Table of Contents
- Unique properties of Smoothed Hankel functions
- Smooth Hankel functions and the HkL family
- Two-center integrals of smoothed Hankels
- Smoothed Hankels for positive energy
- Other Resources
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 are solutions of the Helmholtz wave equation
Solutions are products of radial functions and spherical harmonics , Here is a compound index for the quantum numbers. Radial functions are spherical Hankel or Bessel functions. We will focus on the Hankel functions:
where is the spherical Hankel function of the first kind. Hankel (Bessel) functions are regular (irregular) as thus Hankel functions are exact solutions of the Schrödinger equation in a flat potential with appropriate boundary conditions for large . For small , the situation is reversed with Bessel functions being regular. Hankel and Bessel functions vary as and when .
When envelope functions are augmented with partial waves in spheres around atoms, the irregular part of is eliminated. Thus augmented Hankel functions can form exact solutions to the Schrödinger equation in a muffin-tin potential.
Smooth Hankel functions are regular for both large and small . The Figure below compares ordinary Hankel functions (dashed lines) to smooth ones for . Red, green, and blue correspond to .
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 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 , are a superset of smoothed Hankel functions ; they also incorporate the family of (polynomial)(gaussians). The and the are defined in reference 1, and many of their properties derived there. The are a family of functions with and angular momentum . They are members of the general class of functions which are determined from a single radial function by
with is a polynomial in , so is meaningful to talk about . It is written in terms of conventional spherical harmonics as
Just as the product of two spherical harmonics can be expanded in Clebsh Gordan coefficients and spherical harmonics, so can the product of two spherical harmonic polynomials:
is nonzero only when is an even integer, so the r.h.s. is also a polynomial in , as it must be.
Functions are defined through the radial function (Ref 1, Eq. 6.5):
is parameterized by energy and smoothing radius ; their significance will will become clear shortly. The extended family is defined through powers of the Laplacian acting on :
In real space must be generated recursively from . However, the Fourier transform of has a closed form (Ref 1, Eq. 6.35). The differential operator becomes a multiplicative operator in the reciprocal space so
By taking limiting cases we can see the connection with familiar functions, and also the significance of parameters and .
This is the Fourier transform of , and is proportional to the spherical Hankel function of the first kind, . For general the relation is
which is Eq. 2.
and : .
This is the Fourier transform of a Gaussian function, whose width is defined by . For general we can define the family of generalized Gaussian functions
is a simple gaussian with an extra normalization:
Comparing cases 1 and 2 with Eq. (10), evidently 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, is a convolution of a Hankel function and a gaussian. For , behaves as a Hankel function and asymptotically tends to . For it has structure of a gaussian; it is therefore analytic and regular at the origin, varying as . Thus, the singularity of the Hankel function is smoothed out, with determining the radius for transition from Gaussian-like to Hankel-like behavior. Thus, the smoothing radius determines the smoothness of , and also the width of generalized gaussians .
By analogy with Eq. (9) we can extend the family with the Laplacian operator:
The second equation shows that has the structure (polynomial of order in ). These polynomials are related to the generalized Laguerre polynomials of half-integer order in . 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 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 Eq. (9), we obtain the useful relations
This shows that is the solution to the Helmholz operator 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). is also the solution to the Schrödinger equation for a potential that has an approximately gaussian dependence on (Ref 1, Eq. 6.30).
Two-center integrals of smoothed Hankels
One extremely useful property of the is that the product of two of them, centered at different sites and , can be integrated in closed form. The result a sum of other , evaluated at the connecting vector . This can be seen from the power theorem of Fourier transforms
and the fact that can be expressed as a linear combination of other , or their energy derivatives. This is readily done from the identity
Comparing the first identity and the form Eq.~(10) of , it can be immediately seen that the product of two with different energies can be expressed as a linear combination of two . The second identity applies when the have the same energy; the product will involve the energy derivative of some . For higher , is expanded as a linear combination of using the expansion theorem for spherical harmonics, Eq. (5). In detail,
which can be written as
This last equation is a linear combination of with smoothing radius given as shown.
Using the power theorem the two-center integrals can be directly evaluated:
This can be written as
The special case must be handled using the limiting form of Eq. (18). Differentiation of Eq. (10) with respect to energy results in
The two-center integral now becomes
If we consider a further limiting case, namely , Eq. (22) simplifies to
and the two-center integral simplifies to
When and the are generalized Gaussian functions of the type Eq. (12), scaled by ; 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 in contradistinction to usual convention for
and restricts . According to usual conventions is defined as
We can define for any energy
Then is real and positive if , while is real and positive if .
The smoothed Hankel orbitals for and are real:
To extend the definition to any energy we define as:
The following relations are useful:
Then for , is real and and are also real.
The difference in ordinary and smoothed Hankels is
For , is real and = . The difference in ordinary and smoothed Hankels is
Both are real, though and are complex.
- Many mathematical properties of smoothed Hankel functions and the 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)