Use Of Localized Orbitals Research Articles

The use of local orbitals in multireference calculations

CAS-SCF-type algorithms based on molecular orbitals that preserve their physical nature during the iterative process have been proposed recently by our groups. Our approach is based on the iterative partial diagonalization of the one-body reduced density matrix. If localized guess orbitals are used, the locality property is kept by the final orbitals. The use of local orbitals in multiference calculations has several advantages. It can be used to reduce the number of active orbitals in CAS-SCF calculations on large systems, and in general to have a better control on the physical nature of the active space. The reduction from a complete to a selected reference space is also possible in the case of configuration interaction calculations. The technique is illustrated through applications to the description of bond breaking and n → π* excited states in conjugated systems. The efficiency of selection of local excitations is shown on a magnetic complex.

Massively parallel linear-scaling algorithm in an ab initio local-orbital total-energy method

Similar to the manner of S. Itoh et al. [Comp. Phys. Commun. 88 (1995) 173], we report implementation of a massively parallel linear-scaling algorithm into an ab initio tight-binding method called F ireball [Phys. Rev. B (2001)]. The use of local-orbitals yields a very sparse Hamiltonian matrix which facilitates using a linear-scaling algorithm to obtain the electronic band-structure energy. The general functional form of Kim et al. [Phys. Rev. B 52 (1995) 1640], which minimizes a functional to obtain the electronic band-structure energy, has been parallelized utilizing the conjugate gradient method. The results of this approach are reported here. In addition, the use of “fireball” wavefunctions, where the wavefunctions are explicitly zero beyond some cutoff, allows for pre-generating all integrals describing two- and three-center interactions. The computation of these integrals is then an easily parallelizable problem for which the results are reported. Both integral generation and the linear-scaling optimization procedures are parallelized using the standard MPI message passing interface mixed with an OpenMP strategy.

Comparison of two ways to decompose intermolecular interactions for hydrogen-bonded dimer systems

In this work we test two ab initio methodologies which allow the decomposition of the total intermolecular interaction energy into physically meaningful contributions, namely, the symmetry adapted perturbation theory (SAPT) and the use of localized orbitals within a Møller–Plesset perturbation scheme. The accuracy of the two different methods is compared to supermolecular results, within MP2 and coupled-cluster theory within single and double excitations, with perturbative estimates of the amplitudes of triple excitations [CCSD(T)]. Some relations between the different approaches are conjectured from theoretical considerations, and are confirmed by numerical results. The corresponding calculations have been performed for three model dimers: two NH3⋯H2O dimers, with NH3 acting once as a proton acceptor and once as a proton donor, and the NH4+⋯H2O considered as a prototype of the ion–molecule interaction. We may conclude that third-order terms in SAPT help significantly to reproduce the Hartree–Fock induction and the relaxed, total dispersion in the LMP2 decomposition.

Physical and mathematical content of coupled-cluster equations. III. Model studies of dissociation processes for various reference states

The structure and physical significance of the full set of solutions to coupled-cluster (CC) equations at various stages of the dissociation process and the impact of the choice of reference functions on these solutions have been studied for the first time. The equations for the CC method involving double excitations (CCD) are obtained for the P4 model consisting of two H2 molecules in a rectangular nuclear configuration determined by a geometry parameter α. We consider equations for the reference states |ΦA〉, |ΦQ〉, and |ΦB〉 corresponding to the lowest, highest, and intermediate Hartree–Fock (HF) energies, respectively. The first two states provide a size-consistent description of the dissociation process. For the compact-molecule geometries (α<10.0) the sets of complete solutions to the standard CCD equations [based on molecular orbitals (MOs) of D2h symmetry] in the spin–orbital and spin–symmetry-adapted versions always consist of 20 and 12 entries, respectively. For |ΦA〉 and |ΦB〉 in the dissociation limit (α→∞) only for the latter version the solutions can be attained by homotopy method. In this case we have reformulated the standard spin–symmetry-adapted CCD equations to a version based on the use of localized orbitals (LO) which is extremely simple and can be solved analytically providing an understanding of the unexpected peculiarities of the solutions for α→∞. For |ΦA〉 and |ΦQ〉, there are only two regular solutions. For the remaining 10 solutions, the CCD wave functions are meaningless despite the fact that the corresponding CCD energies are equal to the exact values.

Ab initio correlation corrections to the Hartree-Fock quasi band-structure of periodic systems employing Wannier-type orbitals

A size-consistent ab initio formalism to calculate correlation corrections to ionization potentials as well as electron affinities of periodic systems is presented. Our approach is based on a Hartree-Fock scheme which directly yields local orbitals without any a posteriori localization step. The use of local orbitals implies non-zero off-diagonal matrix elements of the Fock operator, which are treated as an additional perturbation and give rise to localization diagrams. Based on the obtained local orbitals, an effective Bloch Hamiltonian is constructed to second order of perturbation theory with all third-order localization diagrams included. In addition, the summation of certain classes of diagrams up to infinite order in the off-diagonal Fock elements as well as the Epstein-Nesbet partitioning of the full Hamiltonian are discussed. The problem of intruder states, frequently encountered in many-body perturbation theory, is dealt with by employing the theory of intermediate Hamiltonians. As model systems we have chosen cyclic periodic structures up to an oligoethylene ring in double-zeta basis; however, the theory presented here straightforwardly carries over to infinite periodic systems.

Extremely localized molecular orbitals (ELMO): a non-orthogonal Hartree-Fock method

A new optimization method for extremely localized molecular orbitals (ELMO) is derived in a non-orthogonal formalism. The method is based on a quasi Newton-Raphson algorithm in which an approximate diagonal-blocked Hessian matrix is calculated through the Fock matrix. The Hessian matrix inverse is updated at each iteration by a variable metric updating procedure to account for the intrinsically small coupling between the orbitals. The updated orbitals are obtained with approximately n2 operations. No n3 processes such as matrix diagonalization, matrix multiplication or orbital orthogonalization are employed. The use of localized orbitals allows for the creation of high-quality initial “guess” orbitals from optimized molecular orbitals of small systems and thus reduces the number of iterations to converge. The delocalization effects are included by a Jacobi correction (JC) which allows the accurate calculation of the total energy with a limited number of operations. This extension, referred to as ELMO(JC), is a variational method that reproduces the Hartree-Fock (HF) energy with an error of less than 2 kcal/mol for a reduced total cost compared to standard HF methods. The small number of variables, even for a very large system, and the limited number of operations potentially makes ELMO a method of choice to study large systems.

Influence of the electronic core polarization on the electric-field gradients in solids

In conventional band-structure calculations it is normally assumed that closed electronic shells have a spherically symmetric charge density. As a consequence of this approximation, these core states give no contribution to the electric-field gradient (efg) at the nuclear site. In the present paper two equivalent methods for the computation of the actual contribution of closed electron shells to the efg are presented. In the first method the potential of the nuclear quadrupole moment is considered as a perturbation for the core electrons, which causes a polarization of the core states, i.e., a deviation of the core-charge density from spherical symmetry. In this case the core contribution to the efg can be calculated with the help of the Sternheimer function \ensuremath{\gamma}(r). The second method considers the nonspherical parts of the effective crystal potential near the nucleus under consideration as a perturbation for the core electrons. These two methods yield identical results for the interaction energy of the nuclear quadrupole moment with the calculated efg. They are compared with alternative treatments of energetically high-lying core states within the framework of the full-potential linearized-augmented-plane-wave method (semicore calculations and use of local orbitals). As test cases we calculate the efg at the nearest-neighbor sites of a substitutional Ni (Fe) atom in Cu (Al) and the efg at a regular lattice site in hexagonal Mg. Additionally, results for the relaxed atomic positions, the efg, and the asymmetry parameters around a substitutional Pd (V) atom in Cu (Al) are presented.

Møller-Plesset correlation energies in a localized orbital basis using a Laplace transform technique

In electronic structure calculations requiring the handling of large amounts of integrals, storage requirements can often be reduced through the use of localized orbitals which gives rise to sparse integral arrays. However, conventional Moller-Plesset perturbation theory is constrained to canonical orbitals due to the explicit use of orbital energies in the energy expressions, and it is therefore not straightforward to reduce the storage requirements through such orbital localization. This work shows how the constraint of canonical orbitals can be lifted using a Laplace transform technique, and investigates the reduction in storage requirement that can result from the localization of orbitals made possible by such an approach.

Considerations regarding the local treatment of Laplace transform MPPT

The Laplace transform approach of Møller-Plesset perturbation theory (MPPT) by Häser and Almlöf has been investigated for its suitability for localized orbitals. The most critical step in this algorithm, the Laplace transformation of the localized orbitals, leads to a destruction of the physical meaningful localization. However, this downfall does not lead to a decrease of negligible LCAO coefficients in the time critical four index transformation of the electron repulsion integrals. Therefore, the use of localized orbitals or more generally the use of orbitals with a small number of significant LCAO coefficients (mathematically localized orbitals) is a useful refinement in the Laplace transform method of MPPT.

MRD‐CI studies of vertical excitation energies of unsaturated hydrocarbon molecules

AbstractSystematic MRD‐CI calculations using the AM1 Hamiltonian have been carried out for two polyenes and eight aromatic hydrocarbons ranging from benzene to ovalene (C32H14). Twenty singlet–singlet excitation energies in these compounds were calculated and compared with experimental data and ab initio STO‐3G results. On an absolute scale, the AM1/MRD‐CI approach underestimates the excitation energies to states with dominant covalent character by an average of 1.1 eV, whereas the errors for ionic states are between −1.0 and 1.0 eV. The STO‐3G calculated data are much too high by ≈ 1 eV and ≈ 5 eV, respectively. The inclusion of σπ‐correlation effects through second‐order Epstein–Nesbet perturbation theory combined with the use of localized orbitals leads to a significant improvement of the ab initio calculated state energies. In an analogous AM1 treatment, negligible corrections for the σπ correlations are found, which is attributed to the implicit account in the parameters and approximation of the semiempirical Hamiltonian. The possible error sources of the calculational methods are discussed. © 1994 by John Wiley & Sons, Inc.

Band structure and optical properties of MnO

Both the valence and lowest conduction bands of MnO have been performed by the LCAO method thanks to the use of localized orbitals obtained in a self-consistent manner. The imaginary part of the relative dielectric function has been computed by evaluating separately the various band-to-band transitions. The absorption edge has been attributed to 2p → 3d transitions.

Coupled-cluster studies. II. The role of localization in correlation calculations on extended systems

Abstract With the help of a newly developed program for the solution of the coupled-cluster doubles (CCD) equation the use of localized orbitals has been tested

Necessary conditions for the calculation of useful electrostatic isopotential maps from CNDO/INDO wavefunctions

The calculation of reliable electrostatic isopotential maps from CNDO or INDO wavefunctions while preserving the ZDO approximation fails because the individual atoms are treated as if they were spherical charge distributions. The resulting potential maps could be adequately interpreted on the basis of classical electrostatics. Approaches to the determination of useful maps include (a) the use of localized orbitals with a ZDO approximation applied among the localized orbitals as long as the localization procedure is defined with respect to the connectivity of the molecule and has no reference to a set of laboratory coordinates or (b) relaxation of the ZDO approximation in the determination of the potential by including the 3-center potential integrals. In either case, the preservation of rotational invariance takes paramount importance.

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.

Algorithm and program for conformational analysis of molecules by the perturbation-configuration method by use of localized orbitals

Algorithm and program for conformational analysis of molecules by the perturbation-configuration method by use of localized orbitals

On the use of localized orbitals for determination of contracted Gaussian basis sets in ab initio molecular calculations

On the use of localized orbitals for determination of contracted Gaussian basis sets in ab initio molecular calculations

Self-Consistent-FieldXαCluster Method for Polyatomic Molecules and Solids

This paper describes a practical self-consistent-field (SCF) method of calculating electronic energy levels and eigenfunctions, adapted for polyatomic molecules and solids. The one-electron Schr\"odinger equation is set up for a so-called "muffin-tin" approximation to the true potential, spherically symmetrical within spheres surrounding the various nuclei, constant in the region between the spheres, spherically symmetrical outside a sphere surrounding the molecule. The method of solving this equation is a multiple-scattering method, equivalent to the Korringa-Kohn-Rostoker (KKR) method often used for crystals. Once the eigenfunctions and eigenvalues of this problem are determined, one assumes that the orbitals of lowest eigenvalue are occupied, up to a Fermi level. From the resulting charge densities, one can compute a total energy, using a statistical approximation for the exchange correlation. This approximation has an undetermined factor $\ensuremath{\alpha}$ (whence the name $X\ensuremath{\alpha}$ method). The spin orbitals and occupation numbers are varied to minimize this total energy, resulting in one-electron equations. The value of $\ensuremath{\alpha}$ for an isolated atom is determined by requiring that the total energy, using the statistical approximation, should equal the precise Hartree-Fock energy. This leads to very accurate spin orbitals. In a molecule or crystal, one uses the $\ensuremath{\alpha}'\mathrm{s}$ characteristic of the various atoms within the atomic spheres, and a suitable average in the region between. The computer programs for making these self-consistent calculations, for such radicals and polyatomic molecules as S${\mathrm{O}}_{4}^{\ensuremath{-}2}$, Cl${\mathrm{O}}_{4}^{\ensuremath{-}}$, Mn${\mathrm{O}}_{4}^{\ensuremath{-}}$, and S${\mathrm{F}}_{6}$ have been worked out and calculations made. They are more than 100 times as fast as comparable programs using the LCAO (linear-combination-of-atomic-orbitals) method, and the results appear to be in better agreement with experiment than such LCAO results. For calculating the frequencies of optical transitions, one must make a self-consistent calculation, not for the initial or final state, but for what we call the transition state, in which occupation numbers are halfway between the initial and final states. Then it can be proved that the differences of eigenvalues of the $X\ensuremath{\alpha}$ method are more accurate than Hartree-Fock energy values, in that they take account of the modification or relaxation of the orbitals in going from the initial to the final states. These transition states, for a crystal, involve a localized perturbation at the site of the excited atom. The multiple-scattering method is adapted to the use of such perturbed crystals, as well as to isolated molecules, and to perfect crystals. It results, in such problems as x-ray absorption, in the use of localized orbitals rather than bandlike functions. The method is adapted to the calculation of magnetic problems, by use of a spin-polarized version of the method. The method can also be used for calculations of cohesive energy of crystals, and has been used successfully for several types of metals. Unlike most other SCF methods, long-range correlation is automatically included, so that the energy of a system as a function of internuclear positions automatically reduces to the proper values at infinite internuclear distances.

Use of perturbation methods for the study of configuration interaction effects

Localized SCF orbitals are obtained by using the self-energy criteria in the framework of semiempirical methods (Pariser-Parr, Pople-Segal). The delocalisation problem is discussed. These localized orbitals (both occupied and virtual) are used for the calculation of the second-order correlation energy. Qualitative and quantitative features are discussed. The use of localized orbitals allows a deep reduction of the significant terms in the correlation energy.