Efficient implementation of the analytic second derivatives of Hartree–Fock and hybrid DFT energies: a detailed analysis of different approximations

In this paper, various implementations of the analytic Hartree–Fock and hybrid density functional energy second derivatives are studied. An approximation-free four-centre implementation is presented, and its accuracy is rigorously analysed in terms of self-consistent field (SCF), coupled-perturbed SCF (CP-SCF) convergence and prescreening criteria. The CP-SCF residual norm convergence threshold turns out to be the most important of these. Final choices of convergence thresholds are made such that an accuracy of the vibrational frequencies of better than 5 cm^?1 compared to the numerical noise-free results is obtained, even for the highly sensitive low frequencies (<100–200 cm^?1). The effects of the choice of numerical grid for density functional exchange–correlation integrations are studied and various weight derivative schemes are analysed in detail. In the second step of the work, approximations are introduced in order to speed up the computation without compromising its accuracy. To this end, the accuracy and efficiency of the resolution of identity approximation for the Coulomb terms and the semi-numerical chain of spheres approximation to the exchange terms are carefully analysed. It is shown that the largest performance improvements are realised if either Hartree–Fock exchange is absent (pure density functionals) and otherwise, if the exchange terms in the CP-SCF step of the calculation are approximated by the COSX method in conjunction with a small integration grid. Default values for all the involved truncation parameters are suggested. For vancomycine (176 atoms and 3593 basis functions), the RIJCOSX Hessian calculation with the B3LYP functional and the def2-TZVP basis set takes ～3 days using 16 Intel® Xeon® 2.60GHz processors with the COSX algorithm having a net parallelisation scaling of 11.9 which is at least ～20 times faster than the calculation without the RIJCOSX approximation.     

双原子分子激发态势能的自旋相关局域Hartree-Fock密度泛函理论方法

基于自旋相关局域Hartree-Fock (SLHF)势函数，本文提出了一种计算双原子分子激发态势能的密度泛函理论(DFT)方法，并将该方法应用于和的激发态势能曲线的计算。在只考虑交换能的情况下，本文的DFT计算结果与文献中精确方法和Hartree-Fock (HF)方法的结果符合的非常好，说明采用SLHF势函数作为交换势的DFT方法是一个很好的计算激发态势能的方法。本文还计算和探讨了电子的关联势函数和关联能，发现传统的近似方法在较大核间距的情况下大大低估了电子的关联能.     

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.     

The Fourier space restricted Hartree-Fock method for the electronic structure calculation of linear poly(tetrafluoroethylene)

Building on the pioneering work of Jean-Marie André and working in the laboratory he founded, the authors have developed a code called FT-1D to make Hartree-Fock electronic structure computations for stereoregular polymers using Ewald-type convergence acceleration methods. That code also takes full advantage of all line-group symmetries to calculate only the minimal set of two-electron integrals and to optimize the computation of the Fock matrix. The present communication reports a benchmark study of the FT-1D code using polytetrafluoroethylene(PTFE) as a test case. Our results not only confirm the algorithmic correctness of the code through agreement with other studies where they are applicable, but also show that the use of convergence acceleration enables accurate results to be obtained in situations where other widely-used codes(e.g., PLH and Crystal) fail. It is also found that full attention to the line-group symmetry of the PTFE polymer leads to an increase of between one and two orders of magnitude in the speed of computation. The new code can therefore be viewed as extending the range of electronic-structure computations for stereoregular polymers beyond the present scope of the successful and valuable code Crystal.     

On variational approximations in quantum molecular dynamics

The Dirac-Frenkel-McLachlan variational principle is the basic tool for obtaining computationally accessible approximations in quantum molecular dynamics. It determines equations of motion for an approximate time-dependent wave function on an approximation manifold of reduced dimension. This paper gives a near-optimality result for variational approximations. It bounds the error in terms of the distance of the exact wave function to the approximation manifold and identifies the parameters that control the deviation of the variational approximation from the best approximation on the manifold.

     

Calculation of ground and excited states for atoms with an open <Emphasis Type="Italic">f</Emphasis>-shell in the Hartree-Fock-Roothaan approximation

Application of the Roothaan-Bagus method (Hartree-Fock atomic theory) for atoms with one open fⁿ-shell is considered. Energy values for lanthanide atoms in the ground and excited states are calculated by minimization methods of the first and second orders within the limits of Roothaan-Hartree-Fock atomic theory using orbital exponents of Slater-type atomic orbitals. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 75, No. 6, pp. 776–781, November–December, 2008.     

Non-existence of a minimizer to the magnetic Hartree-Fock functional

In the presence of an external magnetic field, we prove absence of a ground state within the Hartree-Fock theory of atoms and molecules. The result is established for a wide class of magnetic fields when the number of electrons is greater than or equal to 2Z + K, where Z is the total charge of K nuclei. Positivity properties are instrumental in the proof of this bound for the maximal ionization.      

Application of the Dyson boson expansion method to the treatment of the pairing force

The treatment of the separable pairing interaction in the context of the BRST formalism and in the Dyson boson expansion method is discussed. The approach is based on the use of the vacuum expectation value of the boson number operator to define a suitable mean field. Received: 10 April 2000 / Accepted: 6 July 2001     

The halo of the exotic nucleus 11Li: a single Cooper pair

If neutrons are progressively added to a normal nucleus, the Pauli principle forces them into states of higher momentum. When the core becomes neutron saturated, the nucleus expels most of the wave function of the last neutrons outside to form a halo, which, because of its large size, can have a lower momentum. It is an open question how nature stabilizes such a fragile system and provides the glue needed to bind the halo neutrons to the core. Here, we show that this problem is similar to that of the instability of the normal state of an electron system at zero temperature solved by Cooper, a solution which is at the basis of BCS theory of superconductivity. By mimicking this approach using, aside from the bare nucleon-nucleon interaction, the long wavelength vibrations of the nucleus ¹¹Li, the paradigm of halo nuclei, as tailored glues of the least bound neutrons, we are able to obtain a unified and quantitative picture of the observed properties of ¹¹Li. Received: 9 May 2001 / Accepted: 17 November 2001     

Ab Initio Calculation of Room Temperature Ionic Liquid 1-Ethyl-3-Methyl-Imidazolium Chlorocuprate (Ⅰ)

The Hartree-Fock method has been employed to investigate the electronic structures of EMIM (1-ethyl-3-methyl-imidazolium ), CuCl2-, Cu2Cl3-, CuCl32-, EMIM -CuCl2-, EMIM -Cu2Cl3-,and EMIM -CuCl32- pairs. Full optimization and frequency analyses of EMIM , CuCl2-, Cu2Cl3-, CuCl32-,eight initial EMIM -CuCl2-, six EMIM -Cu2Cl3-, and four EMIM -CuCl32- geometries have been carried out using Gaussian-94 soft-package at 6-31 G(d,p) basis set level for hydrogen, carbon, nitrogen, chlorine atoms and Hay-Wadt effective core potential for copper atom. The electronic structures of lowest energy of EMIM -CuCl2-, EMIM -Cu2Cl3-, EMIM -CuCl32-, single EMIM , CuCl2-, Cu2Cl3-, and CuCl32- have been comparatively studied. The calculated results showed that EMIM -CuCl2- pair conformer of lowest energy was five ring parallel to Cl-Cu-Cl with 3.2 (A) distance, EMIM -CuCl32- pair conformer of lowest energy was five ring parallel to CuCl32-plane with 3.4 (A) distance, and the optimized EMIM -Cu2Cl3- pair conformer of lowest energy was five ring perpendicular to Cl-Cu-Cl-Cu-Cl plane with 3.0 (A) distance between the terminal Cl atoms and the 5-ring of EMIM . The cohesion between cations and anions analyses suggested that all stationary points are minimum because of no appearing of imaginary frequency.The assigned frequencies were in agreement with the experimental report. The low energy of interaction because of the bulky asymmetry of EMIM and the charge dispersion of cation and anion leads to the low melting point of the ionic liquids, EMIM -CuCl2-, EMIM -Cu2Cl3-, and EMIM -CuCl32-. The interaction energy of EMIM -CuCl2-, EMIM -Cu2Cl3-, and EMIM -CuCl32- is 309.0 k J/mol, 316.8 k J/mol, and 320.2 k J/mol, respectively. The relationship of interaction energy via distance between cations and anions was also investigated by single point energy scan.