J. Phys. Chem. 1994,98, 6936-6941
6936
Vibration Mixing in Terms of Normal Modes Tsuneo Hirano,'*+Tetsuya Taketsugu,# and Yasuyuki Kurita#*l Department of Chemistry, Faculty of Science, Ochanomizu University, 2-1 - 1 Otsuka, Bunkyo-ku, Tokyo 1 1 2, Japan, and Department of Industrial Chemistry, Faculty of Engineering, University of Tokyo, 7-3- 1 Hongo, Bunkyo- ku, Tokyo 1 1 3, Japan Received: January 20, 1994'
For the understanding and prediction of molecular properties and chemical reactions in terms of vibration interaction, two methods, Le., vibration mixing and vibration mapping, are proposed. Equations for vibration mixing, derived by a perturbational method, can predict qualitatively how the normal modes of fragment molecules mix with each other in the interacting system. Vibration mapping can show quantitatively how the vibration mixinn in the interacting system occurs. For demonstrations, these two methods were applied to the HF-NH3 and &CO-H20 systems:
Introduction Recent progress in molecular orbital (MO)calculations has made it possible to calculate the force constants as the second derivativesof the potential energy surface. Through the diagonalization of the mass-weighted force constant matrix (Le., Hessian matrix) calculated at stationary points on the potential surface, normal vibration vectors (eigenvectors)can be obtained with normal frequency (calculated from eigenvalues). Normal vibration vectors correspond to the directions of normal coordinates and serve as a basis for the description of the nuclear wave function. Stationary points on the potential surface are characterized according to the number of negative eigenvalues (imaginary frequency modes). Usually, vibrational states of a molecular system are described in terms of normal coordinates since normal vibrations are independent of each other at the equilibrium position. Various perturbations, however, invoke vibrational interaction among normal vibrations, which may be referred to as vibration mixing. Vibration mixing plays an important role in various chemical phenomena such as chemical reaction, energy dissipation into solvent, intra- and intermolecular vibrational resonance, intramolecular vibrational energy redistribution (IVR), Fermi resonance, etc. For the understanding of these phenomena, vibrational interaction should be examined from various viewpoints. There are many corresponding features between nuclear vibrational states and electronic states. For example, Fermi resonancebetween two vibrational states correspondsto avoided crossing between two electronic states. We usually start with a harmonic approximation for vibrational states and a HartreeFock approximation for electronic states. Normal modes of vibration correspond to molecular orbitals although the electron and vibrational quanta are the fermion and boson, respectively. For quantitative discussion, the electronic correlation effect and anharmonicity need to be taken into account for electronic and vibrational states, respectively. Through formal correspondence between normal modes and molecular orbitals, we propose two novel methods to describe vibration mixing in terms of normal modes. First, equations of vibration mixing for the vibrationally interacting system are presented based on perturbation theory. ~~
~
* To whom correspondence should be addressed. t Ochanomizu University.
University of Tokyo.
1 Presentaddress: TakarazukaResearch Center, SumitomoChemical Co.,
Ltd., Takatsukasa,Takarazuka, Hyogo 665, Japan. 0 Abstract published in Advance ACS Abstructs, April 1, 1994.
For the basic formulas of the mixing of normal modes under perturbation, in short 'vibration mixing",' we borrow formulas of orbital mixing.2" As is the case for the orbital mixing rule,3 the vibration mixing rule derived from the formulas can be used for qualitative interpretation and prediction of the vibration mixing. This is a kind of pictorial quantum chemistry, and rigorous computercalculationis not neededoncethe normal modes of each component molecule are known. Next, vibration mapping is proposed for the quantitative description of vibration mixing in a vibrationally interacting system. Mapping of a normal mode of the interacting system can be made to the normal modes of the component molecules separated at infinite distance, i.e.,the molecular system with no interaction. Such mapping gives a quantitative description of the mixing mechanism, or deformationmechanism, in other words, of normal modes of each isolated molecule. For demonstration, we apply these methods to HF-NH3 and H*CO-H20 systems. The present methods,Le.,vibration mixing and vibration mapping,hold for vibration mixing at any perturbed states unless the geometrical change upon vibrational interaction is fatal for the desired accuracy of the conclusion. Throughout this paper, we sometimesrefer to the normal modes of translation and rotation inclusively by the term normal vibration since the normal modes of translation and rotation are also the eigenvectors of the Hessian matrix of the system and intermixing among eigenvectors is invoked as the consequence of the perturbation.
Under the Born-Oppenheimer approximation,5movement of atoms in a molecule or molecular systemcan be treated separately from the electronicstate by solving each eigenequation,although both are connected through the potential determined by nuclei and electrons. Nucleus movements around their equilibrium points in geometry are approximately described as harmonic oscillations. State functions for such oscillations are specified by the vibrational quantum numbers and are expressed in terms of normal coordinates. The normal coordinate describes a set of synchronized nuclear displacements in the direction of the corresponding normal mode. For a molecule consisting of N atoms, a set of mass-weighted associated with the ith normal coordinate Q, normal modes {L,] ( i = 1, 2, ..., 3N), including those for the translational and rotational movements of a molecule as a whole, can be determined as the eigenvectors of the mass-weighted Cartesian force constant matrix (Hessian matrix) H
Q022-3654/94/209~-6936$Q4.50/00 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 28, 1994 6931
Vibration Mixing in Terms of Normal Modes
HL = LAD Herea6 L~ L = I
(I = unit matrix)
ht? = LO,? HA: Lii (= h i B )
(1)
(2)
(10)
Force constant matrices HA,PA, and H A Bin~eqs 9 and 10 above are defined as shown below in relation to the force constant matrices HO and HABfor the noninteracting and interacting systems, respectively,
(3)
(AD)ij = Xisij
(4)
where E is the adiabatic potential energy of the molecule, x,, is the pth Cartesian displacement coordinate (three each for an atom), and m,,is the mass of the nucleus associated with x,,. Li is the ith normal vibration mode, which can be expressed in terms of the unit Cartesian displacement vector e,, for x,, (cf. eWT e, = &a)
:
AD is the eigenvalue matrix in diagonal form as shown in eq 4. The normal frequency vi is related to the eigenvalue X i as \
A:1
112
vi = -
2* For normal modes of translation and rotation, Xi = 0. 1. Equation of Vibration Mixing. For the perturbational treatment of the vibrationally interacting system, the following correspondence in form between the normal modes and molecular orbitals is helpful (cf. eqs 1, 2, and 5 with their counterparts). Namely, nucleus motions electronicstates * Fock matrix F H in F T = SCQ when S = I molecular orbital pi 4 u atomic orbital x,, e, cc molecular orbital energy q xi c,
In addition, Li belongs to one of the irreducible representations of the molecular symmetry just as a molecular orbital does. This formal correspondence assures us that the perturbation formulas already developed for the orbital mixing” are also applicable in a similar way to the vibration mixing. It follows that, for the A.-B interacting system, the ith normal mode of the fragment A, Lii, with eigenvalue changes into L’A~with X‘A~under the perturbation of the fragment B according to the following equations:’
Vibration mixing equations, Le., eqs 7 and 8 can be applied only when the perturbation is small. Thus, these equations lose their validity in cases where the conformation of the molecular system changes drastically. When some normal modes of A and B fragments are degenerate in eigenvalue, e.g., the case of interaction between translation modes of A and B fragments = x i k = O), eqs 7 and 8 should be applied to the correct zeroth order normal modes under the perturbation. Degenerate cases will be discussed in the next section. In eq 7, the 2nd term is a term for the mixing of normal modes within the fragment A, the 3rd is a term for the mixing from the fragment B, and the 4th is a term for the mixing within the fragment A through the interaction of each normal mode with that of the fragment B. In reference to the naming for the orbital mixing,3 the 2nd and 4th terms may be called static and dynamic vibration mixing, respectively. The 3rd term is direct vibration mixing, which corresponds to direct orbital interaction, such as HOMO-LUMO interaction, dealt with in the frontier electron theory.8 Let us consider the direct interaction (the 3rd term in eq 7) of the normal modes of reacting systems A and B under mutual perturbation. For the qualitative consideration of vibration mixing, the sign and the approximate magnitude of the mixing coefficient for Lik are important. The magnitudes of mixing coefficients depend on the denominator and the numerator. The denominator - hik suggests that the greater mixing is expected from the normal mode L:k having the closer eigenvalue to X i p The numerator h i Btells, according to a group theoretical consideration, whether or not vibration mixing occurs between Lii and Lik modes. When the perturbed Hessian matrix belongs to a total symmetry of the interacting system, hkBhas a nonzero value only when Lii and Lik belong to the same symmetry representation. The numerator hkBcan be written as follows (cf. eqs 3, 5 , 10, and 11 for notation):
(Aii
xii
Aii
Here, Lik and X i k are the kth normal mode and eigenvalue for the fragment B, respectively, and
h Si = L v (HA- H l ) Lii = Li: HAL:, - Xii6ij (= h ’i,.) (9)
Special caremust be taken for thecontributions from thosenormal modes involving hydrogen atoms in the interaction site moiety, e.g., theC-H, N-H, or 0-H stretching mode, since the numerator h f becomes large due to the light mass of hydrogen even for the cases where the denominator is large and Ltk in the numerator is small. 2. Degenerate Case. The denominator of perturbed terms in eqs 7 and 8 has a zero value in a degenerate case. For example, when LOAi and LOBk are degenerate, the corresponding denominator in eqs 7 and 8 becomes zero since hii = Degeneracy occurs very often, since all of the translation and rotation modes of A and B species are degenerate in frequency, having an eigenvalue of zero. In cases of such degenerate modes, thecorrect
&.
6938
The Journal of Physical Chemistry, Vol. 98, No. 28. 1994
Hirano et al.
I’ A’
I
I’
AY
L
-:
A
,
I
B
Figure 1. Schematic representation of the coordinate axes for the A-B interacting system, where G stands for the center of mass of the A-B
iy .
interacting system.
zeroth order normal modes of the interacting system should be employed for perturbational equations. In order to obtain the correct zeroth order normal modes, the secular determinant of the order of 12 should be solved for the determination of the unitary transformation matrix necessary to make all h;, and :h; (eqs 9 and 10) for these degenerate modes be zero. For the Cartesian coordinate of the A-B interacting system, let us define the x-axis as to connect the centers of mass of A and B molecules as shown in Figure 1. The notations of Txuand Rx,, stand for the translation and rotation modes of the X molecule or molecular system along and about the u-axis, respectively. Among the correct zeroth order normal modes of the A-B interacting system,six modes should correspond to the translation and rotation modes of the A-B interacting system as a whole and make no contribution to the A-B interaction. Group theoretical examination of zeroth order normal mode vectors leads us to classify these 12 originally degenerate modes into 4 independent groups:
A’
L
I mode
Figure 2. Schematic representation of Lb, and LiB,correct zeroth order modes. The heavy arrows denote the direction of the movement of each molecule.
modes, T A and ~ Lie,, respectively, as follows:
where mx is the molecular mass for the X molecule. Equation 15 represents the translation of the A-B interacting system as a whole along the axis connecting the centers of mass of A and B molecules. Equation 16 represents the correct zeroth order translation resulting in the collision of A and B molecules. 3. Vibration Mapping. Since (L$) and for the molecules A and B in each isolated state are orthonormal, (LAW)for the A-B interacting system can be expanded as a linear combination of (L$) and {Lid as
(gk)
and
From the discussion on h;, and ht: (see Appendix, for example, for the case of h+kTN), it can be concluded that the original modes belonging to one group do not mix with those belonging to the other groups under the assumption of two-body forces. Intuitive examination of the figure, Figure 2, for example, for a given combination of degenerate normal modes will also be helpful toderive this conclusion. Thus, the six correct zeroth order modes, LiBl,...,LiM,can be constructed within each group. It should be noted that the eigenvalues of these six modes remain zero. The Lie’ mode as a combination of TAXand Th makes A and B molecules close to each other on the line connecting A and B molecules. The other modes, Lie2,...,Lies, make A and B molecules face each interaction site to each other by rotation to maximize the orbital interactions at the interaction sites (cf. maximum overlap between HOMO and LUM0).9 For example, the Lis, and Lie2 modes are given in Figure 2. The most effective and important interaction mode should be LiBl which will result in the direct collision of the interaction sites of A and B molecules. From the secular determinant constructed from TAX and T B ~we, obtain the correct zeroth order
This procedure is called “mapping,” and the mapping coefficients and Mkl tell how the unperturbed normal modes LL and bkeach contribute as an ingredient to the normal mode LAW. From the orthonormality of the basis vectors (e,), the mapping coefficient matrix M can be obtained as
where the numbering of the columns of M is for the normal modes of the A-B interacting system and the numbering of the rows is for the normal modes of the isolated fragments A and B in this order. Thus, elements in the ith column of M are the mapping coefficients for LAWin eq 17. It should be noted that the normalized mass-weighted modes, i.e., normal modes, for the translation and rotation should be included in this mapping. The geometrical restriction for the mapping of normal modes of the A-B interacting system to those of the fragments A and B is more moderate than the geometrical restriction for the mapping of molecular orbitals since the basis for the vibration mapping is a set of displacement vectors. Thus, unless the
The Journal of Physical Chemistry, Vol. 98,No. 28, 1994 6939
Vibration Mixing in Terms of Normal Modes
PH
%:
F
H
c3axis
uH
Y
TABLE 2: Mapping Coefficients for A1 Normal Vibration Modes of the HF-NHJ Coordination System in Terms of A1 Basis Modes. Vibrational Frequency V I of Each Normal Mode (Scaled Down by a Factor of 0.89) Is Also Given in cm-1 in Parentheses basis LHF-NH~,I LHF-NH~,ZLHF-NH3.3 LHF-NH3.4 normal mode (213) (1098) (3298) (3616) GF-NH,,I
1.000
GH3.1
-0.01 1 0.001 0.006
gH3,2 GF.1
0.01 1 1.ooo
-0.008 -0.004
-0.001 0.008 1.000 0.023
-0.006 0.004 -0.023 1.000
L
Figure 3. Equilibrium structure of the HF-NH3 coordination system
(9,symmetry). The coordinate axes are also shown.
TABLE 1: Normal Modes and Each Vibrational Frequency V I of HF and NH3. Normal Frequencies Are AU Scaled Down by a Factor of 0.89 normal mode character (symmetry) uilcm-1 HF str (AI)
3998
GH3,39
bend (AI) str (AI) bend (E)
GH3.5’GH3,6
str (E)
1018 3297 1612 3419
GF,1
GH3A GH3,2
GH3A
geometries of the fragments A and B in the interacting system do change to a large extent, the vibration mapping is always possibleas a first approximation. Even such approximate mapping can still give a valuable insight for the understanding of the ingredients of a complex normal mode. When vibration mapping is performed, the coordinate system should be taken for both isolated and interacting systems in common. As long as the geometrical symmetry of each molecule is retained during interaction, the common coordinate system can be easily determined. Geometrical symmetry, however, often changes under perturbation. For example, H2CO and HzO lose their symmetries when they are involved in the interacting system (see the following section). In such cases, a useful geometrical arrangement for the vibration mapping is to make each isolated molecule’s axes of inertia superpose upon the axes of inertia of each fragment in the interacting system.
Calculations For demonstration of the present methods, the HF-NH3 and H~CO-HZOmolecular systems were studied. Ab initio molecular orbital calculations were performed for each molecular system with 6-31G** basis sets at the Hartree-Fock level using the Gaussian 90 program.1° Normal vibration analyses were carried out for each equilibrium geometry. Normal frequencies calculated at the RHF/6-31G** were all scaled down by a factor of 0.89.” Vibration mapping was performed based on formulas described in the previous section.
Results and Discussion 1. HF-NHJ System. Figure 3 shows a schematic illustration of the equilibrium geometry of the HF-NH3 molecular system having C3, symmetry. The x-axis is set to lie along the C3 axis as is shown. The calculated equilibrium distance between the H atom of HF and the N atom of NH3 is 1.84 A, and the geometry of each fragment molecule in the supermolecule is proved to be almost unchanged from that of each isolated molecule. The stabilization energy of the complex relative to the isolated molecules is calculated to be 11.82 kcal/mol. Table 1 gives normal frequencies of the normal vibration modes of HF and NH3 molecules with each irreducible representation in parentheses. HF and NH3 have 1 and 6 normal vibration modes, respectively, and the HF-NH3 molecular system has 12
normal vibration modes. The original translation and rotation modes of HF and NH3, 11 in total, change into 5 vibration, 3 translation, and 3 rotation modes upon interaction as is discussed for the degenerate case. Since the HF-NH3 molecular system has C3, symmetry, normal modes for the HF-NH3 system can be classified into three groups according to their irreducible representations:
Here, translation and rotation modes, shown without superscript 0, are all degenerate. These degenerate modes should be mixed in with each other in the group to be transformed to the correct zeroth order normal modes under the perturbation. According to the vibration mixing equations, the mixing occurs only among those modes belonging to the same irreducible representation. Here, we consider the mixing among normal modes having A1 symmetry, as an example. For the discussion of the vibration mixing among A1 normal modes, there are 5 normal modes in eq 19 to be considered as the basis modes for the perturbation, two of which are degenerate. As is discussed for the degenerate cases in the previous section, T H Fand ~ T N Htranslation ~~ modes should be transformed into the translation mode of the complex as a whole along the x-axis, T H F - N H(see ~ , ~eq 15), and a vibration mode, LiF-NH3,1 (see eq 16), as the correct zeroth order basis modes before perturbation equations are applied. Substituting mHF = 20.0 and mNH3= 17.0 amu into eqs 15 and 16, we obtain
The vibration mode LiF-NH,,, represents the movement of each fragment to collide with each other. Since the translation mode as a whole, i.e., THF-NH,~, does not contribute to vibration interaction, it should be excluded from the basis mode group for the vibration mixing. Thus, the basis modes for AI vibration become
Table 2 shows results of vibration mapping of A1 normal modes of the interacting system in terms of basis modes in eq 24. Each normal frequency is also given in parentheses. Since the HFNH3 coordination complex has no A1 basis modes having close frequency to each other, the mapping coefficients shown in this table are all small. Thus, the vibration interaction is proved to
Hirano et al.
6940 The Journal of Physical Chemistry, Vol. 98, No. 28, 1994
be weak in this coordination complex. It should be noted that the mapping matrix becomes to be anti-Hermitian. 2. H2CO-H20 System. Table 3 gives harmonic frequencies for normal vibration modes of H2CO and H20 molecules with each irreducible representation in parentheses. H2CO and H20 have 6 and 3 normal vibration modes, respectively, and the H2CO-H20 molecular system has 15 normal vibration modes. For the H2CO-H20 system, two optimized equilibrium structuresI2 were located as shown in Figure 4, a and b. We refer to each structure as SIand St[,respectively. The SIretains Cb symmetry while the SI, has Cl symmetry. The hydrogen bond does not exist in SIbut does in SII.The nearest distances between H2CO and H 2 0 molecules are 2.80 and 2.12 A for SIand SI[, respectively (Figure4), and stabilizationenergiesfor the complexation relative to isolated systems is calculated to be 2.84 and 5.25 kcal/mol for SIand SII,respectively. As shown in Table 3, there are two pairs of normal modes which have frequencies close to each other:
H
H"
Figure 4. Equilibrium structures of the H~CO-HZOcoordination system: (a) SI (CZ,symmetry) and (b) S11 (Cl symmetry).
TABLE 3: H2CO and H20 Basis Normal Modes with Each Since two modes in each pair belong to different symmetry Vibrational Frequency ui. Normal Frequencies Are All Scaled representations,vibration mixing within each pair will be invoked Down bv a Factor of 0.89 only by nontotally symmetric perturbation. VI/ VI/ In the case of SI,the perturbed Hessian matrix belongs to the basis normal mode cm-1 basis normal mode cm-' total symmetryrepresentationof Cbsince St retains Cb symmetry. Thus, the mixing within the pairs shown in eq 25 is symmetryforbidden. In addition,even for the other symmetry-allowedcases, thevibration mixing should be weaksince the original frequencies are widely separated from each other and the spatial distance between H2CO and H2O fragments is large. In the case of SI[,however, no symmetry is found in the interaction between H2CO and H2O molecules. Thus, the electric permeability 3-6) of an enzyme where partial charges symmetry of the perturbed Hessian matrix becomes CI,and are located on the inside wall of the pocket. Dynamic vibration vibration mixing among all normal modes can occur. Especially, mixing may also be important in the enzyme reaction, since in strong vibration mixing is expected for those pairs listed in eq 25 the enzyme-substrate (ES) complex the coupling between the since normal modes in each pair have similar frequencies. As the low-frequency vibration modes of the peptide wall and the results of vibration mapping of the normal modes of the coordination system onto L~2co,l,L~2co,s,L\20,1, and g20,3substrate may be an important driving force for the reaction. These topics will be discussed elsewhere. basis modes, we obtain Appendix Taking Th and TAy for an example, we will demonstrate that the two degenerate modes belonging to different groups defined in eq 14 do not mix in with each other. We assume that the force acting on each atom is a two-body force. Th and T Aare ~ defined as
Thus, it is shown that strong vibration mixing'occurswithin each pair as is predicted by the above symmetry consideration.
Concluding Remarks We proposed two methods, i.e., vibration mixing and vibration mapping, to predict and understand the mechanism of vibration interaction in terms of normal modes. The vibration mixing, derived by a perturbational approach, is the counterpart of the orbital mixing24 and vice versa. The vibration mapping gives sufficiently useful information for the vibration mixing mechanism. As a demonstration, our method was applied for the HFNH3 and H2CO-H20 coordination systems. In this paper, two types of vibration mixing which correspond to the static and dynamic orbital mixing will be left with only a few comments. Staticvibration mixing may be important for the reaction under the electric field from the nearby partial charges, esmiallv for the reaction in the hvdroDhobic Docket (effective
(a-2)
where mN, XM, and y~ are the mass and the unit vectors along the x- and y-axes for the ith atom of A, respectively, NA is the number of atoms belonging to A, and M A is the total mass of A molecule ( M A= Epm,). From eqs 3 and 9, A+-,,J& becomes
Vibration Mixing in Terms of Normal Modes
The Journal of Physical Chemistry, Vol. 98, No.28, 1994 6941 coordinate system, a($)A/aXA is approximately zero as is known for a two point-mass system with central force. Thus,
Here, -(aEAB/ayN) is the y-component of the force acting on the atom A,, (F,,)A/, which is the sum of the intramolecular forces from the A fragment itself, ($)A/,and the intermolecular forces from the B fragment, Thus, eq a-4 becomes
Zp($)A/
Since is the sum of an internal force, it is zero regardlessof axM, i.e., an infinitesimalchangeof x ~ Thechange . of (F;>+,for axAi can be neglected for i # j because the infinitesimalchangeof x . will ~ scarcelyaffect theelectronicstate around the jth atom. Equation a-5 becomes
Here, XAis the displacement of the A molecule as a whole in the x-direction and is the y-component of the total force from the B fragment acting on the A fragment. For our Cartesian
(q!A
References and Notes (1) In an exact sense, it should be called 'normal-mode mixing," since mixing occurs among the normal modes of vibration, translation,and rotation. We, however, prefer the term "vibrationmixing" to strw the image of nucleus movements in contrast to the electronic viewpoint in the 'orbital mixing." (2) Libit, L.;Hoffmann, R. J . Am. Chem. Soc. 1974,%, 1370. (3) Imamura, A.; Hirano, T. J. Am. Chem. Soc. 1975,97,4192. (4) Inagaki, S.;Fujimoto, H.; Fukui, K. J. Am. Chem. Soc. 1976,98, 4054. (5) Born, M.; Oppenheimer, R. Ann. Phys. 1927, 84,457. ( 6 ) Superscript T used hereafter denota the transpose of a vector or matrix. (7) We noticed that a similar formulation using the bra-ket formalism is shown in a book by Ezawa et a/.: Ezawa, H.; Nakamura, K.; Yamamoto, Y. RIKlGAKU (Mechanics); Tokyo-tosyo: Tokyo, 1984. (8) (a) Fukui, K.; Yonezawa, T.; Shingu, H. J. Chem. Phys. 1952, 20, 722. (b) Fukui, K.; Yonezawa, T.; Nagata, C.; Shingu, H. J. Chem. Phys. 1954,22,1433. (c) Fukui, K.; Yonezawa, T.; Nagata, C . Bull. Chem. Soc. Jpn. 1954,27,423.(d) FuLUi, K.; Yonezawa, T.; Nagata, C. 1. Chem. Phys. 1957, 27, 1247. (e) Fukui, K. Forrschr. Chem. Forsch. 1970, IS, 1. (f) Fukui, K. Ace. Chem, Res. 1971, 4, 57. (9) Fukui, K.; Fujimoto, H. Bull. Chem. Soc. Jpn. 1969, 42,3399. (10) Frisch, M. J.; Head-Gordon, M.; Trucks, 0. W.; Foresman, J. B.; Schlegel, H. B.; Raghavachari, K.;Robb,M. A.; Binkley, J. S.;Gonzalez,C.; DeFrets, D. J.; Fox, D. J.; Whiteside, R. A.; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R. L.; Kahn, L. R.; Stewart, J. J. P.; Topiol, S.; Pople, J. A. Gaussian 90,Revision J; Gaussian, Inc.: Pittsburgh, PA, 1990. (1 1) (a) The scale-down factor 0.89 was taken from the calibrationof the 6-310**resultsforH20withexperimentaifrequencies. Miyajima,T.; Kurita, Y.; Hirano, T. J. Phys. Chem. 1987,91,3954.(b) Hamada. Y.; Tanaka, N.; Sugawara,Y.; Hirakawa, A. Y.; Tsuboi, M.; Kato, S.; Morokuma, K. J. Mol. Spectrosc. 1982,96,313. (c) S a c k , S.;Farnell, L.; Riggs, N. V.; Radom, L. J. Am. Chem. Soc. 1984,IW,5047. (12) Lewell, X.Q.;Hillier, I. H.; Field, M. J.; Morris, J. J.; Taylor, P. J. J . Chem. Soc., Faraday Trans. 2 1988,84,893.
