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.     

General approach for the construction of hybrid orbitals

A general approach to construct hybrid orbitals based on the projection of both the subspace of atomic orbitals and the subspace of hybrid orbitals is presented. It is shown that the projection must be associated with a chain of groups to ensure the independence of the method with respect to the selected coordinate system. The eigenfunction method to obtain symmetry projected functions turns out to be the natural projection technique to establish the present approach. The proposed method allows construction of hybrid orbitals in a systematic and general way, even in the hypothetical case where irreducible representations are contained more than once in the reduction of the space of hybrid orbitals. To illustrate our approach, a hybridization d²sp³ for octahedral molecules is discussed. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

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.     

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.     

Maximum bond order hybrid orbitals

Summary Based on the simplified calculation scheme of the maximum bond order principle and the basic idea of the maximum overlap symmetry orbital method, a simple procedure is suggested for constructing systematically the bonding hybrid orbitals, called maximum bond order hybrid orbitals, for a given molecule from the first-order density matrix obtained from a molecular orbital calculation. As an example, the proposed procedure is performed for some typical small molecules by use of the density matrix obtained from CNDO/2 calculation. It is shown that the bonding hybrid orbitals constructed by using the procedure are extremely close to those by using the natural hybrid orbital procedure and in good agreement with chemical intuition, and that the proposed procedure can be performed more easily than the natural hybrid orbital procedure and can given simultaneously the values of the maximum bond order for all bonds in molecules.The project was supported by National Natural Science Foundation of China and also supported partly by Foundation of Hubei Education Commission     

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.     

On the suitability of strictly localized orbitals for hybrid QM/MM calculations

In the QM/MM method we have developed (LSCF/MM), the QM and the MM parts are held together by means of strictly localized bonding orbitals (SLBOs). Generally these SLBOs are derived from localized bond orbitals (LBOs) that undergo tails deletion, resulting in a nonpredictable change of their properties. An alternative set of SLBOs is provided by the extremely localized molecular orbitals (ELMOs) approach, where the orbitals are rigorously localized on some prefixed atoms without tails on the other atoms of the molecule. A comparative study of SLBOs arising from various localization schemes and ELMOs is presented to test the reliability and the transferability of these functions within the Local Self-Consistent Field (LSCF) framework. Two types of chemical bonds were considered: C--C and C--O single bonds. The localized functions are obtained on the ethane and the methanol molecules, and are tested on beta-alanine and diethyl ether molecules. Moreover, the various protonation forms of beta-alanine have been investigated to illustrate how well the polarity variation of the chemical bond can be handled throughout a chemical process. At last, rotation energy profiles around C--C and C--O bonds are reproduced for butane and fluoromethanol. Energetic, geometric, as well as electronic factors all indicate that ELMO functions are much more transferable from one molecule to another, leading to results closer to the usual SCF reference than any other calculations involving any other localized orbitals. When the shape of the orbital is the most important factor then ELMO functions will perform as well as any other localized orbital.     

Localized orbitals and the Fermi hole

The relationship between localized orbitals and the Fermi hole is demonstrated with contour maps of the Fermi hole in the water molecule. These contour maps indicate the presence of regions in which the Fermi hole is relatively stable, regions in which the shape of the Fermi hole changes rapidly, and regions in which the Fermi hole follows the probe electron smoothly. If a single orbital dominates any region of space, the Fermi hole resembles that orbital for any position of the probe electron in the dominated region.     

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.     

Symmetry coordinates and group orbitals from basis functions

The gradients of the basis functions of group theory are vector-valued basis functions. When one of the components of such a gradient is evaluated at atomic positions, and the values are summed over a set of equivalent atoms, the result represents a symmetry coordinate and/or a group orbital.     

Extremely localized nonorthogonal orbitals by the pairing theorem

Using the concepts of L?wdin pairing theorem, a method is developed to calculate extremely localized, but nonorthogonal, sets of molecular orbitals and their strictly localized counterparts. The method is very suitable to study to what extent a given model of bonding in a given molecule can be considered adequate from the point of view of the actual LCAO-MO (Hartree Fock or DFT) wave function and is expected to be useful for doing local approximations of electron correlation.     

Mixing of generalized parity in molecular orbitals

The concept of generalized parity is introduced. It allows mixing of different symmetries in molecular orbitals in the framework of the Parity Mixing in Orbitals method. An extension of this SCF calculational scheme is also discussed and the relevant secular equations are reported.Alexander von Humboldt Fellow. On leave from the Institute Rudjer Bokovi, Zagreb, Croatia, Yugoslavia.     

Where are the angles? Angular dependence (and independence) of orbitals and functions

Certain conditions under which a collection of atomic orbitals becomes angularly independent are considered. The results are applied to molecules of chemical interest. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Simplified box orbitals: A spatially restricted alternative to the slater‐type orbitals

We introduce a new type of spatially restricted basis function (zero beyond a characteristic r₀ value of the radial coordinate) that makes it possible to obtain, in nonconfined systems, similar results to STO functions. This is important because the use of this kind of functions enables the exact application a sort of zero differential overlap approximation to calculate properties of large systems. Our functions are a modification of the BO‐xZ box orbitals introduced by Lepetit et al. First, we replaced these orbitals by a function that is easier to obtain and generalize, that we named “simplified box orbital” (SBO), and we have shown some advantages of the SBO over BO and standard STO functions. Second, we obtained Gaussian developments for both the original BO orbitals and the new SBO orbitals. In this way, it becomes possible to manage our SBO orbitals with standard quantum chemistry software like GAUSSIAN or similar programs. © 2014 Wiley Periodicals, Inc.     

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.     

Icosahedral hybrid orbitals

A set of twelve equivalent icosahedral hybrid orbitals pointing from the centre to the corners of a regular icosahedron has been obtained. Such hybrids can be used to explain the geometry of twelve-coordinate complexes of a rare-earth atom. Using group theoretical considerations, it is shown that these hybrids can be constructed by linear combination of one s, three p, five d and three f-orbitals. Bearing in mind that the twelve hybrids have identical shape but are oriented differently in space, their mathematical expressions have been obtained by applying geometrical transformations to the sp³d⁵f³ hybrid pointing along the positive z-axis. In order to obtain elegant mathematical expressions, the x, y and z axes have been chosen to be coincident with three orthogonal binary axes of the icosahedron.     

Reference program for molecular calculations with Slater-type orbitals

A program for computing all the integrals appearing in molecular calculation with Slater-type orbitals is reported. The program is mainly intended as a reference for testing and comparing other algorithms and techniques. An analysis of the performance of the program is presented, paying special attention to the computational cost and the accuracy of the results. Results are also compared with others obtained with Gaussian basis sets of similar quality. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1284–1293, 1998     

Electron trajectories in molecular orbitals

The time-dependent Schrödinger equation can be rewritten so that its interpretation is no longer probabilistic. Two well-known and related reformulations are Bohmian mechanics and quantum hydrodynamics. In these formulations, quantum particles follow real, deterministic trajectories influenced by a quantum force. Generally, trajectory methods are not applied to electronic structure calculations as they predict that the electrons in a ground-state, real, molecular wavefunction are motionless. However, a spin-dependent momentum can be recovered from the nonrelativistic limit of the Dirac equation. Therefore, we developed new, spin-dependent equations of motion for the quantum hydrodynamics of electrons in molecular orbitals. The equations are based on a Lagrange multiplier, which constrains each electron to an isosurface of its molecular orbital, as required by the spin-dependent momentum. Both the momentum and the Lagrange multiplier provide a unique perspective on the properties of electrons in molecules.     

Pseudosymmetry analysis of molecular orbitals

We introduce a pseudosymmetry analysis of molecular orbitals by means of the newly proposed irreducible representation measures. To do that we define first what we consider as molecular pseudosymmetry and the relationships of this concept with those of approximate symmetry and quasisymmetry. We develop a general algorithm to quantify the pseudosymmetry content of a given object within the framework of the finite group algebra. The obtained mathematical expressions are able to decompose molecular orbitals by means of the irreducible representations of any reference symmetry point group. The implementation and usefulness of the pseudosymmetry analysis of molecular orbitals is demonstrated in the study of σ and π orbitals in planar and nonplanar polycyclic aromatic hydrocarbons and the t_2g and e_g character of the d‐orbitals in the [FeH₆]^3? anion in its high spin state along the Bailar twist pathway. © 2013 Wiley Periodicals, Inc.