at f OR⫽1. However, the distribution of the torsional angle ␶ in OSPP⫹LJ liquids showed rather equal probability 关Fig. 7共b兲兴, with increasing preference of planer structure ␶ ⫽0 as A ␶ increases from 0 to 0.2. This suggests that the torsional barrier may be easily passed in liquid state at ambient temperatures. The corresponding structural data in liquid state are not available.




10859

TABLE II. Averaged fractions ¯␣ i (i⫽1,2,3,⭓4) of NO2 ’s to form monomers, dimers, 3-mers, more than 3-mers, equilibrium constants for N2 O4 2NO2 , and free energy of dissociation in reactive liquid N2 O4 at 273 K. ⬁ denotes that ⌬G l* cannot be obtained because ␣ 1 ⫽0. OSPP⫹LJ potentials are for D e ⫽D e0 ⫺D eab , 0⭐A ␪ ⭐0.5, ␪ lim⫽60°, and 0⭐A ␶ ⭐0.5. The experimental data are also listed. Monomer

Dimer

3-mer

⭓4-mer

0.097 0.534 0.949 0.300 0.713 0.978 0.016 0.060 0.174 0.640 0.986 0.001 0.013 0.136 0.618 0.982

0.071 0.271 0.051 0.359 0.269 0.022 0.984 0.922 0.816 0.360 0.014 0.999 0.987 0.864 0.382 0.018

0.213 0.167 0.000 0.307 0.017 0.000 0.000 0.018 0.010 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.618 0.028 0.000 0.033 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

4.1 87.9 1.0⫻10⫺3 1.4⫻10⫺2 1.2⫻10⫺1 2.8 135 7.1⫻10⫺6 7.1⫻10⫺4 7.5⫻10⫺2 2.5 107

0.0007 0.0013 0.033 0.000 0.002 0.045

0.9993 0.9987 0.967 1.000 0.998 0.955

0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.0007 0.014 0.000 0.0007 0.005

1.000 0.9993 0.986 1.000 0.9993 0.995

0.000 0.000 0.000 0.000 0.000 0.000

System

T共K兲

Monomer

Pure N2 O4

273.1 275.7 298 298 298

0.0005 0.0006

A␪

A␶

K c 共mol/l兲

⌬G l* 共kJ/mol兲

1.7⫻10⫺2 2.3⫻10⫺1 2.0⫻100

9.3 3.3 -1.6

1.2⫻10⫺4 1.2⫻10⫺2 1.2⫻100

20.6 10.1 ⫺0.5

1.8⫻10⫺6 7.1⫻10⫺6 4.4⫻10⫺3 0 1.0⫻10⫺5 8.0⫻10⫺3

2.9⫻10⫺5 1.2⫻10⫺4 7.2⫻10⫺2 0 1.7⫻10⫺4 1.3⫻10⫺1

23.7 20.6 6.0 ⬁ 19.8 4.6

0.000 0.000 0.000 0.000 0.000 0.000

0 1.8⫻10⫺6 7.8⫻10⫺4 0 1.8⫻10⫺6 1.1⫻10⫺4

0 2.9⫻10⫺5 1.3⫻10⫺2 0 2.9⫻10⫺5 1.8⫻10⫺3

⬁ 23.7 9.9 ⬁ 23.7 14.3

Dimer

Kx

K c 共mol/l兲

⌬G * l 共kJ/mol兲

Ref.

0.9995 0.9994

2.83⫻10⫺7 3.14⫻10⫺7

4.60⫻10⫺6 5.13⫻10⫺6 1.77⫻10⫺4 1.78⫻10⫺4 0.30⫻10⫺4

27.9 27.9 21.4 21.4 25.8

9 10 17 17 17

Kx

D e ⫽D e0 0.0

0.2

0.4

0.5

0.4

0.5

0.4

0.5

0.1 0.3 0.5 0.1 0.3 0.5 0.0 0.1 0.2 0.3 0.5 0.0 0.1 0.2 0.3 0.5 D e ⫽D e1 0.0 0.1 0.2 0.0 0.1 0.2 D e ⫽D eab 0.0 0.1 0.2 0.0 0.1 0.2

36.0

Experimental

N2 O4 /C6 H6 N2 O4 /CCl4 N2 O4 /CH3 CN

Encouraged by the ability of the OSPP⫹LJ potential to reproduce the observed structure as well as chemical equilibrium, let us study the completely dimerized liquid NO2 in more detail as a model of real liquid N2 O4 . Here we shall focus on liquid NO2 for D e ⫽D 1e , A ␪ ⫽0.5, ␪ lim⫽60°, A ␶ ⫽0.1. B. Calculation of equilibrium constant from the potential of mean force

In order to simulate accurately the equilibrium constant of N2 O4 2NO2 liquid, we have to calculate the free energy difference, ⌬G l* , between N2 O4 and separated NO2 pair in the extreme limit of pure N2 O4 as solvent. The potential of mean force w(r c ), which represents the relative free energy of the system as a function of the N–N distance r c 共the reaction coordinate兲, is given by21

w 共 r c 兲 ⫽⫺kT ln g 共 r c 兲 ,

共12兲

where g(r c ) represents the probability of occurrence of each value of r c . Importance sampling has been used to cover an adequate range of r c . 21 An arbitrary NO2 pair was chosen as reactant in the initial condition IN2, and a series of simulations is run with biasing functions u(r c ) that center the sampling in different overlapping regions of r c 共windows兲. Distribution function was obtained by averaging over configurations during 16 – 48 ps after equilibration for 16 ps. The true distribution function g(r c ) in each range can be recovered from the resultant distribution function g ⬘ (r c ) via g(r c )⫽ng ⬘ (r c )exp (u(r c )/kT), where n is the normalization constant. The resultant individual potential of mean force ⫺kT ln g⬘(rc)⫺u(rc)⫹constant from different windows are then spliced together to yield the overall continuous w(r c ) by adjusting each constant. Calculated potential of


10860



rium constant.3,9 Simulated ⌬G * gives K c ⫽5.0⫻10⫺6 l mol/l, a very dilute solution of NO2 in solvent N2 O4 . All the 250 NO2 molecules in the simulation should form 125 N2 O4 at this equilibrium constant, and this is what is observed. The calculated ⌬G * l in liquid NO2 /N2 O4 is substantially higher than the change in free energy of N2 O4 →2NO2 in the gas phase: ⌬G g ⫽9.2 kJ/mol at 273.1 K,13 and ⌬G g ⫽4.8 kJ/mol at 298.15 K (⌬H⫽57.2 kJ/mol is lowered by the large entropy increase ⌬S⫽175.8 J K⫺1 mol⫺1 at 298.15 K, where pure gas at 1 atm is the reference state兲.22 This seems to be the reason why dissociation is very much less in liquid phase than the one in the gas phase. IV. CONCLUDING REMARKS

FIG. 7. Time-averaged structural distribution of the stable NO2 pair in OSPP⫹LJ liquids (D e ⫽D e1 , A ␪ ⫽0.5, A ␶ ⫽0.1); 共a兲 P( ␪ )/sin␪ for rocking angle 共deg兲; 共b兲 P( ␶ ) for torsional angle 共deg兲; 共c兲 P(r c ) for N–N distance 共Å兲.

mean force for D e ⫽D 1e , A ␪ ⫽0.5, A ␶ ⫽0.1, is shown in Fig. 8. It shows a minimum at r c ⫽1.85 Å of depth 27.7 kJ/mol and a barrier of height 6.3 kJ/mol at about r c ⫽2.44 Å, the latter is presumably due to the solvent effect. The calculated free energy difference of dissociation in liquid phase ⌬G * l ⫽27.7 kJ/mol at 273 K agrees with ⫽27.9 kJ/mol derived from the experimental equilib⌬G * l

FIG. 8. Calculated potential of mean force of NO2 dimer in reactive liquid N2 O4 at 273 K 共thick curve兲 for D e ⫽D e1 , A ␪ ⫽0.5, ␪ lim⫽60°, A ␶ ⫽0.1 is shown as a function of the reaction coordinate r c 共N–N distance兲. It shows a minimum at r c ⫽1.85 Å of depth 27.7 kJ/mol and a barrier of height 6.3 kJ/mol at about r c ⫽2.44 Å , the latter is due to the solvent effect. OSPP⫹LJ potential for D e0 between NO2 molecules in the equilibrium orientation is shown for comparison 共thin curve兲.

In order to study the equilibrium properties of N2 O4

2NO2 in liquid state, ab initio MO calculation has been carried out to elucidate NO2 -NO2 potential, and an orientation-sensitive pairwise potential 共OSPP兲 which can reproduce highly anisotropic character of covalent bonding between N–N has been formulated. The OSPP potential between NN is parameterized by the well depth D e and by two anisotropy factors for intermolecular rocking and torsional angles. The reactive liquid N2 O4 is modeled as liquid NO2 which interact with OSPP⫹LJ, a sum of OSPP potential between N–N atoms and Lennard-Jones potentials between N–O and O–O atoms. The range of suitable anisotropy factors which reproduces the orientation dependence of ab initio potential and observed torsional barrier height has been derived to be 0.5⭐A ␪ ⭐1.0 and 0.05⭐A ␶ ⭐0.15. The general method of computing the force, the first derivatives of the orientation dependent atom–atom potential with respect to relevant atomic Cartesian coordinates has been given. This is an important step for studying the reaction equilibrium and dynamics in liquid state. Molecular dynamics simulation of reactive liquid N2 O4 has been carried out by using the OSPP⫹LJ potential between NO2 molecules, and we succeeded to form N2 O4 molecules of varying lifetimes from 1 ps to 10 ns. The model potential Morse⫹LJ in our preceding papers was found to overestimate the attractive forces between NN and hence 3-mers and 4-mers were often formed. Equilibrium liquid N2 O4 2NO2 is found to be formed for OSPP⫹LJ of considerable anisotropy A ␪ ⫹A ␶ ⭓0.4⫺0.5, and the fractions of the two species varies with changes of anisotropy factors. The observed dominance of N2 O4 in pure liquid state is reproduced in simulated liquid NO2 for OSPP⫹LJ of large rocking anisotropy 0.5⭐A ␪ ⭐1, small torsional anisotropy A ␶ ⭐0.1, and deep well depth D 1e ⬃D ab e . This accords with the range of anisotropy factors that reproduces the orientation dependence of ab initio potentials and observed torsional barrier height. The equilibrium constant for dissociation reaction has been derived by computing the potential of mean force as a function of the N–N distance r c 共the reaction coordinate兲. The OSPP potential for D e ⫽D 1e , A ␪ ⫽0.5, ␪ lim ⫽60°, and A ␶ ⫽0.1, is found to reproduce the observed liquid phase equilibrium properties fairly well. The dissociation and association dynamics in pure liquid state will be the subject in a forthcoming paper.




APPENDIX A: F „cos ␾ ijk … IN TORSIONAL SUBFACTOR f ␶ „ ␾ ijk ’s, ␶ ijkl ’s…

The factor F(cos ␾ i jk )F(cos ␾ jkl ) is introduced in Eq. 共8兲 in order to avoid the blow-up of the atomic Cartesian derivatives of f ␶ in the limit that either ␾ i jk or ␾ jkl approaches 0 or ␲ . In that case a finite change of the torsional angle ␶ i jkl may be caused by an infinitesimal displacement of an end atom i or l. The dependence of the torsional potential must thus be suppressed by any factor vanishing at the limit ␾ i jk and/or ␾ jkl →0 or ␲ . The function F(cos ␾ i jk ) is so assumed to be given by F 共 cos ␾ i jk 兲 ⫽ 共 1⫺cos2 ␾ i jk 兲关共 1⫺b cos ␾ i jk 兲 2 ⫺b 2 兴 ,

共A1兲

d cos ␪ j ⫽⫺

1 tj

dt j ⫽

1 tj

兺

i⬍i ⬘

共A2兲

␾ i jk ⫽ ␾ eq i jk ,

and the that it takes the maximum value 1 at minimum value 0 at ␾ i jk ⫽0 or ␲ . A simpler factor sin ␾ i jk sin ␾ jkl which vanishes if either ␾ i jk or ␾ jkl approaches 0 or ␲ has been introduced by Ermer23 in the calculation of torsional barriers.

冊

d cos ␾ i jk ⫹cos ␪ j dt j .

d cos ␤ i ji ⬘ .

共B6兲

共B7兲

Combining Eqs. 共B6兲, 共B7兲 leads to 1 cos ␪ j ⵜ cos ␾ i jk ⫺ 2 ⵜ ji cos ␤ i ji ⬘ , t j ji t j i ⬘ ⫽i 共B8兲

兺

ⵜ ji cos ␪ j ⫽⫺

and ⵜ jk cos ␪ j ⫽⫺

␾ eq i jk ,

i

Taking the squares of Eq. 共B5兲 and differentiating, we have

where 2 b⫽⫺cos ␾ eq i jk /sin

冉兺

10861

1 ⵜ cos ␾ i jk . t j jk

共B9兲

The gradients of cos ␾ i jk are calculated as follows. Differentiating both sides of the identity, r jk r ji cos ␾ i jk ⬅r jk •r ji , we have ⵜ jk 共 r jk r ji cos ␾ i jk 兲 ⫽e jk r ji cos ␾ i jk ⫹r jk r ji ⵜ jk cos ␾ i jk . 共B10兲 Using the relation ⵜ jk (r jk •r ji )⫽r ji and rearranging Eq. 共B10兲 yields

APPENDIX B: THE FIRST DERIVATIVES OF OSPP POTENTIAL

ⵜ jk cos ␾ i jk ⫽

In the program of molecular dynamics simulation, the first derivatives of OSPP potential with respect to all the relevant atomic Cartesian coordinates must be calculated. To describe the required potential derivatives compactly, it is convenient to introduce the differential vector operator, ⵜ pq ⬅

冋

册

⳵ ⳵ ⳵ , ⳵ x pq ⳵ y pq ⳵ z pq

x pq ⫽x q ⫺x p ,y pq ⫽y q ⫺y p ,z pq ⫽z q ⫺z p .

共B2兲

The derivative of the rocking factor Eq. 共7兲 for 0⭐ ␪ j ⭐ ␪ lim is written in the form ⵜ pq f ␪ 共 ␪ j 兲 ⫽2A ␪ 共 1⫺cos ␪ lim兲 ⫺2 共 cos ␪ j ⫺cos ␪ lim兲 ⵜ pq cos ␪ j .

共B3兲

ⵜ ji cos ␾ i jk ⫽

tj 1 ⫽⫺ 兩 tj兩兩 tj兩

兺i cos ␾ i jk ,

共B4兲

ⵜ ji cos␤ i ji ⬘ ⫽

冉

⫽ m j ⫹2

兺 i⬍i

⬘

cos ␤ i ji ⬘

sin ␶ i jkl ⫽

冊

,

e ji • 共 e jk ⫻ekl 兲 , sin ␾ i jk sin ␾ jkl

cos ␶ i jkl ⫽ 共B5兲

and ␤ i ji ⬘ is the angle formed be the vectors e ji and e ji ⬘ . Rewriting Eq. 共B4兲 as t j cos ␪ j ⫽⫺ 兺 i cos ␾ i jk , and taking the differentials of both sides, we have

1 共 e ⫺e cos␤ i ji ⬘ 兲 . r ji ji ⬘ ji

共B13兲

共B14兲

where e ji •(e jk ⫻ekl ) is the volume of the parallelepiped formed by the unit vectors, e ji , e jk , and ekl , and

1/2

1/2

共B12兲

Substituting Eqs. 共B11兲–共B13兲 into Eqs. 共B8兲–共B9兲, and then into Eq. 共B3兲, all the Cartesian derivatives of the rocking subfactors are calculated. Let’s then calculate the derivatives of the torsional subfactor. Since the variable range of a torsional angle covers the whole angular region from 0 to 2 ␲ , any given torsional angle can be uniquely determined only by calculating both the sine and cosine of that angle. To accomplish this, we use the relations

where t j ⫽ 兩 tj兩 is given by t j ⫽ 关共 e ji ⫹e ji ⬘ ⫹ . . . 兲 • 共 e ji ⫹e ji ⬘ ⫹ . . . 兲兴

1 共 e ⫺e cos ␾ i jk 兲 . r ji jk ji

Similarly, it holds for cos␤ i ji ⬘ that

From the definition of the rocking angle ␪ j , and using Eq. 共4兲, we have cos ␪ j ⫽⫺e jk •

共B11兲

Exchanging the suffices ji and jk leads to

共B1兲

where x pq , y pq and z pq are the components of the vector between the atoms p and q, defined by

1 共 e ⫺e cos ␾ i jk 兲 . r jk ji jk

e ji •e jk ⫹cos ␾ i jk cos ␾ jkl . sin ␾ i jk sin ␾ jkl

共B15兲

Differentiation of the torsional factor f ␶ ( ␾ i jk ’s,␶ i jkl ’s兲 with respect to cos n ␶ i jkl and cos ␾ i jk gives F 共 cos ␾ i jk 兲 F 共 cos ␾ jkl 兲 n n ⳵f␶ ⫽ 兩 f ␶兩p␶ , ⳵ cos n ␶ i jkl 2m j m k

共B16兲

and


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the third terms by using Eqs. 共B11兲 and 共B12兲. Noting the relations ⵜ i j ⫽⫺ⵜ ji , we substitute ⵜ pq in Eq. 共B19兲 by ⵜ i j , ⵜ jk , and ⵜ kl to obtain

⳵f␶ F 共 cos ␾ jkl 兲 dF 共 cos ␾ i jk 兲 ⫽ ⳵ cos ␾ i jk 2m j m k d cos ␾ i jk ⫻

兺n 兩 f ␶n兩共 1⫹ p ␶n cos n ␶ i jkl 兲 ,

共B17兲

ⵜ i j f ␶⫽

respectively, where dF(cos ␾ i jk )/d cos ␾ i jk in Eq. 共B17兲 is obtained from Eqs. 共A1兲 and 共A2兲 as

ⵜ jk f ␶ ⫽

dF 共 cos ␾ i jk 兲 ⫽⫺2 兵 cos ␾ i jk 关共 1⫺b cos ␾ i jk 兲 2 ⫺b 2 兴 d cos ␾ i jk ⫹b 共 1⫺b cos ␾ i jk 兲共 1⫺cos ␾ i jk 兲其 .

⳵f␶ ⳵f␶ ⵜ ␶ ⫺ ⵜ cos ␾ i jk , ⳵ ␶ i jkl i j i jkl ⳵ cos ␾ i jk ji ⳵f␶ ⳵f␶ ⵜ jk ␶ i jkl ⫹ ⵜ cos ␾ i jk ⳵ ␶ i jkl ⳵ cos ␾ i jk jk ⫺

2

共B18兲

⫽

兺n

冉冉

冉

冊

ⵜ kl f ␶ ⫽

共B19兲

By using the relations cos 2 ␶ i jkl ⫽2 cos2 ␶ i jkl ⫺1 and cos 3 ␶ i jkl ⫽(4 cos2 ␶ i jkl ⫺3)cos ␶ i jkl , differentiation of cos n ␶ i jkl with n⫽ 1, 2 and 3 gives ⵜ pq cos ␶ i jkl ⫽⫺sin ␶ i jkl ⵜ pq ␶ i jkl ,

共B20兲

ⵜ pq cos 2 ␶ i jkl ⫽⫺4 cos␶ i jkl sin ␶ i jkl ⵜ pq ␶ i jkl ,

共B21兲

ⵜ pq cos 3 ␶ i jkl ⫽⫺3 共 4 cos ␶ i jkl ⫺1 兲 sin ␶ i jkl ⵜ pq ␶ i jkl . 共B22兲 2

The derivatives of the torsional angle defined by the atom sequence i jkl are calculated by Wilson’s formulas as24 ei j ⫻e jk r ji sin2 ␾ i jk

ⵜ jk ␶ i jkl ⫽

共B23兲

,

冉

冊

1 cos ␾ i jk cos ␾ jkl e ⫻e ⫹ e ⫻e , r jk sin2 ␾ i jk i j jk sin2 ␾ jkl jk kl 共B24兲

and ⵜ kl ␶ i jkl ⫽

e jk ⫻ekl r kl sin2 ␾ jkl

共B28兲

B. W. McClelland and K. Hedberg, J. Chem. Phys. 56, 4541 共1972兲. W. F. Giauque and J. D. Kemp, J. Chem. Phys. 6, 40 共1938兲; H. K. Roscoe and A. H. Hind, J. Atmos. Chem. 16, 257 共1993兲; A. J. Vosper, J. Chem. Soc. A 1970, 625. 3 P. Gray and P. Rathbone, J. Chem. Soc. 1998, 3550. 4 ¯ and H. Katoˆ, Mol. Phys. 66, 1183 共1989兲. T. Kato 5 ¯ , M. Oobatake, K. Machida, and S. Hayashi, Mol. Phys. 77, 177 T. Kato 共1992兲. 6 ¯ , S. Hayashi, M. Oobatake, and K. Machida, J. Chem. Phys. 100, T. Kato 2777 共1994兲. 7 ¯ , J. Chem. Phys. 105, 4511 共1996兲; 108, 6611 共1998兲. T. Kato 8 ¯ , S. Hayashi, and K. Machida, J. Mol. Liq. 65Õ66, 405 共1995兲. T. Kato 9 C. M. Steese and A. G. Whittaker, J. Chem. Phys. 24, 776 共1956兲; C. M. Steese, ibid., 24, 780 共1956兲. 10 D. W. James and R. C. Marshall, J. Phys. Chem. 72, 2963 共1968兲. 11 A. J. Vosper, J. Chem. Soc. A 1970, 2191. 12 C. H. Bibart and G. E. Ewing, J. Chem. Phys. 61, 1284 共1974兲. 13 J. Chao, R. C. Wilhoit, and B. J. Zwolinski, Thermochim. Acta 10, 359 共1974兲. 14 B. R. Brooks, R. E. Bruccoleri, B. D. Olafson, D. J. States, S. Swaminathan, and M. Karplus, J. Comput. Chem. 4, 187 共1983兲. 15 A. Vedani, J. Comput. Chem. 9, 269 共1988兲. 16 A. D. Beemann, J. Comput. Phys. 20, 130 共1976兲. 17 T. F. Redmond and B. B. Wayland, J. Phys. Chem. 72, 1626 共1968兲; A. Boughriet, M. Wartel, and J. C. Fischer, J. Electroanal. Chem. 190, 103 共1985兲. 18 P. Groth, Acta Chem. Scand. 17, 2419 共1963兲; B. S. Cartwright and J. H. Robertson, Chem. Commun. 共London兲 3, 82 共1966兲. 19 A. Kvick, R. K. McMullan, and M. D. Newton, J. Chem. Phys. 76, 3754 共1982兲. 20 B. W. McClelland, G. Gundersen, and K. Hedberg, J. Chem. Phys. 56, 4541 共1972兲. 21 W. J. Jorgensen, Acc. Chem. Res. 22, 184 共1989兲; J. Chandrasekhar, S. F. Smith, and W. L. Jorgensen, J. Am. Chem. Soc. 106, 3049 共1984兲. 22 P. W. Atkins, Physical Chemistry 共Oxford University Press, Oxford, 1990兲, p. 107. 23 O. Ermer, Bonding 27, 161 共1976兲. 24 E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, Molecular Vibrations 共McGraw-Hill, New York, 1955兲, p. 61. 2

⳵f␶ ⵜ cos ␾ jkl . ⳵ cos ␾ jkl pq

ⵜ i j ␶ i jkl ⫽

⳵f␶ ⳵f␶ ⵜ ␶ ⫹ ⵜ cos ␾ jkl , ⳵ ␶ i jkl kl i jkl ⳵ cos ␾ i jk kl

1

⳵f␶ ⫹ ⵜ cos ␾ i jk ⳵ cos ␾ i jk pq ⫹

共B27兲

respectively.

⳵f␶ ⵜ cos n ␶ i jkl ⳵ cos n ␶ i jkl pq

冊冊

⳵f␶ ⵜ cos ␾ jkl , ⳵ cos ␾ i jk kl

and

The total gradient of the torsional subfactor is given by ⵜ pq f ␶ 共 ␾ i jk ⬘ s, ␶ i jkl ⬘ s 兲

共B26兲

.

共B25兲

On calculating the right-hand side of Eq. 共B19兲, the first term is obtained by using Eqs. 共B20兲–共B22兲, while the second and
