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5: Diagram demonstrating the main class inheritance relationships between classes defined in the DSL. Each box represents a class, named above the dashed line. Below the dashed line, are attributes and methods associated with each class, presented in the format attribute : type = default value and method(arg1,arg2): return type. For clarity, only a selection of methods and attributes for each class are included, and Python default methods are only shown where they have been overloaded for that particular class. Orange arrows indicate inheritance, with the arrow pointing from child class to parent class.
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The evaluation of molecular integrals is a vital but computationally expensive part of electronic structure calculations. This computational expense is particularly problematic for the explicitly correlated methods, in which complicated and numerous integrals over more than two-electrons must be evaluated. The successful R12/F12 methods overcome th...
Citations
... The Coulomb (exchange) interaction for few-center few-electron systems has been a long standing difficulty due to the divergence issue and the occurrence of the nuclear separation. Many notable analytical and numerical approaches were accomplished in the evaluation of 2c-2e integrals for STOs, GTOs, B-spline, Sturmian functions etc. [1][2][3][4][5][6][7][8][9] In 1927, Burrau set forth the concept of confocal elliptic coordinates being intrinsically suited for the 2c integrals with the orbital centers as foci using molecular H 2 + ion. [10] Sugiura pioneered the 2c Coulomb integral using the Neumann expansion in the elliptic coordinates incurring the first order perturbation treatment of H 2 molecule by Heitler and London with molecular wave functions as the linear combination of the product of atomic orbitals of electrons at different nuclei. ...
... For α a = α b , substituting α ab = α a and a 0 ab ¼ 0 [eq. (34)], (3)(4)(5)(6)(7)(8) For α a ¼ 6 α b , (3)(4)(5)(6)(7)(8)(9) with ...
Energetics of two‐center two‐electron (2c–2e) systems carry challenges in theoretical understanding of Schrödinger equation (SE) for well‐known divergence of Coulomb interactions and nuclear separation (R) in modified H‐like AOs, Slater type orbitals (STOs), Gaussian type orbitals (GTOs), B‐spline, Sturmian function and etc. employed to VBT and MOT. Certain elegant computational and analytical techniques were developed for STO, GTO and other square integrable basis set within Born‐Oppenheimer (BO) approximation. STOs and GTOs have an essential limitation of absence of radial nodes. Thus, analytical treatment has become an urge for H‐like AOs. We have considered the diatomic molecules only for the sake of simplicity. Employing Sheffer identity in associated Laguerre polynomial/Whittaker‐M function forms of H‐like AOs and transforming integrals into elliptic coordinates with two nuclei on two foci furnishes exact, analytical and simple Coulomb integrals (Js) in terms of R. Lah number originated from Ln-1 ${L_n^{ - 1} }$ for nuclear coordinates only due to Sheffer identity shows that energetics of diatomic molecules can be anticipated as extremum function of R. Therefore, the optimization of potential energy surface (PES) of electrons as gradient of R may lead to σ‐bond formation. In this paper, we have developed diagonal Js for bound states of H2 molecule.
... Another benefit of code generation is its ability to cover the feature space of an integral engine efficiently, such as support for integrals and their derivatives over dozens of quantum operators utilized in electronic structure, integrals over Gaussian spinors, and so on. 64 The level of sophistication of integral code generation 40,42,46,65 can range from ad hoc tools to embedded or standalone DSLs equipped with custom compilers (e.g., for Libint 61 ). ...
To improve the efficiency of Gaussian integral evaluation on modern accelerated architectures FLOP-efficient Obara-Saika-based recursive evaluation schemes are optimized for the memory footprint. For the 3-center 2-particle integrals that are key for the evaluation of Coulomb and other 2-particle interactions in the density-fitting approximation the use of multi-quantal recurrences (in which multiple quanta are created or transferred at once) is shown to produce significant memory savings. Other innovation include leveraging register memory for reduced memory footprint and direct compile-time generation of optimized kernels (instead of custom code generation) with compile-time features of modern C++/CUDA. High efficiency of the CPU- and CUDA-based implementation of the proposed schemes is demonstrated for both the individual batches of integrals involving up to Gaussians with low and high angular momenta (up to $L=6$) and contraction degrees, as well as for the density-fitting-based evaluation of the Coulomb potential. The computer implementation is available in the open-source LibintX library.
... Even if limited to the three-centre case, it is worth mentioning that May has also developed recurrence relations for two types of three-electron integrals [283]. These recurrence relations were implemented by Womack using automatically-generated code [284]. Recently, we have developed recurrence relations for three-and four-electron integrals for generic correlation factors [263,285]. ...
In this memoir, after a brief introduction of the various quantum chemistry methods considered here, I summarise some of the projects we have been working on in the last ten years.
First, I describe succinctly several studies we have been doing on model two-electron systems.
In particular, we introduced a novel class a quasi-exactly solvable systems that we have been studying exhaustively in several papers.
Following this work, we shed new lights on the universality of correlation effects in two-electrons systems.
The discovery we have made are discussed.
Second, I present the work we performed on the mathematical properties of a new family of uniform electron gases as well as the development of a new class of density functionals based on this new paradigm.
Third, new recurrence relations for three- and four-electron integrals as well as their fundamental integrals and upper bounds are discussed.
In particular, our strategy to bound rigorously and efficiently these many-electron integrals is presented.
This new way of calculating many-electron integrals represents an interesting alternative to what is currently done in explicitly-correlated methods.
Finally, I present two lines of research we have been pursuing recently, namely the development of explicitly-correlated configuration interaction methods and a self-consistent correction to enforce the electron-nucleus cusp in molecular orbitals.
... 57 These recurrence relations were implemented by Womack using automatically-generated code. 58 Recently, we have developed recurrence relations for three-and four-electron integrals for generic correlation factors. 59? ...
We report the three main ingredients to calculate three- and four-electron integrals over Gaussian basis functions involving Gaussian geminal operators: fundamental integrals, upper bounds, and recurrence relations. In particular, we consider the three- and four-electron integrals that may arise in explicitly-correlated F12 methods. A straightforward method to obtain the fundamental integrals is given. We derive vertical, transfer and horizontal recurrence relations to build up angular momentum over the centers. Strong, simple and scaling-consistent upper bounds are also reported. This latest ingredient allows to compute only the $\order{N^2}$ significant three- and four-electron integrals, avoiding the computation of the very large number of negligible integrals.
Theoretical studies of two-center one-electron (2c-1e) small microcluster are associated with hurdles in Schroedinger equation (SE) born out of divergence of Coulomb interactions and nuclear separation (R). The SE deals with morphologically altered H-like AOs, Slater type orbitals (STO), Gaussian type orbitals (GTO), B-spline, Sturmian function and etc in both VBT and MOT calculations. Few elegant computational and analytical methods are available for STO, GTO and other square integrable trial wavefunction under Born-Oppenheimer approximation. Even so, analytical treatment for H-like AOs has become very necessary. Utilizing Sheffer identity in associated Laguerre polynomial/Whittaker-M H-like AOs and adopting elliptic coordinates provide exact, analytical and simple 2c-1e Coulomb exchange interactions (Ks) and overlap integrals as functions of R with different scaling factors associated with electrons. The energetics of diatomic molecule is evident to be the function of R with extrema as Lah number moderated L_n^(-1) for nuclear coordinates.
To improve the efficiency of Gaussian integral evaluation on modern accelerated architectures, FLOP-efficient Obara-Saika-based recursive evaluation schemes are optimized for the memory footprint. For the 3-center 2-particle integrals that are key for the evaluation of Coulomb and other 2-particle interactions in the density-fitting approximation, the use of multiquantal recurrences (in which multiple quanta are created or transferred at once) is shown to produce significant memory savings. Other innovations include leveraging register memory for reduced memory footprint and direct compile-time generation of optimized kernels (instead of custom code generation) with compile-time features of modern C++/CUDA. Performance of conventional and CUDA-based implementations of the proposed schemes is illustrated for both the individual batches of integrals involving up to Gaussians with low and high angular momenta (up to L = 6) and contraction degrees, as well as for the density-fitting-based evaluation of the Coulomb potential. The computer implementation is available in the open-source LibintX library.
