The fragment molecular orbital method for geometry optimizations of polypeptides and proteins

The fragment molecular orbital method (FMO) has been used with a large number of wave functions for single-point calculations, and its high accuracy in comparison to ab initio methods has been well established. We have developed the analytic derivative of the electrostatic interaction between far separated fragments and performed a number of restricted Hartree-Fock (RHF) geometry optimizations using FMO and ab initio methods. In particular, the alpha-helix, beta-turn, and extended conformers of a 10-residue polyalanine were studied and the good FMO accuracy was established (the rms deviations for the former two forms were about 0.2 A and for the latter structure about 0.001 A). Met-enkephalin dimer was used as a model for the polypeptide binding and computed at the 3-21G and 6-31G* levels with a similar accuracy achieved; the error in the binding energy predictions (FMO vs ab initio) was 1-3 kcal/mol. Chignolin (PDB: 1uao) and an agonist polypeptide of the erythropoietin receptor protein (emp1) were optimized at the 3-21(+)G level, with the rms deviation from ab initio of about 0.2 A, or 0.5 degrees in terms of bond angles. The effect of solvation on the structure optimization was studied in chignolin and the Trp-cage miniprotein construct (PDB:1l2y), by describing water with TIP3P. The computed structures in gas phase and solution are compared to each other and experiment.     

Estimating the hessian for gradient-type geometry optimizations

Optimization methods that use gradients require initial estimates of the Hessian or second derivative matrix; the more accurate the estimate, the more rapid the convergence. For geometry optimization, an approximate Hessian or force constant matrix is constructed from a simple valence force field that takes into account the inherent connectivity and flexibility of the molecule. Empirical rules are used to estimate the diagonal force constants for a set of redundant internal coordinates consisting of all stretches, bends, torsions and out-of-plane deformations involving bonded atoms. The force constants are transformed from the redundant internal coordinates to Cartesian coordinates, and then from Cartesian coordinates to the non-redundant internal coordinates used in the specification of the geometry and optimization. This method is especially suitable for cyclic molecules. Problems associated with the choice of internal coordinates for geometry optimization are also discussed.Fellow of the Alfred P. Sloan Foundation, 1981–83     

The effect of numerical error on the reproducibility of molecular geometry optimizations

Geometry optimization is one of the most often applied techniques in computational drug discovery. Although geometry optimization routines are generally deterministic, the minimization trajectories can be extremely sensitive to initial conditions, especially in case of larger systems such as proteins. Simple manipulations such as coordinate transformations (translations and rotations), file saving and retrieving, and hydrogen addition can introduce small variations ( approximately 0.001 A) in the starting coordinates which can drastically affect the minimization trajectory. With large systems, optimized geometry differences of up to 1 A RMSD and final energy differences of several kcal/mol can be observed when using many commercially available software packages. Differences in computer platforms can also lead to differences in minimization trajectories. Here we demonstrate how routine structure manipulations can introduce small variations in atomic coordinates, which upon geometry optimization, can give rise to unexpectedly large differences in optimized geometries and final energies. We also show how the same minimizations run on different computer platforms can also lead to different results. The implications of these findings on routine computational chemistry procedures are discussed.     

Invariance criteria and symmetry conservation rules for geometry optimizations

The objective of this paper is to analyze the behaviour of some minimization methods such as steepest descent method, generalized Newton and quasi-Newton methods under transformations of the variables of the function to be minimized. Energy and molecular coordinates are the function and the variables, respectively, in the case of geometry optimizations. Invariant levels are shown to be decisive for the area where the minimization methods can be successfully employed without rescaling of the coordinates. Specific conditions for symmetry conservation are worked out in context of invariant levels. Symmetry making, breaking and conservation are shown with working examples of geometry optimizations and calculation of energy minimum paths on the basis of certain kinds of molecular coordinates.     

MP6方法在分子构型优化中的应用

用多体微扰方法MP6对三个非氢原子组成的分子体系,分别在cc-pVDZ、cc-pVTZ、和cc-pVQZ基组下,进行了构型优化计算,并利用Dunning提出的经验公式把结果外推到CBS极限。由于各种误差的相互抵消,MP2的构型优化结果较MP6或MP4的好。用耦合簇方法(CCSD(T))在同样条件下进行了优化计算,结果比微扰方法更具有优势。     

Basis sets for geometry optimizations of second-row transition metal inorganic and organometallic complexes

Modest-sized basis sets for the second-row transition metal atoms are developed for use in geometry optimization calculations. Our method is patterned after previous work on basis sets for first-row transition metal atoms. The basis sets are constructed from the minimal basis sets of Huzinaga and are augmented with a set of diffuse p and d functions. The exponents of these diffuse functions are chosen to minimize both the difference between the calculated and experimental equilibrium geometries and the total molecular energies for several second-row transition metal inorganic and organon etallic complexes. Slightly smaller basis sets, based on the same Huzinaga minimal sets but augmented with a set of diffuse s and p functions rather than diffuse p and d functions, are also presented. The performance of these basis sets is tested on a wide variety of second-row transition metal inorganic and organometallic complexes and is compared to pseudopotential basis sets incorporating effective core potentials.     

Elongation method with intermediate mechanical and electrostatic embedding for geometry optimizations of polymers

The elongation method with intermediate mechanical and electrostatic embedding (ELG-IMEE) is proposed. The electrostatic embedding uses atomic charges generated by a charge sensitivity analysis (CSA) method and parameterized for three different population analyses, namely, the Merz–Singh–Kollman scheme, the charge model 5, and the atomic polar tensor. The obtained CSA models were tested on two model systems. Test calculations show that the electrostatic embedding provides several times of decrease in the difference of energies of testing and reference calculations in comparison with the conventional elongation approach (ELG). The mechanical embedding is implemented in a combination of the conventional elongation method and the ONIOM approach. Moreover, it was demonstrated that the geometry optimization with the ELG-IMEE reduces the errors in the optimized structures by about one order in root-mean-square deviation, when compared to ELG.     

Quantum chemical geometry optimizations in proteins using crystallographic raw data

A method is developed for the combination of quantum chemical geometry optimizations and crystallographic structure refinement. The method is implemented by integrating the quantum chemical software Turbomole with the crystallographic software Crystallography and NMR System (CNS), using three small procedures transferring information between the two programs. The program (COMQUM-X)is used to study the binding of the inhibitor N-methylmesoporphyrin to ferrochelatase, and we show that the method behaves properly and leads to an improvement of the structure of the inhibitor. It allows us to directly quantify in energy terms how much the protein distort the structure of the bound inhibitor compared to the optimum vacuum structure (4-6 kJ/mol). The approach improves the standard combined quantum chemical and molecular mechanics (QC/MM) approach by guaranteeing that the final structure is in accordance with experimental data (the reflections) and avoiding the risk of propagating errors in the crystal coordinates. The program can also be seen as an improvement of standard crystallographic refinement, providing an accurate empirical potential function for any group of interest. The results can be directly interpreted in standard crystallographic terms (e.g., R factors or electron density maps). The method can be used to interpret crystal structures (e.g., the protonation status of metal-bound water molecules) and even to locally improve them.     

Genetic algorithms: A robust scheme for geometry optimizations and global minimum structure problems

A Genetic Algorithm for Geometry Optimizations (GALGO) program has been developed to study the efficiency of this method of finding global minimum structures. Using a semiempirical tight-binding potential, the behavior of different genetic algorithm (GA) operators has been tested for the linear chain isomer of a C₈ cluster. An optimum set of parameters for the GA operators is proposed for this problem and afterward is used to obtain the global minimum structure of rare-gas atomic clusters of up to 13 atoms using the 12–6 Lennard-Jones interatomic pair potential. © 1995 by John Wiley & Sons, Inc.     

Ground and excited state CASPT2 geometry optimizations of small organic molecules

A method for computing second-order multiconfigurational perturbation theory (CASPT2) energy gradients numerically has been implemented and applied to a range of elementary organic chromophores, including 1,3 butadiene, acrolein, and two protonated Schiff bases. Geometries of ground and excited states-as well as conical intersections-are compared with the corresponding CASSCF structures, illustrating the effect of including the correction for dynamical electron correlation. It is shown that the differences between the two methods are not readily categorized, but that, while individual changes in bond lengths can be quite large ( approximately 0.01-0.02 A), the natures and CASPT2 energetics of the structures remain similar. Exceptions to this tend to be systems that have a strong ionic character and that are not well described at the CASSCF level. Basis set effects (double- vs. triple-zeta) were examined for a limited number of examples, and found to be quite dramatic at both levels of theory.     

Models of molecular geometry

Although the structure of almost any molecule can now be obtained by ab initio calculations chemists still look for simple answers to the question "What determines the geometry of a given molecule?" For this purpose they make use of various models such as the VSEPR model and qualitative quantum mechanical models such as those based on the valence bond theory. The present state of such models, and the support for them provided by recently developed methods for analyzing calculated electron densities, are reviewed and discussed in this tutorial review.     

Geometry optimizations of open-shell systems with the fragment molecular orbital method

The ability to perform geometry optimizations on large molecular systems is desirable for both closed- and open-shell species. In this work, the restricted open-shell Hartree-Fock (ROHF) gradients for the fragment molecular orbital (FMO) method are presented. The accuracy of the gradients is tested, and the ability of the method to reproduce adiabatic excitation energies is also investigated. Timing comparisons between the FMO method and full ab initio calculations are also performed, demonstrating the efficiency of the FMO method in modeling large open-shell systems.     

Excited state geometry optimizations by analytical energy gradient of long-range corrected time-dependent density functional theory

An analytical excitation energy gradient of long-range corrected time-dependent density functional theory (LC-TDDFT) is presented. This is based on a previous analytical TDDFT gradient formalism, which avoids solving the coupled-perturbed Kohn-Sham equation for each nuclear degree of freedom. In LC-TDDFT, exchange interactions are evaluated by combining the short-range part of a DFT exchange functional with the long-range part of the Hartree-Fock exchange integral. This LC-TDDFT gradient was first examined by calculating the excited state geometries and adiabatic excitation energies of small typical molecules and a small protonated Schiff base. As a result, we found that long-range interactions play a significant role even in valence excited states of small systems. This analytical LC-TDDFT gradient was also applied to the investigations of small twisted intramolecular charge transfer (TICT) systems. By comparing with calculated ab initio multireference perturbation theory and experimental results, we found that LC-TDDFT gave much more accurate absorption and fluorescence energies of these systems than those of conventional TDDFTs using pure and hybrid functionals. For optimized excited state geometries, LC-TDDFT provided fairly different twisting and wagging angles of these small TICT systems in comparison with conventional TDDFT results.     

Random optimizations for direct heteronuclear correlations

A novel optimization method is described for the acquisition of direct one‐bond heteronuclear correlations. The RDSQC (Randomly optimized Direct correlation Single Quantum Coherence) experiment utilizes an optimization based on the randomly ordered sampling of a range of couplings. The random order of the l/(2*(¹J_CHmin)) delays removes the signal dependency on a single type of apodization, thus eliminating a significant portion of the F₁ artifacts induced in the accordion‐optimized ADSQC experiment. Compared to the statically optimized GHSQC, the randomly optimized data maintains the desired signal intensity in most cases, with a small loss for the weakly coupled proton‐carbon pairs and significant gains for the more strongly coupled pairs. Compared to the accordion‐optimized ADSQC data, the randomly optimized data afforded similar signal‐to‐noise without the F₁ modulated artifacts simplifying spectral interpretation.     

Retrieval of crystallographically-derived molecular geometry information

The crystallographically determined bond length, valence angle, and torsion angle information in the Cambridge Structural Database (CSD) has many uses. However, accessing it by means of conventional substructure searching requires nontrivial user intervention. In consequence, these valuable data have been underutilized and have not been directly accessible to client applications. The situation has been remedied by development of a new program (Mogul) for automated retrieval of molecular geometry data from the CSD. The program uses a system of keys to encode the chemical environments of fragments (bonds, valence angles, and acyclic torsions) from CSD structures. Fragments with identical keys are deemed to be chemically identical and are grouped together, and the distribution of the appropriate geometrical parameter (bond length, valence angle, or torsion angle) is computed and stored. Use of a search tree indexed on key values, together with a novel similarity calculation, then enables the distribution matching any given query fragment (or the distributions most closely matching, if an adequate exact match is unavailable) to be found easily and with no user intervention. Validation experiments indicate that, with rare exceptions, search results afford precise and unbiased estimates of molecular geometrical preferences. Such estimates may be used, for example, to validate the geometries of libraries of modeled molecules or of newly determined crystal structures or to assist structure solution from low-resolution (e.g. powder diffraction) X-ray data.     

Time-dependent density functional theory gradients in the Amsterdam density functional package: geometry optimizations of spin-flip excitations

An implementation of time-dependent density functional theory (TDDFT) energy gradients into the Amsterdam density functional theory program package (ADF) is described. The special challenges presented by Slater-type orbitals in quantum chemical calculation are outlined with particular emphasis on details that are important for TDDFT gradients. Equations for the gradients of spin-flip TDDFT excitation energies are derived. Example calculations utilizing the new implementation are presented. The results of standard calculations agree well with previous results. It is shown that starting from a triplet reference, spin-flip TDDFT can successfully optimize the geometry of the four lowest singlet states of CH₂ and three other isovalent species. Spin-flip TDDFT is used to calculate the potential energy curve of the breaking of the C?CC bond of ethane. The curve obtained is superior to that from a restricted density functional theory calculation, while at the same time the problems with spin contamination exhibited by unrestricted density functional theory calculations are avoided.