Ödön Farkas

Geometry optimization with QM/MM, ONIOM, and other combined methods. I. Microiterations and constraints

Hybrid energy methods such as QM/MM and ONIOM, that combine different levels of theory into one calculation, have been very successful in describing large systems. Geometry optimization methods can take advantage of the partitioning of these calculations into a region treated at a quantum mechanical (QM) level of theory and the larger, remaining region treated by an inexpensive method such as molecular mechanics (MM). A series of microiterations can be employed to fully optimize the MM region for each optimization step in the QM region. Cartesian coordinates are used for the MM region and are chosen so that the internal coordinates of the QM region remain constant during the microiterations. The coordinates of the MM region are augmented to permit rigid body translation and rotation of the QM region. This is essential if any atoms in the MM region are constrained, but it also improves the efficiency of unconstrained optimizations. Because of the microiterations, special care is needed for the optimization step in the QM region so that the system remains in the same local valley during the course of the optimization. The optimization methodology with microiterations, constraints, and step‐size control are illustrated by calculations on bacteriorhodopsin and other systems.

Methods for optimizing large molecules. II. Quadratic search

Geometry optimization has become an essential part of quantum-chemical computations, largely because of the availability of analytic first derivatives. Quasi-Newton algorithms use the gradient to update the second derivative matrix (Hessian) and frequently employ corrections to the quadratic approximation such as rational function optimization (RFO) or the trust radius model (TRM). These corrections are typically carried out via diagonalization of the Hessian, which requires O(N3) operations for N variables. Thus, they can be substantial bottlenecks in the optimization of large molecules with semiempirical, mixed quantum mechanical/molecular mechanical (QM/MM) or linearly scaling electronic structure methods. Our O(N2) approach for solving the equations for coordinate transformations in optimizations has been extended to evaluate the RFO and TRM steps efficiently in redundant internal coordinates. The regular RFO model has also been modified so that it has the correct size dependence as the molecular syst...

Methods for optimizing large molecules. Part III. An improved algorithm for geometry optimization using direct inversion in the iterative subspace (GDIIS)

The geometry optimization using direct inversion in the iterative subspace (GDIIS) has been implemented in a number of computer programs and is found to be quite efficient in the quadratic vicinity of a minimum. However, far from a minimum, the original method may fail in three typical ways: (a) convergence to a nearby critical point of higher order (e.g. transition structure), (b) oscillation around an inflection point on the potential energy surface, (c) numerical instability problems in determining the GDIIS coefficients. An improved algorithm is presented that overcomes these difficulties. The modifications include: (a) a series of tests to control the construction of an acceptable GDIIS step, (b) use of a full Hessian update rather than a fixed Hessian, (c) a more stable method for calculating the DIIS coefficients. For a set of small molecules used to test geometry optimization algorithms, the controlled GDIIS method overcomes all of the problems of the original GDIIS method, and performs as well as a quasi-Newton RFO (rational function optimization) method. For larger molecules and very tight convergence, the controlled GDIIS method shows some improvement over an RFO method. With a properly chosen Hessian update method, the present algorithm can also be used in the same form to optimize higher order critical points.

Peptide models. XXXIII. Extrapolation of low-level Hartree-Fock data of peptide conformation to large basis set SCF, MP2, DFT, and CCSD(T) results. The Ramachandran surface of alanine dipeptide computed at various levels of theory

At the dawn of the new millenium, new concepts are required for a more profound understanding of protein structures. Together with NMR and X‐ray‐based 3D‐stucture determinations in silico methods are now widely accepted. Homology‐based modeling studies, molecular dynamics methods, and quantum mechanical approaches are more commonly used. Despite the steady and exponential increase in computational power, high level ab initio methods will not be in common use for studying the structure and dynamics of large peptides and proteins in the near future. We are presenting here a novel approach, in which low‐ and medium‐level ab initio energy results are scaled, thus extrapolating to a higher level of information. This scaling is of special significance, because we observed previously on molecular properties such as energy, chemical shielding data, etc., determined at a higher theoretical level, do correlate better with experimental data, than those originating from lower theoretical treatments. The Ramachandran surface of an alanine dipeptide now determined at six different levels of theory [RHF and B3LYP 3‐21G, 6‐31+G(d) and 6‐311++G(d,p)] serves as a suitable test. Minima, first‐order critical points and partially optimized structures, determined at different levels of theory (SCF, DFT), were completed with high level energy calculations such as MP2, MP4D, and CCSD(T). For the first time three different CCSD(T) sets of energies were determined for all stable B3LYP/6‐311++G(d,p) minima of an alanine dipeptide. From the simplest ab initio data (e.g., RHF/3‐21G) to more complex results [CCSD(T)/6‐311+G(d,p)//B3LYP/6‐311++G(d,p)] all data sets were compared, analyzed in a comprehensive manner, and evaluated by means of statistics.

Methods for geometry optimization of large molecules. I. An O(N2) algorithm for solving systems of linear equations for the transformation of coordinates and forces

The most recent methods in quantum chemical geometry optimization use the computed energy and its first derivatives with an approximate second derivative matrix. The performance of the optimization process depends highly on the choice of the coordinate system. In most cases the optimization is carried out in a complete internal coordinate system using the derivatives computed with respect to Cartesian coordinates. The computational bottlenecks for this process are the transformation of the derivatives into the internal coordinate system, the transformation of the resulting step back to Cartesian coordinates, and the evaluation of the Newton–Raphson or rational function optimization (RFO) step. The corresponding systems of linear equations occur as sequences of the form yi=Mixi, where Mi can be regarded as a perturbation of the previous symmetric matrix Mi−1. They are normally solved via diagonalization of symmetric real matrices requiring O(N3) operations. The current study is focused on a special approac...

Peptide models XV. The effect of basis set size increase and electron correlation on selected minima of the ab initio 2D-Ramachandran map of For-Gly-NH2 and For-L-Ala-NH2

A total of eleven basis sets from 3-21G to 6-3111++G(d,p) have been used at the HF and MP2 levels of theory for geometry optimizations of the global, γl, (φ = −75 °, Ψ = +75 °) and the second lowest, βl, (φ = −150 °, Ψ = + 150 °) minimum energy conformations of the l enantiomer of HCONHCH(CH3)CONH2. The results showed that due to fortuitous cancellation of correlation and basis set effects, the HF/3-21G energy-difference of these conformers agrees well with the MP2/6-311++G(d,p) energy difference, while the HF/6-311++G(d,p) energy difference converges erroneously toward zero. The other legitimate conformers were optimized at the HF/3-21G, HF/6311++G(d,p), and MP2/6-311++G(d,p) levels of theory. The results showed that one of the minima disappeared at HF/6-311++G(d,p) and one more of the minima did not occur at the MP2/6-311++G(d,p) level of theory. The correlation and basis set effects stabilized the higher energy conformers.

Peptide models XXIV: An ab initio study on N-formyl-L-prolinamide with trans peptide bond. The existence or non-existence of αL and ϵL conformations

Abstract N-formyl-L-prolinamide was subjected to geometry optimization at three levels of theory: HF/3-21G, HF/6-31G (d) and B3LYP/6-31G (d). At all three levels of computation the global minimum was γ L (inverse γ -Turn) backbone conformation with two ring-puckered forms “UP” and “DOWN”. At HF/3-21G level of theory three backbone conformations were found γ L , ϵ L , and α L . At higher levels of theory the ϵ L , and α L conformations disappeared. The “UP” and “DOWN” ring-puckered forms, in the γ L backbone conformation, led to practically identical vibrational spectra at the B3LYP/6-31G (d) level of theory.

Peptide models XIX: Side-chain conformational energy surface E = ƒ(χ1,χ2) and amide I vibrational frequencies of N-formyl-l-phenylalaninamide (For-Phe-NH2) in its γL or γinv or C7eq backbone conformation

In a study of cross sections of the E = ƒ(χ1,χ2) side-chain conformational potential energy surface of the γL or c7eq backbone conformation of For-l-Phe-NH2, it was found that there are three conformations (g +, a and g −) due to rotation about the Cαχ1 Cβ bond. It should be emphasised that the γL backbone conformation is conserved during rotation about χ1. However, there is only one unique conformation along the rotation about the Cβχ2 Ph bond. The CH2Ph group showed greater stabilisation, with respect to hydrogen (Gly), than the CH3 (Ala) or CH2OH (Ser) substituents. The hydrogen-bonded CO (amide 1) vibrational frequency is split into two bands due to the coupling of the CO stretching and NH2 scissoring modes of motion. The other carbonyl, not involved in hydrogen bonding, has a characteristic single IR band with a relatively high frequency. The orientation of the Ph group has no appreciable effect on these vibrational frequencies.

PEPTIDE MODELS XXII. A CONFORMATIONAL MODEL FOR AROMATIC AMINO ACID RESIDUES IN PROTEINS. A COMPREHENSIVE ANALYSIS OF ALL THE RHF/6-31+G CONFORMERS OF FOR-L-PHE-NH2

Abstract Phenylalanine is expected to have conformational features similar to the other three naturally occurring aromatic amino acid residues: Tyr, Trp and His. Previous ab initio structure determinations resulted in 19 different conformers of HCO–L–Phe–NH2 at the RHF/3–21G level of theory. The present work summarises the results of a comprehensive analysis incorporating RHF/3–21G, RHF/6–31+G*, RHF/6–31+G*//RHF/3–21G and B3LYP/6–311++G**//RHF/3–21G data as some of the previously established minima have vanished with the larger basis set, three out of the 19 stationary points having migrated during the optimisation. On top of that, the conformational building unit of the right-handed helix-like (αL) conformation and that of the polyproline II (eL) conformation are still missing from the E=E(φ,ψ) surface. The optimised ab initio structures are also analysed in the context of –Phe– conformers taken from a large X-ray database on non-homologous proteins incorporating a total of 158 664 amino acid residues.

Peptide models XXIII. Conformational model for polar side‐chain containing amino acid residues: A comprehensive analysis of RHF, DFT, and MP2 properties of HCO‐L‐SER‐NH2

Geometric and energetic properties of a diamide of serine, HCO‐NH‐L‐CH(CH2OH)CO‐NH2, are investigated by standard methods of computational quantum chemistry. Similarly to other amino acid residues, conformational properties of HCO‐L‐Ser‐NH2 can be derived from the analysis of its E=E(ϕ,ψ;χ1,χ2) hypersurface. Reoptimization of 44 RHF/3‐21G conformers at the RHF/6‐311++G** level resulted in 36 minima. For all conformers, geometrical properties, including variation of H‐bond parameters and structural shifts in the torsional space, are thoroughly investigated. Results from further single‐point energy calculations at the RHF, DFT, and MP2 levels, performed on the entire conformational data set, form a database of 224 energy values, perhaps the largest set calculated so far for any single amino acid diamide. A comprehensive analysis of this database reveals significant correlation among energies obtained at six levels of ab initio theory. Regression parameters provide an opportunity for extrapolation in order to predict the energy of a conformer at a high level by doing explicit ab initio computations only for a few selected conformers. The computed conformational and relative energy data are compared with structural and occurrence results derived from a nonhomologous protein database incorporating 1135 proteins.