- Email: [email protected]
Expansion approach to photodissociation dynamics. Effect of anharmonicity
Volume 78, number 2
1 March 1981
CHEMICALPHYSICS LETTERS
EXPANSION APPROACH TO PHOTODISSOCIATION
DYNAMICS. EFFECT OF ANHARMONICITY
Kazuo TAKATSUKA and Mark S. GORDON Department
of Chemistry,
North Dakota State Umversity,
Fargo, Norrh Dakota 58105.
USA
Received 28 April 1980; in final form 24 November 1980
The effect of anharmoniciry m the rexdual molecuIe A-B in the photodissociation process A-B-C + h v - A-B + C IS euammed. It IS shown that even if the transitIon probabihty for the V-T energy tran fer process A-B (U = n) + C + A-B I~ can be dramatically modified. 6 = n’) + C is not appreciably affected by the anharmomcity, tile partial linewidth r3fn
1. Introduction
transfer [6,7], it is found [6] that, particularly
An important question with regard to the photodecomposition reaction
..4-B-C --f A-B + C concerns the extent to which the mcorporation harmonicity
0) of an-
in the fragment
A-B modifies predicted behavior. Typically, A-B is treated as a harmonic oscillator for simplicity, and this is often reasonable. Anharmonicity effects m A-B can be manifested in two ways: (i) The V-T energy transfer process A-B(u=n)+C+A-B(u=n’)+C
(2)
on the repulsive surface may be modified. This may in turn modify the final-state interaction [ 11, even within the scheme of the quasi-diatomic model [Z]. (ii) The Fran&--Condon overlap integral, which appears in photodissociation theories [3,4]. may be influenced through the deformation of the wavefunction of the scattering process, eq. (2). Since the above V-T process is a multichannel collision problem, one must consider the higher vibrational levels of A-B, and these are not well represented by harmonic functions. Some studies of anharmonicity effects have been camed out. Shapiro et al. 151 have incorporated anharmonicity only in the A-B-C parent molecule in a
calculation of the partial linewidth r&/-‘for the transition (i, a) + V; m), where i, f and Q, m denote electron-
ic and vibrational states, respectively_ For V-T 328
at low
energy, the transition probabilities of the harmonic and anharmonic oscillators are nearly the same qualitatively. It is reasonable to expect significant changes in I-$-/
energy
to accompany large anharmonicity effects on the corresponding V-T energy transfer process. It is shown below, however, that even when the V-T transition probability is nor significantly modified by anharmonicity, rim can change dramatically.
2. Computational
outline
Let xa, Xb (= 0) and xc be the coordinates of a linear A-B-C molecule. The usual transfor.mation [8] among them gves a ham&or&n for the repulsive surface
where r is Ixb - xal and R is the distance between C and the centroid of A-B. is taken to be VR = D exp (--ax,)
The repulsive potential VR
,
(4)
and a Morse function is used to represent the anharmonic A-B potential: V&
= De I1 - exp (--m)] 2 .
(5)
The hamiltonian f+ describes the scattering process of eq. (2), including complete interchannel coupling
Volume 78, number 2
CHEMICAL
1 March 1981
PHYSICS LETTERS
(inelasticity). Approximate scattering wavefunctions X,, are used to obtain partial linewidths l?Lrn by a method described earlier [9,10]. Am is expanded in a basis set:
Table 1 Molecular parameters a) HCNti
1 c+) -~-
rCH = 2.011 bohr wr = 3311.5 cm-’
where S, and C, approach asymptotically the Bessel and Neumann functions, respectively and behave regulariy at R = --m. c: and Frb are eigenfunctions of Morse and harmonic oscillators, respectively. The K matrix and the coefficients dFb, which are determined by the variational principle [ 111, represent the strength of interchannel coupling. It is assumed that (1) the photoemission deactivation from the excited states to the ground state can be ignored and (2) the photoabsorption process is represented by the appropriate Franck-Condon integrals. Each $E is expanded in a basis of 25 eigenfunctions of the corresponding harmonic force field. FranckCondon integrals W, are evaluated using a recursion formula derived in a previous paper [9,10] _We consider the dissociation of HCN occurring only from the vibrationkss state (&)_ For I$, we assume a harmonic potential.
3. Results Recent experimental evidence obtained by Kondow et al. [12] implies that the predissociating state of HCN at 1218 A is a 3s Rydberg state with a bent geometry. Ab initio calculations are also consistent with a non-linear predissociating state [13 ] . Furthermore, it appears that both HCN and DCN predissociate at 1295A [14]. Nonetheless, we have for simplicity chosen the linear direct dissociation of HCN, DCN, and TCN to illustrate the effect of anharmonicity in A-B. The reaction scheme may be expressed as HCN(jT%+)
‘2 _
CN(B 2Z+) f H(l 2S).
(7)
The spectroscopic constants necessary for the calculation are listed in table l_ An important feature of the potential parameters is that both the Iength and strength of the CN bond
/CN = 2.174 bohr wa = 2096.7 cm-r
CN(B*X+) ‘CN = 2.174 bohr De = 0.2308 hartree 1161 wexe = 23.1 cm-l b)
we = 2164.1
cm-’
repulsive surfaces o = 3.535 bohr-’ [l] electronic energy difference between the evcded and ground states = 0.3175 hartree [9,10] a) AU quantities not specifkaUy noted are taken from ref. b) Estimated from D,given
in ref. 1161 and we m ref. [15].
almost unchanged throughout the reaction: rCN e 2.17 bohr and w = 2100 cm-l. One expects the effect of anharmonicity on spectral properties to increase as geometric deformations due to the photoabsorption increase. Since the distortion in this case is small, it might be expected that the effect of anharmonicity in the process described by eq. (7) will be small. Furthermore, the species CN(B2e) has a deep potential well (De= 0.2308 ‘hartree). Again, this would argue for a small anharmonic effect. Indeed, wexe/oe for this system is = 0.01. All of this suggests only a small effect on are
Table 2 Probability of the V-T energy transfer. Comparison of harmonic and anharmonic results HCN Pal anharm.
harm. Pm anharm. harm. P23 anharm. harm. Pw anharm. harm.
0.264 0.271 0.294 0.292 0.223 0.196 0528 0.194
DCN (0) a)
(0) (0) (0) (0) (0) (-1) (-1)
0 257 0.281 0.245 0.248 0.126 0.996 0.626 0.184
TCN (0) (0) (0) (0) (0) (-1) (-2) (-3)
0.207 0.243 0.176 0.184 0.644 0.481 0.661 b)
(0) (0) (0) (0) (-1) (-1) (-3)
a) 0.264 (0) = 0.264 x loo_ b)TLre fourth channel is not open.
329
Volume
78, number
CHEMICAL
2
Tabie 3 Effect of the anharmomcity
-
___-_ __
_____.___c
-..I
-
TOf1 anharm. 7:
on yc
__
----
D = 12 86”) --_- - .-
0.544
(3) 0.360 (-2) harm. f4 anharm TlO 0.142 (-1) harm. 0 448 (-5) r Io anharm. 0 329 (0) harm 0.284 (0) __ __.-_-.____--_--_... a) Pre-ekponential
D=50
0.893 0.978 0.298 0.261 0.394 0.171 0.119 0 562 0 332 0.337
factor of the repulsive surface,
Table 4 Effect of the anharmomcity ___~__. -___. -_ ____-
fl anharm. ri0 harm. YlO f2 anharm.
(-1) (-1) (-2) (-2) (-4) (-4) (-6) (-8) (1) (1)
anharm. harm. 7: anharm. harm. r 1. anhxm. harm.
. - -.--
(0) (0) (0) (-1) (0) (-2) (-2) (-7) (0) (0) ---.--I
-----
0.139 O-180 0 463 0.713 0 853 0.421 0.549 0.274 0.150 0.150
(0) (0) (-2) (-2) (-4) f-4) (-7) (-9) (3) (3)
0.980 (-- 1) 0.114 (0) 9.259 (-2) 0.286 (-2) 0.189 (-4) 0.107 (-4) 0.562 (-8) 0.437 (-10) 0.102 (1) 0.106 (1)
the transitron probabhty for the V-T transltion probabrlity in eq. (2). That this is the case may be seen in table 2, where the transition probabilities P<,i+l (i = 0, 1,2,3) for the f&I colhsion process of eq. (2) are tabulated_ Although P3,4(A) and P3,4(H) (A = anhar-
manic,
H = harmonic)
Table 6 Relative Franck-Condon --Uf
1 2 3 4
factors -L_
different
for HCN.
D = 12.86 a)
harm.
0.157 0.322 0.624 0.161
0.298 0.661 0.182 0.220
(0) (-1) (0) <--I)
from
i(v~l~,)lz/l(u~t~~12
(0) (0) (-1) (-4)
factor of the repulsive
(0)
0.772 0.384
(-2) (-4)
0.215 0.636
(-3) (-2)
0.144
(-8) (-4)
0.340
(-11) (-5)
(0) (0)
0.162 0.161
(3) (3)
0.417 0.438
(0) (0)
0.289
(-2) (-5)
0.439
is not open.
D = 190
anhann.
harm.
0.135 0.795 0.795 0 277
0.250 0.257 0.105 0.253
surface, eq. (4).
0 886 (-1)
0.108 (0) 0.172 (-2)
-
D=50
anharm.
a) Pre-exponential
330
are appreciably
(0)
one another, they are not important for I$c (nnz= 0, 1,2, 3,4)_ The remaining Pi i+l in table 2 are similar for the two potentrals. ’ In spite of the foregomg considerations, I’{” undergoes considerable change when anharmonicity is included_ The values of rLm (= rc/I$ are listed in tables 3-5. The modifkatron is largest for D = 12.86. This is not surprising, since as D increases, the repulsive surface moves away from the carbon atom. Note also that the Iinewrdth rzo 1s affected to a much smaller extent than are the individual linewidths. Table 6 contains the relative Franck-Condon factors (FCFs) Icu~lh,~,)12/i(~61~)12 (m = 1,2,3,4) for HCN. The FCFs do not exclusively determine I’{?, since in the computation the FC overlaps are mixed coherently with the coefficients of the K-matrix elements. Qualitatively, however, for D = 12.86, the partial lmewrdths seem to be dominated by the FCFs. Note that the anharmonicity effect in the FCF is due only to the local distortion in the scattering wavefunction in
.-__ -__________p-__I_ D=50 D = I90 D = 12.86
anhnrm. 0.739 2 harm. 0.140 I-2 anharm. 0.156 7iO 0.200 haml. anharm 0 607 2 0.189 harm. r,,f4 anharm 0 125 harm. 0.842 rio anharm. 0.460 0 226 harm. ____. --___
O-569 0.101 0.106 a) 0 586 0 296
a) The fourth chumcl
eq. (4).
0.151
0.203 (00) 0.437 (-2)
0.136 (0) 0 260 (-1)
rio
D = 190
D=50
0 117 (1) 0.499 (0)
f3 harm.
lin for DCN dlssoclstlon on -rro
_~___ __. .-._-_l_^~_--
f?n for TCN dissociation on -yio
D = 12.86 -------
D = 190 _..__-___1_ (0) (0) (-2) (-2) (-3) (-4) (-5) (-7) (3) (3) _I-
1 IMarch 1981
-_________-
-------
0.102 0.126 0 361 0.432 0.132 0.362 0.117 0.148 0.128 0.12s _.-
0.115 (0)
harm. h anharm r1v
LETTERS
Table 5 Effect of the anharmomclty
for HCN dissociation
_-.__-- -
0.103 (0) 0.747 (0) 0 665 (-1)
harm. anharm
PHYSICS
(-2) (-2) (-3) C-6)
(-2) (-1) (-2) (-6)
anharm.
harm.
0.543 0.147 0.177 0.792
0.362 0.580 0.303 0.568
(-2) (-2) (-3) (-7)
(-2) (-2) (-3) (-7)
Volume 78, number 2
CHEMICAL PHYSICS LETTERS
the Franck-Condon region, while the V-T transfer probability is determined by the “phase shifts” in the scattering wavefunctions of the asymptotic region. Therefore, when a local distortion in X, is not accornparried by an asymptotic disturbance, is it quite likely in the non-tunneling mechanism that T$” can drastically change without large modification in the V-T transfer probability. On the other hand, if the dissociation occurs by the tunneling mechanism (large D), we do not expect the wavefunction to be greatly distorted in the Franck-Condon region. So, the anharmonicity effect might not be so rmportant in the tunneling mechanism if the V-T process remains almost unchanged. At D = 12.86 (non-tunneling mechanism) the anharmonicity effect becomes larger as H is substituted by D and T- l?io(A)/f’;O(H) = 1 .16,2.04, and 1.98 for HCN, DCN. and TCN, respectively. The difference in the sensitivity of each molecule to anharmonicity is reflected in the isotope effect: l?io(DCN)/riO(HCN) = 0.796 for the harmonic potential and Fjo(DCN)/ Fio(HCN) = 1.40 for the anharmonic potential. This result suggests the importance of anharmonicity even for qualitative conclusions regarding the isotope effect. In the non-tunneling mechanism it is difficult to prechct the turnover in the isotope effect, since it is not clear how the local distortion in h,, depends on the isotopic substitution. In contrast, when the tunneling mechanism [ 1,4,10] applies (e.g., D = 190), one can easily predict Fio(TCN) < l?iO(DCN) < Fio(HCN), since the penetration of the wavefunctron into the potential barrier is larger for the lighter atom.
large contributions from <,y(l-> and C,, (n # m) [see eq. (6)]. For example, +, has a large contribution from 5yhb and @hb. The effect_ of anharmonicity on the Franck-Condon overlap
AcknowIedgement This work was supported by National Science Foundation Grant CHE78-18070. The computer time made available by the NDSU Computer Center is gratefully acknowledged_
References III Y-B. Band and K.F. Freed, Chem Phys Letters 28 (1974) 328; J. Chem. Phys. 63 (1975) 3382 [21 K-E. Holdy and L.C. Klotz, J. Chem. Phys. 52 (1970)
131 I41
4. Concluding remarks It is necessary for any dynamical theory to include anharmonicity effects when the anharmonicity itself (o,x,/x,) IS large or when large geometry distortIons occur on photoabsorption. It has been shown in this paper that re can also be strongly modified by the anharmonicity even if the transition probability of the corresponding V-T process is virtually unchanged. Finally, it is noteworthy that the strength of anharmonicity effects increases with the importance of interchannel coupling. In the case of strong interchannel coupling (e.g., HCN, DCN, and TCN at the wavelength considered in this paper), the wavefunction X, has
1 March 1981
[Sl [61 171 ISI I91 II01 [III
4588; M. Shapiro and R D. Levine, Chem. Phys. Letters 5 (1970) 499. S.A. Rice, Excited states, Vol. 2 (Academic Press, New York, 1975) p. 111; W-M. Gelbart, Ann. Rev. Phys. Chem. 28 (1977) 323. K.F. Freed and Y.B. Band, Exited states, Vol. 3 (Academic Press, New York, 1977) p_ 109. M. Shapiro, Chem. Phys. Letters 46 (1977) 442, J.A. Beswick, M. Shapiro and R. Sharon, 3. Chem. Phys. 67 (1977) 4045. D. Secrest, Ann. Rev. Phys. Chem 24 (1973) 379. R J. Gordon, J. Chem. Phys. 65 (1976) 4945; 67 (1977) 5923. and references therem. D. Secrest and B.R. Johnson, J. Chem . Phys 45 (1966) 4556. K. Takatsuka and MS. Gordon, Expansion Approach to Photodissociation Dynamics, I, J. Chem. Phys., to be pubLished K. Takatsuka and MS. Gordon, Expansion Approach to Photodissociation Dynamics. II, J. Chem. Phys , to be published. K. Takatsuka and T. Fueno, Phys. Rev. A19 (1979) 1011,1018.
331
Volume 78, number 2
CHEMICAI. PHYSICS LE’ITERS
[ 121 T. Kondow, T. Nagata and K. Kuchitsu, private communiC-&Oil.
[X 3 J G.J. Vazques and J -F. Gouyet, Chem. Phys. Letters 65
(1979) 51.5; P.J. Hay, pnvate communzation. (14] J.P. Simons, Gas Kinetics and Energy Transfer, Vol. 2, Specldlist Periodica Reports (Chemxcal Society, London, 1977) p. 58;
332
1 March 1981
&f.N.R. Ashfold, &f.T. MacPherson and 3-P. Simons, Cbem. Phys. Letters 55 (1978) 84. IIs] G. Her&erg, Spectra of diatomic molecules (Van Nostrand, Princeton, 1950); Ehctrffnic structure of polyatomic molecules Wan Nostrand, Princeton, 1965). 1161 G. Das, T. Jams and A.C. W&i. J. Chem. P&s. 61 (1974) 1274. [17] K. Takatsuka and MS. Gordon, unpublished.
1 March 1981
CHEMICALPHYSICS LETTERS
EXPANSION APPROACH TO PHOTODISSOCIATION
DYNAMICS. EFFECT OF ANHARMONICITY
Kazuo TAKATSUKA and Mark S. GORDON Department
of Chemistry,
North Dakota State Umversity,
Fargo, Norrh Dakota 58105.
USA
Received 28 April 1980; in final form 24 November 1980
The effect of anharmoniciry m the rexdual molecuIe A-B in the photodissociation process A-B-C + h v - A-B + C IS euammed. It IS shown that even if the transitIon probabihty for the V-T energy tran fer process A-B (U = n) + C + A-B I~ can be dramatically modified. 6 = n’) + C is not appreciably affected by the anharmomcity, tile partial linewidth r3fn
1. Introduction
transfer [6,7], it is found [6] that, particularly
An important question with regard to the photodecomposition reaction
..4-B-C --f A-B + C concerns the extent to which the mcorporation harmonicity
0) of an-
in the fragment
A-B modifies predicted behavior. Typically, A-B is treated as a harmonic oscillator for simplicity, and this is often reasonable. Anharmonicity effects m A-B can be manifested in two ways: (i) The V-T energy transfer process A-B(u=n)+C+A-B(u=n’)+C
(2)
on the repulsive surface may be modified. This may in turn modify the final-state interaction [ 11, even within the scheme of the quasi-diatomic model [Z]. (ii) The Fran&--Condon overlap integral, which appears in photodissociation theories [3,4]. may be influenced through the deformation of the wavefunction of the scattering process, eq. (2). Since the above V-T process is a multichannel collision problem, one must consider the higher vibrational levels of A-B, and these are not well represented by harmonic functions. Some studies of anharmonicity effects have been camed out. Shapiro et al. 151 have incorporated anharmonicity only in the A-B-C parent molecule in a
calculation of the partial linewidth r&/-‘for the transition (i, a) + V; m), where i, f and Q, m denote electron-
ic and vibrational states, respectively_ For V-T 328
at low
energy, the transition probabilities of the harmonic and anharmonic oscillators are nearly the same qualitatively. It is reasonable to expect significant changes in I-$-/
energy
to accompany large anharmonicity effects on the corresponding V-T energy transfer process. It is shown below, however, that even when the V-T transition probability is nor significantly modified by anharmonicity, rim can change dramatically.
2. Computational
outline
Let xa, Xb (= 0) and xc be the coordinates of a linear A-B-C molecule. The usual transfor.mation [8] among them gves a ham&or&n for the repulsive surface
where r is Ixb - xal and R is the distance between C and the centroid of A-B. is taken to be VR = D exp (--ax,)
The repulsive potential VR
,
(4)
and a Morse function is used to represent the anharmonic A-B potential: V&
= De I1 - exp (--m)] 2 .
(5)
The hamiltonian f+ describes the scattering process of eq. (2), including complete interchannel coupling
Volume 78, number 2
CHEMICAL
1 March 1981
PHYSICS LETTERS
(inelasticity). Approximate scattering wavefunctions X,, are used to obtain partial linewidths l?Lrn by a method described earlier [9,10]. Am is expanded in a basis set:
Table 1 Molecular parameters a) HCNti
1 c+) -~-
rCH = 2.011 bohr wr = 3311.5 cm-’
where S, and C, approach asymptotically the Bessel and Neumann functions, respectively and behave regulariy at R = --m. c: and Frb are eigenfunctions of Morse and harmonic oscillators, respectively. The K matrix and the coefficients dFb, which are determined by the variational principle [ 111, represent the strength of interchannel coupling. It is assumed that (1) the photoemission deactivation from the excited states to the ground state can be ignored and (2) the photoabsorption process is represented by the appropriate Franck-Condon integrals. Each $E is expanded in a basis of 25 eigenfunctions of the corresponding harmonic force field. FranckCondon integrals W, are evaluated using a recursion formula derived in a previous paper [9,10] _We consider the dissociation of HCN occurring only from the vibrationkss state (&)_ For I$, we assume a harmonic potential.
3. Results Recent experimental evidence obtained by Kondow et al. [12] implies that the predissociating state of HCN at 1218 A is a 3s Rydberg state with a bent geometry. Ab initio calculations are also consistent with a non-linear predissociating state [13 ] . Furthermore, it appears that both HCN and DCN predissociate at 1295A [14]. Nonetheless, we have for simplicity chosen the linear direct dissociation of HCN, DCN, and TCN to illustrate the effect of anharmonicity in A-B. The reaction scheme may be expressed as HCN(jT%+)
‘2 _
CN(B 2Z+) f H(l 2S).
(7)
The spectroscopic constants necessary for the calculation are listed in table l_ An important feature of the potential parameters is that both the Iength and strength of the CN bond
/CN = 2.174 bohr wa = 2096.7 cm-r
CN(B*X+) ‘CN = 2.174 bohr De = 0.2308 hartree 1161 wexe = 23.1 cm-l b)
we = 2164.1
cm-’
repulsive surfaces o = 3.535 bohr-’ [l] electronic energy difference between the evcded and ground states = 0.3175 hartree [9,10] a) AU quantities not specifkaUy noted are taken from ref. b) Estimated from D,given
in ref. 1161 and we m ref. [15].
almost unchanged throughout the reaction: rCN e 2.17 bohr and w = 2100 cm-l. One expects the effect of anharmonicity on spectral properties to increase as geometric deformations due to the photoabsorption increase. Since the distortion in this case is small, it might be expected that the effect of anharmonicity in the process described by eq. (7) will be small. Furthermore, the species CN(B2e) has a deep potential well (De= 0.2308 ‘hartree). Again, this would argue for a small anharmonic effect. Indeed, wexe/oe for this system is = 0.01. All of this suggests only a small effect on are
Table 2 Probability of the V-T energy transfer. Comparison of harmonic and anharmonic results HCN Pal anharm.
harm. Pm anharm. harm. P23 anharm. harm. Pw anharm. harm.
0.264 0.271 0.294 0.292 0.223 0.196 0528 0.194
DCN (0) a)
(0) (0) (0) (0) (0) (-1) (-1)
0 257 0.281 0.245 0.248 0.126 0.996 0.626 0.184
TCN (0) (0) (0) (0) (0) (-1) (-2) (-3)
0.207 0.243 0.176 0.184 0.644 0.481 0.661 b)
(0) (0) (0) (0) (-1) (-1) (-3)
a) 0.264 (0) = 0.264 x loo_ b)TLre fourth channel is not open.
329
Volume
78, number
CHEMICAL
2
Tabie 3 Effect of the anharmomcity
-
___-_ __
_____.___c
-..I
-
TOf1 anharm. 7:
on yc
__
----
D = 12 86”) --_- - .-
0.544
(3) 0.360 (-2) harm. f4 anharm TlO 0.142 (-1) harm. 0 448 (-5) r Io anharm. 0 329 (0) harm 0.284 (0) __ __.-_-.____--_--_... a) Pre-ekponential
D=50
0.893 0.978 0.298 0.261 0.394 0.171 0.119 0 562 0 332 0.337
factor of the repulsive surface,
Table 4 Effect of the anharmomcity ___~__. -___. -_ ____-
fl anharm. ri0 harm. YlO f2 anharm.
(-1) (-1) (-2) (-2) (-4) (-4) (-6) (-8) (1) (1)
anharm. harm. 7: anharm. harm. r 1. anhxm. harm.
. - -.--
(0) (0) (0) (-1) (0) (-2) (-2) (-7) (0) (0) ---.--I
-----
0.139 O-180 0 463 0.713 0 853 0.421 0.549 0.274 0.150 0.150
(0) (0) (-2) (-2) (-4) f-4) (-7) (-9) (3) (3)
0.980 (-- 1) 0.114 (0) 9.259 (-2) 0.286 (-2) 0.189 (-4) 0.107 (-4) 0.562 (-8) 0.437 (-10) 0.102 (1) 0.106 (1)
the transitron probabhty for the V-T transltion probabrlity in eq. (2). That this is the case may be seen in table 2, where the transition probabilities P<,i+l (i = 0, 1,2,3) for the f&I colhsion process of eq. (2) are tabulated_ Although P3,4(A) and P3,4(H) (A = anhar-
manic,
H = harmonic)
Table 6 Relative Franck-Condon --Uf
1 2 3 4
factors -L_
different
for HCN.
D = 12.86 a)
harm.
0.157 0.322 0.624 0.161
0.298 0.661 0.182 0.220
(0) (-1) (0) <--I)
from
i(v~l~,)lz/l(u~t~~12
(0) (0) (-1) (-4)
factor of the repulsive
(0)
0.772 0.384
(-2) (-4)
0.215 0.636
(-3) (-2)
0.144
(-8) (-4)
0.340
(-11) (-5)
(0) (0)
0.162 0.161
(3) (3)
0.417 0.438
(0) (0)
0.289
(-2) (-5)
0.439
is not open.
D = 190
anhann.
harm.
0.135 0.795 0.795 0 277
0.250 0.257 0.105 0.253
surface, eq. (4).
0 886 (-1)
0.108 (0) 0.172 (-2)
-
D=50
anharm.
a) Pre-exponential
330
are appreciably
(0)
one another, they are not important for I$c (nnz= 0, 1,2, 3,4)_ The remaining Pi i+l in table 2 are similar for the two potentrals. ’ In spite of the foregomg considerations, I’{” undergoes considerable change when anharmonicity is included_ The values of rLm (= rc/I$ are listed in tables 3-5. The modifkatron is largest for D = 12.86. This is not surprising, since as D increases, the repulsive surface moves away from the carbon atom. Note also that the Iinewrdth rzo 1s affected to a much smaller extent than are the individual linewidths. Table 6 contains the relative Franck-Condon factors (FCFs) Icu~lh,~,)12/i(~61~)12 (m = 1,2,3,4) for HCN. The FCFs do not exclusively determine I’{?, since in the computation the FC overlaps are mixed coherently with the coefficients of the K-matrix elements. Qualitatively, however, for D = 12.86, the partial lmewrdths seem to be dominated by the FCFs. Note that the anharmonicity effect in the FCF is due only to the local distortion in the scattering wavefunction in
.-__ -__________p-__I_ D=50 D = I90 D = 12.86
anhnrm. 0.739 2 harm. 0.140 I-2 anharm. 0.156 7iO 0.200 haml. anharm 0 607 2 0.189 harm. r,,f4 anharm 0 125 harm. 0.842 rio anharm. 0.460 0 226 harm. ____. --___
O-569 0.101 0.106 a) 0 586 0 296
a) The fourth chumcl
eq. (4).
0.151
0.203 (00) 0.437 (-2)
0.136 (0) 0 260 (-1)
rio
D = 190
D=50
0 117 (1) 0.499 (0)
f3 harm.
lin for DCN dlssoclstlon on -rro
_~___ __. .-._-_l_^~_--
f?n for TCN dissociation on -yio
D = 12.86 -------
D = 190 _..__-___1_ (0) (0) (-2) (-2) (-3) (-4) (-5) (-7) (3) (3) _I-
1 IMarch 1981
-_________-
-------
0.102 0.126 0 361 0.432 0.132 0.362 0.117 0.148 0.128 0.12s _.-
0.115 (0)
harm. h anharm r1v
LETTERS
Table 5 Effect of the anharmomclty
for HCN dissociation
_-.__-- -
0.103 (0) 0.747 (0) 0 665 (-1)
harm. anharm
PHYSICS
(-2) (-2) (-3) C-6)
(-2) (-1) (-2) (-6)
anharm.
harm.
0.543 0.147 0.177 0.792
0.362 0.580 0.303 0.568
(-2) (-2) (-3) (-7)
(-2) (-2) (-3) (-7)
Volume 78, number 2
CHEMICAL PHYSICS LETTERS
the Franck-Condon region, while the V-T transfer probability is determined by the “phase shifts” in the scattering wavefunctions of the asymptotic region. Therefore, when a local distortion in X, is not accornparried by an asymptotic disturbance, is it quite likely in the non-tunneling mechanism that T$” can drastically change without large modification in the V-T transfer probability. On the other hand, if the dissociation occurs by the tunneling mechanism (large D), we do not expect the wavefunction to be greatly distorted in the Franck-Condon region. So, the anharmonicity effect might not be so rmportant in the tunneling mechanism if the V-T process remains almost unchanged. At D = 12.86 (non-tunneling mechanism) the anharmonicity effect becomes larger as H is substituted by D and T- l?io(A)/f’;O(H) = 1 .16,2.04, and 1.98 for HCN, DCN. and TCN, respectively. The difference in the sensitivity of each molecule to anharmonicity is reflected in the isotope effect: l?io(DCN)/riO(HCN) = 0.796 for the harmonic potential and Fjo(DCN)/ Fio(HCN) = 1.40 for the anharmonic potential. This result suggests the importance of anharmonicity even for qualitative conclusions regarding the isotope effect. In the non-tunneling mechanism it is difficult to prechct the turnover in the isotope effect, since it is not clear how the local distortion in h,, depends on the isotopic substitution. In contrast, when the tunneling mechanism [ 1,4,10] applies (e.g., D = 190), one can easily predict Fio(TCN) < l?iO(DCN) < Fio(HCN), since the penetration of the wavefunctron into the potential barrier is larger for the lighter atom.
large contributions from <,y(l-> and C,, (n # m) [see eq. (6)]. For example, +, has a large contribution from 5yhb and @hb. The effect_ of anharmonicity on the Franck-Condon overlap
AcknowIedgement This work was supported by National Science Foundation Grant CHE78-18070. The computer time made available by the NDSU Computer Center is gratefully acknowledged_
References III Y-B. Band and K.F. Freed, Chem Phys Letters 28 (1974) 328; J. Chem. Phys. 63 (1975) 3382 [21 K-E. Holdy and L.C. Klotz, J. Chem. Phys. 52 (1970)
131 I41
4. Concluding remarks It is necessary for any dynamical theory to include anharmonicity effects when the anharmonicity itself (o,x,/x,) IS large or when large geometry distortIons occur on photoabsorption. It has been shown in this paper that re can also be strongly modified by the anharmonicity even if the transition probability of the corresponding V-T process is virtually unchanged. Finally, it is noteworthy that the strength of anharmonicity effects increases with the importance of interchannel coupling. In the case of strong interchannel coupling (e.g., HCN, DCN, and TCN at the wavelength considered in this paper), the wavefunction X, has
1 March 1981
[Sl [61 171 ISI I91 II01 [III
4588; M. Shapiro and R D. Levine, Chem. Phys. Letters 5 (1970) 499. S.A. Rice, Excited states, Vol. 2 (Academic Press, New York, 1975) p. 111; W-M. Gelbart, Ann. Rev. Phys. Chem. 28 (1977) 323. K.F. Freed and Y.B. Band, Exited states, Vol. 3 (Academic Press, New York, 1977) p_ 109. M. Shapiro, Chem. Phys. Letters 46 (1977) 442, J.A. Beswick, M. Shapiro and R. Sharon, 3. Chem. Phys. 67 (1977) 4045. D. Secrest, Ann. Rev. Phys. Chem 24 (1973) 379. R J. Gordon, J. Chem. Phys. 65 (1976) 4945; 67 (1977) 5923. and references therem. D. Secrest and B.R. Johnson, J. Chem . Phys 45 (1966) 4556. K. Takatsuka and MS. Gordon, Expansion Approach to Photodissociation Dynamics, I, J. Chem. Phys., to be pubLished K. Takatsuka and MS. Gordon, Expansion Approach to Photodissociation Dynamics. II, J. Chem. Phys , to be published. K. Takatsuka and T. Fueno, Phys. Rev. A19 (1979) 1011,1018.
331
Volume 78, number 2
CHEMICAI. PHYSICS LE’ITERS
[ 121 T. Kondow, T. Nagata and K. Kuchitsu, private communiC-&Oil.
[X 3 J G.J. Vazques and J -F. Gouyet, Chem. Phys. Letters 65
(1979) 51.5; P.J. Hay, pnvate communzation. (14] J.P. Simons, Gas Kinetics and Energy Transfer, Vol. 2, Specldlist Periodica Reports (Chemxcal Society, London, 1977) p. 58;
332
1 March 1981
&f.N.R. Ashfold, &f.T. MacPherson and 3-P. Simons, Cbem. Phys. Letters 55 (1978) 84. IIs] G. Her&erg, Spectra of diatomic molecules (Van Nostrand, Princeton, 1950); Ehctrffnic structure of polyatomic molecules Wan Nostrand, Princeton, 1965). 1161 G. Das, T. Jams and A.C. W&i. J. Chem. P&s. 61 (1974) 1274. [17] K. Takatsuka and MS. Gordon, unpublished.