6413
J. Phys. Chem. 1989, 93, 6413-6417 The functional form of the decrease shows similarities with the power laws observed in the close vicinity of critical points. Visual indication of density fluctuations (blue color of scattered white light) is present in a large region around the saturation pressure curve. The interfacial tension at the saturation pressure curve was too low to be detected. The interfacial tension can be mapped by drawing lines of constant interfacial tension in the phase diagram.
Interfacial laser light scattering is well suited to measure interfacal tension at elevated pressures and temperatures in gas/oil model systems.
A,
Acknowledgment. We thank Jan Stensen for helpful comments and Norsk Hydro for the permission to publish these results. Registry No. CH,, 74-82-8; propane, 74-98-6; decane, 124-18-5.
Intraframework Potential Energy Function 0f Zeolites. 2. (SiAIO,Na), Type Na-A Zeolite Kyoung Tai No, Yea Young Huh, Department of Chemistry, Soong Si1 University, Sang Do 1- 1 Dong, Dong Jak Gu, Seoul, Korea
and Mu Shik Jhon* Department of Chemistry, Korea Advanced Institute of Science and Technology, P.O. Box 150 Cheongyang-Ni, Seoul 130-650, Korea (Received: July 19, 1988; In Final Form: March 24, 1989)
Potential energy functions for (SiA104Na),,Atype zeolites have been obtained by using a constraint method. The net atomic charges of Na atoms bound to the framework were obtained as 8Na(I) = 0.52 and BNa(l1) = 0.74. The minimum-energy path and the role of water molecules during the dealumination have teen investigated with these potential functions near the equilibrium structure. The harmonic potential parameters of Si-0 and A1-0 are obtained and are compared with those of the average value of the T-0 bond. Morse type potential functions are also obtained for Si-0 and A1-0 bonds through the optimization procedure. And these Morse type potential functions are applied for the initial stage of dealumination study. The potential energy surface for the dealumination and dissociation of the O(1)-A1 bond is stabilized by introducing one water molecule at the 8-ring, and the surface becomes flat compared with the surface without a water molecule.
I. Introduction For the theoretical investigation of the physical and chemical properties of zeolites,’-’ physically realistic potential energy functions, suitable for the representation of the zeolite framework, are necessary because these potential energy functions contain all the information on both the static and dynamic properties of zeolites. Gordy’s* and Badger’s9 rules may be used for the cal(1) (a) De Lara, E. C. Mol. Phys. 1972, 23, 355. (b) De Lara, E. C.; Delaval, Y. J . Phys. Chem. 1974, 78, 2180. (c) De Lara, E. C.; Tan, T. N. J . Phys. Chem. 1976, 80, 1917. (d) De Lara, E. C.; Geisse, J. V. J . Phys. Chem. 1976,80, 1922. (2) (a) Ogawa, K.; Nitta, M.; Aomura, K. J . Phys. Chem. 1978,82, 1655. (b) Ogawa, K.; Nitta, M.; Aomura, K. Zeolites 1981, I, 169. (c) Nitta, M.; Ogawa, K.; Aomura, K. Zeolites 1981, I, 30. (3) (a) No, K. T.; Chon, H.; Ree, T.; Jhon, M. S . J . Phys. Chem. 1981, 85, 2065. (b) No, K. T.; Jhon, M. S.J . Korean Chem. SOC.1979, 23,374. (c) Kong, Y. S.; Jhon, M. S.; No, K. T. Bull. Korean Chem. SOC.1985, 6, 57. (d) Choi, K. J.; No, K. T.; Jhon, M. S. Bull. Korean Chem. SOC.1987, 8, 158. (e) Song, M. K.; Chon, H.; Jhon, M. S.; No, K. T. J . Mol. C a r d 1988,47,73. (f) No, K. T.; Bae, D. H.; Jhon, M. S.J . Phys. Chem. 1986, 90, 1772. (8) No, K. T.; Seo, B. H.; Jhon, M. S. Theor. Chim. Acra, in press. (h) No, K. T.; Kim, J. S.; Jhon, M. S. Theor. Chim. Acra, in press. (i) No, K. T.; Seo, B. H.; Park, J. M.; Jhon, M. S.J . Phys. Chem. 1988,92,6783. (4) Takaishi, T.; Hosoi, H. J . Phys. Chem. 1982, 86, 2089. (5) (a) Baker, M. D.; Godber, J.; Ozin, G. A. J . Phys. Chem. 1985, 89, 2299. (b) Ozin, G. A.; Baker, M. D.; Godber, J.; Shihua, W . J . Am. Chem. Soc. 1985, 107, 1995. (c) Baker, M. D.; Godber, J.; Ozin, G. A. J . Am. Chem. SOC.1985, 107, 3033. (6) (a) Blackwell, C. S.J . Phys. Chem. 1979, 83, 3251. (b) Blackwell, C. S. J . Phys. Chem. 1979, 83, 3275. (7) Demotis, P.; Suffritti, G. B.; Quartieri, S.;Fois, E. S.; Gamba, A. J . Phys. Chem. 1988, 92, 867.
0022-3654/89/2093-6413$01.50/0
culation of the stretching harmonic potential function of Si-0 and Al-0 bonds, and the force constants between ions and oxygen atoms of the framework can be calculated from Brodskii’slo relation. These potential sets are insufficient for the investigation of the relative stabilities of the atoms, for the description of the experimentally obtained crystal structure, and of the dynamic properties of the zeolite framework. If a physically realistic potential energy function is used for the calculation of minimum-energy geometry of the framework, the minimum-energy structure must correspond to the observed crystal structure. Recently, No et al.’l proposed a potential energy function set suitable for the description of the stability, net force, and force constant of atoms located at the T,O,Na-A type zeolite framework, and a molecular dynamics calculation1*of the Na ions was performed with these potential functions. The simulated diffusion coefficient of Na ion was well in accord with experimental data. Although the crystal structure and diffusion of the Na ions of A type zeolite could be explained by the potential function in which A1 and Si are not distinguished, for the explanation of the more important properties of zeolite as catalysis, for example, acidity, active site, and dealumination of zeolite framework, the potential function must distinguish A1 and Si and contain physically realistic A1-0 and S i 4 bond potential functions, not a harmonic function (8) Gordy, W . J . Phys. Chem. 1946, 14, 305. (9) Badger, R. M. J . Chem. Phys. 1934, 2, 128. (10) Brodskii, I. A,; Zhdanov, S. P. Proc. Int. ConJ Zeolites, 5rh 1980, 234. (11) No, K. T.; Kim, J. S.; Huh, Y. Y.; Kim, W. K.; Jhon, M. S . J . Phys. Chem. 1987, 91, 740. (12) Shin, J. M.; No, K. T.; Jhon, M. S . J . Phys. Chem. 1988, 92, 1012.
0 1989 American Chemical Society
6414
The Journal of Physical Chemistry, Vol. 93, No. 17, 1989
No et al.
form as in our previous paper.” The purpose of this work is to obtain the potential energy function of (SiA10,Na)A type zeolite and calculate the physical quantitites of each atom in the zeolite framework. By use of this potential function, the initial stage of the dealumination procedure can be investigated. This potential function could be used not only for the calculations of the properties of the ions bound to the zeolite framework but also for the test of thermal stability and phase transition of the framework using computer simulations such as molecular dynamics or Monte Carlo.
character, Na atoms are not included in the electronegativity equalization. Therefore, the potential parameter sets were obtained at several aNa)s. For the rapid convergency of the Coulombic lattice sums, the summation of the electric fields in zeolite framework was carried out for all the (SO2) and (AIO2Na) units within the cubic crystal ( R = 2a).” Since A zeolite could be divided into S i 0 2 and A102Na units and the dipole moments calculated for those units oriented in several directions, the net dipole moments converge very rapidly as R increases. ( b ) Polarization Energy ( V p l ) .
11. Method The geometry of the framework (SiA104Na), was taken from the X-ray crystallographic study by Pluth and SmithI3 of dehydrated Na-A zeolite. The space group is Fm3c ( a = 24.555 A), at 350 O C and 10 Torr. In this framework, there are eight kinds of atoms located in different potential fields, with different environments, namely, AI, Si, 0(1), 0 ( 2 ) ,0 ( 3 ) , Na(I), Na(II), and Na(1II). Na(I1) atoms are randomly distributed among the 8-fold degenerate positions of the 8-ring. I . Determination of Parameters of Potential Energy Functions. Since each atom located in a crystal must correspond to a minimum in the potential energy surface, the atom located at the equilibrium position in the crystal does not feel any net force. Therefore, the parameters of potential energy function are obtained by the minimization of the net force on the atoms.
vpo, = -y2Cai[(C7$* + (CZr;)2] i j#i j#i
N
N
(4)
where ai and 7; represent the polarizability of ith atom and the electric field in the x direction of the ith atm created by thejth atom. ( c ) Dispersion and Repulsion Energy (Vd-r). For the nonbonding interaction of 0-0 and 0 - N a atom pairs, a LennardJones (6-1 2 ) type potential function was used. Vd-r = 4cij[(uij/rij)’2 - (uij/rij)6]
(5)
where qi and aij are Lennard-Jones potential parameters and rii is an interatomic distance. (6)Stretching Potential Energy of Si-0 and A1-0 Bonds. For the bonding atomic pairs, Si-0 and AI-0, harmonic or Morse type potential functions were used. VT-0 = VT-o =
where
N
y2CI ]>iCKij(rij - roij)2
CDe(ij)(e-2.(.s08,- 2e-4’ 114’u‘)
(6) (7)
i ]>i
and n and N are respectively the number of parameters and atoms located in different environments. V is the potential energy function, and a,, qj, and qoj represent the potential parameter of ith atom, the geometrical parameter of jth atom, and that in the crystal at equilibrium, respectively. In this work, Cartesian coordinates were used for 9;s. The determination of potential parameters for (SiAIO,Na), type A zeolite is very difficult compared with (T204Na), type A zeolite, especially when the Morse type potential functions are replaced by the harmonic function of Si-0 and AI-0 bonds. The number of potential parameters for (SiA104Na), zeolite is almost twice that of (T204Na),zeolite. The potential parameters of (T204Na), zeolite are used as the initial value for the determination of (SiA104Na), zeolite during the optimization procedure. Since the forces calculated from electrostatic, polarization, dispersion, and bond potential have different orders in their magnitudes, the changes in the potential parameters are quite sensitive to the slope of the potential function in which those parameters are included; in many cases it is quite difficult to obtain a good potential parameter set for the flat potential functions. Therefore, during the optimization, it is practically useful to multiply some factors by the slope of the flat potential functions. 2. Potential Energy Functions. The stabilization energy of the framework was represented as a sum of several terms. ( a ) Electrostatic Energy (Vel). (3)
The net atomic charges are assumed as point charges and are obtained by using Sanderson’s electronegativity equalization condition14 and Huheey’s electronegativity set.I5 Since the bonding between the Na atom and framework has an ionic (13) Pluth, J. J.; Smith, J. V. J . Am. Chem. SOC.1980, 102, 4704. (14) (a) Sanderson, R. T. J . Chem. Educ. 1945.31, 2. (b) Sanderson, R.
T. Chemical Periodiciry; Reinhold Publishing: New York, 1960. (IS) Huheey, E. J . Phys. Chem. 1965, 69, 3284. (16) Ozin, G. A.; Baker, M . D.; Godber, J. J . Phys. Chem. 1984,88,4902.
where Kij and roijare the harmonic potential parameters, De, a, and rolijare the Morse potential parameters, and rij is an interatomic distance. Although the harmonic potential is convenient for the calculation of equilibrium properties, for example, in a vibrational study, the Morse type potential is necessary for the description of bond dissociation and dealumination. The potential parameters to be refined by the constraint method are uW, uSNa, eo+, and t ~ in -the ~Lennard-Jones ~ potential and six kT+’s and six roT+’s in the harmonic potential (or two De’s, six aT+’s, and six rT4’s in the Morse potential). 3. Determination of the Net Atomic Charge of Nu. For the determination of the net atomic charges of the Na atoms, the vibrational frequencies of Na(I), Na(II), and Na(II1) were calculated as a function of 6Na and were compared with the far-IR spectra of Na atoms bound to A type zeolite. The local-mode approximation was used in the normal-mode calculation, and the internal coordinate force constants between framework atoms and N a were calculated as a function of tiNa with the formula in the Appendix. Since the coupling between framework and framework-Na vibrational motion is small, the stretching and bending coordinates of the framework are assumed to be rigid. The error coming from the local-mode approximation was discussed in a previous paper.” 4 . Study on the Initial Stage of Dealumination. By use of the potential energy functions, the changes in stabilization energies and geometries of the framework as the AI-0 bond is elongated were calculated. The geometrical parameters used for the calculation of the minimum-energy path of dealumination are shown in Figure 1. Since the O( 1) atom can be easily moved out from framework toward the center of the &ring compared with other oxygen atoms,” the dissociation of the A1-0( 1) bond was investigated for both dehydrated and hydrated zeolites. The nonbonding interaction energies of water-framework and water-Na pairs were calculated by using Kitaigorodskii potential functions.” The distance rSH of 0-H in a water molecule bound to the framework is 0.96 A, and -0.64e was used for the net atomic charge of the oxygen atom in the water molecule. (17) (a) Caillet, J.; Claverie, P. Acta Crysrallogr. 1975, ,431, 448. (b) Huron, M. J.; Claverie, P. J . Phys. Chem. 1972, 76, 2123.
The Journal of Physical Chemistry, Vol. 93, No. 17, 1989 6415
Potential Energy Function of Na-A Zeolite
n
TABLE I: Crystal Structure Parameters of (SiAI04Na),,A Type Framework Obtained by Energy Minimization and Compared with X-ray Diffraction Results (in Parentbeses)’Vb 1.54 (1.54) 3.85 (3.85) 3.35 (3.35) 2.6 1 (2.59) 1.94 (1.93) 3.71 (3.69) 0.82 (0.82) -3.58 (-3.54)
Figure 1. Geometrical parameters used for the calculation of the minimum-energy path of dealumination near the equilibrium position. The parameters are the positions of a water molecule and a Na(I1) located in the 8-ring window. The water molecule is assumed to be rigid.
‘All values are in
6.14 (6.14) 6.14 (6.14) 6.14 (6.14) 6.12 (6.14) 7.58 (7.58) 3.71 (3.69) 6.14 (6.14) -3.57 (-3.54)
3.92 (3.92) 1.60 (1.60) 0.09 (0.08) 2.59 (2.56) 4.82 (4.82) 3.71 (3.69) 0.95 (0.95) -0.02 (0.00)
*The origin is the center of the a-cage.
(in Kcalimole)
2.8
0
2.6
2.4 x
M
u
CI
w
g
1.9
%-O(
3), r oA1-0( 1-) H
N
4 .A
/
1.7
-100
.ri Y
n
m m Li
1.5
\ -Na( I) -200
0
0.4
0.6
0.8
-270
N e t Atomic Charge of Na
Figure 2. Refined parameters of the potential energy functions plotted against 8Ns values.
111. Results and Discussion In Figure 2 the potential energy parameters of the (SiA104Na), type model are plotted against the net atomic charge of Na (bNa) and are compared with those of the (T,04Na), type model. The harmonic potential parameters depend linearly on bNa and decrease as bNa increases. The harmonic potential parameter roT4 is between rosi4 and roA14. Since the maximum value of the net force on each atom,f= ( d V / d q o k ) ,is less than 0.0015 mdyn/mol, the minimum-enery geometry on the potential energy surface agrees well with the crystal structure as listed in Table I. The stabilization energy of each atom at the minimum-energy geometry is plotted as a function of bNa (Figure 3). The stabilization energies of Na and oxygen atoms show the following tendency: Na(1) > Na(I1) > Na(II1) and O(2) > O(3) > O(1). This preference series for the Na atoms agrees with X-ray diffraction studies. The stabilization energies of AI and Si are not so sensitive to BNa. The second derivatives of the potential energy with respect to the Cartesian coordinates of each atom are obtained at the minimum-energy geometries and are plotted against bNa (Figure 4). The force constant of Na(II1) is larger than that of Na(II), although the binding energy of Na(II1) is smaller than that of Na(I1). This means that the Na(I1) may easily move
% -Al
-300
0.4
0.5
0.6 0.7 0.8 N e t Atomic C h a r g e o f N a
Figure 3. Stabilization energies of the atoms calculated at X-ray positions.
around the eight-membered ring but may not be easily removed from the ring. Since both the stabilization energy and force constant of Na(1) are large, for hydrated A type zeolite, Na(1) is the only cation site which can be identified with certainty. The force constant toward the center of the 8-ring and the stabilization energy of O( 1) are relatively small compared with those of O(2) and O(3) atoms. In Figure 5, the vibrational frequencies of the Na ions are plotted against tiNa. These are the modes of Na(1) bound to the 6-ring (C3”;E, AI), of Na(I1) bound to the 8-ring (Cs;A’, A‘, A”), and of Na(II1) bound to the 4-ring (Cb; B,, AI, B,). The far-IR spectra of Na(1) and Na(I1) vibrations were observed at 212 and 180 cm-l, respectively.I6 The 212 cm-’ band can be assigned to the E mode of Na(I), and the net atomic charge of Na(1) is obtained as bNa = 0.52; the 180-cm-I band can be assigned to the A’ mode of Na(II), and the net atomic charge of Na(I1) is obtained as bNa = 0.74. Therefore, the average net atomic charge, ( ~ : r = I , I I ~ N a ( r ) b N a ( r ~ ) / ( ~ r = I , I I n N a (is r ) )obtained ~ as 0.581. The PO-
rhe Journal of Physical Chemistry, Vol. 93, No. 17, 1989
6416
No et al. TABLE II: Refined Potential Parameters' ( SiA104Na).c
AI
Si
1.783 (1.760)d 1.735 (1.760) 1.775 (1.760) 3.036 (3.488) 3.285 (3.488) 2.858 (3.488) 2.637 2.485 (2.385) (2.614) 0.231 0.246
1.590 (1.605) 1.571 (1.605) 1.595 (1.605) 4.938 (3.488) 5.708 (3.488) 4.775 (3.488)
(T204Na)n()
1.659 1.63 1 1.685 4.125 4.300 4.035 2.701 2.483 0.242 0.230
'OT-O(I) ro T-O( 2) r0T-O(3) KT-O(I) KT-O(2) KT-O(3) "0-0 "0-Na Q-0 Q-Na
A, Kin mdyn/A, u in A, and c in kcal/mol. bThese potential parameters are refined at 6Na = 0.625 with the (T204Na),,Atype framework. this study, bNa is determined as 0.581 in the (SiA104Na),A type framework, and the potential parameters are obtained. dValues in parentheses are taken from the harmonic model of ref 7. For details see ref 7; this value is ro in the harmonic potential. " r o in
TABLE 111: Positions of Water Molecules Bound to Na(I1) and of Na(I1) Determined by Energy Minimization"
Na(I1) c
I
I1
0.4
0.5
H 0 H
0. 6 11.7 0.8 Net A t o m i L C h a r g e o f Na
Figure 4. Cartesian coordinate force constants of each atom calculated at X-ray positions and plotted against
X
Y
-0.731 (-0.822) 1.660 0.744 0.782
6.139 (6.139) 6.153 6.162 6.172
binding energy
Z 0.996 (0.955)b -0.078 -0.363 -1.322
-86.99 -28.70
a Binding energies are in kcal/mol and coordinates are in position of Na(I1) in dehydrated zeolite A.
I I
A. bThe
I
6-b
v
x
2
200
II
-
a 3 5
Dehydrated
I
Hydrated
I
I
/
i
(in
Kcal )
6-d
Dehydrated -10
'
,*/
O/
0
0.4
0.5
0.6 0.7 0.8 Net Atomic Charge of Na
Figure 5. The frequency of each normal mode plotted against havalues.
tential parameters and minimum-energy geometries obtained at this ,S are listed in Table I1 and Table I, respectively. The averaged potential parameters of Si-0 and A1-0 bonds nearly correspond to those of T-O potential parameters of the (T,O,Na),, model. I ' The positions of Na(I1) and the water molecule bound to it were determined by using the potential parameter set in Table I1 and the Kitaigorodskii potential functions." The binding energy of water is obtained as 29 kcal/mol, and the geometries are listed in Table 111. Since the force constant of Na(I1) is small, the position of Na(I1) is changed much upon hydration and cannot be determined by X-ray diffraction study with certainty. If the net atomic charge of Na is assumed to be 1, the theoretically obtained heat of hydration in accord with the calorimetric measurement of heat of immersioni8 was obtained when the di-
+
1.5 'A1-0( 1 j 1.7
1.9
? . I
2.3
-I0
t L
1.65
1.70 1.75 " s , - O ( l J ' '~I-OC 1 j J 2
Figure 6. Minimum-energy paths of dealuminations occurred in dehydrated (a) and hydrated (b) frameworks, the changes in bond lengths, r s , q I )and rAla(l),are depicted during the dealuminations (c), and the changes in stabilization energy along the minimum-energy paths are
plotted against '/2(rsi-Ocl) + ~ A I - o ( ~ ) ) . electric constant in the a-cage was assumed to be 3.5 in the electrostatic energy calculation. Since most of the hydration energy is contributed by the electrostatic energy, the electrostatic energy was overestimated by more than 3 times. The heat of immersion is changed from 24 to 22 kcal/mol when the degree of presaturation changes from 0.33 to 0.41, and this range cor(18) Barrer, R. M.; Cram, P.J. Adu. Chem. Ser. 1974, No. 102.
Potential Energy Function of Na-A Zeolite 100
ro
= 1.590A
Si-0( 1) asi-o( ) = 1. 386A-1 50
De= 185.0 Kcal/mole
0
3.0
, 0 . 4
-50
-1oc
-15c
//
v
Figure 7. Morse type energy curves of Si-O(l) and A1-0(1) bonds. The potential depths of the potentials in ref 7 are 95 kcal/mol for the S i 4 bond and 85 kcal/mol for the AI-0 bond.
responds to the hydration of Na(I1)'s. Therefore, 28 kcal/mol is still large compared with the experimental value. Figure 6a,b shows the minimum-energy configuration of the framework as the A1-0( 1) bond is elongated toward the 8-ring center. Especially, Figure 6b shows the role of the water molecule during the A1-0(1) bond elongation. The water molecule bound to Na(I1) stabilizes the framework by 15 kcal/mol compared with the dehydrated case. As shown in Figure 6c, elongation of the A1-O(1) bond is favored upon hydration. The ratio of bond elongation, drs*l)/drm(l), becomes smaller upon hydration and decreases as the elongation proceeds. This may be explained from the shapes of the potential energy curves of Si-( 1) and A l a ( 1) bonds as shown in Figure 7. Both the second derivatives and potential depth of the Si-( 1) potential curve are larger than those of the A1-0(1) curve. At the same time the water molecules bound to Na(I1) lower the minimum-energy path of dealumination by about 15 kcal/mol with respect to that of dehydrated framework. In Figure 6d, the energy along the minimum-energy path is plotted against I/2(rsi4(l) + rA14(I)),because the contribution of these bonds, rSi4(,) and rA14(l),is predominant to
The Journal of Physical Chemistry, Vol. 93, No. 17. 1989 6417 the dealumination reaction coordinate. The stabilization energy of the dehydrated zeolite at its equilibrium position was used as a reference state. The stabilization energy curve of the dealumination with a water molecule has a flat minimum compared with that without a water molecule. Therefore, the mobility of 0 ( 1 ) may increase upon hydration. Althouth the classical potential function may give some idea of the dealumination near the equilibrium position, quantum mechanical calculations must be used for the full description of this phenomenon. The role of ions and protons, acid-catalyzed condition, may be very important for more mobility and stabilization of the atoms involved in dealumination. Although the potential function cannot describe the detailed potential energy surface for the reactions, dealumination and phase transition of the framework, it contains an effective A1-0 and Si-0 bonding potential function in the framework. With the potential function, the dynamical properties could be calculated not only at minimum-energy geometry, lattice vibrational calc ~ l a t i o n ,but ~ ' also around the minimum-energy geometry within the classical potential limit having a two-body potential additivity approximation. The stability of A zeolite framework as a function of temperature will be studied with this potential function and computer simulations. Acknowledgment. This work was supported by the Soong Si1 University, and all the numerical calculations were carried out at the Computer Center of Soong Si1 University.
Appendix Forces and Force Constants of the Morse Type Potential Energy Represented in Cartesian Coordinates. N fiim
= C K m n ( r m n - romn)(qin - G i m ) / r m n n#m
N
kii"
-
=
C k m n ( 1 - r o m n / r m n + romn(qin - qimI2/rmP)
n#m
Fim = N
(amn(q;- qim)/rmn(e-2a~(rm-rom") - e-amn("n-rOnn) 1)
-2Dei n#m
N
kii"
= 2Dei n#m
(e-2a,(r,-r0m3
am,[(-l/rmn
+ (q? - qim)2/rm,3)x
- e-a,,,,,(rmn-rom))+ a m n ( q i n - qim)'/rm,Z + (2e-2am(rm-romn)- e-andrmn-r'mm
Internal Coordinate Force Constant.
9I
