J. Phys. Chem. 1982, 86, 2182-2187
2182
Calculation B included both types of correlation simultaneously. Based on the results of calculation A, we expect the calculation B configuration lists to underestimate the 2a - 4a correlation and ouerestimate the 2a - 2a correlation difference. Thus, calculation B should yield an approximate lower limit to the binding energy. Indeed, the two correlation effects cancel almost exactly. These two calculations predict a binding energy in the range 88-99 phartrees with a probable numerical error of f 2 phartrees. We certainly do not claim that these preliminary results represent that the limit of correlation in these systems, but they do indicate the size of the effects. Interestingly, as percentages of the total binding energy, the correlation effects in the LiH- excited state are roughly the same as in the ground state. Hence, based on this single case, correlation would appear to be no more nor no less important for the excited states of these negative ions than for the ground states. These examples have indicated the utility of our numerical MCSCF method. It is clear that this method is no replacement for the basis set method, especially since it can only be applied to diatomics. Rather, we view it as a complementary approach. The interaction between the two can be synergistic, as we have tried to illustrate here, thereby leading to an overall improvement in our collective ability to make theoretical chemical predictions of quantitative accuracy.
TABLE VIII: MCSCF Binding Energy Calculations for LiH- First Excited States AERHF + AE*u-2o + ‘ZU-4U
Calculation A 86 Fhartrees --11 phartrees 24 phartrees
-
- 99 phartrees
LiH‘
LiH
Calculation B 1u2(2u2t 3 0 ’ + 50’ t 2030 + 3a5o)4u t 10’20(607u + l n 2 n ) + 1u24u25ut lO’(20* + 5u2)3u l u 2 ( 2 u 2+ 30’ t 50’ t 2030 + 305u)
8.003248 hartrees
- 8.003160 hartrees 88 phartrees
A direct determination of the correlated binding energy would require separate correlated calculations on the ion and neutral to an absolute accuracy of a few microhartree. This is clearly impractical. Instead, we performed less accurate calculations, correlating both systems in exactly the same way. Two different versions of this procedure were tried to assess its reliability. Our results are shown in Table VIII. In all calculations orbitals la-4a were held frozen, with 3a and 4a being obtained from frozen core calculations on the LiH- ground state and excited state, respectively. All other orbitals were optimized for the second eigenvalue of the Hamiltonian. In calculation A, we attempted to compute the 2a - 4a correlation energy and 2a - 2a correlation energy difference separately using sequences of configuration lists of increasing length. The 2a - 4a correlation energy is about 25% of the total binding energy, and the 2a - 2a correlation energy difference is a negative quantity of comparable magnitude. Adding these to the Hartree-Fock energy difference we obtain an estimate of approximately 99 phartrees for the binding energy.
Acknowledgment. I acknowledge my collaborators, Dr. P. A. Christiansen and Dr. Ludwik Adamowicz. They deserve most of the credit for our work. Thanks are due to many of my colleagues for numerous helpful discussions. I particularly thank Professor C. F. Fischer for her assistance, especially in running numerous atomic test problems for us, and Professor F. E. Harris for providing us with access to excellent computational facilities. This research has been generously supported by the National Science Foundation under Grant CHE 7608768.
Theory of Poiyatomic Photodissociation. Adiabatic Description of the Dissociative State and the Translation-Vibration Interaction Vladimlr 2. Kresln*+ and William A. Lester, Jr.” National Resource for Computatlon in Chemistry, Lawrence Berkeley Laboratoiy, University of California, Berkeley, California 94720 (Received: August 7, 1981)
We develop an adiabatic method for the evaluation of the nuclear wave function describing the final dissociative state. The vibrational frequencies of the fragments are found to depend on the interfragment distance. We discuss the role of the translation-vibration interaction in polyatomic photodissociation.
I. Introduction The photodissociation of a polyatomic molecule is a radiation-induced transition from a ground or a quasidiscrete state to a dissociative (D) state. Both the former case (direct photodissociation) and the latter case (indirect *Permanent address: Materials and Molecular Research Division, Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720. t Also Department of Chemistry, University of California, Berkeley, CA. 0022-3654/82/2086-2182$01.25/0
photodissociation)are characterized by a final D state that contains discrete and continuous parts. To describe the dynamics of polyatomic dissociation including photofragment energy distributions requires an indepth understanding of the D state. In this paper we direct our consideration of polyatomic photodissociation to a detailed characterization of this state. For concreteness we limit attention to indirect photodissociation, but the present development is also applicable to direct photodissociation. The D state is usually described by a nuclear wave function which is a product of two independent functions, @ 1982 American Chemical Society
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Theory of Polyatomic Photodissociation
one dependent only on the interfragment distance, and the other dependent solely on internal coordinates; see, e.g., recent reviews’ and the discussion in ref 2. This separated variables approximation (SVA) is strictly valid only in the asymptotic region. Almost 50 years ago in a classic paper, Landau3 raised the problem of the description of the D state. He noted that a rigorous description of this state should contain the adiabatic coupling of the bound and continuous parts of the wave function. More recently in a significant paper, Band and Freed4 introduced the notion that the internal coordinates are adiabatically dependent on a reaction coordinate which asymptotically becomes the interfragment distance. In this paper we develop a rigorous description of the D state following an adiabatic formulation. In section I1 we describe the method of evaluation of the photodissociation rate and consider the D state explicitly in section 111. In section IV we discuss the effect of the translation-vibration interaction and its role in determining the final D state. 11. Background In indirect photodissociation the absorption of light leaves the molecule in an intermediate quasidiscrete (Q) state of finite lifetime. The Q state undergoes a radiationless transition to a final D state. Radiationless transitions are associated with deviations from the Born-Oppenheimer adiabatic approximation and are described by the corresponding operator, namely (see, e.g., ref 5 and 6)
Here 4Dand 4Qare the nuclear wave functions of the D and Q states, ( Qi)are the normal coordinates of the Q state, and Li are the electronic factors given by
where $#Q(?,.@ are the electronic wave functions and do denotes the equilibrium configuration qf the Q Gate. The correctness of the approximation Li(R) N Li(Ro)is connected with the Franck-Condon approximation and the existence of a small parameter 6 AIL. Here A is the vibrational amplitude and L is the bond length. It is, of course, necessary to take into account the degeneracy of the Q and D states? Note also, that because a polyatomic system has several nuclear degrees of freedom, the Q D transition can occur even when both states belong to the same potential energy surface. The D state is characterized by the vibrational and rotational distribution of the molecular fragments. Both distributions are governed by the state of the nuclear subsystem. For this reason we shall focus on the nuclear part of the matrix element (5). To evaluate this factor one must, of course, first specify the nuclear wave functions I#JQ and 4D. The wave function for the Q state usually can be described by the harmonic approximation
-
4Q = n @ v , ( Q i ) i
en” = C(Yi
The notation a/adli, implies that d/d& operates_only on the electronic wave function and, similarly, d/dRll,,,ue operates only on the nuclear wave function. The summation is over nuclei 1. As a transition into the continuous spectrum, the Q D transition probability is given by (see, e.g., ref 7)
-
where HbQ,the coupling matrix element, is
and pE is the density of states. Introducing the Condon approximation permits the following simplification (see also ref 8)
HbQ= CLiNi i
(4)
i
(7)
+ 1/2)hq
where Qi are the normal coordinates, and 4JQi) is a harmonic oscillator wave function with normal frequency wi. The applicability of the harmonic approximation can be tested by consideration of the experimental data. If the absorption spectrum is a set of equidistant bands, the harmonic approximation should be applicable (see, e.g., the investigation of Miller et al.’O of the photodissociation of C2N2). If there is variable spacing between absorption bands, then it is necessary to include anharmonicity corrections. The function @ also contains any rotational dependence. The evaluation of the function 4Dpresents a more difficult problem. We now turn to this problem. 111. The Dissociative State A . Nuclear Hamiltonian. Statement of the Problem. The nuclear wave function 4Dis a solution of the nuclear Schrodinger equation
where
HN4D
=E4D
(8)
where the Hamiltonian HNis given by (1) (a) S. Rice in “Excited States”, Vol. 2,E. Lim, Ed.,Academic Press, New York, 1975,p 111. (b) W. Hase, “Dynamicsof Molecular Collisions”, p. b., Plenum Press, 1976,p 121. (c) K. Freed and Y. Band, ‘Excited States”, E. Lim, Ed., Academic Press, New York, 1977,p 110. (d) W. Gelbart, Annu. Reu. Phys. Chem., 28, 323 (1977). (e) J. Simons in “Gas Kinetics and Energy Transfer”, Vol. 2,P. B. Ashmore and R. J. Donovan, Senior reporters, Burlington House, London, 1977,p 58. (0 M. Okabe, “Photochemistry of Small Molecules”, Wiley, New York, 1978. (9) S. Leone, Adu. Chem. Phys., in press. (2)J. Beswick and W. Gelbart, J. Phys. Chem., 84,3148 (1980). (3)L. Landau, Phys. Z . Sowjetonion, 1, 88 (1932). (4)Y. Band and K. Freed, J. Chem. Phys., 68, 1292 (1978). (5)R. Kubo, Phys. Reu., 86, 929 (1952). (6)J. Ziman, R o c . Cambridge Phil. Soc., 51, 707 (1955). (7)L. Landau and E. Lifshitz, “Quantum Mechanics”, Pergamon Press, New York, 1976. (8)Y. Band and K. Freed, J. Chem. Phys., 63, 3382 (1975).
HN =
?A
+ P B + U(d)
(9)
Here ?A and P B are the kinetic eGergy operators of fragments A and B and U(R) = En(R) is tbe potential energy of nuclear motion. The function E,(R) is the potential energy surface for the nth electronic state. It is convenient to write the Hamiltonian HNin the form H N
=
Pint+ ?; + U(p,qf,q?)
(10)
(9)C. Caplan and M. Child, Mol. Phys., 23, 249 (1972). (10)G.Miller, W. Jackson, and J. Halpern, J. Chem. Phys., 71,4625 (1979).
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Kresin and Lester
where The interfragment distance p does not appear as an explicit expansion variable. The term HZ can be reduced to the form where q f and gy ar_eth_eintern-. coordinates of fragments A and B, ij = R B - R A ( R A and R B correspond to the centers of mass of the separate fragments), and 1.4 = MAW/(MA MB)is the reduced mass of the total system. The potential energy is conveniently expressed in terms of p, q f , and 48. If, for instance, we consider the simplest polyatomic system, a triatomic molecule abc which can dissociate into atom a and diatomic fragment bc, we obtain
+
where (eiz]are the normal coordinates (dependent on p ) of the Zth fragment. For photodissociation of a triatomic molecule, we obtain (see eq 13)
where where ij = p i i (ii is a unit vector directed from atom a to the center of mass of the fragment bc), 1 . 4 ~= mbmc/(mb+ m,) is the reduced mass of the fragment, ij = gs' (q is the distance between atoms b and c, s' is a unit vector directed from atom b to atom c). Unlike the Q state, the potential energy for the D state cannot be expanded in a series of the deviations of all variables from equilibrium, because p corresponds to translational motion. Hence, variables are not separable and the function 4D(p,qf,q7)generally cannot be well approximated as a product 4 T ( p ) D~(qi $ A~,qlB). In the asymptotic region this separable product form is valid because the potential energy no longer depends on p. Because of the Franck-Condon principle, the main contribution to the coupling matrix element H'w (see eq 5) is not connected with the asymptotic region, but comes from a region p pot where po corresponds to the equilibrium configuration of the precursor molecule. Hence, it is necessary to take into account the interfragment interaction, that is, the dependence of the potential energy on p. This interaction is not small, in general, and hence the deviation from the SVA is not governed by a small parameter. For this reason, the contribution of higher-order terms might be large and then, correspondingly, the SVA zeroth-order approximation would be unacceptable. Hence a general expression for 4D(see below) can differ noticeably from the zerothorder SVA. We present here a new approach to the determination of @D. Note that because it is often a good approximation to separate rotational from translational and vibrational motion, we do so here and consider first the latter motions of the fragments. B. Slow Translational Motion (mw > Wb). When the Here recoil energy is considerably larger than the vibrational energy, we can again follow the adiabatic approach by neglecting in this case the kinetic energy of internal motion. This approach yields where 4FU= 4F(ij,Q)4;m (27) (39) P = (2P[€(QO)- ~(P,qo)111/2 The function @(ij,q) satisfies the equation
or (see eq 35)
Using the semiclassical approximation, we obtain the following expression for the radial part of 4;: where
P2(P,4) =
The function; ;4
2 / 4 m- U(p,q)l
(31)
of two functions. One depends only on ij and the other only on q. Hence, we conclude that the SVA is valid if internal motion is slower than relative motion. One should note that the vibrational part of the SVA wave function and the function 4"Lb(q)of eq 36 differ in the value of the frequency. The 6VA frequency is given by WSVA wF = [rF-'(a2UAS(q)/dQ2)q=~oF]1/2 (41) where UAS(q) = U(Z,q)lpm.Here W F and qOFare values for the isolated fragment, while the frequency w, is given by eq 34. When eq 31 is used, it is possible to obtain the approximation relation between wc and wSVA. Indeed, we can write
satisfies the following equation:
We can see from (32) that @;(Q) wave function
(40) P = (%[E - U(p,qo) - (u + '/2)hwJ11/2 Note that 4FUof eq 36 has the form of the zeroth-order separated variables approximation (SVA). It is a product
is the harmonic oscillator
(33)
(13)B. Geilikman, Sou. Phys. JETP, 2, 509 (1956);J. Low Temp. Phys., 4,189 (1971). (14)K. Freed and W. Gelbart, Chem. Phys. Lett., 10, 187 (1971). (15)H. Longuet-Higgins, Adu. Spectrosc., 2, 429 (1961).
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2186
Using eq 31 and 34, we obtain w; = wF2[1
+ A]
Kresin and Lester
(43)
where =
(pFpwF2)-’a2b02(q)) /aq21q=qo
= q - q&J)
Let us estimate the value of X
-
(pFpwF2)-1P02/qo2
where qo E L is the bond length. Hence, one can write
-
( ~ o ~ / P ~ /w~ Fw )F(A~~ ) ( A ~ / L ~ )
where A is the vibrational amplitude. Because ”FA2 h we obtain N
a@
cy I ( A p / A p i b
)
-
(44)
Note that the “clamped” translational function approximation dictates that 6 is a small parameter (see also discussion following eq 6), and consequently that X is also small. Thus a large value of a is usually compensated by a small value of 6 with the results that w, WF. E. Intermediate Case and Further Discussion. We have been considering two limiting cases: a > 1 (sections IIIC and IIID). For the intermediate case a 1, it is possible to write the appropriate solution as a linear combination of the functions (20) and (27)l’ with coefficients that are functions of a. The “clamped” translational function approximation (CTFA) (previous section) enables one to obtain another form of the general solution. The construction of this solution is facilitated by carrying out a direct comparison of the “clamped” limiting case a >> 1with the case a
