J . Phys. Chem. 1986, 90. 2030-2038
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A Vibrational Variational Hyperspherical Approach to the Stretching States of Triatomic
ABA Molecules J. Manz* and H. H. R. Schor Lehrstuhl fur Theoretische Chemie, Technische Universitat Munchen, 0-8046 Garching, West Germany and Departamento de Quimica, ICEX, Universidade Federal de Minas Gerais, 30 000 Belo Horizonte, Brazil (Received: September 19, 1985)
The stretching states of ABA molecules are evaluated by the vibrational variational hyperspherical (VIVAH) method as follows: (i) The symmetric and antisymmetric stretching vibrations are described by motions along the hyperspherical radial r and angular cp coordinates, respectively. (ii) The vlth symmetric and u3th antisymmetric stretching excitations are represented by zero-order wave functions Q,,(cp;r). These are evaluated by the diagonal-corrected vibrational ( q r ) Rulu3(r) adiabatic hyperspherical (DIVAH) method with Born-Oppenheimer-type separation of radial and angular motions. (iii) = ~ t i I I u 3VIVAH ~ c u I I u\k,l,u3f 3DIVAH ~,tiIti3 Finally, VIVAH energies and wave functions are evaluated by using the variational ansatz with coefficients cVIVAH determined by diagonalization of the stretching Hamiltonian in the DIVAH basis. The VIVAH technique is an extension of the DIVAH method, accounting for the diabatic couplings of all u p 3 states. The method is applied to the Rosen-Thiele-Wilson coupled Morse-oscillator model of ABA molecules by using Kulander’s modified Hedges-Reinhardt parameters for a challenging intermediate coupling case. The results are in excellent agreement with the local-mode approach developed by Wallace, Child, and others. Most ABA stretching states may be classified as local or/and hyperspherical modes, using the Hose-Taylor criteria. At low energies, the hyperspherical vlv3 modes correspond to familiar u,v3normal modes. The VIVAH approach generalizes the concept of normal modes to even highly excited ABA (u,c3) molecules.
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1. Introduction The purpose of this paper is to present the vibrational variational hyperspherical (VIVAH) method for the evaluation of the vibrational stretching states of triatomic molecules. The VIVAH method is complementary to the traditional normal-’ and locaL2 mode techniques. There are several motivations for developing this alternative: First, it is well-known that normal modes provide adequate, compact descriptions of vibrational states at low energies. However, at higher energies, the traditional normal-mode picture is often inappropriate, Le. highly excited vibrational eigenstates require very extended expansions in terms of normal-mode basis functions. Clearly, one would like to generalize the familiar normal-mode approach to more compact representations of both low- and higher-energy states. As we shall show below, a useful generalization is provided by the VIVAH method. Second, the alternative local-mode approach, developed by W a l l a ~ eChild , ~ and co-workers: and several others5 (see also ref 2 and 6), often yields very compact representations of accurate vibrational wave functions even at higher energies, in particular for “local” modes where the vibrational energy is localized classically in essentially a single bond. However, as we shall show below, there are some highly excited molecular vibrational states which are well localized, yet they require rather extended expansions in terms of local-mode basis functions. The complementary VIVAH approach provides more appropriate compact representations of these “hyperspherical” modes. Third, an approximate alternative to the normal-’ and local-2 mode theories has already been developed very recently, Le. the diagonal-corrected vibrational adiabatic hyperspherical (DIVAH) In special cases, the DIVAH theory yields accurate, simple representations of vibrational bound states and resonances in terms of hyperspherical modes, in particular for heavy-lightheavy atom systems such as ABA = IHI8~I0 or C1HCL9 For these systems, the large mass ratio m A / m B>> 1 suggests a BornOppenheimer-type separation of slow heavy atom and fast hydrogen motions, described by vibrations along essentially the hyperspherical radius r and angle cp, respecti~ely.~.~ However, for smaller mass ratios m A / m B5 1, the DIVAH results turn out to be less accurate due to “non-Born-0ppenheimer”-typecouplings of radial and angular motions.s Clearly, the intermediate m4/mB *Present address: Insitut fur Physikalische Chemie, Universitat, Marcusstr. 9-1 l , D-8700 Wurburg, West Germany.
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5 1 and strong mA/me 0.1, are listed in descending order of IcI. The expansion bases are indicated below, but superscripts MORSE or DIVAH are omitted in this table. * h = hyperspherical, 1 = local, n = nonlocalized. The type is 1 or/and h if the dominant LCSPMd or/and VIVAHe expansions coefficients have values c2 > 0.5, otherwise the type is n. Quantum numbers for non-l or non-h type states are given in brackets. 'The uIu3 = 90 state is missing, see text. 'All quantum numbers are assigned according to dominant expansion coefficient, except for states labeled " j " , see text.
k
Q,",CSPM
=
nmt /
*A :' H
LCSPM *MORSE Cn,m,t,nmt n'm't
C CDIVAH QDIVAH q'" CVIVAH ulu3
(4.13e) (4.130
With criteria 4.8-4.13, Table I, a and b, yields 10 gerade plus 9 ungerade bound states with preferable or exclusive hyperspherical and 6 gerade plus 5 ungerade states with preferable or exclusive local-mode character, respectively. Using exclusively the local or the hyperspherical mode approaches would leave 2 gerade plus 2 ungerade or 6 gerade plus 0 ungerade “nonlocalized” states with ambiguous assignments. With both approaches, almost all of these hypothetically “chaotic” states actually turn out to be regular; the local and hyperspherical mode approaches are therefore (nearly) complementary. Both approaches are necessary to classify similar numbers of alternative molecular vibrational states. The single state of type n, eq 4.13d, Le. (nm+) = (14+), (0103) = (60) in Table Ia, should also be regular according to a theorem of Kosloff and Rice, i.e. chaotic bound quantum states do not exist (in contrast with classical bound states or resonance^).^^ But obviously, a third alternative basis set, neither local nor hyperspherical, is required to demonstrate the regular character of this single ”quasi-n” level. Incidentally, also the differences of are relatively large, indicating that local and hyand perspherical bases should be replaced, or complemented by a third basis.
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5. Conclusions The VIVAH theory of ABA molecular stretching vibrations has been presented as a variational hyperspherical extension of the DIVAH t e c h n i q ~ e , ~taking - ~ into account the diabatic couplings of unperturbated hyperspherical basis functions I D I V A H . The numerical directions for use of the VIVAH technique are (23) R. Kosloff and S. A. Rice, J. Chem. Phys., 74, 1340 (1981).
summarized as steps a-h at the end of section 3; the computational efforts for the diabatic coupling, steps f-h, are relatively marginal in comparison with the fundamental DIVAH technique, steps a-e, section 2. The VIVAH theory describes molecular ABA vibrations by using zero-order basis functions
*$yAH(cp,r) = Nr-”2Rula3(r) %3(cp;r)
(2.6)
for the vlth symmetric stretch along r and u3th antisymmetric stretch along cp. This adiabatic separation of vibrations along cp and r resembles the decoupling of electronic q and nuclear Q degrees of freedom in zero-order Born-Oppenheimer states
%kQ) = V X q ) *?(Qd
(5.1)
for the 0th rovibrational excitation in the nth electronic state. Therefore, the DIVAH technique, i.e. steps a-e, correspond to the Born-Oppenheimer a p p r ~ x i m a t i o n . By ~ ~ analogy, it should be possible to adapt the additional steps f-h of the VIVAH technique, section 3, for evaluation of molecular rovibronic states by exact diabatic coupling of Born-Oppenheimer states qun
=
Cca,n,,un*an
BO
(5.2)
a’n’
By the present experience, the numerical efforts for evaluating the vibronic coupling coefficients (5.2) should be small once all the Born-Oppenheimer states are available. In conclusion, the VIVAH technique is complementary to alternative non-BornOppenheimer techniques, see, e.g., ref 2 5 . The VIVAH technique has been applied to the Thiele-Wilson coupled Morse-oscillator model12 by using Kulander’s M H R parameter^.'^ The resulting numbers of ABA gerade (+) and ungerade (-) bound states, energies, and wave functions have been compared with those of the familiar local-mode approach.24 The very good agreement of hyperspherical VIVAH and local LCSPM mode results obtained in section 4 may be considered as a successful test of the VIVAH technique, in particular since Kulander’s model13 provides a challenging intermediate coupling case, as VIVAH as well indicated by considerable energetic DIVAH LCSPM shifts from zero to “infinite” order levels, as Morse see Figure 2, as well as extended VIVAH and LCSPM expansions of selected vibrational states listed in Table I, a and b. The ABA vibrational stretching states are classified as hyperspherical (h) or local (1) modes if the dominant VIVAH or LCSPM expansion coefficients satisfy the Hose-Taylor 50% criteria (4.9) or (4.8), respectively, see Table I, a and b. The number of h and 1 states is about equal, including states with exclusive h or 1 as well as others with both h and 1 character; therefore, the hyperspherical and local-mode approaches complement each other in an equivalent way. In any case, they are preferable to more extended normal-mode expansions of highly excited states, cf. e.g. ref 5. Note in the present case, there remains only a single state with extended local and hyperspherical expansions, violating both criteria (4.8) and (4.9), i.e. the nonlocalized (type “n”) state nm+ = (14+), uIu3= (60) listed in Table Ia and plotted in Figure 3. The present classification of hyperspherical (h), local (l), or nonlocalized (n) vibrational states for the Thiele-Wilson modelI2 is entirely analogous to, and in fact has been motivated by, Hose and Taylor’s classification of restricted precessor (“Q’”),quasiperiodic librator (“QII”), or nonlocalized (type “N”) states, respectively, for the Hcnon-Heiles’ systemi5 (see also ref 18-20). This analogy suggests that the H&non-Heiles Q’ and Q” quantum numbers may also be related to each other, e.g. by the total number of quanta and nodal patterns of frontier lobes of wave functions,
-
-
(24) J. Manz, R. Meyer, and J. Romelt, Chem. Phys. Lett., 96,607 (1983); J. Manz and J. R. Romelt, Nachr. Chem., Tech. Lab., 33, 210 (1985). (25) E. F. van Dishoeck, M. C. van Hemert, A. C. Allison, and A. Dalgarno, J . Chem. Phys., 81, 5709 (1984); M. Baer, Top. Current Phys., 33, 117 (1983); D. Stahel, M. Leoni, and K. Dressler, J. Chem. Phys., 79, 2541 (1983); J. Romelt, Int. J. Quantum Chem., 24, 627 (1983); 0. Halkjaer and J. Linderberg, Int. J. Quantum Chem., Quantum Chem. Symp., 13, 475 ( 1979).
J . Phys. Chem. 1986, 90, 2038-2043
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in a way similar to the present simple relations of ~ 1 and ~ nm* 3 quantum numbers, cf. eq 4.11 and 4.12. One may therefore consider the present application of the Hose-Taylor criteriaI5 as an extension from one model providing exclusively resonances (Hitnon-Heiles) to a more realistic one providing molecular bound states as well as resonances (Thiele-Wilson). Extrapolating the present results,16 one should be able to correlate the distinguished ABA hyperspherical modes with quantum numbers u1u3 = Ou, with the ground states of effective radial potentials U",(r),see Figure 1. This conclusion is closely related to Greene and Jungen's26 prediction of H 2 0 stretching states correlating with ground states of gD3(r)(neglecting the diagonal correction in eq 2.9). Further candidates of ~ 1 =~ Ou,3 hyperspherical modes have been reviewed in ref 16. The present analysis demonstrates, however, that the class of hyperspherical modes is by no means restricted to the case ~ 1 =~Ou,,3 cf. Table I; an "extreme" counterexample is u1u3= 80, see plotted in Figure 3. It is of course a challenge to detect highly excited hyperspherical modes experimentally. One possibility is based on the fact that the wave functions of, e.g., n m f = Omf local modes extend
(typically) into the potential valleys, whereas for u1u3 = Ou3 hyperspherical modes, they often extend across the potential ridge. As a consequence, dipole moment transition matrix elements will strongly depend on the type (1 or h) of initial and final states, implying perhaps the possibility of selective excitations of local or hyperspherical modes. This conclusion is supported by typical direct overtone excitations of CH, OH, etc. bonds which tend to populate local modes, see the analysis in ref 2-6. In contrast, a multiphoton study of the Htnon-Heiles system yields preferential excitation of Q' modes, corresponding to hyperspherical modes.27 At very high energies, one should observe mode-selective molecular dynamics, i.e. slow hyperspherical but fast local-mode decay of ABA r e s ~ n a n c e s . ' ~ ~ ' ~
(26) C. H. Greene and C. Jungen, Abstracts of Papers, Twelfth International Conference on the Physics of Electronic and Atomic Collisions, S. Datz, Ed., Gatlinburg, TN, 1981, p 1019.
(27) R. E. Wyatt, G. Hose, and H. S. Taylor, Phys. Reu. A , 28, 815 (1983). (28) J. Manz, Comments A t . Mol. Phys., 17, 91 (1985).
Acknowledgment. We thank Dr. K. C. Kulander for stimulating discussions and Prof. G. L. Hofacker for his kind hospitality and continuous support. Generous financial support by the Deutsche Forschungsgemeinschaft, the Fonds der Chemischen Industrie and the CNPq (Brazil) are also gratefully acknowledged. The computations were carried out on the CYBER 175 of the Bayerische Akademie der Wissenschaften.
The Motecular and Electronic Structure of Perfluoro-1,&butadiene David A. Dixon Central Research and Development Department, Experimental Station, E . I . du Pont de Nemours & Company, Wilmington, Delaware 19898 (Received: October 15, 1985)
The electronic structure of perfluoro- 1,3-butadiene has been determined from ab initio molecular orbital calculations. A double {basis set augmented by a set of polarization functions on carbon (DZ + Dc) has been employed. A number of geometries on the torsional potential surface were gradient optimized. The minimum energy structure is a skew-cis structure with )I = 58.4O. The s-trans structure (optimized) is 1.8 kcal/mol and the s-cis structure (optimized) is 5.7 kcal/mol higher in energy than the optimum structure. The conformational potential was partially determined. The potential is quite harmonic near the minimum and very anharmonic for > 100'. The form of the potential has been used to explain the difference in the theoretical and experimental barrier heights. The vibrational frequencies and infrared intensities have been calculated analytically with a 6-31G*(C) basis set. These quantities are compared with experiment. The analysis of the molecular orbitals is consistent with the spectroscopic evidence.
+
Introduction The structure of 1,3-b~tadiene'-~ and perfluoro-1,3-butadiene1@I4have been the subject of a number of recent studies. The (1) (a) Almenningen, A.; Traeteberg, M. Acta Chem. Scand. 1958, 12, 221; (b) Kuchitsu, K.; Fukuyama, T.; Morino, Y .J. Mol. Struct. 1968,1, 463. (2) Lipnick, R. L.; Garbisch, E. W., Jr., J . Am. Chem. SOC.1973, 95,6370. (3) Carriera, L. A. J . Chem. Phys. 1975, 62, 3851. (4) Dung, J. R.; Bucy, W. E.; Cole, A. R. H., Can. J . Phys. 1976, 53, 1832. (5) Squillacote, M. E.; Sheridan, R. S.; Chapman, 0. L.; Anet, F. A. L. J . Am. Chem. SOC.1979, 101, 3651. (6) Furukawa, Y.; Takeuchi, H.; Harada, I.; Tasumi, M. Bull. Chem. SOC. Jon. 1983. - r ~- 56. 392. (7) Bock, C. W.; George, P.; Trachtman, M.; Zanger, M. J . Chem. S O ~ . Perkin Trans 2 1979, 26. (8) Bock, C. W.; George, P.; Trachtman, M. Theor. Chim.Acta 1984,64, 293. (9) Bruelet, J.; Lee, T. J.; Schaefer, H. F., 111 J . Am. Chem. SOC.1984, 106, 6250. (10) Albright, J. C.; Nielsen, J. R. J. Chem. Phys. 1957, 26, 370. (11) Brundle, C. R.; Robin, M. B. J . Am. Chem. SOC.1970, 92, 5550. (12) Chang, C. H.; Andreassen, A. L.; Bauer, S. H. J . Org. Chem. 1971, 36, 920. (13) Wurrey, C. J.; Bucy, W. E.; Durig, J. R. J. Chem. Phys. 1977, 67, 2765. ~~
1
~
~
1
~
1,3-butadiene moiety is the simplest conjugated system and as such is a model for other types of linear conjugated polyenes. The perfluoro derivative is the simplest fluorocarbon with a conjugated double bond and is thus a model for linear perfluoro conjugated polyenes, e.g., perfluoropolyacetylene. 1,3-Butadiene has two energy minima, a trans form and a twisted form which we define about the C-C bond of as skew-cis with a torsion angle, 30-40°,8!9 with s-cis defined as 0' and s-trans as 180'. The barrier between the two skew-cis structures is at the s-cis geometry and is very small, between 0.48 and 0.79 kcal/mol. This result comes from theoretical calculations since the experimental results are not precise enough to define the structure in this region. The cis-skew conformation is 3.15 kcal/mol above the most stable s-trans structure, again from theory.8 The experimental estimate of this difference is 873 cm-I (2.5 kcal/mol)., Perfluorobutadiene has also been investigated for a number of years. The initial vibrational studies'O showed that the molecule does not have the trans form, but could not distinguish between the s-cis and skew-cis forms. Brundle and Robin" interpreted
+,
(14) Choudhury, T.; Scheiner, S., J . Mol. Struct. (Theochem) 1984, 109, 313.
0022-3654/86/2090-2038$01.50/00 1986 American Chemical Society
