Multidimensional Canonical Integrals for the Asymptotic Evaluation of the S-Matrix in Semiclassical Collision Theory BY J. N. L. CONNOR Dept. of Chemistry, University of Manchester, Manchester M13 9PL Receiced 29th December, 1972 The uniform asymptotic evaluation of multidimensional integrals for the S-matrix in semiclassical collision theory is considered. A concrete example of a non-separable two-dimensional integral with four coalescing saddle points is chosen since it exhibits many of the features of more general cases. It is shown how a uniform asymptotic approximation can be obtained in terms of a non- separable two-dimensional canonical integral and its derivatives. This non-separable two-dimen- sional canonical integral plays a similar role to the Airy integral in one-dimensional integrals with two coalescing saddle points.The uniform approximation is obtained by applying to the two- dimensional case the asymptotic techniques introduced by Chester et al. for one-dimensional integrals. An exact series representation is obtained for the canonical integral by means of complex variable techniques. The series representation can be used to evaluate the canonical integral for small to moderate values of its arguments, whilst for large values of its arguments existing asymptotic tech- niques may be used. 1 . INTRODUCTION An important goal in the semiclassical theory of inelastic and reactive molecular collisions is the derivation of integral representations for the S-rnatri~.’-~ The integral representations may often be evaluated by asymptotic techniques, in which certain canonical integrals play an important role.Consider, for example, the one-dimensional integral 11 = @x) exp ; x)ldx, (1.1) which arises in the semiclassical analysis of the non reactive collinear atom + diatom collision problem.’b* lC* 2c* 2d The main contribution to the integral (1.1) arises from the saddle points of f(a; x), that is, the points at which af(a; x)/dx = 0. Physically, each saddle point is associated with a real or complex classical trajectory, and the parameter a corresponds to the final quantum number of the transition (the initial quantum number is fixed by the initial conditions). The position of the saddle points depends on the parameter a and for a certain value of a, two saddle points may coincide.For this situation, the asymptotic method of Chester et al.’ provides a uniform approximation for the integral (1.1). The first two terms of this uniform asymptotic expansion were derived by the author and Marcus,2C* and in the case where f”(a ; xl)> 0 and f”(a ; x,) < O when the saddle points x1 and x2 are real, are given by52 MULTIDIMENSIONAL CANONICAL INTEGRALS The subscripts indicate that g(x) andf"(a ; x ) are to be evaluated at the appropriate saddle point, and Ai( -5) and Ai'( -[) are the Airy function and its derivative respec- t i ~ e l y . ~ The formula (1.2), used in conjunction with real or complex trajectories,2g accurately reproduces the exact results of Secrest and Johnson8 over a range of ;s loo to = 10-l'. In the one-dimensional example described above, the Airy integral is the canonical integral, and the usefulness of eqn (1.2) and (1.3) is dependent on the fact that the properties of the Airy integral are known and its values are tab~lated.~ When the S-matrix is represented by a multidimensional integral, the derivation of uniform asymptotic expansions is more difficult.For certain special cases the Airy integral is again the canonical integral, for example, when the phase is separable 2h or in the case of two nearly coincident saddle points in n-dimensions.6 In general, however, an n-dimensional integral requires an n-dimensional canonical integral for its uniform asymptotic evaluation. Very little work has been done on this problem of obtaining uniform asymptotic expansions or of investigating the properties of multidimensional canonical integral^.^" This present paper is devoted to this problem.A concrete example of a two-dimensional integral with four nearly coin- cident saddle points is studied. This example is typical of more general ones. Uniform approximations were introduced into the semiclassical theory of elastic scattering by Berry.g In Section 2 it is shown how a uniform asymptotic approximation for this two- dimensional integral with four nearly coincident saddle points may be obtained in terms of the non-separable canonical integral aJ (1.4) 1 w, 5, q) = - (27c)2 1; oo du du exp [i(3u3 + i u 3 + 5u + 5u + quo)], -a3 and its derivatives. This is achieved by applying to the two-dimensional integral the techniques introduced by Chester et aL5 for one-dimensional integrals.An exact series representation for the canonical integral (1.4) is obtained in Section 3 with the help of complex variable techniques. The series can be written as the sum of products of derivatives of the Airy function. For small to moderate values of the parameters c, c, and q, the series can be summed numerically to obtain an exact value of U((, 5, q). For large values of these parameters, existing asymptotic techniques (e.g., the ordinary saddle point or stationary phase method) can be used. Finally, Section 4 discusses the general n-dimensional case and points out some of the difficulties that are encountered as the number of dimensions increases. 2. UNIFORM ASYMPTOTIC EVALUATION OF A TWO-DIMENSIONAL INTEGRAL WITH FOUR NEARLY COINCIDENT SADDLE POINTS The integral under consideration is J - c o J - a ~ The functionf(x, y ) depends on two parameters a = (a1, u2), but they have not been indicated explicitly in order to simplify the notation.For an atom+rigid rotator collision, these two parameters correspond to the final rotational quantum number and final relative orbital angular momentum quantum number.'" It is assumed that the integral (2.1) possesses four saddle points (xi, y l ) with i = 1, 2, 3, 4, which for a certain value of the parameters coalesce or come close together at (xo, yo).J . N. L. CONNOR 53 2.1 NON-UNIFORM ASYMPTOTIC APPROXIMATIONS Suppose that in the vicinity of the point (xo,yo) where the four saddle points coalesce or come close together, f ( x , y ) and g(x, y ) have the Taylor expansions Ax, Y ) = f(x0, Yo) +fx(xo, Yaw- xo) +fv(xo, Yo)(Y -Yo) +fxY(XO, Yo>(x-xo)(Y -Yo) + 4fucx(xo, YO>(X - xol3 + 4fyyY(xo, Y O X Y (2.2) and g(x, Y ) (2.3) Then it is clear that in this approximation the integral (2.1) can be expressed in terms of U (eqn (1.4)) and its derivatives Us, Uc and U,,.This is achieved by a change of origin and normalisation of the highest order terms in the expansions (2.2) and (2.3). This approximation is, however, only valid when all four saddle points are close together. Note that eqn (2.2) is not the only Taylor series expansion that is possible for four coalescing saddle points. For each of the remaining cases, a uniform asymptotic approximation expressed in terms of a different canonical integral can be obtained by applying the techniques described in this paper.On the other hand, when the saddle points coalesce in pairs but with the pairs well separated, the asymptotic approximation to the integral (2.1) is d x o , Yo) +gx(xo, Yo)(=--Xd +gy(xo, Yo)(Y -Yo) + gxAx0, Yo>@ - Xo)(Y -Yo). where and 4x61 = f 2 - f 1 , 4 x t 3 = f 4 - j . 3 - The saddle points have been labelled so that ( x , , y , ) coalesces with ( x 2 , y 2 ) and (x3, y 3 ) coalesces with (x4, y4). When 9W*)( - xij) is replaced by its asymptotic values of zero or unity, eqn (2.4) becomes 9. A t I2 - 2xi C 2 exp (if;.), the ordinary saddle point or stationary phase resu1t,lo and where the sum is over all contributing saddle points. The asymptotic expansion (2.6) is valid when all four saddle points are well separated.The Taylor expansions (2.2) and (2.3) are valid when all four saddle points lie close together, and eqn (2.4) is valid when the saddle points coalesce in pairs, but with the pairs well separated. The problem now is to obtain a unifarm asymptotic expansion which reduces to the non-uniform results of eqn (2.2)-(2.6) in the appro- priate limits. This is done in the next subsection.54 MULTIDIMENSIONAL CANONICAL INTEGRALS 2.2 UNIFORM ASYMPTOTIC APPROXIMATION A uniform asymptotic approximation is obtained for the integral (2.1) by applying to the two-dimensional case the procedure used by Chester et aL5 for one-dimensional integrals. We map f ( x , y ) onto a cubic polynomial in the variables u and v : f(x, y ) = ~ ~ ~ + ~ v ~ + ~ u + ~ u + q u v + A .(2.7) Notice that eqn (2.7) is an exact transformation and not the first few terms of a Taylor series expansion (as in eqn (2.2)). The transformation (2.7) is one-to-one if the saddle points on either side of (2.7) correspond The saddle points (ui, v,) are to be labelled so that their distribution in the (u, u) plane corresponds as closely as possible with the distribution of saddle points in the (x, y) plane. The saddle points are determined by the solutions of the equations and Except in the separable case where q- = 0, the solutions u1 = u,((, r, q) and vi = ul([, r, q-) are too complicated in form to be written down explicitly, but this is of no consequence for the development of the theory given below. u4 + 2[u2 + q3u +p + t q 2 = 0, u4 + 2tV2 + q3v + t2 +[q2 = 0.If the saddle points (2.8) are substituted into eqn (2.7), there results f(xi, yi) = 31413 + + I : + C U ~ + t ~ i + quivi + A, i = 1,2, 3,4. (2-9) Equation (2.9) represents four non-linear equations in the unknowns (, 5, q and A. These equations can be solved (in principle at least) for 5, e, q and A in terms of the saddle points off(x, y). The integral (2.1) now becomes I 2 = 1 du 1 O0 dvg(u, v)J(u, v ) exp [i(4u3 + 3v3 + [u + t v + quv + A)], (2.10) where J(u, v ) is the Jacobian of the transformation (2.7). It is not difficult to show by means of the chain rule that at a saddle point the Jacobian satisfies the equation -03 (fuufvv - f 3 u i , vi> = J2(ui, U i U x x j y y - .L$Xxi, Yi), 4uivi - q2 = J2(ui, VXfxxfyy-.f~)(xi, Yi).or with the help of eqn (2.7), (2.11) (2.12) The first four terms of the asymptotic expansion for I2 are obtained by writing g(u, v)J(u, u) = p+qu+rv+suu. If the saddle points (2.8) are inserted into eqn (2.12), there results g(ui, vi)J(Ui, ~ r ) = P+qui+rvi +SUiUiy i = 1, 2, 3,4, (2.13) which represent four hear equations and these can be solved (by Cramer's Rule for example) for the unknowns p , q, r and s. If the expansion (2.12) is substituted into the integral (2. lo), it follows that I, - (271)2(p U - iqUc - ir U, - isU,,} exp (iA), (2.14)J . N . L . CONNOR 55 where is the canonical integral defined by eqn (1.4). Eqn (2.14) is the uniform asymptotic expansion for four nearly coincident saddle points that we have been seeking. The quantities p , q, r and s are found from eqn (2.13) and 5, c, q and A from eqn (2.9).It is not too difficult to verify that the uniform asymptotic approximation (2.14) reduces to the non-uniform results (2.4)-(2.6). This is done by replacing U and its derivatives by their appropriate asymptotic expansions (found by application of eqn (2.4) and (2.6) to the canonical integral (1.4)), and using eqn (2.11) and (2.13). As this is a straightforward calculation, details of it will be omitted. Iff@, y ) and g(x, y ) are expanded in the Taylor series (2.2) and (2.3), this allows [, 5, q-, and A in eqn (2.7) and p , q, r and s in eqn (2.12) to be identified in terms of the Taylor coefficients. It is then readily shown that the uniform asymptotic approxi- mation (2.14) reduces to the non-uniform result contained in eqn (2.2) and (2.3) for four nearly coincident saddle points. u = w, 5, v ) 3.SERIES REPRESENTATION FOR THE CANONICAL INTEGRAL In the previous section it was shown how a uniform asymptotic approximation (eqn (2.14)) for a two-dimensional integral with four coalescing saddle points could be expressed in terms of the canonical integral (1.4) and its derivatives. The problem now is to evaluate the canonical integral. When the saddle points are well separated or coalesce in pairs with the pairs well separated, the asymptotic formulae (2.4) and (2.6) can be applied to the canonical integral. When however, the saddle points lie close together, corresponding to the Taylor expansion (2.2), exact values of the canonical integral are required.In this section an exact series representation for the canonical integral is obtained with the help of complex variable techniques similar to those used in obtaining a series representation for the Airy integral. l1 The canonical integral (1.4) is first transformed into the form O3 1: du 1: dv exp [i f(u, -v)] + 1 " m, r, v ) = (-{Io du 1 dv exp [if(., v)l+ 0 1: du 1: dv exp [if( - u, v)] + 1: du 103 dv exp [if(-., -v)]), (3.1) 0 where now f(u, v) denotes f(u, v) = +u3 + 4v3 + 5u + c v 4- quv. Consider in the first term of eqn (3.1), the integral over u. Since the integrand contains no singularities, a contour integral along the closed contour O+R+R exp (in/6)+0 will be zero by Cauchy's theorem.12 The contribution along the arc R to R exp (in/6) can be shown to tend to zero as R+oo by Jordan's Lemma.If a similar transformation is applied to the integral over v, the first term in eqn (3.1) becomes = exp (in/3)G([ exp (i2n/3), 5 exp (i2n/3), q exp (i5n/6)), where G(a, p, r) = -2 1; d x J r dyexp [-3x3-~y3+~x+By+yxy]. (3.2) ( 2 456 MULTIDIMENSIONAL CANONICAL INTEGRALS A similar analysis can be applied to the remaining terms in eqn (3.1) except that where the integrand is of the form exp (-i3u3), the closed contour O+R+ R exp (-in/6)+0 is used instead. The canonical integral becomes u(C, 5 , q ) = exp (i71/3)G(5 exp (i2n/3), t exp (i2n/3), 11 exp (iSn/6))+ G(C exp (i2n/3), t exp (- 2?1/3), - iq) + G(C exp ( - i2n/3), t exp (i2n/3), - iq) + exp (-in/3)G(C: exp (-i21t/3), < exp (-i2n/3), q exp (in/6)). (3.3) In the integral (3.2) the cross term can now be expanded in a power series and the series integrated term by term to give 1 " ...n where K(a, n) = exp (ax - +x3)xn dx.(3.5) 1: If eax in eqn (3.5) is also expanded in a power series and integrated term by term, it is found that If the expansions (3.4) and (3.6) are substituted into eqn (3.3), then By recognizing that the series (3.7) can be written in the form O3 q" d"Ai([) d"Ai(C) U((, 5, q) = c i-" - - ___ . n=O n! d r dt" Eqn (3.7) and (3.8) are the series representations of the canonical integral we have been seeking. Numerical summation of these series can be used to obtain exact values of the canonical integral for small to moderate values of c, and q. Because of the relation Ai"(r) = cAi(5) the series (3.8) can be written in terms of the Airy function and its first derivative alone, with higher derivatives absent.The result (3.8) can also be derived from eqn (3.3) and (3.4) be recognizing that K(a, n) is the nth derivative (to within a constant factor) of the integral 1 J" exp (ax-+x3) dx, 2zi which is defined and whose properties are discussed by Jeffreys and Jeffreys. AnJ . N. L. CONNOR 57 alternative derivation of the results (3.7) and (3.8) can be given using convergence factors,13 a method which avoids the complex variable techniques used in the present paper. It can be checked that the series converges for all q, C, and f. 4. DISCUSSION The uniform asymptotic approximation (2.14) is derived from the mapping (2.7) which in turn is suggested by the Taylor series expansion (2.2).The analysis is not restricted to this special case however. When a different Taylor series expansion for four coalescing saddle points is valid, similar techniques can be applied to obtain a uniform asymptotic approximation in terms of a different canonical integral. It is also clear that the techniques described in this paper can be applied to higher dimensional integrals. As the number of dimensions increases, however, so do the number of possible canonical integrals. It is necessary to examine the Taylor expan- sion in the neighbourhood of those points where a number of saddle points coalesce to ensure that the correct canonical form is chosen. If an incorrect choice is made, the remainder term in the asymptotic expansion may become of the same order of magnitude as that of the leading terms, or worse, the remainder term may overwhelm the leading terms.The latter can easily happen, for example in the case of a classically inaccessible transition when the contribution from the leading terms is exponentially small. l (a) W. H. Miller, J. Chem. Phys., 1970, 53, 1949 ; (b) 1970, 53, 3588 ; (c) Chem. Phys. Letters, 1970,7,431; (d) Acc. Chem. Res., 1971, 4, 161 ; (e) J. Chem. Phys., 1971,54,5386 ; (f) C. C Rankin and W. H. Miller, J. Chem. Phys., 1971, 55, 3150 ; (g) W. H. Miller, J. Chem. Phys., 1972,56, 38 ; (h) 1972, 56,745 ; (i) W. H. Miller and T. F. George, J. Chem. Phys., 1972, 56, 5637 ; ( j ) 1972,56, 5668 ; (k) T. F. George and W. H. Miller, J. Chem. Phys., 1972,56,5722. ; ( I ) 1972, 57, 2458; J.D. Doll and W. H. Miller, J. Chem. Phys., 1972 57, 5019. (a) R. A. Marcus, Chem. Phys. Letters, 1970, 7, 525 ; (b) J. Chem. Phys., 1971, 54, 3965 ; (c) J. N. L. Connor and R. A. Marcus, J. Chem. Phys., 1971,55,5636 ; (d) W. H. Wong and R. A. Marcus, J. Chem. Phys., 1971, 55, 5663 ; (e) R. A. Marcus, J. Chem. Phys., 1972, 56, 311 ; (f) 1972,56,3548 ; (g) J. Stine and R. A. Marcus, Chem. Phys. Letters, 1972,15,536 ; (h) R. A Marcus, J. Chem. Phys., 1972,57,4903. The asymptotic formula derived in this reference for a two dimensional integral with four coalescing saddle points (and a similar one given by Miller in ref. (l(e))) is only valid for a separable f(x,y) = f'(x)+f&) when all four saddle points are close together. A detailed discussion is given in ref.(13). (a) R. D. Levine and B. R. Johnson, Chem. Phys. Letters, 1970, 7, 404; (6) 1971, 8, 501 ; (c) R. D. Levine, Mol. Phys., 1971,22,497 ; (d) J. Chem. Phys., 1972,56,1633. (a) P. Pechukas, Phys. Rev., 1969,181, 166, 174; (b) J. C. Y. Chen and K. M. Watson, Phys. Rev., 1969,188,236 ; (c) R. E. Olson and F. T. Smith, Phys. Rev. A, 1971,3,1607 ; (d)B. C. Eu, J. Chem. Phys., 1972, 57, 2531 ; (e) M. D. Pattengill, C. F. Curtiss and R. B. Bernstein, J. Chem. Phys., 1971, 54, 2197; cf) D. R. Bates, Comment Atom Mol. Phys., 1971, 3, 23; (g) D. Richards, J. Phys. B., 1972, 5, L58; (h) K. F. Freed, J. Chem. Phys., 1972, 56, 692; (i) P. Pechukas and J. P. Davis, J. Chem. Phys., 1972, 56,4970; (j) R. A. La Budde, Chem. Phys. Letters, 1972, 13, 154 ; (k) M. A. Wartell and R. J. Cross, J. Chem. Phys., 1971,55,4983. ( I ) J. B. Delos and W. R. Thorson, Phys. Rev. A, 1972, 6,720. (a) C. Chester, B. Friedman and F. Ursell, Proc. Camb. Phil. Soc., 1957,53,599 ; (6) B. Friedman J. Soc. Ind. Appl. Math., 1959, 7, 280; (c) F. Ursell, Proc. Camb. Phil. SOC., 1965, 61, 113; (d) 1972,72,49. J. N. L. Connor, Mol. Phys., 1973, 25, 181. H. A. Antosiewicz, Natl. Bur. Std. (US.) Appl. Math. Ser., 1964, 55, 446$ edited by M. Abramowitz and I. A. Stegun. (a) M. V. Berry, Proc. Phys. Soc., 1966, 89,479 ; (b) J. Phys. B., 1969, 2, 381 ; (c) M. V. Berry and K. E. Mount, Rep. Prog. Phys., 1972,35, 315. lo (a) B. Nagel, Arkiv Fys., 1964,27,181 ; (b) M. Born and E. Wolf, Principles of Optics (Pergamon London, 1965, 3rd revised edn.) Appendix 111, pp. 753-754; (c) N. Chako, J. Inst. Math. * D. Secrest and B. R. Johnson, J. Chem. Phys., 1966,45,4556.58 MULTIDIMENSIONAL CANONICAL INTEGRALS Applics., 1965, 1, 372 ; ( d ) R. M. Lewis, Asymptotic Solutions oj’Diferentia1 Equations and their Applications, (1964, Publication No. 13 of the Mathematics Research Center, U.S. Army, University of Wisconsin, edited by C. H. Wilcox, Wiley), pp. 104-106 ; (e) D. S. Jones, General- ized Functions, (McGraw-Hill, N.Y., 1966), p. 344. l 1 H. Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics (Cambridge Univ. Press, 3rd Edn., 1956), p. 508. l 2 V. I. Smirnov, A Course ofHigher Mathematics, Vol. 111, Part 2, (Pergamon, London, 1964), Chap IV. l 3 J . N. L. Connor, Mol. Phys., 1973, in press.