Extremely Localized Molecular Orbitals Research Articles

Extracting Quantitative Information at Quantum Mechanical Level from Noncovalent Interaction Index Analyses.

The noncovalent interaction (NCI) index is nowadays a well-known strategy to detect NCIs in molecular systems. Even though it initially provided only qualitative descriptions, the technique has been recently extended to also extract quantitative information. To accomplish this task, integrals of powers of the electron distribution were considered, with the requirement that the overall electron density can be clearly decomposed as sum of distinct fragment contributions to enable the definition of the (noncovalent) integration region. So far, this was done by only exploiting approximate promolecular electron densities, which are given by the sum of spherically averaged atomic electron distributions and thus represent too crude approximations. Therefore, to obtain more quantum mechanically (QM) rigorous results from NCI index analyses, in this work, we propose to use electron densities obtained through the transfer of extremely localized molecular orbitals (ELMOs) or through the recently developed QM/ELMO embedding technique. Although still approximate, the electron distributions resulting from the abovementioned methods are fully QM and, above all, are again partitionable into subunit contributions, which makes them completely suitable for the NCI integral approach. Therefore, we benchmarked the integrals resulting from NCI index analyses (both those based on the promolecular densities and those based on ELMO electron distributions) against interaction energies computed at a high quantum chemical level (in particular, at the coupled cluster level). The performed test calculations have indicated that the NCI integrals based on ELMO electron densities outperform the promolecular ones. Furthermore, it was observed that the novel quantitative NCI-(QM/)ELMO approach can be also profitably exploited both to characterize and evaluate the strength of specific interactions between ligand subunits and protein residues in protein-ligand complexes and to follow the evolution of NCIs along trajectories of molecular dynamics simulations. Although further methodological improvements are still possible, the new quantitative ELMO-based technique could be already exploited in situations in which fast and reliable assessments of NCIs are crucial, such as in computational high-throughput screenings for drug discovery.

Initial Maximum Overlap Method Embedded with Extremely Localized Molecular Orbitals for Core-Ionized States of Large Systems

Despite great advances in X-ray absorption spectroscopy for the investigation of small molecule electronic structure, the application to biosystems of experimental techniques developed within this research field remains a challenge. To partially circumvent the problem, users resort to theoretical methods to interpret or predict the X-ray absorption spectra of large molecules. To accomplish this task, only low-cost computational strategies can be exploited. For this reason, some of them are single Slater determinant wavefunction approaches coupled with multiscale embedding techniques designed to treat large systems of biological interest. Therefore, in this work, we propose to apply the recently developed IMOM/ELMO embedding method to the determination of core-ionized states. The IMOM/ELMO technique resulted from the combination of the single Slater determinant Δself-consistent-field-initial maximum overlap approach (ΔSCF-IMOM) with the QM/ELMO (quantum mechanics/extremely localized molecular orbital) embedding strategy, a method where only the chemically relevant region of the examined system is treated at fully quantum chemical level, while the rest is described through transferred and frozen extremely localized molecular orbitals (ELMOs). The IMOM/ELMO technique was initially validated by computing core-ionization energies for small molecules, and it was afterwards exploited to study larger biosystems. The obtained results are in line with those reported in previous studies that applied alternative ΔSCF approaches. This makes us envisage a possible future application of the proposed method to the interpretation of X-ray absorption spectra of large molecules.

Three-Layer Multiscale Approach Based on Extremely Localized Molecular Orbitals to Investigate Enzyme Reactions.

Quantum mechanics/molecular mechanics (QM/MM) calculations are widely used embedding techniques to computationally investigate enzyme reactions. In most QM/MM computations, the quantum mechanical region is treated through density functional theory (DFT), which offers the best compromise between chemical accuracy and computational cost. Nevertheless, to obtain more accurate results, one should resort to wave function-based methods, which however lead to a much larger computational cost already for relatively small QM subsystems. To overcome this drawback, we propose the coupling of our QM/ELMO (quantum mechanics/extremely localized molecular orbital) approach with molecular mechanics, thus introducing the three-layer QM/ELMO/MM technique. The QM/ELMO strategy is an embedding method in which the chemically relevant part of the system is treated at the quantum mechanical level, while the rest is described through frozen ELMOs. Since the QM/ELMO method reproduces results of fully QM computations within chemical accuracy and with a much lower computational effort, it can be considered a suitable strategy to extend the range of applicability and accuracy of the QM/MM scheme. In this paper, other than briefly presenting the theoretical bases of the QM/ELMO/MM technique, we will also discuss its validation on the well-tested deprotonation of acetyl coenzyme A by aspartate in citrate synthase.

QM/ELMO: A Multi-Purpose Fully Quantum Mechanical Embedding Scheme Based on Extremely Localized Molecular Orbitals

The development of computationally advantageous methods for the study of large systems is a long-standing research topic in theoretical chemistry. Among these techniques, a prominent place is certainly occupied by the multiscale embedding strategies, from the well-known QM/MM (quantum mechanics/molecular mechanics) methods to the latest and promising fully quantum mechanical approaches. In this Feature Article, we will briefly review the recently proposed QM/ELMO (quantum mechanics/extremely localized molecular orbital) scheme, namely a new multiscale embedding strategy in which the most chemically relevant region of the investigated system is treated at fully quantum chemical level, while the remaining part (namely, the environment) is described by means of transferred extremely localized molecular orbitals that remain frozen throughout the computation. Other than highlighting the theoretical bases, here we will also review the main results obtained through all the currently available variants of the novel method. In particular, we will show how the QM/ELMO embedding scheme has been successfully exploited to perform both ground and excited state calculations, reproducing the results of corresponding fully quantum mechanical computations but with a much lower computational cost. A first application to crystallography will be also discussed, and we will describe how the QM/ELMO approach has been recently coupled with the Hirshfeld atom refinement technique to accurately determine the positions of hydrogen atoms from X-ray diffraction data. Given the reliability and quality of the obtained results, future applications of the current versions of the QM/ELMO embedding strategy to different types of chemical problems are to be expected in the near future. Moreover, further algorithmic improvements and methodological developments are also envisaged, such as the development of a polarizable QM/ELMO scheme accounting for the effects of the QM region on the ELMO subsystem or the use of the new embedding approach in the context of quantum crystallography to perform unprecedented accurate refinements of macromolecular crystal structures.

Extending Libraries of Extremely Localized Molecular Orbitals to Metal Organic Frameworks: A Preliminary Investigation

Libraries of extremely localized molecular orbitals (ELMOs) have been recently assembled to reconstruct approximate wavefunctions of very large biological systems, such as polypeptides and proteins. In this paper, we investigate for the first time the possibility of using ELMO transferability to also quickly obtain wavefunctions, electron densities, and electrostatic potentials of three-dimensional coordination polymers such as metal organic frameworks (MOFs). To accomplish this task, we propose a protocol that, in addition to exploiting the usual exportability of extremely localized molecular orbitals, also takes advantage of the novel QM/ELMO (quantum mechanics/extremely localized molecular orbital) approach to properly describe the secondary building units of MOFs. As a benchmark test, our technique has been applied to the well-known metal organic framework HKUST-1 ({Cu3(BTC)2}n, with BTC=1,3,5-benzenetricarboxylate) to quickly calculate electrostatic potential maps in the small and large cavities inside the network. On the basis of the obtained results, we envisage further improvements and applications of this strategy, which can be also seen as a starting point to perform less computationally expensive quantum mechanical calculations on metal organic frameworks with the goal of investigating transformation phenomena such as chemisorption.

A Step toward the Quantification of Noncovalent Interactions in Large Biological Systems: The Independent Gradient Model-Extremely Localized Molecular Orbital Approach.

The independent gradient model (IGM) is a recent electron density-based computational method that enables to detect and quantify covalent and noncovalent interactions. When applied to large systems, the original version of the technique still relies on promolecular electron densities given by the sum of spherically averaged atomic electron distributions, which leads to approximate evaluations of the inter- and intramolecular interactions occurring in systems of biological interest. To overcome this drawback and perform IGM analyses based on quantum mechanically rigorous electron densities also for macromolecular systems, we coupled the IGM approach with the recently constructed libraries of extremely localized molecular orbitals (ELMOs) that allow fast and reliable reconstructions of polypeptide and protein electron densities. The validation tests performed on small polypeptides and peptide dimers have shown that the novel IGM-ELMO strategy provides results that are systematically closer to the fully quantum mechanical ones and outperforms the IGM method based on the crude promolecular approximation, but still keeping a quite low computational cost. The results of the test calculations carried out on proteins have also confirmed the trends observed for the IGM analyses conducted on small systems. This makes us envisage the future application of the novel IGM-ELMO approach to unravel complicated noncovalent interaction networks (e.g., in protein-protein contacts) or to rationally design new drugs through molecular docking calculations and virtual high-throughput screenings.

Analysis of crystal field effects and interactions using X-ray restrained ELMOs

The purpose of this study is to assess the capability of X-ray restrained molecular wavefunction methods to reproduce the perturbation due to the crystal field.The effect of the crystal environment on the molecular charge densities has been thoroughly investigated experimentally and theoretically. However, the same analysis in terms of molecular orbitals is unprecedented. We specifically analysed the extremely localized molecular orbitals (ELMOs) because they allow for a valence bond like interpretation that is closer to the typical chemical reasoning.For a set of test molecules, we calculated first principles wavefunctions (called primary wavefunctions) and the corresponding scattered intensities (primary structure factors), which we afterwards used for X-ray restrained Hartree-Fock and ELMO calculations. This mimics the typical procedure of modelling wavefunctions from experimental X-ray diffraction, having the primary wavefunction as benchmark.This study follows a previous investigation concerning the effects of electron correlation, where it was shown that those subtle features could be retrieved only to a minor extent. The electric field generated by surrounding molecules is instead more substantial. Therefore, retrieving this information should be more successful and, in case, extremely useful for supramolecular chemistry and crystal engineering. The work also provides useful indications to experimentalists who want to exploit the potentialities offered by the X-ray restrained wavefunction methods.

Fast and Accurate Quantum Crystallography: From Small to Large, from Light to Heavy.

The coupling of the crystallographic refinement technique Hirshfeld atom refinement (HAR) with the recently constructed libraries of extremely localized molecular orbitals (ELMOs) gives rise to the new quantum-crystallographic method HAR-ELMO. This method is significantly faster than HAR but as accurate and precise, especially concerning the free refinement of hydrogen atoms from X-ray diffraction data, so that the first fully quantum-crystallographic refinement of a protein is presented here. However, the promise of HAR-ELMO exceeds large molecules and protein crystallography. In fact, it also renders possible electron-density investigations of heavy elements in small molecules and facilitates the detection and isolation of systematic errors from physical effects.

NCI-ELMO: A New Method To Quickly and Accurately Detect Noncovalent Interactions in Biosystems.

Accurate determinations of noncovalent interactions in biological systems are fundamental to rationalize the structure and to get insights into the functions and the dynamics of macromolecules. Here we propose a new tool for the efficient identification of noncovalent interactions in proteins. The noncovalent interaction (NCI) method, a well-established strategy to detect noncovalent interactions in chemical systems, is coupled with the libraries of extremely localized molecular orbitals (ELMOs), which allow instantaneous reconstruction of quantum mechanically rigorous electron distributions of polypeptides and proteins. Test calculations performed on different interactions show that the new NCI-ELMO strategy always outperforms the original NCI method based on the promolecular approximation, while it is in close agreement with original NCI analyses based on fully quantum mechanical calculations. The new technique allows for unraveling the network of interactions in biological systems and to rapidly monitor their evolutions with time, with possible consequences on the design of new drugs.

Libraries of Extremely Localized Molecular Orbitals. 3. Construction and Preliminary Assessment of the New Databanks.

The fast and reliable determination of wave functions and electron densities of macromolecules has been one of the goals of theoretical chemistry for a long time, and in this context, several linear scaling techniques have been successfully devised over the years. Different approaches have been adopted to tackle this problem, and one of them exploits the fact that, according to the traditional chemical perception, molecules can be seen as constituted of recurring units (e.g., functional groups) with well-defined chemical features. This has led to the development of methods in which the global wave functions or electron densities of macromolecules are obtained by simply transferring density matrices or fuzzy electron densities associated with molecular fragments. In this context, we propose an alternative strategy that aims at quickly reconstructing wave functions and electron densities of proteins through the transfer of extremely localized molecular orbitals (ELMOs), which are orbitals strictly localized on small molecular units and, for this reason, easily transferable from molecule to molecule. To accomplish this task we have constructed original libraries of ELMOs that cover all the possible elementary fragments of the 20 natural amino acids in all their possible protonation states and forms. Our preliminary test calculations have shown that, compared to more traditional methods of quantum chemistry, the transfers from the novel ELMO databanks allow to obtain wave function and electron densities of large polypeptides and proteins at a significantly reduced computational cost. Furthermore, notwithstanding expected discrepancies, the obtained electron distributions and electrostatic potentials are in very good agreement with those obtained at Hartree-Fock and density functional theory (DFT) levels. Therefore, the results encourage to use the new libraries as alternatives to the popular pseudoatom-databases of crystallography in the refinement of crystallographic structures of macromolecules. In particular, in this context, we have already envisaged the coupling of the ELMO databanks with the promising Hirshfeld atom refinement technique to extend the applicability of the latter to very large systems.

Libraries of Extremely Localized Molecular Orbitals. 1. Model Molecules Approximation and Molecular Orbitals Transferability.

Despite more and more remarkable computational ab initio results are nowadays continuously obtained for large macromolecular systems, the development of new linear-scaling techniques is still an open and stimulating field of research in theoretical chemistry. In this family of methods, an important role is occupied by those strategies based on the observation that molecules are generally constituted by recurrent functional units with well-defined intrinsic features. In this context, we propose to exploit the notion of extremely localized molecular orbitals (ELMOs) that, due to their strict localization on small molecular fragments (e.g., atoms, bonds, or functional groups), are in principle transferable from one molecule to another. Accordingly, the construction of orbital libraries to almost instantaneously build up approximate wave functions and electron densities of very large systems becomes conceivable. In this work, the ELMOs transferability is further investigated in detail and, furthermore, suitable rules to construct model molecules for the computation of ELMOs to be stored in future databanks are also defined. The obtained results confirm the reliable transferability of the ELMOs and show that electron densities obtained from the transfer of extremely localized molecular orbitals are very close to the corresponding Hartree-Fock ones. These observations prompt us to construct new ELMOs databases that could represent an alternative/complement to the already popular pseudoatoms databanks both for determining electron densities and for refining crystallographic structures of very large molecules.

Libraries of Extremely Localized Molecular Orbitals. 2. Comparison with the Pseudoatoms Transferability.

Due to both technical and methodological difficulties, determining and analyzing charge densities of very large molecular systems represents a serious challenge that, in the crystallographers community, has been mainly tackled by observing that the so-called pseudoatoms of the electron density multipole expansions are reliably transferable from molecule to molecule. This has led to the construction of pseudoatoms databanks that have allowed successful refinements of crystallographic structures of macromolecules, while taking into account their corresponding reconstructed electron distributions. A recent alternative/complement to the previous approach is represented by techniques based on extremely localized molecular orbitals (ELMOs) that, due to their strict localization on small molecular fragments (e.g., atoms, bonds, and functional groups), are also in principle exportable from system to system. The ELMOs transferability has been already tested in detail, and, in this work, it has been compared to the one of the pseudoatoms. To accomplish this task, electron distributions obtained both through the transfer of pseudoatoms and through the transfer of extremely localized molecular orbitals have been analyzed, especially taking into account topological properties and similarity indexes. The obtained results indicate that all the considered reconstruction methods give completely reasonable and similar charge densities, and, consequently, the new ELMOs libraries will probably represent new useful tools not only for refining crystal structures but also for computing approximate electronic properties of very large molecules.

Extracting Extremely Localized Molecular Orbitals from X-ray Diffraction Data

The accurate determination of electron densities in crystals from high-resolution X-ray diffraction data has become more and more important over the years. The existing techniques to accomplish this task can be subdivided into two great families: the multipole models and the wave function-based strategies. The former, which are the most widely used, are essentially linear scaling and allow an easy chemical interpretation of the obtained molecular charge densities, but they are also characterized by some drawbacks, such as the possible presence of unphysical negative regions in the resulting electron distributions. On the contrary, the latter always provide quantum mechanically rigorous electron densities, but they are more computationally expensive and, above all, the ease of chemical interpretation is almost completely lost. In this context, in order to combine the easy chemical interpretability of the multipole models with the quantum mechanical rigor of the wave-function based methods, we have recently extended the X-ray constrained wave function approach proposed by Jayatilaka in the framework of a quantum chemistry technique for the a priori determination of Extremely Localized Molecular Orbitals (ELMOs), namely we have developed a new strategy that allows to extract from X-ray diffraction data a single Slater determinant built up wit Molecular Orbitals strictly localized on small molecular fragments (e.g., atoms, bonds or functional groups). Preliminary tests have shown that the determination of X-ray constrained ELMOs is really straightforward. Furthermore, given the reliable transferability of the obtained Molecular Orbitals, we are constructing new ELMOs databases that can be used as alternative to the existing pseudo-atoms libraries for refining crystallographic structures and electron distributions of large systems. A detailed comparison between the new technique and the multipole models is also currently under investigation.

X-ray Constrained Extremely Localized Molecular Orbitals: Theory and Critical Assessment of the New Technique.

Following the X-ray constrained wave function approach proposed by Jayatilaka, we have devised a new technique that allows to extract molecular orbitals strictly localized on small molecular fragments from sets of experimental X-ray structure factors amplitudes. Since the novel strategy enables to obtain electron distributions that have quantum mechanical features and that can be easily interpreted in terms of traditional chemical concepts, the method can be also considered as a new useful tool for the determination and the analysis of charge densities from high-resolution X-ray experiments. In this paper, we describe in detail the theory of the new technique, which, in comparison to our preliminary work, has been improved both treating the effects of isotropic secondary extinctions and introducing a new protocol to halt the fitting procedure against the experimental X-ray scattering data. The performances of the novel strategy have been studied both in function of the basis-sets flexibility and in function of the quality of the considered crystallographic data. The tests performed on four different systems (α-glycine, l-cysteine, (aminomethyl)phosphonic acid and N-(trifluoromethyl)formamide) have shown that the achievement of good statistical agreements with the experimental measures mainly depends on the quality of the crystal structures (i.e., geometry positions and thermal parameters) used in the X-ray constrained calculations. Finally, given the reliable transferability of the obtained Extremely Localized Molecular Orbitals (ELMOs), we envisage to exploit the novel approach to construct new ELMOs databases suited to the development of linear-scaling methods for the refinement of macromolecular crystal structures.

A Hylleraas functional based perturbative technique to relax the extremely localized molecular orbital wavefunction

A way to reduce the computational cost associated with the study of large molecules exploits the transfer of extremely localized molecular orbitals (ELMOs). Unfortunately, due to the reduction in the number of variational parameters, the ELMO description is not as accurate as the Hartree-Fock one, although it is qualitatively correct in most of the cases. Therefore, in order to overcome this significant drawback, we propose a perturbative ELMO approach exploiting the Hylleraas functional. Preliminary tests have been performed and the results are promising for future applications to large systems.

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.

Frozen core orbitals as an alternative to specific frontier bond potential in hybrid Quantum Mechanics/Molecular Mechanics methods

The use of Frozen Core Orbitals (FCOs) at the frontier atom is proposed as an alternative to the specific frontier bond potential in the Local-SCF/MM method. The calculations illustrate that frontier bond length is correctly recovered without adding any specific force-field potential. It is shown that the same accuracy is reached for calculations using FCOs arising either from an atomic (ROHF) or from a molecular calculation using Extremely Localized Molecular Orbitals (ELMOs). In addition FCOs do not modify the description of the remaining of the QM subsystem from both an energetic and/or geometric point of view.

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.

A novel extremely localized molecular orbitals based technique for the one-electron density matrix computation

The ‘nearsightedness’ of electronic structure is an underlying principle in many of the linear scaling methods recently developed to study large systems. Among them, there are strategies based on the transfer of orbitals strictly localized on molecular fragments, such as the extremely localized molecular orbitals (ELMOs). Unfortunately, due to the non-orthogonal nature of these orbitals, the density matrix calculation is computationally demanding, so preventing a straightforward application to very large molecules. In this Letter, we show how this problem can be overcome by a proper application of the ‘Divide and Conquer’ strategy to the ELMO approach.

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.