# Jigsaw Puzzle Orbitals

### Preliminaries

Jigsaw Puzzle orbitals are constructed out of smooth Hankel functions and generalized gaussian orbitals. The reader is advised to read at least the introductory part of this documentation of smooth Hankel functions.

### Special properties of Jigsaw Puzzle Orbitals

Jigsaw Puzzle Orbitals (JPOs) have a number of highly desirable properties of a basis set.

• They are short ranged and atom centered, with pure $Y_{lm}$ character on the augmentation boundary where they are centered. Thus, they serve as good projectors for special subspaces, e.g. correlated atoms.
• They are smooth everywhere. This greatly facilitate their practical implementation.
• They have an exponentially decaying asymptotic form far from a nucleus, as low-lying eigenfunctions do.
• They are tailored to the potential. Inside or near an augmentation site, the Schrödinger equation is carried by almost entirely by a single function. Thus they form a nearly optimum basis set to solve the Schrödinger equation over a given energy window.

In the figure below, the JPO envlope functions are shown for s and p orbitals in a 1D model with two atom centers.

The solid parts depict the interstitial region, where the envelope functions carry the wave function. Dashed lines depict augmenented regions where the envelope is substituted by partial waves — numerical solutions of the Schrödinger equation. It is nevertheless very useful that the envelope functions are smooth there, since sharply peaked envelope functions, require many plane waves to represent the smooth charge density $n_0$

At points where the envelope functions and augmented functions join, the function value is unity on the head site ($V_{1p}$ or $V_{1s}$ on the left, and $V_{2p}$ or $V_{2s}$ on the right) and zero on the other site. (By unity we refer to the radial part of the partial wave; the full wave must be multiplied by $Y_{lm}$, which for p orbitals is $\pm 1$ depending on whether the point is right or left of center.) Moreover, the kinetic energy is tailored to be continuous at the head site and vanish at the other site.

These two facts taken in combination are very important. Consider the Schrödinger equation near an augmentation point. Inside the augmentaion region, the partial wave is constructed numerically, and is very accurate. It is not quite exact: the partial wave is linearized, and the potential which constructs the partial wave is taken to be the spherical average of the actual potential. But it is well established that errors are small: the the LAPW method, for example, considered to be the gold standard basis set for accuracy, makes the same approximation. Since the kinetic energy is continuous across the boundary, the basis function equally well describes the Schrödinger on the other side of the boundary, at least very near the boundary.

This alone is not sufficient to make the basis set accurate. Tails in some channel from from heads centered elsewhere will contribute to the eigenfunction. They can “contaminate” the accurate solution of the head. However, consider the form of the Schrödinger equation:

By construction both value and kinetic energy of all basis functions $V_{Rl}$ vanish except for the single partial wave that forms the head. Thus any linear combination of them will yield a nearly exact solution of the Schrödinger equation locally.

In the 1D model described above, a the JPO basis was applied to double-exponential potential well shown in black in the Figure below. The kinetic energy of a traditional smooth Hankel (green) shows a discontinuity at the augmentation boundary; with JPO’s the discontinuity disappears (red). The JPO kinetic energy is everywhere very close to the exact solution (the exact potential and kinetic energies lie on top of each other).

In many respects, JPOs are nearly ideal basis functions: they are close to being as compact as possible for solving the Schrödinger equation in a given energy window. JPO’s have two important drawbacks, however:

• They are complex objects, complicating their augmentation and assembling of matrix elements. We can however make use analytic properties of JPO envelope functions to greatly amelioriate the increase in computational cost in making matrix elements.
• There is no analytic form for products of two of them, as there are for plane waves and guassian orbitals. However we can make them in the same way as we do for traditional Questaal orbitals based on smooth Hankel functions.

### Other Resources

1. Many mathematical properties of smoothed Hankel functions and the $H_{kL}$ 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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