Atomic Functions: Quantum & Materials Models

• Atomic functions are mathematical constructs that define wave functions, densities, and basis sets used for precise quantum and materials modeling.
• They enable efficient variational calculations, node-less approaches, and optimized auxiliary functions for improved numerical and computational performance.
• They integrate into machine learning frameworks, enhancing physical accuracy and efficiency in modeling atomic-scale phenomena.

Atomic functions refer to a range of mathematical constructs directly associated with the description, analysis, and computation of electronic, structural, and response properties of atoms within quantum mechanical and condensed matter frameworks. This includes wave functions, densities, basis functions, and auxiliary constructs used for computation, modeling, and data analysis in atomic-scale systems. The technical implementation and theoretical roles of atomic functions are diverse, from representing physical wavefunctions and densities to serving as optimized elements in basis set expansions, pseudopotential schemes, and message-passing architectures for materials modeling.

1. Atomic Wave Functions and Coalescence Behavior

Atomic wave functions $\Psi(\mathbf{x})$—particularly those that arise as solutions to the Schrödinger or Dirac-Coulomb equations—encode the full quantum information of electrons in atoms. Their local behavior, especially at coalescence points (e.g., electron-nucleus or electron-electron contact), is governed by universal short-distance physics. This is made precise using the operator product expansion (OPE) in nonrelativistic effective field theory (EFT). Huang, Jia, and Yu provided an exact all-orders OPE for the product of electron and nucleus field operators in the Coulomb–Schrödinger theory:

$$\psi(\mathbf{x})\,N(\mathbf{0}) = \left[1 - mZ\alpha|\mathbf{x}|\right][\psi N](\mathbf{0}) + \left[1 - \tfrac12 mZ\alpha|\mathbf{x}|\right]\,\mathbf{x}\cdot[\nabla\psi N](\mathbf{0}) + \ldots$$

This yields, upon taking the expectation in a bound state, the Kato cusp condition for $S$-wave states:

$$\left.\frac{\partial\Psi}{\partial r}\right|_{r\to0} = -\,mZ\alpha\,\Psi(0)$$

The OPE formalism generalizes to multi-particle coalescence, providing a systematic framework for the universal features of atomic wavefunctions at short distances. This approach both explains the universality of the cusp condition and highlights the breakdown of such universality at higher orders in $|\mathbf{x}|/a_0$ (Huang et al., 2018).

2. Atomic Basis Functions in Wavefunction Expansions

In variational and electronic structure calculations, atomic functions most often refer to the basis elements used to expand the multi-electron wavefunction. For few-electron atoms, fully correlated semi-exponential or gaussoidal basis functions depending on all interparticle distances offer extreme compactness and accuracy. For a three-electron atom, basis functions can be written as

$$\phi_k(\{r_{ij}\}) = r_{12}^{n_1^{(k)}}\,r_{13}^{n_2^{(k)}}\,r_{23}^{n_3^{(k)}}\,r_{14}^{m_1^{(k)}}\,r_{24}^{m_2^{(k)}}\,r_{34}^{m_3^{(k)}} \exp[-a_k r_{14} - b_k r_{24} - c_k r_{34}]$$

with each $(a_k, b_k, c_k)$ freely optimized, affording basis flexibility not present in uniform-exponent Hylleraas expansions. As a result, highly accurate energies can be obtained with $N\approx 40-60$ terms, where traditional expansions require thousands. This framework is systematically extendable to systems with more electrons and other bound-state properties (Frolov, 2010, Frolov et al., 2010).

3. Node-Less and Node-Reduced Atomic Functions

Node-less atomic wave functions (also termed "node-reduced") are constructed to yield the full information of atomic eigenstates while eliminating the radial nodes of higher principal quantum number states. Given the radial Schrödinger equation, a hierarchy of node-less functions $u_n(r)$ is obtained via:

$(H - E_n) u_n(r) = -u_{n-1}(r),$

with $$\psi(\mathbf{x})\,N(\mathbf{0}) = \left[1 - mZ\alpha|\mathbf{x}|\right][\psi N](\mathbf{0}) + \left[1 - \tfrac12 mZ\alpha|\mathbf{x}|\right]\,\mathbf{x}\cdot[\nabla\psi N](\mathbf{0}) + \ldots$$0 (ground state).

These functions encode the complete scattering properties and phase shifts of the full set of atomic states; the physical interpretation is that the node-less function “absorbs” the effects of Pauli repulsion by the core. By inverting the node-less Schrödinger equation, one obtains a semi-local effective potential $$\psi(\mathbf{x})\,N(\mathbf{0}) = \left[1 - mZ\alpha|\mathbf{x}|\right][\psi N](\mathbf{0}) + \left[1 - \tfrac12 mZ\alpha|\mathbf{x}|\right]\,\mathbf{x}\cdot[\nabla\psi N](\mathbf{0}) + \ldots$$1 that transparently represents core-exclusion effects, suitable for direct use in pseudopotentials, projector-augmented wave (PAW), and embedding calculations. The node-less hierarchy is mathematically complete and connected to Slater-type orbitals by their asymptotic and near-origin behaviors (Blöchl et al., 2012).

4. Physically Optimized Atomic Basis Functions in Machine Learning

Recent developments integrate explicit atomic physics into graph-based machine learning models for materials. The LCAONet architecture employs hydrogen-like atomic functions

$$\psi(\mathbf{x})\,N(\mathbf{0}) = \left[1 - mZ\alpha|\mathbf{x}|\right][\psi N](\mathbf{0}) + \left[1 - \tfrac12 mZ\alpha|\mathbf{x}|\right]\,\mathbf{x}\cdot[\nabla\psi N](\mathbf{0}) + \ldots$$2

where $$\psi(\mathbf{x})\,N(\mathbf{0}) = \left[1 - mZ\alpha|\mathbf{x}|\right][\psi N](\mathbf{0}) + \left[1 - \tfrac12 mZ\alpha|\mathbf{x}|\right]\,\mathbf{x}\cdot[\nabla\psi N](\mathbf{0}) + \ldots$$3 are hydrogenic radial functions (with true nuclear charge, no adjustable effective charge), and $$\psi(\mathbf{x})\,N(\mathbf{0}) = \left[1 - mZ\alpha|\mathbf{x}|\right][\psi N](\mathbf{0}) + \left[1 - \tfrac12 mZ\alpha|\mathbf{x}|\right]\,\mathbf{x}\cdot[\nabla\psi N](\mathbf{0}) + \ldots$$4 are selected to match the atom's electron configuration. These functions are pruned by element and enter as the physical “channels” in the message-passing layers of three-body GNNs, yielding substantial gains in prediction accuracy and parameter efficiency on chemically diverse datasets. Ablation studies confirm the superiority of Coulomb-shaped radial forms and occupation-based pruning over geometric or Bessel-based embeddings (Nishio et al., 2024).

5. Atomic Density Functions and Derived Measures

Atomic functions in density-based frameworks refer to the one-electron spin-summed density:

$$\psi(\mathbf{x})\,N(\mathbf{0}) = \left[1 - mZ\alpha|\mathbf{x}|\right][\psi N](\mathbf{0}) + \left[1 - \tfrac12 mZ\alpha|\mathbf{x}|\right]\,\mathbf{x}\cdot[\nabla\psi N](\mathbf{0}) + \ldots$$5

In the multi-configuration Hartree–Fock (MCHF) framework, the density can have non-spherical components in open-shell systems, expanded via spherical harmonics. Spherical averaging is achieved by averaging over all $$\psi(\mathbf{x})\,N(\mathbf{0}) = \left[1 - mZ\alpha|\mathbf{x}|\right][\psi N](\mathbf{0}) + \left[1 - \tfrac12 mZ\alpha|\mathbf{x}|\right]\,\mathbf{x}\cdot[\nabla\psi N](\mathbf{0}) + \ldots$$6, $$\psi(\mathbf{x})\,N(\mathbf{0}) = \left[1 - mZ\alpha|\mathbf{x}|\right][\psi N](\mathbf{0}) + \left[1 - \tfrac12 mZ\alpha|\mathbf{x}|\right]\,\mathbf{x}\cdot[\nabla\psi N](\mathbf{0}) + \ldots$$7. Analytical manipulation leads to the radial distribution $$\psi(\mathbf{x})\,N(\mathbf{0}) = \left[1 - mZ\alpha|\mathbf{x}|\right][\psi N](\mathbf{0}) + \left[1 - \tfrac12 mZ\alpha|\mathbf{x}|\right]\,\mathbf{x}\cdot[\nabla\psi N](\mathbf{0}) + \ldots$$8 and cumulative electron number.

Similarity analysis and information-theoretic measures (e.g., Kullback–Leibler divergence with respect to a noble-gas prior) applied to atomic densities reveal shell structure, electron correlation effects, relativistic contraction, and periodicity in the periodic table. The quantum similarity index and information-overlap measures quantify these trends across elements (Borgoo et al., 2011).

6. Variationally and Algorithmically Optimized Atomic Auxiliary Functions

Atomic functions also denote auxiliary basis functions optimized for numerical purposes—such as density-fitting (DF) or resolution-of-identity (RI) approximations in molecular calculations. These are generally-contracted Gaussians

$$\psi(\mathbf{x})\,N(\mathbf{0}) = \left[1 - mZ\alpha|\mathbf{x}|\right][\psi N](\mathbf{0}) + \left[1 - \tfrac12 mZ\alpha|\mathbf{x}|\right]\,\mathbf{x}\cdot[\nabla\psi N](\mathbf{0}) + \ldots$$9

$S$0

constructed to minimize weighted errors in the Coulomb metric for one-electron product densities. The error functional includes dynamical (correlation) and exchange weights derived from orbital energies. Optimization procedures involve diagonalization in the Coulomb metric and line search or gradient-based tuning of exponents. The compactness and numerical stability of these auxiliary functions are validated across all elements and are essential for maintaining accuracy and tractability in large-scale correlated calculations (Laikov, 2020).

7. Atomic Pair Functions and Structural Analysis

Atomic functions in the context of condensed-matter and diffraction analyses are present in the definition and computation of atomic pair distribution functions (PDFs). The real-space atomic PDF

$S$1

is computed from the reduced structure function $S$2 and reflects the probability of finding a pair of atoms at separation $S$3 relative to random distribution. In textured polycrystalline samples, the atomic pair distributions are averaged over the orientation distribution function (ODF), $S$4, encoding all crystallite misorientations. The determination of $S$5 from scattering data is critical for texture and local structure analysis in polycrystalline and nanostructured materials (Gong et al., 2018).

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References

• (Huang et al., 2018) Deciphering the coalescence behavior of Coulomb-Schrödinger atomic wave functions from an operator product expansion.
• (Frolov, 2010) Compact and accurate variational wave functions of three-electron atomic systems constructed from semi-exponential radial basis functions.
• (Frolov et al., 2010) Compact variational wave functions for bound states in three-electron atomic systems.
• (Blöchl et al., 2012) Node-less atomic wave functions, Pauli repulsion and systematic projector augmentation.
• (Nishio et al., 2024) LCAONet: Message-passing with physically optimized atomic basis functions.
• (Borgoo et al., 2011) Atomic density functions: atomic physics calculations analyzed with methods from quantum chemistry.
• (Laikov, 2020) Optimization of atomic density-fitting basis functions for molecular two-electron integral approximations.
• (Gong et al., 2018) Atomic pair distribution functions (PDFs) from textured polycrystalline samples: fundamentals.