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.     

Divide-and-conquer local correlation approach to the correlation energy of large molecules

A divide-and-conquer local correlation approach for correlation energy calculations on large molecules is proposed for any post-Hartree-Fock correlation method. The main idea of this approach is to decompose a large system into various fragments capped by their local environments. The total correlation energy of the whole system can be approximately obtained as the summation of correlation energies from all capped fragments, from which correlation energies from all adjacent caps are removed. This approach computationally achieves linear scaling even for medium-sized systems. Our test calculations for a wide range of molecules using the 6-31G or 6-31G( * *) basis set demonstrate that this simple approach recovers more than 99.0% of the conventional second-order Moller-Plesset perturbation theory and coupled cluster with single and double excitations correlation energies.     

An efficient fragment-based approach for predicting the ground-state energies and structures of large molecules

An efficient fragment-based approach for predicting the ground-state energies and structures of large molecules at the Hartree-Fock (HF) and post-HF levels is described. The physical foundation of this approach is attributed to the "quantum locality" of the electron correlation energy and the HF total energy, which is revealed by a new energy decomposition analysis of the HF total energy proposed in this work. This approach is based on the molecular fractionation with conjugated caps (MFCC) scheme (Zhang, D. W.; Zhang, J. Z. H. J. Chem. Phys. 2003, 119, 3599), by which a macromolecule is partitioned into various capped fragments and conjugated caps formed by two adjacent caps. We find that the MFCC scheme, if corrected by the interaction between non-neighboring fragments, can be used to predict the total energy of large molecules only from energy calculations on a series of small subsystems. The approach, named as energy-corrected MFCC (EC-MFCC), computationally achieves linear scaling with the molecular size. Our test calculations on a broad range of medium- and large molecules demonstrate that this approach is able to reproduce the conventional HF and second-order Moller-Plesset perturbation theory (MP2) energies within a few millihartree in most cases. With the EC-MFCC optimization algorithm described in this work, we have obtained the optimized structures of long oligomers of trans-polyacetylene and BN nanotubes with up to about 400 atoms, which are beyond the reach of traditional computational methods. In addition, the EC-MFCC approach is also applied to estimate the heats of formation for a series of organic compounds. This approach provides an appealing approach alternative to the traditional additivity rules based on either bond or group contributions for the estimation of thermochemical properties.     

Electrostatic field-adapted molecular fractionation with conjugated caps for energy calculations of charged biomolecules

An electrostatic field-adapted molecular fractionation with conjugated caps (EFA-MFCC) approach is implemented for treating macromolecules with several charge centers. The molecular fragmentation is performed in an "electrostatic field," which is described by putting point charges on charge centers, directly affecting the Hamiltonians of both fragments and conjugated caps. So the present method does not need truncation during the calculation of electrostatic interactions. Our test calculations on a series of charged model systems and biological macromolecules using the HF and B3LYP methods have demonstrated that this approach is capable of describing the electronic structure with accuracy comparable to other fragment-based methods. The EFA-MFCC approach is an alternative way for predicting the total energies of charged macromolecules with acyclic, loop, and intersectional loop structures and interaction energies between two molecules.     

Quasiclassical trajectory calculations on the photodissociation of CF2CHCl at 193 nm: product energy distributions for the HF and HCl eliminations

Quasiclassical trajectory calculations were carried out to determine product energy distributions for the HCl and HF eliminations that take place in the photodissociation of 2-chloro-1,1-difluoroethylene at 193 nm. The trajectories were initiated at the transition states of the HCl and HF elimination channels under microcanonical, quasiclassical conditions, and were propagated with the energies and gradients taken directly from density functional theory calculations. Good agreement with experiment is found, except for the translational energy distribution of the HF elimination channel and the average vibrational energy of the HCl fragment. Possible sources of disagreement are discussed.     

A new algorithm for molecular fragmentation in quantum chemical calculations

In this study, we present a "black-box" method for fragmenting a molecule with a well-defined Kekulé or valence-bond structure into a significant number of smaller fragment molecules that are more amenable to high level quantum chemical calculations. By taking an appropriate linear combination of the fragment energies, we show that it is possible in many cases to obtain highly accurate total energies when compared to the total energy of the full molecule. Our method is derived from the approach reported by Deev and Collins, but it contains significant unique elements, including an isodesmic approach to the fragmentation process. Using a method such as that described in this work it is in principle possible to obtain very accurate total energies of systems containing hundreds, if not thousands, of atoms as the approach is subject to massive parallelization.     

Generalized energy-based fragmentation approach for computing the ground-state energies and properties of large molecules

We present a generalized energy-based fragmentation (GEBF) approach for approximately predicting the ground-state energies and molecular properties of large molecules, especially those charged and polar molecules. In this approach, the total energy (or properties) of a large molecule can be approximately obtained from energy (or properties) calculations on various small subsystems, each of which is constructed to contain a certain fragment and its local surroundings within a given distance. In the quantum chemistry calculation of a given subsystem, those distant atoms (outside this subsystem) are modeled as background point charges at the corresponding nuclear centers. This treatment allows long-range electrostatic interaction and polarization effects between distant fragments to be taken into account approximately, which are very important for polar and charged molecules. We also propose a new fragmentation scheme for constructing subsystems. Our test calculations at the Hartree-Fock and second-order M?ller-Plesser perturbation theory levels demonstrate that the approach could yield satisfactory ground-state energies, the dipole moments, and static polarizabilities for polar and charged molecules such as water clusters and proteins.     

Higher order alchemical derivatives from coupled perturbed self-consistent field theory

We present an analytical approach to treat higher order derivatives of Hartree-Fock (HF) and Kohn-Sham (KS) density functional theory energy in the Born-Oppenheimer approximation with respect to the nuclear charge distribution (so-called alchemical derivatives). Modified coupled perturbed self-consistent field theory is used to calculate molecular systems response to the applied perturbation. Working equations for the second and the third derivatives of HF/KS energy are derived. Similarly, analytical forms of the first and second derivatives of orbital energies are reported. The second derivative of Kohn-Sham energy and up to the third derivative of Hartree-Fock energy with respect to the nuclear charge distribution were calculated. Some issues of practical calculations, in particular the dependence of the basis set and Becke weighting functions on the perturbation, are considered. For selected series of isoelectronic molecules values of available alchemical derivatives were computed and Taylor series expansion was used to predict energies of the "surrounding" molecules. Predicted values of energies are in unexpectedly good agreement with the ones computed using HF/KS methods. Presented method allows one to predict orbital energies with the error less than 1% or even smaller for valence orbitals.     

A self-consistent coulomb bath model using density fitting

A self-consistent Coulomb bath model is presented to provide an accurate and efficient way of performing calculations for interfragment electrostatic and polarization interactions. In this method, a condensed-phase system is partitioned into molecular fragment blocks. Each fragment is embedded in the Coulomb bath due to other fragments. Importantly, the present Coulomb bath is represented using a density fitting method in which the electron densities of molecular fragments are fitted using an atom-centered auxiliary basis set of Gaussian type. The Coulomb bath is incorporated into an effective Hamiltonian for each fragment, with which the electron density is optimized through an iterative double self-consistent field (DSCF) procedure to realize the mutual many-body polarization effects. In this work, the accuracy of interfragment interaction energies enumerated using the Coulomb bath is tested, showing a good agreement with the exact results from an energy decomposition analysis. The qualitative features of many-body polarization effects are visualized by electron density difference plots. It is also shown that the present DSCF method can yield fast and robust convergence with near-linear scaling in performance with increase in system size.     

Implementation of divide-and-conquer method including Hartree-Fock exchange interaction

The divide-and-conquer (DC) method, which is one of the linear-scaling methods avoiding explicit diagonalization of the Fock matrix, has been applied mainly to pure density functional theory (DFT) or semiempirical molecular orbital calculations so far. The present study applies the DC method to such calculations including the Hartree-Fock (HF) exchange terms as the HF and hybrid HF/DFT. Reliability of the DC-HF and DC-hybrid HF/DFT is found to be strongly dependent on the cut-off radius, which defines the localization region in the DC formalism. This dependence on the cut-off radius is assessed from various points of view: that is, total energy, energy components, local energies, and density of states. Additionally, to accelerate the self-consistent field convergence in DC calculations, a new convergence technique is proposed.     

A new quantum method for electrostatic solvation energy of protein

A new method that incorporates the conductorlike polarizable continuum model (CPCM) with the recently developed molecular fractionation with conjugate caps (MFCC) approach is developed for ab initio calculation of electrostatic solvation energy of protein. The application of the MFCC method makes it practical to apply CPCM to calculate electrostatic solvation energy of protein or other macromolecules in solution. In this MFCC-CPCM method, calculation of protein solvation is divided into calculations of individual solvation energies of fragments (residues) embedded in a common cavity defined with respect to the entire protein. Besides computational efficiency, the current approach also provides additional information about contribution to protein solvation from specific fragments. Numerical studies are carried out to calculate solvation energies for a variety of peptides including alpha helices and beta sheets. Excellent agreement between the MFCC-CPCM result and those from the standard full system CPCM calculation is obtained. Finally, the MFCC-CPCM calculation is applied to several real proteins and the results are compared to classical molecular mechanics Poisson-Boltzmann (MM/PB) and quantum Divid-and-Conque Poisson-Boltzmann (D&C-PB) calculations. Large wave function distortion energy (solute polarization energy) is obtained from the quantum calculation which is missing in the classical calculation. The present study demonstrates that the MFCC-CPCM method is readily applicable to studying solvation of proteins.     

Scission shapes of pair fragments in asymmetric and symmetric fission modes

Scission shapes composed of two touching spheroids in asymmetric and symmetric fission modes have been deduced from static potential calculations with the experimental fragment total kinetic energy and excitation energies for a typical fragment pair of ¹⁰³Nb and ¹³⁰Sn in the proton-induced fission of ²³²Th. It was found that the fragment deformation of the heavy fragment ¹³⁰Sn in the symmetric fission mode was extremely large compared with that in the asymmetric fission and those of the complementary fragment ¹⁰³Nb. Such scission shapes also provide internal excitation energies of pair fragments in the two fission modes. The deduced total internal excitation energies of complementary fragments for the two fission modes are nearly the same as the excitation energy of the fissioning nucleus. The results suggest that the two fission modes are strongly characterized by the degrees of fragment deformation of the heavy fragments, not by the total internal excitation energies at scission.     

On the application of the counterpoise correction for the basis set superposition error in geometry optimization calculations of molecular systems: some inconsistent results

It is shown that the conjecture that the total energy for a given molecular or supermolecular system is affected by basis set superposition error (BSSE) leads to inconsistent results. While the calculations of interaction energies, dissociation energies, or energy barriers depend on the fragments (reactants, products) involved in their definitions and, consequently, are affected by BSSE, the total energies of molecular or supermolecular systems do not depend on any virtual fragment partition and are, therefore, BSSE free. Contribution to the Serafin Fraga Memorial Issue.     

香豆素衍生物的荧光发射能计算及XC泛函的合理选择

采用含时密度泛函理论(TDDFT)与单激发组态相互作用(CIS)处理相结合的计算方案对香豆素系列15种已知荧光化合物的发射能进行了系统考察. 结果表明, 发射能与吸收能一样, 其计算值主要取决于交换-相关(XC)泛函的选择. 只要泛函选用得当, 在使用较小基组的TDDFT/6-31G(d)//CIS/3-21G(d)理论水平上即可使绝大部分化合物的实验发射能在精度达0.16 eV以内得以重现. 与吸收能计算不同的是, 无法选用单一的一种泛函来对全系列化合物的发射能作出满意的理论预测. 激发态无明显电荷转移的、7位上有氨(或胺)基取代或有氮原子相连的化合物, 其适用泛函为不含Hartree-Fock(HF)交换能的纯泛函OLYP和BLYP. 而激发态发生较大程度电荷转移的、3 位上有共轭取代基的衍生物, 其适用泛函则为含20%的HF交换成分的混合泛函B3LYP. 因此, 发射能计算中的XC泛函选择, 应同时考虑取代基团效应以及激发态的电子结构特征. 其中, 发射能计算值受XC泛函中HF交换能比例的影响十分敏感. 文中还对激发能计算中的溶剂效应校正方案和激发态几何优化精度的影响进行了讨论.     

Proper Choice of XC Functionals and Calculations of Fluorescence-Emitting Energies for Coumarin Derivatives

A scheme of time-dependent density functional theory (TDDFT) combined with single-excitation configuration interaction (CIS) approach was employed to make a detailed investigation of the emitting energy for fifteen well-known coumarin derivatives. The results showed that the predicted emitting energies as well as the absorption ones were dominated mainly by the exchange-correlation (XC) functional to be used. So long as a functional is properly chosen, the experimental emitting energy of most derivatives can be accurately reproduced within 0.16 eV by a calculation at the TDDFT/6-31G(d)//CIS/3-21G(d) theoretical level. It was found that, nevertheless, the hybrid functional, B3LYP, well predicted the absorption energies for all the fifteen coumarin derivatives but none of the functionals could work equally well for the emitting energy calculations. Two pure functionals, OLYP and BLYP, yield good emitting energies for the 7-aminocoumarins or derivatives with a N atom connected to 7-position, which exhibit inconspicuous charge transfer (CT) in their excited states, whereas the B3LYP hybrid functional, with 20% Hartree-Fock (HF) exchange energy, performs significantly better than OLYP and BLYP for those 3-substituted coumarins with larger CT in excited states. Thus, in comparison with the absorption energies, the selection of proper functionals for the emitting energy calculations becomes more complex. In all probability, it is effective and doable to choose an XC-functional with alterable fraction of HF exchange energy according to the composition and structure characteristics of molecule.     

Communication: variational many-body expansion: accounting for exchange repulsion, charge delocalization, and dispersion in the fragment-based explicit polarization method

A fragment-based variational many-body (VMB) expansion method is described to directly account for exchange repulsion, charge delocalization (charge transfer) and dispersion interactions in the explicit polarization (X-Pol) method. The present VMB/X-Pol approach differs from other fragment molecular orbital (FMO) techniques in two major aspects. First, the wave function for the monomeric system is variationally optimized using standard X-Pol method, as opposed to the iterative update procedure adopted in FMO. Second, the mutual polarizations in the dimeric terms are also variationally determined, whereas single-point energy calculations of the individual dimers embedded in a static monomer field are used in FMO. The second-order (two-body) VMB (VMB2) expansion method is illustrated on a series of water hexamer complexes and one decamer cluster, making use of Hartree-Fock theory, MP2, and the PBE1 and M06 density functionals to represent the monomer and dimer fragments. The computed binding energies are within 2 kcal/mol of the corresponding results from fully delocalized calculations. Energy decomposition analyses reveal specific dimeric contributions to exchange repulsion, charge delocalization, and dispersion. Since the wave functions for one-body and all two-body terms are variationally optimized in VMB2 and X-Pol, it is straightforward to obtain analytic gradient without the additional coupled-perturbed Hartree-Fock step. Thus, the method can be useful for molecular dynamics simulations.     

The Generalized Energy‐Based Fragmentation Approach with an Improved Fragmentation Scheme: Benchmark Results and Illustrative Applications

We propose an improved fragmentation scheme for the generalized energy‐based fragmentation (GEBF) approach, which improves the accuracy of the GEBF approach in total energy calculations and intermolecular interactions. The main modification is to introduce some two‐fragment‐centered primitive subsystems, which are neglected in the previous GEBF implementation. Numerical calculations demonstrate that the present GEBF approach can provide more accurate ground‐state energies and intermolecular interactions. The present GEBF approach with the M06‐2X functional and the cc‐pVTZ basis set are employed to investigate the structures and binding energies in two dimeric species, which are related to pseudopolymorphism of a phenyleneethynylene‐based π‐conjugated molecule. A comparison of the binding free energies in a dimeric species and its corresponding model without C? H???F contacts reveal that the substitution of fluorine atoms weakens the binding of monomers in the dimeric species formed by intermolecular O? H???O hydrogen bonds, but strengthens the binding in the dimer formed by the π–π stacking interaction. Therefore, the C? H???F contacts in these two dimeric species are demonstrated to play a less significant role.     

Fragment occupations in partition Density Functional Theory

The exact ground-state energy and density of a molecule can in principle be obtained via Partition Density Functional Theory (PDFT), a method for calculating molecular properties from Kohn-Sham calculations on isolated fragments. For a given choice of fragmentation, unique fragment densities are found by requiring that the sum of fragment energies be minimized subject to the constraint that the fragment densities sum to the correct molecular ground-state density. We investigate two interrelated aspects of PDFT: the connections between fragment densities obtained via different choices of fragmentation, for which we find "near-additivity", and the nature of their corresponding fragment occupations. Whereas near-integer occupations arise for very large inter-fragment separations, strictly integer occupations appear for small inter-fragment separations. Cases where the fragment chemical potentials cannot be equalized lead to fragment occupations that lock into integers.     

A generalized higher order kernel energy approximation method

We present a general mathematical model that can be used to improve almost all fragment‐based methods for ab initio calculation of total molecular energy. Fragment‐based methods of computing total molecular energy mathematically decompose a molecule into smaller fragments, quantum‐mechanically compute the energies of single and multiple fragments, and then combine the computed fragment energies in some particular way to compute the total molecular energy. Because the kernel energy method (KEM) is a fragment‐based method that has been used with much success on many biological molecules, our model is presented in the context of the KEM in particular. In this generalized model, the total energy is not based on sums of all possible double‐, triple‐, and quadruple‐kernel interactions, but on the interactions of precisely those combinations of kernels that are connected in the mathematical graph that represents the fragmented molecule. This makes it possible to estimate total molecular energy with high accuracy and no superfluous computation and greatly extends the utility of the KEM and other fragment‐based methods. We demonstrate the practicality and effectiveness of our model by presenting how it has been used on the yeast initiator tRNA molecule, ytRN (1YFG in the Protein Data Bank), with kernel computations using the Hartree‐Fock equations with a limited basis of Gaussian STO‐3G type. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010