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.     

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     

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.     

一种计算量小的构造合理最紧缩定域轨道的方法

提出一种计算量小的构造合理最紧缩定域化轨道的方法.非正交定域轨道(NOLMO)没有“正交化尾巴”,比正交定域轨道(OLMO)更局域化、更紧缩、更具可移植性,从而更适合用于化学问题的理论研究.但若在用变分法确定最大限度局域化NOLMO时单纯取消正交条件而不附加其它的限制,会得到趋于线性相关的不合理结果.提出用强制NOLMO与投影自然键轨道(PNBO)的重心重合代替正交化条件确定合理的最大限度局域化NOLMO(合理最紧缩定域轨道)的方法.对一系列不同类型分子的计算结果表明,用该方法可以得到空间分布合理及线性独立的NOLMO,延伸度与文献已有的最佳结果接近,而计算量大幅度降低.由于构造投影自然键轨道的计算量随分子中原子数目的增加只是线性增长,因此,该方法可以用来构造较大体系的合理最紧缩定域轨道     

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     

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.     

Maximal orbital analysis of molecular wavefunctions

We describe a new way to decompose one-electron orbitals of a molecule into atom-centered or fragment-centered orbitals by an approach that we call “maximal orbital analysis” (MOA). The MOA analysis is based on the corresponding orbital transformation (COT) that has the unique mathematical property of maximizing any sub-trace of the overlap matrix, in Hilbert metric sense, between two sets of nonorthogonal orbitals. Here, one set comprises the molecule orbitals (Hartree–Fock, Kohn–Sham, complete-active-space, or any set of orthonormal molecular orbitals), the other set comprises the basis functions associated with an atom or a group of atoms. We show in prototypical molecular systems such as a water dimer, metal carbonyl complexes, and a mixed-valent transition metal complex, that the MOA orbitals capture very well key aspects of wavefunctions and the ensuing chemical concepts that govern electronic interactions in molecules. © 2019 Wiley Periodicals, Inc.     

定域分子轨道重心和分子结构的关系

本文讨论了定域分子轨道（ＬＭＯ）的重心和分子中化学键之间的关系，建立了利用定域分子轨道重心判断分子中成键情况及其化学键类型的方法，进而在ａｂ ｉｎｉｔｉｏ所得正则分子轨道（ＣＭＯ）定域化的基础上，计算了一些常见分子的ＬＭＯ重心；由此讨论了这些分子的化学键性质及其成键规律，探讨了一些化合物的反应性能。     

Transferable group contributions for a variety of chemical phenomena and compounds

An approximate method for calculating molecular electrostatic potential (MEP) maps and atomic point charge models for large molecules in a reduced computational time is proposed and tested for two widely used basis sets (STO-3G and 6–31G*). The method avoids the molecular orbital calculation of the whole system by expressing its first order electronic density matrix in terms of transferable localized orbitals (TLO), previously determined on model molecules, via a localization process followed by the cutting of the tails, and stored in two databases (one for each basis set). For systems with a canonic electronic structure TLO are made of a single vector, involving either two nuclei (to describe the covalent bond between those atoms) or one nucleus (to describe lone pairs and inner shells). Conversely, delocalized systems require many-center TLO, formed by a suitable number of vectors. Density functions of large chemical compounds can thus be built up automatically from a code that recognizes which fragments are contained in the system of interest, extracts them from the chosen database, reorders the atoms consistently with the pertinent TLO and places them in the correct position and orientation on the relevant atoms. A great number of chemical groups were parameterized and the efficiency of the method was evaluated on different systems, including aliphatic hydrocarbons. Numerical calculations on several molecules revealed that this approximation brought no significant loss of accuracy with respect to the corresponding Hartree-Fock (HF) values for the examined properties. Although the method is specifically designed to produce approximate wavefunctions, the point charge models obtained by fitting the corresponding MEP represent a viable alternative when ab initio HF calculations are not affordable, and can be used in connection with any popular force field.From the Proceedings of the 28th Congreso de Químicos Teóricos de Expresión Latina (QUITEL 2002)     

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.     

YinYang atom: a simple combined ab initio quantum mechanical molecular mechanical model

A simple interface is proposed for combined quantum mechanical (QM) molecular mechanical (MM) calculations for the systems where the QM and MM regions are connected through covalent bonds. Within this model, the atom that connects the two regions, called YinYang atom here, serves as an ordinary MM atom to other MM atoms and as a hydrogen-like atom to other QM atoms. Only one new empirical parameter is introduced to adjust the length of the connecting bond and is calibrated with the molecule propanol. This model is tested with the computation of equilibrium geometries and protonation energies for dozens of molecules. Special attention is paid on the influence of MM point charges on optimized geometry and protonation energy, and it is found that it is important to maintain local charge-neutrality in the MM region in order for the accurate calculation of the protonation and deprotonation energies. Overall the simple YinYang atom model yields comparable results to some other QM/MM models.     

含双键有机小分子的定域分子轨道的研究

选择22种具有双键的有机小分子(含C、H、O和N原子)作模型分子,应用ab initioSTO-3G和Foster-Boys定域化分子轨道程序,对它们的定域分子轨道进行了研究,得到了定域分子轨道能量,确定了相应分子轨道间的相互作用。对烯类分子的光电子能谱进行了分析,结果较为满意。     

A non-orthogonal Kohn-Sham method using partially fixed molecular orbitals

A density functional theory method using partially fixed molecular orbitals (PFMOs) is presented. The PFMOs, which have some fixed molecular orbital coefficients and are non-orthogonal, are a generalization of the extreme localized orbitals (ELMOs) of Couty, Bayse, and Hall (1997) Theor Chem Acc 97:96. A non-orthogonal Kohn-Sham method with these PFMOs is derived, and is applied to molecular calculations on the ionization potential of pyridine, the energy difference between cis- and trans-butadiene, the reaction barrier height of the cyclobutene-cis-butadiene interconversion, and the potential energy curve of the hydrogen shift reaction of hydroxycarbene to formaldehyde. The PFMO Kohn-Sham method reproduces well the results of the full Kohn-Sham method without having a restriction on the molecular orbital coefficients. The difference is less than 0.1 eV in the ionization potential and about 0.1 kcal/mol in the barrier height and in the potential energy calculations.     

Atom and Bond Fukui Functions and Matrices: A Hirshfeld‐I Atoms‐in‐Molecule Approach

The Fukui function is often used in its atom‐condensed form by isolating it from the molecular Fukui function using a chosen weight function for the atom in the molecule. Recently, Fukui functions and matrices for both atoms and bonds separately were introduced for semiempirical and ab initio levels of theory using Hückel and Mulliken atoms‐in‐molecule models. In this work, a double partitioning method of the Fukui matrix is proposed within the Hirshfeld‐I atoms‐in‐molecule framework. Diagonalizing the resulting atomic and bond matrices gives eigenvalues and eigenvectors (Fukui orbitals) describing the reactivity of atoms and bonds. The Fukui function is the diagonal element of the Fukui matrix and may be resolved in atom and bond contributions. The extra information contained in the atom and bond resolution of the Fukui matrices and functions is highlighted. The effect of the choice of weight function arising from the Hirshfeld‐I approach to obtain atom‐ and bond‐condensed Fukui functions is studied. A comparison of the results with those generated by using the Mulliken atoms‐in‐molecule approach shows low correlation between the two partitioning schemes.     

The behavior of transition metal nitrido bonds towards protonation rationalized by means of localized bonding schemes and their weights

A new computational scheme is applied to rationalize the different protonation behaviors of the nitrido complexes [L'Mn(V)(N)(acac)](+), [LCr(V)(N)(acac)](+), and [LV(V)(N)(acac)](+). L and L' represent the macrocycles 1,4,7-triazacyclononane and its N-methylated derivative, respectively, and acac is the bidentate monoanion pentane-2,4-dionate. The bonds of the complexes are partitioned into bonds to be investigated and bonds of lesser interest. The investigated bonds are the transition metal nitrido bonds M(V)[triple chemical bond]N| (M = Mn, Cr, and V) and the bonds of lesser interest are located in the ligands. The ligand bonds are described by means of the strongly occupied natural bond orbitals. The electrons in the M(V)[triple chemical bond]N| nitrido bonds, however, are treated more accurately. A full configuration interaction procedure is applied in the space spanned by the strongly occupied natural bond orbitals and their corresponding antibonding orbitals. Localized bonding schemes and their weights are obtained for the d(pi)-p(pi) bonds of interest. This is achieved by representing the two-center natural bond orbitals for a d(pi)-p(pi) bond by the one-center natural hybrid orbitals localized at the bond atoms. The obtained bonding schemes are close to orthogonal valence bond structures. Their weights indicate that the nitrido nitrogen in [LV(V)(N)(acac)](+) is more easily protonated than the nitrido nitrogens in [L'Mn(V)(N)(acac)](+) and [LCr(V)(N)(acac)](+). This result is in good accord with experiment.     

Theoretical Study on the Hydrogen Bonding Interactions in Complexes of 5‐Hydroxytryptamine with Water

The energies, geometries and harmonic vibrational frequencies of 1:1 5‐hydroxytryptamine‐water (5‐HT‐H₂O) complexes are studied at the MP2/6‐311++G(d,p) level. Natural bond orbital (NBO), quantum theory of atoms in molecules (QTAIM) analyses and the localized molecular orbital energy decomposition analysis (LMO‐EDA) were performed to explore the nature of the hydrogen‐bonding interactions in these complexes. Various types of hydrogen bonds (H‐bonds) are formed in these 5‐HT‐H₂O complexes. The intermolecular C4H5^5‐HT···O^w H‐bond in HTW3 is strengthened due to the cooperativity, whereas no such cooperativity is found in the other 5‐HT‐H₂O complexes. H‐bond in which nitrogen atom of amino in 5‐HT acted as proton donors was stronger than other H‐bonds. Our researches show that the hydrogen bonding interaction plays a vital role on the relative stabilities of 5‐HT‐H₂O complexes.     

Orbital Energy Order and the Through Bond Interaction in Homonuclear Diatomic Molecules

The homonuclear diatomic molecules are the simplest systems having both the σ framework and the lone pair orbitals n_a and _b for investigating their through space and through bond interaction. The striking orbital energy order n_g～ n_a+ n_b > n_n ～ n_a - n_b has been accounted for by the through bond interaction. However, when the p-content in the lone pair orbitals n_a and n_b decreases, one may have the reverse orbital energy order: n_g < n_g. A reverse orbital energy order has been found in F₂ and Cl₂, whose n_a and n_b are almost pure s-type atomic orbitals. The reverse order also occurs in molecule N₂ when the internuclear distance is larger man 1.5 Å. It is also found that the detail through space and through bond interaction and the eventual orbital energy order for n_g and n_u can be accounted for by the Fock operator within the localized molecular orbital space.