A fragment energy assembler method for Hartree-Fock calculations of large molecules

We present a fragment energy assembler approach for approximate Hartree-Fock (HF) calculations of macromolecules. In this method, a macromolecule is divided into small fragments with appropriate size, and then each fragment is capped by its neighboring fragments to form a subsystem. The total energy of the target system is evaluated as the sum of the fragment energies of all fragments, which are available from conventional HF calculations on all subsystems. By applying the method to a broad range of molecules, we demonstrate that the present approach could yield satisfactory HF energies for all studied systems.     

A fragment molecular-orbital-multicomponent molecular-orbital method for analyzing HD isotope effects in large molecules

We have developed a fragment molecular orbital (FMO)-multi-component MO (MC_MO) method to analyze isotope effect due to differences between the quantum effects of protons and deuterons for large molecules such as proteins and DNA. The FMO-MC_MO method enables the determination of both the electronic and the protonic (deuteronic) wave functions simultaneously, and can directly express isotope effects, including coupling effects between nuclei and electrons. In our calculations of two polyglycines, which serve as prototypes for biological molecules, by this method, we clearly observed the geometrical relaxation induced by the HD isotope effect in the intramolecular hydrogen bonding portions of the molecules. The HD isotope effect on the interfragment interaction energy, including that of the hydrogen bonding parts, was also demonstrated: the hydrogen bond was weakened by replacement of hydrogen with deuterium. We also developed electrostatic potential approximations for use in the FMO-MC_MO calculations, and the accuracy of the energy differences induced by the isotope effect was independent of the approximation level of the FMO-MC_MO. Our results confirmed that the FMO-MC_MO method is a powerful tool for the detailed analysis of changes in hydrogen bonding and interaction energies induced by the HD isotope effect for large biological molecules.     

Local polarizabilities in molecules,based on ab initio Hartree-Fock calculations

A method to separate the total molecular polarizability, calculated in the uncoupled Hartree-Fock approximations, into local contributions is proposed. The method is tested for H₂, H₂O, H₂CO and C₆H₆ and the results are discussed. It is found that the ratio of the polarizability contributions for two atoms in a molecule almost only depends on the type of atoms and is almost independant of molecule.     

The role of Hartree-Fock instability in calculations on electron transitions and polarizability for simple molecules

The stability problem is discussed for the Hartree-Fock solutions for simple molecules; the nonempirical GAUSSIAN-76 program has been supplemented with modules that enable one to monitor the course of the self-consistent calculation by means of stability matrices. The process is illustrated by a calculation on the nitrogen molecule.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 21, No. 1, pp. 94–97, January–February, 1985.     

Fully numerical Hartree-Fock and density functional calculations. II. Diatomic molecules

We present the implementation of a variational finite element solver in the HelFEM program for benchmark calculations on diatomic systems. A basis set of the form is used, where (μ, ν, φ) are transformed prolate spheroidal coordinates, B _n(μ) are finite element shape functions, and are spherical harmonics. The basis set allows for an arbitrary level of accuracy in calculations on diatomic molecules, which can be performed at present with either nonrelativistic Hartree-Fock (HF) or density functional (DF) theory. Hundreds of DFs at the local spin density approximation (LDA), generalized gradient approximation (GGA), and the meta-GGA level can be used through an interface with the Libxc library; meta-GGA and hybrid DFs are not available in other fully numerical diatomic program packages. Finite electric fields are also supported in HelFEM , enabling access to electric properties. We introduce a powerful tool for adaptively choosing the basis set by using the core Hamiltonian as a proxy for its completeness. HelFEM and the novel basis set procedure are demonstrated by reproducing the restricted open-shell HF limit energies of 68 diatomic molecules from the first to the fourth period with excellent agreement with literature values, despite requiring orders of magnitude fewer parameters for the wave function. Then, the electric properties of the BH and N₂ molecules under finite field are studied, again yielding excellent agreement with previous HF limit values for energies, dipole moments, and dipole polarizabilities, again with much more compact wave functions than what were needed for the literature references. Finally, HF, LDA, GGA, and meta-GGA calculations of the atomization energy of N₂ are performed, demonstrating the superb accuracy of the present approach.     

An unconventional scf method for calculations on large molecules

An unconventional SCF method for calculations on large molecules with more than 100 basis functions is described. Storage problems which arise in conventional SCF schemes when storing more than 10⁷ integrals are avoided by repeated calculation of integrals. The resulting increase in computational times is kept at a reasonable level by (a) improving the initial guess, (b) accelerating convergence, (c) employing a recursive construction of the Fock matrix, and (d) eliminating insignificant integrals from the calculation by a density-weighted cutoff criterion. Sample calculations show that, compared with conventional SCF calculations, computational times increase by 25%–75% depending on the basis set and the shape of the molecule.     

A molecular-orbital theoretical classification of reactions of singlet ground-state molecules

The reaction mechanisms of molecular systems in the ground singlet state are discriminated with the triplet stability—instability criterion of the restricted Hartree—Fock (RHF) solution for the entire system and with the phase continuity criterion of the highest occupied orbital involved. The criteria used are discussed in relation to the outcomes from the configuration-interaction approach.     

Efficient evaluation of short-range Hartree-Fock exchange in large molecules and periodic systems

We present an efficient algorithm for the evaluation of short-range Hartree-Fock exchange energies and geometry gradients in Gaussian basis sets. Our method uses a hierarchy of screening levels to eliminate negligible two-electron integrals whose evaluation is the fundamental computational bottleneck of the procedure. By applying our screening technique to the Heyd-Scuseria-Ernzerhof [J. Chem. Phys. 118, 8207 (2003)] short-range Coulomb hybrid density functional, we achieve a computational efficiency comparable with that of standard nonhybrid density functional calculations.     

A statistical approach to resonance energies of large molecules

For large conjugated molecules, resonance energies of SCF MO quality are not available. Here we outline a method of determining molecular resonance energies by combining a graph theoretical approach to aromaticity with a statistical analysis of random Kekule valence structures. The approach involves construction of random Kekule valence forms and subsequent enumeration of conjugated circuits within each such structure.     

Direct selected multireference configuration interaction calculations for large systems using localized orbitals

A selected multireference configuration interaction (CI) method and the corresponding code are presented. It is based on a procedure of localization that permits to obtain well localized occupied and virtual orbitals. Due to the local character of the electron correlation, using local orbitals allows one to neglect long range interactions. In a first step, three topological matrices are constructed, which determine whether two orbitals must be considered as interacting or not. Two of them concern the truncation of the determinant basis, one for occupied/virtual, the second one for dispersive interactions. The third one concerns the truncation of the list of two electron integrals. This approach permits a fine analysis of each kind of approximation and induces a huge reduction of the CI size and of the computational time. The procedure is tested on linear polyene aldehyde chains, dissociation potential energy curve, and reaction energy of a pesticide-Ca(2+) complex and finally on transition energies of a large iron system presenting a light-induced excited spin-state trapping effect.     

Small gaussian basis sets for AB initio calculations on large molecules

Gaussian basis sets of (5s, 2p) for carbon, nitrogen, and oxygen, and (7s, 4p) for phosphorous and sulfur have been developed for ab initio calculations of biological molecules. Double zeta contracted bases are given for all five atoms. Minimum bases are given for carbon, nitrogen and oxygen, and a method is developed for replicating primitives in order to minimize the energy loss when contracting small bases. The contracted bases are applied to formamide and the results are compared with those obtained from other small basis sets.     

An error analysis for Hartree-Fock crystal orbital calculations

Our error analysis shows that numerical instabilities can be always expected in Hartree-Fock crystal orbital calculations using extended atomic basis sets due to the errors caused by improper lattice sum truncations. These instabilities may appear also at relatively large eigenvalues of the overlap matrix and are not related to a linear dependence in the basis set. An infinite linear chain of hydrogen atoms is analysed as an example, exhibiting the numerical instabilities which were encountered also in other solids.     

Coupled Hartree-Fock calculations of molecular hyperpolarizabilities

The second-order SCF perturbation method for the calculation of electrical polarizabilities is extended to the third order for the calculation of hyperpolarizabilities. Exploratory calculations for the hyperpolarizability of FH suggest that the convergence of the third-order theory is comparable to that of the second, and confirm the sensitivity of β to choice of basis set as found by other workers.     

A hierarchic sparse matrix data structure for large-scale Hartree-Fock/Kohn-Sham calculations

A hierarchic sparse matrix data structure for Hartree-Fock/Kohn-Sham calculations is presented. The data structure makes the implementation of matrix manipulations needed for large systems faster, easier, and more maintainable without loss of performance. Algorithms for symmetric matrix square and inverse Cholesky decomposition within the hierarchic framework are also described. The presented data structure is general; in addition to its use in Hartree-Fock/Kohn-Sham calculations, it may also be used in other research areas where matrices with similar properties are encountered. The applicability of the data structure to ab initio calculations is shown with help of benchmarks on water droplets and graphene nanoribbons.     

Thomas-Fermi-Dirac calculations for molecules

The electrostatic potential derived from a solution to the molecular Thomas-Fermi-Dirac equation for F₂ is combined with the exchange potential and modified to give the correct behavior far from the nuclei. One-electron energy levels in this potential are calculated and are in qualitative agreement with SCF orbital energies. Similar computations are carried through for F and Ar, which correspond to the separated and united atoms for F₂. To compensate for errors in the potential, we subtract from molecular orbital energies the difference of TFD and SCF orbital energies for the separated atoms. Now all the orbital energies are correct to a few electron volts.

---

Zusammenfassung Das elektrostatische Potential, das sich für F₂ aus der Thomas-Fermi-Dirac Theorie ergibt, wird mit dem Austauschpotential kombiniert und so modifiziert, daß sich das richtige Verhalten in Kernnähe ergibt. Die berechneten Einelektronenenergien sind in qualitativer Übereinstimmung mit SCF-Werten. Analoge Rechnungen für F und Ar werden ausgeführt und als Grenzfälle für Korrekturen verwendet. Dann ergeben sich alle Orbitalenergien bis auf wenige eV richtig.

---

---

Supported by the National Science Foundation under Grant GP-20718.