Noninteger Slater-Type Orbitals

• Noninteger Slater-Type Orbitals are generalized orbitals permitting real-valued principal quantum numbers, which enhances the variational flexibility in electronic structure theory.
• They enable accurate representation of singular behavior at the nuclear cusp and proper asymptotic decay, tackling computational challenges in multi-center integral evaluations.
• Recent advances establish their completeness and orthonormality, making NSTOs central to high-accuracy quantum chemical and relativistic Dirac–Hartree–Fock calculations.

A noninteger Slater-type orbital (NSTO) is a generalization of the conventional Slater-type orbital in which the principal quantum number is permitted to take positive real (noninteger) values. These functions, of the form $$\chi_{n,\ell,m}(\mathbf{r}) = N_{n,\ell}\,r^{n-1} e^{-\zeta r} Y_{\ell}^{m}(\theta,\phi)$$, with $n \in \mathbb{R}^+$, encapsulate enhanced variational flexibility for atomic and molecular electronic structure theory, as they can better represent singular behavior at the nuclear cusp and the correct asymptotic decay. Their use, however, introduces significant analytical and computational challenges, particularly in the robust and efficient evaluation of relevant multi-center integrals. Recent advances provide a rigorous foundation for their completeness, orthonormality, and efficient integral evaluation, cementing their role in high-accuracy quantum chemical and relativistic Dirac–Hartree–Fock calculations.

1. Mathematical Formulation and Properties

The normalized NSTO centered at position $\mathbf{R}_A$ is given by

$$\chi_{n,\ell,m}(\zeta, \mathbf{r}_A) = N_{n,\ell} \, r_A^{n-1} \exp(-\zeta r_A) Y_\ell^m(\theta_A, \phi_A),$$

where $r_A = |\mathbf{r} - \mathbf{R}_A|$, $n \in \mathbb{R}^+$, $\zeta > 0$, and $Y_\ell^m$ denotes scalar or spinor spherical harmonics as required. The normalization constant follows from

$$\int_0^\infty \int_\Omega |\chi_{n,\ell,m}|^2\, r^2 dr d\Omega = 1,$$

yielding

$$N_{n,\ell} = (2\zeta)^{n+1/2} / \sqrt{\Gamma(2n+1)}.$$

Variants for orthonormalized NSTOs and related exponential-type orbitals (ETOs) take the form

$$N_{n^*,\ell} = \left[\frac{(2\zeta)^{3} \Gamma(n^*-\ell)}{2n^*\Gamma(n^* + \ell + 1)}\right]^{1/2},$$

with $n^* \in \mathbb{R}^+$. Riemann-Liouville-type fractional calculus enters naturally in the construction and transformation properties of these orbitals, connecting them to orthonormal sets of generalized exponential-type orbitals via Laguerre polynomials with noninteger indices (Bağcı et al., 2022).

2. Completeness, Orthonormality, and Basis Set Construction

NSTOs possess a rigorous mathematical foundation as bases for $L^2$ spaces when constructed as follows:

• All basis functions share a fixed fractional part $\nu \in (0,1]$ of the principal quantum number, i.e., $n^* = \nu + k$, $k \in \mathbb{N}_0$.
• All functions observed in the expansion must share the same Hilbert-space weight exponent $\alpha > 0$ for the radial integration measure $\omega(r) = r^{\alpha - 3}$.

The completeness theorem states that the set

$$\left\{\, \Psi_{n^*,\ell,m} \mid n^* = \nu + k,\ k \in \mathbb{N}_0,\ \ell=0,1,\dots, m=-\ell,\dots,\ell \, \right\}$$

forms a basis in the weighted Sobolev space (Bagci et al., 6 Jul 2025), and any square-integrable function may be expanded in this set. Treating each $n^*_i$ or $\alpha_i$ as an independent variational parameter destroys orthonormality and can introduce severe numerical instability (near-linear dependence, overcompleteness). The extended ETO basis of Guseinov can be constructed analytically from NSTOs via explicit closed-form linear combinations, with coefficients determined by generalized Laguerre polynomial identities and transformation matrices (Bağcı et al., 2022).

3. Atomic and Molecular Integrals: Analytical Challenges and Methodologies

The lack of analyticity of $r^{n-1}$ at $r=0$ for noninteger $n$ precludes straightforward power-series expansions. Consequently, the evaluation of overlap, nuclear attraction, kinetic, and especially two-electron Coulomb integrals requires specialized machinery:

One-center Integrals

These integrals can be reduced to expressions involving incomplete gamma and beta functions: $$S_{ij} = N_i N_j \frac{\Gamma(n^*_i+n^*_j + \alpha - 2)}{(\zeta_i + \zeta_j)^{n^*_i+n^*_j + \alpha - 2}}$$ for overlaps, with analogous closed forms for kinetic and nuclear-attraction operators (Bagci et al., 6 Jul 2025). For electron repulsion integrals, hypergeometric-function forms

$R^L_{n,n'}(\zeta, \zeta') = \frac{\Gamma(n+n'+1)}{(\zeta + \zeta')^{n+n'+1}} \Bigg\{ \frac{1}{n+L+1} \, _2F_1[\dots] + \frac{1}{n'+L+1}\, _2F_1[\dots] \Bigg\}$

have been superseded by recurrence-based finite-sum constructions leveraging incomplete beta functions, improving numerical stability and efficiency (Bağcı et al., 2022).

Multi-center and Two-electron Integrals

For general two-center and two-electron integrals, the obstacles of non-analyticity are overcome via the introduction of "relativistic molecular auxiliary functions": $$\mathcal{G}^{n_1, q}_{n_2, n_3}(p_1, p_2, p_3) = \frac{p_1^{n_1}}{\Gamma(n_1+1)} \int_1^\infty \int_{-1}^1 (\xi\nu)^q (\xi+\nu)^{n_2} (\xi-\nu)^{n_3} e^{-p_2 \xi - p_3 \nu} d\xi d\nu.$$ These can be further reduced using recurrences, binomial rearrangements, and incomplete beta functions so that all necessary integrals are ultimately expressed in finite sums and one-dimensional quadratures (Bagci et al., 2018).

Efficient global-adaptive quadrature algorithms and highly stable recurrence relations permit the evaluation of these integrals to 25+ digits of accuracy for all practical ranges of orbital parameters (Bağcı et al., 2014).

4. Computational Strategies and Implementation

Several strategies are available for evaluating NSTO integrals:

• Auxiliary-function recurrence scheme: Recurrences among the auxiliary functions and incomplete gamma or beta functions eliminate numerical instability in regions of parameter space (e.g., small or nearly zero values for noninteger arguments) and avoid direct evaluation of unstable special functions (Bagci et al., 2018, Bağcı et al., 2022).
• Global-adaptive quadrature: For auxiliary functions with unfavorably large parameters or where analytic series are marginally convergent, global-adaptive nested Gauss–Kronrod quadratures achieve uniform precision (Bağcı et al., 2014).
• Zero-variance Monte Carlo (ZVMC): A general approach based on Gaussian approximations and variance reduction by correlated sampling, applicable to arbitrary orbitals, including NSTOs of any exponent. This method yields high-precision results for four-center two-electron integrals and is naturally amenable to parallelization, albeit at the cost of total sample count (Caffarel, 2019).

Performance comparisons indicate that, when optimized, NSTO integral algorithms achieve computational efficiency matching conventional integer-n STO approaches. The main bottleneck for large quantum numbers becomes the evaluation of initial values (hypergeometric or incomplete-beta calls), after which fast recursions dominate (Bagci et al., 2018). For multi-center integrals, the new function families and global-adaptive algorithms vastly outperform traditional one-center expansions or general-purpose numerical cubatures (Bağcı et al., 2014).

5. Applications to Electronic Structure and Relativistic Theory

NSTOs provide enhanced variational efficiency in both non-relativistic and relativistic quantum chemical calculations:

• In Hartree–Fock–Roothaan atomistic expansions, optimized NSTOs yield lower energies than minimal integer-n STO bases, particularly for compact minimal sets. Gains of up to 36 mH have been reported for Ne-like systems in such cases (Bagci et al., 6 Jul 2025).
• In relativistic Dirac–Hartree–Fock settings, the requirement of large–small component pairing and accurate nuclear-cusp behavior are naturally met by NSTOs, which serve as basis elements for four-component spinors. The same analytic and computational infrastructure applies (Bagci, 2016).
• NSTOs are advantageous for heavy-element and highly charged systems, where accurate description of singular and asymptotic regions is essential (Bagci et al., 2018).

It should be noted that the physical interpretation of NSTOs is limited except for the $1s$ hydrogenic case; the principal quantum number loses its direct spectroscopic meaning for noninteger values (Bagci et al., 6 Jul 2025).

6. Basis Set Construction Protocols and Best Practices

When constructing NSTO basis sets for practical calculations, the following protocol is established:

• Fix a single fractional part $\nu$ for all principal quantum numbers in the basis; define $n^* = \nu + k$ for $k$ integer.
• Fix the weight parameter $\alpha$ of the underlying Hilbert space; do not optimize $\alpha$ or $\nu$ independently for individual basis functions, as this leads to loss of orthonormality and possible linear dependence (Bagci et al., 6 Jul 2025).
• Optimize only radial exponents $\zeta$ (and, optionally, a single global $\nu$) for energy minimization.
• Monitor overlap-matrix eigenvalues to avoid near-linear dependence as the basis size increases.
• All segmented protocols (minimal, double-$\zeta$, etc.) extend naturally to NSTOs; maximal energy reductions are observed in minimal bases, with diminishing returns as basis size increases.

Tables of typical basis parameters and results are given in (Bagci et al., 6 Jul 2025), confirming that under these constraints, NSTOs are a robust and variationally efficient alternative to integer-n STOs, fully preserving completeness and linear independence.

7. Connections to Fractional Calculus and Auxiliary Function Theory

Recent theoretical advances connect NSTOs with Riemann-Liouville fractional calculus, both in the construction of orthonormal sets and in the analytical transformation of multivariate auxiliary functions required for molecular integrals (Bağcı et al., 2022). The molecular auxiliary functions themselves are naturally interpreted as fractional integrals over core exponential–polynomial kernels, further integrating the analytical and numerical methodologies for NSTO-based electronic structure calculations.

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References:

For foundational theory and computational methods: (Bagci et al., 2018, Bağcı et al., 2014, Bağcı et al., 2022, Bağcı et al., 2022, Bagci et al., 6 Jul 2025, Bagci, 2016, Caffarel, 2019).