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 distribution 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 distribution 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 hexadecapole 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 between 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. The relation [I 51 can be proved in a straight- ments of symmetry, besides the n-fold rotation forward way by noting that in the body-fixed qxis, it may also be possible to characterize the frame with polar axis along the symmetry axis, multipole components of order I > n by a single only the component Q,, is different from zero, scalar quantity. Thus for a charge distribution with tetrahedral symmetry T,, (e.g., the methane Can. J. Phys. Downloaded from www.nrcresearchpress.com by UNIV MONCTON on 05/19/17 since only this component is completely invariant under rotations about the polar axis. Using this molecule), which has three twofold and four fact and the relation (Rose 1957) threefold rotation axes one finds (Gray 1968) the following nonvanishing body-fixed components Q,,, for 1 = 3 and 4: we obtain [15] from [14]. Alternatively the form of [15] can be seen simply from the following argument. For an axially symmetric charge distribution, the moments in a space-fixed frame de- pend only on orientation 0 = 04 of the sym- where the scalar octupole and hexadecapole mo- metry axis and not on the orientation of the ments are defined by (Stogryn and Stogryn 1966) 508 CAN. J. PHYS. VOL. 54, 1976

where the r dependence is as indicated in order In [19]-[22] the body-fixed frame is chosen to that (b(v) satisfy Laplace's equation with bound- coincide with the three perpendicular twofold ary condition (b(0) finite. The expansion coeffi- axes. cients must transform under rotations in the The first nonvanishing multipole moment for a same way as the C,,,, since (b is a scalar, and also neutral distribution with tetrahedral symmetry is must satisfy (b,, = (-)"(b,,,,* since (b is real. Any the octupole moment. To see, for example, that special symmetry properties of (b(v) must also be the quadrupole moment vanishes, a formal proof reflected in the (b,,,. The (b,,,, are determined by an using group theory can be given: the identity angular integral over (b(r) in the usual way. representation for the group T, does not occur in Alternatively we can relate the (b,,,, to the field the subspace I = 2. Alternatively one can argue and its gradients at the origin by comparing [24] physically that the multipole moments ([l I]) with the Taylor expansion of (b(v) about the are a measure of the nonsphericity of the charge origin : distribution, and vanish for a spherical distribution since

We recall that the quadratic form associated Hence with a second rank tensor such as the moment of inertia or quadrupole moment is in general ellip- (boo = 40 soidal in shape, but clearly reduces to spherical shape here because of the presence of the four threefold symmetry axes. Hence the quadrupole vanishes because the charge distribution is quasi- spherical for a second rank tensor like Q,,,,. and so on, where C(11121: m,m2m) denotes a

For personal use only. For a neutral charge distribution with octa- Clebsch-Gordan coefficient (Rose 1957), and V,, hedral symmetry (e.g., the SF, molecule) the first the spherical components of the gradient opera- nonvanishing multipole moment is the hexadec- tor; Vo = Vz, and V, , = 12-'I2(V, f iV,). The apole. The relations [20] and [22] are valid here, explicit values for the (b,,,, are if the body-fixed frame is chosen to coincide with the fourfold axes. The discussion of the number of independent multipole components is more complicated in the corresponding Cartesian representation (Stogryn and Stogryn 1966). Substituting [24] in [23] we see that V is a sum In the Appendix we discuss the dependence of over all values of I of terms of the form the Q,,,, on the choice of origin.

Can. J. Phys. Downloaded from www.nrcresearchpress.com by UNIV MONCTON on 05/19/17 3. Interaction Energy of a Charge Distribution where Q,, is the multipole moment defined by With a Given External Field [lo]. From [28] we see that the charge interacts The total interaction energy of a charge cloud with the potential, the dipole with the field, the in an external potential field (b(v) is quadrupole with the field gradient, etc. 4. Multipole Interaction Energy Between Two Charge Distributions Choosing an origin inside the distribution we The results of this section can be derived from expand the field (b(r) = (b(rw) in spherical har- those of the last two sections; however, a direct monics derivation proves to be simpler. Consider two nonoverlapping charge distributions with separa- so that All,,,(rlr2R)must vanish unless (I, + tion R = (RR) between the origins of the distri- 1, + I) is even. Because r,,- ' is real the reduced butions. We denote by v, = (r,o,)the position coefficients A,,12(rlr2R)must be real. Also, r,,-' relative to origin 1 of a typical charge el of distri- is invariant under an interchange of the points bution 1, and by v, the corresponding quantity 1 and 2, i.e. under the transformation v, -+ r,, for a charge e, of distribution 2. r, -+ v,, R -+ -R. This leads to The total electrostatic interaction energy of the two charge clouds is given by a sum of terms of the type (e,e,/r,,), where r,, = Iv, + R - r,I is Four more results similar to [32] can be derived the separation between el and e,. We expand the from the invariance of r,,-' under the trans-

function I.,,-'in terms of products of spherical formations (r,r,R) -+ (-R, v,, -v,), (rlRr2), harmonics referred to an arbitrary polar axis, (-r,Rv,), and (-R, -v,, r,). We do not give these relations for they correspond to regions of configuration space where the charge clouds overlap. We now use the nonoverlapping condition (R > r, + r,) and the fact that r,, - ' is a poten- Because r,,-' is invariant under simultaneous tial function to show that the nonvanishing rotation of r,, v,, and R the expansion coefficient A ,,,,, correspond to 1 = 1, + 1,. For r,, # 0 the must obtain a Clebsch-Gordan coefficient, quantity r12-' is a solution of the three Laplace equations in the variables v,, r2,and R. Recalling that the fundamental solutions of Laplace's equation are of the form [4] it follows that the depen- where the factor Alllz,(r,r2R)is independent of dence of the coefficients A,,,,,(r1r2R)on the the m's. To see that radial variables takes the form

where the constants A,,,,, are independent of all is indeed the invariant combination of three the r's. Terms corresponding to the other solu-

For personal use only. spherical harmonics, we note that the sum over tions of Laplace's equation do not occur as they m, and m, in [31] produces a function, F,,,(o,a,), do not satisfy the boundary conditions for the say, which transforms in the same way as C,l,l(R), region R > r, + r,. Thus for example terms in and that the final sum over in gives ~lllFll,lCllll~~,v1 of the type C,l,,,,(ol)/rl"+lare not included which is the invariant combination of two quanti- because they are not finite when I., -+ 0, and ties which transform like C,,,,. terms in R of the type R1Ylll1(R) are omitted for The quantity r,,-' is also invariant under they do not vanish when R -+ co. We next note simultaneous inversion of r,, v,, and R. Recalling that because the dimensions of [33] must be that that C,,,,(- o) = (-)'C,,,,(o),we see that a factor of an inverse length, 1 = 1, + 1,. Putting (-)11+12+1 is introduced into the expansion [29], AI,I,,~~+~,AI,~, we get Can. J. Phys. Downloaded from www.nrcresearchpress.com by UNIV MONCTON on 05/19/17 where 1 = I, + I,. The remaining constants All,,can be found by considering the special case when r,, v,, and R are parallel to the polar axis. Using [5] we see that for this case [34] reduces to

Comparison of [35] with the direct binomial expansion 510 CAN. J. PHYS. VOL. 51. 1976

yields the coefficient in terms of C(11121;000), which is given by (Rose 1957)

Hence we get

(The phase (-)12 in [38] arises from the choice of R as pointing from distribution 1 to 2.) The total interaction energy between the two charge distributions, zij(eiej/rij), is thus given by a sum over all values of I, and I, of terms of the form

where the multipole moments Q,,,, are defined by [lo]. Expression [39] is the general expression for the electrostatic interaction energy between the multipoles of orders I, and I, of distributions 1 and 2 respectively, V, , being the dipole-dipole interaction, V,, the dipole-quadrupole interaction, etc. The fact that only the maximum value of I, I = I, + I, allowed by the triangle relation among I,, I,, and I occurs is characteristic of the multipole expansion of the Coulomb interaction. For other cases, such as intermolecular overlap and dispersion interactions, terms corresponding to smaller values of I also occur (Gray and Van Kranendonk 1966). We note from [15] that for two axially symmetric distributions the general formula simplifies to

where Ri is the orientation of the symmetry axis of distribution i and

For personal use only. and with Q, defined by [17]. In some calculations it is convenient to choose the polar axis along R. Equation 40 then reduces to

To see for example, that V,, is identical to the familiar Cartesian expression

one can write out [42] and [43] explicitly in quadrupole moments are of course easily per- angles using the fact that C(112; m Z 0) is equal formed using the Cartesian components, although

Can. J. Phys. Downloaded from www.nrcresearchpress.com by UNIV MONCTON on 05/19/17 to (2/3)'12 for n7 = 0 and (1/6)'12 for 17% = $- 1. even here some calculations (e.g., computation of A more elegant approach is to use the algebra of matrix elements between angular momentum irreducible spherical tensors (Fano and Racah states) are simpler with spherical components. 1959) to show that [43] takes the form of [40]. For the octupole and hexadecapole moments calculations are probably most easily done with 5. Concluding Remarks spherical components, as they are no more diffi- We conclude by briefly comparing the Car- cult than the dipole case using spherical com- tesian and spherical tensor methods. ponents. The reason is that simple rules exist Many calculations involving the dipole and (Rose 1957; Brink and Satchler 1962) for inte- GRAY 511

grating, differentiating, and rotating arbitrary MAHLER,R. J. 1966. Phys Rev. 152,325. spherical harmonics. Also, general theorems are MAHLER,R. J. JAMES,L. W., and TANTTILA,W. H. 1966. Phys. Rev. Lett. 16, 259. simpler to prove using spherical methods. OKA,T. 1973. Adv. Atom. Mol. Phys. 9. 127. OZIER,I. and Fox, K. 1970. J. Chem. Phys. 52, 1416. Acknowledgment POLL,J. D. and VANKRANENDONK, J. 1961. Can. J. Phys. The author is grateful to Dr. P. J. Stiles for 39, 189. RAMSEY,N. F. 1956. Molecular beams (Oxford University critical comments on the manuscript. Press, London). ROSE,M. E. 1957. Elementary theory of angular momen- ANANTH,S. M., GUBBINS,K. E., and GRAY,C. G. 1974. tum (J. Wiley and Sons, New York). Mol. Phys, 28, 1005. 1958. J. Math. Phys. 37, 215. BEULER,R. B. and HIRSCHFELDER,J. 0. 1951. Phys. Rev. ROSENBERG.A. and BIRNBAUM.G. 1968. J. Chem. Phys. 83,628. 48, 1396. BIEDENBARN,L. C. and BRUSSARD,P. J. 1965. Coulomb 1970. J. Chem. Phys. 52,683. excitation (Oxford Univ. Press, London). SACK,R. A. 1964. J. Math. Phys. 5,260. BRINK,D. M. and SATCHLER,G. R. 1962. Angular rnomen- SHARMA,R. D. and KERN,C. W. 1971. J. Chem. Phys.55, tum (Oxford University Press, London). 1171. BUCKINGHAM,A. D. 1965. Discuss. Faraday. Soc. 40,232. STOGRYN,D. E. and STOGRYN,A. P. 1966. Mol. Phys. 11, - 1970. Physical chemistry, Voi. 4, Ed. by H. 371. Eyring, D. Henderson, and W. Jost (Academic Press, TAKAYANAGI,K. 1963. Prog. Theor. Phys. Suppl. No. New York). 25. CARLSON,B. C. and RUSHBROOKE,G. S. 1950. Proc. VANKRANENDONK, J. 1959. Physica, 25, 1080. Camb. Phil. Soc. 45,626. WANG,T. 1955. Phys. Rev. 99,566. CHIU,Y. 1964. J. Math. Phys. 5,283. FANO,U. and RACAH,G. 1959. Irreducible tensorial sets (Academic Press, New York), p. 163. GIBBS.P. W., GRAY,C. G., HUNT.J. L., REDDI,P. s., Appendix: Origin Dependence of the Q,, TIPPING,R. H., and CHANC,K. S. 1974. Phys. Rev. If the origin of the charge distribution is shifted Lett. 3, 256. GRAY,C. G. 1968. Can. J. Phys. 45, 135. by a, so that the coordinates of the particles 1971. J. Phys. B, 4, 1661. transform into r, + ri - a, the Q,,, in [lo] trans- GRAY,C. G. and STILES,P. J. 1976. Can. J. Phys. 54. 513. forms into GRAY.C. G. and VAN KRANENDONK,J. 1966. Can. J. P~YS.44,241 I. CA.11 Q~nz + C eiIri - aI1c,m(ar,-a) JACKSON.J. D. 1962. Classical electrodvnamics (J. Wilev i and Sons, New York). For personal use only. KARL,G., POLL,J. D.. and WOLNIEWICZ,L. 1975. Can. J. The regular solid harmonic in [All can be ex- ~hys.53, 1781. panded as

where the dash indicates a sum over values of I, and I, such that (I, + I,) = 1, and is given by [38]. Equation A.2 has been derived by Rose (1958) and Chiu (1964); we indicate briefly a simpler proof. Denoting the left hand side of [A.2] by fll,,(r, a), we expand in spherical harmonics Can. J. Phys. Downloaded from www.nrcresearchpress.com by UNIV MONCTON on 05/19/17 where the C coefficient in [A.3] ensures that the right hand side transforms under rotations in the same way as the left. The reduced expansion coefficient f,,,,,(r, a) is found using the same methods as employed in Sects. 2 and 4, and leads quickly to [A.2]. Substituting [A.2] into [A.1] we get

where I, = I - I,. 512 CAN. J. PHYS. VOL. 54, 1976

Thus under origin translations the change in 1- 1 (-)L-fq!

[A.5] Ql + QL Q,,,, involves the Q,,,,'s of lower order. In parti- ,zoll!(l - ll)! a'-l~~~, cular we note the result (Buckingham 1970) that the first nonvanishing multipole moment is in- Hence, for example, the quadrupole moment dependent of origin. Q = Q2 becomes We also note that for an origin translation lA.61 Q + Q - 2ap + a2q through a distance a along the symmetry axis of an axial distribution the scalar multipole moment where p is the dipole moment and q the net [17] transforms into charge of the distribution. For personal use only. Can. J. Phys. Downloaded from www.nrcresearchpress.com by UNIV MONCTON on 05/19/17