Linear dependence in Hylleraas configuration-interaction calculations of He atom

The explicitly-correlated basis sets are much easier to be linearly dependent than the product type bases constructed by one-electron orbitals due to the explicit inclusion of interelectronic coordinates in system wave functions. In this work, we apply Löwdin's canonical orthogonalization method to study the linearly dependent problems arising from the variational calculations based on Hylleraas configuration-interaction (Hy-CI) basis functions. Both the ground and excited states of He atom are calculated with increasingly large basis sets. Our results show that the linear dependence in Hy-CI basis sets can be successfully overcome by employing Löwdin's canonical orthogonalization method, yet without using extended higher-precision arithmetic in numerical implementations. Therefore, the computational effort can be reduced considerably. It is expected that the present method can be applied to other types of explicitly correlated basis functions.     

P.‐O. Löwdin and the International Journal of Quantum Chemistry: A kaleidoscopic agenda for quantum chemistry

This is a article about P.‐O. Löwdin's life, his work in shaping quantum chemistry into a mature discipline at the intersection of mathematics, physics, chemistry, and biology, and his founding of the International Journal of Quantum Chemistry in 1967. Unavoidably, it is, also, a article reflecting our views about the history of quantum chemistry. We attempt to convey the complexities in the becoming of a subdiscipline, like quantum chemistry, where a variety of factors will have to be taken into consideration for a comprehensive understanding of its historical developments: the relations of chemists to the Heisenberg‐Schrödinger formulation of quantum mechanics after 1926, the institutional dynamics centered around the establishment of new courses and chairs, the research agendas and the vying for dominance within the community of quantum chemists, the methodological, and philosophical issues that have never left the quantum chemists indifferent, and, of course, the dramatic role of the computer in transforming the culture for actually practicing quantum chemistry. Furthermore, attracted by American history, culture, and ways of life, Löwdin suggested in the late 1970s that the post‐WWII character of quantum chemistry was dependent on its ability to hub a “scientific melting pot,” much like the United States of America which he viewed as a fusion of people from diverse provenances and cultures. In this article, we attempt to investigate another metaphor, that of the “kaleidoscope.” Löwdin believed that quantum chemistry's strength arose from its ability to nurture a multiplicity of heterogeneous cultural elements/subcultures and practices, interacting with each other, exchanging perspectives and modes of action, which circulated in an increasingly extended network of actors and institutional frameworks. © 2013 Wiley Periodicals, Inc.     

Matrix‐covariant representation of high‐order configuration interaction and coupled cluster theories

We present the closed form of the reduced density matrices (RDMs) of arbitrary order for configuration interaction (CI) wave functions at any excitation level, up to the full CI. A special operator technique due to Bogoliubov is applied and extended. It focuses on constructions of matrix‐covariant expressions independent of the basis set used. The corresponding variational CI equations are given in an explicit form containing the matrices related to conventional excitation operators. A subsequent transformation of the latter to an irreducible form makes it possible to generate the matrix‐covariant representation for coupled cluster (CC) models. Here this transformation is performed for a simplified high‐order CC scheme somewhat reminiscent of the quadratic CI model. A generalized spin‐flip approximation closely related to high‐order CI and CC models is presented, stressing on a possible inclusion of nondynamical and dynamical correlation effects for multiple bond breaking. A derivation of the full CI and simple CC models for systems involving effective three‐electron interactions is also given, thereby demonstrating the capability of the proposed method to deal with complicated many‐body problem. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Proof of the linear independence of properly selected projected spin eigenfunctions

An inductive proof is given of Löwdin's theorem about the linear independence and completeness of a properly selected set of projected spin eigenfunctions.     

On the formulation of spin-free quantum chemistry

A set of spin-free functions ?_i(r),i = 1 …? f, is obtained which form the basis of spin-free quantum chemistry. The ?_i(r) show a one-to-one correspondence to antisymmetric space-spin functions Ψ_i(r, σ) with spin functions constructed according to Löwdin's projector operator method.     

Charge distributions and multipole moments in molecules

The consideration of multipole moments is suggested as a new criterion for the validity of assignments of atomic charges in molecules. The total quadrupole and octupole moments generated by our definition of atomic charges are compared with the exact moments of the underlying wavefunction for various basis sets in selected diatomics. The analysis includes also total overlap and total dipole moment partitioning as well as 1σ MO overlap partitioning. All considerations together allow us to assess the validity of our charge definition as compared to Mulliken's and Löwdin's and the quality of the basis set.     

Evaluation of integrals over STOs on different centers and the complementary convergence characteristics of ellipsoidal‐coordinate and zeta‐function expansions

One‐electron integrals over three centers and two‐electron integrals over two centers, involving Slater‐type orbitals (STOs), can be evaluated using either an infinite expansion for 1/r₁₂ within an ellipsoidal‐coordinate system or by employing a one‐center expansion in spherical‐harmonic and zeta‐function products. It is shown that the convergence characteristics of both methods are complimentary and that they must both be used if STOs are to be used as basis functions in ab initio calculations. To date, reports dealing with STO integration strategies have dealt exclusively with one method or the other. While the ellipsoidal method is faster, it does not always converge to a satisfactory degree of precision. The zeta‐function method, however, offers reliability at the expense of speed. Both procedures are described and the results of some sample calculation presented. Possible applications for the procedures are also discussed. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 1–13, 1999     

Upper and lower bounds and the generalized variation-perturbation approach of many-electron theory

The theory of upper and lower bounds is used in conjunction with the “varied-portions approach” (VPA ) of the “many-electron theory of atoms and molecules” (MET ). Upper and lower bounds equations are derived. It is recognized that if a very natural portion of the full variational expression is made stationary, according to (VPA ), a for of Löwdin's lower bound wave equation results. By using the bracketing theorem it is also possible to give lower bounds to the quantities arising from the (VPA ) of (MET ). This will be useful to study in the future the (MET ) pair correlations and to supplement the computational methods currently available with lower bound estimates as well.     

Comment on “New methods for old Coulomb few‐body problems”

In this Comment on the above‐mentioned paper by F. E. Harris, A. M. Frolov, and V. H. Smith, we briefly review our contributions to development of new methods for solution of the Coulomb four‐body problem. We show that our research group, headed by Professor T. K. Rebane, had a priority in using the fully correlated exponential basis for variational calculations of four‐body systems. We also draw attention to the fact that our group subsequently implemented a more advanced method, which uses highly efficient exponential‐trigonometric basis functions for solution of the same problem. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Development of spin‐dependent relativistic open‐shell Hartree–Fock theory with time‐reversal symmetry (II): The restricted open‐shell approach

An open‐shell Hartree–Fock (HF) theory for spin‐dependent two‐component relativistic calculations, termed the Kramers‐restricted open‐shell HF (KROHF) method, is developed. The present KROHF method is defined as a relativistic analogue of ROHF using time‐reversal symmetry and quaternion algebra, based on the Kramers‐unrestricted HF (KUHF) theory reported in our previous study (Int. J. Quantum Chem., doi: 10.1002/qua.25356 ). As seen in the nonrelativistic ROHF theory, the ambiguity of the KROHF Fock operator gives physically meaningless spinor energies. To avoid this problem, the canonical parametrization of KROHF to satisfy Koopmans' theorem is also discussed based on the procedure proposed by Plakhutin et al. (J. Chem. Phys. 2006 , 125, 204110). Numerical assessments confirmed that KROHF using Plakhutin's canonicalization procedure correctly gives physical spinor energies within the frozen‐orbital approximation under spin–orbit interactions.     

Master formulas for two‐ and three‐center one‐electron integrals involving Cartesian GTO,STO, and BTO

As a first application of the shift operators method we derive master formulas for the two‐ and three‐center one‐electron integrals involving Gaussians, Slater, and Bessel basis functions. All these formulas have a common structure consisting in linear combinations of polynomials of differences of nuclear coordinates. Whereas the polynomials are independent of the type (GTO, BTO, or STO) of basis functions, the coefficients depend on both the class of integral (overlap, kinetic energy, nuclear attraction) and the type of basis functions. We present the general expression of polynomials and coefficients as well as the recurrence relations for both the polynomials and the whole integrals. Finally, we remark on the formal and computational advantages of this approach. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 83–93, 2000     

Mayer’s orthogonalization: relation to the Gram-Schmidt and Löwdin’s symmetrical scheme

A method introduced by Mayer (Theor Chem Acc 104:163, 2000) for generating an orthogonal set of basis vectors, perpendicular to an arbitrary start vector, is examined. The procedure provides the complementary vectors in closed form, expressed with the components of the start vector. Mayer’s method belongs to the family of orthogonalization schemes, which keep an arbitrary vector intact without introducing any non-physical sequence-dependence. It is shown that Mayer’s orthogonalization is recovered by performing a two-step combination of the Gram-Schmidt and Löwdin’s symmetrical orthogonalization. Processor time requirement of constructing Mayer’s orthonormal set is proportional to ～N ², in contrast to the rough ～N ³ CPU requirement of performing either a full Gram-Schmidt or Löwdin’s symmetrical orthogonalization. Utility of Mayer’s orthogonalization is demonstrated on an electronic structure application using perturbation theory to improve multiconfigurational wavefunctions.     

Discrete sets of many‐body Sturmians

In this work, we present a method to obtain two‐particle Coulomb Sturmians Functions (CSF) with an expansion in a set of L² basis functions. In the two‐body case, we recover the exact (discrete) spectrum of the CSFs for negative energies and a discretized approximation for positive ones. Besides, we make use of this method to analyze the two‐independent electron problem as a Generalized Sturmian problem. We propose a discretized version of the wave function in terms of the CSF states, and show that the problem reduces to find numerical coincidences between energy‐dependent eigencharges of the mutually independent one‐electron systems. This expansion methodology includes the continuum information which is lost in the sets used previously in the literature, and is complete when the size of the basis goes to infinity. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

An open‐source framework for analyzing N‐electron dynamics. II. Hybrid density functional theory/configuration interaction methodology

In this contribution, we extend our framework for analyzing and visualizing correlated many‐electron dynamics to non‐variational, highly scalable electronic structure method. Specifically, an explicitly time‐dependent electronic wave packet is written as a linear combination of N‐electron wave functions at the configuration interaction singles (CIS) level, which are obtained from a reference time‐dependent density functional theory (TDDFT) calculation. The procedure is implemented in the open‐source Python program det CI@ORBKIT, which extends the capabilities of our recently published post‐processing toolbox (Hermann et al., J. Comput. Chem. 2016, 37, 1511). From the output of standard quantum chemistry packages using atom‐centered Gaussian‐type basis functions, the framework exploits the multideterminental structure of the hybrid TDDFT/CIS wave packet to compute fundamental one‐electron quantities such as difference electronic densities, transient electronic flux densities, and transition dipole moments. The hybrid scheme is benchmarked against wave function data for the laser‐driven state selective excitation in LiH. It is shown that all features of the electron dynamics are in good quantitative agreement with the higher‐level method provided a judicious choice of functional is made. Broadband excitation of a medium‐sized organic chromophore further demonstrates the scalability of the method. In addition, the time‐dependent flux densities unravel the mechanistic details of the simulated charge migration process at a glance. © 2017 Wiley Periodicals, Inc.     

Analytic evaluation of two‐center STO electron repulsion integrals via ellipsoidal expansion

Extant analytic methods for evaluating two‐center electron repulsion integrals in a Slater‐type orbital (STO) basis using ellipsoidal coordinates and the Neumann expansion of 1/r₁₂ have problems of numerical stability that are analyzed in detail using computer‐assisted algebraic techniques. Some of these problems can be eliminated by use of procedures known in this field 40 years ago but seemingly forgotten now. Others can be removed by use of a formulation suitable for small values of the STO screening parameter. A recent attempt at such a formulation is corrected and extended in a way permitting its practical use. The main functions encountered in the integrations over the ellipsoidal coordinate of the range 1 … ∞ are Bessel functions or generalizations thereof, as pointed out here for the first time. This fact is used to motivate the derivation of recurrence relations additional to those previously known. Novel techniques were devised for using these recurrence relations, thereby providing new ways of calculating the quantities that enter the ellipsoidal expansion. The convergence rate of this expansion and the numerical characteristics of several computational strategies are reported in enough detail to identify the ranges where various schemes can be used. This information shows that recent discussions of the “convergence characteristics of [the] ellipsoidal coordinate expansion” are in fact not that, but are instead discussions of an inability to make accurate calculations of the individual terms of the expansion. It is also seen that the parameter range suitable for use of Kotani's well‐known recursive scheme is more limited than seems generally believed. The procedures discussed in this work are capable of yielding accurate two‐center electron repulsion integrals by the ellipsoidal expansion method for all reasonable STO screening parameters, and have been implemented in illustrative public‐domain computer programs. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

An open‐source framework for analyzing N‐electron dynamics. I. Multideterminantal wave functions

The aim of the present contribution is to provide a framework for analyzing and visualizing the correlated many‐electron dynamics of molecular systems, where an explicitly time‐dependent electronic wave packet is represented as a linear combination of N‐electron wave functions. The central quantity of interest is the electronic flux density, which contains all information about the transient electronic density, the associated phase, and their temporal evolution. It is computed from the associated one‐electron operator by reducing the multideterminantal, many‐electron wave packet using the Slater‐Condon rules. Here, we introduce a general tool for post‐processing multideterminant configuration‐interaction wave functions obtained at various levels of theory. It is tailored to extract directly the data from the output of standard quantum chemistry packages using atom‐centered Gaussian‐type basis functions. The procedure is implemented in the open‐source Python program det CI@ORBKIT, which shares and builds on the modular design of our recently published post‐processing toolbox (Hermann et al., J. Comput. Chem. 2016, 37, 1511). The new procedure is applied to ultrafast charge migration processes in different molecular systems, demonstrating its broad applicability. Convergence of the N‐electron dynamics with respect to the electronic structure theory level and basis set size is investigated. This provides an assessment of the robustness of qualitative and quantitative statements that can be made concerning dynamical features observed in charge migration simulations. © 2017 Wiley Periodicals, Inc.     

A mathematical discussion of Pons Viver's implementation of Löwdin's spin projection operator

Recently, the molecular electronic structure theories for efficiently treating static (or strong) correlation in a black-box manner have attracted much attention. In these theories, a spin projection operator is used to recover the spin symmetry of a broken-symmetry Slater determinant. Very recently, Pons Viver proposed the practical and exact implementation of Löwdin's spin projection operator (Int. J. Quantum Chem. 2019, 119, e25770). In the present study, we attempt to supply mathematical proofs to Pons Viver's proposals and show a condition for establishing Pons Viver's implementation. Moreover, we explicitly derive the (spin projected) extended Hartree-Fock (EHF) equations on the basis of the model of common orbitals (ie, closed-shell orbitals used in the restricted open-shell Hartree-Fock (ROHF) method), which was combined by Pons Viver with the EHF method.