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

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## 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 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.

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.

## 3 Hermite Gaussian bases

*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 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 \(\gamma <3/2\).

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.

The following two propositions will be used in our analysis.

**Proposition 2**

*Proof*

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

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 \(r>1\).

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**

*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 \(\Phi _{k+1}\in H^{5/2+n-\varepsilon }(\mathbb {R}^3)\).

### 3.2 Approximation by Hermite Gaussian bases

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

**Theorem 1**

**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*

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 \(2(\tilde{n}-1)-{\ell }\).

*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.

**Remark 4**

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 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**

**Theorem 3**

**Lemma 1**

*Proof*

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**

*Proof*

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

*Remark 5*

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*

*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.

*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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