J . Phys. Chem. 1985,89, 214-222
214
FEATURE ARTICLE Nonadiabatic Unimolecular Reactions of Polyatomic Molecules M. Desouter-Lecomte, D. Dehareng, B. Leyh-Nihant,?M. Th. Praet, A. J. Lorquet, and J. C. Lorquet* DZpartement de Chimie, Universite de LiZge, Sart- Tilman, B-4000 LiZge, Belgium (Received: April 2, 1984; In Final Form: August 1 , 1984)
The nonadiabatic couplings which arise when two potential energy surfaces of a polyatomic molecule get close in energy can be classified as follows: (A) avoided crossings, (B) genuine intersections(Jahn-Teller and conical), (C) glancing intersections (Renner-Teller interactions). The characteristics of the potential energy surfaces in the adiabatic and diabatic representations are discussed for each case. The three coupling cases differ in the structure of the Hamiltonian matrix. When the latter is written in the diabatic representation, it is meaningful to retain the leading term only in its power series expansion. This gives rise to a so-called minimum-order model which is found to be surprisingly accurate (at least in a restricted zone of nuclear coordinates) when compared to the results of ab initio calculations. The characteristic features of each coupling case can only be understood in a two-dimensional configuration space, Le., when two nuclear degrees of freedom, often with different symmetry properties, are explicitly considered. A simple expression of the nonadiabatic transition probability between two electronic states can be worked out in the framework of the minimum-order models. Two-dimensional extensions of the Landau-Zener formula are obtained, which can be used to study the consequences of the anisotropic properties of the coupling. In the case of avoided crossings, only nuclear trajectories having a well-defined direction are able to bring about surface hopping, wheras there exists two active degrees of freedom for conical intersections. Hence, nonadiabatic processes which are controlled by genuine intersections are expected to take place faster than those controlled by avoided crossings.
I. Introduction The study of the dynamics of unimolecular reactions is a vast undertaking which can be tackled by one of the three leading formulations of theoretical physics, classical, quantum, and statistical, or possibly by an appropriate combination of two of these methods. In any case, a knowledge of the potential energy surface@) involved in the molecular processes under study is required as input. The Born-Oppenheimer approximation' provides precisely such surfaces, or at least a convenient starting point in this problem. However, it is well-known that difficulties are encountered in this respect whenever two Born-Oppenheimer surfaces approach, cross, or touch each other. In such regions of space, the Born-Oppenheimer approximation becomes invalid, and a description in time of the system requires the introduction of the concept of radiationless transition from one surface to another. It is the aim of the present paper to discuss these problems which are of frequent occurrence in such fields as photochemistry, mass spectrometry, molecular collisions, etc. What are the prospects offered by the'theoretical formulations one can envisage? The classical approach leads to the surface hopping trajectory calculations developed by Tully and Preston.2 Since an exhaustive knowledge of the potential energy surfaces has to be obtained, the method is in practice applicable to molecular processes involving three or four atoms only. In spite of great efforts and important successes3 the quantum approach proves to be very difficult and is currently limited to simple models and coupling cases. It seems therefore clear that, for a long time to come, nonadiabatic chemical processes involving four atoms or more will have to be studied by very approximate methods. In particular, the use of statistical approximations is likely to remain necessary. An attempt to incorporate the theory of nonadiabatic coupling in the semiclassical approximation into the transition-state theory has been proposed and appears It is with such a goal in mind that the present account is written. At the simplest level of approximation, the problem can be seen as a systematic stepwise process which requires the sequential Boursier IRSIA.
0022-3654/85/2089-0214$01.50/0
calculation of the following quantities: (i) relevant potential energy surfaces at the ab initio level, (ii) nonadiabatic coupling matrix elements, (iii) transition probabilities, (iv) rate constants. Only the last are observables; the first three quantities should be considered as necessary prerequisites in the calculation of rate constants. A proper discussion of the problem requires a classification of the different types of nonadiabatic interaction which can be encountered. The following one can be proposed: (A) avoided crossings; (B) genuine intersections (encountered in Jahn-Teller interactions and in conical intersections); (C) glancing intersections, Le., contacts between two tangent surfaces (they are encountered in Renner-Teller interactions). Since our main interest is in chemical kinetics, we shall consistently restrict ourselves to the simplest possible level of approximation and shall ignore many of the additional subtleties which are necessary in the interpretation of spectroscopicproblems. An approximation will be adopted which consists in treating the electrons quantum mechanically whereas the nuclear motion is described by a classical trajectory and its quantum nature ignored.z&" This approximation, often referred to as semiclassical, should be distinguished from another semiclassical approach developed by Miller, Heller, and others.12 We shall limit ourselves (1) OMalley, T. F. Adu. At. Mol. Phys. 1971, 7, 223. (2) Preston, R. K.; Tully, J. C. J . Chem. Phys. 1971, 54, 4297. Tully, J. C.; Preston, R. K. Ibid. 1971, 55, 562. Tully, J. C. 'Dynamics of Molecular Collisions", Part B; Miller, W. H., Ed.; Plenum Press: New York, 1976. (3) Gelbart, W. M. Annu. Reu. Phys. Chem. 1977, 28, 323. (4) Zahr, G. E.; Preston, R. K.; Miller, W. H. J . Chem. Phys. 1975, 62, 1127. (5) Kinnersly, S . R. Mol. Phys. 1979, 38, 1067. (6) Raet, M. T.; Lorquet, J. C.; Rageev, G. J . Chem. Phys. 1982,77,4611. (7) Desouter-Lecomte, M.; Sannen, C.; Lorquet, J. C. J . Chem. Phys. 1983, 79, 894. (8) Hellman, H.; Syrkin, J. K. Acra Physicochim. UFSS 1935, 2, 433. (9) Nikitin, E. E. 'Theory of Elementary Atomic and Molecular Processes in Gases";Kearsley, M. J., Trans.; Clarendon: Oxford, 1974. (IO) Delos, J. B.; Thorson, W. R. Phys. Reu. A 1972, 6 , 728. (11) Desouter-Lecomte,M.; Lorquet, J. C. J . Chem. Phys. 1979, 7, 4391.
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 2, 1985 215
Feature Article in the case in which only two electronic states interact. This does not exclude the possible existence of more complicated cases, provided they can be treated as a succession of independent zones of nonadiabatic interaction. Rotation will be ignored, and the molecule will be treated as a collection of (3N - 6) oscillators. However, the decisive simplifying step comes from a decision to retain the minimum number of terms in a series expansion to be defined later. This gives rise to so-called minimum-order models which have already been discussed by various author^.'^-'^ In the present paper, we plan to compare the predictions of these models with the results of ab inito calculations on real molecules. The models will be found to be surprisingly accurate in their predictions, at least in the nonadiabatic coupling zone. In addition, they are simple (since they depend on two parameters at most), practical (because the parameters can be determined with a minimum amount of ab inito calculations), and convenient (because a simple solution to the transition probability problem is available for each of them).
U,(t) =
U,(t) -. = I
-($ ) a l [ H 1 2- i h g Q ] exp
gjCQj) E (42la/dQjl$l)
(2.4)
(2.5)
Let us now consider in turn the two particular choices which bring about a simplification of the problem. ( 1 ) The Adiabatic Basis (+i(q,Q)).By definition, this basis set diagonalizes He’. The coupling then results entirely from the matrix elements (2.5). Given an arbitrary (but real) basis set ( 4 1 , ~this 2 ) ,diagonalization can always be parameterized by an angle O(Q) such thatI9
(J-( -
In the semiclassical approximation adopted here, the nuclei are
+
1
where g(Q) is a vector whose components are
11. Electronic Basis Sets
treated as classical particles executing a trajectory Q ( t ) in a (3N - 6)-dimensional configuration space. The electrons are described by a time-dependent wave function which is expanded8-” in a hitherto unspecified basis set of orthonormal electronic functions, noted ($j(q,Q)).If Nd is the dimension of the basis set, one has
(Hll - H22)dt
cose(Q) -sine(Q)
)( )
sine(Q) cose(Q)
@I
&
(2.6)
with O(Q) = (1/2) arctan [ ~ H I ~ ( Q ) / H -~ Hii(Q)I ~(Q)
(2.7)
The angle O(Q) plays a fundamental role in the theory, as will be seen later on. From eq 2.1 it is easily demonstrated that ($2lvQl$l)
where Hkl = ($k/Hcl($l) are the matrix elements of the usual electronic Hamiltonian. The informative quantities are the time-dependent amplitudes aj(t) to be in electronic statej at time t . Substituting into the time-dependent SchrGdinger equation, one gets a set of Ndcoupled differential equations for these amplitudes:
=
dt
=
3y ( j
&)Qj
= VQ.Q
(2.3)
where VQ represents the vector operator ( d / d Q I ,..., d/dQzhi6). The theory is greatly simplified if the summation (2.1) is restricted to two terms, Le., if Nd can be taken equal to 2. Let us assume this to be the case, and let us consider an arbitrary two-dimensional basis set consisting of two real functions Equation 2.2 then becomes Miller, W. H. J. Chem. Phys. 1970,53, 1949; 1971,55, 3150. Heller, Chem. Phvs. 1975. 62. 1544 and references therein. Longuet-Higgins, H. C.Adu. Spectrosc. 1961, 2, 429 Liehr, A. D. J. Phys. Chem. 1963,67, 389,471. Carrington, T. Acc. Chem. Res. 1974, 7, 20. Davidson, E. R. J. Am. Chem. SOC.1977, 99, 397. Davidson, E. R. J. Phys. Chem. 1983, 87, 4183. KBppe.1, H.; Cederbaum, L. S.;Domcke, W. J. Chem. Phys. 1982, 77,
(2.8)
(2) Diabatic Basis Sets (Xi(q,Q)).As an alternative choice, we define a diabatic basis set (Xi(q,Q))as diagonalizing the V Qoperator,8*20Le., by the equation (XilVQlxj) = 0
(2.9)
In the two-state case, this leads (upon substitution into eq 2.5 and 2.8) to a well-known condition: ~QO(Q) =
In a case in which no nonadiabatic coupling exists, the BornOppenheimer approximation is valid. Nd can be taken equal to one; Le., expansion (2.1) reduces to a single term. If c o w i n g takes place, the expansion is necessary. Two kinds of matrix elements are seen to be responsible for the variation of the amplitudes: the off-diagonal elements of the electronic Hamiltonian and those of a differential operator which can be written
- VQO(Q)
(+ZlvQl#l)
($zlv~l+~) = g(Q)
(2.10)
The same equation is obtained, at least in the two-state case, if the quantum nature of the nuclear motion is taken into account.2’,22 However, it has been shown by Mead and T r ~ h l a r ~ ~ that, for a polyatomic molecule, eq 2.10 has a solution only if the dimension of the electronic basis set is rigorously limited to Nd = 2. If the two states on which attention is focused are extracted from a large secular equation of dimension Nd (as is always the case in an ab initio calculation), a rigorous solution exists only if all the matrix elements of VQbetween any of the two adiabatic states under study and the remaining ones are equal to zero. This is of course never strictly the case in the whole range of internuclear distances, but the troublesome matrix elements are often found to be negligibly small in the range of internuclear distances dominated by a two-state nonadiabatic interaction. Under such conditions, it is acceptable to use eq 2.10 to obtain approximate diabatic states which are satisfactory for all practical purposes. A more physical way of interpreting eq 2.9 is to associate diabatic states xi with a well-defined electronic configuration (e.g., ionic, covalent, Rydberg, etc.).’ In quantum chemical language this implies that a diabatic wave function is a single Slater determinant or a linear combination of Slater determinants with coefficients which remain constant throughout the entire zone of nonadiabatic interaction. Then, even if eq 2.10 is not exactly (19) Cohen-Tannoudji, C.; Diu, B.; Laloe, F. “Mecanique quantique”; Hermann: Paris, 1977; Vol. I, p 418. (20) Smith, F. T. Phys. Reu. 1969, 179, 111. (21) Baer, M. Chem. Phys. Lett. 1975, 35, 112. Baer, M. Chem. Phys. 1976, 15, 49. (22) Chapuisat, X.; Nauts, A.; Dehareng, D. Chem. Phys. Le??.1983, 95, 139. (23) Mead, C. A.; Truhlar, D. G. J Chem. Phys. 1982, 77, 6090.
216 The Journul of Physical Chemistry, Vol. 89, No. 2, 1985
soluble, it is nevertheless possible to define a diabatic wave function by requiring its derivative to be as small as possible. Diabatic states are then as smooth functions of the internuclear distances as the physics allows. A practical method of satisfying this criterion in the framework of ab initio calculations has been proposed by HendekovkZ4
Desouter-Lecomte et al. Multiplication on the left and on the right by the two electronic functions ( k Jand (11 leads to a power series of the matrix elements of the electronic Hamiltonian: 3N-6
H d Q ) = (klwI(q,Qo)ll)+
EJ
(klHj’(q,Qo)ll)Qj +
111. Potential Energy Surfaces The adiabatic potential energy surfaces E , (J/ilH”lIJ/i)are obtained as the eigenvalues of the diabatic matrix Hv E (xAH“‘lxj). In the two-state problem f
This expansion can be carried out whatever the basis of electronic functions. However, even if the best diabatic states which can be obtained are not strictly independent of the nuclear disEi(Q) = tances, the matrix elements of the electronic Hamiltonian in the ( ~ / ~ ) [ H I I +( QWQ)1 ) ( 1 / 2 ) [ W Q Y + ~ H I Z ~ ( Q ) I ~ diabatic ’~ basis can be expected to be smooth functions of the (3.1) internal coordinates. It is therefore meaningful to truncate expansion (4.3) after its leading term.15 This would not be apwith propriate for the adiabatic basis set which is a very sensitive (3.2) U ( Q=)H22(Q) - HII(Q) function of the nuclear geometry. Historically, this model was first proposed by Zenerz6 for a By definition, the intersection locus M(Q)= 0 is called the diatomic molecule and by Tellerz7 in the case of a conical inseam between the two diabatic surfaces. As stated by the tersection. However, for a polyatomic molecule, the simplest well-known noncrasing rule, adiabatic surfaces cannot cross unless situation is encountered in avoided crossings, and we are going two conditions are simultaneously fulfilled: to examine them first. The case of conical, Jahn-Teller, and AH(Q) = 0 and H 1 2 ( Q )= 0 (3.3) Renner-Teller interactions requires additional considerations which will be dealt with in subsequent sections. This circumstance gives rise to a classification of the nonadiabatic coupling cases. V. Avoided Crossings ( 1 ) The adiabatic potential energy surfaces never cross or touch Potential Enera Surfaces. We restrict ourselves for simplicity in the whole configuration space because H12# 0 everywhere to a two-state interaction in a two-dimensional configuration space along the seam. This situation gives rise to an uuoided crossing. [Q,Q,,]. The two electronic states belong everywhere to the same For the other cases, the conditions ( 3 . 3 ) are fulfilled at certain irreducible representation, and there is no accidental degeneracy particular points of configuration space. However, the circumanywhere. If the matrix elements of the electronic Hamiltonian stances which give rise to the degeneracy can be different and one in the diabatic representation are expanded according to eq 4.3, has to distinguish several cases. one gets, up to first order ( 2 ) The two adiabatic surfaces cross and are discontinuous at their crossing point. The wave functions also exhibit a discontinuity. This case gives rise to a genuine intersection and is further In the linear approximation, the diabatic surfaces are planes subdivided into two subcases. If the discontinuity is accidental, which cross along a particular straight line called the seam. The a so-called conical intersection is obtained. If the degeneracy is problem is Simplified if the expansion is made about a point located enforced by symmetry, one has a Juhn-Teller interaction. on that line. With such a choice of coordinates the zero-order ( 3 ) If the two adiabatic surfaces are degenerate but remain term vanishes in the expansion of the diagonal elements continuous everywhere, a glancing intersection is obtained. The wave functions are differentiable at the point of contact. This case corresponds to the Renner-Teller interaction. The Longuet-Higgins theorem25provides a criterion to deterbut there is no reason for this term to vanish in the case of an mine the particular situation one is dealing with. It says, “If an off-diagonal element, and we truncate immediately after it.z8 An adiabatic wave function J/(q,Q)changes its sign after a one-diavoided crossing is then said to obey the minimal-order model if mensional closed loop in configuration space, there exists necthe Hamiltonian matrix in the diabatic representation has the form essarily within the loop a point where J/ is discontinuous. Hence, the corresponding state must be degenerate with another one at that point”. The inverse is, however, not true. Surfaces can be degenerate at a particular point although the wave functions The seam between diabatic surfaces then obeys the following remain continuous. This is precisely the property which distinequation: guishes glancing (Renner-Teller) from genuine (conical and Jahn-Teller) intersections, as will be seen in sections VI, VII, and Qy = -[(a2 - ~ 1 ) / ( b 2 h ) l Q x I -(a/b)Qx (5.3) VIII. The shape of the corresponding adiabatic surfaces is that of IV. Minimum-Order Models a hyperbolic cylinder oriented along the seam (Figure 1 ) : AE Let us expand the electronic Hamiltonian operator in a power is minimum and constant along that particular direction. Adiaseries around some particular point Qo. batic and diabatic surfaces coincide asymptotically except along the direction of the seam. Coupling Functions. In the minimal-order model one has, from eq 2.7 6(Qx7Qy)= ( 1 / 2 ) arctan [2ao/(uQx +
where the primed quantities represent derivatives of the Hamiltonian operator. For example
The loci along which 6 remains constant (tan 26 = c) are a set of straight lines obeying the equation QY
(24)HendekoviE, J. Chem. Phys. Lett. 1982, 90, 193. HendekoviE,J. Croat. Chem. Acta 1983,56, 375. (25)Longuet-Higgins,H.C.Proc. R. Soc. London,Ser. A 1975,344,147.
(5.4)
= -(a/b)Qx + 2ao/bc
(5.5)
(26) Zener, C.Proc. R. SOC.London,Ser. A 1932,137,696. (27)Teller,E. J . Phys. Chem. 1937,41, 109. (28)Stine, J. R.;Muckerman, J. T. J. Chem. Phys. 1976,65, 3975.
The Journal of Physical Chemistry, Vol. 89, No. 2, 1985 217
Feature Article
fix
I \ d=37O
ii 6.0
X
Figure 1. Schematic view of the adiabatic and diabatic potential energy
\
surfaces in the neighborhood of an avoided crossing in the minimumorder model.
2.01
X
,/"
/
Figure 3. Ab initio calculated coupling matrix element (2la/aRIl) between two 2Alstates of the CH2' ion for a constant value of the valence angle a: crosses, numerical results; full line, fit to the Lorentzian pre-
dicted by the minimum-order model.
'i
RIA)
I
35
LO
45 dldrg.150
Figure 2. Avoided crossing between two 2A1states of the CH2' ion. Ab initio calculated contour map of the adiabatic energy gap (in atomic units) as a function of the symmetrical bond stretch coordinate R and valence angle a.
The seam corresponds to the locus along which 8 = a / 4 . It provides a convenient reference direction. The angle 8 is a continuous function of the coordinates. In conformity with the Longuet-Higgins theorem:5 the adiabatic functions do not change their sign after an arbitrary closed loop in the plane Q,,Q,. The coupling vector g has a component which is equal to zero along all the parallels to the reference direction, since B is then constant (eq 2.10). Its components in every other direction have a Lorentzian shape with a height which is maximum for a displacement perpendicular to the seam. This has important consequences on the transition probability. Ab Initio Calculations. Two 2Al states of the CH2+ion exhibit a clear case of avoided crossing which has been detected by Sakai et al.29and further studied by the present authors. The contour map of the adiabatic energy difference A E is given in Figure 2 as a function of the symmetrical bond stretching R and the angle of bending a. In conformity with the model, the contours are a set of straight lines which are all parallel to the seam between the diabatic surfaces. The coupling functions g can also be calculated a b initio by numerical differentiation of the configuration interaction wave f u n c t i ~ n . ~They ~ . ~ are ~ found to exhibit a Lorentzian shape with an area equal to r / 2 , as they should. A typical example is given in Figure 3. The value of the angle 8 is obtained by a two-dimensional numerical integration of the g functions:
(5.6) It has been checked that the value of 8 at a particular point is independent of the path leading to that point. The map of the 8 function is given in Figure 4. The agreement with the predictions of the minimum-order model is very good in the range a = [35O,43O]. The model ceases to be valid for larger values of a. (29) Sakai, S.;Kato, S.; Morokuma, K.; Kusunoki, J. J. Chem. Phys. 1981,
75, 5398.
(30) Galloy, C.; Lorquet, J. C. J . Chem. Phys. 1977, 67, 4672. (31) Dehareng, D.; Chapuisat, X.;Lorquet, J. C.; Galloy, C.; Ragccv, G. J . Chem. Phys. 1983, 78, 1246.
A
44
0"
1.8
I
I
I
1 - 8
35
I
I
I
I
Lo
I
I
I
I
I
,
1
,
45 &(de;)
Figure 4. Map of the 0 function characterizing the avoided crossing between two *Al states of the CH2+ion. The symbols indicate the values obtained by a two-dimensional integration of ab initio coupling matrix elements. They have been fitted to straight lines, as predicted by the minimum-order model.
VI. Degeneracy and Symmetry The study of cases in which degeneracies are encountered (Le., conical, Jahn-Teller, and Renner-Teller interactions) brings about additional complications, as might be expected. A nondegenerate electronic adiabatic wave function $(q,Q) must be a continuous function of the nuclear coordinates Q. On the other hand, if +(q,Q) becomes degenerate at some point Qo,it may (but also may not) become discontinuous a t that point. If it does (as in conical and Jahn-Teller intersections), the coupling function g becomes infinite at Qo,whereas the angle 8 becomes discontinuous a t that point and multivalued modulo r elsewhere. It is easily seen from eq 2.6 that such a multivaluedness of 8 brings about a mutivaluedness of the wave function and an indeterminateness of its sign. The latter is at the origin of the sign reversal after a closed loop in configuration space predicted by the LonguetHiggins theorem. The sign reversal in the electronic wave function necessarily brings about a corresponding reversal of the sign of the nuclear wave function, so that the total wave function remains single valued. This point is discussed in ref 23 and 32, where an optional (32) Mead, C. A. J . Chem. Phys. 1983, 78, 807.
218
The Journal of Physical Chemistry, Vol. 89, No. 2, 1985
procedure to make nuclear and electronic wave functions single valued is indicated. This is certainly a valuable point in treatments in which the quantum nature of the nuclear motion is taken into account, but it need not appear in the simpler semiclassical method developed here. When possible, symmery considerations lead to a simplification of the expansion (4.3). The matrix elements of H/ (or H i ) appear only if coordinate Q, (respectively the product Q,Qj)transforms as the product k*l of the electronic wave functions under the symmetry operations of the appropriate point group. Let us call Q the total configuration space and Qo the subspace where the adiabatic energy surfaces are degenerate. At points Qo,the appropriate point group is SO;it may correspond to a higher symmetry than the point group S appropriate to the neighboring conformations. The wave functions, adiabatic or diabatic, can be classified according to the irreducible representations of either Soor S. Likewise, it turns out to be convenient to choose for Q generalized coordinates (i.e., normal or symmetry coordinates) which belong to irreducible representations of S,. For the optimum use of symmetry considerations, it is appropriate to adopt complex nuclear coordinates and to define a basis set of complex electronic function^.^^*'^ This will be done in the next section.
VU. Jahn-Teller and Conical Intersections There exists a great similarity between these two kinds of interactions. The difference lies in a loss of symmetry when going from the Jahn-Teller to the conical coupling, so that an essential degeneracy in the former case becomes accidental in the latter. Jahn-Teller Intersection. Let us consider a triatomic molecule made up of three identical atoms. The reference configuration Qothen belongs to point group D3*. Three sets of nuclear coordinates will be introduced. (i) The set of real normal coordinates Q, = (Q1, Q,, Q,), where Qlis the symmetrical stretch (Al’ representation) and Q, and Qy are a set of doubly degenerate normal coordinate^'^ (E’ representation). (ii) A set of complex coordinates Q, = (Q,,Q+,Q-)defined by
(iii) A set of polar coordinates Qp = P =
+
(Qxz
4 = arctan
