2802
J . Phys. Chem. 1991, 95, 2802-2810
with cold CCI2radicals, and presumably the additional amount comes from internal energy already present in the nascent CC12. Because the CC12starts with a wide range of internal energies, little can be said about its dissociation dynamics, but since a large fraction of its available energy goes into translation, the dissociation likely occurs directly from an electronically excited state. Summary
We have generated CC13 radicals by photolysis of CCll at 193
nm in a pulsed radical beam source and then dissociated the CC13 at 308 nm. The only primary reaction channel observed was the production of CC12 and C1 with a relatively low translational energy release. Some of the CC12absorbed a second photon and
dissociated to CCI and C1. In addition to CCI, radicals and undissociated CCb in the beam, C12was also produced, presumably from the recombination of C1 atoms in the supersonic expansion. There was no evidence for any other species produced in the source in quantities detectable with a mass spectrometer.
Acknowledgment. This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division, of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098. Professor W. M. Jackson thanks the Miller Institute of the University of California for financial support. Registry NO. CCI,, 3170-80-7; CI, 22537-15-1; CC12, 1605-72-7.
Consistent Force Field Modeling of Matrix- Isolated Molecules: Site Dependence of Normal Vibrations of gauche - 1,2-Dllluoroethane:Ar,,, R. Gunde,? H. J. Keller, T.-K. Ha, and H. H. Gunthard* Physical Chemistry Laboratory, ETH Zentrum, CH-8092 Zurich, Switzerland (Received: October 1 , 1990)
Results of consistent force field modeling of the normal modes of a system built from one molecule, gauche-1,2-difluorcethane (gDFE), substituted for the center atom of a cube-shaped crystal fragment consisting of 365 argbn atoms will be reported. Earlier modeling work established the existence of three structurally different locally stable self-consistent substitutional defects: GHI, GH21, and GH22. For these systems and for two self-consistent matrix models Ar365and Ar364-V(0,0,0) (vacancy in center of Ar365),the normal-mode problems were solved by numerical computation of all 1 116 (1095, 1092) normal modes. The modes of the three gDFE:Ar!@ systems are found to classify as (i) typical internal vibrations of gDFE positively shifted in frequency by 2-30 cm-l relative to those of the free substituent, (ii) about 10 typical moleculematrix (A-M) interaction modes arising from normal modes of Arw through strong perturbation by the substituent, featuring frequency shifts from =65 to 150 cm-I, and (iii) matrix modes forming a set of over 1000 remaining modes extending from =70 to 3 cm-' with highly irregular density distribution of frequency (singly excited vibrational states) and symmetry species. Both internal and A-M modes form spectra, which are characteristic for each of the three defect configurations, supporting the commonly accepted interpretation of unexpected splittings in empirical vibrational matrix spectra as 'site splittings". For the two defects GH21 and GH22 the electric polarization caused by the dipole and quadrupole moment of gDFE was computed, considering induced dipole moments of gDFE and all argon atoms. Combining the polarization energies with zero-point energy estimates obtained from normal-coordinate analysis, improved estimates of interconversion energies of the 'sites" GHI, GH21, and GH22 are reported.
1. Introduction The present paper considers models for vibrational spectra of matrix-isolated molecules. In a series of preceding papers consistent force field (CFF) modelings of some static aspects of molecules isolated in rare gas matrices have been reported.14 Motivation for this work originated from numerous phenomena observed constantly in spectra of matrix-isolated molecules, e.g., unexpected splitting patterns ('site splittings"), frequency shifts and line shapes of vibrational transitions, interconversion of conformers by infrared radiation, excitation of chemical reactions, etc.; for reviews the reader is referred to refs 5-9. For most of these phenomena intuitive or qualitative interpretations have been put forward; e.g., already in the first matrix spectra observed1° unexpectedly complex line patterns were interpreted as site splittings. However, relatively few attempts have been published aiming at a more detailed interpretation of particular effects in terms of well-defined models; for a review the reader is referred to ref 9. In our earlier work so far mostly static properties of a system consisting of one molecule, 1,2-difluoroethane (DFE) in either the gauche (g, more stable by -180 cm-I) or trans (t) conformation, substituted for the center atom of an (originally) cubeshaped crystal fragment built from 365 argon atoms (DFE:Ar3a) 'Present address: Laboratory for Chemical Engineering and Industrial Chemistry, ETH Zentrum, CH-8092 Zurich, Switzerland.
were investigated, describing the potential energy of this system as a function of all its (1 116) Cartesian coordinates with respect to the crystallographic coordinate system. By extended searches for substitutional defects with special site symmetry, so far six self-consistent structures were found from potential energy minimization: defects G1 and T1 where the 2-fold axis of gDFE and tDFE is directed along a 4-fold axis (e.g., [OOl] direction, site i l l ) of the cubic crystal fragment, and two pairs of defects G21, G22 and T21, T22, where the 2-fold axis of DFE is parallel to a diagonal 2-fold axis ([Oll] direction say, site el'). The G and T defects feature two and one ("external") degrees of freedom defining orientation of the molecular coordinate system with (1) Gunde, R.; Felder, P.; Gunthard, H. H. Chem. Phys. 1982,643133. (2) Gunde, R.; Gunthard, H. H. Chem. Phys. 1987,111,339; 1988,126, 229. (3) Gunde, R.; Gunthard, H. H. J . Mol. Struct. 1989, 202, 325. (4) Gunde, R.; Ha, T. K.; Gunthard, H. H. Chem. Phys. 1990,145,37. (5) Hallam, H. E., Ed. Vibrational Spectroscopy of Trapped Species; Wiley: New York, 1973. (6) Andrews. L.; Moskovits, M., Eds. Chemistry and Physics of Matrix Isolated Species; North-Holland: Amsterdam, 1989. (7) Clark, R. J. H.; Hater, R. E., Eds. Advances in Infrared and Ramon Spectroscopy; Wiley: Chichester, 1989; Vol. 17. (8) Frei, H.; Pimentel, G. C. Annu. Rev. Phys. Chem. 1985, 36, 491. (9) Allavena, M. In Quantum Theory of Chemical Reactions; Daudel, R., Salem, L., Veillard, A., Eds.; Reidel: New York, 1980; p 129. (10) Pimentel, G. C. Personal communication to one of the authors. (1 1) Hahn, T., Ed. International Tables for Crystallography; Reidel: Dordrecht, 1983; Vol. A, p 663.
0022-3654191 12095-2802%02.50/0 0 1991 American Chemical Society
The Journal of Physical Chemistry, Vol. 95, No. 7, 1991 2803
Normal Vibrations of gauche- 1,2-Difl~oroethane:Ar~~~ TABLE I: Structural Parameters of Defects gDFE:Arwa
external structural aarameters
defect
GH 1 GH21
GH22 GSl
GS21 GS22
iteration sequencesb P10M250, P15M300, P15M250, P10M300 M5000, M55M05, M55W5 P30M40,P35M40,P35M35 PlOM250, P15M250, P15M200 M5500, M55P05, M50P05 P30M60, P25M60, P2OM60
deg 73.5 (7) 69.7 (1) T’,:
74.8 (1) 74.7 (3) 69.49 (1) 74.5 (1)
dCCe:
deg
46.6 (2) -54.0 (2) 26.0 (20) 45.1 (5) -54.7 (3) 22.7 (3)
[mnol [0011 [02-1/22-1/2]
[02-1/22-’/2] [OOII [02-1/22-’/2] [02-1/22-1/2]
Tf[m]e:A -0.128 (+2/0) +0.026 (8)
-0.17 ( 9 ) -0.17 (2)
+0.032 (4) -0.42 (4)
‘gauche- 1,2-Difluoroethanesubstituted for the center atom of cube-shaped crystal fragment ATsbSconsisting of four elementary cells in all three directions, Values marked by a prime and subscript e denote estimate for self-consistent structure (re structure). Notation for iteration sequences used for localization of self-consistent configurations by energy minimization; cf. ref 3. Dihedral angle estimate for self-consistent structures. dAngle of rotation of DFE molecule about its 2-fold axis (coincident with crystallographic direction [mno]). ‘Translation vector of origin of molecular system with respect to crystallographic system is given by [mno]T’l,,lc; cf. Figure 1 and ref 3 for GHI.
0
Figure 1, I ,2-Difluoroethane (DFE):Ar,@schematic geometrical model for G2-type substitutional defect. (C‘,Od: molecule-fixed coordinate frame used for quantum chemical electric moment and polarizability calculation. (Ccs,Ocs):crystallographiccoordinate frame used for consistent force field (CFF) modeling. (PEC, TIolll:external structural parameters defining position of (t‘,Ofl relative to (Ccs,0csl. (Ccs,Od: coordinate frame used for electric polarization energy calculation. df, dl, d,, a,/3, 8, and T are structural parameters of DFE.
respect to the crystallographic system; parameter values for the C-type defects are collected in Table I; cf. Figure 1. Two salient aspects experienced in the search for self-consistent defect structures by C F F modeling should be pointed out: (i) In order to arrive at acceptably reliable structural information of any locally stable nuclear configuration by potential energy minimization, one has to use a number (13) of energy minimization sequences, starting from different suitably chosen initial configurations and converging toward presumably the same minimum-energy configuration. Even for initial configurations lying apparently close to a specific local minimum, behavior of different energy minimization sequences may be widely individual with respect to rate of convergence and the final states reached after break off (either by breakdown of symmetry, stagnation of convergence, etc.), characterized by the final values of potential energy and structural parameters pee, T ~ o l matrix l ~ , atom coordinates, etc., may differ considerably. In order to extract estimates on energetic and structural data of the local minimum (re structure), most often extended extrapolation procedures based on several energy minimization sequences are required, which then produce an approximate equilibrium structure ( r l structure). Nevertheless, in some cases certain structural parameters cannot be determined to high precision. (ii) Use of several very long iteration sequences’* for the same local minimum may not allow one to come sufficiently close to the minimum configuration in order to assert that the potential function is locally well approximated by a quadratic form of the Cartesians. Rather, the potential function seems to possess a more ~
(12) Gunde,
R.;Gunthard, H. H. To be published.
complex form (“orange skin”). This is relevant for the concept of normal (infinitesimal) modes of the gDFE:Ar,M system considered in this paper. Presently, incomplete knowledge of molecule-matrix interaction forces prompted us in all foregoing CFF modelings to express these forces as pair interaction potentials between all atoms (H, C, F) of DFE and all matrix atoms (Ar). Two sets of pair potentials selected on widely different grounds were used: (1) pair potentials derived from combination rules for the parameters of the Buckingham-type potential expression, based on published like-pair parameters (“hard” p~tential);’*’~-’~ (2) potentials for (H,Ar), (C,Ar), and (F,Ar) pairs derived by interpolation of LennardJones-type rare gas unlike-pair potentials, using quantum chemically computed charge orders (gross atomic populations) as interpolation parameters; these potential will be called oft.','^,'^ Setting out from the information available at present on simplest self-consistent models of gDFE:Ar364, we address in this paper some aspects of the normal vibrations of both gDFE and of the matrix models Ar365and Ar364-V(0,0,0) by treating the system as a molecule with 3.372 - 6 = 1100 (1089, 1086) proper infinitesimal modes. In section 2, a modification of the theory of electrical polarization for type G2 defects and computational details will be mentioned, which appear relevant for critical assessment of the results. The latter will be presented in section 3 comprising frequencies of “internal” and part of “matrix” normal modes, as well as information on zero-point energy and electrical polarization energy of the defects GH1, GH21, and GH22. Discussion of the results is put forward in section 4, where it will be shown that the internal modes experience individual (harmonic) up-frequency shifts of the order of 10 cm-’ with respect to the free molecule normal modes. Defects G1 and G22 (which are similar in CFF energy) exhibit similar harmonic shifts but differ markedly from G21; this behavior applies analogously for both hard and soft potentials. The normal modes of the self-consistent matrices Ar365and Ar364-V(0,0,0) are found to extend from a p proximately 65 to 3.8 cm-’. By the substituent the highest matrix modes experience a marked upwardshift to 5 150 cm-’ and form strongly mixed molecule-matrix (A-M) modes, in which, besides DFE, predominantly Ar atoms of the, first coordination sphere take part with large vibrational amplitudes. From normal frequencies estimates of vibrational zero-point energies are obtained, which-together with results of the electrical polarization energy computations-allow correction of the earlier published energies of site interconversions G H l GH21 and GH21 GH22. In section 5 some conclusions and further applications will be mentioned.
-.
-
2. Basic Data and Computational Procedures 2.1. Program System for CFF Computations. In the present work the program system used earlier for determination of self(13) Manz, J.; Mirsky, K. Chem. Phys. 1980.46.457. (14) M.irskaya, K. V. Tefrahedron 1973, 29, 679. (1 5 ) Kitaigorodsky, A. I. Molecular Crystals and Molecules; Academic Press: New York, 1973; p 164. (16) Pollack, G. L. Rev. Mod. Phys. 1964.36.746. (1 7) Klein, M. L.;Venables, J. A. Rure Cas Solids; Academic Press: New York, 1976; Vol. 1.
2804 The Journal of Physical Chemistry, Vol. 95, No. 7,1991
Gunde et al.
TABLE 11: Electric Moments and Polarizability of gDFE in Fml and Estimated r , Defect Coafigurrtioas (Referredto Molecular Frame (Of, 011)"
sequence GH2l M5000 M55M05 M55P05 GH21
7,
deg
34.84 34.85 34.85 34.85
rc estimate
GH22 P30M40
37.30
P35M45
37.34
P35M35 GH22 re estimate
37.35 37.4
Tp,ll], A 0 0.0076 0.0076 0 -0,0056 -0.0056 0 0.0224 0.0224 0 0.0184 0.0184 0
-0.263 -0.263 0 -0.2892 -0.2892 0 -0.2276 -0.2276 0 -0.1202 -0.1202
(PCC,
deg
-53.74
MIb -2.8513
6.5616 1.0525
2.7228 -0.241 1
1.0517
2.7236 -0.2412
1.0517
2.7236 -0.2412
1.0517
2.7236 -0.241 2
0.8660
2.9141 -0.2649
0.8629
2.9192 -0.2653
0.8621
2.9 179 -0.2654
0.8582
2.92 18 -0.2658
0 -54.01
-2.8519 0 -2.8509
-54.0 (2)
-2.8509
28.75
-2.7633
32.68
-2.7618
32.83
-2.761 5
26 (2)
X
-2.7596 0 0
VA(SA(XA))+ VAM(€AM(XAJM)) + ~M(€M(XM)) (1) taken as a function of all 372 Cartesion coordinate vectors. In V
the potential VA of gDFE the potential to internal rotation has been replaced by a term of the form '/JT(7 - 7 e)z, with fT = 8.99375 kcal/rad2 = 0.12497 X IO-" erg/rad2, andthe remainder is expressed as a quadratic form of the internal coordinates given earlier.'* Also, the other two terms of eq 1 have been taken in the same form as earlier1v2with either soft and hard interaction potential VAM. For normal-coordinate analysis in Cartesian coordinates the Hessian matrix
is required, taken for the nuclear configuration of the local minimum representing a particular defect. Use of Cartesian coordinates implies the Hessian to have signature (1 110,6,0); i.e., (18) Huber-Waelchli, P.;Gunthard, H. H. Spctrochim. Acru 1982,37A,
6.1153
0.2637 7.3431
6.1677
0.2796 7.3431
6.1685
0.2799 7.3431
6.1688
0.2800 7.3431
6.1698
0.2803 7.3431
6.5071
-0.5924
electrostatic cgs. CElectricquadrupole moment, unit 1
consistent defect configurations was used, extended for calculation of normal modes in either mass-weighted Cartesian or internal (infinitesimal) coordinates. As a rule, all 1116 normal frequencies and eigenvectors of DFE:Ar364 systems were computed, but analysis of the data (frequency, symmetry, and degeneracy of normal modes and eigenvectors, respectively) was usually restricted to 20-50 highest and 50 lowest modes. By nullifying all molecule-matrix interaction forces the program system may directly be used to compute the self-consistent state and the normal modes of the system Ar,,-V(O,O,O) (Ar3, crystal fragment with vacancy in the center) and by inserting an Ar atom in the center of DFE:Ar,,,, annulling all DFE-Ar interactions and adding all pair interactions between the central Ar and the Ar3,-fragment C F F model and normal modes of the complete Ar363matrix were obtained. 2.2. Potential Energy Function and Normal-Coordinate Analysis. As described in the foregoing work, locally stable nuclear configurations of the DFE:Ar3, system were determined by a self-adapting step length, steepest descent minimization procedure of the potential energy function (5 denotes internal coordinates)'
0.2637 7.3431
6.5081
-0.5967
0 0
6.1153
6.5084
-0.5976
0 0
0.2637 7.3431
6.5092
-0.601 1
0 0
6.1153
6.5616
-0.8105
0 0
0.2636 7.3431
6.5616
-0.8105
0 0
6.1151 6.5616
-0.8105
0
-53.96
aAd
-0.8 144
0
OCf.Figure I . bElectric dipole moment, unit 1 Polarizability tensor of DFE, unit A3.
285.
M'
X
electrostatic cgs.
six eigenvalues should vanish, all other being positive (gDFE:Ar3, systems). The same holds in mass-weighted Cartesians, where the Hessian reads M'/2FeMl/2= Fe (M = mass matrix). In the present work this requirement could be satisfied only in rare cases: often one and two of the improper modes and one of the proper modes turned out to have negative eigenvalues. Partly this deficiency originated from only approximate CFF determination of the re structure of the defects, partly from the principally high condition number K = Xmax/Xmin of Fe.I9 The spectrum of Re is given by o{FJ = [XllFe- X.11 = 0) (3') (1 = unit matrix) and the normal frequencies 'v (cm-I) follow from 3 = X'/2/2izc
(3")
In the case at hand, admitting values of the order of SO.1 cm-I for frequencies of the improper modes, the condition number amounts to From this high condition number one may conclude, taking the numerical precision applied in the present work into account, the improper eigenvalues and eigenvectors in most cases to be badly determined and also to violate occasionally the symmetries expected. Use of the Newton-Raphson minimization in various modifications did not allow to reduce the inaccuracies or to reduce (eliminate) proper negative eigenvalues. It should be mentioned that the second derivatives required in eq 2 were computed from analytical expressions.' 2.3. Structural Parameters. The r,' structural parameters of self-consistent nuclear configurations of the defects GH1, GH21, and GH22 are taken over from earlier work'-3 without modification. Equilibrium external parameters and dihedral angles of the three defects as estimated from energy minimization sequences are listed in Table I. Besides estimated re parameters each energy minimization sequence produces its own final parameters; some of these are collected in Table 11, whereas all others like final matrix coordinates and structural parameters of DFE will not be reproduced here, for obvious reasons. Figure 1 illustrates the coordinate frame, internal structural parameters, and external (19) Schwarz, H. R.; Rutishauser, H.; Stiefel, E.Numerik symmetrischer
Matrizen; Teubner: Stuttgart, 1968; p 22.
The Journal of Physical Chemistry, Vol. 95, No. 7, 1991 2805
Normal Vibrations of gauche- 1,2-Difl~oroethane:Ar~~., structural parameters (qcc,T[oll~) used to define the position of the molecule with respect to crystal coordinate system. 2.4. Electric Polarization Energy of G2-Type Defects. In a preceding paper formulas for computation of the electric polarization of type GI and TI defects of DFE:Ar364systems were derived. Starting from permanent dipole and quadrupole moments of DFE ("point multipole approximation") first a solution for induced dipole moments of all Ar atoms and of DFE within the first-order induced moments-first-order polarizability approximation has been formulated (neglect of higher induced moments and hyperpolarizabilities), which then allowed to one express the polarization energy of the system as a sum of linear and quadratic forms of all induced dipole^.^ Electric moments as a function of the dihedral angle (referred to the molecular frame, Figure 1) were obtained from DZ quality quantum chemical calculations, which served as a data basis for expansion of the multipole components into Fourier series. Values of all components corresponding to the values of the dihedral angle of DFE were then computed from the Fourier series for (i) all final structures emerging from energy minimization sequences listed in Table I and (ii) the estimated re structure of the defects GH21 and GH22. Likewise in Table I1 estimates of the polarof DFE (referred to the molecular frame) are izability tensor dA) included, calculated from a bond polarizability model for the same set of dihedral angles T . ~ A, value ~ ~ of ctM= 1.63 A3 for the scalar polarizability of the Ar atom has been chosen.20 For type G2 defects the previously derived theory of dielectric polarization has to be modified. First in this setting the point symmetry is realized by (1(3) denotes the unit matrix in R3)
e,[oll] = ( E , C2[0ll]) = {l?[;
j}
(4)
CE U CC2 U e 2 E , with ICE1 = ICC21 = 178, ld2El = 8 ( 5 ) Summations over sets CE and e 2 E have to be carried out accordingly. Third, the formula for the induced dipole moment of the matrix atoms now reads4 (&. k and X e 2 E k denote Cartesian coordinate vectors of matrix atoms of set CE and e 2 E ,respectively, writing C2 for C2[011] and 1 for 1 ( 3 ) )
1)~6€k)Ib'6Ek9
-aM
+
+ CzT((C2 [ T ( X 6 E k ) a ( A ) ( 1 + C2)T(XEEk) +
C2)T(XGEk) i#k
c
IGW
l@zW
Numerical solution of the linear system (6'"") is most conveniently carried out using the coordinate system {Z"s,Of} (crystallographic axes with origin shifted to origin of the frame system, cf. Figure 1). This implies the translation TloIl]of all matrix atom coordinate vectors to origin Or and transformation of electric moments and polarizability dA) according to the homogeneous transformation
{efeie;}= {eFSefSeFS)D{e@,],4]
0 -21Rsin 4 D(e&,
4) = 2-ln
1 cos 4 [l -cos#
4
:]
-2lDcUs
(*)
The rotation angle 4 stands for the rotation angle cpcc of molecular basis (2.9 with respect to basis {Ccs}. As a consequence of eq 8, transformation of vectors V and symmetric tensors S of rank 2 from frame {Of,Odto frame {ga,0f} is to be carried out according to vlecsl= DVle'l slecst = Blelb
Second, the set of Ar atoms decomposes into equivalent sets in general site e (site symmetry group 6 )and in site a (site symmetry group e2);2' using the notation of ref 4,one finds the decomposition
V k E [1,178]: 11 - a M [ T ( X 6 E k ) a W ( 1
rank 2 and 3 expressing the electric field in x produced by a dipole AdA1) and quadrupole MA2), respectively, localized at the origin. The total electric energy may be expressed by the induced dipole moments of the matrix atoms, leading to uclt= -$#AI)(a(A))-lp(AI) - j / 2 p ( A I ) ( a ( A ) ) - l p ( A I ) (1 /2aM)(2 ~ ( G E k 9 p ( Q E k 9+ E $@zEkl)p(@zEkI)) (7)
(9)
Table 111 contains dipole and quadrupole and polarizability data referred to {ecs,Of)required in this work. Every energy minimization sequence used to localize the structure of defects GH21 or GH22 leads to slightly different values of final coordinates of the matrix atoms and the parameters cpcc, T~olIl, and T . In contrast to the latter, no attempts so far have been made to obtain estimates of the re structure matrix coordinates by suitable extrapolation p r o c e d ~ r e s .In ~ ~order to provide for information about systematic errors in computations of the polarization energy originating from use of nonextrapolated matrix coordinates, the following steps were adopted: (i) for every energy minimization sequence the electric polarization has been computed by using its specific final configuration; (ii) for each defect GH21 and GH22 computation of the polarization energy is carried out by combining r,' parameters p'cc,c, T\9,!1,e,and 7,' with the final matrix coordinates of every energy minimization sequence used to determine the defect. By these procedures a survey of the errors originating from use of final instead of r,' matrix coordinates should be provided.
3. Results 3.1. CFF Energy and Normal Modes of Matrix Models ArXS T(X6Ek - X6EE) + c 2 T ( c 2 X 6 € k - X6EE)lp'6ELo and Ar3,-V(0,0,0). Some results of CFF modelings of the matrix [ T ( X 6 E k ) a ( A ) T ( X @ 2 E , t ) + T ( X 6 E k - X@,Ek)lp(ezEE') = aM systems Ar365and Ar364-V(0,0,0) are collected in Table IV, and legrl Table V information on the seven highest and nine lowest normal aMT(X,$Ek)hf(AI) + 1/2aMfl-I'3)(X6Ek)hf(A2) (6') modes is listed; both data sets result from the Buckingham-type pair potential. Vk E [ 1981: [I - (YMT(X@zEk)"(A)T(X~,Ek)]p(ezEk9 The following statements should complement the data shown [ T ( X @ 2 E k ) a ( A ) ( 1 + C ~ ) T ( X ~+E(1 ~+ ) aM in Tables IV and V: ltE.6 (i) For both matrix systems and for both pair potentials used c2)T ( X @ 2 E k - X 6 E t ) l p ' 6 E L o energy minimization sequences converge rapidly and monotonically tzk within =120 steps to the final values of potential, first difference aM [T(x@,Ek)(r'A'T(x,z,t) + T ( x @ z E k - &?zE,E)lP(ezEiCo = l@*E.cI and residual gradient given in Table IV, in sharp contrast to the ( Y M T ( & 2 E k ) h f ( A 1 ) + 1 / 2 ( Y ~ ~ - " ~ ) ( X @ ~ ~ k (6") ) h f ( ~ ~ )complicated, often erratic convergence behavior experienced with DFE:Ar3, systems. Furthermore, no negative eigenvalues of pe were observed and the highest improper frequency found with these. where T ( x ) , fl-IJ), x E R(3)/{0,0,0} denote the usual tensors of systems amounted to 10.01 cm-I. (ii) Final values then may safely be considered as close ap(20) Hirschfelder, J. 0.;Curtis, C. F.; Birs, R. B. Molecular Theory 01 proximations of re values of both potential, matrix atom coordinates Gases and Liquids; Wiley & Sons: New York, 1954; p 947. and normal-mode frequencies, for both types of pair potentials. (21) Hahn, T., Ed. International Tables for Crystallography; Reidel: Dordrecht, 1983; Vol. A, p 108 (Wykoff notation). However, the latter lead to different final potential energies. The 16ELl
2806 The Journal of Physical Chemistry, Vol. 95, No. 7, 1991
Gunde et al.
TABLE III: Electric Moments and Polarizability of gDFE Refererd to Coordinate Frame (5a,0# MI b Mc sequence GH21 MSOOO
0
-2.0162 -2.0 I62 0 -2.01 59 -2.01 59
M55M05 M55P05
0
-1.9970
1.0146 0.9985
-1.9848
1.0305 0.9924
-1.9872
1.0275 0.9936
-1.9852
1.0299 0.9926
2.4546
-I .4444 -1.2273
2.7152
-1.2225 -1.3576
2.7246
-1.2136 -1.3623
2.2526
-1.5851 -1 .I 263
-2.01 59 -2.0159 GH2l
0
re estimate
GH22 P30M40
-2.0 I59 -2.01 59 0
P35M45 P35M35 GH22 re estimate
-1.9539 -1.9539 0 -1.9529 -1.9529 0 -1.9527 -1.9527 0 -1.95 13 -1.9513
aCf.Figure I . bElectric dipole moment, unit 1 Polarizability tensor, unit A3.
X
OLAd
-1.0146 -1 3099 0.9985 -1.0305 -1.8029 0.9924 -1.0275 -1 3042 0.9936 -1.0299 -1.8032 0.9926
6.2932
-0.3581 6.8634
6.2885
-0,3551 6.8657
6.2894
-0.3557 6.8653
6.2887
-0.3552 6.8657
1.4444 0.6261 -1.2273 1.2225 0.7599 -1.3576 1.2136 0.7655 -I .3623 1.585 1 0.5339 -1 .I263
7.3070
0.2443 6.3565
7.2551
0.2950 6.3825
7.2531
0.2967 6.3835
7.3385
0.2049 6.3407
electrostatic cgs. cElectric quadrupole moment, unit 1
0.3581 -0.3016 6.8634 0.3551 -0.3041 6.8657 0.3557 -0.3037 6.8653 0.3552 -0.3040 6.8657 -0.2443 0.1527 6.3565 -0.2950 0.1259 6.3825 -0.2967 0.1246 6.3835 -0.2049 0.1664 6.3407 X
electrostatic cgs.
TABLE I V Consistent Force Field (CFF) Data of Matrix Models Aru4 and Arw-V(O,O,O) model
pair
- V,
potential'
kcal/mol
B
569.628 407 546.822 539 565.497 700 542.843 3 16
Ar365
LJ B
Ar364-V(030.0)
LJ
IAll?
kcal/mol step
