Continuous saturation of “dispersion” singularities and application to molecular triplet states

JOURNAL

OF MAGNETIC

RESONANCE

7,207-218

(1972)

ContinuousSaturation of “Dispersion” Singularities and Application to Molecular Triplet States J. MARUANI Cenfre

de Mtkanique

Ondulatoire

Appliquke,

23, Rue du Maroc,

Paris

19Pme,

France

Presented at the Fourth International Symposium on Magnetic Resonance, Israel, August, 1971 Analytic expressions are derived for the shape of nonoverlapping broadened singularities in the magnetic resonance spectra of randomly oriented spin systems. Multilevel systems presenting various amounts of homogeneous and inhomogeneous broadening are considered. Spectral diffusion and rapid-passage effects are neglected. Continuous saturation curves are also calculated for the dispersiontype singularities, thus allowing determination of the homogeneous line width and spin-lattice-relaxation probability at the corresponding critical orientations. Applications to molecular triplet states formed in glasses are presented. INTRODUCTION

has been a widely used technique in relaxation studies, because it is easy to employ with a conventional spectrometer. Bloembergen, Purcell and Pound (1) have shown how the product T, T2 of spin-lattice and spin-spin relaxation times can be extracted from the continuous saturation curve of a single, homogeneously broadened, resonance line. Portis (2) has extended the treatment to inhomogeneously broadened lines consisting of very narrow, noninteracting spinpackets. Castner (3) has considered systems with various amounts of homogeneous and inhomogeneous broadening and derived a method to extract the ratio of the corresponding widths, as well as the product T1 T,, from characteristics of saturation curves. Only two spin levels were explicitly considered in these studies, although applications were often made to multiple-level systems (3). Lloyd and Pake (4) have shown that, in general, one can relate the always measurable saturation factor S to a quantity which generalizes the relaxation probability, W = 1/2T,, of the two-level spin system. The aim of this paper is threefold. 1) To extend Portis’ and Castner’s approach to inhomogeneous broadening to multiple-level systems, including the case studied by Lloyd and Pake (4). 2) To obtain explicit forms of saturation curves for the singularities of polycrystalline spectra as given by the theory of Coope (5). 3) To use the resulting method to get an estimation of relaxation times for molecular triplet states dissolved in a glass, and to draw some conclusions from the observed values. Continuous

saturation

THEORETICAL The absorption signal V given by a standard spectrometer is proportional to the imaginary part x” of the complex dynamic susceptibility of the sample and to the

rotating component H, of the microwave magnetic field inducing transitions, V = Kx” HI, b: 1972 by Academic

Press, Inc.

207

[II

208

MARUANI

where K is a constant depending on the units and on instrument parameters. The first stage is to express x” as a function of H, and of the steady field H or, more conveniently, of the corresponding resonance frequency CO.A saturation curve is given by a plot of V, at a defined value of CO,for increasing H,. Following Portis (2), we consider a line shape as a superposition of noninteracting, homogeneously broadened spin-packets centered on different frequencies, this resulting in inhomogeneous broadening. The susceptibility associated with a single spin-packet will be obtained by equating the following macroscopic and microscopic expressions for the corresponding net absorbed power: Pa = 3w(2H,)* x”,

PI

P,=Aw~~(N,-N,,,)W,& m>n

[31

In the second expression, N,, and N,,, are the populations of states n and m and Wz is the probability per second for microwave-induced absorption or emission, the states being numbered in order of increasing energy. As shown by Lloyd and Pake (4) the basic quantity in relaxation studies is the saturation factor S,,,, between a pair of levels n and m, Sn, = Wn - NJ/W:

[41

- NO,),

where Nz and Nz are the populations of levels n and m when the spin system is in thermal equilibrium with the lattice. If N and Z are the total numbers of spins and states, respectively, and if the energy difference tzw,,, is much smaller than the lattice temperature kTL, we may write NH - N; = (N/Z) (hw,,/kT,).

The combination of Eqs. [4] and [5] will give an expression for N,, - N,,, to be substituted in Eq. [3]. Finally, W,M,will be given by the expression W,M,= P-/h*)

I
-GA,

where M= yhH, s*r*u (the tensor I’ allowing for a possible anisotropy in the effective magnetogyric ratio r) and g,,(w - CO,,) is the density of transitions between states n and m at angular frequency w. We shall thus write

M

WE = 257~~H: w,“, gmnb - w,uJ, with w,“,= I(mlS*r-uln)l*. By combining quantity

Eqs. [2]-[6] and substituting

171

in the resulting expression for x” the

Z*-lNNfi*y* x0 = ___.3 4kT, ’

PI

we obtain

3x0 XI =z(z*

- 1) ccm>n %m&m4WET5n"b

- %I"),

with MC given by Eq. [7]. Two extreme cases will be considered.

CONTINUOUS

SATURATION

OF SINGULARITIES

209

(A) Each spin level comes to thermal equilibrium with the lattice before a spin temperature can be reached; a necessary condition for this is that the resonance line be due to a single pair of levels, since otherwise the populations of the concerned levels could be connected (1). Lloyd and Pake (4) have given the following formula for S,,,, in this case: &,=(l +2wz&zn)-1, PO1 with

Here W,,$,is the probability per second for lattice-induced transitions from initial spin state n to final spin state k. These probabilities are one of the main subjects of theoretical relaxation studies; they are usually given as a function of the magnetic field and the temperature, the form of which depends on the mechanisms involved and the coefficients, on the nature and strength of spin-lattice interactions. C,, in Eq. [ll] is the cofactor of the n-th column element in the k-th row of a specific 2 x Z determinant containing the WA’s (4). The quantity T,,,,, which reduces to the usual spin-lattice-relaxation time for a two-level spin system, defines a relaxation probability (4) when there is no single-defined relaxation time. In the case of a triplet state, the expression for Line (I, 2), for instance, would be

(B) The whole spin system can validly be described by a spin temperature; a Boltzmann distribution of the populations of the spin levels can be reached through normal spin-spin, harmonic and cross relaxation (6). The latter mechanism is also responsible for spectral diffusion within lines (6), and care must be taken when extending Portis’ model to such systems. It should be pointed out that this case can only be considered as a fictitious limiting case, because a spin temperature, strictly speaking, cannot be reached when the microwave field is present (7~). Following a standard procedure (7b) and using the fact that WE = W,M,, one can easily show that S,,,, in such a case would be written, for all pairs of levels (m,n): S=(l+2WMTJl,

P21

with

[I31

((r, is the mean value of w1 = El/h.) The ratio which occurs in the right-hand side of both Eqs. [13] and [14] is a dimensionless quantity which equals 1 for two levels and will be written a,, in general. All spin levels here relax with the same characteristic time T1. For a triplet state with quasiequal spacing of the levels, we would have 1 -=2 TI

J4%lfW3L2;

(

4

WL 31

* 1

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A drastic simplification occurs when only one of the gmn’s, say go, has a significant value at w (that is, when all the lines are well resolved, as is generally the case for triplet states). Equation [13] then reduces to W,M CL~,and Eq. [9] takes the simpler form 3xowo x"(w - wo, H,) = ---.~~ Z(Z2-l)l

-.

..-- PO7Tgo(w- wo> +y*~:B0~~0(~--0)~0~,,0'

[I51

where PO= 4wf (which reduces to unity in the isotropic two-level case) and x0 = 1 in case (A) while T,, o 3 Tl in case (B). So far, we have referred to x” for a single spin-packet. We now introduce inhomogeneous broadening effects. For a given line, the sources of inhomogeneous broadening can be divided into two classes: those which produce an unresolved, bell-shaped broadening of the line, and random orientations in the magnetic field, which produce well-defined singularities in the line shape (5,8). If G and Y are the corresponding shape functions at a given point, the observed shape in the neighborhood of this point can, according to Eq. [l], be represented by the expression V=K(Y

* G * x”)H,,

[161

where x” is the susceptibility for a spin-packet, given by Eq. [15]. This double convolution implicitly assumes that the three kinds of broadening are independent. In particular, spectral diffusion within the line (6, 9) is neglected. This restricts severely the systems with a spin temperature that can be treated by this model. It is customary to attribute to the homogeneous and unresolved inhomogeneous shape functions, go and G, the Lorentzian and Gaussian forms, respectively (1. 2), 1 1 go(w - w”) = 77Aw, 1 + [(w - w&lwJ

’

WJOwhere Q. is the resonance frequency at the maximum of the Gaussian envelope and dw, and dw, are related to the corresponding full-widths at half-height h’ and between points of extremum slopes h by the expressions do, = h;/2 = h,d3/2,

Aw, = h;/zu’l;i

= h&2.

If one introduces the quantities r = (w - sZ,)/Aw,,

r ’ = (coo - sZo)/A~,,

a = Aw,/Aw,,

and uses the fact that at high fields Sz, > w. - Q. wherever G has significant value, one obtains (3) for the term G * x” in Eq. [16]: G * x” E f x”(w - coo,H,) G(w, - ~,)dw, 0

+m

3xoQoPo

= z(Z* - 1) +Aw,

s -m

exp(-a2 ___r ‘*) dr ’ t2+(r-r’)2

m Gb * L

[I71

CONTINUOUS

where

i ’ =

SATURATION

OF SINGULARITIES

21 I

1 + s2 with s2 =~~~:Bo~,hJ-z,o,

[181

T2, ,, = l/dwL.

This is again a convolution of a Gaussian G,, of width b = l/a, and a Lorentzian L,, of width t. The independent saturation of the spin-packets appears in the occurrence of a saturation term in t but not in b. We now consider the spectral singularities arising from a random distribution of orientations of anisotropic systems in a magnetic field. According to Coope’s theory (5), the polycrystalline singularities of a magnetic resonance spectrum belong to one of the four following types: increasing stepwise, decreasing stepwise, logarithmic, and inverse-square root (increasing or decreasing) stepwise. These correspond, respectively, to two-dimensional maxima or minima, saddle points, and axially symmetric onedimensional extrema, all on disjoint branches of resonance-field surfaces; this entails well-defined topological relations between the numbers in which they can occur in the spectrum. The theory also gives the strength C of the singularities as a function of surface curvature at the stationary points: C = TA-“~, where A is the local value of the Hessian. The first derivative of the absorption line shape, with which we shall be concerned, will then closely consist in a superposition of features of shape represented by 2; = K(S * G * x”); H, = KS’ * (G * x”)~ H, u91 where the subscript i refers to the location of the singularity S, if rapid-passage effects (10) are negligible. In the expression [17] for G * x”, the quantities /3,,, Q, dw,, dw, = l/TZ,e and T,,, may depend on the subscript i, and it is assumed’ that they depend only on i. Let us first consider the case of a stepwise (for example, increasing) singularity with the simple Heaviside form (5) S(r - ri) = CL c?(r- ri).

PI

According to Eq. [19], 1; can be written, since 0’ is the &function (Ila): I’, = KCi(G * x”)[ H,.

PI

Since this line-shape derivative is identical to the absorption component itself (Sd, e), Castner’s results (3) can be applied to the saturation curve of the line center, provided there is no significant overlap from other features. ’ A singularity corresponds to a given transition at a canonical orientation of the steady magnetic field with respect to the system axes. The double convolution given by Eq. [19] already implies that none of the parameters occurring in the expressions of G and x” varies significantly in the neighborhood of this canonical orientation and, furthermore, that T,, 0, T2,0, a0 and PO remain constant through the frequency spectrum defined by AwG. However, x” itself may consist of a superposition of sub-spinpackets with different values for these quantities, since these do not depend on the level spacings only [for example, J. H. FREED, J. Chem. Phys. 43,2312 (1965)]. In particular, /I,, may also be a function of the direction of polarization of the microwave field, in the plane perpendicular to H, with respect to the system axes when the spin is not quantized along H [for example, F. K. KNEULCHL AND B. NATTERER, Helc. Phys. Acta 34, 710 (1961); PH. Korr~s AND R. LEFEBVRE, J. Chem. Phys. 39, 393 (1963); R. LEFEBVRE AND J. MARUANI, J. Chem. Phys. 42, 1496 (1965)]. Our simple model, however, should be sufficient in many instances.

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We now consider the case of logarithmic singularities, which occur in nonaxial systems (8~); in the simplest case they are of the form (5) S(r-ri)=-$lnIr-ri]

(r # ri).

L91

According to Eq. [19], 2; can be written (11~):

This is just the Hilbert transform of Vi, given by Eq. [21], and, since the latter is an “absorption” line, it will be a “dispersion” line (5). It should be pointed out at this stage that this formal dispersion, whose saturation behavior we shall study, is different from the true dispersion, which does not follow the Kronig-Kramers relation [23] in the nonlinear region (2) since it may not saturate. Equations [21] and [23] can be given a more practical form by taking account of Eq. [17] and making use of exponential Fourier transforms, P (II). We first use (dropping the subscript i from now on) F(V)

ccP(G F’(U)

* x”) = Fe(G, * L,) = .S”‘(G,)-F’(L,), = 9” Z(Y)

= i sgn(r”) Se(V),

where sgn(P) is the sign of the conjugate variable ?. The expressions for P(GJ and P(L,) are well known and can be found in tables (Ilb). We then apply the inverse transform and use parity properties to obtain new expressions for the original functions from the tables. These expressions (22) can be cast into the condensed form V = (Ds/t) u(ar, at),

[241

U = -(Ds/t)

[251

v(ur, at),

where u and v are the real and imaginary parts, respectively, of the regular function w (13), viz., w(z) = exp(-z2) erfc(-iz) (w = 24+ iv, z = at + iur), and D does not depend on s-defined

in Eq. [18]-as

can be seen from the formula

From V and U given above, we shall define the quantities V;(s) and U;(s) by the expressions

where ar,, t is the abscissa of a defined point, such as an extremum or a half-height point, of u or v for a given value of at. The first expression reduces to Eq. [19] of Castner’s

CONTINUOUS SATURATION OF SINGULARITIES

213

paper (3) at the point ra,t = 0, where V;(s) is maximum for every value of t(s). The most convenient experimental point at which a saturation curve for a dispersion-like singularity can be measured will also be an extremum. We shall therefore be concerned with the two functions

[271

where aFu,f is the abscissa of the maximum of u for given at. The standard saturation curues thus defined have unit slope at the origin for every value of a. Two other quantities of interest are the standard line projles (14) which can be defined by such expressions as

where py, f is a solution of the transcendental equation @p,, f, at) = 40, atIP. All absorption lines given by Eq. [28], which are the so-called Voigt profiles (24), have in common the central maximum and the half-height points, and all dispersion lines given by Eq. [29] have in common the origin and the two extremums. A visual comparison can thus be made between Voigt profiles corresponding to different values of a or between the corresponding dispersion shapes. In practice, we shall consider unsaturated line profiles (t = 1) only. The basic operations involved in the use of Eqs. [26]-[29] (search for maximums and half-height points, changes of scale, etc.) have been set into a program, using Gautschi’s algorithm (1.5) for the evaluation of the functions u and U. A number of different curves of the four preceding types have been drawn, some of which are shown in Figs. 1 and 2. For saturation curves (3) as well as for unsaturated line profiles (14) of absorption lines, the convolution parameter a can be obtained from the ratio of the abscissas of welldefined points on the experimental plot if one knows the theoretical dependence fo this ratio on a. One will choose either one method or the other or both depending on which combination, when possible, gives the best answer. The same procedure can be used for “dispersion” line profiles or saturation curves. Once a is known, a method similar to that proposed by Castner (3) for absorption lines can be used to extract the product r* = /3,,c(~T, , ,, T2, 0 from the saturation curves of dispersion-like singularities. * For the standard saturation curves given by Eqs. [26] and [27], the abscissa S, of the intersection point of the tangents at the origin and ’ The factor /3, a0 is usually omitted by authors using the result derived for the isotropic two-level case. For the quasiforbidden transitions occurring in some anisotropic multilevel spin systems, however, this factor may lead to an unusual angular dependence or greater difficulty in producing saturation. Such effects have sometimes been unduly attributed to T,, 0 [for example, H. SHMZU, J. Chem. Phys. 42,3603 (1965); P. H. H. FISCHER AND A. B. DENISON. Mol. Phys. 17,297 (1969)].

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FIG. 1. Absorption

10-Z 10“

2

(A) and Dispersion (D) standard line profiles for a = lo-*, 0.5, 10’; I = I.

, 5

1

i

;

Ill

2

5

IO?

2

5

IO? 5

FIG. 2. Logarithmic plots of Absorption (A) and “Dispersion” (D) standard saturation curves for a = 10, 1, lo-‘, lo-*, 10e3. From left to right: Al,,, A,, &O-I, DK-I, .410-2, DIO-2. ALO-3, DIN.

at the maximum (which equals the ordinate of the maximum since the slope at the origin is unity) is a computable function of a. Now from Eq. [18] S, = H,,ZY/H,,Z, where H1,2 = H,/s = l/y7 and Hllz, has the same meaning as s, but for the experimental plot of PR or D, vs H, (which requires knowledge of H,). ? can then be obtained from 7 = s,,IYHI,~,,

[301 where s, is deduced from a and H,,2u is directly measured. Figure 3 shows the dependence of s, on a for absorption and “dispersion” saturation curves. The last step is to determine T2,0 = l/dw, from the observed line width. If A: is the full-width at half-height for absorption lines and A; is the separation between extremums for dispersion lines, the ratio AA/h: is a known function of a: hi/hi = p., ,/Fab,,, where

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SATURATION

OF SINGULARITIES

215

pa, 1 and k 1 are the quantities introduced in Eqs. [28] and [27], respectively. This relation allows determination of h: when hi is measured. From Xi one can deduce

FIG. 3. Semilogarithmic plots of s. vs a for Absorption (A) and “Dispersion” (D) standard saturation curves (we have displaced s, for D by l/3 unit upwards to distinguish it clearly from s. for A).

AwL, since Aw,-/~J is a function of a, equal to 1/2p,, t. One can thus use one of the following relations : Am, = Wp,, 1 = U3a, ,, [311 where the functions pa., and ?,, t are given by the plots shown in Fig. 4. One could also obtain Aw, by dividing Au, by a. If a is either very large or very small, one may

FIG. 4. Semilogarithmic

identify hi/2 with either Aw, or 2/ln2Ao,.

plots of p,,,I and ?,,., vs n.

T,, ,, will be deduced from the relation

T,, o = 7’ AwJPo ao.

1321

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MARUANI

The last type of singularity considered in Coope’s theory may occur in nonaxial (5) as well as in axial systems (da-c); it can be written S(r - ri) = Ci O(r - ri)/(r ~ ri)1/2.

[331

An “analytic” expression for 2: in Eq. [19] can readily be obtained, in the same manner as previously used to derive Eqs. [24] and [25], by applying the relations (I I) 9yy)

= 27$F(Z),

c@-‘(~) a P(S).P(Gb)

-.Fe(L,),

and S’(S) = C(l - isgnJ)/2/r’1”2. The final result can be cast into the condensed form T = (h/t)

v(ar, at),

1341

where v is @r/8) Ii2 times the sum of the real and imaginary parts of the confluent hypergeometric function (16) $(3/4,1/2;z*) (z = ar + iar), related to the Weber function D-3,2 by $(3/4,1/2;22) = 2 3’4exp(z2/2) D_3,2(ZG). Relations similar to those defining the standard saturation curves and standard line profiles for dispersion-like singularities could also be introduced here. However, the minimum will lead to a different choice than the maximum in the present case, because the line shapes are strongly dissymmetric, as can be seen from Eq. 1331.Unfortunately, results comparable to those described above cannot be derived, as currently available tables (17) do not include all complex variables. A possible approximation to these results would be to consider the type of line in question as intermediate, in all respects, to the types previously studied. We shall not encounter this line profile in the applications presented here. APPLICATIONS

The method previously described allows determination of the homogeneous line width 2/T, and spin-lattice relaxation probability 1/2T, for the different lines and canonical orientations of a randomly oriented spin system. Strict conditions for its applicability are that: 1) the lines or the singularities which arise from them do not overlap; 2) the homogeneous broadening be Lorentzian, the unresolved inhomogeneous broadening be Gaussian, and the singularities be of one of the simple forms given by Eqs. [20] and [22]. One effect of spectral diffusion would be to produce homogeneous saturation of non-Lorentzian line shapes (6). Moreover, the singularities which occur in real cases are often distorted by additional terms coming from the first derivatives (5). The theory does not take such effects into account; 3) the spectrum must be the first derivative of the absorption, without rapid-passage effects (10); 4) a multiple-level system must belong to one of the limiting cases (A) and (B) if one wishes to assign definite expressions to a and T,. Among the paramagnetic systems which may approximately fulfill these conditions, the triplet states of conjugated molecules are of particular interest. A very strong axial

CONTINUOUS

SATURATION

OF SINGULARITIES

217

anisotropy of T, has been reported (28) for the flm = 1 transitions of the photoexcited triplet state of pyrene-d-10 in various crystal hosts. Two questions arise: 1) whether such a strong anisotropy would develop in an isotropic medium such as a glass; 2) whether nonaxial anisotropy (19) could also be observed. Experiments were carried out by C. Chachaty, at the Centre d’Etudes Nucleaires de Saclay, on perdeuterated naphthalene, anthracene, phenanthrene and biphenyl dissolved in an ethanol glass. These systems being nonaxial, all Am = 1 singularities are of the “absorption” or “dispersion” types. Evidence has been found for the possibility of complete anisotropy of T,, but the relaxation times in these systems were much less anisotropic than for pyrene in mixed crystals: the ratio between the extreme values of T, was about 1: 2. instead of I : 10 as in the crystals. Although the triplets studied there were different from that studied by Fischer and Denison (18), these results seem to stress the importance of the role played by the structure of the matrix in relaxation processes. Detailed results from this study are reported elsewhere (20). CONCLUSIONS

We have shown how the continuous saturation behavior of the polycrystalline spectra of multilevel spin systems can be a source of information on relaxation processes. Parameters characterizing the relaxation along the critical orientations can be obtained, allowing determination of the anisotropy of the relaxation, though not of its angular dependence. It appears that this very anisotropy may depend drastically on the “crystallinity” of the host medium. ACKNOWLEDGMENTS I wish to thank the many people whose readiness to discuss helped me to clarify some of these ideas. Professors J. Neveu and G. Petiau are especially acknowledged for suggesting the mathematical procedures used. Mr. J. Fouquet helped in writing the computer program.

REFERENCES 1. N. BLOEMBERGEN, E. M. PURCELL, AND R. V. POUND, Phys. Rev. 73,679 (1948). 2. A. M. PORTlS,PhJ’s. Rev. 91, 1071 (1953). 3. T. G. CASTNER, Phys. Rev. 115,1506 (1959). 4. J. P. LLOYD AND G. E. PAKE, Phys. Rev. 94,579 (1954). 5. J. A. R. COOPE, Chem. Phys. Lett. 3, 589 (1969). 6. N. BLOEMBERGEN, S. SHAPIRO, P. S. PERSHAN, AND J. 0. ARTMANN, Phys. Rev. 114, 445 (1959). 7. A. ABRAGAM, “The Principles of Nuclear Magnetism,” (a) Chapter V; (b) Chapter IX, Section I, B, The Clarendon Press, Oxford, 1961. 8. (a) G. E. PAKE, J. Chem. Phys. 16, 327 (1948); (b) B. BLEANEY, Proc. Phys. Sot. (London) A 63, 407 (1950); (c) N. BLOEMBERGEN AND T. J. ROWLAND, Acta Met. 1, 731 (1953); (d) J. A. WEIL AND H. G. HECHT, J. Chem. Phys. 38,281 (1963); E. (e) WASSERMAN, L. C. SNYDER, AND W. A. YAGER, J. Chem. Phys. 41,1763 (1964). 9. A. M. PORTIS, Phys. Rev. 104,584 (1956). 10. M. WEGER, Bell. System Tech. J. 39, 1013 (1960). 11. (a) L. SCHWARTZ, “Methodes mathematiques pour les sciences physiques,” Hermann, Paris, 1965; (b) A. ERDELYI, W. MAGNUS, F. OBERHET~INGER, AND F. G. TRICOMI, “Tables of Integral Transforms,“Vol. I, McGraw-Hill, New York, 1954. Seealso: W. MAGNUS, F. OBERHETTINGER, AND R. P. SONI, “Formulas and Theorems for the Special Functions of Mathematical Physics,” 3rd ed., Ch. XI, Springer-Verlag, New York, 1966. 12. M. BORN, “Optik,” Section 93, Springer, Berlin, 1933.

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13. V. N. FADDEEVA AND N. M. TERENT’EV, “Tables of Values of the Function w(z) for Complex Argument,” Pergamon Press, New York, 1961 (first published in Russian in 1954). 14. H. C. VAN DE HULS-I AND J. J. M. REESINCK, Astrophys. J. 106,121 (1947). 15. W. GAUTSCHI, Comm. ACM 12,635 (1969); SIAM J. Numer. Anal. 7,187 (1970). 16. A. ERDELYI, W. MAGNUS, F. OBE~T~INGER, AND F. G. TRICOMI, “Higher Transcendental Functions,” Vol. I, Ch. VI, McGraw-Hill, New York, 1953. 17. J. C. P. MILLER, “Tables of Weber Parabolic Cylinder Functions,” Her Majesty’s Stationery Office, London, 1955; K. A. KARPOV, “Tablitsy Functsii Vebera” (3 vol.), Publishing House of the Academy of Sciences, Moscow, 1959-1968. 18. A. B. DENSON AND P. H. H. FISCHER, “Proc. XV. Coll. AMPERE,” p. 455, North-Holland, Amsterdam, 1968; P. H. H. FISCHER AND A. B. DENEON, Mol. Phys. 17,297 (1969). 19. J. MARUANI, Chem. Phys. Lett. 7,29 (1970). 20. C. CHACHATY AND J. MARUANI, C.R. Acud. Sci. (Paris), B 273,1119 (1971).