Localization of virtual orbitals

The application of the MBPT in the localized representation requires that both the occupied and the virtual orbitals obtained by the canonical HF equation should be localized. The localization of the occupied orbitals is straightforward in general by any localization method. It is shown that by using Boys' method the localized virtual orbitals are spatially well separated and transferable not only in minimal basis sets.     

Modification of virtual orbitals

The utility of modifying the virtual orbitals of the Fock operator by introducition of an additional potential is discussed. A particularly convenient form for computational implementations is obtained, and improved methods for the practical solution of the secular problem are recommended.     

Localized virtual and occupied molecular orbitals

This report describes the generation of localized from canonical molecular orbitals such that the method (1) be consistently applicable to occupied bond, lone pair and unoccupied orbitals and (2) permit symmetry related orbitals in molecules of two-fold or higher symmetry. Minimization of populations completely external to each local pair region effectively meets these criteria. Finally, conventional strategy for finding the global extremization point is costly in time and memory to implement; a much more efficient numerical search procedure for the global extremum is described. Results for ethylene, butadiene and benzene are presented to clarify the difficulties and their resolution.     

Quasiparticle virtual orbitals in electron propagator calculations

The computational limits of accurate electron propagator methods for the calculation of electron binding energies of large molecules are usually determined by the rank of the virtual orbital space. Electron density difference matrices that correspond to these transition energies in the second-order quasiparticle approximation may be used to obtain a virtual orbital space of reduced rank that introduces only minor deviations with respect to the results produced with the full, original set of virtual orbitals. Numerical tests show the superior accuracy and efficiency of this approach compared to the usual practice of omission of virtual orbitals with the highest energies.     

Localization of molecular orbitals on fragments

A non‐iterative algorithm for the localization of molecular orbitals (MOs) from complete active space self consistent field (CASSCF) and for single‐determinantal wave functions on predefined moieties is given. The localized fragment orbitals can be used to analyze chemical reactions between fragments and also the binding of fragments in the product molecule with a fragments‐in‐molecules approach by using a valence bond expansion of the CASSCF wave function. The algorithm is an example of the orthogonal Procrustes problem, which is a matrix optimization problem using the singular value decomposition. It is based on the similarity of the set of MOs for the moieties to the localized MOs of the molecule; the similarity is expressed by overlap matrices between the original fragment MOs and the localized MOs. For CASSCF wave functions, localization is done independently in the space of occupied orbitals and active orbitals, whereas, the space of virtual orbitals is mostly uninteresting. Localization of Hartree–Fock or Kohn–Sham density functional theory orbitals is not straightforward; rather, it needs careful consideration, because in this case some virtual orbitals are needed but the space of virtual orbitals depends on the basis sets used and causes considerable problems due to the diffuse character of most virtual orbitals. © 2012 Wiley Periodicals, Inc.     

Calculation of second-order properties with improved virtual orbitals

¹³C---H nuclear spin coupling and ¹³C shielding constants of CH₄, C₂H₆, C₂H₄, C₂H₂ are calculated by means of the second-order perturbation theory and MO LCAO wavefunctions. The aim of this paper is to test the ability of the recently proposed virtual orbitals to accelerte the convergence of the sum over states, compared to the calculation with canonical MO's. The ¹³C---H nuclear spin coupling constants show a significant improvement, the changes of the ¹³C shielding are insignificant.     

Pipek–Mezey localization of occupied and virtual orbitals

Recent advances in orbital localization algorithms are used to minimize the Pipek–Mezey localization function for both occupied and virtual Hartree–Fock orbitals. Virtual Pipek–Mezey orbitals for large molecular systems have previously not been considered in the literature. For this work, the Pipek–Mezey (PM) localization function is implemented for both the Mulliken and a Löwdin population analysis. The results show that the standard PM localization function (using either Mulliken or Löwdin population analyses) may yield local occupied orbitals, although for some systems the occupied orbitals are only semilocal as compared to state‐of‐the‐art localized occupied orbitals. For the virtual orbitals, a Löwdin population analysis shows improvement in locality compared to a Mulliken population analysis, but for both Mulliken and Löwdin population analyses, the virtual orbitals are seen to be considerably less local compared to state‐of‐the‐art localized orbitals. © 2013 Wiley Periodicals, Inc.     

Modifications of virtual orbitals in the limited CI calculations for electron-rich molecules

The efficiency of modified virtual orbitals (MVO) of ionic type and of approximate orthogonalized natural orbitals (ONO) in the CI-SD calculations was studied for O₃ and SO₂ molecules and compared with the commonly used canonical virtual orbitals (CVOs). The systems studied represent a class of electron-rich molecules, in which the number of valence electron pairs exceeds substantially the number of formal chemical bonds. We found that the modified orbitals of the types studied appear to be less effective for these systems than in the similar calculations for the AH_n type molecules. Physical reasons for this difference were discussed. The evolution of spatial properties of virtual orbitals within the modification process was analyzed. © 1996 John Wiley & Sons, Inc.     

Optimization of virtual orbitals in the framework of a multiconfiguration spin-coupled wave function

A new method is introduced for the optimization of nonorthogonal virtual orbitals for use in general multiconfiguration spin-coupled wave functions. The use of a number of highly effective approximations greatly reduces the computational effort involved, the most important being the use of a second-order perturbation expression for the energy and an approximate expression for the elements of the Hessian. As a result, the overall scheme scales very favourably with respect to the numbers of active electrons and of basis functions, making it suitable for the accurate study of large systems. Benchmark calculations are presented for the dissociation of LiH(X¹Σ⁺) and Li₂(X¹Σ⁺ _g ) using a highly compact four-configuration wave function. Standard spin-coupled valence bond expansions in the same virtual space are required to be significantly larger before equivalent results are obtained. The results are shown to compare very favourably with full valence complete active space self-consistent field calculations using an identical basis, and binding energies are within 4% of the values obtained from full configuration interaction calculations in the same basis set. Received: 10 June 1997 / Accepted: 7 October 1997     

Effect of modified virtual orbitals on local Coulomb correlation holes in FCN

Local Coulomb correlation hole distribution functions may be used to assess the extent to which electron correlation effects are present in large scale SCF + CI wave functions. From a set of modified virtual orbitals, ordered according to their interaction with the SCF configuration, we have constructed a limited SCF + CI wave function with improved convergence characteristics with respect to that formed from the canonical virtual orbital set. These wave functions, of the same size yet with different energies, have been used to examine the range and depth of local Coulomb correlation holes in FCN. In all cases, the depth of the local Coulomb hole is no more than 10% or so of that of the corresponding Fermi hole, and the range Fermi correlation is generally less than that of Fermi correlation. This is particularly marked in the high density regions around the nuclei. The significance of our results is discussed in relation to a recent proposal for the incorporation of Coulomb correlation into the local exchange method.     

Intramolecular interactions in acyclic and macrocyclic compounds containing nucleotide bases

Acyclic and cyclic counterparts containing thymine and two 3,6-dimethyluracil fragments bridged by methylene chains have been prepared and studied by UV and NMR spectroscopy; in water the uracil units of the acyclic counterpart form an intramolecular stack but arrange in a linear array in chloroform while the fragments of the uracilophane form an intramolecular stack both in chloroform and water; uracil units of bis(3,6-dimethyluracil-1-yl)butane and the macrocyclic counterpart form a stack in chloroform.     

pH/R f diagrams of nucleotide bases

The chromatographic mobilities of uracil, cytosine, thymine, orotic acid, adenine, guanine, hypoxanthine, and xanthine as functions of the pH of buffer systems in the pH range from 0 to 12 have been studied by absorption chromatography on paper. Changes in R _f are observed comparatively rarely in the regions of the pK_a values.     

Calculation of the brueckner orbitals and generalized natural orbitals

The diagrammatic-perturbation approach for the construction of the one-particle Hermitian pseudoeigenvalue problem determining the Brueckner orbitals and/or generalized natural orbitals is elaborated.     

Thoeretical study of the tautomerism of nucleotide bases in the gas phase and in aqueous media

The relative stabilities of the tautomers of nucleotide bases in the gas phase and in aqueous media were estimated by the self-consistent field (SCF) MO method within the-electron approximation. It is shown that the solvation energy, which depends substantially on the order of the tautomers with respect to their energies, must be taken into account for the accurate evaluation of the most stable tautomeric structure. The probabilities of spontaneous mutations in the gas phase and in aqueous media were estimated. The ionization potentials and electron affinities of the normal and rare tautomeric structures were calculated.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 1, pp. 94–99, January, 1978.     

Comparison of explicitly correlated local coupled-cluster methods with various choices of virtual orbitals

Explicitly correlated local coupled-cluster (LCCSD-F12) methods with pair natural orbitals (PNOs), orbital specific virtual orbitals (OSVs), and projected atomic orbitals (PAOs) are compared. In all cases pair-specific virtual subspaces (domains) are used, and the convergence of the correlation energy as a function of the domain sizes is studied. Furthermore, the performance of the methods for reaction energies of 52 reactions involving 58 small and medium sized molecules is investigated. It is demonstrated that for all choices of virtual orbitals much smaller domains are needed in the explicitly correlated methods than without the explicitly correlated terms, since the latter correct a large part of the domain error, as found previously. For PNO-LCCSD-F12 with VTZ-F12 basis sets on the average only 20 PNOs per pair are needed to obtain reaction energies with a root mean square deviation of less than 1 kJ mol(-1) from complete basis set estimates. With OSVs or PAOs at least 4 times larger domains are needed for the same accuracy. A new hybrid method that combines the advantages of the OSV and PNO methods is proposed and tested. While in the current work the different local methods are only simulated using a conventional CCSD program, the implications for low-order scaling local implementations of the various methods are discussed.     

On the use of localized orbitals and restricted alternant orbitals in chemical systems

An analysis of the transformation of localized orbitals into restricted alternant orbitals is proposed. This approach has the advantage of expressing the wave-function in an orbital product while some electron correlation is introduced permitting the study of dissociation reactions. All applications of the orbital technique may be made as easily as with RHF, but with the additional possibility of studying chemical radicals. Some illustrations of this fact are shown for the molecules HF, H₂O, NH₃, CH₄, C₂H₆ and for the dissociation reactions of CH₄ and C₂H₆ generating CH₃ radicals.     

Improved virtual orbitals in perturbative calculations: The case of an average double-hole potential model

The possibility of using modified virtual orbitals generated by an average double-hole potential model in perturbative calculations of correlated potential energy curves is tested in a semi-empirical framework. Comparison is made with the results of conventional perturbative calculations.     

Usefulness of modified virtual orbitals in multireference CI procedure illustrated by calculations on lithium clusters

The maximization of the exchange interaction between the canonical Hartree–Fock virtual and occupied orbitals leads to a transformed set of virtual orbitals which are well suited as one-electron functions for CI calculations. The procedure, generally known for a long time is seldom applied, despite its simplicity and very low computational demand. However, it is found to be particularly useful in the case of multireference CI, since an improved energy is obtained with a considerable shortening of the CI expansion. Moreover, in the final CI wave function, several configurations appear with considerable weight, thus allowing an easy choice of additional configurations to be inserted in the definition of a new zero-order wave function. The efficiency of the computational procedure is discussed for the case of a Li₆ cluster of D_3h symmetry and for the NaCO and PdCO complexes. Results are reported for the relative stability of four different geometrical arrangements of the Li₆ cluster.