J. Phys. Chem. 1995,99, 15694-15698
15694
Angular and Hyperangular Momentum Coupling Coefficients as Hahn Polynomials Vincenzo Aquilanti,* Simonetta Cavalli, and Dado De Fazio Dipartimento di Chimica, Universith di Perugia, I-06123 Perugia, Italy Received: July 11, 1995; In Final Form: August 21, 1 9 9 9
The relationship is investigated between Hahn coefficients, Le., normalized Hahn polynomials of a discrete variable, and generalized 3j symbols, which extend the algebra of quantum mechanical vector coupling to (hyper)angular momenta, in particular allowing f o r j values multiple of l/4. The calculation of these coefficients is illustrated, both directly from the defining generalized hypergeometric series 3F2( 1) and from three-term recursion relationships, the latter particularly useful for large values of the entries. Their role is outlined as matrix elements for the overlap of alternative hyperspherical harmonics (timber coefficients). The semiclassical limit is also investigated, with reference to their use as discrete analogs of hyperspherical harmonics.
1. Introduction The extension of the concept of angular momentum to spaces mathematically larger than the physical one leads to the introduction of hyperspherical harmonics as orthonormal expansion basis for a variety of problems, some of which are reviewed in ref 1. For these harmonics an algebra needs to be developed, similar to the one describing couplings and recouplings of ordinary angular momenta. However, these extensions require the introduction of analogs of physical angular momenta which can take values other than integer or half-integer quantum numbers. Extensive literature, authoritatively reviewed for example in refs 2 and 3, has established the role of angular momentum algebra within a much wider context-relating fields such as the theory of discrete orthogonal polynomials and of finite difference equations. The vector coupling coefficients-or the closely related Wigner's 3j symbols-appearing in the quantum mechanical theory of angular momentum can be cast as finite sums (generalized hypergeometric functions 3F2( 1) of unit argument with five integer parameters4). Symmetries and other properties of vector coupling coefficients5 have been discussed in terms of symmetries and other properties of hypergeometric function~.~~' These particular 3F2(l) functions can also be identified as Hahn polynomial^,^^^ more general than vector coupling coefficients for which the defining parameters of the 3F2( 1) function are restricted to be all integers. Hahn polynomials, of interest in numerical analysis, have been extensively studied,8-10and their relationship to vector coupling coefficients has been established." Vector coupling coefficients generalized to angular momenta multiples of l/4 appear as elements of unitary transformations between hyperspherical harmonics related to alternative coupling schemes of angular m ~ m e n t a ' ~ and .'~ between the wave functions of the multidimensional hannonic oscillator in hyperspherical and Cartesian coordinate^.^^^'^ In our application^,'^ Hahn polynomials serve as discrete analogs of hyperspherical harmonics. In ref 16 Raynal has generalized the definition of 3j symbols to arbitrary values of the involved momenta in terms of generalized hypergeometric functions 3F2( 1) of unit argument. In this paper we find it useful to establish the connection between the generalized 3j symbols and Hahn polynomials, which we propose to normalize as in eq 6 below and to call
Hahn coefficients. Recursion properties of Hahn coefficients, useful for evaluating matrix elements as well as for their explicit calculation for large values of the entries, are presented in section 3. The role of these coefficients as elements of orthogonal matrices relating alternative sets of hyperspherical harmonics is considered in section 4 and illustrated by establishing the connection between alternative wave functions for the 4-dimensional hydrogen atom. In section 5 a limiting relationship between Hahn coefficients and Jacobi polynomials provides the basis for their use as discrete analogs of hyperspherical harmonics. Conclusions follow in section 6.
2. Hahn Polynomials and Generalized 3j Symbols Hahn polynomials Qn(x,a,j3,N) are defined explicitly as generalized hypergeometric sums:9 Qn(x,a,/3,N) = ,Fz(-n,-x,n+a+j3+l;a+l,-N;1) (1)
In (1) x is the discrete variable, n is the degree of the polynomial, and both vary in the range Q 5 n, x 5 N (N being a positive integer). Both parameters a and j3 are greater than - 1. The polynomials defined above constitute an orthonormal set with respect to both n and x : N
and N
where w(x) is the weight function w(x) =
(4)
and nnthe norm of the polynomial
+ a + j3 + l)!j3! (n + a)!(n + a + j3)! (2n + a + /3 + l)]/[(N - n ) ! (N + a + ,8 + n + l)! a! ( a + /3)! (n + p)! n! ( a + j3 + l)] (5)
nn= [N! (N
We find it useful to introduce the notation:
@Abstractpublished in Advance ACS Abstracts, September 15, 1995.
QQ22-365419512Q9915694$09.0010
+ + + +
( a + x ) ! @ + N - x ) ! ( a +j3 l)! N ! x ! a! j3! (N - x ) ! (N a j3 l)!
0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 42, 1995 15695
Angular Momentum Coupling Coefficients
3F2(j
TABLE 1: Hahn Coefficients for N = 1 V n=O
n=l
- -1 2
-j3 - m3, -j3 +mi
+ 1,
+ + 1,
j , m, -j2 -j3
+ m,;
-j, 1
+ m,;) (7)
values-and their “projections” m’s-which are multiples of appear: this case of interest for our applications is examined below. The definition of Hahn coefficients (see eq 6) allows their explicit computation exhibiting them as elements of an orthonormal matrix of order N 1, whose labels for rows and columns are n and x ; for fixed N these coefficients depend on the parameters a and /3. The dependence of on a and /3 for the simplest cases N = 1 and N = 2 is illustrated in Tables 1 and 2, respectively: v = N/2 - x is preferred as a label over the discrete variable x , in order to emphasize its role as the projection of a (hyper)angular momentum, as detailed in section 5.
+
e:;$”
to represent Hahn coefficients with the orthonormal properties ( 2 ) and (3). These coefficients are elements of an orthonormal matrix of dimension (N 1) x (N l ) , which for fixed a, p, N differ from zero only when n and x take the integer values 0, 1 , ...,N. Below, we find it useful to set v = N/2 - x, in order to emphasize the role of this discrete variable as the projection of an angular momentum (see eq 9 and section 5 ) ; its range is -N/2 Iv INl2. The definition of 3j symbols with generalized momentaj and projections m (eq 29 of ref 16) is also given in terms of a hypergeometric series in eq 7 , where j l - ml is required to be an integer and identified with n. Here we remark that when the parameters of the hypergeometric function are all integers: this equation coincides with one-explicitly Racah’s-form of the ordinary 3j symbols: Comparing the generalized 3j symbols defined in eq 7 with the Hahn coefficients, Le., with the polynomials multiplied by the proper normalization factor, as given in eq 6, yields the relationship
+
+
til 1, i13
3. Recursion Relations and Other Properties Some properties of the generalized 3j symbols stem directly from those of Hahn polynomial^.^ In particular, in the following we give the three-term recursion relations when either one of the projection or one of the (hyper)angular momenta changes by unity: these relationships are important for evaluating matrix elements and can also serve to develop an algorithm for evaluating the generalized 3j symbols for large values of their arguments. Hahn coefficients obey a three-term recursion relation? which can be written
=
exp[in(2n
+ j3 - x)](2n + a + j3 + 1 )
-112
aJ,N
Q,,
(8)
with
where
+ a + j3 + l ) ( n + a + 1)(N - n ) + a + j3 + 1)(2n + a +8, + 2) (N + n + a + j3 + l)(n + j3)n (12) (2n + a + j3 + 1)(2n + a + j3) A(n) = [ { ( n + a + B)(n + a ) ( N - n + 1)(N + n + a + j3 + l ) ( n + j3)n}/{(2n + a + ~ ) ~ ( +2 an + j3 + 1)(2n + a + j3 - 1)}11’2( 1 3 ) B(n) = x -
- a +, B
mi-
N a m2=y-y-x
m3=-m,-m2
or conversely
n=jl-m,
x=j3+m3
N=j2+j3-ml
a = j 2 - j 3 + m , j3 =j 3 - j 2
(10)
+ m,
where n, x, and N are integers. From eq 9 we see that ml Ij l 5 j2 j , and -j3 5 m3 5 j3 - /3. Equation 8 establishes a relation between the theory of orthogonal polynomials in discrete and the theory of vector coupling coefficients with generalized momenta. In the particular case of integer values for a and p this equation is seen to exhibit the connection between ordinary Wigner’s 3j symbols and Hahn polynomialsI0 when each of the j ’ s and m’s are allowed to take only integer or half-integer values. From eq 9 we see that when a and 8 are also allowed to be half-integers, then hyperangular momenta for which j ’ s
+
(n
(2n
In this case, eqs 8 and 9 transform the above expression into a recursion relation for generalized 3j symbols extending one known for 3j symbols (cf. eq 6a in ref 17):
+ l ) fmil
)
j 2 j 3 = 0 (14) m2 m3 where A ~ I=) [(2jl 1)(2j1- 1)]’”2jlA(n),AGI -t 1) = [(2j1 1)(2jl 3)]’/2(2j1 2)A(n l ) , and B(j1) = (2j1 f 1)u1+ 1)2jlB(n). The angular momenta and their projections in the above recursion relation can be any multiple of l/4.
j,A(j,
+
+
+ +
+
15696 J. Phys. Chem., Vol. 99, No. 42, 1995
Aquilanti et al.
TABLE 2: Hahn Coefficients for N = 2 V
n=O
(
+
n=2
n=l
-(
(a+2 ) ( a + 1) (a+ j 3 2 ) ( a + j 3 + 3)
-1
2 ( a + 2 ) @ + 1) 2 ) ( a +j3 4)
(a+ j 3
+
+
-(
2@
(a + j 3 1
+;
((a
(
t;i '++: 1
2(a+ 1 ) @ + 2 )
(a +j3
2)
+ 2 ) ( a +/3 + 4)
(
1
(a+2 ) ( a + 1) (a + j 3 + 3)(a+/3 + 4)
The Hahn coefficients also satisfy a difference e q ~ a t i o n , ~ which can be written
j
150+S/2
100
50 +&I2
44 - Si2
-94
8
where
D(x) = n(n
+ 2 ) ( a + 2)
+ 3 ) ( a+ j 3 + 4)
Po Ob
+ a + /3 + 1) - (a+ x + 1)(N- x ) x(N+p-x+
C(x) = [ x ( a
P o O 6
1) (16)
+ x)(N - x + 1)(N + j3 - x + 1)]''2
P o O
e
0
e
(17)
Inserting eqs 8 and 9 into eq 11 yields the three-term recursion relation for generalized 3j coefficients:
.0.5
I
-1.0
'- -
50.0
P
o
L-'
100.0
150.0
--A 200.0
250.0
Figure 1. Ordinary and generalized 3j coefficients as functions of one
formally equivalent to the recursion relation for Wigner's 3j symbols (eq 9a in ref 17) but valid also for angular momenta and projections that are multiples of l/4. In eq 18 C(m2) = C(x l), C(m2 1) = C(x), and D(m2) = -D(x). Together with the orthogonality conditions of generalized 3j symbols, which follow from those for Hahn polynomials (see eqs 2 and 3), the recurrence relationships (14) and (18) prove useful to calculate matrix elements explicitly in the hyperquantization te~hnique.'~ We have developed an algorithm for accurate evaluation of generalized 3j coefficients with large entries, based on the recursion relationships obtained above and pattemed after one introduced by Schulten and Gordon." The algorithm has been tested against direct summation of the defining series; its extensive use has verified that a wealth of these quantities enjoy properties closely related to those of ordinary 3j symbols. As an example, Figure 1 compares the j dependence of a generalized 3j symbol with those of its nearest ordinary 3j neighbors. The Hahn coefficient appearing in eq 8 is a terminating hypergeometric series which degenerates to a single term when the degree of the polynomial n and/or the discrete variable x vanish. In these particular cases the generalized 3j symbol takes an explicit form that generalizes one known for Wigner's 3j coefficients and corresponds to the cases ml =jl and m3 = -j3 for n = 0 and x = 0, respectively:
+
+
expiin(2n
+ /3>1(2n+ a + B + 1)-1/2Jn,wo
with w(x) and nngiven by eqs 4 and 5, respectively.
of the angular momenta, j , for its whole allowed range. Open circles and squares correspond to 6 = 0 and 1, respectively, therefore implying only integer or half-integer quantum numbers appropriate to ordinary Wigner's 3j coefficients. Full triangles represents values of generalized coefficients corresponding to 6 = '/2.
4. Hahn Coefficients as Overlaps of Alternative Hyperharmonics The timber coefficients in the theory of hyperspherical harmonics'0s'2serve to transform sets of hyperspherical hannonics pertaining to altemative coupling schemes of (hyper)angular momenta. In general, these coefficients are generalized hypergeometric functions of unit argument, identified with Hahn coefficients in particular cases: the parameters a and amount then to quantum numbers. An important application occurs in the theory of the hydrogen atom whose wave functions in momentum space are related to hyperspherical harmonics by Fock's projection. Ordinary angular momentum algebra describes altemative coupling schemes for the physical case of three spatial dimensions. Here the role of Hahn polynomials has been studiedI8 corresponding in this case to ordinary 3j symbols. However, the hydrogen atom in higher mathematical dimensions has recently received much attention.Ig In particular, let us now consider the 4-dimensional hydrogen atom wave function Idlm) in hyperpolar coordinate^,'^ where n is the principal quantum number (not to be confused with the degree n of the Hahn polynomial).' By extending known techniques for the 2DI and 3D cases, we have established the orthogonal transformation leading to the wave functions of the 4-dimensional hydrogen atom in parabolic coordinates Inln2lm) (where nl and n2 are parabolic quantum numbers) by Fourier transformation into momentum space, where a correspondence emerges with altemative 5-dimensional hyperspherical harmonics. Such
Angular Momentum Coupling Coefficients
J. Phys. Chem., Vol. 99, No. 42, 1995 15697
a transformation matrix can be written in terms of Hahn coefficients: "-1
+
For example, when n - 1 = 1 1 only I = 1 , 1 -t 1 are allowed. Comparing the Hahn coefficients of the above expression with those given in Table 1 , we are lead to identify a = B = 1 '12. In this simple case the matrix elements in Table 1 are equal to the elements of a matrix corresponding to a rotation by nI4. More complicated cases arise for higher dimensions.
+
.'s
Lt-.
,
,
,
0.75
5. The Semiclassical Limit of Hahn Coefficients and Their Use as Discrete Analogs of Hyperharmonics Recently, we have developed a technique to solve the SchrWnger equation numerically for few-body system^.'^^^^ Our recipe starts by establishing the connection between slicing the angular variable into N 1 intervals and the integer or halfinteger v value which counts the slices; see eq 21. The basic ingredients utilize Hahn polynomials as the discrete analogs of hyperspheric al harmonics. Let us consider now the high N limit of Hahn coefficients, physically corresponding to the semiclassical limit of generalized 3j symbols. The Jacobi and Hahn polynomials are related by the following limiting relation*
+
where
N - h N+1
2~ N+1
cos e = -- For the normalization factor in eq 20 see Edmonds in ref 5 , eq 4.1.22. The vector model gives insight into the interpretation of the above formula in terms of (hyper)angular momenta. We look at N/2 as an artificial .angular momentum whose magnitude for large N. The vector N/2 [N/2(N/2 1 ) ] I I 2 GZ N/2 processes along the quantization axis with a projection v. The orientations of the vector N/2 divide the range of cos 8 variable into N 1 slices with the projection v labeling the lattice points. The orthonormality conditions, eqs 2 and 3, and the finite difference, eq 11, may be viewed as discrete analogs of the orthonormality, completeness, and three-term recurrence relations for the Jacobi polynomials. Therefore, Hahn polynomials amount to discrete analogs of Jacobi polynomials defined on a grid of points. Exploiting the limiting relation in eq 20, hyperspherical harmonics that are products of classical Jacobi polynomialsI3 or their particular cases are represented by products of Hahn coefficients. In particular, if we take a = ,L? equal to integer values, harmonics involve associated Legendre polynomials whose discretized analogues are ordinary angular momentum coupling coefficients.20 In such cases, eq 20 can be used and interpreted as the semiclassical limit of angular momentum coefficients.2' However, when a = /3 takes halfinteger values, the Jacobi polynomials have Gegenbauer polynomials as particular and eq 20 can be interpreted as the semiclassical limit of the hyperangular momenta coupling coefficients of interest here. In Figure 2 such a relationship is
+
+
+
-0.25
0.25
0.75
4.25
-0.75
0.25
-2
0.75
case or z V / ~ t 1 ) Figure 2. Illustration of the relationship between Hahn polynomials 1)/2]"2Q~1/2.'+'""% and Gegenbauer polynomials d,f'/': [ ( N {2'+1/21
