Translation matrix elements for spherical Gauss

Translation matrix elements for spherical Gauss-Laguerre basis functions

Jürgen Prestin and Christian Wülker∗ 1

1

,2

Institute of Mathematics, University of Lübeck, Germany Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD, USA 2

May 20, 2018

Abstract Spherical Gauss-Laguerre (SGL) basis functions, (l+1/2) type Ln−l−1 (r2 )rl Ylm (ϑ, ϕ), |m| basis of the space L2 on R3 with

i.e., normalized functions of the

≤ l < n ∈ N, constitute an orthonormal polynomial radial Gaussian weight exp(−r2 ). We have recently

described reliable fast Fourier transforms for the SGL basis functions. The main application of the SGL basis functions and our fast algorithms is in solving certain three-dimensional rigid matching problems, where the center is prioritized over the periphery. For this purpose, so-called SGL translation matrix elements are required, which describe the spectral behavior of the SGL basis functions under translations. In this paper, we derive a closed-form expression of these translation matrix elements, allowing for a direct computation of these quantities in practice.

2010 Mathematics Subject Classication: Key words and phrases:

MSC 65D20 and MSC 33F05

Spherical Gauss-Laguerre (SGL) basis functions, translation, three-

dimensional rigid matching, computational harmonic analysis

∗

Email: [email protected]

2

J. Prestin and C. Wülker

1 Introduction Spherical Gauss-Laguerre (SGL) basis functions (Def. 2.1) constitute an orthonormal polynomial basis of the space L2 on R3 equipped with the radial Gaussian weight function exp(−r2 ). We have recently described reliable fast Fourier transforms for the SGL basis functions [Prestin and Wülker, 2017]. These algorithms allow for a fast computation of SGL Fourier coecients for spectral analysis. A main application of the SGL basis functions and our SGL Fourier transforms is the fast solution of certain three-dimensional rigid matching problems, where the center is prioritized over the periphery (Prob. 2.3). For this purpose, so-called SGL translation matrix elements are required. These elements describe the spectral behavior of the SGL basis functions under translations in R3 . In this paper, we derive a closed-form expression of these translation matrix elements (Thm. 4.2). This allows for a direct computation of the SGL translation matrix elements when required in practice. In the derivation of the closed-form expression of the SGL translation matrix elements, we make use of the fact that the SGL basis functions are akin to so-called Gaussian-type orbital (GTO) basis functions in the unweighted space L2 (R3 ), which themselves carry an exponential radial decay factor exp(−r2 /2) [Ritchie and Kemp, 2000]. These functions are nowadays extensively made use of in biomolecular recognition simulation such as protein-protein docking (see, e.g., [Ritchie and Kemp, 2000] or [Ritchie et al., 2008, Sec. 2.1]). Ritchie [2005] has already established a closed-form expression of the GTO translation matrix elements. In principle, we can apply the same approach to the SGL case. However, as a key tool, Ritchie introduces a special class of Hankel transforms, referred to as the spherical Bessel transform [Ritchie, 2005, Sec. 2.3]. The GTO basis functions are eigenfunctions of these transforms, a fact that crucially simplies the derivation of the GTO translation matrix elements. This is not the case with the SGL basis functions. In fact, the spherical Bessel transform of Ritchie is not even well-dened in the SGL case, as the underlying integrals diverge. For this reason, we introduce a weighted spherical Bessel transform (Def. 3.7) by adding a Gauss-Weierstrass convergence factor to the unweighted spherical Bessel transform used by Ritchie. While this ensures convergence of the corresponding integrals, it also causes additional technical diculties we must solve. The remainder of this paper is organized as follows: In Section 2, we give a brief overview on SGL basis functions. We illustrate their application, and show how our fast SGL Fourier transforms can be used in this context. In Section 3, we present the theoretical tools to derive the closed-form expression for the SGL translation matrix elements. The derivation itself is presented in Section 4.

2 Background Let k · k2 denote the standard Euclidean norm on R3 . We consider the weighted L2 space

H :=

f : R → C : f Lebesgue measurable and 3

Z

2 −kxk22

|f (x)| e R3

endowed with the inner product

Z hf, giH := R3

and induced norm k · kH :=

p h·, ·iH .

2

f (x)g(x)e−kxk2 dx,

f, g ∈ H,

dx < ∞ ,


3

SGL basis functions, evolutionarily related to the GTO basis functions in the unweighted space introduced by Ritchie and Kemp [2000], are orthogonal polynomials in the weighted L2 space H . They arise from a particular construction approach in spherical coordinates. These are dened as radius r ∈ [0, ∞), polar angle ϑ ∈ [0, π], and azimuthal angle ϕ ∈ [0, 2π), being connected to Cartesian coordinates x, y , and z via (cf. Fig. 1)

L2 (R3 )

x = r sin ϑ cos ϕ, y = r sin ϑ sin ϕ, z = r cos ϑ. In this paper, we write x = [r, ϑ, ϕ] when r, ϑ, and ϕ are spherical coordinates of x (potential ambiguity is not a problem). By S2 := {x ∈ R3 : kxk2 = 1} we denote the two-dimensional unit sphere. The SGL basis functions are now dened as follows.

Denition 2.1 (SGL basis functions).

The SGL basis function of orders n ∈ N, l ∈ {0, . . . ,

n − 1}, and m ∈ {−l, . . . , l} is dened as Hnlm : R3 → C,

Hnlm (r, ϑ, ϕ) := Nnl Rnl (r)Ylm (ϑ, ϕ),

where

s Nnl :=

2(n − l − 1)! Γ(n + 1/2)

is a normalization constant, Ylm is the spherical harmonic of degree l and order m [Dai and Xu, 2013, Sec. 1.6.2], while the radial part Rnl is dened as (l+1/2) Rnl (r) := Ln−l−1 (r2 )rl , (l+1/2)

Ln−l−1 being a generalized Laguerre polynomial [Andrews et al., 1999, Sec. 6.2].

Theorem 2.2 [Prestin and Wülker, 2017, Cor. 1.3]. The SGL basis functions constitute an orthonormal polynomial basis of the space H . Thus, every function f ∈ H can be approximated arbitraly well with respect to k · kH by linear combinations of the SGL basis functions. For a translation vector t ∈ R3 , we dene the

translation operator T (t) via

T (t)[f ](x) := f (x − t). Further, for a given rotation R ∈ SO(3) in R3 , we dene the

rotation operator R via

Rf (x) := f (R−1 x). The main application of the SGL basis functions is the fast solution of the following threedimensional rigid matching problem, where due to the present weight function, the center is prioritized over the periphery:

Problem 2.3. t∈

R3

(2.1)

For admissible functions f and g , maximize with respect to R ∈ SO(3) and the absolute value of the weighted overlap integral Z 2 I(R, t) := T (t)[Rf ](x)g(x)e−kxk2 dx. R3

4


x0

z

r0

x ϑ0

r

ϑ

ϕ = ϕ0 y

x S2 Figure 1.

Spherical coordinates in

from the point

x

R3 .

x0 r, ϑ, and ϕ

The point

with spherical coordinates

with spherical coordinates by a translation along the

r 0 , ϑ0 , z axis.

and

ϕ0

arises

Problem 2.3 can be tackled in the following way: Assume that f, g ∈ H are bandlimited functions with bandwidth B ∈ N, i.e., hf, Hnlm iH = hg, Hnlm iH = 0 for n > B . This means that

f =

X

g =

X

|m|≤l 0 be given. For admissible functions f : (0, ∞) → R, we dene the weighted spherical Bessel transform as f˜l (γ; β) :=

r Z ∞ 2 2 f (ξ)jl (βξ)ξ 2 e−γξ dξ, π 0

β ∈ (0, ∞).

For a large class of functions, including the radial part of the SGL basis functions, a corresponding inversion formula can be derived (see [Watson, 1995, Sec. 14.4]). This inversion formula allows for a pointwise reconstruction of such functions from their spherical Bessel transform.

Lemma 3.8 (Inversion formula). Let l ∈ N0 and γ > 0 be given. Let further f : (0, ∞) → R be continuous and of bounded variation on every bounded subset of (0, ∞), and let f satisfy Z

∞

2

|f (ξ)|ξ e−γξ dξ < ∞.

0

Then f can be recovered pointwise from its Bessel transform via γξ 2

f (ξ) = e

r Z ∞ 2 f˜l (γ; β)jl (βξ)β 2 dβ, π 0

ξ ∈ (0, ∞).

In the derivation of the SGL translation matrix elements in the upcoming Section 4, technical diculties arise from the introduction of the Gauss-Weierstrass weight to the spherical Bessel transform. Specically, when solving these problems, we encounter the hypergeometric series:

Denition 3.9 (Hypergeometric series).

For p, q ∈ N with p ≤ q + 1 and parameters a0 , ..., ap−1 ∈ C; b0 , ..., bq−1 ∈ C \ {0, −1, −2, . . . }, the hypergeometric series p Fq (a0 , ..., ap−1 ; b0 , ..., bq−1 ) : C → C is dened as the power series [Andrews et al., 1999, Eq. 2.1.2] (3.8)

p Fq (a0 , . . . , ap−1 ; b0 , . . . , bq−1 ; z) :=

∞ X (a0 )k · · · (ap−1 )k z p k=0

with the

(b0 )k · · · (bq−1 )k p!

,

Pochhammer symbol (rising factorial) (c)k := c(c + 1)(c + 2) · · · (c + k − 1),

c ∈ C, k ∈ N0 .

z ∈ C,


9

Remark 3.10. By denition, the order of the parameters a0 , ..., ap−1 and b0 , ..., bq−1 is irrelevant. If one of the parameters a0 , ..., ap−1 is a non-positive integer, then by denition of the Pochhammer symbol, the hypergeometric series reduces to a polynomial. In this work, we will only encounter the special cases 1 F1 and 2 F1 . The series 1 F1 is called Kummer's (conuent hypergeometric) function. It is absolutely and uniformly convergent on bounded subsets of the complex plane [Andrews et al., 1999, Thm. 2.1.1]. The function 2 F1 , commonly referred to as the (ordinary or Gaussian ) hypergeometric function, will appear only as a polynomial in this work. Finally, we state two auxiliary lemmas for later use. The rst lemma is well-known in the theory of nite dierence equations.

Lemma 3.11. Let

p

be a polynomial of degree less than n ∈ N. Then n X n p(k) = 0. (−1)k k k=0

Proof. See [Levy and Lessman, 1992, Sec. 1.4], for instance.

Lemma 3.12. Let

n ∈ N0 .

Then Γ(n + 1/2) =

√

π (n + 1)n 4−n .

Proof. This follows from the functional equation Γ(z + 1) = zΓ(z), z ∈ C \ {0, −1, −2, . . . } [Andrews et al., 1999, Eq. 1.1.6] and the fact that Γ(1/2) =

√

π [Andrews et al., 1999, Eq. 1.1.22].

4 SGL translation matrix elements We now derive the closed-form expression for the SGL translation matrix elements. To this end, let |m| ≤ l < n ∈ N and |m0 | ≤ l0 < n0 ∈ N be given. Moreover, x ν > 0 and set x0 := x − νez = [r0, ϑ0, ϕ] for x = [r, ϑ, ϕ]. With the dominated convergence theorem, we nd that (mm0 ) Tnn0 ll0 (ν)

Z ∞Z πZ = lim

γ→0+

0

0

2π

0 2

2

Rnl (r0 )Ylm (ϑ0, ϕ)e−γ(r ) Rn0 l0 (r)Yl0 m0 (ϑ, ϕ)e−r dϕ sin ϑ dϑ r2 dr.

0

By Lemma 3.8, the rst part of Lemma 3.3, and with Remark 3.5, we get for r0 > 0 (4.1) 0

0

Rnl (r )Ylm (ϑ , ϕ)e

−γ(r0 )2

r Z ∞ 2 ˜ nl (γ; β)jl (βr0 )Ylm (ϑ0, ϕ)β 2 dβ = R π 0 r Z ∞ ∞ l+l00 X X 2 (ll00 |m|) ˜ nl (γ; β) = R (−1)k Ak jk (βν)jl00 (βr)Yl00 m (ϑ, ϕ)β 2 dβ. π 0 00 00 l =0 k=|l−l |

It is [Erdélyi, 1954, Sec. 8.9, Eq. 5] (4.2)

n−l−1

˜ nl (γ; β) = Nnl (γ − 1) R γ n+1/2

β

l

(l+1/2) Ln−l−1

l+3/2 β2 −β 2 /4γ 1 e , 4γ(1 − γ) 2

0 < γ < 1.

10


For γ = 1, the formula reads as [Erdélyi, 1954, Sec. 8.9, Eq. 2]

2n−l−1/2 Nnl 2n−l−2 −β 2 /4 1 β e . (n − l − 1)! 2

˜ nl (1; β) = R

(4.3)

(mm0 )

Formula (4.2) shows us particularly that after inserting (4.1) into the above formula for Tnn0 ll0 (ν), changing the order of integration is legitimate. By Fubini's theorem and the second part of Lemma 3.3, we thus obtain (4.4)

(mm0 ) Tnn0 ll0 (ν)

r Z ∞Z ∞ ∞ X 2 ˜ nl (γ; β) = lim R γ→0+ π 0 0 00

l =0

00

l+l X

(ll00 |m|)

(−1)k Ak

δl0 l00 δmm0

k=|l−l00 | 2

× jk (βν)jl00 (βr)β dβ Rn0 l0 (r)r2 e−r dr 2

0

= δmm0

l+l X

k

(−1)

k=|l−l0 |

(ll0 |m|) Ak lim γ→0+

Z |0

∞

˜ nl (γ; β) R ˜ n0 l0 (1; β)jk (βν)β 2 dβ , R {z } (nn0 ll0 )

=: Ik

(γ;ν)

(nn0 ll0 )

where the limγ→0+ Ik (γ; ν) is yet to be investigated. The two Kronecker symbols δl0 l00 and δmm0 are due to the orthonormality of the spherical harmonics. Because of the remaining (|m|) (mm0 ) Kronecker symbol δmm0 , we shall write Tnn0 ll0 (ν) = Tnn0 ll0 (ν). Due to Lemma 3.6, we have to consider only the case when l − l0 + k is even in the following.

Lemma 4.1. Let

l − l0 + k

be even and set µ := n0 + (l − l0 + k)/2 ∈ N. Then √

(nn0 ll0 ) lim Ik (γ; ν) γ→0+

π Nnl Nn0 l0 4 (n0 − l0 − 1)! n−l−1 X (−1)j n − 1/2 Γ(µ + j + 1/2) (nn0 ll0 k) × Cj (ν), j! n−l−1−j Γ(k + 3/2) n−l−1 k

= (−1)

ν

j=0

where (nn0 ll0 k) Cj (ν)

=

n−µ X p=0

2 F1 (j, −n

+ µ + p; −n + µ − j + 1; −1)(−µ + k − j + 1)p −j − p ν 2p . p!(k + 3/2)p n−µ−p

0 0

Proof. By inserting (4.2) and (4.3) into Ik(nn ll ) (γ; ν), we derive (nn0 ll0 ) Ik (γ; ν)

2n0+l−l0+2j+1 n−l−1 Nnl Nn0 l0 X (−1)j n − 1/2 1 = (−1) 0 0 (n − l − 1)! j! n−l−1−j 2 j=0 Z ∞ 2 −1 0 0 × γ −n−j−1/2 (1 − γ)n−l−j−1 e−β (1+γ )/4 jk (βν)β 2n +l−l +2j dβ , |0 {z } n−l−1

(n0 ll0 k)

=: Jj

(γ;ν)

where we have also used the closed-form expression of the generalized Laguerre polynomials [Andrews et al., 1999, Eq. 6.2.2]. Now we solve the integral on the right-hand side by using


11

[Erdélyi, 1954, Sec. 8.6, Eq. 14]: (n0 ll0 k) Jj (γ; ν)

r =

µ+j+1/2 γ π Γ(µ + j + 1/2) 0 0 22n +l−l +2j−1/2 2 Γ(k + 3/2) 1+γ γ 2 γ k − 1+γ ν 2 ×ν e ν , 1 F1 − µ + k − j + 1; k + 3/2; 1+γ

where 1 F1 denotes Kummer's conuent hypergeometric function (cf. Def. 3.9). Now we investigate the limit of

√

(nn0 ll0 ) Ik (γ; ν)

π Nnl Nn0 l0 − γ ν2 γ µ−n e 1+γ 0 0 4 (n − l − 1)! n−l−1 X (−1)j n − 1/2 Γ(µ + j + 1/2) (1 + γ)−µ−j−1/2 × j! n−l−1−j Γ(k + 3/2) j=0 γ n−l−j−1 2 × (1 − γ) ν 1 F1 − µ + k − j + 1; k + 3/2; 1+γ n−l−1 k

= (−1)

ν

as γ tends to zero from above. Since the right-hand side contains the critical factor γ µ−n , we distinguish between three dierent cases. (nn0 ll0 )

If k > 2(n − n0 ) − l + l0, then µ − n > 0, and thus limγ→0+ Ik

lim 1 F1 − µ + k − j + 1; k + 3/2;

(4.5)

γ→0+

γ ν2 1+γ

(γ; ν) = 0, since by (3.8),

= 1.

If, on the other hand, k = 2(n − n0 ) − l + l0, then µ − n = 0, and

√

(nn0 ll0 ) lim Ik (γ; ν) γ→0+

π Nnl Nn0 l0 4 (n0 − l0 − 1)! n−l−1 X (−1)j n − 1/2 Γ(µ + j + 1/2) , × j! n−l−1−j Γ(k + 3/2) n−l−1 k

= (−1)

ν

j=0

which also follows from (4.5). If k < 2(n − n0 ) − l + l0, then µ − n < 0. In this case, we cannot make use of (4.5) directly. However, we have the power-series representation −j−p

(1 + γ)

−j

(1 − γ)

=

∞ X −j − p q=0

q

2 F1 (j, −q; 1

− q − p − j; −1)γ q ,

with the hypergeometric function 2 F1 (cf. Def. 3.9), derived from the Cauchy product of the binomial series of (1 + γ)−j−p and (1 − γ)−j [Abramowitz and Stegun, 1972, Eq. 3.6.8]. This series is absolutely convergent for |γ| < 1. Rearranging the summands, we thus obtain together with (3.8) for suciently small 0 < γ < 1

√

(4.6)

(nn0 ll0 ) Ik (γ; ν)

γ π Nnl Nn0 l0 − 1+γ ν2 e Γ(k + 3/2)−1 0 0 4 (n − l − 1)! ∞ X ν 2p (nn0 ll0 k) p+q−n+µ × (1 + γ)−µ−1/2 (1 − γ)n−l−1 Dpq γ , p!(k + 3/2)p

n−l−1 k

= (−1)

ν

p,q=0

12


where we set (4.7)

0 0

(nn ll k) := Dpq

n−l−1 X

n − 1/2 (−1)j Γ(µ + j + 1/2) n−l−1−j j! j=0 −j − p × (−µ + k − j + 1)p 2 F1 (j, −q; 1 − q − p − j; −1). q

(nn0 ll0 k)

Before taking the limit γ → 0+, we now prove that Dpq = 0 for p + q < n − µ. To this end, we rst use Lemma 3.12 to rewrite n − 1/2 Γ(n + 1/2) = (n − l − 1 − j)!Γ(l + j + 3/2) n−l−1−j

=

4−n+l+j+1 (n + 1)n . (n − l − 1 − j)!(l + j + 2)l+j+1

Again by Lemma 3.12, we nd that

Γ(µ + j + 1/2) = The fact that

µ ≥ 1 + l0 +

√

π (µ + j + 1)µ+j 4−µ−j .

l − l0 + k l + l0 + |l − l0 | ≥ 1+ ≥ 1+l 2 2

then reveals √ n − 1/2 π (n + 1)n (µ + j + 1)µ+j µ−n+l+1 Γ(µ + j + 1/2) = (4.8) 4 n−l−1−j (n − l − 1 − j)! (l + j + 2)l+j+1 √ π (n + 1)n (l + j + 3/2)µ−l−1 42µ−n . = (n − l − 1 − j)! By denition of the hypergeometric function 2 F1 (Def. 3.9), we get

q 1 X −j − p (4.9) (−1)s (−q)s (1 − p − q + s − j)q−s (j)s . F (j, −q; 1 − j − q − p; −1) = 2 1 q q! s=0

Inserting (4.8) and (4.9) into (4.7), changing the order of summation, and expanding with (n − l − 1)! yields

√

(nn0 ll0 k) Dpq

q

π (n + 1)n 1 X = 4 (−1)s (−q)s (n − l − 1)! q! s=0 n−l−1 X j n−l−1 × (−1) (l+j +3/2)µ−l−1 (−µ+k−j +1)p (1−p−q+s−j)q−s (j)s . j 2µ−n

j=0

Since (l + j + 3/2)µ−l−1 (−µ + k − j + 1)p (1 − p − q + s − j)q−s (j)s can be seen as a polynomial (nn0 ll0 k) of degree p + q + µ − l − 1 evaluated at j , we nd that Dpq = 0 for p + q < n − µ by Lemma 3.11, as claimed. Therefore, the double series in (4.6) can be seen as a power series in γ with convergence radius one, which as such converges uniformly on all compact subsets of [0, 1). This allows for a term-by-term limit formation γ → 0+. Because all summands in (4.6) with


13

p + q > n − µ vanish, we are left with the summands where p + q = n − µ. This means that (nn0 ll0 ) lim Ik (γ; ν) γ→0+

√ n−µ ν 2p π Nnl Nn0 l0 X (nn0 ll0 k) Dp,(n−µ)−p . ν 0 0 4 (n − l − 1)! p!(k + 3/2)p

n−l−1 k

= (−1)

p=0

In summary, applying Lemma 4.1 to (4.4), we obtain the following closed-form expression for the SGL translation matrix elements:

Main theorem 4.2 (SGL translation matrix elements). The SGL translation matrix elements possess the closed-form expression √ (|m|) Tnn0 ll0 (ν)

n−l−1

= (−1)

0

l+l X π Nnl Nn0 l0 (ll0 |m|) k (−1)k Ak ν 0 0 4 (n − l − 1)! 0 k=|l−l |

n−l−1 X (−1)j n − 1/2 Γ(µ + j + 1/2) (nn0 ll0 k) Cj (ν), × j! n−l−1−j Γ(k + 3/2)

ν > 0,

j=0 0

0 0

|m|) where the coecients A(ll are those of Lemma 3.3, while µ and the functions Cj(nn ll k) are k 0 0 given in Lemma 4.1 (setting Cj(nn ll k) := 0 whenever l − l0 + k is odd). 0 0

Remark 4.3. Note that the hypergeometric function 2 F1 in Cj(nn ll k) is, in fact, a polynomial.

Acknowledgments The authors would like to thank Sabrina Kombrink, Vitalii Myroniuk, and Nadiia Derevianko for scientic discussion and helpful comments on the manuscript.

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