Calculation of localized molecular orbitals

Two techniques are presented for reducing the effort required to determine localized orbitals based on the energy or density criteria. The first, population weighted editing of the electronic repulsion integrals, reduces the effort required for each iteration of the localization procedure. The second, damping/extrapolation of the transformation matrix, reduces the number of iterations required to reach convergence. Numerical results are provided for methane and formaldehyde (for the editing method) and for carbon monoxide and boron fluoride (for the damping/extrapolation technique).     

Optimal group symmetric localized molecular orbitals

Summary The concept and generating method of optimum group symmetric localized molecular orbitals (OSLMOs) are proposed. The OSLMOs have strong points of orthogonality, equivalence and symmetry, and they are simultaneously as close to the classical VB structure as possible. By using the OSLMOs as one-electron orbitals the multiconfigurational correlation calculations are reduced. The scheme is also a valuable popularization and development to hybridization theory.     

Quadratically convergent calculation of localized molecular orbitals

Two iterative procedures for the transformation of canonical self-consistent field molecular orbitals to intrinsic localized molecular orbitals are proposed. A first-order method based on a series of (n × n) unitary transformations may be applied to orbitals which are far from convergence. The second method, based on Newton's method, yields quadratic convergence. Numerical results based on Boys' criterion are presented for water, carbon monoxide, boron fluoride, nitric oxide, and methylacetylene. A composite method may be used to obtain rapid convergence for large molecules for which it is not practical to calculate the entire hessian matrix. The performance of the composite method is demonstrated by application to the dinitrogen tetroxide molecule. Highly converged localized molecular orbitals may be obtained for most molecules with five to eight first-order iterations followed by three or four iterations based on either the second-order or composite method.     

Extremely localized molecular orbitals: theory and applications

Orbitals that are extremely localized on molecular fragments represent a powerful tool for a number of purposes: to cite a few examples, they allow to reduce strongly the complexity of calculations on large systems and are easily transferable from one molecule to another, providing a suitable and efficient way to build up the electronic structure of large molecules. Recently, we have developed efficient algorithms to determine extremely localized molecular orbitals (ELMOs), which will be reviewed in this paper. As a rigorous localization is strictly connected to a reduction in the number of variational parameters, which reflects into an increased value of the associated energy with respect to the Hartree Fock value, we have developed a number of strategies to relax the wavefunction built up using transferred localized orbitals. The extreme localization has also been exploited in connection with the “Divide and Conquer” technique to determine the electron densities of large polypeptides assembled from orbitals computed on small model molecules. Moreover, we will discuss the recent application of the ELMOs in the framework of the hybrid QM/MM methods to describe the frontier region. We will also show that the ELMOs can be used to extract chemical interpretations from numerical results. A variety of applications will be presented.     

Construction of molecular orbitals localized on polyatomic fragments

A simple method of localizing molecular orbitals on polyatomic molecular fragments is proposed; the method allows one to separate orbitals in the structural units of extended molecules. The method is illustrated by semiempirical calculations of the binuclear bridged complexes [(NH₃)₅Rupy-(C₂H₂)_n-py-Ru(NH₃)₅]⁵⁺ (n = 0,1,2,3). One of possible application is construction of orbital bases for calculations by the configuration interaction method with limited sets of active MOs. Translated fromZhumal Strukturnoi Khimii, Vol. 39, No. 4, pp. 571–578, July–August, 1998.     

Transferability of some properties of localized molecular orbitals

The transferability of the first and second order electric moments of localized orbitals have been shown. Some moment characteristics have also been investigated. The standard deviation of first order localized moments is less than 2.0%. The standard deviation of second order electric moment components are larger: although they do not exceed 4.0% for bond orbitals, for lone pair ones they vary about 1.5–10.1%.     

Transferability of some properties of localized molecular orbitals

The localized molecular orbitals of some related ten- and eighteen-electron systems have been studied. The transferability of the kinetic, self-interaction, Coulomb and exchange interaction energies on localized orbitals have been shown. The standard deviation of the kinetic and of the interaction energies (including exchange) are less than 2.5% except for lone pair orbitals of the oxygen atoms where the standard deviation is close to 4%.     

An efficient method for constructing nonorthogonal localized molecular orbitals

A new method for constructing nonorthogonal localized molecular orbitals (NOLMOs) is presented. The set of highly localized NOLMOs is obtained by minimization of the spread functional starting from an initial set of canonical orthogonal molecular orbitals. To enhance the stability and efficiency, the centroids of the NOLMOs are constrained to be those of the corresponding orthogonal localized molecular orbitals (OLMOs), which are obtained with the Boys criterion in advance. In particular, these centroid constraints make the optimization for each NOLMO independent of the others, which is an attractive feature for application to large systems. The minimization with the constraints incorporated through the multiplier-penalty function method is stable and efficient in convergence. While exhibiting the classical bonding pattern in chemistry and sharing a spatial distribution similar to that of the corresponding OLMOs, the obtained NOLMOs are more compact than the corresponding OLMOs with about 10%-28% reduction in the value of the spread functional and devoid of the troublesome "orthogonalization tails."     

Optimum hybrid orbitals in localized orbitals

Using the criterion for maximizing the projection of localized bond orbitals onto the space spanned by the occupied MO 's, a method for constructing hybrid orbitals of a molecule is described. For illustration purposes the method is applied to single-determinant closed shell wave functions, calculated by means of ab initio and semiempirical procedures, for the molecules of methane, acetylene, ethylene, ethane, propylene, butadiene, ammonia and hydrogen cyanide. The predictions of hybridization are briefly discussed.     

Study on the construction of satisfactory nonorthogonal localized molecular orbitals

Comparing to orthogonal localized molecular orbitals (OLMO), the nonorthogonal localized molecular orbitals (NOLMO) exhibit bonding pictures more accordant with those in the traditional chemistry. They are more contracted, so that they have a better transferability and better performances for the calculation of election correlation energies and for the linear scaling algorithms of large systems. The satisfactory NOLMOs should be as contracted as possible while their shapes and spatial distribution keep in accordance with the traditional chemical bonding picture. It is found that the spread of NOLMOs is a monotonic decreasing function of their orthogonality, and it may reduce to any extent as the orthogonality descends. However, when the orthogonality descends to some point, the shapes and spatial distribution of the NOLMOs deviate drastically from the traditional chemical bonding picture, and finally the NOLMOs tend to linear dependence. Without the requirement of orthogonalization, some other constrain     

Optimal virtual orbitals to relax wave functions built up with transferred extremely localized molecular orbitals

Extremely localized molecular orbitals (ELMOs), namely orbitals strictly localized on molecular fragments, are easily transferable from one molecule to another one. Hence, they provide a natural way to set up the electronic structure of large molecules using a data base of orbitals obtained from model molecules. However, this procedure obviously increases the energy with respect to a traditional MO calculation. To gain accuracy, it is important to introduce a partial electron delocalization. This can be carried out by defining proper optimal virtual orbitals that supply an efficient set for nonorthogonal configurations to be employed in VB-like expansions.     

Unravelling the origin of intermolecular interactions using absolutely localized molecular orbitals

An energy decomposition analysis (EDA) method is proposed to isolate physically relevant components of the total intermolecular interaction energies such as the contribution from interacting frozen monomer densities, the energy lowering due to polarization of the densities, and the further energy lowering due to charge-transfer effects. This method is conceptually similar to existing EDA methods such as Morokuma analysis but includes several important new features. The first is a fully self-consistent treatment of the energy lowering due to polarization, which is evaluated by a self-consistent field calculation in which the molecular orbital coefficients are constrained to be block-diagonal (absolutely localized) in the interacting molecules to prohibit charge transfer. The second new feature is the ability to separate forward and back-donation in the charge-transfer energy term using a perturbative approximation starting from the optimized block-diagonal reference. The newly proposed EDA method is used to understand the fundamental aspects of intermolecular interactions such as the degree of covalency in the hydrogen bonding in water and the contributions of forward and back-donation in synergic bonding in metal complexes. Additionally, it is demonstrated that this method can be used to identify the factors controlling the interaction of the molecular hydrogen with open metal centers in potential hydrogen storage materials and the interaction of methane with rhenium complexes.     

On a definition of bond energies based on localized molecular orbitals

A theoretical model is presented for defining bond energies based on localized molecular Orbitals. These bond energies are obtained by rearranging the total SCF energy including the nuclear repulsion term to a sum over orbital and orbital interaction terms and then to total orbital terms, which can be interpreted as the energies of localized orbitals in a molecule. A scaling procedure is used to obtain a direct connection with experimental bond dissociation energies. Two scale parameters are employed, the C-C and the C-H bond dissociation energy in C₂H₆ for A-B and C-H type bonds, respectively. The implications of this scaling procedure are discussed. Numerical applications to a number of organic molecules containing no conjugated bonds gives in general a very satisfactory agreement between experimental and theoretical bond energies.     

Pseudopotentials and localized molecular orbitals. Application to the methane molecule

A pseudopotential method is used for the calculation of the localized molecular orbitals of the ground state of the methane molecule. Good agreement with calculations involving all the electrons and with experiment is obtained.     

Electrons are transported through phenylene-ethynylene oligomer monolayers via localized molecular orbitals

We report excellent correlations between the first negative threshold potentials (V(TH)s) for electric conduction, electrochemical potentials, and computed lowest unoccupied molecular orbital energies in a series of phenylene-ethynylene oligomers bearing a sulfur-based anchoring unit and different electroactive substituents on the central benzene ring. The theoretical and electrochemical results strongly suggest that the peaks observed in the i-V curves have a true molecular origin and are associated with distinct unoccupied molecular levels of the compounds that are strongly localized on the central ring (except for compound I). This localization might account for the existence of a long-lived radical-anion state that permits lateral electron hopping and leads to charge trapping and storage.     

On the use of local basis sets for localized molecular orbitals

Two procedures are discussed for the direct variational optimization of localized molecular orbitals which are expanded in local subsets of the molecular basis set. It is shown that a Newton-Raphson approach is more efficient than an iterative diagonalization scheme. The effect of the basis-set truncation on the quality ofab-initio SCF results is investigated for Be, Li₂, HF, H₂O, NH₃, CH₄ and C₂H₆.     

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 n ² operations. No n ³ 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. Received: 30 December 1996 / Accepted: 5 June 1997