Computation and analysis of molecular Hartree—Fock momentum intracules

An efficient general algorithm for the computation of molecular momentum intracule densities from Hartree-Fock wavefunctions using Gaussian basis functions is described. The momentum intracules for a number of systems are examined, and comparison with their position space counterparts discussed.     

Parallelization of analytical Hartree—Fock and density functional theory Hessian calculations. Part I: parallelization of coupled-perturbed Hartree—Fock equations

Solving the coupled-perturbed Hartree-Fock (CPHF) equations is the most time consuming part in the analytical computation of second derivatives of the molecular energy with respect to the nuclei. This paper describes a unique parallelization approach for solving the CPHF equations. The computational load is divided by the nuclear perturbations and distributed evenly among the computing nodes. The parallel algorithm is scalable with respect to the size of the molecule, i.e. the larger the molecule, the greater the parallel speedup. The memory storage requirements are also distributed among the processors, with little communication among the processors. The method is implemented in the Q-Chem software package and its performance is discussed. This work represents the first step in a research project to parallelize analytical frequency calculations at Hartree-Fock and density functional theory levels.     

Existence of Hartree–Fock excited states for atoms and molecules

For neutral and positively charged atoms and molecules, we prove the existence of infinitely many Hartree–Fock critical points below the first energy threshold (that is, the lowest energy of the same system with one electron removed). This is the equivalent, in Hartree–Fock theory, of the famous Zhislin–Sigalov theorem which states the existence of infinitely many eigenvalues below the bottom of the essential spectrum of the N-particle linear Schrödinger operator. Our result improves a theorem of Lions in 1987 who already constructed infinitely many Hartree–Fock critical points, but with much higher energy. Our main contribution is the proof that the Hartree–Fock functional satisfies the Palais–Smale property below the first energy threshold. We then use minimax methods in the N-particle space, instead of working in the one-particle space.     

Solutions of the Dirac–Fock Equations for Atoms¶and Molecules

The Dirac-Fock equations are the relativistic analogue of the well-known Hartree-Fock equations. They are used in computational chemistry, and yield results on the inner-shell electrons of heavy atoms that are in very good agreement with experimental data. By a variational method, we prove the existence of infinitely many solutions of the Dirac-Fock equations "without projector", for Coulomb systems of electrons in atoms, ions or molecules, with Z h 124, N h 41, N h Z. Here, Z is the sum of the nuclear charges in the molecule, N is the number of electrons.     

A mathematical and computational review of Hartree–Fock SCF methods in quantum chemistry

We present a review of the fundamental topics of Hartree–Fock theory in quantum chemistry. From the molecular Hamiltonian, using and discussing the Born–Oppenheimer approximation, we arrive at the Hartree and Hartree–Fock equations for the electronic problem. Special emphasis is placed on the most relevant mathematical aspects of the theoretical derivation of the final equations, and on the results regarding the existence and uniqueness of their solutions. All Hartree–Fock versions with different spin restrictions are systematically extracted from the general case, thus providing a unifying framework. The discretization of the one-electron orbital space is then reviewed and the Roothaan–Hall formalism introduced. This leads to an exposition of the basic underlying concepts related to the construction and selection of Gaussian basis sets, focusing on algorithmic efficiency issues. Finally, we close the review with a section in which the most relevant modern developments (especially those related to the design of linear-scaling methods) are commented on and linked to the issues discussed. The paper is intentionally introductory and rather self-contained, and may be useful for non-experts intending to use quantum chemical methods in interdisciplinary applications. Moreover, much material that can be found scattered in the literature has been put together to facilitate comprehension and to serve as a handy reference.     

Comparison of ring currents evaluated consistently at density functional and Hartree–Fock levels

Ring-current maps give an immediate visualisation of aromaticity on the magnetic criterion–by which a cyclic system that supports diatropic (paratropic) current induced by a perpendicular magnetic field is aromatic (anti-aromatic). Calculations of maps with the ipsocentric choice of origin are made in the 6‐31G** basis set at Hartree–Fock (HF) and density functional (DFT) levels (PW91 and B3LYP functionals) on porphyrin, porphycene, orangarin, sapphyrin and hexabenzocoronene. In these systems, DFT and HF approaches produce optimal geometries with different point-group symmetries and/or different patterns of bond alternation. The ring-current maps derived with all four combinations of methods indicate that the main features of the current (global nature, direction, estimated strength) survive in systems with symmetry-breaking, but that choice of geometry is more critical for the detail of the current than is the electronic-structure method.     

线性π—共轭系统的Hartree—Fock解及其稳定性

本文利用π电子近似ＰＰＰ方法计算了一类线性π共轭系统ＨＦＲ方程的对称性和对称性破缺解．计算中对双电子积分采用了四种不同形式以比较电子间相互作用对能谱、组态及相跃迁的影响．结果表明，当耦合作用较强时，Ｐｅｉｅｒｌｓ畸变似难以出现，但ＣＤＷ、ＳＤＷ在一定条件下仍然存在，并且对作用力程较短的情况ＳＤＷ比ＣＤＷ易于发生，而对作用力程较长的情况ＣＤＷ比ＳＤＷ易于发生．     

周期系统的介电矩阵—Hartree—Fock自洽场方法

从 Hartree-Fock 自洽场得到了周期性媒质中包含交换作用的介电矩阵,并应用它讨论了半导体超晶格的等离激元色散关系和金属钠的直接带间跃迁。数值计算表明:在有限波矢情形,交换效应的修正是重要的.     

A comparison of finite basis set and finite difference Hartree—Fock calculations for the open-shell (X 2Σ+) molecules BaF and YbF

A comparison is made of the accuracy with which the total electronic energy can be calculated by using the finite basis set approach (the algebraic approximation) and the finite difference method in calculations employing the Hartree—Fock model for the open shell ground (X ²Σ⁺) states of the fluorides BaF and YbF. The convergence of the calculations carried out within the algebraic approximation is monitored by employing systematically constructed basis sets of increasing size. The difference between the finite basis set and finite difference Hartree—Fock energies is 2.6μE _h for BaF and 2.8μE _h for YbF. Dipole moments determined within the algebraic approximation are also compared with the corresponding finite difference expectation values.     

Resolution of Unity of Generalized Bargmann Spaces by means of Displaced Fock states

We resolve the unity of generalized Bargmann spaces in C^n[J.Math.Phys,41(2000)3507] by means of displaced Fock states.     

Energy Spectrum Symmetry of Heisenberg Model in Fock Space

We extend the BCS paring model with equally spaced energy levels to a general one-dimensional spin-1/2 Heisenberg model The two well-known symmetries of the Heisenberg model, i.e. permutational and spin-inversion symmetries, no longer exist. However, when jointing these two operations together, we find a new symmetry of energy spectrum between its subspace n and subspace L - n of the Fock space. A rigorous proof is presented.     

Differentiable programming and density matrix based Hartree–Fock method

Differentiable programming is an emerging programming paradigm that allows people to take derivative of an output of arbitrary code snippet with respect to its input. It is the workhorse behind several well known deep learning frameworks,and has attracted significant attention in scientific machine learning community. In this paper, we introduce and implement a density matrix based Hartree–Fock method that naturally fits into the demands of this paradigm, and demonstrate it by performing fully variational ground state calculation on several representative chemical molecules.     

Reconciling phase diffusion and Hartree–Fock approximation in condensate systems

Despite the weakly interacting regime, the physics of Bose–Einstein condensates is widely affected by particle–particle interactions. They determine quantum phase diffusion, which is known to be the main cause of loss of coherence. Studying a simple model of two interacting Bose systems, we show how to predict the appearance of phase diffusion beyond the Bogoliubov approximation, providing a self-consistent treatment in the framework of a generalized Hartree–Fock–Bogoliubov perturbation theory.     

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.     

A New Decomposition Approach of Dirac Brueckner Hartree—Fock G Matrix for Asymmetric Nuclear Matter

Asymmetric nuclear matter is investigated by the Dirac Brueckner Hartree-Fock (DBHF) approach with a new decomposition of the Dirac structure of nucleon self-energy from the G matrix.It is found that the isospin dependence of the scalar and vector potentials is relatively weak,although boty potentials for neutron (proton) become deep (shallow) in the neutron-rich nuclear matter.The results in asymmetric nuclear matter are rather different from those obtained by a simple method,where the nucleon self-energy is deduced from the single-particle energy.The nuclear binding energy as a function of the asymmetry parameter fulfils the empirical parabolic law up to very extreme isospin asymmetric nuclear matter in the DBHF approach.The behaviour of the density depandence of the asymmetry energy is different from that obtained by non-relativistic approaches,although both give similar asymmetry energy at the nuclear saturation density.