J . Phys. Chem. 1990, 94,5493-5496 anticipated for both the reaction enthalpies and enthalpies of activation computed at the MP4 level.
IV. Conclusions The major conclusions to be gleaned from this work are as follows: 1. Analysis of the total electron density in terms of bent bond lengths reveals little or no evidence of rcomplex character in any of the cyclopropene rings. The M-C bonds consistently bend away from the ring, not inwards as would be anticipated in a true ?r-complex. Fluorine substitution at the metal center shortens (lengthens) the M-C (C=C) bonds. 2. Intrinsic vibrational frequencies and force constants show that the thermodynamic stability of the c-[MX2C2H2]metallacyclopropenes decreases in the order M = C > Si > Ge > Sn and that fluorine substitution at the metal center strengthens (weakens) the M-C ( C 4 ) bonds.
+
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5493
3. The reactions MX2 H m H c-[MH2C2H2]proceed without a barrier for X = H. However, fluorine substitution at the metal center induces substantial barriers to the formation of all the corresponding difluorometallacyclopropenes. 4. Based on MP2/3-21G(d)//RHF/3-21G(d) relative energies, the experimentally observed instability of stannacyclopropene is due to both the overwhelming stability of the reactants leading to ring formation and the relative weakness of the Sn-C bonds. Acknowledgment. This work was supported by grants from the National Science Foundation (NSF CHE86-40771) and the Air Force of Scientific Research (AFOSR 87-0049). The calculations were performed on the North Dakota State University IBM 3081/D32 computer, on a VAX 8530 (provided with the aid of DoD grant 87-0237), a microVAX 11, a Celerity 1260D, a VAXstation 3200 (all purchased with the aid of NSF grant CHE85-11697), and on the Cray XMP located at the San Diego Supercomputer Center.
Ab I nltlo Calculations of Vibrational Spectra of Nonrigid Molecules Petr &nky,* Institute of Organic Chemistry and Biochemistry, Czechoslovak Academy of Sciences, 166 10 Prague 6, Czechoslovakia
Vladimir Spirko,* J . Heyrovskjj Institute of Physical Chemistry and Electrochemistry, Czechoslovak Academy of Sciences, 182 23 Prague 8, Czechoslovakia
B. Andes Has, Jr., and Lawrence J. Scbaad Department of Chemistry, Vanderbilt University, Nashville, Tennessee 37235 (Received: November 1, 1989; In Final Form: March 26, 1990)
The standard quantum chemical treatment (the calculation of gradients, Hessians, and perhaps higher derivatives followed by the harmonic and perturbational treatments of normal modes) of vibrational spectra cannot be applied to nonrigid molecules. and cyclobutadiene (C4H4), with strongly nonharmonic We present two examples of nonrigid molecules, propargylene (C3H2) potential surfaces to which a special treatment must be applied. The two examples are used as a basis for our discussion of the procedure, common features, and computational aspects of the treatments of nonrigid molecules by ab initio calculations.
Introduction Since the pioneering work of Pulay' enormous progress has been made in the analytical ab initio evaluation of energy derivatives. The availability of GAUSSIAN?CADPAC, and other programs noted in ref 3 now makes the computation of harmonic IR and Raman spectra routine. The only limiting factors are the computer efficiency and the cost. Impressive progress has also been made in the analytical calculation, at both the SCF and post-Hartree-Focklevels, of higher energy derivatives3needed in theoretical approaches to vibrational spectra going beyond the harmonic approximation. Programs for such anharmonic treatments are also now available. They are based on either second-order perturbation theory" or variational approaches? and they require cubic (1) Pulay, P. Mol. fhys. 1969, 17, 197. (2) Binkky, J. S.; Frisch, M. J.; Raghavachari, K.; DeFrees, D. J.; Schlegel. H. B.; Whiteside, R. A.; Fluder, E. M.; Seeper, R.;Pople, J. A. GAUSSIANa2 and its later versions, Carnegie-Mellon University. Pittsburgh, PA. (3) Jsrgensen, P., Simons, J., Eds. Geometrical Derivatives of Energy Surfocrs and Molecular h p e r t f e s ;D. Reidel: Dordrecht, The Netherlands, 1986. (4) Mills, 1. M.In Molecular Specirmcopy: Modern Research; Rao, K. N., Mathews, C. W., Eds.; Academic Pres: New York. 1972.
and quartic force constants as input data. One might therefore be inclined to believe that the problems of quantum chemical applications in vibrational spectroscopy are essentially solved. The aim of our paper is to show that this is not the case. Progress in experimental techniques makes it now possible to measure vibrational spectra of reactive species with unusual bonding. This is just the area where the spectroscopists need help from quantum chemists. Unfortunately many such molecules are nonrigid, and the "standard" approach noted above is not applicable to them. In the following section we analyze briefly the standard approach and its limitations. We then report two examples of nonrigid molecules, propargylene and cyclobutadiene, to which a special treatment had to be Finally, in the last section we formulate some general features of computational strategy used in treatments of nonrigid molecules.
Standard Approach and Its Limitations The standard approach may be defined as consisting of the four steps: (i) geometry optimization, (ii) analytical evaluation of second derivatives at the optimum geometry, (iii) calculation of ( 5 ) Romanowski, H.; Bowman, J. M.; Harding, L.B. J. Chem. fhys. I=, 82, 4155.
0022-3654/90/2094-5493%02.50/00 1990 American Chemical Society
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Cdrsky et al.
The Journal of Physical Chemistry. Vol. 94, No. 14, 1990
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cm” Figure 3. (a) UMP2/6-31G** harmonic infrared spectrum of propargylene. Positions of bands are represented by bars whose heights give the intensity relative to the strongest band. (b) UMP2/6-31G** infrared spectrum of propargylene. Calculated antisymmetric CCH bend and C C stretch frequencies and intensities were obtained from a theoretical anharmonic model, whereas the other seven modes were treated in a harmonic approximation. (c) Experimental spectrum of propargylene. All three spectra originate from ref 6.
truncated at the fourth derivatives, cannot reproduce the potential well in Figure 1. Figure 2 represents a typical case of two conformations separated by a low potential barrier. For ammonia, as an example, only the ground and first excited harmonic states lie below the inversion barrier. Even though the harmonic potential is a good approximation for a single well (in this case), its use in a standard perturbation or variational approach cannot give the observed splitting of the vibrational levels.
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Figure 1. UMP2/6-31G** potential6 for the antisymmetric C C H bending in propargylene, HC=C==CH. This is an example of a nonharmonic potential for which the harmonic treatment fails and for which the harmonic oscillator eigenfunctions are a poor basis set in variational or perturbation treatments.
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Figure 2. GVB/4-31G potential’ for the antisymmetric CC stretching in cyclobutadiene. This is an example of a double-well potential. The harmonic approach cannot be applied unless the barrier is several times higher than the respective harmonic frequency.
the harmonic vibrational spectrum by means of the FG method, and perhaps (iv) calculation of cubic and quartic force constants at the optimum geometry followed by perturbational4 or variational treatment.s The standard approach so defined is applicable to well-behaved rigid molecules for which the optimum geometry is a minimum on the potential surface lying well below the dissociation limits and separated from other minima by high potential barriers. The minima must be deep enough to accommodate several levels for each vibrational mode. We also require that the potential surface have a near-harmonic behavior around the minima. Figures 1 and 2 represent typical situations where these conditions are not satisfied. In Figure 1 the potential is flat around the reference configuration, and the difference
vp= v - v,
(1)
between the true (V)and harmonic (V,) potentials, which might be treated as perturbation, is so large that perturbation theory cannot be applied. Such flat potentials were found, for example, for the antisymmetric motion of hydrogen in the hydrogen-bonded systemsn9[F.-H-F]- and [HOI-H.-OH]-. Standard variational calculations do not help here either because a Taylor expansion, ( 6 ) Maiu, G.; Rcisenauer, H. P.;Schwab, W.; &sky, P.;Spirko, V.;Ha, B. A., Jr.; Schaad, L. J. J. Chem. Phys. 1989, 91,4163. (8) Janssen, C. L.; Allen, W. D.; Schaefer, H.F., 111; Bowman, J. M.
Chem. Phys. Lett. 1986, 131, 352. (9) Spirko, V.;Kraemer, W.P.;eejchan, A. J. Mol. Spectrosc., in press.
Examples of Nonharmonic Treatments: Propargylene and Cyclobutadiene Propargylene has been prepared by matrix isolation experiments along with two other C3H2isomers, cyclopropenylidene and vinylidenecarbene.lOJ1 The measured IR spectra of the latter two were well reproduced by theoretical harmonic spectra from ab initio calculations.”~12 The agreement was so good that the structures of cyclopropenylidene and vinylidenecarbene were unambiguously confirmed. The problem of propargylene was more difficult since, as seen in Figure 3, its harmonic spectrum6J3did not match the observed spectrum. We thought that this might be due to the nonrigidity of the molecule. Inspection of the UMP2/6-3 1G** potential surface shows6 that the nonrigidity of the system is most associated with the antisymmetric CCH bend which is coupled with the antisymmetric CC stretch. We decided therefore to pick out these two coordinates, to compute the potential surface for them, and then to compute the vibrational frequencies of this surface. These were obtained by a numerical solution of the one-dimensional Schrijdinger equation for the effective potential, which was the antisymmetric CCH bending potential corrected adiabatically a t each point by the energy of the antisymmetric CC stretch. The remaining seven modes were treited separately in a harmonic approximation. The composite theoretical spectrum obtained in this way and presented in Figure 3b shows a considerable improvement over the spectrum in Figure 3a obtained by a harmonic treatment of all nine modes. The only unsatisfactory feature in this theoretical spectrum is the CH stretch (10) Reismauer, H.P.;Maier, G.; Riemann, A,; Hoffmann, R. W. Angew. Chem., Int. Ed. Engl. 1984. 23, 641. (1 1) Maier, G.; Reisenauer, H.P.;Schwab, W.;h s k y , P.;Hess, B. A., Jr.; Schaad, L. J. J . Am. Chem. SOC.1987, 109, 5183. (12) Lee, T. J.; Bunge, A,; Schaefer, H. F., Ill J . Am. Chem. SOC.1985, 107, 137. (13) DeFrees, D. J.; McLean, A. D. Astrophys. J . 1986, 308, L?1
Vibrational Spectra of Nonrigid Molecules region. It gives two allowed transitions for the C H stretches, whereas experimentally only a single band was observed. The disagreement is most likely due to the harmonic treatment of the C H stretches. The second example concerns the automerization of cyclobutadiene. The energy profile along the automerization coordinate is shown in Figure 2. The top of the automerization barrier corresponds to a square Da structure which is about 10 kcal/mol (Le., 3500 cm-l) higher in energy than the optimum rectangular structure. At first sight the barrier seems to be high, and one might think the use of a single-well potential and the harmonic approximation are adequate for the interpretation of the infrared spectrum of cyclobutadiene. Indeed, the agreement between the calculated and observed infrared spectrum of cyclobutadiene and its isotopic derivatives is very good (for a review see ref 14). On the other hand, the observed high rate of the cyclobutadiene automeri~ation'~ confirms previous assumptions about tunneling through the automerization barrier. This implies the 4 (Raman active) vibrational levels should be split. The only way to account for this splitting is to treat cyclobutadiene as a nonrigid molecule and to apply a nonharmonic approach to a double-well potential. Many other examples may be found in the literature, showing that the harmonic approximation fails with nonrigid molecules and that the availability of a general computational method for nonharmonic treatment becomes more and more topical. Unfortunately, for practical reasons each particular case so far requires a special treatment, depending on the size of the molecule, the number and nature of nonhindered motions, and their coupling with the other modes. Still, there exist some common features of these treatments, and we comment on them in the following section.
Common Features and Computational Aspects of the Nonharmonic Treatments of Nonrigid Molecules We comment first on the potential function. A nonharmonic treatment of a nonrigid molecule requires an explicit expression for the whole potential surface. A single-point evaluation of energy and its derivatives is not enough for this purpose. Unfortunately calculation of many data points is impractical because it requires considerable human effort and is very demanding of computer time if a highly correlated wave function and a large basis set are to be used. Moreover fitting the calculated potential surface may be very difficult. Therefore the efficiency and elegance of the analytical evaluation of energy derivatives should be exploited at least in part. One should realize that the analytical calculation of energy derivatives is feasible not only at the potential surface minimum but also at a maximum (transition structure). Ideally, it would be profitable to find an empirical continuous function for the interpolation of energy derivatives calculated at all of the extrema since this would reduce considerably the number of data points needed for the construction of a potential surface. We also consider it useful to make a technical remark. In the direction of nonhindered motions the potential energy surface has to be calculated up to much higher limits of the energy than is usual in ordinary quantum chemical applications. It should be realized that these limits stand for the infinite energy in the numerical solution of the vibrational Schrijdinger equation. Having the potential surface, our next task is to identify the nonhindered motions that are associated with transitions over low-lying potential barriers. These motions should be treated accurately. Since the only rigorous and general method for the solution of the vibrational Schriidinger equation is numerical integration, the number of such motions is limited to one or two. The rest must be assumed to be of harmonic or near-harmonic behavior. For example, on our UMP2/6-31G** surface of propargylene6 we have found two minima that correspond to structures la and l b and were separated by a low barrier of about 400 (14) Arnold, B. R.; Michl, J. In Kinetics and Spectroscopy of Carbenes and Biradicals; Platz, M., Ed.; Plenum Press: New York, in press. (IS) Orendt, A. M.; Arnold, B. R.; Radziszewski, J. G.; Facelli, J. C.; Malsch, K. D.; Strub, H.;Grant, D. M.; Michl, J. J . Am. Chem. Soc. 1988, 110, 2648.
The Journal of Physical Chemistry, Vol. 94, No. 14, I990 5495 H 'C-C-C-H
H
lb
18
H
\c=c=c/
/H
H-C-C-C
I C
cm-' from structure IC. The "reaction path" la F? l c e l b may be described by a combination of the coordinates R1 and R2:
R1 = (2)-'/2(81 - 82)
(2)
R2 = (2)-1/2(r1- r2)
(3) where O1 and P2 are the CCH angles, and rl and r2 are the CC bond lengths. R1 = R2 = 0 corresponds to structure IC. The energy profile along R1 (for R2 = 0) is shown in Figure 1. Since motions along RI and R2 are coupled, they both have to be used in the nonharmonic treatment. In case of a double-well potential it is profitable to define a coordinate for which the respective potential curve passes through the top of the barrier and close to the two minima. Our automerization coordinate for cyclobutadiene was defined as Rl
=
!4(r12
- r23 + r34 - r14)
(4)
where the ru are the four C-C bond lengths. Then, in general, we may extend the nonharmonic treatment of the double-well potential problem for other coordinates. In our nonharmonic treatment of cy~lobutadiene~ we assumed that the automerization represented by coordinate R1is primarily coupled with the motions for the same geometry (Ag in the D a point group), so we included in our treatment the symmetric CC stretch and symmetric CCH bending coordinates
where D is the equilibrium CC bond length in the square transition structure, and /3's are the C C H bond angles. It is convenient in the proper vibrational treatment to make a transformation of coordinates. First we express the potential ..., energy surface in linearized mass-dependent coordinates SI, S,, instead of curvilinear mass-independent coordinates R , , ..., R,. This is achieved by expanding the potential energy Vas a Taylor series in the linearized coordinatesI6 V(RI, ..., R,) = V(Sl)
+ b ' i ( S l ) S i+ j/2kkFi,(Sl)S,Sj+ ... r=Z
i=2j=2
(7) where SIis the coordinate of the nonhindered motion. Next we express SIas an angular coordinate, even if SI represents a stretching motion. This does not make any complication. For example, in cyclobutadiene the automerization coordinate can be expressed as 2D(sin ( a / 2 )-cos ( a / 2 ) ) s, = sin ( a / 2 ) - cos (a/2) (8) where a is the angle between the diagonals of rectangular cyclobutadiene. The other coordinates for "hindered" motions are readily transformed by means of the G F matrix method to normal coordinates S = LQ
(9)
and dimensionless normal coordinates Q,= h'/Zh;'/4qi
(10)
The purpose of transformations (7),(9), and (10) is to express the potential function in coordinates q, that simplify the kinetic energy operator. Having the potential function in a suitable form, the next task is to obtain the kinetic energy part of the Hamiltonian. It is profitable to keep a close analogy with the archetype of these (16) Hoy,A. R.; Mills, I. M.; Strey, G. Mol. Phys. 1972, 24, 1265.
J . Phys. Chem. 1990, 94, 5496-5498
5496
problems, the well-understood nonrigid bender model of ammonia,” and to use a Hamiltonian of the Hougen-Bunker-Johns (HBJ) type.’* This explains why we always express the nonhindered motion by means of an angular coordinate. Next it is necessary to select the computational method. Basically, our choice is limited to the variational method or the adiabatic approach. If we decide upon a variational calculation, it is best to split the Hamiltonian as H = Ho + HI,where all terms coupling the “large amplitude” and “harmonic” motions are collected in H,.Homay be constructed as a HBJ-type Hamiltonian: R
which gives us the eigenfunctions that are then used as a basis set for the variational calculation. Since in practical applications to polyatomic molecules such variational calculations are hardly feasible for more than four modes, it is normally necessary to look for a less rigorous approach. The “adiabatic” approach seems to be well suited for this purpose. In this approach we also distinguish between “large amplitude” and “harmonic” motions, but in addition we introduce the condition of “adiabaticity”, which means that “harmonic” (“fast”) motions must be considerably higher in energy than the “large amplitude” (“slow”) motions. The idea is of course the same as that of slow nuclei and fast electrons in electronic structure theory, and we make use of this analogy to give the adiabaticity condition a semiquantitative expression. In ammonia, for example, the highest vibrational frequencies are of about 3500 cm-I, compared to the energy of the lowest excited singlet state of about 50000 cm-I. The same ratio of energies is satisfied for a pair of vibrational frequencies of 200 and 2900 cm-I. For some types of vibrational motion the energy separation may be considerably smaller and we may-on the basis of experience accumulated in the literature-still expect a weak interaction. When the adiabatic approximation is applied, the “harmonic” motions may be treated separately, and we may obtain for their vibrational energy an explicit expression U(Sl, ...,Smvrrcl, ...,u,) as a function of coordinates SI,..., S, for large amplitude motions and quantum numbers of the “harmonic” modes, v , + ~ , ..., v,. In the simplest case, U(Sl, ..., S,,,V,+~,..., u,) may be evaluated in the harmonic approximation. If a potential energy fit also contains cubic (and higher) force constants for coordinates (17) Spirko, V. J . Mol. Specrrosc. 1983, 101, 30. (18) Hougen, J. T.; Bunker, P. R.; Johns, J. W. C. J. Mol. Specrrosc. 1970, 34, 136.
SH1, ..., S,, the anharmonicity may be accounted for by per-
turbation theory. The energy of “harmonic” modes for the preselected quantum numbers ulrcI, ..., v, may be then added to a pure potential V(Sl, ..., SJ, and the resulting effective potential may be used for a rigorous treatment of large amplitude motions. We use this potential for the construction of the HBJ-type Hamiltonian, and the respective Schrijdinger equation is solved numerically to give a manifold of vibrational levels for a set of preselected quantum numbers vrrcl,..., v,. The condition of adiabaticity was well satisfied for propargylene. The 0 1 frequency for the large amplitude CCH bending mode was about 100 cm-I, and that for the near harmonic CC stretching was about 1400 cm-I. We considered this difference large enough for a safe use6 of the adiabatic approach. The situation with cyclobutadiene was different. The 0 1 frequencies corresponding to coordinates (4)-(6) were 1630, 1420, and 1220 cm-I. The small differences between these frequencies preclude the use of the adiabatic approach, and we have therefore treated’ cyclobutadiene by a variational calculation. To summarize this section, we list a sequence of steps that should be followed in a theoretical approach to a nonrigid molecule: (i) Find all extrema on the potential surface and calculate analytically for them second and, if possible, higher energy derivatives. (ii) Calculate the energy at additional points, and get a fit for the whole potential surface. (iii) Inspect all the minima, and estimate the number of vibrational levels they can accommodate. Identify the nonhindered large amplitude motions that are most associated with the anharmonicity of the system. (iv) Design a suitable set of coordinates, and express the potential function in these coordinates. Identify in the potential function the terms representing the large amplitude motions, the harmonic potential for the other modes, and the coupling between the kinds of motions. (v) Drop the coupling terms from the potential function and add the corresponding kinetic energy terms to get Ho. Obtain the basis set functions. These are typically numerical solutions for the large amplitude motions and harmonic oscillator eigenfunctions for the other modes. (vi) Perform a variational calculation to mix these harmonic basis functions. If it is not feasible because of too large a number of modes, apply the adiabatic approximation. Registry No. Propargylene, 2008-19-7; cyclobutadiene, 1 120-53-2.
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Nonnuclear Attractors in the Li, Molecule J. Cioslowskif Department of Chemistry and Supercomputer Computations Research institute, Florida State University, Tallahassee, Florida 32306-3006 (Received: November 1, 1989; In Final Form: February 2, 1990) Topological features of the electron density in the Li2 molecule, calculated at the HF/6-311+G* and MP2/6-311+G* levels, are described and rationalized within the catastrophe theory. Depending on the internuclear distance, the density exhibits zero, one, or two nonnuclear maxima. The phase diagrams for the electron density are preented and the bifurcation points are characterized by the respective critical exponents for the differences in the positions of extrema, electron densities, and the electron density Laplacians at the extrema. Introduction Since the electron density of an isolated atom falls off expo. nentially at large distan=,l* one a n easily demonstrate that the electron density of any homonuclear diatomic molecule is expected to possess no maxima other than the cusps at nuclei, provided the ‘Camille and Henry Dreyfus Foundation New Faculty Awardee.
internuclear distance R is sufficiently large. Moreover, it can be proven rigorouslylb that the electron density in the H2+molecular ion has no maxima other than the nuclear CUSPS for all values of (1) (a) Alrichs, R.; Hoffmann-Ostenhof, M.; Hoffmann-Ostenhof, T.; Morgan 111, J. D. Phys. Rev. 1981, ,423,2106. (b) Hoffmann-Ostenhof, T.; Morgan 111, J. D. J . Chem. Phys. 1981, 75, 843.
0022-3654/90/2094-5496$02.50/0 0 1990 American Chemical Society
