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.     

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.     

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.     

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.     

Fast noniterative orbital localization for large molecules

We use Cholesky decomposition of the density matrix in atomic orbital basis to define a new set of occupied molecular orbital coefficients. Analysis of the resulting orbitals ("Cholesky molecular orbitals") demonstrates their localized character inherited from the sparsity of the density matrix. Comparison with the results of traditional iterative localization schemes shows minor differences with respect to a number of suitable measures of locality, particularly the scaling with system size of orbital pair domains used in local correlation methods. The Cholesky procedure for generating orthonormal localized orbitals is noniterative and may be made linear scaling. Although our present implementation scales cubically, the algorithm is significantly faster than any of the conventional localization schemes. In addition, since this approach does not require starting orbitals, it will be useful in local correlation treatments on top of diagonalization-free Hartree-Fock optimization algorithms.     

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.     

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.     

Localized charge distributions

A localized molecular orbital has been found to extend slightly and regularly into regions away from the chemical bond which contains most of its charge cloud. This was made the basis for a method of transferring localized orbitals among similar molecules. Each localized orbital induces a set of so-called molecule invariant fragments consisting of one bond fragment and collections of geminal fragments, vicinal fragments, and third and fourth neighbor fragments. Localized orbital expansion coefficients in a hybrid basis can be calculated for these molecule invariant fragments without solving any equations or performing any laborious computations. The present work is an application to acylic hydrocarbons. The results are based on the analysis of 33 INDO-SCF molecular orbital wavefunctions in the localized representation. Computational methods for obtaining close approximations to localized orbitals are also discussed. The application of a suggested pseudo-eigenvalue localization method and its accompanying self-consistent iteration process are found to not converge.     

An efficient linear scaling procedure for constructing localized orbitals of large molecules based on the one-particle density matrix

We have developed a linear-scaling algorithm for obtaining the Boys localized molecular orbitals from the one-particle density matrix. The algorithm is made up of two steps: the Cholesky decomposition of the density matrix to obtain Cholesky molecular orbitals and the subsequent Boys localization process. Linear-scaling algorithms have been proposed to achieve linear-scaling calculations of these two steps, based on the sparse matrix technique and the locality of the Cholesky molecular orbitals. The present algorithm has been applied to compute the Boys localized orbitals in a number of systems including α-helix peptides, water clusters, and protein molecules. Illustrative calculations demonstrate that the computational time of obtaining Boys localized orbitals with the present algorithm is asymptotically linear with increasing the system size.     

A localized molecular-orbital assembler approach for Hartree-Fock calculations of large molecules

We describe an alternative fragment-based method, the localized molecular-orbital assembler method, for Hartree-Fock (HF) calculations of macromolecules. In this approach, a large molecule is divided into many small-size fragments, each of which is capped by its local surroundings. Then the conventional HF calculations are preformed on these capped fragments (or subsystems) and the canonical molecular orbitals of these systems are transferred into localized molecular orbitals (LMOs). By assembling the LMOs of these subsystems into a set of LMOs of the target molecule, the total density matrix of the target molecule is constructed and correspondingly the HF energy or other molecular properties can be approximately computed. This approach computationally achieves linear scaling even for medium-sized systems. Our test calculations with double-zeta and polarized double-zeta basis sets demonstrate that the present approach is able to reproduce the conventional HF energies within a few millihartrees for a broad range of molecules.     

Fast orbital localization scheme in molecular fragments resolution

The method for localizing orbitals on a set of predefined molecular fragments is introduced. Regional localized molecular orbitals (RLMO) are obtained through block diagonalization of the one-electron density matrix and further refinement of the resulting eigenvectors. The algorithm is fast and reliable, as is illustrated by a few examples. Potential applications range from conceptual insight into a chemical bonding to reduced scaling computational techniques. RLMOs are particularly well suited for fragmentation computational methods and for exploiting the locality of electronic correlation in post-HF methods.     

Local Hartree–Fock orbitals using a three‐level optimization strategy for the energy

Using the three‐level energy optimization procedure combined with a refined version of the least‐change strategy for the orbitals—where an explicit localization is performed at the valence basis level—it is shown how to more efficiently determine a set of local Hartree–Fock orbitals. Further, a core–valence separation of the least‐change occupied orbital space is introduced. Numerical results comparing valence basis localized orbitals and canonical molecular orbitals as starting guesses for the full basis localization are presented. The results show that the localization of the occupied orbitals may be performed at a small computational cost if valence basis localized orbitals are used as a starting guess. For the unoccupied space, about half the number of iterations are required if valence localized orbitals are used as a starting guess compared to a canonical set of unoccupied Hartree–Fock orbitals. Different local minima may be obtained when different starting guesses are used. However, the different minima all correspond to orbitals with approximately the same locality. © 2013 Wiley Periodicals, Inc.     

A theory of molecules in molecules

SCF wave functions have been calculated using a minimal atomic basis set of Gaussian lobe functions for the para-, meta-, and ortho-forms of the molecules C₆H₄XY, where X, Y can be either of CN, OH, or F. It is found that in all cases the total energies increase in the sequence meta-, para-, ortho-compound. For the molecules containing the CN group the energy differences are extremely small (0.1–1 kcal/mole) for the other molecules they are one to two orders of magnitude larger. The reliability of these results is discussed. The theory of molecules in molecules is applied to these cases. The wave function of C₆H₄XY is constructed from the fragments C₆H₅X and HY by transferring some of the localized orbitals of the wave functions of the fragments and recalculating the orbitals in the region of interaction. For the molecules containing the CN group the energy differences are too small so that they are not correctly reproduced except by the most exact calculations, which involve no approximations other than the transfer of localized orbitals. For the other molecules satisfactory results are obtained.     

The Fock Matrix Analysis for Atomic Orbitals in Molecular Orbitals I. A New Look on the Covalent Bond in H2?

The character of the molecular orbitals can be better accounted for in terms of molecular adapted atomic orbitals and the Fock matrix expanded in these atomic orbital sets. A clean‐cut and unique criterion for the diradicals and the covalent bonds can be given for the molecular orbitals in both restricted and unrestricted Hartree‐Fock wavefunctions. Instead of the picture that overlap charge migrates into the bonding region, the new analysis displays another picture that the charge densities for the electrons with α and β spins give rise to two opposite spin density shifts. If the α one shifts from atom A toward atom B then it is vice versa for the β one. The spin density shifts proceed until the bonding molecular orbitals form.     

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."     

An ab initio method for approximation of the frozen molecular fragment

An ab initio method for calculation on many-electron molecular systems with the approximation of the inactive part of a molecule by frozen molecular fragment is presented. In the following method the SCF calculations are performed in two series. First the molecular orbitals resulting from the first SCF calculation (modest basis set) are localized. In the second SCF run, the basis set is extended for the active part of the molecule, while molecular orbitals of the inactive part, selected from the localized set, are kept frozen. The results are in good agreement with the extended basis set calculation.     

Transferability of Electronic Interaction Energies Between Localized Orbitals

Localized molecular orbitals have been shown to be transferable among structurally-related molecules. We further show that the electronic energies between localized orbitals are transferable and their magnitude can be estimated. A set of parameters of group interaction energies are established from the analysis of small molecules. Use of these parameters is demonstrated successfully for large molecules.