Smooth Hankel Functions

Table of Contents

Unique properties of Smoothed Hankel functions

Smooth Hankel functions are convolutions of ordinary Hankel functions and Gaussian functions and are regular at the origin. Ordinary Hankel functions H¯L\bar H_L are solutions of the Helmholtz wave equation

(Δ+ε)H¯L(ε,r)=4πδ(r)(1)\left(\Delta+\varepsilon\right)\bar H_L(\varepsilon,{\mathbf{r}}) = -4\pi \delta({\mathbf{r}}) \quad\quad\quad\quad (1)

Solutions are products of radial functions and spherical harmonics YL(r^)Y_L(\hat{\mathbf{r}}), Here LL is a compound index for the m\ell{}m quantum numbers. Radial functions are spherical Hankel or Bessel functions. We will focus on the Hankel functions:

H¯L(ε;r)=iκ¯+1h(1)(iκ¯r)YL(r^)whereκ¯2=ε(2)\bar H_L(\varepsilon;{\mathbf{r}})=-i^\ell{\bar\kappa}^{\ell+1}h_\ell^{(1)}(i{\bar\kappa}{}r)Y_L(\hat{\mathbf{r}}) \quad \mathrm{where} \quad {\bar\kappa}^2 = -\varepsilon \quad\quad\quad\quad (2)

where h(1)(z)h_\ell^{(1)}(z) is the spherical Hankel function of the first kind. Hankel (Bessel) functions are regular (irregular) as rr \rightarrow \infty thus Hankel functions are exact solutions of the Schrödinger equation in a flat potential with appropriate boundary conditions for large rr. For small rr, the situation is reversed with Bessel functions being regular. Hankel and Bessel functions vary as r1r^{-\ell-1} and rr^{\ell} when r0r \rightarrow 0.

When envelope functions are augmented with partial waves in spheres around atoms, the irregular part of H¯L\bar H_L is eliminated. Thus augmented Hankel functions can form exact solutions to the Schrödinger equation in a muffin-tin potential.

Smooth Hankel functions HLH_L are regular for both large and small rr. The Figure below compares ordinary Hankel functions H¯L(ε,r)\bar H_L(\varepsilon,{\mathbf{r}}) (dashed lines) to smooth ones HLH_L for ε=0.5\varepsilon{=}-0.5. Red, green, and blue correspond to =0,1,2\ell=0,1,2.

Ordinary and Smooth Hankel functions

Smooth Hankels are superior to ordinary ones, first because real potentials are not flat so there is scope for improvement on the Hankel functions as the basis set.

Also the fact that the HLH_L are everywhere smooth can greatly facilitate their implementation. In the present Questaal implementation the charge density is kept on a uniform mesh of points. Sharply peaked functions require finer meshes, and some smoothing would necessary in any case.

Finally also have a sensible asymptotic form, decaying exponentially as real wave functions do when far from an atom. Thus they have better shape than gaussian orbitals do.

Smooth Hankels have two big drawbacks as a basis set. First, they are more complicated to work with. One center expansions of ordinary Hankels (needed for augmentation) are Bessel functions. A counterpart does exist for smooth Hankels, but expansions are polynomials related to Laguerre polynomials. The expansion is cumbersom and introduces an extra cutoff in the polynomial order.

Second, gaussian orbitals hold an enormous advantage over both ordinary and smooth Hankels, namely that the product of two of them in real space can be expressed as another gaussian (plane waves have a similar property). There exist no counterpart for Hankels, so an auxiliary basis must be constructed to make the charge density and matrix elements of the potential. The Questaal suite uses plane waves for the auxiliary basis.

Smooth Hankel functions and the HkL family

Methfessel’s class of functions HkLH_{kL}, are a superset of smoothed Hankel functions HLH_L; they also incorporate the family of (polynomial)×\times(gaussians). The HLH_L and the HkLH_{kL} are defined in reference 1, and many of their properties derived there. The HkL(r)H_{kL}({\mathbf{r}}) are a family of functions with k=0,1,2,...k=0,1,2,... and angular momentum LL. They are members of the general class of functions FL(r)F_L({\mathbf{r}}) which are determined from a single radial function by

FL(r)=ΥL()f(r)(3)F_L({\mathbf{r}}) = \Upsilon_L(-\nabla) \, f(r) \quad\quad\quad\quad (3)

ΥL(r)\Upsilon_L({\mathbf{r}}) with r=(x,y,z){\mathbf{r}}=(x,y,z) is a polynomial in (x,y,z)(x,y,z), so is meaningful to talk about ΥL()\Upsilon_L(-\nabla). It is written in terms of conventional spherical harmonics as

ΥL(r)=rYL(r^)(4)\Upsilon_L({\mathbf{r}}) = r^\ell Y_L (\hat{\mathbf{r}}) \quad\quad\quad\quad (4)

Just as the product of two spherical harmonics can be expanded in Clebsh Gordan coefficients CKLMC_{KLM} and spherical harmonics, so can the product of two spherical harmonic polynomials:

YK(r^)YL(r^)=MCKLMYM(r^)ΥK(r)ΥL(r)=MCKLMrk+mΥM(r)(5)\begin{array}{rlr} Y_K(\hat{\mathbf{r}})Y_L(\hat{\mathbf{r}}) & = \sum_M C_{KLM} Y_M({\hat{\mathbf{r}}}) & \\ \Upsilon_K({\mathbf{r}})\Upsilon_L({\mathbf{r}}) & = \sum_M C_{KLM} r^{k+\ell-m} \Upsilon_M({\mathbf{r}}) & \qquad (5) \end{array}

CKLMC_{KLM} is nonzero only when k+m{k+\ell-m} is an even integer, so the r.h.s. is also a polynomial in (x,y,z)(x,y,z), as it must be.

Functions HL(r)H_{L}({\mathbf{r}}) are defined through the radial function h(r)h(r) (Ref 1, Eq. 6.5):

HL(ε,rs;r)=ΥL()h(ε,rs;r)(6)h(ε,rs;r)=12r(u+(ε,rs;r)u(ε,rs;r))(7)u±(ε,rs;r)=eκ¯r[1erf(rsκ¯2rrs)](8)\begin{array}{rlr} H_{L}(\varepsilon,r_s;{\mathbf{r}}) & = \Upsilon_L(-\nabla) h(\varepsilon,r_s;r) & \qquad (6) \\ h(\varepsilon,r_s;r) & = \frac{1}{2r}\left(u_+(\varepsilon,r_s;r) - u_{-}(\varepsilon,r_s;r)\right) & \qquad (7) \\ u_{\pm}(\varepsilon,r_s;r) & = e^{\mp{\bar\kappa}{}r}\left[1-\mathrm{erf}\left(\frac{r_s{\bar\kappa}}{2}\mp{}\frac{r}{r_s}\right)\right] & \qquad (8) \end{array}

HLH_{L} is parameterized by energy ε\varepsilon and smoothing radius rsr_s; their significance will will become clear shortly. The extended family HkL(r)H_{kL}({\mathbf{r}}) is defined through powers of the Laplacian acting on HL(r)H_{L}({\mathbf{r}}):

HkL(r)=ΔkHL(r)(9)H_{kL}({\mathbf{r}}) = \Delta^k H_{L}({\mathbf{r}}) \quad\quad\quad\quad (9)

In real space HkLH_{kL} must be generated recursively from hh. However, the Fourier transform of HkLH_{kL} has a closed form (Ref 1, Eq. 6.35). The differential operator becomes a multiplicative operator in the reciprocal space so

H^kL(ε,rs;q)=4πεq2ΥL(iq)(q2)kers2(εq2)/4(10)\hat{H}_{kL}(\varepsilon,r_s;\mathbf{q}) = -\frac{4\pi}{\varepsilon-q^2}\Upsilon_L(-i\mathbf{q})\,(-q^2)^k\,e^{r_s^2(\varepsilon-q^2)/4} \quad\quad\quad\quad (10)

By taking limiting cases we can see the connection with familiar functions, and also the significance of parameters ε\varepsilon and rsr_s.

  1. k=0k=0 and rs=0r_s=0: H^00(ε,0;q)=4π/(εq2)\hat{H}_{00}(\varepsilon,0;\mathbf{q})=-{4\pi/(\varepsilon-q^2)}
    This is the Fourier transform of H00(ε,0;r)=exp(κ¯r)/rH_{00}(\varepsilon,0;r)=\exp(-{\bar\kappa}{}r)/r, and is proportional to the =0\ell=0 spherical Hankel function of the first kind, h0(1)(z)h_0^{(1)}(z). For general LL the relation is

    H0L(ε,0;r)=HL(ε,0;r)=H¯L(ε,r)=iκ¯+1h(1)(iκ¯r)YL(r^)H_{0L}(\varepsilon,0;{\mathbf{r}})=H_{L}(\varepsilon,0;{\mathbf{r}})=\bar H_{L}(\varepsilon,{\mathbf{r}})=-i^\ell{\bar\kappa}^{\ell+1}h_\ell^{(1)}(i{\bar\kappa}{}r)Y_L(\hat{\mathbf{r}})

    which is Eq. 2.

  2. k=1k=1 and ε=0\varepsilon=0: H^10(0,rs;q)=4πers2q2/4\hat{H}_{10}(0,r_s;\mathbf{q})=-{4\pi} e^{-r_s^2q^2/4}.
    This is the Fourier transform of a Gaussian function, whose width is defined by rsr_s. For general LL we can define the family of generalized Gaussian functions

    G0L(ε,rs;r)=ΥL()g(ε,rs;r)=(1πrs2)3/2eεrs2/4(2rs2)ler2/rs2YL(r^)G_{0L}(\varepsilon,r_s;{\mathbf{r}}) = \Upsilon_L(-\nabla) g(\varepsilon,r_s;r) = \left(\frac{1}{\pi r_s^2}\right)^{3/2} e^{\varepsilon r_s^2/4} \left(2r_s^{-2}\right)^l e^{-r^2/r_s^2} Y_L(\hat{\mathbf{r}})

    gg is a simple gaussian with an extra normalization:

    g(ε,rs;r)=(1πrs2)3/2eεrs2/4er2/rs2(11)g(\varepsilon,r_s;r) = \left(\frac{1}{\pi r_s^2}\right)^{3/2} e^{\varepsilon r_s^2/4}e^{-r^2/r_s^2} \quad\quad\quad\quad (11)

Comparing cases 1 and 2 with Eq. (10), evidently H^L(q)\hat{H}_L(\mathbf{q}) is proportional to the product of the Fourier transforms of a conventional spherical Hankel function of the first kind, and a gaussian. By the convolution theorem, HL(r){H_L}({\mathbf{r}}) is a convolution of a Hankel function and a gaussian. For rrsr\gg r_s, HL(r){H_L}({\mathbf{r}}) behaves as a Hankel function and asymptotically tends to HL(r)rl1exp(εr)YL(r^)H_L({\mathbf{r}})\to r^{-l-1}\exp(-\sqrt{-\varepsilon}r)Y_L(\hat{\mathbf{r}}). For rrsr\ll r_s it has structure of a gaussian; it is therefore analytic and regular at the origin, varying as rlYL(r^)r^lY_L(\hat{\mathbf{r}}). Thus, the rl1r^{-l-1} singularity of the Hankel function is smoothed out, with rsr_s determining the radius for transition from Gaussian-like to Hankel-like behavior. Thus, the smoothing radius rsr_s determines the smoothness of HLH_L, and also the width of generalized gaussians GLG_L.

By analogy with Eq. (9) we can extend the GLG_{L} family with the Laplacian operator:

GkL(ε,rs;r)=ΔkG0L(ε,rs;r)=ΥL()Δkg(ε,rs;r)(12)GkL(ε,rs;r)=ΥL()(1r2r2r)kg(ε,rs;r)(13)G^kL(ε,rs;q)=ΥL(iq)(q2)kers2(εq2)/4(14)\begin{array}{rlr} G_{kL}(\varepsilon,r_s;{\mathbf{r}}) & = \Delta^k\, G_{0L}(\varepsilon,r_s;{\mathbf{r}}) = \Upsilon_L(-\nabla) \Delta^k g(\varepsilon,r_s;r) & \qquad (12) \\ G_{kL}(\varepsilon,r_s;{\mathbf{r}}) & = \Upsilon_L(-\nabla)\left(\frac{1}{r}\frac{\partial^2}{\partial r^2}r\cdot\right)^k g(\varepsilon,r_s;r) & \qquad (13) \\ \hat{G}_{kL}(\varepsilon,r_s;\mathbf{q}) & = \Upsilon_L(-i\mathbf{q})(-q^2)^k e^{r_s^2(\varepsilon-q^2)/4} & \qquad (14) \end{array}

The second equation shows that GkLG_{kL} has the structure (polynomial of order kk in r2r^2)×GL\times G_L. These polynomials are related to the generalized Laguerre polynomials of half-integer order in r2r^2. They obey a recurrence relation (see Ref 1, Eq. 5.19), which is how they are evaluated in practice. They are proportional to the polynomials PkLP_{kL} used in one-center expansions of smoothed Hankels around remote sites (see Ref 1, Eq. 12.7).

Differential equation for smooth Hankel functions

Comparing the last form Eq. (14) to Eq. (10) and the definition of HkLH_{kL} Eq. (9), we obtain the useful relations

Hk+1,L(ε,rs;r)+εHkL(ε,rs;r)=4πGkL(ε,rs;r)(15)(Δ+ε)HkL(ε,rs;r)=4πGkL(ε,rs;r)(16)\begin{array}{rlr} H_{k+1,L}(\varepsilon,r_s;{\mathbf{r}})+\varepsilon H_{kL}(\varepsilon,r_s;{\mathbf{r}}) & = - 4\pi G_{kL}(\varepsilon,r_s;{\mathbf{r}}) & \qquad (15) \\ \left(\Delta+\varepsilon\right)H_{kL}(\varepsilon,r_s;{\mathbf{r}}) & = -4\pi G_{kL}(\varepsilon,r_s;{\mathbf{r}}) & \qquad (16) \end{array}

This shows that HkLH_{kL} is the solution to the Helmholz operator Δ+ε\Delta+\varepsilon in response to a source term smeared out in the form of a gaussian. A conventional Hankel function is the response to a point multipole at the origin (see Ref 1, Eq. 6.14). HkLH_{kL} is also the solution to the Schrödinger equation for a potential that has an approximately gaussian dependence on rr (Ref 1, Eq. 6.30).

Two-center integrals of smoothed Hankels

One extremely useful property of the HkLH_{kL} is that the product of two of them, centered at different sites r1{\mathbf{r}}_1 and r2{\mathbf{r}}_2, can be integrated in closed form. The result a sum of other HkLH_{kL}, evaluated at the connecting vector r1r2{\mathbf{r}}_1-{\mathbf{r}}_2. This can be seen from the power theorem of Fourier transforms

H1(rr1)H2(rr2)d3r=(2π)3H^1(q)H^2(q)eiq(r1r2)d3q(17)\int H^*_1({\mathbf{r}}-{\mathbf{r}}_1) H_2({\mathbf{r}}-{\mathbf{r}}_2) d^3r = (2\pi)^{-3}\int \hat{H}^*_1(\mathbf{q}) \hat{H}_2(\mathbf{q}) e^{i\mathbf{q}\cdot({\mathbf{r}}_1-{\mathbf{r}}_2)} d^3q \quad\quad\quad\quad (17)

and the fact that H^k1L1(q)H^k2L2(q)\hat{H}^*_{k_1L_1}(\mathbf{q})\hat{H}_{k_2L_2}(\mathbf{q}) can be expressed as a linear combination of other H^kL(q)\hat{H}_{kL}(\mathbf{q}), or their energy derivatives. This is readily done from the identity

1(ε1q2)(ε2q2)=1ε1ε2[1ε2q21ε1q2]limε2ε11(ε1q2)2(18)\frac{1}{(\varepsilon_1-q^2)(\varepsilon_2-q^2)}=\frac{1}{\varepsilon_1-\varepsilon_2} \left[ \frac{1}{\varepsilon_2-q^2} - \frac{1}{\varepsilon_1-q^2} \right] \longrightarrow\lim_{\varepsilon_2\to\varepsilon_1} \frac{1}{(\varepsilon_1-q^2)^2} \quad\quad\quad\quad (18)

Comparing the first identity and the form Eq.~(10) of H^p0(q)\hat{H}_{p0}(\mathbf{q}), it can be immediately seen that the product of two H^p0(q)\hat{H}_{p0}(\mathbf{q}) with different energies can be expressed as a linear combination of two H^p0(q)\hat{H}_{p0}(\mathbf{q}). The second identity applies when the H^p0(q)\hat{H}_{p0}(\mathbf{q}) have the same energy; the product will involve the energy derivative of some H^p0(q)\hat{H}_{p0}(\mathbf{q}). For higher LL, ΥL1(iq)ΥL2(iq)\Upsilon^*_{L_1}(-i\mathbf{q})\Upsilon_{L_2}(-i\mathbf{q}) is expanded as a linear combination of ΥM(iq)\Upsilon^*_{M}(-i\mathbf{q}) using the expansion theorem for spherical harmonics, Eq. (5). In detail,

H^k1L1(ε1,rs1;q)H^k2L2(ε2,rs2;q)=(4π)2(ε1q2)(ε2q2)ΥL1(iq)ΥL2(iq)(q2)k1+k2ers12(ε1q2)/4+rs22(ε2q2)/4(19){\hat{H}^*_{k_1L_1}}(\varepsilon_1,r_{s_1};\mathbf{q}) {\hat{H}_{k_2L_2}}(\varepsilon_2,r_{s_2};\mathbf{q}) = \frac{(4\pi)^2}{(\varepsilon_1-q^2)(\varepsilon_2-q^2)} \Upsilon^*_{L_1}(-i{\mathbf{q}})\Upsilon_{L_2}(-i{\mathbf{q}}) (-q^2)^{k_1+k_2}\,e^{r_{s_1}^2(\varepsilon_1-q^2)/4+r_{s_2}^2(\varepsilon_2-q^2)/4} \quad\quad (19)

which can be written as

(4π)2ε1ε2[ers12(ε1ε2)/4e(rs22+rs12)(ε2q2)/4ε2q2ers22(ε2ε1)/4e(rs12+rs22)(ε1q2)/4ε1q2]×(q2)k1+k2i21MCL1L2MΥM(iq)(q2)(1+2m)/2\frac{(4\pi)^2}{\varepsilon_1-\varepsilon_2} \left[ \frac{e^{r_{s_1}^2(\varepsilon_1-\varepsilon_2)/4}e^{(r_{s_2}^2+r_{s_1}^2)(\varepsilon_2-q^2)/4}}{\varepsilon_2-q^2}- \frac{e^{r_{s_2}^2(\varepsilon_2-\varepsilon_1)/4}e^{(r_{s_1}^2+r_{s_2}^2)(\varepsilon_1-q^2)/4}}{\varepsilon_1-q^2} \right]\times (-q^2)^{k_1+k_2} i^{2\ell_1} \sum_M C_{L_1L_2M} \Upsilon_{M}(-i{\mathbf{q}}) (-q^2)^{(\ell_1+\ell_2-m)/2}

where rs2=rs12+rs22r_{s}^2=r_{s_1}^2+r_{s_2}^2.

This last equation is a linear combination of HkLH_{kL} with smoothing radius rsr_s given as shown.

Using the power theorem the two-center integrals can be directly evaluated:

Hk1L1(ε1,rs1;rr1)Hk2L2(ε2,rs2;rr2)d3r=1(2π)3H^k1L1(ε1,rs1;q)H^k2L2(ε2,rs2;q)eiq(r1r2)d3q(20)\int{H^*_{k_1L_1}}(\varepsilon_1,r_{s_1};{\mathbf{r}}-{\mathbf{r}_1}) {H_{k_2L_2}}(\varepsilon_2,r_{s_2};{\mathbf{r}}-{\mathbf{r}_2})\, d^3r = \frac{1}{(2\pi)^3} \int {\hat{H}^*_{k_1L_1}}(\varepsilon_1,r_{s_1};{\mathbf{q}}) {\hat{H}_{k_2L_2}}(\varepsilon_2,r_{s_2};{\mathbf{q}})\, e^{i{\mathbf{q}}\cdot\left({\mathbf{r}}_1-{\mathbf{r}}_2\right)} d^3q \quad\quad (20)

This can be written as

(1)14πε1ε2MCL1L2M×[ ers22(ε2ε1)/4Hk1+k2+(1+2m)/2,M(ε1,rs;r1r2)ers12(ε1ε2)/4Hk1+k2+(1+2m)/2,M(ε2,rs;r1r2) ](-1)^{\ell_1}\frac{4\pi}{\varepsilon_1-\varepsilon_2} \sum_M C_{L_1L_2M} \times \big[ {\ e^{r_{s_2}^2(\varepsilon_2-\varepsilon_1)/4}{H_{k_1+k_2+{(\ell_1+\ell_2-m)/2},M}}(\varepsilon_1,r_{s};{\mathbf{r}}_1-{\mathbf{r}}_2)} {- e^{r_{s_1}^2(\varepsilon_1-\varepsilon_2)/4}{H_{k_1+k_2+{(\ell_1+\ell_2-m)/2},M}}(\varepsilon_2,r_{s};{\mathbf{r}}_1-{\mathbf{r}}_2)} \ \big]

The special case ε1=ε2=ε\varepsilon_1=\varepsilon_2=\varepsilon must be handled using the limiting form of Eq. (18). Differentiation of Eq. (10) with respect to energy results in

H˙^kLH^kLε=1εq2H^kL+(rs2/4)H^kL.(21)\hat{\dot{H}}_{kL} \equiv \frac{\partial\hat{H}_{kL}}{\partial\varepsilon} = -\frac{1}{\varepsilon-q^2}\hat{H}_{kL} + (r_s^2/4) \hat{H}_{kL}. \quad\quad\quad\quad (21)

The two-center integral now becomes

Hk1L1(ε,rs1;rr1)Hk2L2(ε,rs2;rr2)d3r=(1)14πMCL1L2M×[H˙k1+k2+(1+2m)/2,M(ε,rs;r1r2)(rs2/4)Hk1+k2+(1+2m)/2,M(ε,rs;r1r2)](22)\begin{array}{cl} \int{H^*_{k_1L_1}}(\varepsilon,r_{s_1};\mathbf{r}-\mathbf{r}_1) H_{k_2L_2}(\varepsilon,r_{s_2};\mathbf{r}-\mathbf{r}_2)\, d^3r = & \\ (-1)^{\ell_1}{4\pi} \sum_M C_{L_1L_2M} \times \big[ {\dot{H}_{k_1+k_2+{(\ell_1+\ell_2-m)/2},M}}(\varepsilon,r_{s};\mathbf{r}_1-\mathbf{r}_2) - (r_s^2/4) {H_{k_1+k_2+{(\ell_1+\ell_2-m)/2},M}}(\varepsilon,r_{s};\mathbf{r}_1-\mathbf{r}_2) \big] & \qquad (22) \end{array}

If we consider a further limiting case, namely ε1=ε2=0\varepsilon_1=\varepsilon_2=0, Eq. (22) simplifies to

H˙^kLlimε01q2H^kL+(rs2/4)H^kL=H^k1,L+(rs2/4)H^kL(23)\hat{\dot{H}}_{kL} \longrightarrow\lim^{\varepsilon\to 0} - \frac{1}{-q^2}\hat{H}_{kL} + (r_s^2/4) \hat{H}_{kL} = - \hat{H}_{k-1,L} + (r_s^2/4)\hat{H}_{kL} \quad\quad\quad\quad (23)

and the two-center integral simplifies to

Hk1L1(0,rs1;rr1)Hk2L2(0,rs2;rr2)d3r=(1)1(4π)MCL1L2M×Hk1+k21+(1+2m)/2,M(0,rs;r1r2).(24)\int {H}^*_{k_1L_1}(0,r_{s_1};{\mathbf{r}}-{\mathbf{r}_1}) {H}_{k_2L_2}(0,r_{s_2};{\mathbf{r}}-{\mathbf{r}_2})\, d^3r = (-1)^{\ell_1}(-{4\pi}) \sum_M C_{L_1L_2M} \times H_{k_1+k_2-1+{(\ell_1+\ell_2-m)/2},M}(0,r_{s};{\mathbf{r}}_1-{\mathbf{r}}_2). \qquad (24)

When ε=0\varepsilon=0 and k1k\ge 1 the HkLH_{kL} are generalized Gaussian functions of the type Eq. (12), scaled by 4π-4\pi; see Eq. (16). Eq. (24) is then suitable for two-center integrals of generalized Gaussian functions.

Smoothed Hankels for positive energy

The smooth Hankel functions defined in Ref. 1 for negative energy also apply for positive energy. We demonstrate that here, and show that the difference between the conventional and smooth Hankel functions are real functions.

Ref. 1 defines κ¯{\bar\kappa} in contradistinction to usual convention for κ\kappa

κ¯2=εwithκ¯>0{\bar\kappa}^2 = -\varepsilon \quad \mathrm{with}\quad {\bar\kappa}>0

and restricts ε<0\varepsilon {\lt} 0. According to usual conventions κ\kappa is defined as

κ=ε,Im(κ)0.\kappa = \sqrt\varepsilon, \quad Im(\kappa) \ge 0.

We can define for any energy

κ¯=iκ.\bar\kappa = -i \kappa .

Then κ¯\bar\kappa is real and positive if ε<0\varepsilon \lt 0, while κ\kappa is real and positive if ε>0\varepsilon \gt 0.

The smoothed Hankel ss orbitals for =0\ell=0 and =1\ell=-1 are real:

h0s(r)=(u+u)/2rh1s(r)=(u++u)/2κ¯\begin{array}{rl} h^s_0 (r) & = (u_+ - u_-) / 2r \\ h^s_{-1}(r) & = (u_+ + u_-) / 2{\bar\kappa} \end{array}

To extend the definition to any energy we define U±U_{\pm} as:

U±=e±iκrerfc(r/rs±iκrs/2)U_{\pm} = e^{\pm i\kappa r}\, \mathrm{erfc}\left(r/r_s\pm{i\kappa r_s}/2\right)

The following relations are useful:


Then for ε<0\varepsilon \lt 0, iκi\kappa is real and U+=2eiκru+U_+ = 2e^{i\kappa r} - u_+ and U=uU_- = u_- are also real.

The difference in ordinary and smoothed Hankels is

h0h0s=[eiκru+/2+u/2]/r=[U+/2+U/2]/rh1h1s=[eiκru+/2u/2]/κ¯=[U+/2U/2]/(iκ)\begin{array}{rl} h_0 - h^s_0 & = [e^{i \kappa r} - u_+/2 + u_-/2] /r \\ & = [U_+/2 + U_-/2] /r \\ & \\ h_{-1} - h^s_{-1} & = [e^{i \kappa r} - u_+/2 - u_-/2] /{\bar\kappa} \\ & = [U_+/2 - U_-/2] /(-i \kappa) \end{array}

For ε>0\varepsilon>0, κ\kappa is real and U+U_+ = UU_-^*. The difference in ordinary and smoothed Hankels is

h0h0s=[U+/2+U/2]/r=Re(U+)/rh1h1s=[U+/2U/2]/(iκ)=Im(U+)/κ\begin{array}{rl} h_0 - h^s_0 & = [U_+/2 + U_-/2] /r = \mathrm{Re}(U_+) /r \\ h_{-1} - h^s_{-1} & = [U_+/2 - U_-/2] / (-i \kappa) = -\mathrm{Im} (U_+) / \kappa \end{array}

Both are real, though h0h_0 and h1h_{-1} are complex.

Other Resources

  1. Many mathematical properties of smoothed Hankel functions and the HkLH_{kL} family are described in this paper: E. Bott, M. Methfessel, W. Krabs, and P. C. Schmid, Nonsingular Hankel functions as a new basis for electronic structure calculations, J. Math. Phys. 39, 3393 (1998)

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