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Enhancement of reactivity through near-resonant radiative excitation of an overtone: model study of an isomerization by hydrogen tunneling
399
Chemical Physics 115 (1987) 399404 North-Holland, Amsterdam
ENHANCEMENT OF REACTIVITY THROUGH NEAR-RESONANT RADIATIVE EXCITATION OF AN OVERTONE: MODEL STUDY OF AN ISOMERIZATION BY HYDROGEN TUNNELING Alessandro LAM1 Istituto di Chimica Quantistica ed Energetica Molecolare de1 CNR, via Risorgimento
35, 56100 Piss, Italy
Received 27 February 1987
The enolization of 2-methyl acetophenone derivatives through hydrogen tunneling is simply modeled to study the possibility of increasing the reaction yield by exciting the v = 7 C-H overtone. Vibrational relaxation is assumed to occur via Fermi resonances transforming one C-H quantum into two quanta of the H bending. The role of the quasi-continuum of C-C modes is taken into account via an effective hamiltonian. It is shown that the only way to increase significantly the reaction yield is using light detuned a few hundred cm -r from the overtone level, as suggested by Tannor et al.
1. Introduction In the past years there has been growing experimental and theoretical interest in excitation of high-energy localized vibrational states by radiation absorption. The overtones of A-H stretching (A = C, 0, N) have been extensively studied experimentally [l-5] and successfully interpreted by means of the local mode picture [6-91, originally proposed by Henry and Siebrand [6]. The radiationless decay of such states, which is the main source of the broadening of spectral lines, has been interpreted in terms of coupling (essentially kinetic) of the initial state to trees of other near-degenerate states [8,9]. For example it seems quite well established that the overtones of the C-H stretching decay by transforming one quantum in the stretching into two quanta in the bending involving the same H [5,8,9]. This idea has been incorporated in previous models of isomerization after excitation of a C-H overtone [lo-121 which have influenced the present model. The above Fermi resonances are the firts step in the energy redistribution of the localized excitation to the whole molecule, which takes place in times ranging from 10 to 1000 fs. The rapid energy spread among many modes
does not encourage chemists to use selective vibrational excitation of a given bond to increase the rate of a chemical reaction involving the breaking of that bond [13]. The aim of the present paper is to study theoretically the possibility of overcoming this difficulty by a suitable choice of the radiation characteristics. The reaction selected for that purpose is the following isomerization by hydrogen tunneling (enolization) [14,15]: CH3
CH3
which is an example of an important class of reactions (R, and R, are generic substituents introduced in order to avoid degeneracy of the C-H oscillator involved). I will show, by use of a simple model for the above reaction, that the excitation by two or more frequencies, which has been recently proposed as an efficient way for reducing the vibrational relaxation rate [ll], does not work at all in the large-molecule limit considered here, due to the large density of states at the overtone energy. An alternative method based on a near-resonant
0301-0104/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
400
A. Lumi / Reactivity enhancement through overtone excitation
monochromatic source [16] is shown to be quite effective in increasing the isomerization yield.
2. Theory Reaction (1) is modeled as in fig. 1. The diabatic picture by Siebrand is used for H tunneling, which is supposed to be one-dimensional. The energy threshold for the reaction can be estimated from the existing data to be around the u = 6 C-H overtone. In the present paper I consider the excitation of the u = 7 overtone. The excited overtone is taken to decay irreversibly into the products or to transfer energy into other states, as will be shown. The irreversibility of the decay towards the product can be attributed to the rapid energy redistribution occurring in the isomer and could be analyzed in terms of sequential couplings, as will be done for the initial excited overtone. This is, however, unnecessary, since the control one can achieve acting on the radiation source concerns only the first steps, starting from the u = 7 state (i.e. the “doorway state”). The doorway state u(str) = 7 is assumed to be coupled to the state 1u(str) = 6, u(bend) = 2) = 16, 2). The latter is coupled to the 1.5,4) state. Both ]6,2) and 15,4) are allowed to decay irreversibly into a quasi-continuum of C-C modes. The isomerization can then be simply modeled by a three-level system (fig. 2) described by the effec-
-
.I
w
lg~=Iop> Fig. 2. The levels included in the model. The excited overtone decays directly into the isomerization quasi-continuum (on the right) or relaxes, via Fermi resonances, into the quasi-continuum of C-C modes (on the left). The 17, 0) overtone is excited from the ground state 18) via photon absorption.
tive hamiltonian, -iy, Herr G
v,, i
v,, -& + J-G
G
53
v,, VzJ , -iy, + E3
(2)
where 11) = 17, 0); 12) = 16, 2) and 13) = 15, 4). The various quantities appearing in eq. (1) have been estimated as follows. The width y1 is assumed to be lo-’ cm-‘, which gives an isomerization rate of about 2 x 10” s-l, as can be estimated from Siebrand’s approach [14]. The width y2 is considered a free parameter to be determined so as to have a spectral width of the u = 7 overtone of the same order as measured for analogous cases [4] (in the following y2 is taken to be 22.5 cm-‘). The width y3 is calculated on the assumption that the difference in rate of 12) and 13) is determined uniquely by the density of bath modes at E2 and E,, YJY~ = dE,)/dE,). This ratio of densities is evaluated Marcus-Rice formula [17]: P(G)/P(E,)
Fig. 1. The isomerization reaction is due to the tunneling of H from the C-H oscillator (left) to the H-O oscillator (right).
= [(K+
&)/(E,
from
the
+ &)I N-1,
(3)
where N is the number of oscillators and E, is the zero-point energy. For the present case I assume that N = 9 and that the frequencies are distributed
A. Lami
/ Reactivity
enhancement
uniformly between 900 and 1500 cm-‘, so that E, = 5400 cm-‘. The energies E,, E,,E3 and the couplings Vi, and V& have been computed assuming that the two local modes can be described by the hamiltonian (the most relevant kinetic terms are included P81): H=
-$c,,-$- fCslr~$ - %qj&j
where S and B are the stretching and bending internal coordinates and Gss, GBB and G,, are the c~r~nate-dependent G-matrix elements [19],
G,, = --pc sin(B + KHC)/rc_,; =kd(rC-H
overtone excitation
401
acterizes the decay of the excited overtone in the frame of the simple model discussed previously, in which the main appro~mation is the complete identification of the C-H stretching with the reaction coordinate. In the following I compare the isomerization yields for various kinds of excitation with the state 11) acting as doorway for the interaction with the field(s). For that purpose the radiation-molecule coupling is treated perturbatively, whereas the intramode coupling is considered exactly (the use of perturbation theory is fully justified even for quite strong laser sources). One has ( ]g) is the ground state) H(r) =&S
W(t),
J& = Heff + @I) (g[,
G.W=PH+PC;
GBB
through
+ s>’
(51
(7)
w(t) = w,,(f>tl>(gi
+ h-c.
In first order (taking ti = 1)
+ k/‘&C
U(t) = U”(t) -iJrUo(t-
+p&~~-H+S)-2+T&
t’)W(t)U”(t’)
dt’
0
-2cos(B+~CHC)/(r~_,+S)r,_,]
(8)
and (since Es= 0)
and the Morse potentials are
qg(t)= -i
Ui( S) = 13,(e-2s1s - 2 eKpts), 1>, = 0.2 au;
& = 0.8601 au-‘;
/3*= 1.0118 au-‘.
- t’)~,~(t’)
dl’.
(9)
interaction of the molecule with a polychromatic field is written as
The
U2(B)=D2(e-2B2B-2e-82B), D, = 13 au;
‘C$(t
I0
(6)
The choice of a Morse potential for the bending is certainly not the best one (a polynomial would give a best fit of experimental frequencies), but it is the simplest giving analytical results and taking into account ~armo~city. The accuracy of results is sufficient for the present purposes. As a consequence of the choice of h~ltoni~ (4) the only coupling between zero-order states (products of Morse wavefunctions) is kinetic (as in most previous papers [SJl]). The following results have been obtained from a numerical analysis (the differentiation is, however, analytical), retaining the full dependence of G,,, G,, and G,, on the internal coordinates): E, = 19777 cm-‘; E, = 20470cm-‘; E3= 20965cm-‘; VIZ= - 102 cm-‘; V,, = -291 cm-‘. The effective h~lto~~ (2) completely char-
W,,(t) = cEa~pa co&t) 4
= CA, cos(w,t). a (10)
Furthermore q:( t - r’) = C q,C,,
e-ir*(t-r’f,
k
(11)
where C is the orthogon~ tr~sfo~ation which diagonalizes He" and the E are the complex eigenvalues. Substituting into (9) and performing a rotating-wave approximation one has:
(12) The population of the continuum coupled to state 1j) (j = 1, 2, 3) at time t, l+(t), is B’(t) = 2y4-45&‘)
I2dt’
(131
402
A. Lami / Reactivity enhancement through overtone excitation
(see appendix). B,(t) + B3(1) gives the fraction of molecules which are relaxed at time t, whereas B,(t) is the fraction of isomerized molecules. Hence the reaction yield, I;(t), is P(t)=B,(#)/lB,tt)+B,(t)~B,(t)f.
(14)
I consider now various kinds of excitation. (i) Suppose that the source is mo~~~omatic. Eq. (12) gives f&(t)
“A~C,*kc,ke-‘“:~~-i’*‘. WI k k
After a characteristic time given by the lifetime of the most stable eigenstate of Neff, qs becomes constant so that, for long times,
interpreted by observing that the vibrational relax;ltion of the overtone requires a partial transfer of the excitation from the stretching C-H to another internal degree of freedom, i.e. the bending, whereas the isomer&ion is direct (the stretching is the reaction coordinate). Going out of the resonance the first step becomes the creation of a virtual excitation whose lifetime is not sufficient to permit an effective trans~ssion of the vibration to the bending, as required for relaxation. (ii) In the case of a continuum of frequencies with the same amplitudes and phases, from eq, (lo), follows
and q,(t) Since &&cik = L$, it happens that using an w such that o - ek is quite independent of k, the yield can be made close to 1. This can be done using near-resonant excitation, as shown in fig. 3, where the isomerization yield is plotted as a function of the dettming A from the doorway state, A = Eg -t- w - El_ The plot has not been extended too far from resonance (where eq. (14) gives F = 1) because in that region the model does not maintain its full validity, since (1) other doorway states must be included and (2) the rotating-wave appro~mation becomes questionable. Fig. 3 shows a si~fi~tive e~~rnent of the yield even in the restricted range considered. This effect should be
Fig. 3. The yield of the isomer&&on as a function of the detuning d (see fig. 2), for the near-resonant monochromatic excitation discussedin the text.
= A c Cj&+lk e-““‘. k
In this way, corresponding to a 6 pulse in the time domain, one First excites the doorway state, which then decays by vibrational relaxation (the preponderam? channel) and by isomerization. The reaction yield is, for such case, F = 0.01, i.e. quite low with respect to the most favorable situation shown in fig. 3. (iii) The third case considered is that of an excitation source containing a few frequencies. In fact Hofme and ~utc~son [ll] pointed out that if the overtone is a linear ~mbination of various molecular eigenstates, one can use polyc~omatic light to prepare the coherent superposition corresponding to the overtone. I show that this argument does not apply to the large-molecule case considered here. The reason is that the molecular eigenstates must be replaced in the present case by the decaying eigenstates of Heft, so that every linear combination of them will still’ relax (i.e. relaxation can not be eliminated by interference). The calculation can be performed starting from eq. (12) and considering a long-time excitation, so that transients disappear. The remaining oscillations are averaged out to give a time-independent yield. The results for some three-frequency excitations, corresponding to the energy difference among the real part of the three eigenvalues of fi
A. Lmni / Reactivity enhancement through overtone excitation
and the ground state, are: -42 -=-=
-42
4
A3
1:
I;= 0.011;
A -LIm(r,), Al 2A
A,
+_. 3
_ -p 16)
Im(e,)
’
ImkJ II-44
A2 I&3) -=_* A, Im(r,)
F = 0.010; * ’
F=OiX1’
The above reaction yields compare quite rmfavorably with the results obtained for near-resonant excitation (fig_ 3). Notice that the calculations in ref. [ll] were performed with quantized fields, whereas here classical fields have been used. This difference is, however, only apparent since the two treatments give exactly the same results using an identical approximation (i.e. rotating wave here and neglect of non-resonant transitions in the quitted version). In concluding this section I mention for completeness that the use of a strong resonant field has been recently suggested as a tool for reducing vibrational relaxation [20]. The same idea has been previously discussed by Gigolo and the author 1213, but its application to the control of chemical reactions seems difficult, since in many cases the intensity required can be estimated to be strong enough to open new channels through multiphoton absorption from the overtone up to new excited states.
3. Conclusion A simple model of isome~zation through hydrogen tunneling stimulated by overtone excitation has been discussed. This is an example of competition between two channels, i.e. the reactive channel and vibrational relaxation. The latter redistributes the energy among the various modes on a short time scale and does not permit taking advantage of the local excitation for enhancing the reactivity. I have compared various ways of exciting the overtone and shown that in large molecules (i.e. when the states into which the excited
403
overtone decays form a dense ma~fold~ only Ned-resonant excitation permits a si~fi~t enhancement of reactivity. The model used is based on the assumption that the doorway state for radiation excitation is the C-H overtone itself. The same assumption has been used by others and has lead to a successful interpretation of the overtone spectra [4,8]. The decay of the excited overtone has been described in an effective hamiltonian formalism, taking into account the quasi-continuum of the modes of the molecule [22], and assuming that the initial step in the decay involves a deuce ~teraction of the C-H stretching with the H bending. Matrix elements have been calculated assuming Morse potentials for the two modes and retaining the full dependence of the G matrix on the internal coordinates. The results obtained indicate that the enolization yield of acetophenone derivatives can be significantly enhanced by exciting the u = 7 C-H overtone with a frequency detuned from the resonance by at least 500 cm-l. The applicability of the present model is, however, not restricted to the particular reaction considered. It seems to be sufficiently simple and general to be useful as a guideline in analyzing a whole class of isomerizations by hydrogen tunneling (for large molecules), since the mechanism of competition between motion along the reaction coordinate and relaxation is basically the same. As a conclusion I notice that the model discussed is the simplest one containing the basic ingredients. Its predictions should be interpreted as semiquantitative. The improvements necessary for more quantitative results should include a more careful dete~nation of the actual reaction path, with the inclusion of all the internal coordinates involved, and the use of more accurate potential energy surfaces.
Comments by unknown referees have been very useful in ameliorating the text and are greatefully acknowledged.
A. L.umi / Reactivity enhancement through overtone excitation
404
Appendix
[2] R.G. Bray and M.J. Berry, J. Chem. Phys. 71 (1979) 4909.
Suppose that one of the discrete levels in fig. 2, say 1j), is coupled to the quasi-continuum { ) v)}. The Schrodinger equation for the quasi-continuum part is id, = E,,C,,+ Vv,Cj, which can be solved to give C”(t) = -iJ,I
e -iE@“‘VV.Cj(t’)
dt’
and , c,(t)
12 = Jd dt’Jb
dt”
e-i&(t’-f”)
x Iy,, I2 q.*(t”)q.(t’).
The total population of the quasicontinuum at time t is P,(t)
= c I w> Y
{ IV)}
I 2.
Assuming that the coupling Kj is essentially independent of V, and going to the continuum limit one has -iEAr'-r")
=
2T~2&(t’
_ ,rJ)
=
2y,6(t’-
t”).
ClYjl'e
Y
Hence I’,(t)
=2yj&$‘)12
dt’.
References [l] R.J. Hayward, Spectry.
B.R. Henry 46 (1973) 207.
and
W. Siebrand,
J. Mol.
[3] K.V. Reddy, D.F. Heller and M.J. Berry, J. Chem. Phys. 76 (1982) 2814. [4] H.L. Fang and R.L. Swofford, J. Chem. Phys. 72 (1980) 6382. [5] H.L. Fang, D.M. Meister and H.L. Swofford, J. Phys. Chem. 88 (1984) 405. [6] B.R. Henry and W. Siebrand, J. Chem. Phys. 49 (1968) 1860. [7] M.L. Sage, J. Chem. Phys. 80 (1984) 2872. PI E.L. Sibert III, W.P. Reinhardt and J.T. Hynes, J. Chem. Phys. 81 (1982) 1115. J.T. Hynes and W.P. Reinhardt, J. Phys. 191 J.S. Hutchinson, Chem. 90 (1986) 3528. WI T. Uzer and J.T. Hynes, Chem. Phys. Letters 113 (1985) 483. Chem. Phys. Letters 124 1111 T.A. Hohne and J.S. Hutchinson, (1986) 181. Chem. Phys. WI T. Uzer, J.T. Hynes and W.P. Reinhardt, Letters 117 (1985) 600. 1131 J.M. Jasinski, J.K. FrisoIi and C.B. Moore, J. Chem. Phys. 79 (1983) 1312; D. Klenerman and R.N. Zare, Chem. Phys. Letters 130 (1986) 190: F.F. Crim, Ann. Rev. Phys. Chem. 35 (1984) 657. P41 W. Siebrand, T.A. Wildman and M.Z. Zgierski, J. Am. Chem. Sot. 106 (1984) 4083; 106 (1984) 4089. 1151 J.C. Scaiano, Chem. Phys. Letters 73 (1980) 319; K.H. GrehmaM, H. Weller and E. Tatter, Chem. Phys. Letters 95 (1983) 195; R. Heag, J. Wirz and P.J. Wagner, Helv. Chim. Acta 60 (1977) 2595. WI D.J. Tannor, M. Blanc0 and E.J. Heller, J. Phys. Chem. 88 (1984) 6240. [I71 R.A. Marcus and O.K. Rice, J. Phys. Colloid Chem. 55 (1951) 894; P.J. Robinson and K.A. Holbrook, in: Unimolecular reactions (Wiley-Interscience, New York, 1972) p. 131. principles of quantum WI E.C. Kimble, in: The fundamental mechanics (Dover, New York, 1958) p. 237; E.L. Sibert III, J.T. Hynes and W.P. Reinhardt, J. Phys. Chem. 87 (1983) 2032. 1191 E.B. Wilson, J.C. Decius and P.C. Cross, Molecular vibrations (McGraw-Hill, New York, 1955). PO1 S. Mukamel and K. Shan, Chem. Phys. Letters 117 (1985) 489. 1211 P. Grigolini and A. Lami, Chem. Phys. 30 (1978) 61. P21 S. Mukamel and K. Shan, J. Phys. Chem. 89’(1985) 2447.
Chemical Physics 115 (1987) 399404 North-Holland, Amsterdam
ENHANCEMENT OF REACTIVITY THROUGH NEAR-RESONANT RADIATIVE EXCITATION OF AN OVERTONE: MODEL STUDY OF AN ISOMERIZATION BY HYDROGEN TUNNELING Alessandro LAM1 Istituto di Chimica Quantistica ed Energetica Molecolare de1 CNR, via Risorgimento
35, 56100 Piss, Italy
Received 27 February 1987
The enolization of 2-methyl acetophenone derivatives through hydrogen tunneling is simply modeled to study the possibility of increasing the reaction yield by exciting the v = 7 C-H overtone. Vibrational relaxation is assumed to occur via Fermi resonances transforming one C-H quantum into two quanta of the H bending. The role of the quasi-continuum of C-C modes is taken into account via an effective hamiltonian. It is shown that the only way to increase significantly the reaction yield is using light detuned a few hundred cm -r from the overtone level, as suggested by Tannor et al.
1. Introduction In the past years there has been growing experimental and theoretical interest in excitation of high-energy localized vibrational states by radiation absorption. The overtones of A-H stretching (A = C, 0, N) have been extensively studied experimentally [l-5] and successfully interpreted by means of the local mode picture [6-91, originally proposed by Henry and Siebrand [6]. The radiationless decay of such states, which is the main source of the broadening of spectral lines, has been interpreted in terms of coupling (essentially kinetic) of the initial state to trees of other near-degenerate states [8,9]. For example it seems quite well established that the overtones of the C-H stretching decay by transforming one quantum in the stretching into two quanta in the bending involving the same H [5,8,9]. This idea has been incorporated in previous models of isomerization after excitation of a C-H overtone [lo-121 which have influenced the present model. The above Fermi resonances are the firts step in the energy redistribution of the localized excitation to the whole molecule, which takes place in times ranging from 10 to 1000 fs. The rapid energy spread among many modes
does not encourage chemists to use selective vibrational excitation of a given bond to increase the rate of a chemical reaction involving the breaking of that bond [13]. The aim of the present paper is to study theoretically the possibility of overcoming this difficulty by a suitable choice of the radiation characteristics. The reaction selected for that purpose is the following isomerization by hydrogen tunneling (enolization) [14,15]: CH3
CH3
which is an example of an important class of reactions (R, and R, are generic substituents introduced in order to avoid degeneracy of the C-H oscillator involved). I will show, by use of a simple model for the above reaction, that the excitation by two or more frequencies, which has been recently proposed as an efficient way for reducing the vibrational relaxation rate [ll], does not work at all in the large-molecule limit considered here, due to the large density of states at the overtone energy. An alternative method based on a near-resonant
0301-0104/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
400
A. Lumi / Reactivity enhancement through overtone excitation
monochromatic source [16] is shown to be quite effective in increasing the isomerization yield.
2. Theory Reaction (1) is modeled as in fig. 1. The diabatic picture by Siebrand is used for H tunneling, which is supposed to be one-dimensional. The energy threshold for the reaction can be estimated from the existing data to be around the u = 6 C-H overtone. In the present paper I consider the excitation of the u = 7 overtone. The excited overtone is taken to decay irreversibly into the products or to transfer energy into other states, as will be shown. The irreversibility of the decay towards the product can be attributed to the rapid energy redistribution occurring in the isomer and could be analyzed in terms of sequential couplings, as will be done for the initial excited overtone. This is, however, unnecessary, since the control one can achieve acting on the radiation source concerns only the first steps, starting from the u = 7 state (i.e. the “doorway state”). The doorway state u(str) = 7 is assumed to be coupled to the state 1u(str) = 6, u(bend) = 2) = 16, 2). The latter is coupled to the 1.5,4) state. Both ]6,2) and 15,4) are allowed to decay irreversibly into a quasi-continuum of C-C modes. The isomerization can then be simply modeled by a three-level system (fig. 2) described by the effec-
-
.I
w
lg~=Iop> Fig. 2. The levels included in the model. The excited overtone decays directly into the isomerization quasi-continuum (on the right) or relaxes, via Fermi resonances, into the quasi-continuum of C-C modes (on the left). The 17, 0) overtone is excited from the ground state 18) via photon absorption.
tive hamiltonian, -iy, Herr G
v,, i
v,, -& + J-G
G
53
v,, VzJ , -iy, + E3
(2)
where 11) = 17, 0); 12) = 16, 2) and 13) = 15, 4). The various quantities appearing in eq. (1) have been estimated as follows. The width y1 is assumed to be lo-’ cm-‘, which gives an isomerization rate of about 2 x 10” s-l, as can be estimated from Siebrand’s approach [14]. The width y2 is considered a free parameter to be determined so as to have a spectral width of the u = 7 overtone of the same order as measured for analogous cases [4] (in the following y2 is taken to be 22.5 cm-‘). The width y3 is calculated on the assumption that the difference in rate of 12) and 13) is determined uniquely by the density of bath modes at E2 and E,, YJY~ = dE,)/dE,). This ratio of densities is evaluated Marcus-Rice formula [17]: P(G)/P(E,)
Fig. 1. The isomerization reaction is due to the tunneling of H from the C-H oscillator (left) to the H-O oscillator (right).
= [(K+
&)/(E,
from
the
+ &)I N-1,
(3)
where N is the number of oscillators and E, is the zero-point energy. For the present case I assume that N = 9 and that the frequencies are distributed
A. Lami
/ Reactivity
enhancement
uniformly between 900 and 1500 cm-‘, so that E, = 5400 cm-‘. The energies E,, E,,E3 and the couplings Vi, and V& have been computed assuming that the two local modes can be described by the hamiltonian (the most relevant kinetic terms are included P81): H=
-$c,,-$- fCslr~$ - %qj&j
where S and B are the stretching and bending internal coordinates and Gss, GBB and G,, are the c~r~nate-dependent G-matrix elements [19],
G,, = --pc sin(B + KHC)/rc_,; =kd(rC-H
overtone excitation
401
acterizes the decay of the excited overtone in the frame of the simple model discussed previously, in which the main appro~mation is the complete identification of the C-H stretching with the reaction coordinate. In the following I compare the isomerization yields for various kinds of excitation with the state 11) acting as doorway for the interaction with the field(s). For that purpose the radiation-molecule coupling is treated perturbatively, whereas the intramode coupling is considered exactly (the use of perturbation theory is fully justified even for quite strong laser sources). One has ( ]g) is the ground state) H(r) =&S
W(t),
J& = Heff + @I) (g[,
G.W=PH+PC;
GBB
through
+ s>’
(51
(7)
w(t) = w,,(f>tl>(gi
+ h-c.
In first order (taking ti = 1)
+ k/‘&C
U(t) = U”(t) -iJrUo(t-
+p&~~-H+S)-2+T&
t’)W(t)U”(t’)
dt’
0
-2cos(B+~CHC)/(r~_,+S)r,_,]
(8)
and (since Es= 0)
and the Morse potentials are
qg(t)= -i
Ui( S) = 13,(e-2s1s - 2 eKpts), 1>, = 0.2 au;
& = 0.8601 au-‘;
/3*= 1.0118 au-‘.
- t’)~,~(t’)
dl’.
(9)
interaction of the molecule with a polychromatic field is written as
The
U2(B)=D2(e-2B2B-2e-82B), D, = 13 au;
‘C$(t
I0
(6)
The choice of a Morse potential for the bending is certainly not the best one (a polynomial would give a best fit of experimental frequencies), but it is the simplest giving analytical results and taking into account ~armo~city. The accuracy of results is sufficient for the present purposes. As a consequence of the choice of h~ltoni~ (4) the only coupling between zero-order states (products of Morse wavefunctions) is kinetic (as in most previous papers [SJl]). The following results have been obtained from a numerical analysis (the differentiation is, however, analytical), retaining the full dependence of G,,, G,, and G,, on the internal coordinates): E, = 19777 cm-‘; E, = 20470cm-‘; E3= 20965cm-‘; VIZ= - 102 cm-‘; V,, = -291 cm-‘. The effective h~lto~~ (2) completely char-
W,,(t) = cEa~pa co&t) 4
= CA, cos(w,t). a (10)
Furthermore q:( t - r’) = C q,C,,
e-ir*(t-r’f,
k
(11)
where C is the orthogon~ tr~sfo~ation which diagonalizes He" and the E are the complex eigenvalues. Substituting into (9) and performing a rotating-wave approximation one has:
(12) The population of the continuum coupled to state 1j) (j = 1, 2, 3) at time t, l+(t), is B’(t) = 2y4-45&‘)
I2dt’
(131
402
A. Lami / Reactivity enhancement through overtone excitation
(see appendix). B,(t) + B3(1) gives the fraction of molecules which are relaxed at time t, whereas B,(t) is the fraction of isomerized molecules. Hence the reaction yield, I;(t), is P(t)=B,(#)/lB,tt)+B,(t)~B,(t)f.
(14)
I consider now various kinds of excitation. (i) Suppose that the source is mo~~~omatic. Eq. (12) gives f&(t)
“A~C,*kc,ke-‘“:~~-i’*‘. WI k k
After a characteristic time given by the lifetime of the most stable eigenstate of Neff, qs becomes constant so that, for long times,
interpreted by observing that the vibrational relax;ltion of the overtone requires a partial transfer of the excitation from the stretching C-H to another internal degree of freedom, i.e. the bending, whereas the isomer&ion is direct (the stretching is the reaction coordinate). Going out of the resonance the first step becomes the creation of a virtual excitation whose lifetime is not sufficient to permit an effective trans~ssion of the vibration to the bending, as required for relaxation. (ii) In the case of a continuum of frequencies with the same amplitudes and phases, from eq, (lo), follows
and q,(t) Since &&cik = L$, it happens that using an w such that o - ek is quite independent of k, the yield can be made close to 1. This can be done using near-resonant excitation, as shown in fig. 3, where the isomerization yield is plotted as a function of the dettming A from the doorway state, A = Eg -t- w - El_ The plot has not been extended too far from resonance (where eq. (14) gives F = 1) because in that region the model does not maintain its full validity, since (1) other doorway states must be included and (2) the rotating-wave appro~mation becomes questionable. Fig. 3 shows a si~fi~tive e~~rnent of the yield even in the restricted range considered. This effect should be
Fig. 3. The yield of the isomer&&on as a function of the detuning d (see fig. 2), for the near-resonant monochromatic excitation discussedin the text.
= A c Cj&+lk e-““‘. k
In this way, corresponding to a 6 pulse in the time domain, one First excites the doorway state, which then decays by vibrational relaxation (the preponderam? channel) and by isomerization. The reaction yield is, for such case, F = 0.01, i.e. quite low with respect to the most favorable situation shown in fig. 3. (iii) The third case considered is that of an excitation source containing a few frequencies. In fact Hofme and ~utc~son [ll] pointed out that if the overtone is a linear ~mbination of various molecular eigenstates, one can use polyc~omatic light to prepare the coherent superposition corresponding to the overtone. I show that this argument does not apply to the large-molecule case considered here. The reason is that the molecular eigenstates must be replaced in the present case by the decaying eigenstates of Heft, so that every linear combination of them will still’ relax (i.e. relaxation can not be eliminated by interference). The calculation can be performed starting from eq. (12) and considering a long-time excitation, so that transients disappear. The remaining oscillations are averaged out to give a time-independent yield. The results for some three-frequency excitations, corresponding to the energy difference among the real part of the three eigenvalues of fi
A. Lmni / Reactivity enhancement through overtone excitation
and the ground state, are: -42 -=-=
-42
4
A3
1:
I;= 0.011;
A -LIm(r,), Al 2A
A,
+_. 3
_ -p 16)
Im(e,)
’
ImkJ II-44
A2 I&3) -=_* A, Im(r,)
F = 0.010; * ’
F=OiX1’
The above reaction yields compare quite rmfavorably with the results obtained for near-resonant excitation (fig_ 3). Notice that the calculations in ref. [ll] were performed with quantized fields, whereas here classical fields have been used. This difference is, however, only apparent since the two treatments give exactly the same results using an identical approximation (i.e. rotating wave here and neglect of non-resonant transitions in the quitted version). In concluding this section I mention for completeness that the use of a strong resonant field has been recently suggested as a tool for reducing vibrational relaxation [20]. The same idea has been previously discussed by Gigolo and the author 1213, but its application to the control of chemical reactions seems difficult, since in many cases the intensity required can be estimated to be strong enough to open new channels through multiphoton absorption from the overtone up to new excited states.
3. Conclusion A simple model of isome~zation through hydrogen tunneling stimulated by overtone excitation has been discussed. This is an example of competition between two channels, i.e. the reactive channel and vibrational relaxation. The latter redistributes the energy among the various modes on a short time scale and does not permit taking advantage of the local excitation for enhancing the reactivity. I have compared various ways of exciting the overtone and shown that in large molecules (i.e. when the states into which the excited
403
overtone decays form a dense ma~fold~ only Ned-resonant excitation permits a si~fi~t enhancement of reactivity. The model used is based on the assumption that the doorway state for radiation excitation is the C-H overtone itself. The same assumption has been used by others and has lead to a successful interpretation of the overtone spectra [4,8]. The decay of the excited overtone has been described in an effective hamiltonian formalism, taking into account the quasi-continuum of the modes of the molecule [22], and assuming that the initial step in the decay involves a deuce ~teraction of the C-H stretching with the H bending. Matrix elements have been calculated assuming Morse potentials for the two modes and retaining the full dependence of the G matrix on the internal coordinates. The results obtained indicate that the enolization yield of acetophenone derivatives can be significantly enhanced by exciting the u = 7 C-H overtone with a frequency detuned from the resonance by at least 500 cm-l. The applicability of the present model is, however, not restricted to the particular reaction considered. It seems to be sufficiently simple and general to be useful as a guideline in analyzing a whole class of isomerizations by hydrogen tunneling (for large molecules), since the mechanism of competition between motion along the reaction coordinate and relaxation is basically the same. As a conclusion I notice that the model discussed is the simplest one containing the basic ingredients. Its predictions should be interpreted as semiquantitative. The improvements necessary for more quantitative results should include a more careful dete~nation of the actual reaction path, with the inclusion of all the internal coordinates involved, and the use of more accurate potential energy surfaces.
Comments by unknown referees have been very useful in ameliorating the text and are greatefully acknowledged.
A. L.umi / Reactivity enhancement through overtone excitation
404
Appendix
[2] R.G. Bray and M.J. Berry, J. Chem. Phys. 71 (1979) 4909.
Suppose that one of the discrete levels in fig. 2, say 1j), is coupled to the quasi-continuum { ) v)}. The Schrodinger equation for the quasi-continuum part is id, = E,,C,,+ Vv,Cj, which can be solved to give C”(t) = -iJ,I
e -iE@“‘VV.Cj(t’)
dt’
and , c,(t)
12 = Jd dt’Jb
dt”
e-i&(t’-f”)
x Iy,, I2 q.*(t”)q.(t’).
The total population of the quasicontinuum at time t is P,(t)
= c I w> Y
{ IV)}
I 2.
Assuming that the coupling Kj is essentially independent of V, and going to the continuum limit one has -iEAr'-r")
=
2T~2&(t’
_ ,rJ)
=
2y,6(t’-
t”).
ClYjl'e
Y
Hence I’,(t)
=2yj&$‘)12
dt’.
References [l] R.J. Hayward, Spectry.
B.R. Henry 46 (1973) 207.
and
W. Siebrand,
J. Mol.
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