Molecule intrinsic minimal basis sets. I. Exact resolution of ab initio optimized molecular orbitals in terms of deformed atomic minimal-basis orbitals

A method is presented for expressing the occupied self-consistent-field (SCF) orbitals of a molecule exactly in terms of chemically deformed atomic minimal-basis-set orbitals that deviate as little as possible from free-atom SCF minimal-basis orbitals. The molecular orbitals referred to are the exact SCF orbitals, the free-atom orbitals referred to are the exact atomic SCF orbitals, and the formulation of the deformed "quasiatomic minimal-basis-sets" is independent of the calculational atomic orbital basis used. The resulting resolution of molecular orbitals in terms of quasiatomic minimal basis set orbitals is therefore intrinsic to the exact molecular wave functions. The deformations are analyzed in terms of interatomic contributions. The Mulliken population analysis is formulated in terms of the quasiatomic minimal-basis orbitals. In the virtual SCF orbital space the method leads to a quantitative ab initio formulation of the qualitative model of virtual valence orbitals, which are useful for calculating electron correlation and the interpretation of reactions. The method is applicable to Kohn-Sham density functional theory orbitals and is easily generalized to valence MCSCF orbitals.     

Symmetry and equivalence restrictions in electronic structure calculations

A simple method for obtaining MCSCF orbitals and CI natural orbitals adapted to degenerate point groups, with full symmetry and equivalence restrictions, is described. Among several advantages accruing from this method are the ability to perform atomic SCF calculations on states for which the SCF energy expression cannot be written in terms of Coulomb and exchange integrals over real orbitals, and the generation of symmetry-adapted atomic natural orbitals for use in a recently proposed method for basis set contraction.     

An intrinsic localization procedure for active CAS SCF orbitals

An intrinsic localization criterion for the active (valence) orbitals of a CAS SCF wavefunction is presented. The localization criterion is based on minimization of the energy of a “perfect pairing” configuration. Equations for carrying out the localization in terms of an exponential transformation are developed. The technique can easily be incorporated into any MC SCF program. The CAS SCF wavefunction obtained using these localized active orbitals corresponds to a full VB calculation where the VB structures are built from orthogonal “molecule adapted minimal basis set atomic orbitals” and thus offers an interpretational advantage over the use of canonical CAS SCF orbitals. The method is applied to the 1,3-dipole, nitrone.     

An ab initio method for approximation of the frozen molecular fragment

An ab initio method for calculation on many-electron molecular systems with the approximation of the inactive part of a molecule by frozen molecular fragment is presented. In the following method the SCF calculations are performed in two series. First the molecular orbitals resulting from the first SCF calculation (modest basis set) are localized. In the second SCF run, the basis set is extended for the active part of the molecule, while molecular orbitals of the inactive part, selected from the localized set, are kept frozen. The results are in good agreement with the extended basis set calculation.     

Molecule intrinsic minimal basis sets. II. Bonding analyses for Si4H6 and Si2 to Si10

The method, introduced in the preceding paper, for recasting molecular self-consistent field (SCF) or density functional theory (DFT) orbitals in terms of intrinsic minimal bases of quasiatomic orbitals, which differ only little from the optimal free-atom minimal-basis orbitals, is used to elucidate the bonding in several silicon clusters. The applications show that the quasiatomic orbitals deviate from the minimal-basis SCF orbitals of the free atoms by only very small deformations and that the latter arise mainly from bonded neighbor atoms. The Mulliken population analysis in terms of the quasiatomic minimal-basis orbitals leads to a quantum mechanical interpretation of small-ring strain in terms of antibonding encroachments of localized molecular-orbitals and identifies the origin of the bond-stretch isomerization in Si4H6. In the virtual SCF/DFT orbital space, the method places the qualitative notion of virtual valence orbitals on a firm basis and provides an unambiguous ab initio identification of the frontier orbitals.     

An efficient self-consistent field method for large systems of weakly interacting components

An efficient method for removing the self-consistent field (SCF) diagonalization bottleneck is proposed for systems of weakly interacting components. The method is based on the equations of the locally projected SCF for molecular interactions (SCF MI) which utilize absolutely localized nonorthogonal molecular orbitals expanded in local subsets of the atomic basis set. A generalization of direct inversion in the iterative subspace for nonorthogonal molecular orbitals is formulated to increase the rate of convergence of the SCF MI equations. Single Roothaan step perturbative corrections are developed to improve the accuracy of the SCF MI energies. The resulting energies closely reproduce the conventional SCF energy. Extensive test calculations are performed on water clusters up to several hundred molecules. Compared to conventional SCF, speedups of the order of (N/O)2 have been achieved for the diagonalization step, where N is the size of the atomic orbital basis, and O is the number of occupied molecular orbitals.     

Extended floating spherical Gaussian basis sets for molecules. Alternative correlating orbitals for molecular energy calculations

A procedure previously described for representing large basis SCF results in terms of a smaller floating spherical Gaussian orbital (FSGO) basis set is generalized to apply to the virtual orbitals from the SCF calculation. This provides a method for systematically reducing the dimensions of the virtual space or replacing the virtual orbitals with a simpler, compact basis set. The method is illustrated by application to Lill.     

Orbital relaxation in the Rydberg series of the lithium atom. An excited state SCF calculation

Self-consistent-field (SCF ) calculations for a series of Rydberg states (1s²ns)²S of the Li atom are performed using the generalized Brillouin theorem (GBT) method. The calculated energy is a proper upper bound to the excited state energy. The SCF term values of the Rydberg states are almost the same as those of the frozen-core approximation ones. The orbital behavior shows that the core is slightly expanded by the penetration of the Rydberg orbitals, and the higher Rydberg orbitals can be very well represented by the modified hydrogen-like orbitals.     

Localized orbitals based on the fermi hole

The Fermi hole provides a direct (non-iterative) method for tansforming canonical SCF molecular orbitals into localized orbitals. Except for simple overlap integrals required to maintain orthogonality, this method requires no integrals over orbitals or basis functions. This method is demonstrated by application to a furanone (C₄H₄O₂), methylacetylene, and boron trifluoride. The results of these calculations are compared to those determined by the orbital centroid criterion of localization.     

An iterative process to calculate the SCF open shell orbitals

A method is described for calculating SCF open shell orbitals. In comparison with the coupling operator method, a greater velocity of convergency of the iterative process is obtained by taking into account not only the correct variational conditions, but also the best variations of orbitals step by step.     

Development of parallel density functional program using distributed matrix to calculate all-electron canonical wavefunction of large molecules

We developed a new parallel density-functional canonical molecular-orbital program for large molecules based on the resolution of the identity method. In this study, all huge matrices were decomposed and saved to the distributed local memory. The routines of the analytical molecular integrals and numerical integrals of the exchange-correlation terms were parallelized using the single program multiple data method. A conventional linear algebra matrix library, ScaLAPACK, was used for matrix operations, such as diagonalization, multiplication, and inversion. Anderson's mixing method was adopted to accelerate the self-consistent field (SCF) convergence. Using this program, we calculated the canonical wavefunctions of a 306-residue protein, insulin hexamer (26,790 orbitals), and a 133-residue protein, interleukin (11,909 orbitals) by the direct-SCF method. In regard to insulin hexamer, the total parallelization efficiency of the first SCF iteration was estimated to be 82% using 64 Itanium 2 processors connected at 3.2 GB/s (SGI Altix3700), and the calculation successfully converged at the 17-th SCF iteration. By adopting the update method, the computational time of the first and the final SCF loops was 229 min and 156 min, respectively. The whole computational time including the calculation before the SCF loop was 2 days and 17 h. This study put the calculations of the canonical wavefunction of 30,000 orbitals to practical use.     

Equivalence-restricted open-shell SCF theory

A method is presented for generating open-shell equivalence-restricted SCF orbitals in high-symmetry situations using Roothaan–Hartree–Fock programs which are adapted for lower symmetry.     

Méthode des Biorbitales: Application Préliminaire au Calcul de l'Energie de l'Etat Fondamental de l'Atome de Beryllium

The ground-state electronic energy of Be is calculated using the method of biorbitals (SCF –BI ). In this method the wave function is represented by an antisymmetrized product of identical pair functions. The basic set used to develop the biorbitals consists of the Watson s and p orbitals. The pair function is presumed to describe a singlet pair state. The energy associated with this function is minimized using a steepest descent procedure. A value of 0.0414 a.u. was found for the correlation energy, which is 44% of the total correlation energy. The SCF –BI method is compared with the CI method. The relationships are established between the expansion coefficients of both methods. The occupation numbers of orbitals are calculated.     

An ab initio study of the ground and excited states of p-benzoquinone

The results of anab initio SCF calculation for the ground state and CI calculations for the excited states of p-benzoquinone are presented and discussed. A minimum basis set of Slater type orbitals was employed and the CI calculations were performed by considering single excitations from valence to virtual SCF molecular orbitals. The convergence of the calculated excitation energies is studied as a function of the number of orbitals used in the CI calculations. These calculations explain quite well the experimental results.     

Symmetry adaptation of configuration basis in MCSCF method

Summary A novel approach of space symmetry adaptation is developed for multiconfigurational (MC) functions in fully optimized reaction space and complete active space SCF calculations. The bonded tableau and two box symmetric tableau are basic representations (rep) of configuration functions; the group symmetric localized orbitals are used as one-electron orbitals. The method is proposed for generating a complete and orthonormal set of MC single excited functions. The redundant variable in MCSCF can be eliminated by symmetry adaptation.     

Valence type vacant orbitals for configuration interaction calculations

A method is proposed to determine the valence type vacant orbitals, which are suitable for CI calculations and for the initial guess orbitals in MC SCF calculations. The method was applied to calculate the ionization energies of series of molecules and to draw the potential energy curves of various states of N₂ and N⁺₂.     

On the use of frozen orbitals in molecular orbital cluster calculations: Cl on Si(111)

By using the corresponding orbital transformation a quantitative criterion is found for the MOs that can be kept fixed during calculations using a perturbative SCF procedure. Results obtained keeping fixed a subset of molecular orbitals are quantitatively in agreement with those obtained by using the canonical SCF procedure but with a considerable saving of computer time in the SCF step.     

One electron orbitals intrinsic to the reduced hamiltonian

One electron orbitals are determined from the reduced hamiltonian by a simple one-step diagonalization. These reduced hamiltonian orbitals (RHO's) are uniquely determined and virtual orbitals obtained in this procedure are on a par with filled orbitals. These RHO's appear well suited for CI calculations. Minimum basis set calculations are presented for H₂O and compared with similar SCF studies.     

Computation scheme for optimizing multiconfigurational wave-functions

A computation scheme is proposed to determine the wave-functions of molecular systems within the framework of the CMC SCF theory and the APSG SCF approach. The orbital optimization is carried out by the refined first-order one-electron Hamiltonian method. Explicit expressions of the first and second energy derivatives are obtained. In the suggested scheme all the calculations are based on using the matrices of the “partial” Coulomb and the exchange operators constructed over the orbitals at the current iteration cycle.     

Coupled multiconfiguration self-consistent field (MC SCF) perturbation theory

The analytical form of the perturbation theory for the MC SCF method of Veillard and Clementi is presented. The appropriate second-order energy functional which takes into account the self-consistency requirements, leads to a set of coupled first-order perturbed equations determining the perturbed configuration coefficients and orbitals. The second-order energy formula derived from this functional can be given a clear physical interpretation. The present analytical approach is compared with the finite perturbation MC SCF scheme. The possibility of the approximate solution of the coupled MC SCF perturbation equations is also discussed and the so-called uncoupled procedures are devised. In the limit of the single determinant wave function the present formulae are shown to be equivalent to the appropriate Hartree-Fock perturbation results. The differences between the one-configuration SCF and the MC SCF approach are illustrated by the calculation of the electric dipole polarizability of. HZ in the CNDO/2 approximation. It is shown that the one-configuration SCF approaches cannot account for the correct asymptotic properties of the second-order energy for large internuclear distances. This feature of the SCF perturbation theories does not depend on the specific approximations of the CNDO/2 scheme and is corrected by using the MC SCF perturbation theory.