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Spherical Tensor Approach to Multipole Expansions. I Spherical tensor approach to multipole expansions. I. Electrostatic interaction~l~~ C. G. GRAY Department of Physics, University of Grtelph, Glrelpl~,Ontrrrio Received July 25, 1975 Using spherical harmonic expansions, the electrostatic field due to a given charge distribution, the interaction energy of a charge distribution with a given external field, and the electrostatic interaction energy of two charge distributions are decomposed into multipolar components. Extensive use is made of symmetry arguments. Comparisons with the Cartesian tensor method are also given. Au moyen de developpements en harmoniques spheriques, on obtient le champ d'une distribu- tion de charge donne. les composantes multipolaires de I'energie d'interaction d'une distribution de charge avec un champ exterieur donne, ainsi que de I'energie d'interaction electrostatique de deux distributions de charge. On fait un usage considerable des arguments de symetrie. On donne aussi des comparaisons avec la methode des tenseurs Cartksiens. [Traduit par le journal] Can. J. Phys.. 54,505 (1976) 1. Introduction In the next section we discuss the classic 'one- Multipole interactions have been reviewed a center' expansion problem, i.e. the decomposi- number of times in recent years in the Cartesian tion of the potential field due to a given charge tensor formalism (Buckingham 1965, 1970). Al- distribution into multipolar contributions. The though some individual topics have been dis- solution to this problem leads to a natural intro- cussed (Jackson 1962; Brink and Satchler 1962; duction of the spherical multipole components Rose 1957), there appears, however, to be no Q,,. We give the general properties of the Q,,, comprehensive review from the spherical har- and special properties for charge distributions of For personal use only. monic viewpoint. The use of spherical harmonic symmetrical shape. or spherical tensor methods is now very wide- Section 3 deals with the interaction energy of spread and as examples of electrostatic calcula- a charge distribution with an externally applied tions from various fields we mention the follow- field. This interaction is decomposed into multi- ing: hyperfine interactions (Ramsey 1956), polar terms. The relation with the corresponding Coulomb excitation of nuclei (Biedenbarn and Cartesian development is discussed. Brussard 1965), molecular rotational excitation The 'two-center' expansion problem is taken cross sections (Gray and Van Kranendonk 1966; up in Sect. 4 where the multipole-multipole inter- Takayanagi 1963; Oka 1973), rotational energy action energy for two charge distributions is de- bands in solid hydrogen (Van Kranendonk 1959), veloped. This problem was first solved for the molecular vibrational relaxation (Sharma and general case in spherical tensors by Carlson and Kern 1971), structure factors and thermody- Rushbrooke (1950) and has subsequently been namic properties of molecular fluids (Ananth discussed from a number of points of view (Beuler and Hirschfelder 1951 ; Rose 1958 ; Sack Can. J. Phys. Downloaded from www.nrcresearchpress.com by UNIV MONCTON on 05/19/17 et al. 1974), collision induced absorption (Poll and Van Kranendonk 1961 ; Gray 1971). Similar 1964; Gray 1968). methods can be employed in magnetostatic and electrodynamic problems (cf. following paper 2. Field Outside a Given Charge Distribution (Gray and Stiles 1976)). For simplicity we consider a discrete charge distribution. The electrostatic potential at a point 'This paper is dedicated to Professor H. L. Welsh on R = (R, 9+) (R, Q) due to the charge ei at the occasion of his retirement from active teaching at the University of Toronto. ri = (rimi) is ei/r, where r = IR - ril. Choosing ZResearchsupported by the National Research Council an arbitrary coordinate frame with its origin of Canada. within the distribution we expand the function 506 CAN. J. PHYS. VOL. 54, 1976 r - ' in spherical harmonics where the multipole moments Q,,,, of the distribu- tion are defined by where the C,,'s are unnormalized Racah spher- We note that some authors (e.g., Jackson 1962; ical harmonics with Condon and Shortley phases Van Kranendonk 1959; Gray 1968) include (Brink and Satchler 1962). Since r - ' is invariant a complex conjugation and/or a factor of under simultaneous rotation of ri and R the ex- ((21 + 1)/4n)'12 in the definition [lo]. For a con- pansion must take the form tinuous distribution of charge the definition [lo] is replaced by We note that A,(riR) must be real and symmetric where p(r) is the charge density at the source in ri and R, as r-' is real and symmetric in ri point r. The moments [lo] with I = 0, 1, 2, 3, 4, andR.Forr # 0,r-' = IR - ril-'isasolution etc. correspond to the charge, dipole, quadrupole, of Laplace's equation in the variables ri and R. octupole, hexadecapole, etc. By writing out the At field points R outside the charge distribution ri'~,nl(mi)in the definition [lo] explicitly in terms (R > 7,) the dependence of the expansion coeffi- of xi, y,, and zi, we find the following relations cients on the radial variables is with the Cartesian components of charge 13 1 A,(riR) = (ril/~"'')A, where A, is a factor independent of ri and R. The dipole relation [3] follows from the fact that the funda- p = 1eiri mental solutions of Laplace's equation are of the i form and quadrupole 1 [4I rlClrrI(m); Clm(~)Ir"' ' Q = eiT(3riri - r:l): and from the boundary conditions that r-' re- i main finite as ri + 0 and vanish as R + GO. Qoo = 4 For personal use only. The remaining numerical constants A, can be obtained by considering a special case, e.g., when Qio = P, ri and R are both parallel to the polar axis. Using t51 C1,,1(04) = 6m0 we see that for this case [2] becomes and so on. In recent years effects arising from higher (I > 2) permanent and transition multipole mo- On the other hand a direct binomial expansion ments of nuclei (Brink and Satchler 1962; Wang yields 1955; Mahler et al. 1966; Mahler 1966) and molecules (Stogryn and Stogryn 1960; Gray Can. J. Phys. Downloaded from www.nrcresearchpress.com by UNIV MONCTON on 05/19/17 197 1 ; Ozier and Fox 1970; Rosenberg and Birn- baum 1968, 1970) have been detected experi- Comparison of [6] and [7] gives A, = 1. Thus mentally. For example the permanent octupole moment of methane and the permanent hexa- decapole moment of sulphur hexaflouride have been invoked in an interpretation of collision The total electrostatic potential at R due to all induced infrared absorption observed in these the charges in the distribution is thus given by a gases. The hexadecapole moment of the hydrogen sum over all values of I of terms of the type molecule has recently been measured experi- mentally from the collision induced absorption spectrum (Gibbs et al. 1974), and has also beeh GRAY 507 calculated theoretically (Karl et al. 1975). Also, distribution about the symmetry axis (i.e. not on results of NQR experiments on Sb have been the third Euler angle x). The quantities Q,,,, and interpreted in terms of the permanent nuclear C,,,, which transform identically, are now func- hexadecapole moment (Wang 1955). tions of the same variable o and must therefore A number of properties of the Q,,, are im- differ by at most a constant factor, Q, say. To find mediately evident from the definition [lo]. For the value of Q, we compare the expressions [15] a given 1 there are (21 + 1) independent com- and [lo] for Q,, in the body-fixed frame. We see ponents in general. Under inversion we have that Q,,, -+ (-)'QIn,, and under complex conjugation [I31 Qlln*= (-)"'Qiii where E r -m. where P,(cos 0) is the Legendre polynomial. The The Q,, transform under rotations like the scalar multipole moment Q, is often referred to spherical harmonics C,,,,. In particular, choosing as 'the' multipole moment of order 1. Thus the a coordinate frame fixed relative to the charge quadrupole moment is given by distribution, we have the following relation be- tween the components Q,,f, in the space-fixed frame and the components Q,,, in the body-fixed frame : In the literature one also finds definitions of Q which differ from [I81 by factors of 2 and/or e, and one should also note that in quantum me- chanical discussions what is somitimes meant where D,,,,,' denote the representation coefficients (Rose 1957; Brink and Satchler 1962) by 'the' for the rotation group in the convention of Rose quadrupole moment is the quantity (J, M = (1957) and ($0~)denote the Euler angles of the JlQ2,1J, M = J). We note that the Q, are zero rotation carrying the space-fixed frame into co- for odd I if the charge distribution has inversion incidence with the body-fixed frame. symmetry. The number of independent multipole com- For [I51 to hold the charge distribution need ponents for fixed 1is reduced from (21 1) if the + have only an n-fold symmetry axis, where tz > 1. charge distribution is of symmetrical shape. For For personal use only. This follows again from the fact that only Q,, is example, an axially symmetric distribution has nonzero in the body-fixed frame. (For an t7-fold only one independent component for each 1. This axis only the Q,,,, for rn = n x ($- integer or zero) follows from the result that for an axial distribu- are different from zero, since only these com- tion the Q,,, take the form ponents are invariant under rotations of (2x/tz).) Thus [I51 is valid for the quadrupole moment Q2,,, of the NH, molecule which has a threefold where Q, is a constant and o denotes the orienta- axis. tion of the symmetry axis of the charge distribu- If the charge distribution has additional ele- tion.
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