Numerische Mathematik

, Volume 128, Issue 1, pp 137–165 | Cite as

Error estimates for Hermite and even-tempered Gaussian approximations in quantum chemistry

Article

Abstract

Atomic-like basis functions provide a natural, physically motivated description of electronic states, among which Gaussian-type orbitals are the most widely used basis functions in molecular simulations. This paper aims at developing a systematic analysis of numerical approximations based on linear combinations of Gaussian-type orbitals. We derive a priori error estimates for Hermite-type Gaussian bases as well as for even-tempered Gaussian bases. Numerical results are presented to support the theory.

Mathematics Subject Classification

41A25 65N35 65Z05 

1 Introduction

The subject of electronic structure theory is the modelling of many-electron systems in chemistry and physics, which enables the investigation and prediction of properties of molecules and materials in condensed phases. Among the different models of electronic structure theory, for large-scale systems Kohn–Sham density functional theory (DFT) [21, 27] achieves so far the best compromise between accuracy and computational cost, and has become the most widely used electronic structure model in chemistry and materials science. Let us consider a closed shell system with \(M_\mathrm{n}\) nuclei of charges \(\{Z_1,\ldots ,Z_{M_\mathrm{n}}\}\), located at the positions \(\{\mathbf{R}_1,\ldots ,\mathbf{R}_{M_\mathrm{n}}\}\), and an even number \(M_\mathrm{e}\) of electrons in the non-relativistic setting. The ground state solutions of the system can be obtained by solving the following Kohn–Sham equation: Find \(\lambda _i\in \mathbb {R}, 0\ne \phi _i\in H^1(\mathbb {R}^3)\), for \(i=1,2,\ldots ,M_\mathrm{e}/2\), such that \( \int _{\mathbb {R}^3}\phi _i\phi _j = \delta _{ij}\) and
$$\begin{aligned} \left( -\frac{1}{2}\Delta + v_\mathrm{eff}[\rho ] \right) \phi _i = \lambda _i\phi _i\quad \text{ in }~\mathbb {R}^3, \quad i=1,2,\ldots , M_\mathrm{e}/2, \end{aligned}$$
(1)
where \( \{\lambda _i\}_{i=1}^{M_\mathrm{e}/2}\) are the lowest \(M_\mathrm{e}/2\) eigenvalues, \( \rho (\mathbf{r})=2\sum \nolimits _{i=1}^{M_\mathrm{e}/2}|\phi _i(\mathbf{r})|^2\) is the electron density, and \(v_\mathrm{eff}[\rho ]=v_\mathrm{ext} + v_\mathrm{H}[\rho ] + v_\mathrm{xc}[\rho ]\) is the effective potential where
$$\begin{aligned} v_{\text {ext}}(\mathbf{r})=-\sum _{k=1}^{M_n}\frac{Z_k}{|\mathbf{r}-\mathbf{R}_k|},\quad v_\mathrm{H}[\rho ](\mathbf{r}) =\int \limits _{\mathbb {R}^3}\frac{\rho (\mathbf{r'})}{|\mathbf{r}-\mathbf{r}'|}\,\mathrm{d}\mathbf{r}' \end{aligned}$$
are the electrostatic potential generated by the nuclei and the Hartree potential for interactions between electrons, respectively, and \(v_{\text {xc}}[\rho ]\) is the exchange-correlation potential [32].
A second important example is the Hartree–Fock method [19]. It is derived from the high-dimensional electronic Schrödinger equation by assuming that the wavefunction is an antisymmetrized tensor product of single-electron orbitals. This leads to a nonlinear problem for these orbitals that has the same basic structure as (1), but with effective potential (for closed-shell systems) \(v_\mathrm{eff}[\rho ,\tau ]=v_\mathrm{ext} + v_\mathrm{H}[\rho ] + v_\mathrm{K}[\tau ]\), where the exchange potential \(v_\mathrm{K}[\tau ]\) is a nonlocal operator defined for a given function \(\psi \) by
$$\begin{aligned} v_\mathrm{K}[\tau ] \psi = -\int \limits _{\mathbb {R}^3} \frac{\tau (\mathbf {r},\mathbf {r}')}{|\mathbf {r}-\mathbf {r}'|} \psi (\mathbf {r}')\,\mathrm{d}\mathbf {r}' \, \end{aligned}$$
with the density matrix \(\tau (\mathbf {r},\mathbf {r}') = \sum _{i=1}^{M_\mathrm{e}/2}\phi _i(\mathbf{r})\,\phi _i(\mathbf{r}')\). The orbitals \(\phi _i\) obtained from the Hartree–Fock model usually serve as the starting point for better approximations of the true wavefunction by so called Post-Hartree–Fock methods.

The nonlinear eigenvalue problems for both DFT and Hartree–Fock models are commonly solved by a self-consistent field iteration (SCF) algorithm [30, 32]. In each iteration, a new Hamiltonian is constructed from a trial electronic state, and a linear eigenvalue problem is then solved to obtain the eigenfunctions for the lowest eigenvalues. In each step, the algorithm requires discretization of the problem on a finite-dimensional space spanned by a set of basis functions.

The choice of these basis functions, which ultimately determines the approximation quality of the solutions, is therefore crucial. Methods based on linear combinations of atomic orbitals are the most widely used methods in chemistry, which capture the essence of the atomic-like features of molecules and provide an intuitive description of electronic states. Among these, Gaussian-type orbitals (GTO) are used in the overwhelming majority of computations. They were first adopted by Boys [6], see also, e.g., [33, 38]. The Gaussian basis functions can be written as products of a radial function and a spherical harmonic angular function as
$$\begin{aligned} \psi (\mathbf{r})=R(r)Y_{lm}(\hat{\mathbf{r}}_i), \end{aligned}$$
(2)
where \(\mathbf{r}_i=\mathbf{r}-\mathbf{R}_i\), \(r=|\mathbf{r}|\), \(\hat{\mathbf{r}}=\mathbf{r}/r\), and \(Y_{lm}(\hat{\mathbf{r}})\) denotes the spherical harmonic functions on \(S^2\). The radial part \(R(r)\) depends only on the radial coordinate \(r\) and contains a Gaussian term \(\exp (-\eta r^2)\) with fixed or variable parameter \(\eta \). A great advantage of Gaussian bases is that required matrix elements can be computed analytically.

At a first glance, Gaussian orbitals appear to be a poor basis set since in principle, they have the wrong behavior both close to the nuclei and very far from them [28]. However, with appropriately chosen exponents \(\eta \), Gaussian approximations have fast convergence rates in practical calculations. Over the years, many good and flexible Gaussian basis sets have been proposed and some highly developed codes based on them are widely used in quantum chemistry, e.g., GAUSSIAN, MolPRO and PSI. The parameters \(\eta \), which are crucial for the quality of resulting approximations, are usually determined by physical insight and experience. So far, it has not been possible to construct a single, universal Gaussian basis set that is applicable under all circumstances. Therefore, understanding why Gaussian basis functions have good or bad performance from a theoretical point of view is of great importance for assessing the accuracy of calculations, and may eventually allow to design more reliable and efficient basis sets.

The purpose of this paper is to give a systematic error analysis for certain Gaussian approximations and derive error estimates for the approximation of more general functions in the \(H^1\)-norm. To our knowledge, there are only a few previous works devoted to the numerical analysis of atomic-like orbital bases, and in particular, to a priori error estimates with respect to the number of basis functions \(N\).

We consider two basic types of Gaussian basis functions. The first are Hermite-type GTOs, comprised of a polynomial and a Gaussian factor, where the parameter \(\eta \) is kept fixed. Klahn and Morgan [25] studied in the convergence of expansions of the ground state of the hydrogen atom in such a Hermite Gaussian basis, and showed that the energy error goes as \(N^{-3/2}\). By optimization of the exponent, Klopper and Kutzelnigg [26] showed numerically that a rate of \(N^{-2}\) is achievable. Here, an important issue is the regularity of the wavefunctions. It was shown in [16, 22] that the exact electron densities are analytic away from the nuclei and satisfy certain cusp conditions at the nuclear positions, which leads to such rather unsatisfactory algebraic convergence rates. Thanks to recent results by Flad et al. [13], higher regularity in weighted Sobolev spaces for eigenfunctions of Schrödinger type eigenvalue problems can be employed in our analysis, which yields better convergence rates for atomic-like basis functions.

The second class of Gaussian basis functions that we consider are so-called even-tempered GTOs, which can be regarded as a simple instance of the type of bases typically used in practice. In this case, there are no additional polynomial factors, and efficient approximations are achieved mainly by careful adjustment of the parameters \(\eta \). As a pioneering contribution to a theoretical understanding of their practical efficiency, Kutzelnigg [28] gave an error estimate for the expansion of \(1/r\) and \( e^{-\alpha r}\) in terms of even-tempered Gaussians, with respect to the Chebyshev norm and to the energy expectation value (see also [29, 30]). He obtained an error decay of the form \( e^{-c\sqrt{N}}\). Braess [7] and Braess and Hackbusch [8] studied exponential sum approximations of \(1/\sqrt{r}\) and \( e^{-\alpha \sqrt{r}}\) (corresponding to Gaussian approximations of \(1/r\) and \( e^{-\alpha r}\)) in a weighted \(L^1\)-norm as well as in the Chebyshev norm. Although the underlying technique based on best approximation by exponential sums is quite different, an \( e^{-c\sqrt{N}}\) behavior of the error was found in these cases as well. The existing results are thus all based on one-dimensional expansion of the radial part of hydrogen-like 1s wavefunctions, and the error estimates are mainly in \(L^{\infty }\)-norm or for the ground state energy expectation value, which do not lead to clear conclusions for practical calculations.

In this work, we obtain error estimates in \(H^1\)-norm—which is the relevant error measure in a wide range of problems—for approximation of functions with radial parts of the form \(r^n e^{-\gamma r}\), \(n\in \mathbb {N}_0\), \(\gamma >0\), by even-tempered Gaussians. A central tool in our treatment are integral transforms and sinc approximation theory. To our knowledge, this is the first such approximation result for functions with an additional polynomial factor, and in this particular norm. The polynomial term also leads to a possible construction of approximations for more general wavefunctions: roughly speaking, our result states that for wavefunctions that can be efficiently approximated by short sums of Slater-type orbitals (STOs), which reflect the known properties of molecular wavefunctions much more closely than GTOs, one can also obtain rapidly convergent expansions by even-tempered GTOs. As in the mentioned previous works on approximations of this type, we obtain an error decay of the form \( e^{-c\sqrt{N}}\), that is, substantially faster convergence rates than for the Hermite-type basis functions; however, this comes at the price of very restrictive conditions on the approximands.

The remainder of this paper is arranged as follows: in the following section, we present the basic model problem and the type of error estimates that we shall consider. In Sect. 3, we consider approximation by Hermite-type GTOs; as an ingredient for these results, we also present regularity results for eigenfunctions of Schrödinger-type equations. In Sect. 4, we treat approximation by even-tempered GTOs. In Sect. 5, we present some numerical experiments.

2 Preliminaries

Throughout this paper, we shall denote by \(C\) a generic positive constant which stands for different values at its different occurrences. For \(\mathbf{r}\in \mathbb {R}^3\), we shall denote by \(r=|\mathbf{r}|\) and \(\hat{\mathbf{r}}=\mathbf{r}/r\). Moreover, we shall denote \(\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\) by \(\sum _{\ell m}^{\infty }\), and \(\sum _{\ell =0}^{L}\sum _{m=-\ell }^{\ell }\) by \(\sum _{\ell m}^L\) for simplicity.

As a model problem, we shall consider the following Schrödinger-type linear eigenvalue problem, which can be viewed as a linearization of (1): find \(\lambda \in \mathbb {R}\) and \(0\ne u\in H^1(\mathbb {R}^3)\) such that \(\Vert u\Vert _{L^2(\mathbb {R}^3)}=1\) and
$$\begin{aligned} a(u,v)=\lambda (u,v)\quad \forall ~v\in H^1(\mathbb {R}^3), \end{aligned}$$
(3)
where the bilinear form \(a:H^1(\mathbb {R}^3)\times H^1(\mathbb {R}^3)\rightarrow \mathbb {R}\) is defined by
$$\begin{aligned} a(u,v)=\frac{1}{2}\int \limits _{\mathbb {R}^3}\nabla u\cdot \nabla v + \int \limits _{\mathbb {R}^3}v_{\text {eff}}\,u \,v, \end{aligned}$$
(4)
and the effective potential \(v_\mathrm{eff}\) is smooth with exception of the singularities at the nuclear positions. As the analysis in this paper can be carried out for each nucleus separately, in the remainder of this work we consider a single nucleus located at the origin.
It is known that under certain restrictions on \(v_\mathrm{eff}\), the eigenfunctions of (3) decay exponentially as \(r\rightarrow \infty \) (see, e.g. [2]), that is,
$$\begin{aligned} |u(\mathbf {r})| \le Ce^{-\alpha r} \end{aligned}$$
(5)
with some parameter \(\alpha \). This exponential decay property can be proven for the electronic Schrödinger equation [22, 40], for the Hartree–Fock equations [31], and also for some Kohn–Sham models [1, 3]. In what follows, we make the assumption that the functions we want to approximate satisfy (5).
Let \(V_{N}\) be the finite-dimensional space spanned by a set of Gaussian basis functions. The Galerkin approximation of the ground state solution of the linear Schrödinger-type equation (3) is determined by the variational problem: find \(\lambda _{N}\in \mathbb {R}\) and \(0\ne u_{N}\in V_{N}\) such that \(\Vert u_{N}\Vert _{L^2(\mathbb {R}^3)}=1\) and
$$\begin{aligned} a(u_{N},v)=\lambda _{N}(u_{N},v) \quad \forall ~v\in V_{N}. \end{aligned}$$
(6)
Using standard estimates [4, 40], we have that the \(H^1\)-error of the finite-dimensional approximation is estimated by
$$\begin{aligned} \Vert u-u_{N}\Vert _{H^1(\mathbb {R}^3)}\le C\inf _{v_{N}\in V_{N}} \Vert u-v_{N}\Vert _{H^1(\mathbb {R}^3)}. \end{aligned}$$
(7)
Since a direct calculation leads to
$$\begin{aligned} |\lambda _{N}-\lambda | = |a(u_{N}-u,u_{N}-u) - \lambda _N(u_{N}-u,u_{N}-u)| \le C\Vert u-u_{N}\Vert ^2_{H^1(\mathbb {R}^3)}, \end{aligned}$$
we have that the error of eigenvalue approximations goes as the square of the corresponding eigenfunction error.
For the nonlinear Kohn–Sham and Hartree–Fock problems as in (1), we have discretized problems of the form
$$\begin{aligned} \left\{ \, \begin{array}{lcl} (H[\rho _N,\tau _N]\,\phi _{i,N},v) = \displaystyle (\lambda _{i,N}\phi _{i,N},v) \quad \forall ~v\in V_{N}, \quad i=1,\ldots ,M_\mathrm{e}/2,\\ \displaystyle \int \limits _{\mathbb {R}^3}\phi _{i,N}\phi _{j,N} = \delta _{ij}, \end{array} \right. \end{aligned}$$
where \(\rho _N(\mathbf{r})=2\sum _{i=1}^{M_\mathrm{e}/2}|\phi _{i,N}(\mathbf{r})|^2\), \(\tau _N(\mathbf {r},\mathbf {r}') = \sum _{i=1}^{M_\mathrm{e}/2}\phi _{i,N}(\mathbf{r})\,\phi _{i,N}(\mathbf{r}')\), and
$$\begin{aligned} H[\rho _N,\tau _N]=-\frac{1}{2}\Delta +v_{\text {eff}}[\rho _N,\tau _N] \end{aligned}$$
with \(v_\mathrm{eff}[\rho ,\tau ]\) defined as in the introduction.
Using similar arguments as in [10] (see also [9]), for Kohn–Sham equations we can prove the estimate
$$\begin{aligned} \sum _{i=1}^{M_\mathrm{e}/2}\Vert \phi _i-\phi _{i,N}\Vert _{H^1(\mathbb {R}^3)} \le C\sup _{1\le i\le M_\mathrm{e}/2} \inf _{v_{N}\in V_{N}}\Vert \phi _i-v_{N}\Vert _{H^1(\mathbb {R}^3)} \end{aligned}$$
(8)
under certain assumptions (including the coercivity assumption on the tangent space, the regularity assumption on the exchange-correlation term, and the completeness assumption on the limit of the finite dimensional space). Due to the continuity of the energy functional, for the error in the ground state energy we obtain (cf. [9, 10])
$$\begin{aligned} |E(\{\phi _{i,N}\})-E(\{\phi _i\})| \le C\sum _{i=1}^{M_\mathrm{e}/2} \Vert \phi _i-\phi _{i,N}\Vert ^2_{H^1(\mathbb {R}^3)}. \end{aligned}$$
Therefore we only need to obtain estimates of the right hand sides of (7) and (8) for specified \(V_{N}\).

3 Hermite Gaussian bases

The simplest GTO basis functions may be written in the following form
$$\begin{aligned} \psi ^\mathrm{GTO}_{n{\ell }m}(\mathbf{r}) = c_n r^n e^{-\zeta r^2}\, Y_{\ell m}(\hat{\mathbf{r}}), \quad n-{\ell }=0,2,4,\ldots , \end{aligned}$$
(9)
where \(\zeta \) is a fixed exponent and \(c_n\) are normalization constants. This can be regarded as a type of spectral approximation. However, note that in our context, we cannot expect to achieve spectral convergence rates due to the cusps at the nuclear positions (see, e.g., the analysis in [14, 15, 16, 22]).

3.1 Regularity

In our analysis, we rely on regularity results in weighted Sobolev spaces for Schrödinger-type eigenvalue problems as developed in [13]. This type of analysis has been introduced to investigate singularities of boundary value problems in conical domains with corners and edges, see [5, 12, 17] for further details. In our case the geometry is fairly simple, but the singular electrostatic potential generated by the nuclei still fits perfectly into this framework.

We define the space
$$\begin{aligned} H_\mathrm{cone}^k(\mathbb {R}^3) =\left\{ u\in H^k_\mathrm{loc}(\mathbb {R}\times S^2)|_{\mathbb {R}_+\times S^2}: (1-\omega )u\in H^k(\mathbb {R}^3)\right\} , \end{aligned}$$
(10)
where \(\omega (r)\) is a smooth cut-off function, that is, \(\omega =1\) near zero and \(\omega =0\) outside some neighborhood of zero. Further, we define the \(k\)th weighted Sobolev space with index \(\gamma \) by
$$\begin{aligned} \mathcal {K}^{k,\gamma }(\mathbb {R}^3)=\left\{ u(\mathbf{r})\in L^2(\mathbb {R}^3):r^{|\alpha |-\gamma }\partial ^{\alpha }u \in L^2(\mathbb {R}^3)\quad \forall ~|\alpha |\le k\right\} , \end{aligned}$$
(11)
where \(k\in \mathbb {N}\) and \(\gamma \in \mathbb {R}\). The difference between the spaces \(H^k(\mathbb {R}^3)\) and \(\mathcal {K}^{k,\gamma }(\mathbb {R}^3)\) only lies in the introduction of the weight functions \(r^{|\alpha |-\gamma }\).
Note that neither of (10) nor (11) is really appropriate for our purposes. Instead, we consider the combination
$$\begin{aligned} \mathcal {K}_\mathrm{cone}^{k,\gamma }(\mathbb {R}^3):= \omega \mathcal {K}^{k,\gamma }(\mathbb {R}^3)+(1-\omega )H_\mathrm{cone}^k(\mathbb {R}^3), \end{aligned}$$
(12)
which provides the appropriate behavior in the limits \(|\mathbf{r}|\rightarrow 0\) and \(|\mathbf{r}|\rightarrow \infty \) simultaneously.
We now consider subspaces \(\fancyscript{K}^{k,\gamma }(\mathbb {R}^3)\) of \(\mathcal {K}_\mathrm{cone}^{k,\gamma }(\mathbb {R}^3)\), defined by
$$\begin{aligned} \fancyscript{K}^{k,\gamma }(\mathbb {R}^3)\!=\!\left\{ u\in \mathcal {K}_\mathrm{cone}^{k,\gamma }(\mathbb {R}^3):u(\mathbf{r})\!-\!\omega (r)\sum _{j=0}^n c_{j}(\hat{\mathbf{r}})r^{j}\in \mathcal {K}_\mathrm{cone}^{k,\gamma +n}(\mathbb {R}^3),\quad \forall ~n\in \mathbb {N}\right\} \!. \end{aligned}$$
(13)
Here each \(c_{j}\) belongs to the finite dimensional subspace \(M_j=\mathrm{span}\{Y_{\ell m},0\le \ell \le j,|m|\le \ell \} \subset C^{\infty }(S^2)\) and \(Y_{\ell m}\) denote the spherical harmonics on \(S^2\).
On a more intuitive level, this means that these spaces consist of functions with asymptotic expansions of the type
$$\begin{aligned} u(\mathbf{r})\sim \sum _{j\in \mathbb {N}_0} c_{j}(\hat{\mathbf{r}})\,r^{j}\quad \mathrm{as}~r\rightarrow 0, \end{aligned}$$
(14)
where the powers of \(r\) can only be nonnegative integers, and the corresponding remainders belong to higher order weighted Sobolev spaces.

We shall use such weighted Sobolev spaces with asymptotic type (14) within a particular range of parameters \( \gamma \), which motivates the following definition.

Definition 1

A function \(u\) is called asymptotically well behaved if \(u\in \fancyscript{K}^{\infty ,\gamma }(\Omega )\) for \(\gamma <3/2\).

For the regularity results of this section, we make the assumption that the effective potential in (4) is of the form
$$\begin{aligned} v_{\text {eff}}(\mathbf{r})=-\frac{Z}{|\mathbf{r}|} + \rho *\frac{1}{|\mathbf{r}|} + \alpha _\mathrm{K} v_\mathrm{K}[\tau ] + \alpha _s v_s(\mathbf{r}),\quad \alpha _\mathrm{K}, \alpha _s \in \mathbb {R} \end{aligned}$$
(15)
where \(v_s\), as well as the orbitals used in forming \(\rho \) and \(\tau \), are assumed to be asymptotically well behaved functions.

The following regularity result for eigenfunctions of (3) will be used in our analysis.

Proposition 1

If \(u\) is an eigenfunction of (3) with potential of the form (15) satisfying (5), then \(u\) is asymptotically well behaved.

The proof for the case \(\alpha _\mathrm{K}=1\), \(\alpha _s=0\), corresponding to the Hartree–Fock model, is given in [13, Theorem 1]; in the other cases, which correspond to Kohn–Sham models (including hybrid functionals), one can follow the lines of this proof to obtain the result in an analogous manner.

Remark 1

For Kohn–Sham equations, the case of local density approximations (LDA) corresponds to \(\alpha _\mathrm{K}=0\), \(\alpha _s=1\) and \(v_s = v_\text {xc}[\rho ]\). Note, however, that the assumption of such \(v_s\) being asymptotically well behaved to our knowledge is not known for usual LDA exchange-correlation potentials used in practice. As a consequence, the above regularity result may not be directly applied to Kohn–Sham equations in the LDA setting. A further investigation of different density functionals concerning this condition is beyond the scope of this work. Note, however, that the results directly apply to the Hartree–Fock model.

Denote by \(\mathcal {Y}_{\ell m}\) the solid harmonics
$$\begin{aligned} \mathcal {Y}_{\ell m}(\mathbf{r}) = r^{\ell } Y_{\ell m}(\hat{\mathbf{r}}). \end{aligned}$$
Since the solid harmonics \(\mathcal {Y}_{\ell m}(\mathbf{r})\) have explicit Cartesian expressions (see, e.g. [19]) as
$$\begin{aligned} \mathcal {Y}_{\ell m}(\mathbf{r})&= \mathcal {Y}_{\ell m}(\mathbf{r}(x,y,z)) = \mathcal {P}_{\ell m}(x,y,z) \nonumber \\&= N_{\ell m}(x+\mathrm{sgn}(m)iy)^{|m|} \sum _{t=0}^{({\ell }-|m|)/2}C_{t}^{{\ell }|m|}(x^2+y^2)^t z^{{\ell }-2t-|m|} \end{aligned}$$
(16)
with the constants \(N_{\ell m}\) and \(C_t^{\ell |m|}\), they yield a significant simplification for numerical integrations in computations (see [19]). Therefore, the solid harmonics are always used as the angular parts of the atomic bases instead of spherical harmonics \(Y_{\ell m}(\hat{\mathbf{r}})\).

The following two propositions will be used in our analysis.

Proposition 2

If \(u\) is an eigenfunction of (3) with potential of the form (15) satisfying (5), then
$$\begin{aligned} u\in C^{\infty }(\overline{\mathbb {R}}_+\times S^2) \end{aligned}$$
(17)
and it can be expanded by solid harmonics as
$$\begin{aligned} u(\mathbf{r})=\sum _{\ell m}^{\infty } R_{\ell m}(r)\mathcal {Y}_{\ell m}(\mathbf{r}) \end{aligned}$$
(18)
with \(R_{\ell m}\in C^{\infty }(\overline{\mathbb {R}}_+)\).

Proof

The proof of (17) is given in [13, Proposition 1], and therefore it only remains to prove (18).

Since \(u\) is asymptotically well behaved, we have the expansion
$$\begin{aligned} u(\mathbf{r}) = \omega (r)\sum _{j=0}^k r^j\alpha _j(\hat{\mathbf{r}})+\Phi _{k+1}(\mathbf{r}), \end{aligned}$$
(19)
where \(\Phi _{k+1}\in \fancyscript{K}_\mathrm{cone}^{s,\gamma }(\mathbb {R}^3)\) for \(\gamma <\frac{3}{2}+k\) and any \(s\in \mathbb {R}_+\). Using the definition in (11), we can derive that for any \(t\in \mathbb {R}_+\), there exists sufficiently large \(k\), such that \(\Phi _{k+1}\in H^{t}(\mathbb {R}^3)\).

The asymptotic type (14) shows that for each \(j\) only a finite number of spherical harmonics with \({\ell }\le j\) can contribute, i.e. \(\alpha _j(\hat{\mathbf{r}})\in \mathrm{span}\{Y_{\ell m}(\hat{\mathbf{r}}),~0\le {\ell }\le j\}\). This proves that the first part of (19) can be represented by \(\sum _{\ell m}^{\infty }R^a_{\ell m}(r)\mathcal {Y}_{\ell m}(\mathbf{r})\) with \(R^a_{\ell m}(r)\in C^{\infty }(\overline{\mathbb {R}}_+)\) and \(R^a_{\ell m}(r)=0\) for \(r>1\).

For the second part of (19), we define
$$\begin{aligned} R^b_{\ell m}(r)=\frac{1}{r^{\ell }}\int \limits _{S^2}\Phi _{k+1}(\mathbf{r})Y_{\ell m}(\hat{\mathbf{r}})\,\mathrm{d}\hat{\mathbf{r}}, \end{aligned}$$
and have
$$\begin{aligned} \Phi _{k+1}(\mathbf{r})=\sum _{\ell m}^{\infty }R^b_{\ell m}(r)\mathcal {Y}_{\ell m}(\hat{\mathbf{r}}). \end{aligned}$$
Note that (17) implies that \(R^b_{\ell m}\in C^{\infty }(\mathbb {R}_+)\), and it is only necessary to prove that \(R^b_{\ell m}\) is smooth at the origin.

Using a direct calculation and the expression of (16), we have that if \(f(r)\) has a singularity at \(r=0\) such that \(f\notin C(\overline{\mathbb {R}}_+)\), then \(f(r)\mathcal {Y}_{\ell m}(\mathbf{r}) \notin H^{3/2+{\ell }}(\mathbb {R}^3)\). Since for any \({\ell }\in \mathbb {Z}\) and \(s\in \mathbb {R}_+\), there exists sufficiently large \(k\) such that \(\Phi _{k+1}\in H^{3/2+{\ell }+s}(\mathbb {R}^3)\), we can obtain that \(R^b_{\ell m}(r)\in C^{s}(\overline{\mathbb {R}}_+)\).

Taking \(R_{\ell m}(r)=R^a_{\ell m}(r)+R^b_{\ell m}(r)\), we obtain (18) with \(R_{\ell m}(r) \in C^{\infty }(\overline{\mathbb {R}}_+)\). This completes the proof. \(\square \)

Proposition 3

If \(u\) is an eigenfunction of (3) with potential of the form (15) satisfying (5), then for any \(n\in \mathbb {N}\), there exist smooth functions \(\alpha _j(\hat{\mathbf{r}})\in \mathrm{span}\{Y_{\ell m}(\hat{\mathbf{r}}),~{\ell }\le j\} \subset C^{\infty }(S^2),~j=0,1,\ldots ,n\), such that
$$\begin{aligned} u(\mathbf{r})-\omega (r)\sum _{j=0}^n r^j\alpha _j(\hat{\mathbf{r}}) \in H^{5/2+n-\varepsilon }(\mathbb {R}^3). \end{aligned}$$
(20)

Proof

Again we use the asymptotic expansion of \(u\) as (19). For any \(s\in \mathbb {R}\), there exists a sufficiently large \(k\), such that \(\Phi _{k+1}\in H^s(\mathbb {R}^3)\). Therefore, for any \(n\in \mathbb {N}\), there exists \(k\) such that \(\Phi _{k+1}\in H^{5/2+n-\varepsilon }(\mathbb {R}^3)\).

Since a direct calculation and interpolation theory imply that
$$\begin{aligned} r^k Y_{\ell m}(\hat{\mathbf{r}}) \in H^{k+3/2-\varepsilon }(B(0,1)) \end{aligned}$$
for \(k\ge {\ell }\) and \(B(0,1)\equiv \{x:|x|\le 1\}\), we have
$$\begin{aligned} \omega (r)\sum _{j=n+1}^k r^j\alpha _j(\hat{\mathbf{r}})\in H^{5/2+n-\varepsilon }(\mathbb {R}^3). \end{aligned}$$
This completes the proof. \(\square \)

3.2 Approximation by Hermite Gaussian bases

We now turn to approximation results based on the above regularity estimates.

Theorem 1

Let \( V_{N}=\mathrm{span}\{\psi ^\mathrm{GTO}_{n{\ell }m}:0\le n\le N, ~{\ell }=n,n-2,\ldots ,n-2\left\lfloor \frac{n}{2}\right\rfloor ,~-{\ell }\le m\le {\ell }\}\). If \(u\) is an eigenfunction of (3) with potential of the form (15) satisfying (5), then
$$\begin{aligned} \inf _{v_{N}\in V_{N}} \Vert u-v_{N}\Vert _{H^1(\mathbb {R}^3)} \le CN^{-\frac{3}{4}+\varepsilon }. \end{aligned}$$
(21)
If the basis functions (9) have odd orders \(n-{\ell }\) included in the basis set for \(n\le K<N\), that is, \( V^K_{N}=V_{N} \oplus \mathrm{span}\{\psi ^\mathrm{GTO}_{n{\ell }m}:0\le n\le K, ~{\ell }=n-1,n-3,\ldots ,n-1-2\left\lfloor \frac{n-1}{2}\right\rfloor ,~-{\ell }\le m\le {\ell }\}\), then
$$\begin{aligned} \inf _{v_{N}\in V^K_{N}} \Vert u-v_{N}\Vert _{H^1(\mathbb {R}^3)} \le CN^{-\frac{3}{4}-\frac{K}{2}+\varepsilon }. \end{aligned}$$
(22)

Remark 2

The comparison of (21) and (22) shows that if we add polynomials with some odd orders \(n-{\ell }\) to the set of basis functions (which yields a formally overcomplete set), the error decreases much faster. Unfortunately, the incorporation of odd-order polynomials makes the evaluation of integrals as difficult as for Slater type basis functions, and is therefore of less practical interest.

Proof

First, we note that the three-dimensional harmonic oscillator (HO) functions are defined by (see, e.g. [19])
$$\begin{aligned} \psi ^\mathrm{HO}_{\tilde{n}{\ell }m}(\mathbf{r})&= R_{\tilde{n}{\ell }}^\mathrm{HO}(r)Y_{\ell m}(\hat{\mathbf{r}}), \quad \tilde{n}-{\ell }=1,2,\ldots , \nonumber \\ R_{\tilde{n}{\ell }}^\mathrm{HO}(r)&= \frac{(2\zeta )^{3/4}}{\pi ^{1/4}}\frac{\sqrt{2^{\tilde{n}+1} (\tilde{n}-{\ell }-1)!}}{\sqrt{(2\tilde{n}-1)!}} \left( \sqrt{2\zeta }r\right) ^{\ell } L_{\tilde{n}-{\ell }-1}^{{\ell }+1/2}(2\zeta r^2)\mathrm{exp}(-\zeta r^2),\qquad \end{aligned}$$
(23)
where the polynomials \(L_k^{\alpha }\) are generalized Laguerre polynomials
$$\begin{aligned} L_{k}^{\alpha }(x) = \frac{x^{-\alpha }e^x}{k!} \frac{d^k}{dx^k}(e^{-x}x^{k+\alpha }). \end{aligned}$$
The HO functions are the eigenfunctions of a three-dimensional isotropic harmonic oscillator
$$\begin{aligned} H=-\frac{1}{2}\Delta + \frac{1}{2}(2\zeta )^2r^2 \end{aligned}$$
with the principal quantum number \(2(\tilde{n}-1)-{\ell }=n\), which corresponds directly to the power of \(r\) in (23).

We observe that the HO functions with \(2(\tilde{n}-1)-{\ell }\le N\) span the same space as \(V_N\). To see this, we note that any HO function \(\psi ^\mathrm{HO}_{\tilde{n}{\ell }m}\) may be written by linear combinations of (9) with powers of \(r\) less or equal to \(2(\tilde{n}-1)-{\ell }\).

If we write the Hamiltonian in Cartesian coordinates as
$$\begin{aligned} H=-\frac{1}{2}\Delta + \frac{1}{2}(2\zeta )^2(x^2+y^2+z^2), \end{aligned}$$
which is separable in the three Cartesian directions, then the solution can be expressed as a product of one-dimensional HO functions in Cartesian coordinates
$$\begin{aligned} \psi ^\mathrm{HO}_{\tilde{n}{\ell }m}(\mathbf{r}(x,y,z))= \varphi ^\mathrm{HO}_{ijk}(x,y,z)=\chi _i^\mathrm{H}(x)\chi _j^\mathrm{H}(y)\chi _k^{H}(z), \end{aligned}$$
where \(\chi _i^\mathrm{H}\) are Hermite functions of degree \(i\), that is,
$$\begin{aligned} \chi _i^\mathrm{H}(x)&= \left( \frac{2\zeta }{\pi }\right) ^{\frac{1}{4}} \frac{1}{\sqrt{2^i i!}} H_i\left( \sqrt{2\zeta }x\right) \exp (-\zeta x^2), \\ H_i(x)&= (-1)^i\exp (x^2)\frac{d^i}{dx^i}\exp (-x^2), \end{aligned}$$
and the quantum number is given by \(i+j+k= n\). Define \(\tilde{\pi }_N:L^2(\mathbb {R})\rightarrow \mathrm{span}\{\chi _k^\mathrm{H},k\le N\}\) satisfying
$$\begin{aligned} (v-\tilde{\pi }_{N}v,v_N)=0 \quad \forall ~v_N\in \mathrm{span}\left\{ \chi _k^\mathrm{H},k\le N\right\} . \end{aligned}$$
Denote \(\tilde{\partial }_x=\partial _x+x\). For integer \(t\), we define
$$\begin{aligned} \tilde{H}^t(\mathbb {R}^3)=\left\{ v\in L^2(\mathbb {R}^3):\tilde{\partial }^{\alpha }v\in L^2(\mathbb {R}^3),\quad 0\le |\alpha |\le t\right\} \end{aligned}$$
with the associated norm \(\Vert v\Vert _{\tilde{H}^t(\mathbb {R}^3)} = \left( \sum _{|\alpha |\le t}\Vert \tilde{\partial }^{\alpha }v\Vert ^2_{L^2(\mathbb {R}^3)} \right) ^{1/2}\). For fractional \(t\), \(\tilde{H}^t(\mathbb {R}^3)\) and \(\Vert \cdot \Vert _{\tilde{H}^t(\mathbb {R}^3)}\) are defined by interpolation theory. From standard spectral analysis results [36], for any \(v\) satisfying \(\tilde{\partial }^s v\in L^2(\mathbb {R})\) and \(s\ge t\) we have
$$\begin{aligned} \Vert \partial ^t (v-\tilde{\pi }_N v)\Vert _{L^2(\mathbb {R})} \le CN^{-(s-t)/2}\Vert \tilde{\partial }^s v\Vert _{L^2(\mathbb {R})},\quad t=0,1. \end{aligned}$$
(24)
Let us define the projection \(\Pi _N:L^2(\mathbb {R}^3)\rightarrow \widetilde{V}_N \equiv \mathrm{span} \{\varphi _{ijk}^\mathrm{HO},0\le i,j,k\le \frac{N}{3}\}\) as a product of three one-dimensional projections by
$$\begin{aligned} \Pi _N=\tilde{\pi }_{\lfloor \frac{N}{3}\rfloor ,x}\circ \tilde{\pi }_{\lfloor \frac{N}{3}\rfloor ,y} \circ \tilde{\pi }_{\lfloor \frac{N}{3}\rfloor ,z}. \end{aligned}$$
Note that since (20) implies in particular that \(u\in H^{5/2-\varepsilon }(\mathbb {R}^3)\), which together with the exponential decay property of \(u\) leads to \(u\in \tilde{H}^{5/2-\varepsilon }(\mathbb {R}^3)\). We can thus derive from (24) that
$$\begin{aligned} \Vert u-\Pi _N u\Vert _{H^1(\mathbb {R}^3)} \le C\left\lfloor \frac{N}{3}\right\rfloor ^{-\left( \frac{5}{2}-\varepsilon -1\right) /2} \Vert u\Vert _{\tilde{H}^{5/2-\varepsilon }(\mathbb {R}^3)} \le CN^{-\frac{3}{4}+\varepsilon }. \end{aligned}$$
(25)
Since \(\widetilde{V}_N\subset \mathrm{span}\{\varphi _{ijk}^\mathrm{HO},0\le i+j+k\le N\}=V_{N}\), we can derive (21) directly from (25).
For a given \(K\), by taking \(\omega (r)\) in (20) that equals \(e^{-\alpha r^2}\) in a neighborhood of the origin and vanishes outside this neighborhood, we can divide \(u\) into two parts
$$\begin{aligned} u(\mathbf{r})=e^{-\alpha r^2}\sum _{j=0}^K r^j\alpha _j(\hat{\mathbf{r}}) + u_s(\mathbf{r}), \end{aligned}$$
such that \(u_s\in H^{5/2+K-\varepsilon }(\mathbb {R}^3)\) according to Proposition 3.
Therefore, we can obtain from (24) that
$$\begin{aligned} \Vert u_s-\Pi _N u_s\Vert _{H^1(\mathbb {R}^3)}\le CN^{-(\frac{3}{2}+K-\varepsilon )/2} \Vert u_s\Vert _{\tilde{H}^{5/2+K-\varepsilon }(\mathbb {R}^3)} \end{aligned}$$
Take \( v_{N}=e^{-\alpha r^2}\sum _{j=0}^K r^j\alpha _j(\hat{\mathbf{r}}) + \Pi _N u_s \in V_N^K\), we have
$$\begin{aligned} \Vert u-v_{N}\Vert _{H^1(\mathbb {R}^3)} = \Vert u_s-\Pi _N u_s\Vert _{H^1(\mathbb {R}^3)} \le CN^{-(\frac{3}{2}+K-\varepsilon )/2}. \end{aligned}$$
This completes the proof. \(\square \)

Remark 3

We mention that it has been shown by Klahn and Morgan [25] that the error of the ground state energy of the hydrogen atom goes as \(N^{-3/2}\) with \(N\) being the number of basis functions. Furthermore, the one-dimensional calculations by Hill [20] and numerical experiments by Klopper and Kutzelnigg [26] show that by adding a finite number of functions with odd order polynomials in (9), the error of energy is improved by a factor \(N^{-2}\) with each additional basis function, which is consistent with our results.

4 Even-tempered Gaussian bases

As we see from Theorem 1, the convergence of approximations by GTO basis functions of the form (9) with only even-order polynomials is relatively slow, while for bases including odd powers of \(r\), the convergence accelerates substantially, since such a basis set can better resolve the singularities of the wavefunctions [28]. However, the integration of basis functions including odd powers of \(r\) is much more complicated than in the case of only even powers, leading to the same difficulties as for STOs in practical computations (see, e.g. [19] for a discussion of GTO integral evaluation).

For this reason, basis functions with variable Gaussian exponents are the most popular choice in practice. Here, the radial parts are described exclusively by means of variable exponents adapted to the particular wavefunction to be approximated, and the only powers of \(r\) introduced are those associated with the angular momentum quantum number \(\ell \) of the spherical harmonics.

This leads to the set of spherical harmonic GTO basis functions
$$\begin{aligned} \psi ^\mathrm{GTO}_{\eta _{n}{\ell }m}(\mathbf{r})&= c_n e^{-\eta _{n}r^2} r^{\ell } Y_{\ell m}(\hat{\mathbf{r}}), \quad n=0,1,\ldots , \end{aligned}$$
(26)
with normalization factors \(c_n\).

Remark 4

Most practical computations are based on basis sets comprised of fixed linear combinations of GTOs,
$$\begin{aligned} \tilde{\psi }_{a{\ell }m}(\mathbf{r}) = \left( \sum _{k=1}^{K_a} \kappa _{ak} e^{-\eta _{k} r^2}\right) r^{\ell } Y_{\ell m}(\hat{\mathbf{r}}), \quad n=0,1,\ldots , \end{aligned}$$
(27)
where \(\kappa _{ak}\) are called contraction coefficients. Such linear combinations are known as contracted GTOs, and the individual Gaussians from which the contracted GTOs are constructed are referred to as primitive GTOs (see, e.g. [19]). In this paper, we focus on the approximation properties of primitive GTOs.

Some sufficient but not necessary conditions for completeness of a basis set (26) have been established in [19, 24]. The criteria are not helpful in practice since they do not guide us towards bases that converge rapidly, but only tell us that it is in principle possible to construct complete basis sets of the simple form (26).

The optimization of Gaussian exponents is a highly nonlinear problem with multiple solutions. Inspection of corresponding results reveals that the ratio between subsequent exponents is approximately constant [19], which suggests a simplified parametrization of the form
$$\begin{aligned} \eta _{n}=\alpha \beta ^n \end{aligned}$$
(28)
with parameters \(\alpha \) and \(\beta \). We shall focus in the following analysis on this specific type of GTOs, which are commonly referred to as even-tempered Gaussian bases.

The main result of this section is the following theorem, which considers the convergence rate of even-tempered Gaussian approximations for functions with finite angular momenta and corresponding radial parts that are products of exponentials and polynomials. This corresponds to linear combinations of STOs, containing exact excited-state wavefunctions of hydrogen-like systems as a special case. Note that also the standard STO-\(k\)G basis sets (see, e.g., [19]) are based on approximations of single STO by linear combinations of \(k\) Gaussians.

Essentially, the result thus states that for wavefunctions that can be efficiently approximated by short sums of STOs, one can also obtain efficient approximations by even-tempered GTOs. However, as can be seen from the proof, the constants in the resulting estimates grow rapidly with increasing polynomial orders and increasing angular momenta.

Theorem 2

Given an integer \(L\). Let \( V_{N}=\mathrm{span}\{\psi ^\mathrm{GTO}_{\eta _{n}{\ell }m} : 0\le n\le 2N,~0\le {\ell }\le L,~-{\ell }\le m\le {\ell }\}\), where \(\eta _{n}\) is of the form (28) with parameters \(\alpha _N,\beta _N\) depending on \(N\). If \(u\) is of the form
$$\begin{aligned} u(\mathbf{r})=\sum _{\ell m}^L P_{\ell m,M}(r) \,e^{-\gamma _{\ell } r}\mathcal {Y}_{\ell m}(\hat{\mathbf{r}}), \end{aligned}$$
(29)
where \(\gamma _{\ell }\) are constants and \(P_{{\ell }m,M}\) are polynomials with degrees no greater than \(M\), then for sufficiently large \(N\), there exist \(\alpha _N\) and \(\beta _N\) such that
$$\begin{aligned} \inf _{v_{N}\in V_{N}} \Vert u-v_{N}\Vert _{H^1(\mathbb {R}^3)} \le C_{M,L}\, e^{-c\sqrt{N}} \end{aligned}$$
(30)
with some constants \(C_{M,L}\) and \(c\).
To prove this theorem, we shall apply sinc quadrature approximation, for details of which we refer to the monograph of Stenger [37]. We now briefly recapitulate the results required for our purposes. For \(d>0\), we introduce the notation
$$\begin{aligned} \fancyscript{D}_d:=\{z\in \mathbb {C}:\mathrm{Im}\, z\in (-d,d)\}. \end{aligned}$$
(31)
We shall apply the following version of [37, Theorem 3.2.1].

Theorem 3

Let \(0< d < D\), and let \(f:\fancyscript{D}_D\rightarrow \mathbb {C}\) be holomorphic on \(\fancyscript{D}_D\). Assume that
$$\begin{aligned} \int \limits _{-\infty }^\infty | f(t \pm i d) | \,\mathrm{d}t < \infty , \end{aligned}$$
and that for any \(\delta \) with \(|\delta | < d\) and \(t\in \mathbb {R}\), \(|f(t + i\delta )| \rightarrow 0\) as \(|t|\rightarrow \infty \). Then for each \(h>0\), introducing the notation
$$\begin{aligned} \varphi ^\pm _d(t) := \frac{e^{-\pi (d\mp it)/h}}{\sin (\pi (t \pm id)/h)}, \end{aligned}$$
(32)
we have
$$\begin{aligned} \int \limits _{-\infty }^\infty f(x)\,\mathrm{d}x - \sum _{k=-\infty }^\infty h\, f(kh) = \frac{i}{2} \int \limits _{-\infty }^\infty f(t - id) \, \varphi ^-_d(t) - f(t+id)\, \varphi ^+_d(t) \,\mathrm{d}t. \end{aligned}$$
To obtain the result of Theorem 2, we first consider Gaussian approximations of products of a monomial and an exponential factor, that is,
$$\begin{aligned} f_{n,\gamma }(r):=r^n e^{-\gamma r},\quad n\in \mathbb {N}_0, \gamma >0. \end{aligned}$$
(33)
Our corresponding approximation results are based on integral representations obtained by the inverse Laplace transform (denoted by \(\mathcal {L}^{-1}\)) of \(t\mapsto f_{n,\gamma }(t^\frac{1}{2})\),
$$\begin{aligned} f_{n,\gamma }(\sqrt{t}) = \int \limits _0^\infty \mathcal {L}^{-1}\left( f_{n,\gamma }(\sqrt{\cdot }),s\right) e^{-s t} \,\mathrm{d}s,\quad t > 0. \end{aligned}$$
As an immediate consequence, we have the Gaussian integral representation
$$\begin{aligned} f_{n,\gamma }(r) = \int \limits _0^\infty \mathcal {L}^{-1}\left( f_{n,\gamma }(\sqrt{\cdot }),s\right) e^{-s r^2} \,\mathrm{d}s,\quad r > 0, \end{aligned}$$
which we will combine with the error representation from Theorem 3. The required inverse Laplace transforms can be obtained by the following recursion.

Lemma 1

For \(n\in \mathbb {N}_0\) and \(\gamma >0\), we have
$$\begin{aligned} \mathcal {L}^{-1} \bigl (r^{n/2} e^{-\gamma \sqrt{r}}\bigl )(s) = \gamma ^{n+1}s^{-\frac{2n+3}{2}}P_{n}(\gamma ^{-2}s)e^{-\frac{\gamma ^2}{4s}}, \end{aligned}$$
where \(P_n\) is the polynomial of degree \(\lceil n/2\rceil \) given by the recursion
$$\begin{aligned} P_0=(2\sqrt{\pi })^{-1},\quad P_1(s)=-(4 \sqrt{\pi })^{-1}(2s-1), \\ P_{n+2}(s) = -\frac{2n+3}{2} s P_n(s) + s^2 P'_n(s) + \frac{1}{4} P_n(s). \end{aligned}$$

Proof

For \(\gamma =1\) and \(n=0,1\), we know
$$\begin{aligned} \mathcal {L}^{-1}\left( e^{-\sqrt{r}}\right) (s)= (2\sqrt{\pi })^{-1} s^{-3/2}e^{-\frac{1}{4s}} \end{aligned}$$
and
$$\begin{aligned} \mathcal {L}^{-1}\left( \sqrt{r}e^{-\sqrt{r}}\right) (s)= -(4\sqrt{\pi })^{-1} s^{-5/2}(2s-1)e^{-\frac{1}{4s}}. \end{aligned}$$
Recall that \(\mathcal {L}^{-1}(\varphi (r))(s)=\Phi (s)\) implies \(\mathcal {L}^{-1}(r\varphi (r))(s)=\Phi '(s)\) provided that the corresponding integrals exist and \(\Phi (0) = 0\) (see [34]). Applying this property recursively, for \(k\in \mathbb {N}_0\) we obtainfor even-order polynomials, as well asfor odd-order polynomials. For \(\gamma >0\), we thus have
$$\begin{aligned} \mathcal {L}^{-1}\left( f_{2k,\gamma }(\sqrt{r})\right) (s)&= \gamma ^{-2k}\mathcal {L}^{-1}\left( f_{2k,1}(\gamma \sqrt{r})\right) (s) \\&= \gamma ^{2k+1}s^{-\frac{4k+3}{2}}P_{2k}(\gamma ^{-2}s)e^{-\frac{\gamma ^2}{4s}} \end{aligned}$$
for even-order polynomials and
$$\begin{aligned} \mathcal {L}^{-1}\left( f_{2k+1,\gamma }(\sqrt{r})\right) (s)&= \gamma ^{-2k-1}\mathcal {L}^{-1}\left( f_{2k+1,1}(\gamma \sqrt{r})\right) (s) \\&= \gamma ^{2k+2}s^{-\frac{4k+5}{2}}P_{2k+1}(\gamma ^{-2}s)e^{-\frac{\gamma ^2}{4s}} \end{aligned}$$
for odd-order polynomials. \(\square \)

We now come to our central result on the approximation of functions of the form (33), where we use the notations \(n!! := n(n-2)(n-4)\ldots (n+1-2\left\lfloor \frac{n+1}{2}\right\rfloor )\) and \([a]_+ = \max \{ a, 0\}\).

Lemma 2

Let \(\ell \in \mathbb {N}_0\), and let \(f_{n,\gamma }(r)\) be defined as in (33) with \(n\in \mathbb {N}_0\) and \(\gamma >0\). Let \(d < \frac{\pi }{2}\). For \(N \ge N_0(n,\gamma ,\ell ,d)\), there exist \(N\)-dependent parameters \(\alpha _N,\beta _N\in \mathbb {R}\) and \( f_N\in \mathrm{span}\{e^{-\eta _n r^2}\}_{n=0}^{2N}\) with \(\eta _n\) as in (28), such that
$$\begin{aligned}&\bigl ||r^\ell \bigl (f_{n,\gamma }(r)-f_N(r)\bigr )\bigr ||_{L^2(\mathbb {R}_+)} + \bigl ||r^\ell \bigl (f_{n,\gamma }'(r)-f'_N(r)\bigr )\bigr ||_{L^2(\mathbb {R}_+)} \nonumber \\&\quad \le K(n,\gamma ,\ell ,d)e^{-\sqrt{2\pi d \mu _\ell N}}, \end{aligned}$$
(34)
where \(\mu _\ell :=\textstyle \lfloor \frac{n}{2}\rfloor + \frac{\ell }{2} + \frac{1}{4}\),
$$\begin{aligned} K(n,\gamma ,\ell ,d)&= \hat{C} \frac{ n!!\, \sqrt{(2\ell + 1)!} }{ 2^\ell (\cos d)^{\frac{3}{4}+ \frac{\ell }{2}} } \Bigg ( \max \{n,\gamma ^{n+1}\}\\&+ \,\gamma ^{-\ell -n+\frac{1}{4}} \left( \frac{4}{\cos d} \right) ^{\frac{7}{4}+\frac{\ell }{2}+n} \lceil \ell /2 + n\rceil ! \Bigg ) \end{aligned}$$
with \(\hat{C}>0\) independent of \(n\), \(\gamma \), \(\ell \), and \(d\), and
$$\begin{aligned} N_0(n,\gamma ,\ell ,d) = (2\pi d)^{-1} \max \bigl \{ 4\mu _\ell \,[\ln (4\gamma ^{-2} (3n+2\ell +2))]_+^2,\, \mu _\ell ^{-1} \ln ^2 2 \bigr \}. \end{aligned}$$

Proof

Lemma 1 yields
$$\begin{aligned} F_{n,\gamma }(s) := \mathcal {L}^{-1}\left( f_{n,\gamma }(\sqrt{r})\right) (s)= \gamma ^{n+1}s^{-\frac{2n+3}{2}}P_{n}(\gamma ^{-2}s)e^{-\frac{\gamma ^2}{4s}}. \end{aligned}$$
(35)
In what follows, we shall write \(f\) and \(F\) in place of \(f_{n,\gamma }\) and \(F_{n,\gamma }\), respectively, to simplify notations. By a variable substitution, we obtain
$$\begin{aligned} f(r) = \int \limits _0^{\infty }F(s)e^{-sr^2}\,\mathrm{d}s = \int \limits _{-\infty }^{\infty } G(x)e^{-e^x r^2}\,\mathrm{d}x, \end{aligned}$$
where
$$\begin{aligned} G(x) := e^x F(e^x) = e^x \gamma ^{n+1}e^{-\frac{2n+3}{2}x}P_{n}(\gamma ^{-2}e^x)e^{-\frac{\gamma ^2}{4}e^{-x}}. \end{aligned}$$
(36)
With \(h>0\) to be specified later, we define
$$\begin{aligned} f_h(r) := h\sum _{k=-\infty }^{\infty } G(kh)\,e^{-e^{kh} r^2}, \quad f_{Nh}(r) := h\sum _{k=-N}^{N} G(kh)\,e^{-e^{kh} r^2}. \end{aligned}$$
We will obtain an estimate for the total error using
$$\begin{aligned} ||r^\ell (f - f_{Nh})||_{L^2(\mathbb {R}_+)} \le ||r^\ell (f-f_h)||_{L^2(\mathbb {R}_+)} + ||r^\ell ( f_h-f_{Nh})||_{L^2(\mathbb {R}_+)} \end{aligned}$$
and the analogous estimate for \(||r^\ell ( f' - f_{Nh}' )||_{L^2}\). We first consider the \(L^2\)-error estimate for \(\Vert r^{\ell } (f-f_h)\Vert _{L^2(\mathbb {R}_+)}\). We have
$$\begin{aligned} \Vert r^\ell (f-f_h) \Vert ^2_{L^2(\mathbb {R}_+)}&= \int \limits _0^{\infty } r^{2\ell } \left( \,\,\int \limits _{-\infty }^{\infty }G(s)e^{-e^{s}r^2}\,\mathrm{d}s - h\sum _{j=-\infty }^{\infty }G(jh)e^{-e^{jh}r^2}\right) ^2 \,\mathrm{d}r \\&= \int \limits _0^{\infty } \left( \,\,\int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty } G(s)\,G(t)r^{2\ell }e^{-(e^s+e^t)r^2}\,\mathrm{d}s \,\mathrm{d}t \right. \\&\left. +\,h^2\sum _{j=-\infty }^{\infty }\sum _{k=-\infty }^{\infty } G(jh)G(kh)r^{2\ell } e^{-(e^{jh}+e^{kh})r^2}\right. \\&\left. -\,2h\sum _{j=-\infty }^{\infty } G(jh) \int \limits _{-\infty }^{\infty } G(s)r^{2\ell } e^{-(e^{s} + e^{jh})r^2}\,\mathrm{d}s \right) \,\mathrm{d}r, \end{aligned}$$
and with the notation
$$\begin{aligned} \displaystyle T(x) :=\int \limits _0^{\infty } r^{2\ell }e^{-xr^2}\,\mathrm{d}r = c_\ell \, x^{-\frac{1}{2} - \ell },\quad c_\ell := \frac{\sqrt{\pi }(2\ell -1)!}{2^{\ell +1}}, \end{aligned}$$
integration over \(r\) yields
$$\begin{aligned} \Vert r^\ell (f-f_h)\Vert ^2_{L^2(\mathbb {R}_+)}&= \int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty } G(s)\,G(t)\,T(e^s+e^t) \,\mathrm{d}s \,\mathrm{d}t \\&+\, h^2\sum _{j=-\infty }^{\infty }\sum _{k=-\infty }^{\infty } G(jh)\,G(kh)\,T(e^{jh}+e^{kh}) \\&-\,2h\sum _{j=-\infty }^{\infty }\int \limits _{-\infty }^{\infty } G(s)\,G(jh)\,T(e^s+e^{jh})\,\mathrm{d}s. \end{aligned}$$
In terms of the further abbreviations
$$\begin{aligned} q(s,t) := G(t)\,T(e^s+e^t),\quad Q(s) := \int \limits _{-\infty }^{\infty } q(s,t)\,\mathrm{d}t - h\sum _{k=-\infty }^{\infty }q(s, kh) \end{aligned}$$
this can be rewritten as
$$\begin{aligned} \Vert r^\ell (f-f_h)\Vert ^2_{L^2(\mathbb {R}_+)} = \int \limits _{-\infty }^{\infty }G(s) Q(s)\,\mathrm{d}s -h\sum _{k=-\infty }^{\infty }G(kh) Q(kh). \end{aligned}$$
(37)
As a first step, we use Theorem 3 to obtain a representation for the error term \(Q\). Note first that since \(d<\pi /2\), for each \(s\in \mathbb {R}\) the function \(t\mapsto q(s,t)\) is holomorphic in \(\fancyscript{D}_d\). Furthermore, for any \(s + i\delta \) with \(s, \delta \in \mathbb {R}\) and \(|\delta |\le d\), we can obtain the estimate
$$\begin{aligned} \int \limits _{-\infty }^\infty |q(s+i\delta ,t\pm id)| \,\mathrm{d}t&\le c_\ell \int \limits _{-\infty }^\infty \left| e^{s+i\delta } + e^{t\pm id}\right|^{-\frac{1}{2}-\ell } \left|e^{t\pm id} F(e^{t\pm id})\right|\,\mathrm{d}t\nonumber \\&\le \frac{c_\ell }{(\cos d)^{\frac{1}{2}+\ell }} \int \limits _{-\infty }^\infty e^{\left( \frac{1}{2}-\ell \right) t} \left|F(e^{t\pm id})\right| \,\mathrm{d}t, \end{aligned}$$
(38)
where we have again used \(d < \frac{\pi }{2}\).
In order to estimate the right hand side in (38), we need to take a closer look at the expression for \(F\circ \exp \) in (36). From the recursion for the polynomials \(P_n\), it can be seen that the absolute values of the coefficients in these polynomials are bounded by \((2\sqrt{\pi })^{-1} n!!\), and consequently
$$\begin{aligned} \left|F (e^{t\pm id})\right| \le (2\sqrt{\pi })^{-1} \gamma ^{n+1} n!!\, e^{-\frac{\gamma ^2\cos d}{4} e^{-t} } \sum _{k=0}^{\lceil n/2 \rceil } \gamma ^{-2k} e^{-\left( \frac{3}{2}+n-k\right) t}. \end{aligned}$$
(39)
Thus the integral on the right hand side of (38) is finite, and in particular, in the case \(\delta =0\) we obtain a bound independent of \(s\in \mathbb {R}\). Thus we can apply Theorem 3 to \(Q\) to obtain
$$\begin{aligned} Q(s) := \frac{i}{2}\int \limits _{-\infty }^{\infty } \bigl ( q(s,t-id) \varphi ^-_d(t) - q(s,t+id) \varphi ^+_d(t) \bigr ) \,\mathrm{d}t. \end{aligned}$$
(40)
Next, we verify that we can apply Theorem 3 a second time to the product of \(G\) and \(Q\) in (37). The integral in the representation (40) of \(Q(s)\) is holomorphic as a function of \(s\in \fancyscript{D}_d\). This follows because \(q(s,t\pm id)\) is holomorphic as a function of \(s\in \fancyscript{D}_d\) for each \(t\), and in addition
$$\begin{aligned} \int \limits _{-\infty }^{\infty } \bigl | q(s,t\pm id) \varphi ^\pm _d(t) \bigr | \,\mathrm{d}t \le \frac{e^{-\pi d/h}}{\sinh (\pi d/h)} \int \limits _{-\infty }^{\infty } |q(s,t\pm id)|\,\mathrm{d}t, \end{aligned}$$
where the right hand side is uniformly bounded as a consequence of (38). To verify the remaining bounds required for Theorem 3, we estimate
$$\begin{aligned} \int \limits _{-\infty }^\infty |G(s\pm i d) Q(s\pm id)|\,\mathrm{d}s \le \frac{e^{-\pi d/h}}{2 \sinh (\pi d/h)} \mathcal {N}^\pm _d(G,T) \end{aligned}$$
with the notation
$$\begin{aligned} \mathcal {N}^\pm _d(G,T)&:= \int \limits _{-\infty }^\infty \int \limits _{-\infty }^\infty \left|G(s\pm i d)\right| \left( \left|G(t - id)\,T(e^{s\pm id} + e^{t - id})\right| \right. \\&\left. + \left|G(t + id)\,T(e^{s\pm id} + e^{t + id})\right|\right) \,\mathrm{d}s\,\mathrm{d}t. \end{aligned}$$
For the latter expression, using the estimates
$$\begin{aligned} \left|T(e^{t \pm id}+e^{s \pm id})\right|&\le c_\ell (e^t + e^s)^{-\frac{1}{2}-\ell } \le \frac{c_\ell }{2^{\frac{1}{2}+\ell }}\, e^{-\left( \frac{1}{4}+\frac{\ell }{2}\right) t} e^{-\left( \frac{1}{4}+\frac{\ell }{2}\right) s}, \\ \left|T(e^{t \pm id}+e^{s \mp id})\right|&\le \frac{c_\ell }{(\cos d)^{\frac{1}{2}+\ell }} (e^t + e^s)^{-\frac{1}{2}-\ell } \\&\le \frac{c_\ell }{(2\cos d)^{\frac{1}{2}+\ell }} e^{-\left( \frac{1}{4}+\frac{\ell }{2}\right) t} e^{-\left( \frac{1}{4}+\frac{\ell }{2}\right) s}, \end{aligned}$$
we obtain
$$\begin{aligned} \mathcal {N}^\pm _d(G,T) \le \frac{2 c_\ell }{(2\cos d)^{\frac{1}{2}+\ell }} \left( \,\,\int \limits _{-\infty }^{\infty } e^{\left( \frac{3}{4}-\frac{\ell }{2}\right) t} \max \bigl \{|F(e^{t-id})|,|F(e^{t+id})|\bigr \} \,\mathrm{d}t \right) ^2.\nonumber \\ \end{aligned}$$
(41)
Provided that the right hand side is finite, we have thus verified the assumptions of Theorem 3 for the function \(s\mapsto G(s)\,Q(s)\), which yields
$$\begin{aligned} \Vert r^\ell (f\!-\!f_h)\Vert ^2_{L^2(\mathbb {R}_+)}&= -\frac{1}{4}\int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty } \left( \varphi ^-_d(t) \varphi ^-_d(s)G(t\!-\!id)\,G(s\!-\!id)\,T(e^{t-id}\!+\!e^{s-id}) \right. \nonumber \\&\left. +\, \varphi ^-_d(t) \varphi ^+_d(s) G(t-id)G(s+id)\,T(e^{t-id}+e^{s+id})\right. \nonumber \\&\left. +\, \varphi ^+_d(t) \varphi ^-_d(s) G(t+id)G(s-id)\,T(e^{t+id}+e^{s-id})\right. \nonumber \\&\left. +\, \varphi ^+_d(t) \varphi ^+_d(s) \, G(t+id)G(s+id)\,T(e^{t+id}+e^{s+id}) \right) \,\mathrm{d}s \,\mathrm{d}t \nonumber \\&\le \frac{1}{4}\left( \frac{e^{-\pi d/h}}{\sinh (\pi d/h)} \right) ^2 ( \mathcal {N}^+_d(G,T) \!+\! \mathcal {N}^-_d(G,T) ). \end{aligned}$$
(42)
Before completing the required estimate for \(\mathcal {N}^\pm _d(G,T)\) using (41), we consider the analogous argument for the \(H^1\)-seminorm error \(||r^\ell (f'-f_N')||_{L^2(\mathbb {R}_+)}\). Here we approximate
$$\begin{aligned} f'(r)=-2r\int \limits _{-\infty }^{\infty }e^x G(x)e^{-e^x r^2}\,\mathrm{d}x \end{aligned}$$
by
$$\begin{aligned} f'_{Nh}(r) = -2rh\sum _{k=-N}^{N} e^{kh} G(kh)e^{-e^{kh} r^2}. \end{aligned}$$
We can apply the same arguments as in (41) and (44) to \(\tilde{G}(x) := -2e^xG(x)\) and
$$\begin{aligned} \tilde{T}(x) :=\int \limits _0^{\infty } r^{2+2\ell }e^{-xr^2}\,\mathrm{d}r = \tilde{c}_\ell \, x^{-\frac{3}{2} -\ell },\quad \tilde{c}_\ell := \frac{\sqrt{\pi } (2\ell +1)!}{2^{\ell +2}}, \end{aligned}$$
which leads to
$$\begin{aligned} \mathcal {N}_d^\pm (\tilde{G}, \tilde{T}) \le \frac{2\tilde{c}_\ell }{(2\cos d)^{\frac{3}{2}+\ell }} \left( \,\,\int \limits _{-\infty }^{\infty } e^{\left( \frac{5}{4} -\frac{\ell }{2}\right) t} \max \left\{ \left| F(e^{t-id})|, |F(e^{t+id})\right| \right\} \,\mathrm{d}t \right) ^2. \end{aligned}$$
To complete our treatment of \(||r^\ell (f-f_h)||_{L^2(\mathbb {R}_+)}\), \(||r^\ell (f'-f_h')||_{L^2(\mathbb {R}_+)}\), it thus remains to use our information on \(F\) in the estimates for \(\mathcal {N}_d^\pm ({G}, {T})\) and \(\mathcal {N}_d^\pm (\tilde{G}, \tilde{T})\). Concerning the parts of the integrals over the positive real axis, we combine (39) with
$$\begin{aligned} \int \limits _0^\infty e^{\frac{5}{4} - \frac{\ell }{2}} e^{-\left( \frac{3}{2} +n -k\right) x} e^{-\frac{\gamma ^2 \cos d}{4} e^{-x} } \le \left( \frac{1}{4} + \frac{\ell }{2} + n - k \right) ^{-1}. \end{aligned}$$
Concerning the integration over the negative real axis, note that for \(\alpha >0\),
$$\begin{aligned} \int \limits _{-\infty }^0 e^{-\alpha x} e^{-\frac{\gamma ^2 \cos d }{4} e^{-x}} \,\mathrm{d}x&= \int \limits _0^\infty e^{\alpha x} e^{-\frac{\gamma ^2}{4} e^{x}\cos d} \,\mathrm{d}x \\&= \int \limits _1^\infty s^{\alpha - 1} e^{-\frac{\gamma ^2}{4} s \cos d} \,\mathrm{d}s \le \left( \frac{4}{\gamma ^2 \cos d}\right) ^{\alpha } \Gamma (\alpha ). \end{aligned}$$
We apply this to each term in (39) with \(\alpha = \frac{3}{4} + \frac{\ell }{2} + n - k\) for each \(k\), and from (42) we thus obtain
$$\begin{aligned}&||r^\ell ( f-f_h)||_{L^2(\mathbb {R}_+)} + ||r^\ell (f'-f_h')||_{L^2(\mathbb {R}_+)} \nonumber \\&\quad \le C \frac{e^{-\pi d/h}}{\sinh (\pi d/h)} \frac{(\sqrt{c_\ell } + \sqrt{\tilde{c}_\ell })n!!}{(2\cos d)^{\frac{3}{4}+\frac{\ell }{2}}} \sum _{k=0}^{\lceil n/2\rceil } \gamma ^{n-2k+1} \left( \bigl ( \textstyle n - k + \frac{1}{4} + \frac{\ell }{2}\displaystyle \bigr )^{-1}\right. \nonumber \\&\quad \quad \left. + \left( \frac{4}{ \gamma ^2 \cos d} \right) ^{n-k+\frac{3}{4}+\frac{\ell }{2}} \left\lceil \textstyle \frac{\ell }{2}\displaystyle +n-k\right\rceil ! \right) . \end{aligned}$$
(43)
We next consider the errors arising from the truncation of the infinite sums, where we obtain
$$\begin{aligned} \nonumber \Vert r^\ell (f_h-f_{Nh})\Vert ^2_{L^2(\mathbb {R}_+)}&= h^2\sum _{|k|>N}\sum _{|l|>N} G(kh) G(lh) T_\ell (e^{kh}+e^{lh}) \\&\le \frac{h^2 c_\ell }{2^{\frac{1}{2}+\ell }} \left( \,\, \int \limits _{(-\infty ,-N]\cup [N,\infty )}F(e^{xh})e^{\left( \frac{3}{4}-\frac{\ell }{2}\right) xh}\,\mathrm{d}x\right) ^2\qquad \end{aligned}$$
(44)
as well as
$$\begin{aligned} \left\| r^\ell (f'_h-f'_{Nh})\right\| ^2_{L^2(\mathbb {R}_+)} \le \frac{h^2 \tilde{c}_\ell }{2^{\frac{3}{2}+\ell }} \left( \,\, \int \limits _{(-\infty ,-N]\cup [N,\infty )} F(e^{xh})\,e^{(\frac{5}{4}-\frac{\ell }{2})xh}\,\mathrm{d}x \right) ^2.\qquad \end{aligned}$$
(45)
Concerning the decay of \(F\) in (44) and (45), on the one hand, for \(x\ge 0\) we have
$$\begin{aligned} |F(e^x)| \le (2\sqrt{\pi })^{-1} n!! \max \{n,\gamma ^{n+1}\} e^{-(\lfloor \frac{n}{2}\rfloor + \frac{3}{2}) x}, \end{aligned}$$
which leads to the choice of \(\mu _\ell \) in the hypothesis. Setting
$$\begin{aligned} \hat{x}_\ell := 2 \left[ \ln \left( 4\gamma ^{-2} (3n+2\ell +2)\right) \right] _+, \end{aligned}$$
on the other hand, for \(x \ge \hat{x}_\ell \) we obtain
$$\begin{aligned} e^{\left( \frac{3}{4} + n + \frac{\ell }{2}\right) x } e^{-\frac{\gamma ^2 }{4} e^x} \le e^{-\mu _\ell x}. \end{aligned}$$
For all \(N\) with \(Nh \ge \hat{x}_\ell \), we thus obtain
$$\begin{aligned}&\left||r^\ell (f_h-f_{Nh})\right||_{L^2(\mathbb {R}_+)} + \left||r^\ell \left( f_h'-f_{Nh}'\right) \right||_{L^2(\mathbb {R}_+)} \nonumber \\&\quad \le C \frac{n!! \max \{n,\gamma ^{n+1}\} (\sqrt{c_\ell } + \sqrt{\tilde{c}_\ell })}{ 2^{ \ell /2 } \mu _\ell } \, e^{-\mu _\ell N h}. \end{aligned}$$
(46)
In order to balance the \(h\)-dependent exponents in (43) and (46), we now choose \(h := \sqrt{2\pi d / (\mu _\ell N) }\), and use the corresponding \(f_{Nh}\) as the approximation \(f_N\) in the hypothesis. Note that this amounts to choosing
$$\begin{aligned} \alpha _N=e^{-\sqrt{\frac{2\pi dN}{\mu _\ell }}},\quad \beta _N=e^{\sqrt{\frac{2\pi d}{\mu _\ell N}}} \end{aligned}$$
as the parameters (28) of the even-tempered GTO basis. Since by assumption \(N\ge (\ln ^2 2) / (2 \pi d \mu _\ell )\), we have \(e^{-\pi d/ h}/\sinh (\pi d/h) \le 4 e^{-\sqrt{2 \pi d \mu _\ell N}}\) in (43). Furthermore, \(N \ge \hat{x}_\ell ^2 \mu _\ell / (2\pi d)\) ensures \(Nh\ge \hat{x}_\ell \). This completes the proof. \(\square \)

With the above results at hand, we are now ready to prove Theorem 2.

Proof

(Theorem 2 ) Let the even-tempered Gaussian approximation of \(u\) be
$$\begin{aligned} v_{N}(\mathbf{r}) = \sum _{\ell m}^{L}\sum _{k=0}^{2N}c_{k{\ell }m}\chi _k(r)r^{\ell } Y_{\ell m}(\hat{\mathbf{r}}) \in V_{N} \end{aligned}$$
with \(\chi _k(r)=e^{-\eta _k r^2}~(k=0,1,\ldots ,2N)\) and \(c_{k{\ell }m}\) the approximation coefficients. Since for \(u\) we have
$$\begin{aligned} u(\mathbf{r})=\sum _{\ell m}^{L} P_{\ell m,M}(r) e^{-\gamma _{\ell } r}\mathcal {Y}_{\ell m}(\hat{\mathbf{r}}), \end{aligned}$$
we obtain
$$\begin{aligned} \Vert u-v_{N}\Vert _{H^1(\mathbb {R}^3)} \le \sum _{\ell m}^{L} \left\| \left( P_{\ell m,M}(r)e^{-\gamma _{\ell } r}-\sum _{k=0}^{2N}c_{k{\ell }m}\chi _k(r)\right) r^{\ell }Y_{\ell m}(\hat{\mathbf{r}})\right\| _{H^1(\mathbb {R}^3)}.\nonumber \\ \end{aligned}$$
(47)
The Laplacian can be expressed in polar coordinates as
$$\begin{aligned} \Delta =\frac{1}{r^2}\frac{\partial }{\partial r}\left( r^2\frac{\partial }{\partial r} \right) +\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }\left( \sin \theta \frac{\partial }{\partial \theta } \right) +\frac{1}{r^2\sin ^2\theta }\frac{\partial ^2}{\partial ^2\phi }, \end{aligned}$$
where the last two terms multiplied by \(r^2\) are the Laplace–Beltrami operator on sphere \(S^2\). It can thus be verified by a simple calculation (see, e.g., [11]) that
$$\begin{aligned}&\Vert f(r)Y_{\ell m}(\hat{\mathbf{r}})\Vert ^2_{H^1(\mathbb {R}^3)}\\&\quad = \int \limits _0^{\infty }r^2 \left( f^2(r)+\left( \frac{\partial f(r)}{\partial r}\right) ^2 + \frac{{\ell }({\ell }+1)}{r^2}f^2(r)\right) \,\mathrm{d}r \int \limits _{S^2}Y^2_{\ell m}(\hat{\mathbf{r}}) \,\mathrm{d}\hat{\mathbf{r}}. \end{aligned}$$
Therefore, each term of the right hand side of (47) can be estimated as
$$\begin{aligned}&\left||\left( P_{\ell m,M}(r)e^{-\gamma _{\ell } r}-\sum _{k=0}^{2N}c_{k{\ell }m}\chi _k(r)\right) r^{\ell }Y_{\ell m}(\hat{\mathbf{r}})\right||^2_{H^1(\mathbb {R}^3)}\\&\quad = \left||r^{{\ell }+1}\left( P_{\ell m,M}(r)e^{-\gamma _{\ell } r}-\sum _{k=0}^{2N} c_k\chi _k(r)\right) \right||^2_{L^2(\mathbb {R}_+)} \\&\quad \quad + {\ell }^2 \left||r^{\ell }\left( P_{\ell m,M}(r)e^{-\gamma _{\ell } r}-\sum _{k=0}^{2N} c_k\chi _k(r)\right) \right||^2_{L^2(\mathbb {R}_+)} \\&\quad \quad + \left||r^{{\ell }+1}\partial _{r}\left( P_{\ell m,M}(r)e^{-\gamma _{\ell } r}-\sum _{k=0}^{2N} c_k\chi _k(r)\right) \right||^2_{L^2(\mathbb {R}_+)} \\&\quad \quad +{\ell }(1+{\ell }) \left||r^{\ell }\left( P_{\ell m,M}(r)e^{-\gamma _{\ell } r}-\sum _{k=0}^{2N} c_k\chi _k(r)\right) \right||^2_{L^2(\mathbb {R}_+)}. \end{aligned}$$
Applying Lemma 2, where we choose a fixed \(d \in (0,\frac{\pi }{2})\), we obtain
$$\begin{aligned}&\left\| \left( P_{\ell m,M}(r)e^{-\gamma _{\ell }r}-\sum _{k=0}^{2N}c_{k{\ell }m}\chi _k(r)\right) r^{\ell } Y_{\ell m}(\hat{\mathbf{r}})\right\| _{H^1(\mathbb {R}^3)} \nonumber \\&\quad \le C\frac{\sqrt{(2{\ell }+3)!}}{2^{\ell }} M!! \left( \max \left\{ M,\gamma _{\ell }^{M+1}\right\} \right. \nonumber \\&\quad \quad \left. +\, \gamma _{\ell }^{-{\ell }-M} b_d^{\frac{\ell }{2}+M}\lceil {\ell }/2+M \rceil ! \right) e^{-2c\sqrt{N}}, \end{aligned}$$
(48)
where \(b_d := \frac{4}{\cos d}\), \(C\) is a constant depending only on \(d\) and the coefficients of the polynomial \(P_{\ell m,M}\), and \(c\) is a constant determined by \(d\) and \(\mu _{\ell }\) in Lemma 2. Summing up (48) over the indices \(\ell \) and \(m\) in (47), we obtain (30), where
$$\begin{aligned} C_{M,L} \sim (L+1)^2 \frac{\sqrt{(2L+3)!}}{2^L}M!! \Biggl ( \max \{M,\overline{\gamma }^{M+1}\}+ \underline{\gamma }^{-L-M} b_d^{\frac{L}{2}+M} \lceil L/2+M \rceil ! \Biggr ) \end{aligned}$$
with \(\overline{\gamma }=\max \{\gamma _0,\ldots ,\gamma _L\}\) and \(\underline{\gamma }=\min \{\gamma _0,\ldots ,\gamma _L\}\). This completes the proof. \(\square \)

Remark 5

The above theorem gives an error estimate for the approximation of Slater-type functions by even-tempered Gaussians. More general functions can in turn be approximated by sums of STOs
$$\begin{aligned} \psi ^\mathrm{STO}_{n\ell m}(\mathbf{r}) = L_n^{\alpha }(r)\,e^{-\gamma _{\ell } r}Y_{\ell m}(\hat{\mathbf{r}}), \quad n=0,1,\ldots , \end{aligned}$$
(49)
where \(\alpha ,\gamma _{\ell }\) are constants and \(L_n^{\alpha }\) are Laguerre polynomials. Since each \(\psi ^\mathrm{STO}_{n\ell m}\) as defined in (49) is of the form (29) in Theorem 2, from this one can obtain estimates for more general functions, for which the Laguerre function expansion coefficients of radial parts have sufficiently rapid decay.

The regularity result of Proposition 2, however, only yields superalgebraic decay of the Laguerre function coefficients of the eigenfunctions, which is insufficient considering the rapid growth with \(L\) and \(M\) of the constants \(C_{L,M}\) in Theorem 2 (see also our numerical tests in Example 3). In an extension of Theorem 2 to general functions, quite restrictive additional assumptions may thus be required in order to ensure semi-exponential convergence rates of even-tempered GTO approximations.

Remark 6

Kutzelnigg [28, 29, 30] has shown that the error of ground state energy expectation value for the hydrogen atom obtained by certain even-tempered Gaussian bases goes as \(e^{-c\sqrt{N}}\), where \(N\) is the number of basis functions. However, it is not clear whether these results can be extended to \(H^1\)-error estimates for the approximation of general wavefunctions in \(\mathbb {R}^3\).

5 Numerical experiments

All the numerical results in this section are given in atomic units.

Example 1

We simulate a hydrogen atom by Gaussian basis sets as a simple example to support the theory in this paper. Solve the linear eigenvalue problem: find \(\lambda \in \mathbb {R}\) and \(u\in H^1(\mathbb {R}^3)\) such that
$$\begin{aligned} -\frac{1}{2}\Delta u - \frac{1}{|\mathbf{r}|}u=\lambda u. \end{aligned}$$
We consider only the first two states of this system, for which the corresponding exact energies are \(-0.5\) and \(-0.125\).
We first use the Hermite Gaussian basis functions (9) by taking \(\zeta =0.2\) with and without one odd order polynomial. The numerical errors of the first two states energies are presented in Figs. 1 and 2 respectively. We observe that the convergence rates of the energy errors are consistent with our theoretical results.
Fig. 1

Numerical errors of Hermite Gaussian approximations for the first state energy (exact value \(E=-0.5\))

Fig. 2

Numerical errors of Hermite Gaussian approximations for the second state energy (exact value \(E =-0.125\))

We also apply the even-tempered Gaussian basis functions in this example. The numerical errors of the first two state energies are presented in Fig. 3, which decay semi-exponentially and are consistent with our theoretical results. We compare the approximations obtained by Hermite Gaussian basis and even-tempered Gaussian basis in Fig. 4 using the same basis size, say \(N=5\). We observe that the even-tempered Gaussian approximation captures the cusp at the nuclear position much better than Hermite Gaussian approximation, even though neither of these basis functions have the cusp property.
Fig. 3

Numerical errors of even-tempered Gaussian basis for the first two state energies

Fig. 4

Comparison of radial wavefunctions obtained by Hermite Gaussian basis and even-tempered Gaussian basis

Example 2

We use the package Molpro [39] to simulate a water molecule H\(_2\)O using even-tempered Gaussian basis sets. Molpro is a system of ab initio programs for molecular electronic structure calculations, based on which we perform an all-electron DFT calculations with the Slater–Dirac exchange potential [35] and a SCF iteration. For angular momenta, we include \(spdf\) types for the oxygen atom and \(spd\) types for the hydrogen atoms. Since we do not have the analytic solution, the numerical result obtained by the default basis of the package, say the VDZ basis [39], are taken to be the exact one. The numerical errors of the ground state energy per electron are presented in Fig. 5, from which we observe a semi-exponential decay of the ground state energy error.

Fig. 5

Numerical errors per electron of the ground state energy for H\(_2\)O by the Molpro package using even-tempered basis

Example 3

We consider the \(H^1\)-error for the approximation of \(f(r) := r^n e^{-r}\) as in (34), i.e.,
$$\begin{aligned} \varepsilon _N := \left\| r^\ell (f-f_N)\right\| _{L^2(\mathbb {R}_+)}+\left\| r^\ell \partial _r(f-f_N)\right\| _{L^2(\mathbb {R}_+)} \end{aligned}$$
for \(n=0,\ldots ,4\) and \(\ell = 0,2\). Here the approximations \(f_N\) are constructed as in Lemma 2. Note that in this construction, the value of \(h\) depends on the choice of a free parameter \(d > 0\), which also enters in the error estimates. This parameter is chosen as follows: for given \(n\), \(\ell \), and \(N\), we first symbolically evaluate \(\varepsilon _N\) as a function of \(d\) using Maple, and then perform a numerical minimization of \(\varepsilon _N\) with respect to \(d\), using extended precision. The results are shown in Fig. 6. With increasing \(n\) and \(\ell \), we observe the expected growth of the constants in the error estimates, but also an increase in convergence rates.
Fig. 6

\(H^1\)-errors (in double logarithmic scale) for of \(r^n e^{-r}\) in Example 3

6 Concluding remarks

In this paper we have given approximation error estimates in \(H^1\)-norm for two types of Gaussian approximations in electronic structure calculations, Hermite-type Gaussian bases and even-tempered Gaussian bases. The applications we have focused on, Kohn–Sham and Hartree–Fock equations, are nonlinear models in three dimensions, but error estimates of this type may be of use also in the analysis of higher-dimensional wavefunction methods.

Although especially even-tempered Gaussians already lead to efficient approximations, there are some further practical aspects that are not covered by this analysis. There exist many basis sets that are optimized for the approximations of particular molecular systems, and whose properties are thus difficult to capture in an asymptotic error analysis. Furthermore, since typically a relatively large number of Gaussian basis functions are needed to represent the wavefunctions, basis sets are usually constructed from fixed linear combinations of several Gaussians as in the contracted GTOs (27) (e.g. for STO-\(k\)G and Huzinaga basis sets [18, 23]), instead of using single Gaussian basis functions individually. In this paper we have not considered this additional aspect of atomic basis sets used in practice, but restricted ourselves to the basic primitive Gaussian approximations. This work may thus be viewed as a step towards a more complete understanding of Gaussian bases from a numerical point of view.

Notes

Acknowledgments

The research for this paper has been enabled by the Alexander von Humboldt Foundation, whose support for the long term visit of H. Chen at Technische Universität Berlin is gratefully acknowledged.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Markus Bachmayr
    • 1
  • Huajie Chen
    • 2
  • Reinhold Schneider
    • 2
  1. 1.Institut für Geometrie und Praktische MathematikRWTH AachenAachenGermany
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany

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