AICCM

ab initio Cyclic Cluster Model

The Cyclic Cluster Model (CCM) applies periodic boundary conditions directly to a finite cluster that corresponds to a supercell of the solid, surface, or polymer. Unlike the conventional supercell model (SCM), the CCM is a Γ-point approach: integration is carried out in real space within a finite interaction area, and no summation over special k-points is required. The local environment of every atom is replaced by a notional cyclic arrangement of the cluster atoms, closing the cluster into a ring (1D), torus (2D), or hypertorus (3D). The interaction range of every atom is defined by its Wigner–Seitz supercell (WSSC), constructed from the translation vectors of the unit cell and centered at that atom.

Context and Motivation

The CCM occupies a unique position in the landscape of quantum-chemical models for periodic systems. The standard approaches can be compared across two axes: whether they retain a molecular reference (and thus give access to the full hierarchy of quantum-chemical methods) and whether they enforce translational symmetry (avoiding boundary effects). The CCM is the only model that achieves both.

Comparison of four quantum-chemical models for solids: Free Cluster, Embedded Cluster, Periodic, and Cyclic Cluster
Models for the quantum-chemical treatment of solids. The Free Cluster Model and Embedded Cluster Model retain a molecular reference but lack translational symmetry and suffer from boundary effects. The conventional Periodic Model (supercell) enforces translational symmetry but has no molecular reference and therefore only limited access to quantum-chemical methods. The Cyclic Cluster Model combines a molecular reference with periodic boundary conditions and no boundary effects. From the 2013 talk presented at the Mulliken Center for Theoretical Chemistry, University of Bonn.

The scientific motivation for the AICCM work arose within DFG Collaborative Research Area SFB 813 “Chemistry at Spin Centers,” specifically Project C5 on spin centers in molecular solids. The project aimed to investigate stacked arrangements of organic radical cations that may form organic semiconductors with magnetic properties alongside conductivity. Predicting stacking behavior and spin-center interactions from first principles for such molecular crystals requires methods that combine advanced quantum chemistry with periodic boundary conditions, which the CCM is designed to provide.

Because the WSSC of every atom is stoichiometric and ensures electroneutrality, the CCM avoids the divergence of Coulomb lattice sums that plagues the SCM when electron–nuclear attraction and electron–electron repulsion are treated separately. The close similarity between the CCM formalism and molecular quantum chemistry means that sophisticated post-Hartree–Fock methods can in principle be applied to solid-state problems with only moderate modification of existing molecular codes.

Historical Development

The basic ideas of the CCM date back more than 40 years. The name “cyclic cluster model” was first used in a study of the electronic structure of α-quartz in 1982. Implementations at the semiempirical level proliferated through the 1980s and 1990s, and a semiempirical CI implementation (MSINDO-CIS) followed later. Attempts at density functional theory implementations were made within the local density approximation and subsequently within the generalized gradient approximation, but all non-semiempirical calculations encountered problems. The only quantum-chemical program employing periodic boundary conditions via the CCM that remained available prior to our work was the semiempirical code MSINDO. A general ab initio CCM formalism was missing until the work described below.

Chronological timeline of cyclic cluster model implementations from 1970 to 2010
Figure 1. Chronological overview of cyclic cluster model implementations from the first semiempirical codes in 1970 to the DFT implementations around 2010. Reproduced from Peintinger & Bredow, J. Comput. Chem. 2014.

Theory: Cyclic Boundary Conditions and Weighting Schemes

A cluster is chosen as a supercell of the primitive or conventional unit cell, repeated N1 × N2 × N3 times along the lattice vectors. The cyclic Born–von-Kármán boundary conditions are applied directly to this cluster. At the Γ-point, the periodic Bloch orbitals reduce to atomic orbitals, making the CCM basis directly comparable to a molecular one.

The central challenge at ab initio level is the correct treatment of three- and four-center electron repulsion integrals (ERIs) involving basis functions at the borders of WSSCs. Atoms exactly on a WSSC border appear more than once within the interaction range and must be weighted to avoid double-counting. The AICCM introduces new weighting schemes for three- and four-center interactions based on the union of WSSCs, which extends the interaction range consistently while avoiding unphysical contributions.

One-dimensional MNOP cluster showing real and virtual atoms with their atomic orbitals
Figure 2. One-dimensional cluster of a fictive MNOP system. Real atoms M, N, O, P with their atomic orbitals are surrounded by virtual atoms M−, M+, … created by the translation vectors. Interactions are restricted to within each atom’s Wigner–Seitz supercell, with translationally equivalent atoms replacing distant partners. Reproduced from Peintinger & Bredow, J. Comput. Chem. 2014.

Three-center interactions

The nuclear attraction energy involves interactions among up to three centers (two basis function centers and one nuclear center). The AICCM uses the union of the WSSCs of the two basis function centers, WSSC(MN) = WSSC(M) ∪ WSSC(N), as the reference for selecting which nuclear centers C are included. The weighting factor for the integral IμνC is the product of the two-center weight ωμν and the average of the two-center weights ωμC and ωνC. This extends the interaction range from ±½t to ±t, ensuring a consistent treatment.

Wigner-Seitz supercells for three-center nuclear attraction integrals
Figure 3. The interaction of the orbital product μρ− with nuclear centers C that must be included for the nuclear attraction integral Iμρ−C. The union of WSSCs of the two basis function centers defines the interaction region. Reproduced from Peintinger & Bredow, J. Comput. Chem. 2014.

Four-center interactions

The classical Coulomb term and the nonclassical exchange term involve electron repulsion integrals (ERIs) over up to four centers. The four-center weighting scheme is derived analogously from the two-center weights. For the integral (μν|ρσ), the total weight is the product of the two-center weight ωμν, an averaged weight over the cross-pairs ωμρ and ωνρ, and the two-center weight ωρσ. This requires translating the cluster twice in every direction (±2t) to cover the maximum interaction range of ±¾t.

Wigner-Seitz supercells for four-center electron repulsion integrals
Figure 4. WSSCs for orbital products involved in four-center interactions of the reference orbital μ. The union of WSSCs of the first electron pair defines which third-center atoms are included, and their WSSCs define the fourth center. Reproduced from Peintinger & Bredow, J. Comput. Chem. 2014.

Results: Convergence and Validation

The AICCM implementation was validated against the crystalline orbital program CRYSTAL09 using reference calculations on one-dimensional hydrogen chains. As the cluster size is increased (equivalently, as the number of k-points in the Γ-point approach grows), the AICCM total energy per atom converges to the SCM reference value. For a unit cell containing two hydrogen pairs with alternating H–H distances of 0.8 and 1.2 Å and an STO-3G basis set, convergence to within 10−6 Hartree per atom is achieved with a cluster of 40 atoms (10 unit cells). The correspondence between integral tolerances and k-points in CRYSTAL and cluster size in the AICCM is demonstrated to be quantitatively consistent.

One-dimensional H4 cluster with alternating H-H distances
Figure 5. One-dimensional H4 cluster containing two hydrogen pairs with alternating H–H distances of 0.8 and 1.2 Å. This model system was used to validate the AICCM implementation by comparing total energies to CRYSTAL09 reference calculations. Reproduced from Peintinger & Bredow, J. Comput. Chem. 2014.

A second test system, the equidistant cyclic H6 cluster with H–H distances of 1.0 Å, confirms that the crystalline orbital coefficients exhibit perfect symmetry degeneracy, as expected from the analytical form of the Bloch orbitals at the implicit Γ-point k-vectors.

Overlap matrix and linear dependence

A known limitation of the Γ-point approximation is that for small clusters or diffuse basis sets the overlap matrix can become indefinite (develop negative eigenvalues), making the calculation unphysical. This is analogous to linear-dependence problems in the SCM with diffuse functions. The critical eigenvalues of the overlap matrix must be monitored and screened before running CCM calculations; canonical orthogonalization or an increase in cluster size resolves the issue.

Critical eigenvalues of the CCM overlap matrix as a function of H-H distance and cluster size
Figure 6. Critical eigenvalues of the CCM overlap matrix as a function of interatomic distance and cluster size for equidistant hydrogen chains. For small clusters or short interatomic distances the matrix becomes indefinite, requiring screening or a larger cluster. Reproduced from Peintinger & Bredow, J. Comput. Chem. 2014.

The AICCM Program

The AICCM code is an object-oriented, educational quantum-chemical program written in Python and Cython with C/C++ extensions. It is designed as a native calculator for the Atomic Simulation Environment (ASE) and makes use of NumPy, SciPy, and LAPACK routines. The program supports DFTB, DFTB-SCC, restricted and unrestricted Hartree–Fock, and MP2 calculations. Atomic orbitals are expanded in contracted Gaussian basis functions. Convergence accelerators (levelshift, Fock matrix mixing, DIIS) are implemented, as are Mulliken and Löwdin population analyses. Molecular and crystalline orbitals can be visualized with Gabedit. Electron repulsion integrals over Gaussians are computed via the ERI library libint.

The program was developed by Michael F. Peintinger and Thorsten Claff at the Mulliken Center for Theoretical Chemistry, University of Bonn. It served both as the research vehicle for the AICCM publication and as an educational tool for demonstrating how periodic boundary conditions can be incorporated into a molecular quantum-chemical framework.

The formalism is fully general and applies in principle to 1D, 2D, and 3D periodic systems. The 1D results shown above cover nanotubes and nanowires as representative applications. Extension to higher dimensions depends on implementation efficiency. The treatment of charged defects is a particularly attractive application: because the CCM uses a finite cluster with no compensating jellium, charged systems present no fundamental problem, in contrast to the SCM where complicated finite-size correction schemes are required.

The concepts and implementation developed in the AICCM project later informed the design of vibe-qc, an open-source quantum chemistry code developed via AI-assisted vibe-coding that includes a reimplementation of the periodic cyclic cluster model.

Invited Talk: University of Bonn, February 2013

The work was presented as an invited talk at the Mulliken Center for Theoretical Chemistry, University of Bonn, on February 1, 2013, under the title “Quantum Chemistry of Periodic Systems: The Cyclic Cluster Model at ab initio Level.” The 43-slide presentation covers the full theoretical framework from Bloch’s theorem and the Wigner–Seitz cell through the construction of cyclic clusters, the multi-center weighting schemes, and the AICCM implementation, concluding with validation results.

Title slide: Quantum Chemistry of Periodic Systems, The Cyclic Cluster Model at ab initio Level, Michael F. Peintinger, Mulliken Center, University of Bonn, 2013
Title slide from the February 2013 talk at the Mulliken Center for Theoretical Chemistry, University of Bonn.

Selected slides illustrating the key concepts are shown below.

Timeline of CCM implementations from 1970 to 2010
A brief history of the CCM. Implementations from the first semiempirical CNDO codes in 1971 through the DFT-GGA implementation in 2008. As of 2013, the only code employing periodic boundary conditions via the CCM remained the semiempirical program MSINDO.
Construction of the Wigner-Seitz cell in a 2D lattice
Wigner–Seitz cell construction. The WSC is defined as the region of space closer to the origin than to any other lattice point, constructed by drawing perpendicular bisectors between the reference point and its neighbors. It is unique, primitive, and reflects the lattice symmetry.
Construction of cyclic cluster showing MNOP unit cell, virtual atoms, and closure into a ring
Construction of cyclic clusters. Starting from the MNOP reference unit cell, virtual atoms are generated by translations ±t. Each real atom’s interaction range is bounded by its Wigner–Seitz supercell (WSC, half the translation vector). Enforcing the cyclic Born–von-Kármán boundary conditions closes the chain into a ring, replacing distant real-crystal neighbors with the corresponding virtual atoms.
Correspondence between k-point sampling in reciprocal space and cyclic cluster size in real space
k-points versus cyclic cluster size. The conventional unit cell (CUC) in real space corresponds to the irreducible Brillouin zone (IBZ) in reciprocal space. A 3×3 Monkhorst–Pack net in the IBZ corresponds to a 3×3 Wigner–Seitz supercell in the CCM main region. Increasing cluster size implicitly increases the number of k-points sampled.
Three-center nuclear attraction integral weighting using the union of Wigner-Seitz supercells
Three-center nuclear attraction integrals. For the integral (μν|C), the nuclear center C is selected from the union of the WSSCs of the two basis function centers. Three-center interactions require one full translation of the cluster in each direction. The weighting factors are averaged over all relevant two-center weights.
Four-center electron repulsion integral weighting using sequential WSSC unions for electron 1 and electron 2
Four-center electron repulsion integrals. For the ERI (μν|ρσ), the first electron’s basis functions define a WSSC union from which the second electron’s interaction range is drawn. Four-center interactions require two full translations in each direction (±2t). The weighting factor is a product of an averaged factor and two two-center weights.
Comparison of highest occupied crystalline orbital of H8 cyclic cluster versus highest occupied molecular orbital of H8 free cluster
Crystalline versus molecular orbitals. The highest occupied crystalline orbital of the H8 cyclic cluster (top) shows an alternating phase pattern consistent with the periodic boundary conditions and the implicit Γ-point k-vector, while the highest occupied molecular orbital of the free H8 cluster (bottom) reflects purely molecular bonding without translational symmetry.

[Download full slide deck (PDF, 43 slides)]

Key Publication

Journal of Computational Chemistry cover, Vol. 35, 2014 The Cyclic Cluster Model at Hartree–Fock Level
M. F. Peintinger, T. Bredow
Journal of Computational Chemistry, 2014, 35, 839–846.
DOI: 10.1002/jcc.23550

Featured on the cover of J. Comput. Chem., Vol. 35, Issues 11–12, 2014.
[Download PDF]