DISCUSSION

TOF MS

4.5 DISCUSSION

The results presented in the previous section show the utility of performing both temporally and spectrally resolved measurements of photodissociation reactions.

Even though both the temporal and spectral resolution is necessarily compromised, the chosen characteristics are most appropriate for doing spectroscopy of incipient reaction fragments. This discussion section addresses the physical origin of the previously described experimental results.

The discussion will focus on considering three interrelated interpretations of the temporally and spectrally observable effects. An explanation will be provided for the transient (pulsed) feature, which manifests in measurement of the dynamics of OH formation. Any explanation will have to account for both the red and blue-wing absorption features. Finally, the observation that the magnitude of the perturbed fragment linewidth decreases with increasing rotational angular momentum must also be explained.

The most straightforward view of the dynamics of photodissociation results from the understanding that the state vector evolution is driven primarily by the sudden system perturbation. The drastic perturbation arises from the photoinduced electron promotion and the associated change of the electronic PES from being strongly attractive to being purely repulsive. Bersohn and Zewail23^a have described the temporal evolution of the incipient reaction product absorption spectrum fol- lowing ( or convolved with) photoexci tation. Their model treats the dynamics in a classical fashion. The relevant generic repulsive surfaces may be described by23^a

(5.1)

and

V2(R)

=

V20 exp (-R/12)

+ nw 2

^(5.2)

where

nw';f°

is the transition energy to a specified final state for the asymptotically separated (free) fragments. The probe pulse of a given frequency, nwprobe, will be tuned into resonance with an excited state of the S2 ^+- S1 transition, for example.

The perturbed fragment absorption occurs at given internuclear ( reaction coordi- nate) separations R~R* when 1iw

1

^{x < nwprobe <}

nw2 .

This is the case for pulsed excitation with photon energy 1iw

1

^x from an initial bound ground state. In the case where the upper repulsive potential is flat, that is, 1 2 - oo, the expression for the time dependence absorption spectrum becomes23a

where a Lorentzian of half width _I describes the probe pulse energy distribution.

This expression does not include the convolution with the pump-probe system re- sponse function, which is, however, presented in Ref. 23a.

In the present case, the value of E, which is the terminal kinetic energy, is much larger than I and , 2 (i.e., E=nwjx - Do

=

2 ^X 10⁴cm-¹ vs. , 2 '.:::'. 12.5cm-^{1 ).} ff the assumption of 1 2 - oo is applied to the interpretation of the present experiment,

then the amount of potential energy remaining is only of the order less than 20cm-¹.

The authors of Ref. 23a have plotted Eqn. ( 5.3) for a series of values of the fractional portion of the remaining energy ( compared to the initial potential energy). They observe curves similar to Figures 2b, 2c, and 2d when the remaining fraction of the initial potential energy is 0.01, 0.05, and 2::0.10, respectively. Clearly, there is discrepancy between the present measurements of the spectral and temporal dynamics and the calculation conditions necessary to qualitatively simulate the temporal behavior.

The notion that the present experimental observations do not originate sim- ply from the exponential spreading of the V 1 and V 2 potential surfaces may be reinforced. It may be noted that the V ₁ surface for HOOH, ignoring the torsional interactions, has been reported⁴² as

V ₁ = 705e V · exp ( -R/0.266A) (5.4)

Values of t* for Eqn. (5.3) as large as 0.1-lps and OH asymptotic fragment ve- locities in excess of v=3.5A/ps⁴⁰ result in a value of sech² ( vt* /21_{1 )} to be small compared to unity. These conditions imply that the present picosecond probing process is sampling the near asymptotic region of the potential surface. This idea is consistent with the small spectral detuning of the experiments. However, the large value expected for vt/21 (i.e., >5) implies that the observed dynamical evo- lution of the time-resolved signal does not obtain from the exponential separation of the two potential surfaces connected via probe excitation. See Figure 2 of Ref.

23a for further information about the timescales necessary to measure the repulsive dynamical evolution.

The method of Ref. 23a also cannot explain the presence of blue-shifted wing absorption observed in the present study. The model has been extended, however, to give an accounting of a physical basis for the blue absorption. ^23b Even this extended model does not account for the dependence of the width of the perturh.ed spectral line on the degree of rotational excitation.

Khundkar et al.⁴⁷ have formulated a more inclusive theory, which describes the temporal dynamics of photodissociation and extend the description to more explcitly treat spectral domain data. In certain cases, this model reduces to the model of Ref. (23a). The results of Ref. 45 demonstrates the effect the spectral and temporal widths of the pulses ( especially the probe) have on the observable quantities of a pump-probe experiment. Figure 7B.4 of Ref. 47 shows the wing absorption expected from the convolution of Eqn. (5.3) with the system response function by integrating from -oo to oo. As previously stated, the wing is only red-shifted in time. No blue-shifted wing is obtained. The main result of Ref. 47 is an expression for ( a simulation of) the measured spectrum

(5.5)

where ~ is the energy of detuning from resonance in the interval (O,E), and o-s is the spectral width of the probe pulse in units of energy. The temporal width of the response function, O-t, is given in units of v/2L1. Eqn. (5.5) must be convoluted with the probe bandwidth spectral response function for direct comparison with the experiment.

A crucial assumption for the derivation of Eqn. (5.5) is that the absorption at a given spectrally defined energy is proportional to the amount of time the system spends in that spectral/ coordinate region. It is noted in Ref. ( 4 7) that expression (5.5) predicts a tail absorption feature that diminishes in amplitude as the recipro- cal of the energy detuning near the resonance. The difference spectra, obtained by way of this formalism, do show red and blue shifted spectral wings. Even though the qualitative aspects of Figures 6c, 6d, and 7 c may be explained by this formal- ism, quantitative differences remain. Firstly, the amplitude of observed perturbed spectral features in Figure 8 is 10-20% of the on resonance peak amplitude. By contrast, the simulated result via Eqn. (5.5) shows only a 1-2% spectrally per- turbed amplitude. Secondly, the measured ratio of the red to blue wing'amplitudes is about 3:1 whereas the amplitude ratio in the simulations is about half of that.

Finally, the experimentally observed red and blue wings appear to have about the same spectral width, whereas the simulations show that the red-wing is about an order of magnitude broader than the blue wing.

The discussion up to this point has concluded that the spectrally perturbed lineshapes and perturbed fragment induced transient (absorption) features originate from dynamics occuring in the near asymptotic region of the repulsive ¹ Au PES.

It also appears that exponential separation of the two electronic surfaces connected by the probe beam cannot alone explain the effects described in the previous figures of this chapter. A more intuitively satisfying explanation of the dynamical features may be given by considering the effects that long-range, weak ( e.g., dispersive or electrostatic) interactions may have on the temporal and spectral data.

Buckingham⁴⁸ has described the long-range van der Waal interactions⁴⁹ of two dipolar fragments. The expression for the radial and angular potential interaction energy V(R,0) for two such fragments is given by⁴⁸

V(R,0)

=

^{µ1µ2R-3 •} (2 cos 01 cos 02

+

sin 01 sin 82 cos</>)

+ ~

^{µ1 02R-4} [cos 01 (3 cos2 81 - 1)

+

^{2 sin 8}1 sin 82 cos 02 cos¢>]

+ [ ~

^0102R-5

^{(1 -}

^{5 cos2 81 - 5cos2 82}

⁺

^{17 cos2 01 cos2 02}

+

2 sin² 01 sin² 02 cos² ¢,

+

16 sin 81 cos 81 sin 02 cos ⁸² cos¢,)]

+

higher order terms. (5.6)

Here, ¢>

=

</>1

+

^¢>2 is the rotation about the 0-0 internuclear axis. The value of 01 is specified as the angle each OH dipole makes with respect to the 0-0 axis, namely, the OOH bend angle.

The primary mechanism of rotational excitation in photofragmentation was postulated by Klee et al.³⁶¹⁴⁰; the fragment rotational angular momentum arises from the torsional dependence of the ¹ Au surf ace. This conclusion will lead to significantly different values for the multipolar interaction potential than for the

•--

orthogonal projection of the angular momentum resulting from bending motion and

an orthogonal torque applied during the HOOH photodissociation. It is expected that the degree of P(N _{1 ,} N _{2 )} fragment correlation observed will allow for a given OH (N1 =1) fragment to effectively sample different orientational potentials than

An ab-initio concern is that off-resonance wing absorption due to the probabil- ity of the targeted fragment sampling different configurations may be masked by the wing absorption. The latter arises from the simple exponential separation of the rel- evant potential surfaces. It is expected, however, that the long red-wing absorption resulting from this later process will have a smaller transition moment than absorp- tion which originates in the nearly asymptotic region of the ¹ Au _, ²II potential.

This intuitive conclusion arises from the large difference between the magnitude of the transition moments for HOOH ¹ Au .- ¹ A₉ excitation (

~

^{10-20cm2).50 The}

fragments in the near asymptotic region are expected to more closely resemble free OH than HOOH. At closer internuclear separations the incipient fragments more closely resemble HOOH than OH. In addition, the fragments at smaller internuclear separations exist in a spectroscopically selected region for a shorter period of time than in the asymptotic region, so the wing absorption strength will also be reduced on this basis.

The analysis of the long range interactions may be undertaken by assuming that the dissociation process yields two groups of products. The first group arises from the torsion excitation and it will be assumed that those hydroxyl radicals that obtain angular momentum from this mechanism do so with a maximal alignment of JOH with

v.

The second type of fragment interactions will be for the extreme case of .loH 1-

v.

Gericke et al.⁴⁰ have considered the three limiting orientations of .JoH relative to ilHOOH and 'VQH. In their coordinate system the 0-0 recoil axis is

x

and the orientation of iJHOOH is orthogonal to

x

^along

z.

The first limiting orientation is .JoH 1- {µHOOH, voH}, and is therefore along

y.

Second, .loH

II

ilHoOH thus along

z.

Third, .loH

II

voH and along

x.

From experimental evaluation of .• the ,8~(22) (cf . .loH

llv)

and ,8~(02) (cf. the second correlation) and ,8~(20) (cf. theµ

II v

correlation) they conclude⁴⁰ that the expectation value of the square of the angular momentum components are

(Jx ²⁾

~

^{0.61 J(J}

⁺

¹⁾

(Jy 2)

~

⁰

(Jz ²⁾

~

^0.39J(J

⁺

¹⁾

(5.7a)

(5.7b) (5.7c) In a separate calculation they find that the energy for rotational motion parallel, E11, and perpendicualr, E..1-, to

z

is

( E11)

= (

450

±

25)cm-¹ (5.8a)

and

( E..1-)

=

⁽⁶¹⁵

±

25)cm-¹. (5.8b) The wave function of a rigid rotor is characterized by the total angular momentum

J

and its projection M on the space-fixed axis ( e.g. z) of quantization, given in representation

I

JM ). The explcit form is

The probability for finding the rotor oriented along the solid angle element dD.

=

sin 0 d0 d<j) for state

I

J M ) is

The two quantization conditions of interest, along

x

^and

z

respectively are for Y ~ and Y ~ and the probability weighting given by Eqns. ( 5. 7). Therefore the present assumption of quantization about these two axes may be done with some justifica- tion.

Margenau⁴⁹ has described a method for the calculation of the resonance ener- gies for rigid linear dipoles and quadrupoles. The two OH incipient fragments are in states characterized by the quantum number j ₁ m₁ and

h

m 2. The state function that represents the two species interacting in their unperturbed condition is the product of spherical harmonics ( or Legendre polynomials )⁴⁹, namely,

'/Pk= ip(j1 m1, h m2) =P(j1 m1, cos01) · P(h m2, cos02)

· exp i( m1 c/>1 + m2 c/>2) (5.9)

The degeneracy of the state vector implies that the resonance values are obtained by solution of the determinant⁴⁷

(5.10)

However, all of the matrix elements of ^Vij are zero so there is no first order effect.

The nonzero second order effect is of the order of ^{I Vij 1}2 and will not be considered here. The calculation of the quadrupole-quadrupole interaction involves solving the quantity 01 md2 m2

I

Vee

I

j1 md2 m2), and this will be done below.

The symmetry of the second and third terms ( dipole-quadrupole) of Eqn. (5.6) may also eliminate those terms from consideration. Coordinate inversion of ( 0i, cf>i) gives ( ^{1r -} 0i, ^{1r -}

cpi).

The coordinate inversion changes the sign of the second and third terms of Eqn. ( 5.6) ( as well as the first term as implicitly noted in the preceeding paragraph). The integration of ^Vij over all space yields zero values for these two terms since the expressions are even functions. The fourth term for Vee is non-zero. The quadrupole term of Eqn. (5.6) may be rewritten in terms of spherical harmonics expressions.51 •52 In this formulation cos2 0i =

½ [

✓ 161r/5 Y~(i)+l], and sin2 0i =

-½ [

✓ l61r /5 Y~ (i) - 2], where Y{n(i) = Y{n(0i, c/>i) and i=l,2. Therefore

the quadrupole interaction term becomes

5{[ ^~

^{2 ] [}

^~

^{2 ] }}

Vee= 1-

3 y ₅ -

5^{- Y0}(1) -1

+ y ₅ -

5^{- Y0}(2)

+

1

+ ~ 7

[ ff- ^{Y)(l) +} 1] [ ff-Yi(2) + 1]

+ ¼[2- {Sf-YWJ] [2- J16.-/5Y)(2J]

+ ~; {

(Yi(l)

+

Y:₂(1)] (Yi(2)

+

^Y:2]

+

(Yi(l) - Y:₂(1)] [Yi(2) - Y:₂(2)]}

+

³

!1r {

^[Y:1(1) - Yi(l)) [Y:₁(2) - Yi(2))

+

[Y:₁ (1)

+

Yi(l )] [Y:₁ (2)

+

Yi(2))} (5.11) The above equation appears more formidible than the quadrupole term of Eqn. (5.6) but this form becomes easier to manipulate and evaluate. The spherical harmonic addition theorem⁵¹ may be used to evaluate the spatial integrals of the form

fa )

{5.12)

m3

It is known that the normalized eigenfunction Yj ^m has the property

(5.13)

This property allows the evaluation of a form of Eqn. (5.12), which is useful for solution of the above potential interaction. In particular, for the quantization m j

{27r f

^7r

Jo Jo Yjj * Y2m Yjj sin0d0dcp

=

(-1

Y 1

²

1r11r

Yj -j Y 2 m Yjj sin 0d0 dcp

=

-1. [5(2j+1)2]1/2.

(j

^{2 j) ( j_ 2}

( Y

^{41r O O O -J m (5.14)}

The relevant formulation of the Wigner 3-j symbols gives⁵¹

( j 2 j )

= ( _

^l

y

^{+ 1 . 2j (j}

+

1)

O O O [(2j + 3)(2j + 2)(2j + 1) 2j(2j - 1)]112 (5.15) and

( j 2 j ) ~ 2 { ^{n • 2} j(j

+

1) 1 ( 16.

\ - j m j =Dmo· JJ - [(2j+3)(2j+2)(2j+1)2j(2j-1)] 112

J

^{5· )}

Combining these expressions gives a simplified form for Eqn. (5.14)

r27r f

^7r

Jo Jo Yjj *Y2m Yjj sin0d0dcp

=

^(Yjm)

.

=

8 ^{0 •} (-1) . _ J -

m 2j

+

3· (5.17)

This expression is evaluated twice to obtain the full spatially arranged potential interaction energy for the two hydroxyl radicals,

(5.18)

Evaluation requires substitution of the OH ground or excited quadrupole moments, the relevant rotational angular momentum quantum numbers.

Before proceeding with evaluation it should be made clear that the present method of obtaining the quantity

has been adapted from the analogous problem of calculating the linewidths of pres- sure broadened spectral lines as developed by Anderson.⁵³ A review of the subject⁵⁴ has clarified the treatment of Ref. (53). The subject of collision induced broad- ening is closely related to the present study of perturbed fragment spectra follow- ing photodissociation. The subject of collision induced absorption has seen con- siderable activity in far infrared gas phase absorption⁵⁵ and in transiently induced polarizabilities in liquids. ⁵⁶

The dipole and quadrupole moments of ²II and ²E OH have been evaluated by way of ab-initio CI methods. ⁵⁷ The same quantities as well as microwave measure- ments of the rotational levels for the ²II 5sa,b and ²E ssb states have been measured⁵⁸b

and tabulated. ssb The quadrupole moments at the OH equilibrium internuclear sep- aration of

Re =

l.84a0 57 are 0

=

1.82 x 10-²⁶ esu • cm² for the ²II state and 0 = 4.05 x 10-²⁶ esu · cm² for the ²E level.

The transition energy for a given transition is obtained by evaluating Eqn.

(5.18) for the two potential surfaces and for the appropriate values of j. For the case of a transition where 6j=0 and 6m=0, the expression for the interaction energy difference is

6Epert

(5.19)

Some values of this function are presented in Table 2. The most notable trend is the positive value of the (net) interaction energy. This, in turn, means that the associ- ated absorption spectral feature is shifted to higher energy. The blue-shift is seen to decrease with decreasing values of j _{1 (}

=

^N 1 ). This is a trend that would be observed

as a decrease in the width of the blue wing in the experimentally measured spectral (t=O) scans. The average correlated j2 value,

U

2), is obtained from Ref. (36).

The torsional excitation would lead to the condition

h = h-

The experimentally observed perturbed spectral features, especially the blue-wing absorption, is seen to be very similar to the trend expected from product rotational excitation, which results from a torsional dependance for the upper potential surface. The agreement also appears to be more than just qualitative.

As indicated in the above discussion, a significant fraction ( approximately 0.4) of the total rotational energy is associated with the OH fragments wherein

JoH l..

v.

^36•40 It is useful to determine the type of perturbed energy which would arise from these alternative dissociation mechanisms and OH-OH fragment interactions. The method of the calculation will be similar to that presented above. The quantity to be evaluated in the interaction energy is given by

(5.20)

The measurements of Gericke et al.³⁵ show that the expected correlation are j₁

<

_{j 2}

for small j1 and j₁

>

j2 for j₁ 2:: 7. It is also expected that the relative alignment of fragments 1 and 2 will be orthogonal. Furthermore, it will be assumed that fragment 1 will have m1 =0 while for fragment 2 _{m 2 =}

h-

The utilization of the spherical harmonic addition theorem and implicitly the Wigner-Eckart theorem and the proper definitions for the 3-j matrices will yield the single particle orientational distribution. This is given by

(5.21)

This expression may be used in the expression for the quadrupole interaction to evaluate the total interaction energy. After some algebra, one obtains

(ji Oj2 m2

IV I

j1 Oj2 m2}

0 0 R-s. {-64j{j2 - 64jij2

+ 30h +

8j{

+

8j 1

+ 39}

1 2

[ 3(2j1

+

3)(2j1 - 1 )(2h

+

3)]

0102 R-^{5 •} E12 (5.22)

The perturbation energy for .6.j=O and .6.m=O transitions is given by [ 0102(2~) - 01 02(2II)] · R-^{5 •} E12

- (2.0 x 10⁴ cm-¹

A.

⁵) • R-⁵ · E_{12 .} (5.23)

Table 3 contains the calculated value of this expression for j 1, correlated average j2

=

^(j2) and 4.A.,

5A

and 6.A. values for the 0-0 separation. The terms in Table 3 are negative in value, meaning that they will contribute to a red-shifted wing absorption in contrast to the dynamics that produced the data of Table 2.

Also, it is seen that the magnitude of the interaction energy decreases.

The polarization data and anisotropies observed in Figures 8-10 should also be explained. To recall, the anisotropy of the perturbed spectral feature associated with the R2(3) transition is negative during the pump-probe pulse overlap. By contrast, the anisotropy associated with the Q1 (2) transition decayed fron a positive value during the pump-probe pverlap. Different A-doublet levels are probed by the R and Q-branch transitions. The Q-branch probes those transitions for which the lone electron p-orbital is in the 0-H plane of rotation (A-): JoH j_

µQ,

The R-branch transitions then probe .loH

II µQ.

The torsional mechanism for the dissociation is associated with the .loH

II v.

correlation. Since

v

^{j_ µHOOH} for this mechanism then the Q-branch transitions would monitor the correlation ^µHoOH

II

^µQ, The R-branch transitions would monitor the µHOOH .l_ µR correlation.

Hence, if the torsional mechanism is the primary cause of the blue-shifted per- ....

turbed spectral lineshape then it may be naively assumed that the anisotropy would

reflect a positive correlation for the Q-branch transitions and an anti-correlation for the R-branch transitions. This is indeed observed in Figures 8-10. Because the asymptotic values for the parallel and perpendicular data are made equal the anisotropy decays to zero. In actuality, the magnitude of the difference for the data sets would yield the previously measured asymptotic alignment.