Theoretical study on bond-selective reaction of HOD with H atom*

Xu Hong, Xu Xianzhong, Li Zonghe
(Department of Chemistry, Beijing Normal University, Beijing, 100875, China)

Received May 2, 1999; Supported by the National Natural Science Foundation of China.
*The Chinese version of this paper was published in Chemistry Online (Huaxue Tongbao), 1999, (13): 62.

Abstract The bond-selective reaction HOD and H has been studied with the theory of dynamical properties on reaction path (IRC) in molecular internal coordinates constructed by authors. Compared with the reaction HOH and H that has no bond selectivity and the reaction H₂(u=0,1) and OH that has mode selectivity, w_K (frequencies orthogonal to IRC) and B_KL (coupling constants between breaking bond and the other vibrations) of every reaction in internal coordinates have been obtained. In order to reach the object of bond-selective reaction at certain electron state, the theoretic demands including the vibration frequencies w_K of breaking bond and the coupling constants B_KL between breaking bond and the vibrations of other bond (bond angle) in the molecules have been obtained, based on the analysis of bond selectivity on each reaction.
Keywords bond-selective reaction, molecular internal coordinates, HOD.

Changing productive rate, controlling the process of chemical reaction and machining molecules at will are goals of chemists. In many chemical reactions, these goals are partly realized by changing outer conditions (such as temperature and pressure) or finding the right catalyzer through iterative experiments. But all of these can only interfere the reaction superficially and can not reach the goal to clip the molecule following our inclinations, to synthesize the material we want.
    The goal of bond-selective reaction is to machine molecule selectively. Vibration excited mode is the only way of molecular dissociation and many chemical reactions. To reach the goal of bond-selective reaction, it is necessary to study the molecular vibrational excited mode. For the life of local-mode vibrational eigenstates is too short to observe easily, it results in difficulties in the progress of bond-selective reaction.
    The reaction between OH and vibrationally excited H₂, OH+H₂(u=1) H₂O+H, was studied experimentally at 298 K by R. Zellner and W. Steinert in 1981^[1]. They found that the rate constant for the reaction was enhanced a factor of 2 when molecule H₂ excited to vibrational mode v=1. A comparison with the rate constant k (u=0) for the ground-state reaction of reaction OH+H₂(u=1) H₂O+H R(1) leads to

It shows that the reaction has mode selectivity.
    In 90's, Zare et al. studied the reaction of HOD and H experimentally^[2,3]. The reaction has two main passages: one is HOD+HHO+DH R(2.a),the other is DOH+H DO+HH R(2.b). They found that the OD:OH ratio is 1.38+0.14 for the reaction H+HOD(0,0,0) (ground-state HOD). In contrast, this ratio is greater than 25:1 (enhanced the product branching ratios OD/OH more than 18 times)for H+HOD(0,0,1) and less than 1:5.8 (enhanced the product branching ratios OH/OD more than 8 times) for H+HOD(1,0,0) . Here (0,0,1) and (1,0,0) denote the O-H and O-D stretch fundamentals respectively. In 1993, Crim et al. studied the reaction of HOD and H in high excited state experimentally^[4]. The experiment demonstrated that excitation of the third O-H stretching overtone, 4u_OH, favors breaking the O-H bond with product branching ratio OD:OH>100 and excitation of the fourth O-D stretching overtone, 5u_OD, favors breaking the O-D bond with main product OH. All of these indicated that the reaction for HOD and H has bond selectivity.
    A quasiclassical trajectory study of the state-to-state dynamics of H+H₂O OH+H₂ had been done by George C.Schatz et al. in 1984^[5]. The work include a detailed study of reagent normal-mode and local-mode excitation effects, and of product state energy partitioning. Their conclusion is that if energy transfer between vibrational modes and between vibration and rotation were neglected, the reaction rate constant of the bond breaking enhanced rapidly when certain of the local-mode is excited. In 1992, D.C.Clary et al. reported the calculations of vibrationally state-selective cross section for the H+HOD H₂+OD reaction with a quantum-mechanical method^[6]. Desheng Wang et al. presented approximate quantum calculations of cumulative reaction probabilities, rotationally averaged cross sections, and branching ratios for the reactions H+H₂OOH+H₂ and        using the Walch-Dunning-Schatz-Elgersma potential^[7].
    The mass-weighted Cartesians coordinates are used to deal with vibrations in all of the calculations above. Normal-modes of any point along reaction path are expressed in mass-weighted Cartesians coordinates. The physical diagrams of normal modes of polyatomic molecules reaction in mass-weighted Cartesians coordinates are complex, not clear and not easy to understand, which brings limitations to discuss bond-selective reaction. In the calculations above, all of these chemists did not consider the reassignment of excited energy and did not compare the results with experimental data actual. It can not illustrate the situations about transfer of excited bond's energy between vibrational modes in reaction progress.
    To illustrate bond-selective reaction, a set of reaction path theory in internal coordinates is constructed by authors^[8]. We'll use the theory to discuss the selectivity of reaction HOD with H atom.

1 THE THEORY OF REACTION PATH IN MOLECULAR INTERNAL COORDINATES
In 1970, Fukui^[9] defined that the intrinsic reaction coordinates (IRC) is a spatial curve (R(s)) along the perpendicular line of the potential face's tangential plane which connects the points of reactants, transitional state and products. The reaction coordinates got by this method have no relation with the selection of coordinate systems. It has the properties of short ground return^[10] which is constant in the change of internal coordinates.
    The IRC equation in internal coordinates can be expressed as^[11]:

where g denotes the gradient vector with ,
R is station vector in internal coordinates, W is station energy.G is represented with elements:
.

,  where

is atom a's displacement vector.

    The physical meaning of the vector   is as follows: Keep all atoms except atom a in their equilibrium positions.   is in the direction along which a given displacement of atom a will produce the greatest increase of .  The magnitude of   is equal to the increase in   produced by unit displacement of the atom in this most effective direction.
    N is a suitable normalization factor; it can be written as:,
    According to IRC equation, the components of motion vector along IRC (L_F) in internal coordinates are determined as:
,
where R_L is No L internal coordinate of the system, W is the potential of the system, s is intrinsic reaction coordinate.
    From the IRC equation in molecular internal coordinates, vibrational analysis of the various points along IRC can be done and the energy gradient and force constant matrix can be obtained. To get the vibrational mode orthogonal to IRC, we define operator P:
,
where P is a 3N-6 dimensional phalanx.
    We define the operator as: . I is a 3N-6 dimension unit matrix.
    When is used to act on force constant matrix in internal coordinates the projected force constant matrix is got:
(4),
    With GF equation, it can be obtained^[12]:
.
    Solving the equation (5) can get the vibrational frequency w _K and vibrational vector orthogonal to IRC in internal coordinates.
    In terms of differential geometry principles, major perpendicular vector of IRC curve R(s) defined by Fukui is the differential vector of R(s)'s tangential line, which is the differential vector of the motion vector along IRC:

,
here

The curvature of IRC is:.
    So the projected curvature B_KF of each vibrational mode L_K in internal coordinates can be easily expressed as:
.       (7)
    Apparently, B_KF can also be regarded as the coupling constant between the motion vector along IRC and No K vibrational vector orthogonal to IRC in internal coordinates.Similarly, B_KL is defined as the coupling constant between each two vibrational vectors orthogonal to IRC:in which, K,L=1,2,...,3N-7. The finite difference approximation is applied to the solving of the equation of B_KL. Finally, we obtained:
   (8)
    So far ,the expressions of all dynamical properties (L_F,L_K,w _K,B_KF,B_KL) on reaction path in internal coordinates have all been obtained, and we will use them to study the bond-selective reaction.

2 CALCULATIONS AND DISCUSSION
To study the bond-selective reaction    R(2), we studied the dynamical properties of typical mode-selective reaction OH+H₂(u=0,1) HOH+H R(1) and the reaction has no bond selectivity H₂O+H HO+HH R(3) in molecular internal coordinates at the same time. The molecular internal coordinate system of these reactions is defined as figure 1:
Fig.1 The molecular internal coordinate system of three reactions R(1),R(2) and R(3)

where r₁₂, r₂₃, r₃₄ are bond lengths, a₁, a₂ are bond angles, B is dihedral angle (planes 123-234). The molecular internal coordinates are:

where R₁,R₂,R₃ are internal coordinates of the unreacted bond, the breaking bond and the forming bond respectively, R₄,R₅ are internal coordinates of bond angles in reaction and R₆ is internal coordinate of dihedral angle.
    Based on BERNY energy gradient method (at UHF/6-31G** level), the geometries of reactants, transition state and products in the reaction were optimized, and the vibrational analysis of the transition state has been done, using computer programs Gaussian 94.Then starting off from the transition state with GAMESS, based on MOROKUMA numerical analysis method, the intrinsic reaction coordinates (IRC) is obtained. Later, according to the IRC equation based on the theory of reaction path in internal coordinates, L_F (motion vector along IRC), P (projection operator), F^P (projected force constant matrix) can all be gained. Both vibrational vectors and vibrational frequencies orthogonal to IRC in internal coordinates are obtained by solving the eigenequation . The coupling constants B_KL are calculated by using the equation (8).

Fig. 2(a),(b),(c) give the change of harmonic frequencies along reaction coordinates in molecular internal coordinates of the three reactions R(3\1) ,R(2.a) ,R(2.b). Here Q₁ is the change of stretch vibration of unreacted bond (OH or OD); Q₂ is the change of stretch vibration of reacting bond (OH or OD); Q₃ is the change of stretch vibration of forming bond (HH or DH); Q₄ is the change of bend vibration of ₁(HOH or HOD);Q₅ is the change of bend vibration of₂(OH(D)H); Q₆ is the change of out-of-plane (B) vibration of the reaction system..

    Figure 2 shows that the frequencies of vibrational mode Q₁ change little in these reactions. It indicates that Q₁ hardly take part in reactions. The frequencies of Q₂ and Q₃ change greatly in the process of the reaction. It reflects the breaking process of the reacting bond and the forming process of the bond HH(D). The change of Q₄ shows the disappearing process of ₁(HOH or HOD). Q₅ and Q₆ represent the forming and disappearing process of ₂( OH(D)H ) and the reaction plane (B) respectively. Apparently all of these are reasonable.
    Seen from reverse direction, the Figure 2 (a) shows the change of vibrational mode of reaction OH+H₂ HOH+H. It is reasonable apparently too.
    Observing Figure 2 . (a),(b),(c) we can find that the obvious difference among the bond-selective reactions R(2) and R(3) with no bond selectivity is the margin between vibrational frequencies of unreacted bond and reacting bond of reaction. The margin of R(2.a) and R(2.b) is nearly 1000cm^-1, while the vibrational frequencies of R(3)'s unreacted bond and reacting bond are almost equal (the margin is only about 100 cm^-1). The vibrational frequencies margin between bond OH and OD is big, thus results in that the respective excited state can be made easily in the reaction of R(2.a) and R(2.b). However, in the reaction R(3), to excite one predetermined bond OH by laser is notoriously difficult, it brings difficulties to the preparation of local-mode high vibrationally excited eigenstates.

(a)

(b)

(c)

Fig.3 (a) Plots of coupling constants between vibrational mode of breaking bond orthogonal to IRC and others vibrational modes BKL vs. reaction coordinate s of reaction R(3\1) (HOH+H HO+HH) (b) Plots of coupling constants between vibrational mode of breaking bond orthogonal to IRC and others vibrational modes BKL vs. reaction coordinate s of reaction R(2.a)(HOD+H HO+DH) (c) Plots of coupling constants between vibrational mode of breaking bond orthogonal to IRC and others vibrational modes BKL vs. reaction coordinate s of reaction R(2.b)(DOH+H DO+HH)

    Figure 3. (a),(b),(c) give the mutative curve of coupling constants B_KL between breaking bond and the inner molecular bond (bond angle) or the breaking bond and intermolecular (interatomic) bond (bond angle) of the reaction in molecular internal coordinates. Here B₁₂ is the coupling between internal coordinates R1 and R2, representing unreacted bond (D)H¹O²'s stretch vibration and reacting bond O²H(D)³'s stretch vibration respectively; B₂₃ is the coupling constants between internal coordinates R2 and R3, representing the stretch vibration of the breaking bond O²H(D)³ and the stretch vibration of forming bond H(D)³H⁴; B₂₄ is the coupling constants between internal coordinates R2 and R4, representing the stretch vibration of breaking bond O²H(D)³ and the bend vibration of bond angle ₁; B₂₅ is the coupling constants between internal coordinates R2 and R5, representing the stretch vibration of breaking bond O²H(D)³ and the bend vibration of bond angle ₂. The coupling constants B₁₂ and B₂₄ are inner molecular coupling constants and B₂₃,B₂₅ are intermolecular coupling constants.
    In these two kinds of coupling, the greater the coupling between breaking bond and inner molecular bond (bond angle) is, the more it does harm to the reaction; the greater the coupling between breaking bond and intermolecular (interatomic) bond (bond angle) of the reaction is, the more it is conducive to the process of the reaction. The exist of the inner molecular coupling results in that the excited energy transfer to other bonds, not retaining on the excited bond solely, especially when laser is used to excite the breaking bond. The greater the coupling is, the more quickly the energy transfer.
    Fig.3 (a),(b),(c) show that reactions R(2.a) ,R(2.b) and R(3) all have the coupling between breaking bond and inner molecular bond (bond angle), but there is no coupling between breaking bond H-H (which is the molecule itself) and other inner molecular bond (bond angle) in reaction HO+HH HOH+H R(1). The reaction rate will be enhanced 100 times if we excite H₂ to one quantum of vibration u=1 in reaction R(1)^[1], but the reaction R(2) can not be enhanced equally. The product branching ratio of R(2) is enhanced no more than 18 times^[2,3]. In the local-mode reaction at high excited vibrational eigenstates, the exist of the coupling between inner molecular vibrational modes made the action of energy transfer easier. Consequently, the product branching ratio can not be enhanced more than one hundred times by excitation of OH into 4 quanta of vibration u=4_OH or excitation of OD into 5 quanta of vibration u=5_OD^[4].
    The inner molecular coupling divided into two kinds: one is the coupling between breaking bond and inner molecular bond angle such as B₂₄, the other is the coupling between breaking bond and unreacted bond such as B₁₂. Classic transition state theory indicates that the reaction will go on to the products once the transition state is overcome. It can be seen clearly from Fig.3 that B₂₄ are almost same before transition states of the former three reactions, but B₁₂ of them are different greatly. The maximum absolute value of coupling constant B₁₂ is more than 2.00 in the reaction HOH+H HO+HH R(3), no more than 0.78 in the reaction HOD+H HO+DH R(2.a), is only 0.58 in the reaction DOH+H DO+HH R(2.b). According to these, the order of coupling constants that affect bond selectivity of three reactions is B₁₂ R(3) > B₁₂ R(2.a) >B₁₂ R(2.b). It can be deduced that energy transfers quickly to bond H¹O² when bond O²H³ is excited in reaction R(3). Consequently, the energy of reacting bond O²H³ fall down, the reaction has no bond selectivity. When bond OD or bond OH of HOD is excited, the energy transfers slowly for the coupling is little in reaction R(2). So the reaction has bond selectivity. These are consistent with the experimentally qualitative analysis of Bronikowski^[2,3] and Zellner^[4].

3 CONCLUSION
The bond-selective reaction of HOD with H atom has been discussed in this article with the theory of dynamical properties on reaction path (IRC) in molecular internal coordinates constructed by authors. It indicates that in order to break a predetermined bond (bond selective reaction) between an atom or atomic group of certain molecular on one end of predetermined bond and another molecule, atomic or atomic group, to reach the goal of bond-selective by excitation of the chosen bond with laser, these are required on theory:
(1) The bigger the margin between the predetermined bond's vibrational frequency of certain molecule and the ambient bond's (bond angle's) vibrational frequency is, the more benefic it is to the preparation of the bond's local-mode vibrationally excited eigenstates exactly. The margin would be better more than 1000 cm^-1.
(2) The smaller the coupling between the predetermined bond and the other inner molecular bond (bond angle) of the molecule is, the more selective the reaction is, especially the coupling between the predetermined bond and the other inner molecular bond. It would be better that the coupling constant is less than 0.7.

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