Accurate correlation energies in one-dimensional systems from small system-adapted basis functions

We propose a general method for constructing system-dependent basis functions for correlated quantum calculations. Our construction combines features from several traditional approaches: plane waves, localized basis functions, and wavelets. In a one-dimensional mimic of Coulomb systems, it requires only 2–3 basis functions per electron to achieve high accuracy, and reproduces the natural orbitals. We illustrate its effectiveness for molecular energy curves and chains of many one-dimensional atoms. We discuss the promise and challenges for realistic quantum chemical calculations.

Figure 1

First two natural orbitals, labeled by their occupation numbers, of (1D) He. An X marks the location of the nucleus.

Figure 2

One of the scaling functions (solid blue line) and some of the wavelets (dashed lines) of a wavelet basis of type Coiflet-18. These functions are based on a fine grid with spacing $1/32$, and the level parameter $z$ gives the size scale of each function as $2^z/32$. Both the scaling function and rightmost wavelet are at $z=5$.

Figure 3

Energy errors for 1D He, Li, Be, ${\mathrm{H}}_2$, and ${\mathrm{H}}_4$ when evaluated in a basis of $N_f$ exact NOs of greatest occupancy.

Figure 4

Same as Fig. 1 but for 1D ${\mathrm{H}}_4$ at $R=4$. X's mark the locations of the nuclei.

Figure 5

Product plane-wave (PPW) functions for a 1D ${\mathrm{H}}_2$ at $R=3$. Here the box size is $L=7.72$, and marked by pink vertical lines. The upper figure shows the windowing functions $cos\left(k_{1}x\right)$ and $sin\left(k_{1}x\right)$ and the lower figure shows the first three PPWs (the first is just the LDA orbital).

Figure 6

The first five PPWs (red dashed) after orthogonalization compared to the exact natural orbitals (blue) for 1D ${\mathrm{H}}_4$ with $R=3$. Here $L=13.8$. These functions are similar to those found in Ref. [85] from density partitioning.

Figure 7

Finite-basis error of PPWs, to be contrasted with Fig. 3, which has exact NO's.

Figure 8

The function $exp(-0.5*|x+3\left|\right)+exp(-|x-3\left|\right)$ (black dashed) is divided into two orthogonal pieces (red and green solid lines) using wavelets. The wavelet basis used was based on Coiflet-24 with $\mathrm{\Delta }=1$, and the dividing line separating the two cells (dotted line) was $x=0.9$. The small oscillating tails make the two function pieces orthogonal. The two singularities each only appear in one piece because the high momentum wavelets representing the singularities are more and more localized the higher the momentum.

Figure 9

Some of the WLOs for each cell of a 1D ${\mathrm{H}}_4$ chain. Shown are the first two in both the first atom's cell (far left) and third atom's cell. Green vertical lines are drawn midway between each atom and weights of each function are labeled near each curve. The calculation this was taken from was $b=0,\mathrm{\Delta }=1.0,N_J=0$.

Figure 10

Error as a function of bond length $R$ for 1D ${\mathrm{H}}_2$ using both pure PPWs and WLOs, for various values of $J$. The sudden shift is at the Coulson-Fischer point of the LDA calculation, beyond which a broken spin-symmetry solution, with twice as many orbitals, has the lowest energy.

Figure 11

Finite-basis energy error as a function of covariance cutoff $\eta$ for 1D ${\mathrm{H}}_2$ at $R=2$ with $J=\mathrm{\Delta }=1$. Without cutoff, there are 24 functions in the basis. The integer near each point is the number of functions in the basis.

Figure 12

The first two natural orbitals for stretched 1D LiH (X's denote nuclear centers with Li on right). The exact NO's are marked in red, and are indistinguishable from the WLO NO's $\left(\mathrm{\Delta }=J=1\text{and}\eta ={10}^{-4}\right)$, black dotted line, but slightly different from the occupied LDA orbitals (green). Also shown are WLOs with weights above ${10}^{-4}$ (dashed lines). The WLO basis has 11 functions, and an error of 1.04 kcal/mol.