564
J . Phys. Chem. 1984, 88, 564-514
Force-Field Model for the Stretching Anharmonicities of SF6 Burton J. Krohn* University of California, Los Alamos National Laboratory, Theoretical Division, Los Alamos, New Mexico 87545
and John Overend Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455 (Received: May 25, 1983)
A geometrical intramolecular force-field model for the SF6 molecule is used to compute spectroscopic parameters. The model includes a general valence harmonic potential whose constants were determined from spectroscopic analysis, plus explicit anharmonicities in the form of (a) a Morse potential in each stretching S-F bond and (b) a Urey-Bradley interaction between each nearest-neighbor nonbonded pair of F atoms. Problems with the Urey-Bradley potential are recognized and treated. Three of the five Morse and Urey-Bradley parameters are constrained to impose consistency with properties of the stretching bond. The other two parameters are varied to find the best agreement with 10 observed anharmonicities involving the stretching modes, vl, v2, and vg. The fits are good, and the best parameters are used to predict unobserved stretching anharmonicities. Attempts to model the bending motions are less successful.
I. Introduction Since the observations of collisionless multiple-photon dissociation of SF6,1,2UFg,394 and SeF6,’ with the CF4 laser6 in the case of UF6, many spectroscopists have turned their attention to heavy spherical-top molecules, seeking to determine the anharmonicities which shift and split the excited vibrational levels or cause resonances between them. Low-resolution (- 1 cm-’) spectra in the range7s8600-3000 cm-’ gave initial estimates for some of the anharmonicities of SF6. With lasers came detailed spectroscopic analyses of sF6, first of the infrared-active fundamental and v412 bands and then the Raman-active v i and v2 bands; the dominant vibration-rotation structure coefficients derived from and at = -(B, - Bo),are these spectra, such as g = -(3/7)1/2z3, linear functions of cubic potential constant^.^^ Later, high-resolution (laser) infrared spectroscopy of excited states yielded very accurate values for several anharmonicity constants, particularly from the 3v3 overtone manifolds of SF6,16J7UF6,I8 and SiF4,I9 (1) R. V. Ambartsumian, Yu. A. Gorokhov, V. S. Letokhov, and G. N. Makarov, Z h . Eksp. Teor. Fiz., Pis.’ma Red., 21, 375 (1975) [JETP Lett. (Engl. Transl.), 21, 171 (1975)]; Zh. Eksp. Teor. Fiz. 72,440 (1976) [Sou. Phys.-JETP (Engl. Transl.), 44, 231 (1976)l. (2) J. L. Lyman, R. J. Jensen, J. Rink, C. P. Robinson, and S. D. Rockwood, Appl. Phys. Left., 27, 87 (1975). (3) (a) J. J. Tiee and C. Wittig, Opt. Commun., 27, 377 (1978). (b) P. Rabinowitz, A. Stein, and A. Kaldor, ibid., 27, 381 (1978). (4) M. Alexandre, M. Clerc, R. Gagnon, M. Gilbert, P. Isnard, P. Nectoux, P. Rigny, and J.-M. Weulersse, J . Chim. Phys., 80, 331 (1983). (5) J. J. Tiee and C. Wittig, J . Chem. Phys., 69, 4756 (1978). (6) R. S. McDowell, C. W. Patterson, C. R. Jones, M. I. Buchwald, and J. M . Telle, Opt. Lett., 4, 274 (1979). (7) V. V. Bertsev, T. D. Kolomiitseva, and N. M. Tsyganenko, Opt. Spectrosc., 37, 263 (1974). (8) R. S. McDowell, J. P. Aldridge, and R. F. Holland, J . Phys. Chem., 80. 1203 (1976). (9) R. S . MdDowell, H. W. Galbraith, B. J. Krohn, C. D. Cantrell, and E. D. Hinkley, Opt. Commun., 17, 178 (1976). (IO) M. Loete, A. Clairon, A. Frichet, R. S. McDowell, H. W. Galbraith, J.-C. Hilico, J. Moret-Bailly, and L. Henry, C.R Hebd. Seances Acad. Sci., Ser. B, 285, 175 (1977). (11) Ch. J. Bord6, M. Ouhayoun, A. van Lerberghe, C. Salomon, S. Avrillier, C. D. Cantrell, and J. Bordb, in “Laser Spectroscopy IV, Proceedings of the 4th International Conference, Rottach-Eaern. June 1979”. H. Walther and K. W. Rothe, Eds., Springer Series inoptical Sciences, Vol. 21, Springer-Verlag, New York, 1979, pp 142-53. (12) K. C. Kim, W. B. Person, D. Seitz, and B. J. Krohn, J . Mol. Spectrosc., 76, 322 (1979). (1 3) H. Berger, A. Aboumajd, and R. Saint-Loup, J. Phys., Lett., (Orsay, Fr.),38, L-373 (1977): A. Aboumajd, H. Berger, and R. Saint-Loup, J . Mol. Spectrosc., 78, 486 (1979). (14) P. Esherick and A. Owyoung, J . Mol. Spectrosc., 92, 162 (1982). (15) K. T. Hecht, J . Mol. Spectrosc., 5, 355 (1960). (16) A. S. Pine and A. G. Robiette, J . Mol. Spectrosc., 80, 388 (1980). (17) C. W. Patterson, B. J. Krohn, and A. S. Pine, J . Mol. Specfrosc., 88, 133 (1981). (18) G. A. Laguna, K. C. Kim, C. W. Patterson, M. J. Reisfeld, and D. M. Seitz, Chem. Phys. Lett., 75, 357 (1980). ~~
0022-3654/84/2088-0564$01.50/0
+
-
the v2 v4 combination band of CF4,20the 2v1 + v3 band of SF6,” and the v 1 v3 v3 hot band of SF6.” Other hot bands in the and v425 regions of the SF6 spectrum and in the v3 region of UF626have given effective X,, coefficients for i = 3 or 4, but the evaluation of the actual anharmonicities awaits more detailed knowledge of the upper states. SF6transitions between 2150 and 4800 cm-’, correspondingto ternary and higher combinations, have been identified.27 Techniques of double resonance have been used to o b s e r ~ e and ~ ~ a- n~ a~l y ~ e ~2v3 ’ . ~ ~ v3 transitions, and the contour of multiphoton absorption of highly intense laser radiation33is determined by the anharmonic structure of the overtone states. Recently it was found34that SF6 molecules in excited nu3 overtone states emit radiation at the v2 + v6 frequency, suggesting that higher nu3 levels are not “pure”, but in reality resonate with ( n - l)v3 (v2 v6) combination levels through a (236) cubic potential term. To understand and to model such spectroscopic and multiplephoton phenomena r e q ~ i r e ~ ’detailed -~~ knowledge of the anharmonic structure. In excited states both this vibrational structure and the rotational substructure are sensitive functions of the potential energy surface; l 5 even in a binary overtone the ratios
+
v323924
-
+
+
(19) C. W. Patterson and A. S . Pine, J . Mol. Spectrosc., 96, 404 (1982). (20) C. W. Patterson, R. S. McDowell, N. G. Nereson, R. F. Begley, H. W. Galbraith, and B. J. Krohn, J . Mol. Spectrosc., 80, 71 (1980). (21) A. S. Pine and C. W. Patterson, J . Mol. Spectrosc., 92, 18 (1982). (22) P. Esherick, A. J. Grimley, and A. Owyoung, Chem. Phys., 73, 271 (1982). (23) A. V. Nowak and J. L. Lyman, J . Quant. Spectrosc. Radiat. Transfer, 15, 945 (1975). (24) K. C. Kim, R. F. Holland, and H. Filip, Appl. Spectrosc., 32, 287 (1978). (25) W. B. Person and K. C. Kim, J . Chem. Phys., 69, 2117 (1978). (26) B. J. Krohn and K. C. Kim, J . Chem. Phys., 77, 1645 (1982). (27) H. B. Levene and D. S. Perry, J . Chem. Phys., 80, 1772 (1984). (28) P. F. Moulton, D. M. Larsen, J. N. Walple, and A. Mooradian, Opt. Lett., 1, 51 (1977). (29) C. C. Jensen, T. G. Anderson, C. Reiser, and J. I. Steinfield, J . Chem. Phys., 71, 3648 (1979). (30) F. Herlemont, M. Lyszyk, and J. Lemaire, Appl. Phys., 24, 369 (1981). (31) C. Reiser, J. I. Steinfeld, and H. W. Galbraith, J . Chem. Phys., 74, 2189 (1981). (32) M. Dubs, D. Harradine, E. Schweitzer, J. I. Steinfeld, and C. Patterson, J . Chem. Phys., 77, 3824 (1982). (33) S. S. Alimpiev, N. V. Karlov, S. M . Nikiforov, A. M. Prokhorov, E. G. Sartakov, E. M. Khokhlov, and A. L. Shtarkov, Opt. Commun., 3, 309 (1979). (34) W. Fuss, Chem. Phys. Lett., 71,77 (1980). (35) C. C. Jensen, W. B. Person, B. J. Krohn, and J. Overend, Opt. Commun., 20, 275 (1977). (36) C. D. Cantrell and H. W. Galbraith, Opt. Commun., 18, 513 (1976); 21, 374 (1977). (37) H. W. Galbraith and J. R. Ackerhalt in “Laser-Induced Chemical Processes”, J. I . Steinfeld, Ed., Plenum Press, New York, 1981.
0 1984 American Chemical Society
The Journal of Physical Chemistry, Vol. 88, No. 3, 1984 565
Stretching Anharmonicities of SF6 of anharmonic structure constants to the Coriolis splitting parameter (B{Jeff control the qualitative behavior of the spectrum.38 For SF6 a new summary of the anharmonicities has been compiled,39both from new FT-IR data and from previously published work. Our confidence in the values of those parameter^^^ depends on the particular spectra used to derive them and ranges from very certain (for the case of the nu3 ladder constant^"^^^ X33,G33, and T33measured from the v3 and 3v3 manifolds) to doubtful (for the case of X44,where even the identification of the 2v4 v4 hot band is questionable). The successful interpretati~n~l of the multiphoton spectrum33demonstrates the utility of the well-determined nu3 ladder constants. On the other hand, when an analysis requires anharmonicities that are known with much less certainty (or are not available at all), model values must serve instead. For example, a detailed c a l ~ u l a t i o nof~ the ~ v3 - (v2 v6) Fermi resonance used a force-field value of the (236) cubic constant to interpret the v2 + u6 features in the spectrum of SF6 in excited nu3 levels.34 Model potential surfaces are essential to the prediction of electric dipole transitions involving changes of more than one quantum of vibration. Those transition moments depend sign i f i ~ a n t l y on ~ ~the - ~ anharmonicities. ~ They appear explicitly in formulations for the infrared vibration-rotation spectra of the binary overtones of tetrahedral XY4,38and they have been used in the pure-vibrational intensity modeling for the binary transitions of CF4,47the binary combinations4* in SF6 and UF6, and 3v3 overtone manifolds of SF,49,50and UF6.51 The earlier model calculation^^^"^^^ assumed that the second derivativesof the dipole moment made only insignificant contributions to the intensity. More recently the calculations served to estimate second derivatives from experiments2 and to compare them with a b initio valuess3 for SF6. In this work we have tested the capabilities of a set of intramolecular force-field models of SF6to reproduce the 10 stretching-mode anharmonicities that are presently known. The general valence force field (GVFF) for an octahedral xY6 molecule has 18 internal displacement coordinates including the 6 independent X-Y bond stretches Ar, si = ri - rie and the 12 deformations of the Y-X-Y angles Aai = a, - aie,where each equilibrium angle is ale= 90°. Since this molecule of N = 7 atoms has 3 N - 6 = 15 internal degrees of freedom, 3 redundancies exist among the 12 angular displacements. The GVFF quadratic potential contains 11 possible coeffic i e n t ~but, , ~ ~because of the redundancies, only 7 of them can be determined independently. We adopt the conventions5 which assigns zero values to those constants whose two internal coordinates do not share a common bond. The nonvanishing coefficients then include the pure stretch and pure bend ( K , and K J ,
-
+
TABLE I: Explicit Cubic and Quartic Anharmonicities in a Urey-Bradley Force Field for an Octahedral XY, Molecule cubic anharmonicities
no. torin of terma of terinsb
( A RI l 3 ( AR I )’
AR ,
( A RI ) * aa ,, AR,ARR,Aa,,
AR I ( A a I , (AcY,,)~
Vasquez, “Infrared Combination Bands and Vibrational Anharmonicity in Sulfur Hexafluoride”, in preparation. (40) C. W. Patterson, B. J. Krohn, and A. S.Pine, Opt. Lett., 6, 39 (1981). (41) C. W. Patterson and A. S. Pine, Opt. Commun., 44, 170 (1983). (42) D. P. Hodgkinson and A. G. Robiette, Chem. Phys. Lett., 82, 193 (1981). (43) R. K. Heenan, Ph.D. Thesis, University of Reading, Reading, England, 1979. (44) H. Hanson, H. H. Nielsen, W. H. Shaffer, and J. Waggoner, J . Chem. Phys., 27, 40 (1957). (45) F. Legay, Cah. Phys., 12, 416 (1958). (46) C. Secroun, A. Barbe, and P. Jouve, J . Mol. Spectrosc., 45, 1 (1973). (47) W. G. Golden, C. Marcott, and J. Overend, J . Chem. Phys., 68,2081 (1978). (48) W. B. Person and J. Overend, J . Chem. Phys., 66, 1442 (1977). (49) C. Marcott, W. G. Golden, and J. Overend, Spectrochirn. Acta, Part A , 34, 661 (1978). (50) K. Fox, J . Chem. Phys., 68, 2512 (1978). (51) C . Marcott, W. G. Golden, and J. Overend, J. Chem. Phys., 68,2929 11978) \--
- I ’
(52) D. S. Dunn, K. Scanlon, and J. Overend, Spectrochim. Acta, Part A , 38, 841 (1982). (53) K. Scanlon, R. A. Eades, and D. A. Dixon, Spectrochim. Acta, Part A , 38, 849 (1982). (54) C. W. F. T. Pistorius, J . Chem. Phys., 29, 1328 (1958). (55) P. Labonville, J. R. Ferraro, M. C. Wall, S. M. C. Basile, and L. J. Basile, Coord. Chem. Rev., 7, 257 (1972).
6 24 24 12 24 12
no. form of term0
of
(AR,)4
(AR,)’AR, (AR,)2(AR,)2
(AR,)’Aa,, (AR,)~AR,ACY,, ( A R ,) 2 ( ~ ~ , , ) 2 AR I AR ,(A& ,,1
aR,(~a,,)~
(ACX,,)4
teriiisb
6 24 12 24 24 24 12 24 12
AR ,, A R , , and A a , , refer respcctively to stretching displaceiiicnts of two adjacent X-Y bonds and the deforniation of their included angle. The number ot‘ terms is consistent with restricted summations, as in eq I .
’
adjacent-bond and opposite-bond stretching interactions ( K , and K,,), interactions between bending angles in the same plane and in perpendicular planes (Kaa and KO,,),and a stretch-bend interaction (Kra). The harmonic potential has the form
24 k a
C
(Ari)(Aaj) (1)
iJ
in this notation of restricted summations, where each distinguishable term appears only one time. In principle, the knowledge of the six harmonic frequencies w , plus one of the two Coriolis coupling constants (t3,14)determines the seven independent K coefficients in eq 1, as in the cases of SF6,8939UF6, 56 WF6,57and M o F ~ , ~ In ’ low resolution the values of ( i = 3, 4) are deduced from the separation between the points of maximum intensity in the P- and R-branch profiles of the v I fundamental band; 59 in high resolution {, is related to the fitted line spacing parameter n from the equation60s61 (scalar)
N
m
+ nM + pM2 + ...
(2)
(where M = -Jo for the P branch and +(Jo + 1) for the R branch) by60,15
n (38) K. Fox, J . Mol. Spectrosc., 9, 381 (1962). (39) R. S. McDowell, B. J. Krohn, J. L. Lyman, H. Flicker, and M.
quartic anharmonicities
-
N
Bo
+ B, - 2(B{Jeff
N
2B,(1 - l1)
(3)
Ideally the constants 5; satisfy the condition6* (3
+ c4
=
‘/2
(4)
There are three contributions to the anharmonicities of our model. One is implicit anharmonicity arising from the nonlinear transformation of the GVFF potential to normal coordinates. Another is an explicit Morse potential63between bonded pairs of atoms in the molecule; 64 here it is assumed only in each of the six S-F bonds. The model containing these two sources alone has two explicit anharmonic terms, Kr,,(ArJ3and Krrrr(ArJ4; it was (56) R. S. McDowell, L. B. Asprey, and R. T. Paine, J . Chem. Phys., 61, 3571 (1974). (57) R. S. McDowell and L. B. Asprey, J. Mol. Spectrosc., 48, 254 (1973). (58) R. S. McDowell, R. J. Sherman, L. B. Asprey, and R. C. Kennedy, J . Chem. Phys., 62, 3974 (1975). (59) W. F. Edge11 and R. E. Moynihan, J . Chem. Phys., 27, 155 (1957). (60) G. Herzberg, “Infrared and Raman Spectra of Polyatomic Molecules”, Van Nostrand-Reinhold, Princeton, NJ, 1945. (61) B. Bobin and K. Fox, J . Phys. (Paris), 34, 571 (1973). (62) R. S. McDowell, J . Chem. Phys., 41, 2557 (1964); 43, 319 (1965). (63) P. M. Morse, Phys. Rev., 34, 57 (1929). (64) S. Reichman and J. Overend, J . Chem. Phys., 48, 3095 (1968).
566
The Journal of Physical Chemistry, Vol. 88, No. 3, 1984
used in the earlier estimations of the nv3 overtone structure^^^^^^ for SF6 and the intensities of binary combination^^^ and 3u3 overtone transitions in the spectra of SF649and UF6.s1 A third contribution, added to improve the accuracy, is a portion of an explicit Urey-Bradley i n t e r a ~ t i o n between ~ ~ , ~ ~ the 12 nearestneighbor nonbonded pairs of F atoms. Thus, our complete model intramolecular potential consists of a quadratic GVFF portion and a simpler central force field (CFF)55anharmonic portion. The resolution of the Urey-Bradley interactions into GVFF coordinates generates a rich set of explicit cubic and quartic anharmonic terms (see Table I). Modeling the spectroscopic parameters presumes knowledge of the normal-coordinate anharmonicities. The program library at the University of Minnesota contains convenient transformations of the potential energy from a Taylor expansion in general valence coordinates to the normal-coordinate representation through quartic terms. The change in representation is accomplished in two stages. The first routine performs the nonlinear transform a t i ~ of n ~an~ arbitrary potential surface in valence displacement coordinates to mass-adjusted Cartesian difference coordinates. The second routine, written for arbitrary molecular species,68 transforms linearly to dimensionless normal coordinates. The well-known theoretical problems associated with redundant coo r d i n a t e ~ ~ ~have - ’ ~ recently been attacked. The correct general treatment r e q u i r e ~ ~that ~ s ~all~ of the independent redundancy relations be incorporated into a modified potential energy by using the undetermined Lagrange multipliers known as “intramolecular tensions”. A computer algorithm73has correctly propagated the effects of redundancy through quartic terms. The algorithm is independent of any particular form of the GVFF. Application of the appropriately modified to the model CFd7, and SF67scalculations using superposed Morse and Urey-Bradley potentials for the anharmonicities has resulted in improved agreement with experiment. Unfortunately, many of the present computations at Los Alamos were carried out before the latest refinement^^^-^^ were accomplished at the University of Minnesota, so the present work neglects the influence of the redundant coordinates. Yet, as our primary concern here is the stretching vibrations, and the six X-Y bond stretches are truly independent (among themselves and of the bending motions), we expect71this model to agree rather well with the spectroscopy involving modes vl, v2, and v3, As the spectroscopicobservables for spherical tops are identified with the coefficients of scalar or invariant tensor operator^'^*^^ in the Hamiltonian, a formalism is needed to relate them to the parameters of the normal-coordinate potential surface. For the case of tetrahedral XY, molecules the complete second-order vibration-rotation Hamiltonian77was rearranged15into scalar and tensor forms. For the case of octahedral xY6 molecules the presence of four triply degenerate vibrations makes the Hamiltonian somewhat more complicated than it is for XY4. To our knowledge the analogous tensor arrangement of the full vibration-rotation Hamiltonian for xY6, with detailed expressions for the coefficients, has not been published to date. However, useful portions of it have appeared in analytic form, including the cubic (65) H. C. Urey and C. A. Bradley, Phys. Rev., 38, 1969 (1931). (66) T. Shimanouchi (Simanouti), J . Chem. Phys., 17, 245, 734, 848 ( 1949). (67) M. A. Pariseau, I. Suzuki, and J. Overend, J . Chem. Phys., 42,2335 (1965). (68) W. D. Gwinn, J . Chem. Phys., 55, 477 (1971). (69) J. E. Rosenthal, Phys. Rev., 45,538 (1934); 46,730 (1934); 49, 535 (1936). (70) E. B. Wilson, J . Chem. Phys., 9, 76 (1941). (71) E. B. Wilson, J. C. Decius, and P. C. Cross, “Molecular Vibrations”, McGraw-Hill, New York, 1955. (72) B. Crawford and J. Overend, J . Mol. Spectrosc., 12, 307 (1964). (73) C. Marcott, S.D. Ferber, H. A . Havel, A. Moscowitz, and J. Overend, Spectrochim. Acta, Part A , 37, 241 (1981). (74) I. Suzuki and J. Overend, Spectrochim. Acta, Part A , 37, 1093 (1981). (75) I. Suzuki and J. Overend, Spectrochim. Acta, Part A , 38, 767 (1982). (76) J. Moret-Bailly, J . Mol. Spectrosc., 15, 344 (1965). (77) W. H. Shaffer, H. H. Nielsen, and L. H. Thomas, Phys. Rev., 56,895 ( 1939).
Krohn and Overend and quartic potential energy functions,78 the bond-stretching anharmonicity coefficients involving modes vl, v2, and v3,79and some important vibration-rotation coefficient^.^^^^^ These formulations link our calculations to spectroscopy and permit us to use the model and to evaluate it through direct comparisons with experiment. We now compare the present work with the most recent model calculation^^^ for SF6: (i) Both studies have used a GVFF quadratic potential, whose constants were obtained from spectroscopic analysis, and a C F F for the anharmonicity, which superposes a Morse potential in each S-F bond with part of a Urey-Bradley interaction between each nearest-neighbor pair of nonbonded F atoms. The GVFF portion has been improved in the present study. (ii) The earlier work included the effects of redundant coordinates; the present work ignores them. (iii) The previous work fixed the Urey-Bradley and Morse parameters and allowed the quadratic potential constants and intramolecular tensions to vary. The present work fixes the quadratic potential and varies two of the five Morse and Urey-Bradley parameters over an extensive grid but fixes the remaining three parameters to preserve the experimental values of three properties of the stretching S-F bond. In section I1 we present the mathematics of the constrained Urey-Bradley-Morse model, and we discuss and evaluate the model in section 111. 11. Theory
Potential Surface for the sF6 Model. We superpose three source functions to construct the complete model, which is then expanded in valence coordinates about the equilibrium configuration through quartic terms. First we put identical stretching potentials in each of the six S-F bonds, whose length is rr L
where each term in eq 5a is the Morse function63 with a > 0. It is emphasized that, contrary to earlier work,3s~47-49,51,s2,64 we do not take rM to coincide with the true equilibrium bond length re for the SF6molecule. In our model rM is always slightly smaller than re. For the second source we subject each of the 12 pairs of nearest-neighbor F atoms (separated by distance qV)to identical Urey-Bradley interactions,6s,66so that
When the Urey-Bradley method was first introduced,6s attention was focused primarily on the quadratic and higher-degree terms in the expansion of the potential, while the nonvanishing linear terms were ignored. Thus, without additional contributions, the Urey-Bradley model did not describe the equilibrium condition of the molecule. For the present case of an octahedrally symmetric xY6 molecule (as also for the case of a tetrahedral x Y 4 molecule), when one converts to the GVFF representation, the linear contributions in eq 6a from the angular coordinates cancel identically. Only the linear terms in the six independent bond-stretching coordinates, Arr, survive. This work supplies those essential contributions to remove the Ar,terms by adjusting the parameter rMin eq 5b for the S-F bonds, as mentioned above. Here each V , term in eq 6a is a Lennard-Jones potentials1 in the forma2
(78) K. Fox, B. J. Krohn, and W. H. Shaffer, J . Chem. Phys., 71, 2222 (1979). (79) D.P. Hodgkinson, R. K. Heenan, A. R. Hoy, and A. G. Robiette, Mol. Phys., 48, 193 (1983). (80) H. Berger and A. Aboumajd, J . Phys. (Paris),Lett., 42, L-55 (1981). (81) J. E. Lennard-Jones, Proc. Phys. SOC.,London, 43, 461 (1931). (82) F. Fowler and E. Guggenheim, “Statistical Thermodynamics”, Cambridge University Press, London, 1956.
Stretching Anharmonicities of SF6
The Journal of Physical Chemistry, Vol.88,No. 3, 1984 567
1
S-F
B o n d Stretching P o t e n t i a l
I
0 8 t
I
i
0
Figure 1. SF6 molecule, with six of its seven nuclei frozen in the equilibrium configuration, and one F nucleus free to move along the projected S-F bond. The distance between the displaced F nucleus and one of its four nearest neighbors is q = ( r 2 r:)Il2.
+
where n = 12 and qo is the value of qij at which V, attains its absolute minimum. For the third source we superpose that set of quadratic GVFF terms which is required to bring the complete harmonic potential (see eq 1 ) into agreement with experiment. These three sources are equivalent to the observed harmonic force field, plus the Morse and Urey-Bradley potentials (eq 5 and 6 ) whose five parameters, DM,a, rM,W, and qo, are chosen to provide a stable molecule and to match observed properties as closely as possible. Knowledge of the seven harmonic constants in eq 1 and the five anharmonic constants in eq 5 and 6 enables us to model the vibrational energy eigenvalues and the observable spectroscopic parameters. As part of the specificationof the latter five constants, we impose three constraints on the model to force it to be consistent with three observations. To accomplish this, we consider a molecule whose nuclei are frozen in the equilibrium configuration with the exception of one F nucleus, which is free to move in one dimension, along the projected S-F bond (see Figure 1). Taking as our stretching coordinate the displacement from true equilibrium, s = Ar = r - re, we express the total potential energy, V&) as
vT(s)=
vM(s)
+ VU,($)
(7)
(>O)
(8)
We let
d = re-rM
and expand the Morse potential (eq 5) in a Taylor series about s=o
For the Urey-Bradley potential we let
and the expansion of eq 6 about s = 0 is c
6(R” - R6)
s+
re 9(2R12- R6) re2
s2 -
(35R” - 8R6) r,3
s3
- 4R6) + 3(63R12 4r,4
(11) In eq 1 1 the factor 4 represents the four nearest-neighbor, non-
-0.8 0.5
1
1.5
2.5
3.5
r (A) Figure 2. Relative contributions of the Morse and Urey-Bradley components of the potential energy. Parameters for the two components are listed in column C of Table 111. The solid curve is the superposition of the two sources, VT(r) = VM(r) VuB(r). See Figure 3.
+
Urey-Bradley
Function
Morse Function
Figure 3. Component potential functions for the stretching S-F bond. Solid curves represent the functions defined in eq 5 and 6 , whose parameters are listed in column C of Table 111. Dashed and dotted curves indicate polynomial expansions about r = re (see eq 9 and 11) including contributions, respectively, through the quadratic and quartic terms.
bonded F atoms, which affect the “free” F atom in an identical fashion. The complexity of eq 9 and 11 in general, and the appearance of nonvanishing linear terms in particular, are consequences of expansion about the point r = re, which is not at the minimum of either of the two individual potential functions. Figure 2 illustrates the relative contributions from these two explicit sources of anharmonicity, and the polynomial expansions of these functions about r = re are shown in Figure 3. The three experimental facts by which we constrain the model parameters are as follows: (i) The dissociation energy for the F-SF, bond was observeds3 to be Eobsd = 89.9 f 3.4 kcal/mol = 0.624 f 0.024 mdyn-.&/ molecule. This value represents the difference in energy between (a) the SF6molecule in equilibrium and (b) the unassociated F atom at rest plus the SF5molecular fragment, which is relaxed, rather than in the frozen C,, configuration in Figure 1. As we do not have an estimate of this relaxation energy ( E R > 0), we + E R and set E D equal to neglect its contribution to E D = Eobsd Thus, the depth of the model potential VT(s)(represented by the solid curve in Figure 2) is probably somewhat too small. (83) T. Kiang, R. C. Estler, and R. N. Zare, J . Chem. Phys., 70, 5925 (1979).
The Journal of Physical Chemistry, Vol. 88,No. 3, 1984
568
(ii) Electron diffraction measurements have givens4 the value of re = 1.564 8, for the length of the S-F bond in SF6. However, our model assumes the slightly smaller value, re = 1.5561 A, suggested by recent spectroscopic analysis (see section 111). (iii) From the most recent values39of the harmonic frequencies and f3 and l4for SF,, the pure-stretching potential constant K , (see eq 1 ) was determined to be K , = 2.730 mdyn/8,. These three observations directly influence the depth and the first two derivatives of the total model stretching potential, VT(s), at the true equilibrium, s = 0. That is, only the combined potential relates to the observables; the parameters of its two individual components must be inferred. Since the S-F bond-stretching coordinates are independent, the redundancies among the valence angles do not affect the following treatment.’I Substitution of eq 9 and 1 1 in eq 7 , followed by separate comparisons of the terms of degrees 0, 1 , and 2 yields
q = q 1 2 = ( r 1 2 + r 2 2 - 2 r 1 r 2c o s a l 2 )112 Figure 4. Relationship between the separation ( 4 ) between two nonbonded atoms in a triatomic configuration and the valence coordinates ( r l rr2, cyl2) for the two bonds.
We express eq 17 in the form of a quartic equation in one unknown parameter a:
+
+
p4a4 + p3a3 pzaZ po = O
(20)
= ( 7 / 1 2 ) d 2 [ 6 / K+ r 4 d ( P / K r + a/ED) +
(21)
where
VT/T~~=O = -ED =
+ &a2 - d3a3+
(-1
Krohn and Overend
DM
+ 4(R12 - 2R6)W (12)
P4
+ 3 d ( P / K , + a / E D ) + d 2 / E ~ ] (22) pz = 1 / K , + 2d(/3/Kr + a / E D ) + d Z / E D (23)
p3 = - d [ 3 / K r
24(R12- R6) re
w (13)
36(2R1’ - R6) W (14) re2 Equations 12-14 represent three dependencies to relate the five variables, DM, a, rM,W, and qo. We have found that a practical course to follow is to select, in a systematic manner, arbitrary but fixed values for the two characteristic distances rMand qo. These choices fix the values of d and R and reduce the number of unknowns to three, so that we can solve eq 12-14 for the values of DM, a , and Win terms of E,, re, K,, d , and R . (The solution is unique, as discussed below.) For convenience in section I11 we introduce the projected Urey-Bradley bond distance
(see Figure 1 ) ; thus, the selection of rUBis equivalent to that of 40.
The algebraic elimination in eq 12-14 is straightforward and easily programmed on the computer. From eq 13 one has W=
re (2daz - 3d2a3 + y3d3a4)DM (16) 24(R1’ - R 6 )
Substitution of eq 16 into both eq 12 and 14 yields
-=-DM
[
+ ( 2 a d + &)a2 - (3a& + d3)a3+
-1
ED 3
I
-(4ad3 12 = (l/K,)[(l
1
+ d4)a4
+ 2pd)a2 - 3(d + pd2)a3+ 7/(3d2+ 2pd3)a4] (17)
where a = re(R12- 2 R 6 ) / ( 6 ( R 1* R6))
(18)
p = 3(2R12 - R 6 ) / ( 2 r e ( R l-2 R6)I
(19)
(84) V. C. Ewing and L. E. Sutton, Tram. Faraday Soc., 59, 1241 (1963). (85) T. Shimanouchi, I. Nakagawa, J. Hiraishi, and M. Ishii, J . Mol. Spectrosc., 19, 78 (1966). (86) H. Kim, P. A. Souder, and H. H. Claassen, J . Mol. Spectrosc., 26, 46 (1968).
PO = -1 / E D (24) The roots of eq 20 can be determined either by exact methods for solving quartic equations given in textbooks on the theory of algebraic equations or by standard iterative computer programs. In the domain of physically meaningful values of rM C re and ruB > re, and with reasonable values of K , and ED, the four roots of eq 20 include a complex conjugate pair, one real negative root, and one real positive root. Only the positive real root is relevant. After a is derived in this manner, then the coefficients of DM and Win eq 12-14 are all known, and one may then find DM and W by a simultaneous solution of any two of those three linear equations. We note that this algorithm is singular when either rM = re (so that d , p4, and p 3 vanish) or rUB= re (so that a and @ become infinite). However, there appears to be a region, perhaps arbitrarily close to the limit point (rM,ruB)= (re,re)(where both rMand ruBare approaching their limits), in which unique solutions can be derived. This is mentioned further in section 111, in connection with SF6 modeling. At this p i n t 1 1 independent parameters determine the complete model potential. Nine of them come directly from experiment, namely, the seven quadratic constants (including K, for the pure S-F bond stretching), the observed dissociation energy ED = eo^, and the equilibrium bond length re. The remaining two of them, rMand rUB,are selected as discussed in section 111. Three other model parameters, DM, a, and W, are automatically specified by the set of values for r M ,rUB,ED, re, and K,, as discussed above. It is now necessary to transform the Urey-Bradley contribution to the potential energy (eq 6 ) from the C F F coordinates q,, to the GVFF coordinates rr,rJ,and ajJ,so that it can serve as input to the programs6’@ which calculate and use the normal-coordinate surface parameters. As the first step in the conversion to GVFF representation we take the Taylor expansion of I/vB(q) in the vicinity of q = qe = 21/2re(see Figure 1 ) VUB(qe
+ Aq)
VO
+ Vl(Aq) +
V2(Aq)2
+ V3(Aq)3 + V4(Ad4 (25)
From the form of vUV,,(q) in eq 6 with n = 12 the coefficients in eq 25 are Vo = W(R” - 2R6)
(26)
VI = ( 1 2 V / q e ) ( R ” - R 6 )
(27)
- 7R6)
(28)
V3 = (28W/q,3)(13R1Z- 4R6)
(29)
V4 = ( 3 W / q ; ) ( 4 5 5 R l 2 - 84R6)
(30)
Vz = (6 W/q:)( 13R”
where R is defined in eq 10.
The Journal of Physical Chemistry, Vol. 88, No. 3, 1984 569
Stretching Anharmonicities of SF6
Substitution of eq 26-31 and 34-38 in eq 25 finally yields
TABLE 11: Partial Derivatives of q with Respect t o Valence Coordinates" __1;irst Derivatives aq/ari = s i j q ( a q / a a i j ) = rirj sin eij
VuB(Ar,)z Yo + Y1 + Yz + Y3
+ Y4
(39)
where
Yo = W(RI2- 2R6)
Second Derivatives q ( a 2 q / a r i 2= ) rijZ 41 a 2 q / ( a r i a r j ) J= -tijrji q1 a 2 q / ( a r iaeij)J= r j s j i t i j q ( a 2 q / a a i j 2=) -rirjsijsji
(40)
Y, = VIA = -(6W/re)(R12- R6)(Arl+ Ar2 + Ar3)
(41)
(with the Ar3 terms vanishing in the summation over the 12 Urey-Bradley terms, as mentioned earlier in this section)
TI1ird Derivatives q 2 ( a 3 q / a r i 3=) - 3 ~11. . 1r1 . . ~ q 2[ a 3 q / ( a r i 2 a r j ) ]= rij(2sijrji- sjirij) q 2I a 3 q / ( a r i 2a a i j ) ] = -rjtij(2sijsji r i j r j i ) q 3[ a 3 q / ( a r i arj aaij)] = rirj sin aij (2siisji + r i j t j i ) q 2 [ a 3 q / ( a r iaoij2)]= r j 2 [ s i j t j i -2 s ~ ~ ( ~ rijrji) ~ ~ s , ~ ~ q 3 ( a 3 q / a e i i 3 =rirj ) sin aij (3rirjsijsji- - q 2 )
Y2 = V2A2 + VIB
+
+
+
+
= ( 3 W / r 2 ) [ ( 6 R l 2- 3R6)(Ar12 ArZ2 2ArlAr3 2Ar2Ar3) (7RI2- 4R6)(Ar32 2ArlAr2)](42)
+
+
1,'ourth Derivatives q 3 ( a a q / a r i 4=) 3rijz(4sij2- I . . ' ) 431 a4q/(ari3 arj)] = 3rij(2sij$rij - 2sij2tji + tij2rji) q31a4q/(ari2a,;.')] = - 8 s , . iIs . .11r . . 1r1. .11 + 2s,.2f..2 + 2~,.2r..2 - 3 f . .121 t .-1. 2I 11 l i 11 q3[a4q/(ari3 = 3rjrij(2sijrijtji2+2sij2sji - sjirij:j 4'1 a 4 q / ( a r i 2arj aa,)] = tij(2rjsijsji - 5risijfijrji - 4risij'sji + 4r:~;;t;;~)
+
+ 2V2AB + V,C = -(W/r,3)[(35R12- 8R6)(Ar13+ ArZ3)+ (147R1248R6)(Ar12Arz+ Ar1Arz2)+ (105R12- 24R6)(Ar12Ar3 + ArZ2Ar3)+ (258R12 - 78R6)ArlArzAr3+ (126R12 36R6)(ArlAr32+ ArzAr33)+ (55R12- 19R6)Ar33](43) Y, = V4A4 + 3V3A2B+ V2(B2+ 2AC) + V,D Y3
= V3A3
W 4r,4
+
+ 240R6)(Ar13Ar2 + ArIArz3)+ (2394R12504R6)Ar12Ar22 + (756R12- 48R6)(ArI3Ar3+ Arz3Ar3)+
= --[(189RI2 - 12R6)(Ar14 Ar24) (1344R" a See Figure 4 and cq 3 2 and 33. The quantities sij and f i j satiot'y t h e identities sij2 + f l j 2 = 1, s i j f j i + sjiti, = sin cui,. and r .I.lrl.1. -s..s.. - cos 01.. 11j111.
The next step is to expand the C F F displacement Aq = Aq12 (see Figure 4 ) in terms of the valence displacements Arl, Ar2,and Aa12,which we denote, in general, by ARi: Aq r A B + C + D (31)
+
where
a4 A=X--AR, i aRi
1 B=2!
a24 ARiARj c-dRidRj ij
etc. The required partial derivatives are listed in Table 11, in terms of the geometrical quantities66 sij = (ri - rj cos a i j ) / q u tij
(32)
= rj sin aij/qij
(33)
For the case of octahedral xY6 molecules the equilibrium values of ri = rj = re,aij= r / 2 , and sij= tu = 2-'12 simplify the expansion considerably. Temporarily using the notation
Ar3 = r,Aal2
(34)
we have A = ( 1 / 2 1 / 2 ) ( A r+ 1 Arz
+ Ar3)
(3612R12 - 624R6) (ArI2Ar2Ar3 + ArlAr22Ar3)+ (1428RI2 - 168R6)(Ar12Ar32Ar22Ar32)+ (3696R12 - 672R6)ArlAr2Ar32 (1320RI2 228R6)(ArlAr33 Ar2Ar33) (476RI2- 104R6)Ar34](44)
+
Equations 39-44 give the GVFF representation for a single nonbonded nearest-neighbor pair of F atoms. When we consider the 12 pairs coordinated in an octahedral xY6 configuration, each term that contains Ar3 or that mixes Ar, with Ar2 corresponds to one term in a restricted summation represented by Table I. In those cases the cubic and quartic coefficients in the total potential, V,, are taken directly from eq 43 and 44. For example, we have Kr,,(T) = Kr+,(UB) = -(W/r:)(258RlZ - 78R6) (45) and, for the quartic, pure-bending term K,,,,(T)
+
(35)
+ 2ArlAr3 + 2Ar2Ar3)
(36) 1 C= [3(-Ar13- ArZ3 Ar12Ar2 ArIArz2)+ 24(21/2)r2 9(-Ar12- ArZ2 2ArlAr2)Ar3- 3(Ar, + Ar2)Ar32- Ar33] (37) 1 D= [3(3Ar14+ 3Ar24+ 4Ar13Ar2 4Ar1Arz3192(2ll2)r: 14ArlZAr22) + 36(Ar13+ Ar23- Ar12Ar2- ArIArz2)Ar3 42(Ar12 ArZ2- 2ArlAr2)Ar32- 4(Arl Ar2)Ar33 Ar34] (38)
+
+
+
+
+
+
+
+
= K,,,,(UB)
= W( 1 19RI2- 26R6)
(46)
However, for the,pure-stretching terms we must introduce a factor 4 in the total because the interactions of four nearest-neighbor pairs contribute to the stretching potential in each X-Y bond. This gives
B= 1 (Ar12 Arz2- Ar32- 2Ar,Ar2 4( 21/2)r,
+
+ +
Krrr(UB) = -(4W/r,3)(35R12 - 8R6)
(47)
Krrrr(UB) = ( 3 W/r:)(63Rl2 - 4R6)
(48)
Again, the complete potential surface for this model is the superposition of (i) the Morse potential in each S-F bond as expanded in eq 9 with s 2 Ari, (ii) 12 Urey-Bradley F-F interactions, and (iii) the other necessary quadratic terms discussed below. Briefly we consider two categories of terms, pure-stretch (S) and all the others (SB-stretch-stretch, pure-bend, and stretch-bend)
VT(Ar,Aa)= Vs(Ar)+ VsB(Ar,Aa)
(49)
In the model, sources i and ii completely determine V,(Ar)through the quartic terms
Vs(Ar)=
6
(-ED 1=1
+ 0 + KrAr,2+ KrrrArI3+ KrrrrAr14)(50)
570 The Journal of Physical Chemistry, Vol. 88, No. 3, 1984 TABLE 111: Varying parameters of the Geometrical Force-Field Model' --_____-_______ con\ t a n t Ab BC @,d
0.624
I . : D , ~mdyn.A rM, A 'UB, A DM, n1dyn.A
2.335 0.6780 2.1005 11.83 19.1
a, A - '
1 O4 W, mdyn.A 0,
0.624 1 SO86 2.040 0.65 11 2.1065 40.4 20.0
1 SO55
x
0.687 1.5 I33 2.020 0.7079 1.9986 40.2 17.3
Krohn and Overend
DC,/
0.624 1.4738 2.206 0.7020 2.0979 31.1 49.6
-_____
t/,u
1 (', t
0.624 1.4784 2.670 0.7265 2.0923 4.94 40.2
0.687 1.4728 2.206 0.7645 1.9885 31.1 48.2
Gf,6'
0.687 1.4775 2.670 0.7891 1.9878 4.94 39.9
Values of the fixed parameters are listed in Table IX. Point of best agreement (Le.. the siiiallcst u ) for the weighting indicated in eq Point of best agrectnent for the weighting indicatcd in cq 58. The Morse and Urey-Bradley contributions for this case are illustrated in 1:igtire 2. e The specified valucs o f Wand of y, = 2.70 A (see the present eq 15) are taken froiii ref 85. The quality indes u is defined by eq 56. The specified values o t Wand o f q , = 3.09 A are taken froiii ret' 86. E D = 0.624 ii1dyn.A is the measured dissociation energy. EObsd(see ref 83). ED = 0.687 represents a 109, increase. a
56.
TABLE IV: G V F F Model Coefficientsa cubic anharmonicities -
constant
A
B
C
K,.,.,., indyn/A2 K,.,.,.,, mdyn/A2 K,,,, indyn/A K r r t a , nidyn/A K,,, mdyn K,,. n1dyn.A
-5.494 -0.787 -0.897 -2.163 -1.651 -1.105
-5.525 -0.865 - 1,004 - 2.388 -1.828 -1.208
-5.3 15 -0.795 -0.924 -2.195 -1.681 -1.110
_____ constant
quartic anhartnonicitics
K,.,.,.,.. tndyii/A3 K,.,.,.,.,, mdyn/A3 K r r r t r Jindyn/A3 , K,.,.,.,, indyn/A2 Kj-rr,,, nidyn/A2 Krr,,, mdyn/A Krriaa, indyn/A K,.,,. mdsn K,,, 1ndyn.A
A
t3
C
8.393 1.200 2.120 1.079 5.024 3.132 7.981 4.445 2.467
8.279 1.356 2.382 1.245 5.685 3.580 9.01 7 5.031 2.768
7.327 1.250 2.194 1.150 5.239 3.302 8.309 4.6 36 2.549
a These coefficients refer t o restricted summations. The order of thcir appearance is the same as in Table I. Columns A-C correspond to t h e respective column\ in Table 111.
where E D and K, are measured experimentally and formulated in eq 12 and 14, and where KAT) = KAM)
= DH( -a3
+ KrrAUB)
+ 3 d a 4 ) - %(35RI2
- 8R6)
re3
7 = -DHa4 12
(51)
111. SF6 Calculations, Results, and Evaluation
3w
+ 7 ( 6 3 R 1 2 - 4R6) re
from eq 9, 41, and 48. The Morse function ensures that the stretching anharmonicities are consistent with the correct qualitative behavior of the potential, at least in the channel to dissociation by large stretching displacements of a single S-F bond. On the other hand, the terms in T'sB(Ar,Aa) of eq 49 are more difficult to specify, and the Urey-Bradley interactions generate only a subset of the allowed terms. In particular, the interaction between opposite stretching bonds is excluded from this subset, and their corresponding anharmonicities are absent from Table I. Although the coefficients of the cubic and quartic terms in Table I are determined by the values of DH,a, rM,W, and qo (or rUB),a problem exists in the quadratic portion, where Wand qo are overdetermined by the three independent, observed constants, K,,, K,, and K, in eq 1 . Thus, to the quadratic terms with coefficients
- 4R6)
(53)
K,,(UB) = (l8W/r,)(2Rl2 - R6)
(54)
K,,(UB) = (6W/r:)(7Rl2
the form of the explicit terms in our model, along with the imposed constraints on pure stretching, to lead to reasonable values of the fitted parameters, to rather good agreement with known stretching anharmonicities, and probably to fairly reliable predictions of those anharmonic constants involving modes vl,v2, and w3 that have not been measured.
K,(UB) = 3W(7R12 - 4R6)
(55)
(see eq 42) we have added the terms from source iii to "complete" them and also to fill the K,,,K,,, and K,,, terms, which do not relate at all to sources i and 11. Although source iii removes the difficulty in the quadratic portion, it artificially truncates away the unknown cubic and quartic contributions that would correctly "round out" these quadratic terms. This could lead to improper behavior at higher energies, where motions (other than pure S-F stretches) begin to have large amplitudes. Nevertheless, we expect
The two model parameters which are not determined directly from experiment are the characteristic distances rMand rUB(see eq 15). Our method is to vary their values over the lattice points on an extensive grid in the rMrUB plane to find the regions of good agreement with the known parameters of stretching anharmonicity. (Although several coefficients involving bending modes are also known, they are not included in this fit because their agreement is much poorer in general.) At the time of this work the values of 1 1 stretching constants have been obtained from spectra, namely, X11, XU,X13r X33, '333, T33, a1, az, a3, GS,and Z3r. As a dependency relation, discussed below, exists among the six parameters associated with w3 overtones, one degree of freedom is removed and only 10 of these parameters are independent. The quality of the agreement between calculated and observed constants is measured by the dimensionless standard deviation = ([S(xll) + S(xl2) + $(XI,) + YZ(S(x33) + S(G33)) + S(T33) + S(a1) + S(az) + S(a3) + S(Z3s) + S(Z3r)I/9)"2 (56) where each quantity S is the squared fractional deviation, e.g.
S(Xl,) = ((Xll0b" - x1 1 cdcd )/XI
1°bsd)2
(57)
In eq 56 the deviations are weighted equally, with the exception of S(X3,) and S(G3,), which share a weight of 1 because of the dependency. An alternative procedure is to give extra weight to those two constants associated with the symmetrical stretching mode w1
The Journal of Physical Chemistry, Vol. 88, No. 3, 1984 571
Stretching Anharmonicities of SF6
3.5
-
3 3.0 '5[
3.0
a
a
v
V
m
m
3
3
2.5
2.5
2.0
2.0
1 1.48
1.50
1.52
1.54
(A) Figure 5. Contours of agreement between observed and calculated anharmonicities for ED = 0.624 m d y d . Here u is defined as in eq 56, and the curves represent u = 19.1%, 19.5%, 20%, 22%, and 25%. The best agreement is u = 19.09%, at the point corresponding to column A in Tables 111-VI. The points labeled D and E refer respectively to ref 85 and 86, in which the pairs of values (qo,W) are specified. See columns D and E in Table 111. rM
rM ( A ) Figure 7. Contours of agreement for ED = 0.687 m d y d . Here u is defined as in eq 56, and the curves represent u = 17.4%, 18%, 19%, 20%, 22%, and 25%. The best agreement is u = 17.32%, at the point corresponding to column C in Tables 111-VI. The points labeled F and G refer respectively to ref 85 and 86, in which the pairs of values (q,,,W) are specified. See columns F and G in Table 111.
1 A
6
V
2.5
16 Figure 8. Present convention for labeling the nuclei of a hexafluoride molecule (see Table VII).
1.48
1.50 rM
1.52
1.54
(A)
Figure 6. Contours of agreement for ED = 0.624 m d y d . Here u is defined as in eq 58, and the curves represent u = 20.1%, 20.5%, 22%, and 25%. The best agreement is u = 20.03%, at the point corresponding to column B in Tables 111-VI. The points labeled D and E refer to columns D and E in Table 111, except here the respective values of u are 40.9% and 33.3%, consistent with eq 58. was calculated, and the contours of constant u were plotted (see Figures 5 and 6 ) . We have also investigated the sensitivity of these calculations to the value of one of the constraints, E D . The value of EDwas increased by lo%, from 0.624 to 0.687 mdyn.%L. This test seems reasonable in the realization that the unknown relaxation energy of the dissociated S F 5 fragment probably reduces the observed value of E D somewhat, as discussed above. The resulting contours are shown in Figure 7 . Anharmonic energy calculations involve terms in the Hamiltonian containing the Coriolis constants and thus the IkiY coefficients in the transformation from mass-weighted Cartesian displacements (Tk = xk, Yk, z k ) to normal coordinates are requireds7
(mk)"'Tk = ClkI'Qi
(59)
The dependence upon the lk,% appears in the signs of terms containing odd powers of cy; these signs can vary with the choice of phases of the lk% Our values of the 1 coefficients (see Table VI1 and Figure 8) are closely related to the symmetry coordinates of ref 79 except we have chosen t = 30' instead of 120' (as discussed in ref 7 8 ) . Hence, the vibrational anharmonicity formulas of ref 79 are directly applicable to our work. In other present conventions we define
= -AB, = -(B, - Bo)
(60)
for a fundamental transition in mode vl, so that B, N Be - C401+ YzdJ
(61)
I
where d, is the degeneracy of v,. The vibrational structure constants XIJ,GI,, and TIJare consistent with the notation of both ref 15 and 7 9 . We also use the parameters b2, PI, and 6, ( i = 3 or 4) of ref 79, with
PI = 1/3zls 6, =
-1ozi,
(62) (63)
where Z,, and Z,, are used in ref 1 5 . The formulas for the six spectroscopic constants related to the overtones of v3, listed in Table VIII, reveal that only five independent anharmonic potential constants connect them. Thus, a functional relationship exists among them
1
f(X33,G33,T33,ff3,Z3s,Z3t) I0 (87) H.H.Nielsen, Reu. Mod. Phys., 23, 90 (1951).
(64)
The first three equations in Table VI11 will show that a3,Z3#, and
572 The Journal of Physical Chemistry, Vol. 88, No. 3, 1984
Krohn and Overend
TABLE V: Expan\ion Coefficients (em -' ) for the Dimensionless Normal-Coordinate Potential Energya
con \ t an t
obsd
A - 19.03 -43.6 -55.6 - 18.8
+35.6 - 17.2 -4.0 -59.1 - 106.3 -8.3 t35.0 + 18.4 - 12.1
c
B - 19.78 -43.9 -55.3 -20.7 +36.8 -19.1
c a n $1an t
-57.2
-105.0
Dl122 Dl133
D l 1 4 4 DIl~S Dl166
D2222
Dj333~
-101.8
-9.2 +35.1 +20.2 - 12.8
+0.385 +1.86
D,,,,
-18.70 -42.2. -53.3 - 19.0 +35.1 - 17.5 -4.3 -56.2
-5.5
obsd
D3333B
-8.4 +34.0
+3.21 -7.99
D4444A 0,4448
+ 18.7 -12.2
+0.69 -1.73 +0.29 -0.54
D5i55.4
DssssB
- 12.6 + I 1.8 - 12.6
-35.3 -t8.1
+9.3 -16.1 - 14.8 +32.4
-13.2 +13.4 -13.2 -38.5 +9.1
- 12.6
D6bbbA
+12.1 - 12.4 -35.6
D6,,,n
+8.5
D 2 2 33 8
- 18.9
'2244A
-
14.8 +34.8
-
-0.38 + 12-75 +0.26 -0.6 1 -0.84 +2.67 -0.54 +2.08
D2233.4
t9.6 16.3 -14.8 +32.5
+10.0
02244B '2255.4 D225iB D22beA D22bbB
a
A 10.744 +2.95 +3.46 + 1.48 + 1.44 +0.38 1-0.75 + 3.06 -7.64 t 0.24 t0.67
B
C
+0.788 t2.97 +3.40 +1.71 +1.67 t 0.52 +0.74 +2.97 -7.44 +0.27 +0.76
t0.713 +2.64 +2.98 + 1.56
+0.78 -1.93 +0.3 1 -0.55
-0.35 t12.36 +0.39 -0.82 -0.86 +2.92 -0.49 +2.18
+153
+0.45 +0.65 +2.62 -6.70 t0.25 t0.69 +0.7 1 -1.79 t0.30 -0.53 -0.35 t10.93 t0.32 -0.73 -0.81 12.71 -0.50
+2.08
See ref 78 for notation. Colunins A-C corre5pond to the respective C O I U I I I I ~ S i n Table Ill.
TABLE VI: Spectroscopic Parameters' in ern-' _____-________
con\tant
XlIb Xllb XI,
XI,
XI,
ob\d
-0.896' -2.358' -2.902' -1.1 44c -1.15 -0.3625'
-_______
A
B
C
-0.609 -2.487 -2.983 -0.399 -0.423 +0.182
-0.682 -2.644 -3.193 -0.471 -0.533 t0.280 -0.121 -0.227
-0.596 -2.499 -3.0.51 -0.328 -0.361 t0.251 ~-0.172 -0.176
x 2 2
-0.11 1
G22
-0.199
con\ t d n t
X,, ti,4 T,, 104uIb
lO4aZb 104013b IO401,
10401, X2 3
-2.103
T23
-0.105
-1.068 --0.044 -1.657 +0.909 -0.227
-2.210 -0.092 -1.154 -0.052 -1.664 +0.896 -0.222
-2.273 -0.121 -1.028 --0.048 -1.707 +0.902 -0.232
lo4@,
obsd
, +1.1038' -0.397c1f +1.3099g +0.1941'
A t0.186 -0.054 -0.0021
B
C
+0.190 -0.058 -0.0003
t0.202 -0.062 --0,0029
+1.230 -0.223 +1.317 +0.252 +1.248 +0.747
t1.295 --0.216 +1.309 +0.308 +1.304 +0.789
+1.201 -0.264 +1.249 +0.260 +1.258 t0.754
t0.290 +0.300 +0.306 t0.3056g t0.276 +0.267 +0.258 -3.7519g -1.74256d -3.742 -3.720 -3.590 x 3 3 +0.918805d -0.016 +0.002h -0.033 -0.018 C :,, T33 -0.24635td 1046, +0.426' +0.190 t0.058 t0.178 The notation for all constants is consistent with ref 79. (See the present eq 61-63. Colunins A-C correspond to the respective colunins in Table Ill. These quantities were used to dctcrminc u in eq 56 or 58. The other observed parameters were not included in the Fits. ' See ref 39. See ref 17. e See ref 14. This value conics from further analysis (see ref 39) o f data presented in ref 13. See ref 11. I : r o n i r e t 7 9 , 4 = 1 / l ( ~ - - D i . Tlirvnltieof'U=-1.91 X c i i i - ' istakenfroni rci'12,and t h c v a l u e o f p = - 1.96 X isderived inref 39. ' See ref 12. x 2 4
T24
Z,, are linearly related to c l 3 3 , C233, and C3,,. A convenient expression for the dependency is
+ 3633 = , -2w32z31c1332 + 2/3w32z32c2332
X33
+ t/2w?z35c3352 + 3Be3;2
(65)
A similar dependency is not found among the v3 constants of tetrahedral XY, molecule^.'^ However, the four coefficients concerning the v2 overtones are also related X22
+ 3 6 2 2 = -2~1*~2lC122~ + 2C22z2/(9W2)
(66)
and this dependency occurs in both xY6 and XY4 spherical tops. Independent spectroscopic determinations of the six constants of the v3 overtone^'^^",^^ of SF6have yielded the values listed in Table VI. Unfortunately, they do not satisfy eq 65 very well: the left-hand side is +1.01 cm-', while the right-hand side is +1.25 cm-'. In this work we first obtained C133, c233,and C,,, from a3, Z3$,and Z3,. Then D3,,,, was uniquely determined from T33. Finally, the two results of D3333A = 3.13 and 3.29 cm-I followed respectively from the formulas for X 3 , and G33. Their average, D,,,,,= 3.21 cm-I, is the experimental value assumed here.
104h,
104p3b 10463b 104p,
Model parameters which remain constant through all of the calculations in this work include re and the seven quadratic GVFF constants (see Table IX). This also fixes the harmonic frequencies and five Coriolis coupling constants. In actuality the harmonic frequencies ui,the true S;,'s, and quadratic potential constants in Table IX were derived from the observed band centers, effective and 14,Xl,'s, and isotope shift data.39 The parameters in Table IX represent an average of eight calculations of the set of Flusymmetry force constants F33,F3,, and Fd4from different sources of information. Their uncertainties reflect the full dispersion of values in those calculations. The constants Be and re were calculated from eq 61 by using the ground-state valueI7 of B, = Bo = 0.091084 cm-I, the observed values of a l , az,a3,and a4listed in Table VI, and the calculated values of asand a6in column A of Table VI. The uncertainty of 0.000 10 cm-I in Be is the change that occurs in B, when both Clssand c l 6 6 (which c ~ n t r i b u t e ' ~ J ~ to as and ff6, respectively) are set equal to zero. A choice of variables ED,rM,and rUBcompletes the parameter set. Each set determines the three remaining Urey-Bradley and Morse potential constants, as given in Table I11 and all of the GVFF cubic and quartic coefficients from eq 43, 44, 51, and 52
c3
The Journal of Physical Chemistry, Vol. 88, No. 3, 1984 573
Stretching Anharmonicities of SF6
TABLE VII: Coefficientslkir in the Transformation from Mass-Adjusted Cartesian Displacements‘ to Normal Coordinatesb
-
Cartesian
coordinate
Q,
Q2e
Q2f
Pax
Q,,
2,
0.408 25 -0.408 25
0.500 00 -0.500 00
0.288 6 8 -0.288 6 8
0.424 92 0.424 92 0.045 11 0.045 I 1 0.045 11 0.045 1 I -0.794 20
-0.435 07 -0.435 07 0.343 23 0.343 23 0.343 23 0.343 23 -0.387 57
xq x2
x5 x3 X6
*l
Cartesian coordinate
,v2 yr
Q, 0.40825 -0.40825
Qze
Qzf
-0.50000 0.500 00
0.288 68 -0.288 6 8
Y, YI I’4 3’1
23
2,
Q 0.408 25 -0.408 25
Q2e
Q2f
-0.577 35 0.577 35
ZI 24
22
z5 2,
‘ Here each x k 3)’k.
QAv -0.435 07 -0.435 07 0.343 23 0.343 23 0.343 23 0.343 23 -0.387 57
Q3, 0.424 92 0 424 92 0.045 I 1 0.045 I 1 0.045 I I 0.045 I I -0 794 20
-0.435 07 -0.435 07 0.343 23 0.343 23 0.343 23 0.343 23 -0.387 57
Q3V
Y3
Cartesian coordinate
0.424 92 0.424 92 0.045 11 0.045 11 0.045 I I 0.045 11 -0.794 20
or Zk is weighted b y the factor (mk)”’.
See cq 59.
Q,Z
Q ~ Y
425,
-0.500 00 0.500 00
0.500 00 0.500 00 -0.500 00 -0.500 00
Q,,
Q, v
-0.500 00 0.500 00
0.500 00 0.500 00 -0.500 00 -0.500 00
-0.500 00 0.500 00
Q5Z
-0,500 00 0.500 00
QSX
Q S Y
-0 500 00 0.500 0 0
-0.500 00 0.500 00
Q6.X
Q,Z
0.500 00 0.500 00 -0.500 00 -0.500 00
Our convention for labeling the nuclei is illustrated i n 1:igure 8.
TABLE VI11: Dependencies among SpectroscopicCoefficients‘
TABLE 1X: Fixed Parameters of the Geometrical Force-Field Model‘ _ I _ _ _ -
spectroscopic constants w I = 786.99
(8)
ctn - I w 2 = 654.62 ( 9 ) w 3 = 965.85 ( 9 ) w 4 = 622 ( 2 ) w 5 = 530 ( 2 ) w,=352.42(9)
potential constantsb
p, = t 0 . 7 1 I (18)
K,.= 2.730 ( 7 ) m d y n / A
p, = -0.21 1 ( I 7) p,, = +OS91 ( 1 4 ) tS6= -0.380 ( I I ) p,, = t 0 . 7 7 8 ( 6 )
K,.,. = 0.3560 tndyn/AC K,.,., = 0.05 ( 4 ) Indyn/A K , = 1.064 (4) mdyn.A K,, = 0.223 ( 4 ) n1dyn.A K,,’=0.113 ( 3 ) t i l d y ~ n
5,=0.09162(10) Ctll
-I
r e = 1.5561 ( 8 ) A
K,,=0.695
(18)iiidyn
’
Uncertainties in p;ircnthcscs refer to the last digit(s) i n each These constants are defined tor the unrestricted entry. Thi9 coett’icicnt is insensitive to variations sutiitnations in cq I , in t h e int‘rarcd-active synitnetry c o n s t a n t r F3,,, I+‘,,, and I+’44.
(Table IV). Table V presents the complete list of 22 calculated cubic constants for the normal-coordinate potential energy and a partial list of the coefficients of the diagonal quartic terms. A few of these potential constants can be compared with their experimental values, which were derived from spectroscopic parameters by using equations like those listed in Table VIII. In most cases their uncertainties are dominated, not by those in the
measurements of the parameters, but by those of the calculated values of the Ct,’s which enter the formulas. The spectroscopic parameters in Table VI offer the best opportunity for direct comparison with experiment. Columns A-C in Tables 111-VI represent coordinates (rM,rUB) which lie very close to the absolute minima of the respective u surfaces. Two earlier estimate^^^,^^ of the Urey-Bradley parameters W and qo for two interacting F atoms are included in columns D-G of Table 111. (The parameters of ref 85 were used in the most recent SF, model calculation^.^^) The combination of Wand qo with ED (and the eight fixed parameters) in our procedure constrains these interaction potentials to be consistent with the known properties of the F-SF5 bond. The set (W,qO,&) uniquely determines r M ,DM, and a. The contour plots in Figures 5-7 show that both of the earlier models give values of rMthat are a little too small. However, it is remarkable that a two-parameter model has predicted 10 stretching anharmonicities with u < 50% in each calculation. Each contour plot shows an extensive domain in which the rms agreement is better than 25%. In each case the absolute minimum is well-defined but shallow because the minimum value of u is not much less than 25%. As ruB increases, the contours indicate a trough approaching a constant depth, at some fixed value of r M . At the other extreme, when u > uc, where uc is some critical
574
J. Phys. Chem. 1984, 88, 574-580
-
value for any particular set of contours, the contours appear to approach the limiting point, (rM,rUB) (re,re)arbitrarily closely. Such is the case for E D = 0.687 (see Figure 7), where uc lies between 24% and 25%. Closer examination of the 25% contour near (re,re),where re is set at 1.556 1000 A, reveals a continually narrowing region with u < 25%, extending at least to rUBI 1.556 106 0 A and rMI1.556 099 0 A. (Inside that tiny domain about the singularity the contours behave irregularly, probably because of excessive roundoff error in the computer algorithm.) W is a sensitive function of rUBand falls very rapidly with increasing rUB. When rUB= 3.6 A, at the top of Figures 5-7, W m d y d ; even in the change from rUB= 2.040 A (column B in Tables 111-VI) to rUB= 2.335 8, (column A) W decreases nearly by a factor of 4. The calculated values of DM and a are much more stable. In earlier SF6 models35@which did not include the Urey-Bradley interactions a was given the value 1.7 A-1. The presence of the Urey-Bradley terms appears to raise a to the range 2.0-2.1 A-1, The ability of this model to fit to 10 observed stretching anharmonicities with u I20% is not impaired when we use a different scheme for weighting the deviations or change ED by a small amount. In Figures 6 and 7 these two tests showed some shifts of the contours, and slight changes in the depth of the absolute minimum, but no significant qualitative changes. Such stabilities suggest that, even if the model is somewhat inaccurate, it should still predict unknown stretching anharmonicities with fair reliability. Thus, for example, the calculated value of T23= -0.105 cm-l predicts a small sp1itting;O A 9 -l6TZ3 E 1.7 cm-’ between the F,, and F,, vibrational components of the v2 v3 combination state of SF6, with the FZucomponent at higher energy. Since the Coriolis splitting of a rotational J level in each of the two substates would be approximatelyss B13J = 0.063 J, the split rotational sublevels of the two substates should overlap for J 2 13, resulting in the complete mixing of the Fluand F2, sublevels and an irregular band. In fact, this prediction is consistent with a recent scan of the v2 v3 region in moderate r e s o l ~ t i o nwhich , ~ ~ does not reveal any pronounced, regular series of lines. On the other hand, Table VI fails to show consistently good agreement for spectroscopic constants involving the bending modes. Therefore, we cannot
-
+
+
assess the reliability of our calculated value for a coefficient such as c236 = -12.6 cm-l; we note that the value of e 2 3 6 = -6.42 cm-’ was assumed in the v3 - ( v , + v6) Fermi resonance calculation.42943 Similarly, the calculated values of Clssand e166 (and consequently of as and ( Y g ) may not be reliable. This is the reason for our pessimistic error limits on the values of Be and re given in Table IX. In conclusion,we have found that an empirical quadratic GVFF potential for SF6, coupled with simple two-atom (CFF) anharmonic potential functions, and subjected to experimental constraints on the stretching S-F bond, can model the known stretching anharmonicities with reasonable success ( u 5 20%) and can predict unknown stretching anharmonicities. Incompleteness in the bending and stretch-bend anharmonicities is responsible for shortcomings in the calculations involving the lower-frequency modes; undoubtedly anharmonic opposite-bond stretching interactions would increase accuracy in the fit to stretching vibrations. It is evident that improved, more reliable modeling will require additional explicit anharmonic terms.
Note Added in Proof: Very recent models of Halonen and Child for the stretching vibrations in tetrahedral hydrides XH4 and XD19 and octahedral hexafluoridesg0are simpler than ours. They include (i) a two-parameter Morse potential in each bond and (ii) harmonic stretch-stretch interactions (one term for XY4 and two terms for XY6). One can then characterize the vibrations between the “normal” and “local” limits. For the hexafluorides, their fit to higher stretching overtones and combinations converges more slowly in SF6 than in WF6 or UFge90This is attributed in part to stronger kinetic coupling between opposite bonds due to the smaller mass of the sulfur atom. Their conclusions suggest that the addition of an explicit opposite-bond anharmonic interaction to our model may bring substantial improvements, as mentioned above. Acknowledgment. We are grateful to Dr. Jenny Rosenthal, author of ref 69, for her interest in this work and for helpful, stimulating discussions. Registry No. Sulfur hexafluoride, 255 1-62-4
(88) C. W.Patterson, H. W. Galbraith, B. J. Krohn, and W. G. Harter, J. Mol. Spectrosc., 77,457 (1979).
(89) L.Halonen and M. S. Child, Mol. Phys., 46,239 (1982). (90) L.Halonen and M. S. Child, J . Chem. Phys., 79, 559 (1983).
Infrared Reflection-Absorption Spectroscopy of Surface Species: A Comparison of Fourier Transform and Dispersion Methods William G. Golden,* David D. Saperstein, IBM Instruments, Inc., San Jose, California 95110
Mark W. Severson, and John Overend Department of Chemisrry, University of Minnesota, Minneapolis, Minnesota 55455 (Received: May 25, 1983)
It is now possible, with both Fourier transform and dispersion infrared spectrometers, to measure the infrared spectra of monolayer and submonolayer quantities of molecules adsorbed on low-area surfaces either in ultrahigh vacuum or in the presence of high pressures of adsorbing and nonadsorbing gases with no severe restrictions on substrate temperature. This paper compares the various advantages associated with dispersion and Fourier transform infrared reflection-absorption spectroscopy (IRRAS). Discussions of the design and experimentalconsiderations of both approaches are given. The infrared spectra of Langmuir-Blodgett monolayers on silver and carbon monoxide on platinum are shown.
Introduction Vibrational spectroscopy, with its ability to yield vibrational frequencies and oscillator strengths which can be interpreted in terms of adsorbate structure and bonding, is an important addition to modern techniques for the study of low-area surfaces. One of 0022-3654/84/2088-0574$01.50/0
the major difficulties in performing the measurement in a conventional way has been that, because of the small number of adsorbate molecules present in the sample, the signal of interest is extremely small relative to the total amount Of energy striking the detector. Infrared reflection-absorption SPectroscoPY (IR0 1984 American Chemical Society
