Calculation of wave‐functions with frozen orbitals in mixed quantum mechanics/molecular mechanics methods. II. Application of the local basis equation

The application of the local basis equation (Ferenczy and Adams, J. Chem. Phys. 2009 , 130, 134108) in mixed quantum mechanics/molecular mechanics (QM/MM) and quantum mechanics/quantum mechanics (QM/QM) methods is investigated. This equation is suitable to derive local basis nonorthogonal orbitals that minimize the energy of the system and it exhibits good convergence properties in a self‐consistent field solution. These features make the equation appropriate to be used in mixed QM/MM and QM/QM methods to optimize orbitals in the field of frozen localized orbitals connecting the subsystems. Calculations performed for several properties in divers systems show that the method is robust with various choices of the frozen orbitals and frontier atom properties. With appropriate basis set assignment, it gives results equivalent with those of a related approach [G. G. Ferenczy previous paper in this issue] using the Huzinaga equation. Thus, the local basis equation can be used in mixed QM/MM methods with small size quantum subsystems to calculate properties in good agreement with reference Hartree–Fock–Roothaan results. It is shown that bond charges are not necessary when the local basis equation is applied, although they are required for the self‐consistent field solution of the Huzinaga equation based method. Conversely, the deformation of the wave‐function near to the boundary is observed without bond charges and this has a significant effect on deprotonation energies but a less pronounced effect when the total charge of the system is conserved. The local basis equation can also be used to define a two layer quantum system with nonorthogonal localized orbitals surrounding the central delocalized quantum subsystem. © 2013 Wiley Periodicals, Inc.     

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.     

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.     

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     

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.     

An analysis of the complex molecular orbitals method

Numerical results for the ground state of the HN ₂ ⁺ and HCO⁺ molecular ions at their near equilibrium geometry, obtained by the complex molecular orbitals (CMO) method in the extended basis set, are reported. The CMO wavefunction of the HN ₂ ⁺ ion is compared with the CI wavefunction obtained in the same basis set. This reveals the nature of approximations inherent in the CMO method. A peculiar feature of the occupation numbers of the CMO natural orbitals is also explained.Alexander von Humboldt Fellow. On leave from the Institute Rudjer Bokovi, Zagreb, Croatia, Yugoslavia.     

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.     

Quantum molecular mechanics—a noniterative procedure for the fast Ab Initio calculation of closed shell systems

In this article, we advance the foundations of a strategy to develop a molecular mechanics method based not on classical mechanics and force fields but entirely on quantum mechanics and localized electron‐pair orbitals, which we call quantum molecular mechanics (QMM). Accordingly, we introduce a new manner of calculating Hartree–Fock ab initio wavefunctions of closed shell systems based on variationally preoptimized nonorthogonal electron pair orbitals constructed by linear combinations of basis functions centered on the atoms. QMM is noniterative and requires only one extremely fast inversion of a single sparse matrix to arrive to the one‐particle density matrix, to the electron density, and consequently, to the ab initio electrostatic potential around the molecular system, or cluster of molecules. Although QMM neglects the smaller polarization effects due to intermolecular interactions, it fully takes into consideration polarization effects due to the much stronger intramolecular geometry distortions. For the case of methane, we show that QMM was able to reproduce satisfactorily the energetics and polarization effects of all distortions of the molecule along the nine normal modes of vibration, well beyond the harmonic region. We present the first practical applications of the QMM method by examining, in detail, the cases of clusters of helium atoms, hydrogen molecules, methane molecules, as well as one molecule of HeH⁺ surrounded by several methane molecules. We finally advance and discuss the potentialities of an exact formula to compute the QMM total energy, in which only two center integrals are involved, provided that the fully optimized electron‐pair orbitals are known. © 2012 Wiley Periodicals, Inc.     

New program for molecular calculations with Slater‐type orbitals

A new program for computing all the integrals appearing in molecular calculations with Slater‐type orbitals (STO) is reported. This program follows the same philosophy as the reference pogram previously reported but introduces two main changes: Local symmetry is profited to compute all the two‐electron integrals from a minimal set of seed integrals, and a new algorithm recently developed is used for computing the seed integrals. The new code reduces between one and two orders of magnitude the computational cost in most polyatomic systems. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 148–153, 2001     

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.     

Determination of extremely localized molecular orbitals in the framework of density functional theory

Extremely localized molecular orbitals are rigorously localized on only a preselected set of atoms and do not have any tails outside the localization region. The importance of these orbitals lies in their ability to be transferred from one molecule to another one. A new algorithm to determine extremely localized molecular orbitals in the framework of the density functional theory method is presented. This could also be a valuable tool in the quantum mechanics/molecular mechanics methodology where localized molecular orbitals are used to describe covalent bonds across the frontier region. The present approach is used to build up the electron density of thymopentin, a polypeptide constituted by five residues, starting from extremely localized molecular orbitals determined on a set of model molecules. The results obtained confirm good transferability properties for these orbitals.Proceedings of the 11th International Congress of Quantum Chemistry satellite meeting in honor of Jean-Louis Rivail     

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.     

Pattern recognition in molecular quantum mechanics

It is shown that the classical concept of an open system does not encompass quantal systems but has to be replaced by the non-Boolean notion of an entangled system. Molecular, chemical, or biological phenomena can be considered to be reduced to a fundamental theory like quantum mechanics only if the fundamental and the phenomenological theories are formally and interpretatively connected, and if the classifications used in the empirical sciences are shown to follow from a single set of fundamental dynamical laws. These conditions enforce a non-statistical and ontic interpretation of quantum mechanics, hence a non-Boolean calculus of propositions. In this interpretation the notion of a world state is well-defined, its Schmidt-decomposition defines a background-dependent model state for molecular systems and creates the phenomena we can observe. To any molecular system there is associated in an objective way a nonnegative number which we call the integrity. The integrity measures the inherent fuzziness of the system concept in a holistic theory, and is used to define recognizable molecular patterns.     

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     

Trap mechanism based on frontier molecular orbitals of additives in polyethylene insulators: A theoretical study and molecular design strategy

In this work, the energy gaps (E_g) of highest occupied orbitals and lowest unoccupied orbitals, trap energies (E_t(e) and E_t(h)) and excited energies of polyethylene model compound, typical defect compound, acetophonene, and 33 designed additives are obtained using density functional method at B3LYP/6–311+G(d, p) level. The correlation between trapping‐electrons (holes) abilities of additives and molecular frontier orbitals is established, and a new understanding for trap mechanism based on chemical molecular orbital levels is given for the first time which could be used to filter qualitative additives as voltage stabilizers of polyethylene. The role of trap energies and the energy gaps on discussing space charge accumulation and electric breakdown is analyzed in detail. A molecular design strategy for potential additives of cross‐linked polyethylene insulated high‐voltage cable is shown based on conjugation effect, substituents character, and polycyclic aromatic compounds. © 2015 Wiley Periodicals, Inc.     

Synthesizing quantum probability by a single chaotic complex‐valued trajectory

The current trajectory interpretation of quantum mechanics is based on an ensemble viewpoint that the evolution of an ensemble of Bohmian trajectories guided by the same wavefunction Ψ converges asymptotically to the quantum probability . Instead of the Bohm's ensemble‐trajectory interpretation, the present paper gives a single‐trajectory interpretation of quantum mechanics by showing that the distribution of a single chaotic complex‐valued trajectory is enough to synthesize the quantum probability. A chaotic complex‐valued trajectory manifests both space‐filling (ergodic) and ensemble features. The space‐filling feature endows a chaotic trajectory with an invariant statistical distribution, while the ensemble feature enables a complex‐valued trajectory to envelop the motion of an ensemble of real trajectories. The comparison between complex‐valued and real‐valued Bohmian trajectories shows that without the participation of its imaginary part, a single real‐valued trajectory loses the ensemble information contained in the wavefunction Ψ, and this explains the reason why we have to employ an ensemble of real‐valued Bohmian trajectories to recover the quantum probability . © 2015 Wiley Periodicals, Inc.     

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.