Vibration Mixing: An Application to the Prediction of the Transition

Jul 1, 1994 - Vibration Mixing in Terms of Normal Modes. The Journal of Physical Chemistry. Hirano, Taketsugu, Kurita. 1994 98 (28), pp 6936–6941...
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J. Phys. Chem. 1994, 98, 6942-6949

6942

Vibration Mixing: An Application to the Prediction of the Transition Normal Vibration Mode Tsuneo Hirano,’*+Tetsuya Taketsugu,$and Yasuyuki Kurita**g 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 113, Japan Received: January 20, 1994”

Vibration mixing and vibration mapping, which we proposed recently as novel methods to describe vibrational interaction in terms of normal modes, were applied for the prediction and understanding of the transition normal vibration (TNV) in atom molecule type reactions (Ha H-H , Ha HzC-CHz , and H - oxirane ). According to the equations of vibration mixing, we derived vibration mixing rules for the prediction of the T N V which consist of a principle for the selection of the candidate normal modes growing into the TNV and of the estimation of the sign and magnitude of the mixing coefficients for the normal modes to mix in. Thus, the possible TNVs can be predicted by pictorial examination of the normal modes of the reactants. The results are in good agreement with those of vibration mapping.

-

+

+

-

-

+

+

Introduction

Vibration Mixing Rules

Transition normal vibration (TNV), defined here as the normal vibration with imaginary frequency at the transition state of reaction, has gained more and more attention, since it describes the movement of atoms along the intrinsic reaction coordinate (IRC).’ Sometimes there are many transition states along the different IRCs for a given reaction system. When several reaction paths, say, reaction modes, can be anticipated, prediction of the TNV becomes especially important. First Bade9 and then Pearson3 and Salem4presented a method to predict the movement of atoms at the transition state in terms of the second-order Jahn-Teller effect from the inspection of the electronicwave functions at the transition state. For this method, knowledge of the symmetries of the MOs, in most cases the highlying occupied MO (HOMO) and the low-lying unoccupied MO (LUMO),is indispensable. Such MOs can be obtained directly by solving the eigenequation for the electronic state of the transition state or can be estimated in a qualitative way by using the orbital mixing r~le.S-~The orbital mixing rule has been presented for intramolecular polarization by Libit and Hoffman115 and for intermolecular interaction (catalysis) by Imamura and one of the present authors (T.H.)6 and has been extended later for general cases by Fukui et al.7 Use of the orbital mixing rule is sufficient for the qualitative discussion of the movement of atoms in a reaction system and hence for the understanding of thenatureofthereaction mode. Thismethod is, however, indirect. The TNV is a matter of normal vibration, and hence there might be a method to predict the TNV directly from the normal modes of the component molecules in the reaction system, but none, to our knowledge, has been reported. Recently, we proposed two methods, i.e.,vibration mixing and vibration mapping,* to describe vibrational interaction in terms of normal modes. Equations of vibration mixing can be applied for the case where changes of the Hessian matrix are small and the conformation of molecules is not drastically changed under the perturbation. In this paper, our two methods, the vibration mixing rule and vibration mapping, are applied for the TNV of atom molecule type reactions,i.e.,He + H-H, He + H~C==CHZ, and H - + oxirane.

1. General Equation for Vibration Mixing. When molecule B approaches molecule A, the ith normal mode of the fragment A, Lii, with eigenvalue, A: changes into L’A( and A‘A~ under the perturbation of fragment B accordingto the equations of vibration mixing,s

+

* To whom correspondenceshould be addressed. t

Here, NXis the number of atoms of molecule x,L& and hi, are the kth normal mode and eigenvalue of fragment B, respectively, and9

h;i = Li:

(HA - Hi)Li,

hiB = Li,T HABTL i i

(3) (4)

Force constant matrices HA,H i , and HAB~ in eqs 3 and 4 above are defined as shown below in relation to the force constant matrices HO and HAB for the noninteracting and interacting systems, respectively,

2. Case of Atom-Molecule Type Reaction. Let us consider a reaction between atom A and molecule B. The x-axis is defined as connecting atom A and the center of mass of molecule B in thedirection fromatom A to molecule B. Translation and rotation modes of a molecule X along and about the u-axis in the massweighted coordinate system are designated by Txu and RxY, respectively. For example,

Ochanomizu University.

t University of Tokyo. I Presentaddress: TakarazukaResearchCenter,SumitomoChemicalCo., Ltd., Takatsukasa, Takarazuka, Hyogo 665, Japan. *Abstract published in Advance ACS Absfracfs,April 1, 1994.

0022-3654/94/2098-6942%04.50/0 0 1994 American Chemical Society

The Journal of Physical Chemistry, Vol. 98, No. 28, 1994 6943

Prediction of the Transition Normal Vibration Mode where e , and me are a unit vector along the x-axis and the atomic mass of the uth atom of molecule B,and mg is the total mass of molecule B. Atom A has only translation modes, which are degenerate with the translation and rotation modes of B in frequency. As was discussed in our previous paper! such degenerate modes should be transformed into the correct zeroth order normal modes before eqs 1 and 2 are applied. The most important mode among the correct zeroth order normal modes should be a mode obtained as a combination of TI1, and T B ~ , which will result in the direct collision of the reaction sites of molecules A and ! B

A i k in the denominator is of positive value. Thus, the vibration mixing in this case results in the lowering of the AiB, i.e., from zero to a negative value, indicating that the correct zeroth order normal mode LiB grows into the TNV under the perturbation. As shown in eq 7, the mixing ratio of T h to Th in LiB is determined only by the mass ratio of species A and B. Thus, we conclude, it is highly possible that the translation mode of the lighter species, Le., atom A in most cases, along the direction connecting the reaction sites will grow into the TNV by mixing in of the translation mode of the heavier species in the opposite direction. Next, we consider how the kth normal vibration mode of molecule B, will mix in Lip The sign and the approximate magnitude of the mixing coefficient for in eq 8 can be estimated as follows. The denominator, has a negative value since Aik is of positive value. The numerator h t i B ,defined as eq 13, can be approximated as (see Appendix c )

Gk,

where mA and mg are the total mass of atom A and molecule B, respectively. The mixing ratio in this zeroth order normal mode is determined only by the mass ratio of species A and B. Comparing the coefficients for Tkr and T B ~we, can see that the main part of the LiBmode is the translation mode of the lighter species, Le., atom A in most cases. Note that the LiB mode should be of total symmetry because it representsthe movements of atoms along the line connecting atom A and the center of mass of molecule B. The TNV should have a negative eigenvalue (an imaginary normal frequency). Since the correct zeroth order normal mode LiBhas a zero eigenvalue ( X i B = 0), it is a candidate for a normal mode growing into the TNV. Equations 1 and 2 for this LiB mode can be written as

and

since atom A has no vibration modes to mix in LiB. In eqs 8 and 9, summation over k goes over those normal modes which are totally symmetric (T.S.) in the supermolecule system because the mixing of the normal modes occurs within each irreducible representation.* First, we consider the change of eigenvalue under the perturbation. The 2nd term h'AgAg in eq 9, defined as eq 10 in this case, can be transformed as (see Appendix a)

- - mA + mB NE

a2EAB

mAmg

axAaxBo

T

%

--

(14)

The nearly equal sign in the above equation comes from the expectationthat the vibrationalinteractionbetween distant atoms from the reaction site will be small and hence can be neglected. Equation 14 means, therefore, that localvibrationalcharacteristics of the reaction site moiety is the determining factor of h& As is discussed in the Appendix b, the sign of the force constant ~2EAB/axAdxgu for the bond-forming sites near and at the transition state is positive. The sign of can be assumed to always be negativelo because, in the bond-forming process near and at the transition state, the directions of movement of atoms at the reaction sites should be face-to-face. As a result, the sign of the numerator h t i B for the mixing of L& becomes negative (= a negative value of x a positive force constant).'O Thus,aslongas wechoosethephaseofthecomponent normal modes so as to make the movement of two atoms at each reaction site directed in bond-forming fashion, the mixing coefficient for the 2nd term in eq 8 becomes positive:

@EAB -L!r-2,k

(1 1)

In the case where atom A faces the reaction site of molecule B, the largest term in h'AB,Ag should be the one which contains the second derivativeof E A B with respect to the x-coordinates of the atoms at the reaction site, Le., a2EAB/axAdxBa where /3 stands for the reaction site of molecule B. Hence, ~'AB,AB can be approximated as

Near the transition state, a2EAB/axAaxwis positive from the definition of the transition state (eq b-3 in the Appendix b). This means that the ABAB AB in eq 12 is negative. It should also be noted that the sign of the 3rd term of eq 9 is always negative since

The numerator in eq 15 implies that the magnitude of vibration mixing is proportional to &-2,k and inversely proportionalto the root of the mass of the atom at the reaction site of molecule B. Special caremust betaken for thecontributionsfrom thosenormal modes involving the lightest atom, hydrogen, in the reaction site moiety. Thus, we can predict how vibrational normal modes of molecule B, g k , mix with the correct zeroth order mode LiB. Equation 15 also tells that only those normal modes having large components at the reaction site need to be considered; the local symmetry of the normal mode at the reaction site is d6cisively important in vibration mixing, as was previously pointed by us for orbital mixing.6 It follows that each normal mode of a symmetricand antisymmetricpair can contribute in the vibration mixing in almost similar magnitudes and that the result of such mixing almost cancels the out-of-phase parts and leaves the in-

6944 The Journal of Physical Chemistry, Vol. 98, No. 28, 1994

phase parts in an additive way for the motion of atoms along the IRC a t the transition state. In summary, the following process is appropriate for the prediction of the TNV for the reaction of atom A and molecule B. (1) First, the reaction center in molecule B should be determined. This can be done easily in many cases (a) by using the orientation principlellg given by Fukui et al. in the frontier electron theory" or (b) by just assuming the possible reaction sites. For method b, close inspection of the normal vibrations of molecule B will give necessary information, since the site and direction of large components in a normal mode may qualify to be the first candidate to be considered. This is an important and useful process when multireaction modes, Le., different kinds of reaction a t different reaction sites, are anticipated or when we have the need to predict different reaction paths. (2) Second, the translation mode of A directing to the target atoms in molecule B should be chosen. (3) Then, those normal modes of fragment B which have largecomponents at the reaction site and along the bond-forming line (or along the bond-cleaving line) should be chosen. (4) Next, the phase of the thus selected normal modes should be arranged in-phase for bond formation (or for vibrational interaction) between those of reactant and substrate (this is not necessary if one may be careful of the sign of J$&k, however).IO Then, the mixing coefficient becomes always positive (eq 16). The approximate magnitude of mixing can be estimated from the quantity of each variable appearing in the above coefficient. This is the kernel of the vibration mixing rule for the TNV. ( 5 ) Sometimes the contribution from sites other than the reaction center cannot be negligible in the vibration mixing for the TNV. As is pointed out above, this situation arises since the numerator h t i B in the mixing coefficient contains m,-'I2, and hence a contribution from a light atom, hydrogen for example, may not be disregarded. We will discuss this case later for a nucleophilic reaction of oxirane.

Hirano et al. x2

x3

XI

X

b--r12--+------____r13. _ _ _4_ _ _ _ _ _

-

Figure 1. Spatial alignmentand definition of the x-axis assumed for the H2-H1 + .H3 reaction.

Mode 1 (Bvib)

Mode 1' (A-J

4

Mode 2 (B-)

Mode 2'

Mode 3

Mode 3'

b-b

&

8

Mode 4

Mode 5

8

0

Mode 6

B

A

(H2-H1) (.H3 1 Figure 2. Basis sets for thevibration mappingof the H2-Hl-H3 reaction system. Modesmarked withE&bB-,andA-areudforthevibration mixing and mapping, as shown in the text.

Results and Discussion Vibration Mapping

The normal modes {LAB,)for the A-B interacting system can be expanded as a linear combination of {Lii) and {Lo,,) for molecules A and B in each isolated state as?

where M,i and Mki are mapping coefficients and tell how the unperturbed normal modes L L and Lik contribute each as an ingredient to the normal mode LAB^. The mapping coefficient matrix M can be obtained as

O M=[f'.

L;]

T LAB

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. It should be noted that the normalized massweighted modes, Le., normal modes, for the translation and rotation should be included in this mapping. Calculations

Molecular orbitals and normal modes were calculated by an abinitiomethodat the Hartree-Focklevel with STO-3G, 4-31G, or 6-311G** basis sets using the Gaussian 82 program.12 Cartesian force constants were calculated analytically. Each normal mode is shown in mass-weighted coordinates with calculated frequency scaled down by a factor of 0.89.13

1. Vibration Mapping. Let us first consider the mapping of normal modes for the reaction

H2-Hl

-

+ *H3

+

H2***HI.*.H3 H2* Hl-H3

The adiabatic potential surface for this reaction has been extensively inve~tigated.'~Since a qualitative profile of the reaction is sufficient for the present discussion, we performed U H F MO calculations with the 6-311G** basis set. As shown in Figure 1, we assume that H 3 approaches the H2-H1 molecule along the molecular axis of H2-H1 which is defined as the x-axis. The normal modes of the interacting system (A--B) are mapped in terms of normal modes of the isolated H2-H 1 (B) and .H3 (A) fragments shown in Figure 2. In this case the invariance of the spatial symmetries of the fragment molecules during the reaction is assured. In Table 1 are shown the total energy of the interacting system along the IRC, the H2-Hl distance (rlz), the Hl-.H3 distance (r13), the force constants for H2-.H1 (k12) and Hl.-H3 (k13),and the mapping coefficients for the normal mode growing into the TNV at various r13 distances. Figure 3 shows how this normal mode grows into the TNV as the interacting system goes up along the IRC. At r13 = 2.5 A, the normal mode with the wavenumber of 223 cm-1, which will grow into the TNV, is mapped mainly to the normal modes of H2-H1 (BtIam)and -H3 (A,,,,) in a combination such that the H2-H1 and -H3 fragments approach each other. As r13 becomes shorter, the contribution of the vibration mode of molecule B, Bvib, becomes greater and that of the translation mode of molecule A, Atranr, smaller. Positive contribution of the Bvib mode here means that H2 and H1 tend to dissociate. At a certain point around r13 = 1.5 A, the force constant aZEAB/ 8 ~ 1 3 2becomes negative (in other words a2EAB/ax1 8x3 becomes positive), as is intuitively suggested by the change of curvature

The Journal of Physical Chemistry, Vol. 98, No. 28, 1994 6945

Prediction of the Transition Normal Vibration Mode

-

TABLE 1: Total Energies, Force Constants, and Mapping Coefficients' of the Normal Mode Growing into the TNV at Various Stages of the Reaction: H2-Hl -H3

+

distance/A

force constantsdfmdyne A-1

mapping coeff of the normal mode in Fig 3 ~

stag& 0 1

2 3 4 5 6 7 8

9 1oC

20.0 2.5 2.0 1.7 1.6 1.5 1.2 1.15 1.1

1.0 0.934

0.736 0.736 0.737 0.740 0.743 0.746 0.774 0.785 0.799 0.849 0.934

re1 tot energy/kcal mol-' 0.00 0.38 1.73 3.97 5.13 6.57 12.55 13.72 14.87 16.89 17.55

O.OO0

0.020 0.057 0.052 0.023 -0.037 -0.600 -0.74 1 -0.957 -1.239 -1.043

~

~~~

~~

~~

k12

k b

&ram

~truu

6.262 6.255 6.166 5.928 5.760 5.507 3.662 3.072 2.357 0.500 -1.043

-0.001 -0.002 0.005 0.01 1 0.023 0.165 0.234 0.337 0.660 0.866

0.577 0.577 0.577 0.577 0.577 0.569 0.56 1 0.544 0.434 0.289

0.817 0.8 17 0.8 16 0.8 16 0.816 0.805 0.794 0.769 0.6 14 0.408

kl3

See cq 17 for the definition. Reaction stage along the IRC, which is also shown with the corresponding numbering in Figure 3. Total energy for the total energy E. Note that #E/dxIdx3 and #E/t3xlaxz have relative to the reaction stage 0 (r13 = 20 A). Defined as # E / d r 1 3 ~and @ZE/~3r12~ signs opposite to thost of d2E/dr13' and #E/dr122, respectively, as is discussed in the Appendix, section b. Transition state.

Stage (Eigenvalue) 1 (0.037)

*-*

TABLE 2 Mapping Coefficients' for tbe TNV of tbe Reaction H2C=CHt He -, in Terms of C2)4 and He Basis

4-e

+

Modes

H*

C2H4 mapping coeff

basis mode

mapping coeff

3 (0.247)

O.OO0

Tc2bX Tc2by Tc2br

O.OO0

4 (0.284)

0.193 0.253

2 ( 0.136)

basis mod&

o.Oo0 O.Oo0

5 (0.312)

0.107 0.012 0.146 0.017 0.032

6 (0.113) 7 (-0.054) 8 (-0.334)

9 (-1.610) 10 (-3.107)

Figure3. Schematic representationof a normal vibration mode growing into the TNV at various stages given in Table 1. (Associated eigenvalues are given in parentheses in mdyne A-1 amu-1).

of the potential surface near the transition state (see the Appendix b). Accordingly, the eigenvalue for the normal mode growing into the TNV changes its sign from positive to negative at shorter r13 (Figure 3), resulting from the change of the sign of the IRC curvature from positive to negative at a certain point close to the saddle point. Simultaneously, the movement of H2 starts to change its direction opposite to that of the H1 movement. The latter is due to the growing positive contribution of the &b component in the normal mode of imaginary frequency which turns out to be the TNV. These features become clear at the transition state: the major components of the TNV are now the positive &b mode with a little smaller positive A ,, mode, indicating that the bond between H1 and -H3 is going to form with simultaneous bond scission between the H2 and H1 atoms. Since energy is associated with each normal mode, the above description in terms of the vibration mixing is equivalent to thedescriptionthat the translational energy of =H3 (and also of H2-H1) converts mainly to the vibration energy of H2-H1 in such way that H2 and H1 depart from each other and the bond between H1 and -H3 is to be formed. This description coincides completely with the classical image of the

0.163 0.055

basis mode TH.,~ TH.,~ TH.,~

mapping coeff 0.859 O.OO0

0.289

o.Oo0 Q2bY 0.1 18 Rqbr O.OO0

RqH*x

O.OO0 O.OO0

Sce cq 17 for the definition. Mode number in Figure 5, followed by its irreducible representation of symmetry in parentheses. T b and RA"denote the translation and rotation modes of molecule A along and about the u-axis, respectively.

+

H2-H1 eH3 reaction's in terms of the movements of the fragment species on the potential surface. Since the translation mode of molecule B, B, can be regarded as the antisymmetric H2-H1 stretching mode,I6 the above conclusion for the TNV mapping can be interpreted as follows: the TNV arises from the translation mode of the attacking eH3 atom with mixing-in of the symmetric and antisymmetric H2H1 stretching modes. This description will be shown later to be inconformity with thegeneral rule suggested by the perturbational treatment. Another example is shown in Table 2 for the reaction

H2C=CH2 + H e

-

H2C=CH2*-H*

The transition state has already been studied with the 4-31G basis set by Nagase et a1.I' We adopted their transition state geometry to calculate the normal modes of vibration with the same basis sets. The x-axis is chosen to be perpendicular to the molecular plane and the z-axis to pass through the two carbon atoms. The TNV thus calculated is shown in Figure 4. In Table 2, only the result of mapping of the TNV is shown, although the results for the other normal modes are also informative. Figure 5 shows the basis normal modes of ethylene calculated for the equilibriumgeometry. Table 2 shows that three normalvibration pairs of ethylene, which have similar local symmetry at the reaction site (modes 2 (B3,) and 3 (B& modes 6 (A8) and 8 (A8), and modes T C , (B3,) ~ and R C ~ H(Bze)), * ~ mix into the translation modes of H. to form the TNV. As is seen from the pictorial superposition of basis modes given in Figure 5 according to the magnitude of mapping coefficients, the aforementioned mixing

Hirano et al.

6946 The Journal of Physical Chemistry, Vol. 98, No. 28, I994 X

L Figure 4. Calculated TNV for the HzC=CHz

+ H. reaction.

Figure 6. Calculated TNV for the oxirane + H - reaction.

TABLE 3 Mapping Coefficients' for the TNV of the Reaction Oxirane H - -, in Terms of Oxirane and H Basis Modes

+

Mode 1 (lBzu,831.6)

Mode 2 (lBau, 1001.0)

Mode 3 (lBzg, 1034.0)

basis modeb

?-J Mode 4 (IAu, 1038.9)

2 4 3-c Mode 5 (1B3g, 1233.1)

Mode 6 (IAg, 1353.2)

H H H Mode 7

Mode 8

Mode 9

(lBI,,, 1458.0)

(2A,, 1651.0)

(2BIu, 2940.0)

H H Mode 11

Mode 10 (3A,, 2962.6)

(2B3g, 3004.8)

x

Mode 12 (2Bzu,3032.6)

Figure 5. Normal vibrationmodes of ethylene(D2h symmetry)with each irreducible representation and frequency (in cm-I).

results in the bond formation between He and the a-carbon, the downward movement of a-hydrogens suggesting the rehybridization at a-carbon to sp3 type, the movement of a- and 8-carbons in the direction of bond elongation, and the near cancellation of the movement of @-hydrogens. The third example is the nucleophilic ring opening reaction of oxirane by H-ion. All the calculations were performed with the STO-3G basis sets. The x-axis is chosen to be perpendicular to the molecular plane and the y-axis to pass through the two carbon atoms. The calculated TNV of this reaction is shown in Figure 6. The result of mapping of the TNV is given in Table 3 and vibrational basis modes of oxirane having nonzero mapping coefficients are shown in Figure 7. Here again mapping coefficients given in Table 3 show that the TNV consists of the translation modes of H - and the symmetric and antisymmetric pairs having similar local symmetry a t the reaction site (modes 7 (B2) and 8 (A,) and modes 10 (Bz) and 1 1 (AI)). The mixingin of the symmetric and antisymmetric normal modes of similar magnitude in each pair results in the cancellation of the movement of atoms in the &carbon moiety and makes the oxygen atom

1 (1Bd 2 (1Al) 3 (1A2) 4 (1Bd 5 (2Bl) 6 (2A2) 7 (2Bz) 8 (2Al) 9 AI) lO(3B2) 1 1 AI)

oxirane mapping basis modeb*' coeff 0.000 -0.080 0.000 0.099 0.000 0.000 0.410 0.354 -0.007 0.144 0.152

12 (4Bz) 13 (5A1) 14(3A2) 15 (3Bl) Tokane,x

To-nhy Toxirane,r

H-

mapping coeff 0.030 0.034 0.000 0.000 0.000 0.065 0.101 0.039 0.000 0.000

basis mapping mode coeff T H - , ~ O.OO0 T H - , ~ 0.428 TH;~

0.667

a See eq 17 for the definition. Mode number in Figure 7, followed by its irreducible representation of symmetry in parentheses. e TAUand Rau denote the translation and rotation modes of molecule A along and about the u-axis, respectively. move away in the direction of C,-O bond cleavage. The interpretation given above for the vibrational interaction will be discussed again in the perturbational treatment given below. 2. Vibration Mixing. (a) H-H -HReaction. We explain here an example of the application of the vibration mixing rule to the TNV of the reaction

+

+

H2-H1+ *H3 H2* H1-H3 The steps given in the preceding section will be followed. The reaction sites and the direction of approach are obvious from the shape of the LUMO which receives the unpaired electron of .H. Hence, .H3 approaches one of the H atoms, say H1, in the Hz molecule on the line which coincides with the molecular axis. The translation mode of .H along the approaching line should be the candidate normal mode togrow into the TNV. The normal modes of the substrate, H2-H1, are given in Figure 2. The mixing of the translation mode of .H3 radical with those normal modes having large component along the bond formation line should give the TNV. Since all the normal modes under consideration, Le., modes 1 (Bvib), 2 (BtranS),and 1' (Atrans),are given in Figure 2 to be in-phase with respect to the displacement vectors at the reaction sites, the signs of the mixing coefficients become all positive according to the mixing rules given in the previous section. The magnitude of the mixing coefficient for ,,B should be relatively small in comparison with that for &b, since the mixing between the two translation modes obeys the equation for the degenerate case (see eq 7) and the constant mixing ratio18 for such case will be surpassed by the large mixing between the nondegenerate pair on approaching the transition state. The TNV thus estimated isshown in Figure 3, which tells thebond formation between H1 and .H3 with simultaneous bond cleavage between H 2 and H1. +

Prediction of the Transition Normal Vibration Mode

The Journal of Physical Chemistry, Vol. 98, No. 28, 1994 6947

Mode 2 (+0.19)

Mode 2 (lAl, 975.8)

Mode 4 (1B2, 1091.9)

:.-

Mode 7 (2B2, 1213.5)

!+

?.

Mode 3 (4.25)

I

9

1

Mode 10

Mode 8

Mode 9 (3A1, 1357.2)

(2AI, 1264.5)

TY~,(+ 0.16)

(3B2, 1565.5)

P

T, (+ 0.91)

@)

Mode 11 AI, 1629.8)

Mode 12 (4B2, 3206.6)

Mode 13 (5A1, 3223.4)

Figure 8. Mixing of normal modes for the HQ-CH2 + H. reaction. (a) Assumed reaction mode. (b) Candidate normal modes of the reactants in thevibration mixing (mapping coefficientof each normal mode for the TNV is given in parentheses). (c) Calculated TNV,which is essentially the same as that deduced from the vibration mixing rule.

Figure 7. Normal vibration modes of oxirane (Ca symmetry) relevant to the mixing for the TNV, with each irreducible representation and frequency (in cm-1).

The mixing of &b and B- modes can be regarded as the mixing of the symmetric and antisymmetric pair of H2-Hl stretching modes with less amount of the latter at the transition state. Thus, H1 approaches much faster than H2 toward sH3, resulting in the bond cleavage between H2 and H1. All these results are in good agreement with the quantitative results from the vibration mapping. (b) H2C'==CH2 *HReaction. The next exampleof application of vibration mixing rule is the reaction HzC=CHz .H -. The most probable reaction between ethylene and .H is the attack of .H on the spz carbon, C, for example, in H ~ C = B C,Hz. Judging from the shape of the LUMO which receives the unpaired electron from .HI the most probable reaction path is the approach of .H in the plane perpendicular to the molecular plane of ethylene (Figure sa). The candidate normal modes to grow into the TNV are such translation modes of *H that have a large component along the line connecting .H and C,. For the ethylenepart, those normal modes which have a large component along this line are the candidates for the normal modes to mix in the translation modes of qH discussed above. Normal modes shown in Figure 8b are selected from this point of view. The sign and rough magnitude of the mixing coefficient for each normal mode can be determined by using the rule shown in point 4 in the preceding section, 'Case of Atom-Molecule Type Reaction'. The results are qualitatively in good agreement with the results from the mapping analysis shown in Table 2. Thus, informationon normal modes belonging to the fragment molecules, .H and HzC==CHz here, is sufficient for the prediction of the TNV. Cancellation in the antisymmetric region of the symmetric and antisymmetric pair leaves H, going down and HB at almost the same position. The mixing of made 8 (A,) results in the elongation of the C,=C~bond. From theseconsiderations, we can image the TNV for the present mode of reaction just like the one shown in Figure 8c, which is the actual TNV obtained directly by independent molecular orbital calculations. Thus, here again, the vibration mixing rule based on the perturbation theory works well for a priori estimation of the TNV.

+

P

M& 7 (4.41)

+

T ,

(+ 0.79)

@)

+

Figure 9. Mixing of normal modes for the oxirane H - reaction. (a) Assumed reaction mode. (b) Candidate normal modes of the reactants in the vibration mixing (mapping coefficientof each normal mode for the TNV is given in parentheses). (c) Calculated TNV, which is essentially the same as that deduced from the vibration mixing rule.

+

(c) H - Oxirane Reaction. The mechanism of the nucleophilic ring opening reaction of oxirane is well-known experimentally and theoretically. When the distance between H - and oxirane is far, the electrostatic interaction between the negative charge on H - and the partial charges distributed on the oxirane ring favors the back-side attack of H - in the plane of the oxirane ring (Figure 9a). At shorter distances, the orbital interaction becomes the dominant factor in the determination of the reaction path. Since the LUMO of the oxirane is the orbital to receive the electron from H -,the H - ion approaches, tracking the path where the maximum overlap of the LUMO of oxirane with the HOMO of H - is expected. Charge transfer occurs, and the transferred electron is going to pile up at the farthest point, i.e.,

Hirano et al.

6948 The Journal of Physical Chemistry, Vol. 98, No. 28, 1994

the 0 atom, and the increase of the antibonding character of the C,-O bond due to the mixing of the partially filled original LUMO results in the opening of C,-0 bond. Let us consider, however, the case where the mechanism given above is unknown. First, the translation mode of H - may grow into the TNV. Second, from the net charge distribution in the oxirane the H - may approach the C atom in the reaction. Thus, the normal vibrations having a large component at the C atom may be the candidate normal vibration to mix into the translation mode of H The symmetric and antisymmetric pair of normal modes of oxirane in Figure 9b is the most probable ones for the mixing. In addition to these, two other normal modes 4 (B2) and 9 (Al), having a large component at the C atom are found in the set of normal modes of oxirane given in Figure 7. The difference from the other pair depicted in Figure 9b is the sign of the contribution from the neighboring two hydrogen atoms as shown in Figure 7. Contribution from these nearest neighbor hydrogen atoms cannot be neglected since hj?:B in the numerator of mixing coefficient contains the mU-ll2factor as shown in eq 14. For the normal modes 4 (B2) and 9 (AI), the sign of L:,,-2,kand a2EAB/ d X A d X B a are negative and positive, respectively, for the H-...C. interaction and are both negative for the two H --H, interactions. It should be noted that, a t the transition state, the forceconstant d2EAB/dx.q3xg, along the IRCis positive and, hence, that the force constants for the H --H, interaction, not along the IRC, are negative. Thus, in normal mode 4, the negative contribution to from the H --C, interaction may cancel with the positive contributions from the two H --H, interactions, resulting in a minor contribution to the TNV. In fact, the numerical value of a2EAB/axAaxB,,for the H --.C, interaction is 0.265 mdyne A-I amu-l/2, and for two H-.-H, interactions it is -0.218 X 2 mdyne A-1 amu-1/2. A similar situation holds for normal mode 9, and hence this normal mode also contributes by negligible magnitude. For normal modes 7 (B2) and 8 (AI), contribution from H --H, interaction cannot be neglected as well. In these cases, however, the contribution from these a-hydrogen atoms is additive to that from C, and works in the direction to increase the amount of the mixing of these normal modes into the TNV. Thus, the normal modes given in Figure 9b are sufficient for the vibration mixing for the TNV, and we can draw an image of the TNV shown as in Figure 9c, which is the result of the independent exact calculation of theTNV. The signof the mixing coefficient for each normal mode shown in Figure 9b can be determined by the same consideration given in the previous examples: the sign of the mixing coefficient becomes positive for all normal modes given in Figure 9b since the phase of the normal mode shown there has already been arranged to be in-phase with the translation mode of H -. All the conclusions derived here from the vibration mixing rule completely agree with the results of normal vibration mapping discussed previously (cf. Table 3).

in an approximate sense for the transition state or any state along the IRC path. Appendix

( a )Equation 1 1 . By the action-reaction law under no external force, the sum of the x components of all internal forces should be zero for an A-B system, where molecules A and B consist of N A and NB atoms, respectively.

-.

This situation will not change for infinitesimal moveaxA,of atom A,*

Summing up over p,

Similarly,

Using eqs a-3 and a-4, we can transform the hiB,Ae for ~i~ in eq 7 as

This equation can be regarded as the product of (reduced mass)-l and the force constant for the interaction between molecules A and B. When N A = 1, eq a-6 turns out to be eq 11. ( b ) The Sign of the Second Derivative of the Potential Energy. Let us define the Cartesian displacement coordinates of the atoms A and B as X A and X B , the interatomic distance as r, and the equilibrium distance as ro. Since Ir - rol = IXB - x A ( ,

d2E -=-

a2

a2E - a2E axA2 ax;

--

Concluding Remarks We examined our two methods, i.e., vibration mixing and vibration mapping, for the prediction and interpretation of the TNV in atom molecule type reactions (H. H-H , He + H2C=CH2 , and H - oxirane -). Vibration mixing rules for the prediction of the TNV were presented based on the perturbation equations for vibration mixing. For the above three reactions, vibration mixing rules were helpful for the prediction of the TNV. Vibration mapping also gives sufficiently useful information for the reaction mechanism although it holds only

-+

+

+

At the transition state, aZElar2 becomes negative according to the definition of the transition state when r is the main component of the reaction coordinate, and hence

-+

a2E > o axAaxB

(b-3)

The change of the signs of a2Elar2and a2E/axAaxBoccurs at a certain distance r where the curvature of the potential function, e.g., the Morse curve, changes its sign from positive to negative.

Prediction of the Transition Normal Vibration Mode

The Journal of Physical Chemistry, Vol. 98, No. 28, 1994 6949

( c ) Equation 14. For the interaction between molecules A and B, ht:B defined in eq 13 can be written as

h2iB = Li:[HA,T

HB

- H",lL;B

(c- 1)

(c-2) The third term should be zero since@'&.I can be written as &A: through the eigenequation and the inner product TB, is zero due to their orthogonality. By using eq a-1,

g>

Thus,

where 4 stands for a Cartesian coordinate of the 3NB dimension space and m, is the mass of the atom associated with &. When NA = 1 and only dominant contributions, Le., the x components of the contributions from the atoms in the reaction site moiety of molecule B, are picked up, eq c-5 turns out to be eq 14.

References and Notes (1) Fuhi, K. J. Phys. Chem. 1970, 74, 4161. (2) (a) Bader, R.F. W. Mol. Phys. 1960,3, 137. (b) Bader, R. F. W. Cun. J. Chem. 1%2,40, 1164. (3) (a) Pearson, R.G. J . Am. Chem. Soc. 1%9,91,1252. (b) Pearson, R.G. Acc. Chem. Res. 1971,4,152. (c) Pearson. R.G. J. Am. Chem. Soc. 1972, 94, 8287. (4) Salem, L. Chem. Phys. k t t . 1969, 3,99. (5) Libit. L.; Hoffmann, R.J. Am. Chem. Soc. 1974,96, 1370. (6) Imamura, A.; Hirano, T. J . Am. Chem. Soc. 1975, 97,4192. (7) Inagaki, S.;Fujimoto, H.; Fukui, K. J . Am. Chem. Soc. 1976, 98, 4054. (8). Hirano, T.; Taketsup, T.; Kurita, Y. J . Phys. Chem.,previous paper in this issue. (9) Superscript T used hereafter indicates the transpose of a vector or matrix. (10) With no loa in generality, the phase of & can be arranged so as to be in bond-forming fashion, Le., < 0. If the phase of & is determined so 88 to give L: > 0, the s i p of the numerator h$ for the T N V bee" positive and%nce the following discussion concerning the vibration mixing coefficient will become the reverse in sign. (11) (a) Fuhi, K.; Yonezawa, T.; Shingu, H. J. Chem. Phys. 1%2,20, 722. (b) Fuhi, K.; Yonczawa, T.;Nagata, C.; Shinp, H. J . Chem. Phys. 1%4,22,1433. (c) Fuhi, K.; Yonezawa, T.; Nagata, C . Bull. Chem. Soc. Jpn. 1954,27,423. (d) Fukui, K.; Yonezawa, T.; Nagata, C . J . Chem. Phys. 1957, 27, 1247. (e) Fukui, K. Fortschr. Chem. Forsch. 1970, 15, 1. (f) Fuhi, K. Acc. Chem. Res. 1971, 4, 57. (g) Fukui, K.; Fujimoto, H. Bull. Chem. Soc. Jpn. 1%9,42, 3399. (12) Binkley,J.S.;Frisch,M. J.;Raghavachari,K.;DeFrcts,D.;Schlegel, H. B.;Whiteside, R.A.; Fluder, E.; Secger, R.;Pople, J. A. Guwsiun 82, Release H; Camegie-Mellon University: Pittsburgh, PA, 1982. (13) (a) The scaledown factor 0.89 was taken from the calibrationof the 6-31G.O resultsforH2Owithexperimentalfrequencies. Miyajima,T.;Kurita, Y.; Hirano, T. 1 . Phys. Chem. 1987,91,3954. (b) Hamada, Y.; Tanaka, N.; Sugawara,Y.; Hirakawa, A. Y.;Tsuboi, M.; Keto, S.;Momhma, K. J. Mol. Spectrosc. 1982, 96, 313. (c) Saeba, S.; Famell, L.; Rim, N. V.;Radom, L. J . Am. Chem. Soc. 1984,106,5047. (14) (a) Liu, B. J . Chem. Phys. 1973,58, 1925. (b) Siegbahn, P.; Liu, B. J. Chem. Phys. 1978,68, 2457. (15) Glasstone,S.;Laidler,K. J.;Eyring, H. The TheoryofRuteProcesseq McGraw-Hill: New York, 1964. (16) Symmetricand antisymmetricis referred to the q,plane of H2-HI. (17) Nagase, S.; Kem, C . W. J . Am. Chem. Soe. 1980,102,4513. (18) Mixing ratio of Th to Tb is determined by mass ratio of A and B fragmentsaccording to eq 7; in this case the mixing ratio is (fi)2:12(cf. the actual ratio from Table 1, = 1.412:12).