Constrained optimization in cartesian coordinates

Modifications are made to a previously published algorithm for constrained optimization in Cartesian coordinates (J. Comp. Chem., 13 , 240, 1992) to incorporate both fixed and dummy atoms. Standard distance and angle constraints can now be specified with respect to dummy atoms, greatly extending the range of constraints that can be handled. Fixed atoms can be eliminated from the optimization space and so there is no need to calculate their gradients resulting in potentially significant savings of CPU time in ab initio computations. Several examples illustrate the range and versatility of the modified algorithm. © John Wiley & Sons, Inc.     

All cartesian coordinates for the optimized molecule structures

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On the evaluation of cartesian symmetry coordinates in molecular vibrations

A method for evaluating internal symmetry coordinates recently proposed by the authors uses the diagonalization of the G matrix. It is shown in the present note how to find a suitable matrix to be diagonalized in order to obtain cartesian symmetry coordinates, even when the point group of the molecule is unknown.     

The use of internal cartesian coordinates for describing molecular vibrations

An analysis of the influence of isotope substitution on the system of electronic-nuclear equations for an arbitrary molecular system was used as a basis for formulating invariance conditions with respect to isotope substitution of the potential energy surface written in the Cartesian coordinates rigidly bound with the center of mass of the molecule (internal Cartesian coordinates). This property of the potential function obviates the necessity of using curvilinear natural coordinates, which can be replaced by Cartesian coordinates, in theoretical studies of the vibrational spectra of molecules and their isotopomers and in solving the direct and inverse anharmonic problems. An equation for the quantum-mechanical Hamiltonian of a normal molecule in internal Cartesian coordinates was obtained.     

On the general transformation from molecular geometric parameters to cartesian coordinates

This article presents a general method and accurate algorithm for calculating the Cartesian coordinates (x_a, y_a, z_a) from an arbitrary triple of distances r(a,i), angles, θ(a,j,k), or dihedral angles ?(a,l,m,n), specifying the position of the nucleus a relative to nuclei i, ?, n with known Cartesian coordinates. There is a brief discussion of the requirements on the 3N-6 geometric parameters in order for them to determine the shape of an N-atomic molecule.     

Vibrational analyses of silacyclopentanes

The complete infrared and Raman spectra of 1,1-difluoro-1-silacyclopentane and 1,1-dichloro-1-silacyclopentane have been recorded and analyzed. Furthermore, a number of the vibrational frequencies of the silacyclopentane and silacyclopentane-1, 1-d₂ molecules have been reassigned.A normal coordinate calculation for each of these molecules was carried out and this demonstrated the validity of the assignments. Considerable mixing of the modes was found especially where ring vibrations and SiX₂ motions were involved.     

Force-constant computations in cartesian coordinates. Elimination of translational and rotational contributions

A projection method is described for elimination of spurious contributions to calculated cartesian force constants which give rise to non-zero frequencies for translational and rotational modes and which may cause errors in vibrational frequencies. Illustrative calculations for water monomer (in STO—3G) and water dimer (in 4–31G) are discussed. The preference for deuterium-bonded versus hydrogen-bonded isotopomeric water -dimer structures is demonstrated by calculations which satisfy the Teller—Redlich product rule provided that projected force constants are employed.     

Geometry optimization in cartesian coordinates: The end of the Z-matrix?

Geometry optimization directly in Cartesian coordinates using the EF and GDIIS algorithms with standard Hessian updating techniques is compared and contrasted with optimization in internal coordinates utilizing the well known Z-matrix formalism. Results on a test set of 20 molecules show that, with an appropriate initial Hessian, optimization in Cartesians is just as efficient as optimization in internals, thus rendering it unnecessary to construct a Z-matrix in situations where Cartesians are readily available, for example from structural databases or graphical model builders.     

Techniques for geometry optimization: A comparison of cartesian and natural internal coordinates

A comparison is made between geometry optimization in Cartesian coordinates, using an appropriate initial Hessian, and natural internal coordinates. Results on 33 different molecules covering a wide range of symmetries and structural types demonstrate that both coordinate systems are of comparable efficiency. There is a marked tendency for natural internals to converge to global minima whereas Cartesian optimizations converge to the local minimum closest to the starting geometry. Because they can now be generated automatically from input Cartesians, natural internals are to be preferred over Z-matrix coordinates. General optimization strategies using internal coordinates and/or Cartesians are discussed for both unconstrained and constrained optimization. © John Wiley & Sons, Inc.     

Small-amplitude protein conformational dynamics: Second-order analytic relation between cartesian coordinates and dihedral angles

An analytic expression for protein atomic displacements in Cartesian coordinate space (CCS) against small changes in dihedral angles is derived. To study time-dependent dynamics of a native protein molecule in CCS from dynamics in the internal coordinate space (ICS), it is necessary to convert small changes of internal coordinate variables to Cartesian coordinate variables. When we are interested in molecular motion, six degrees of freedom for translational and rotational motion of the molecule must be eliminated in this conversion, and this conversion is achieved by requiring the Eckart condition to hold. In this article, only dihedral angles are treated as independent internal variables (i.e., bond angles and bond lengths are fixed), and Cartesian coordinates of atoms are given analytically by a second-order Taylor expansion in terms of small deviations of variable dihedral angles. Coefficients of the first-order terms are collected in the K matrix obtained previously by Noguti and Go (1983) (see ref. 2). Coefficients of the second-order terms, which are for the first time derived here, are associated with the (newly termed) L matrix. The effect of including the resulting quadratic terms is compared against the precise numerical treatment using the Eckart condition. A normal mode analysis (NMA) in the dihedral angle space (DAS) of the protein bovine pancreatic trypsin inhibitor (BPTI) has been performed to calculate shift of mean atomic positions and mean square fluctuations around the mean positions. The analysis shows that the second-order terms involving the L matrix have significant contributions to atomic fluctuations at room temperature. This indicates that NMA in CCS involves significant errors when applied for such large molecules as proteins. These errors can be avoided by carrying out NMA in DAS and by considering terms up to second order in the conversion of atomic motion from DAS to CCS. © 1995 by John Wiley & Sons, Inc.     

The construction of the Hamiltonian of an isotope substituted molecule in internal cartesian coordinates

A scheme for constructing the Hamiltonian of an isotope substituted molecule in internal Cartesian coordinates is considered. The reliability of the suggested scheme is substantiated by the fulfillment of the Teller-Redlich product rule for some water molecule isotopomers.     

Transfer of molecular property tensors in cartesian coordinates: A new algorithm for simulation of vibrational spectra

A direct transfer of Cartesian molecular force fields (FF) and electric property tensors is tested on model systems and compared to transfer in internal coordinates with an aim to improve simulation of vibrational spectra for larger molecules. This Cartesian transformation can be implemented easily and offers greater flexibility in practical computations. It can be also applied for transfer of anharmonic derivatives. The results for model calculations of the force field and vibrational frequencies for N-methylacetamide show that our method removes errors associated with numerical artifacts caused by nonlinearity of the otherwise required Cartesian to internal coordinate transformation. For determination of IR absorption and vibrational circular dichroism intensities, atomic polar and axial tensors were also transferred in the Cartesian representation. For the latter, which are dependent upon the magnetic dipole operator, a distributed origin gauge is used to avoid an origin dependence. Comparison of the results of transferring ab initio FF and intensity parameters from an amide dimer fragment onto a tripeptide with those from a conventionally determined tripeptide FF document some limitations of the transfer method and its possible applications in the vibrational spectroscopy. Finally, application to determination of the FF and spectra for helical heptapeptide are presented and compared to experimental results. © 1997 by John Wiley & Sons, Inc.     

Vibrational spectra,normal coordinates and infrared intensities of four isotopic phenyl acetates

The i.r. and Raman spectra of phenyl acetate and its deuteromethyl and deuterophenyl derivatives have been recorded. The fundamental frequencies of the four isotopic species have been assigned by referring to the Raman depolarization ratios and isotopic frequency shifts. The normal coordinates have been calculated on the basis of a valence force field. The general valence type force constants have been refined by the least-squares method. The i.r. spectra of the four isotopes have been simulated by using the obtained force field in combination with a suitable set of the atomic charges and their fluxes.     

Vibrational analyses of vinylsulfonamide CH2CHSO2NH2

The structure and conformational stability of vinylsulfonamide CH2CHSO2NH2 were investigated by DFT-B3LYP/6-311+G** and ab initio MP2/6-311+G** calculations. From the calculations the molecule was predicted to exist predominantly in the gauche-syn (vinyl group nearly eclipses one of the SO bonds and the NH2 and the SO2 moieties eclipse each other) conformation with the possibility of low abundance of the cis-syn and the gauche-anti forms. The asymmetric potential function for the internal rotation about CS bond was determined for the molecule. The vibrational frequencies were computed at DFT-B3LYP level for the gauche-syn conformer of the molecule and its d2(C2H3SO2ND2) and d3(C2D3SO2NH2) deuterated species. Normal coordinate calculations were then carried out and the potential energy distributions were calculated for the molecule.     

Normal coordinates analyses of beta-D-allose and alpha-D-talose in the crystalline state

IR and Raman spectra of beta-d-allose, alpha-d-talose and beta-d-allose O-D(5) have been recorded in the 4000-400 cm(-1) and in the 4000-20 cm(-1) regions. These spectra constitute the experimental support that allows to reproduce theoretically the vibrational frequencies and to establish a force field for these saccharides through a normal coordinate analysis. For this purpose, a modified UBSFF has been combined with an intermolecular potential energy function that includes the Van der Walls interactions, the electrostatic terms, and an explicit hydrogen bond function. The initial force field parameters are derived either from those of D-glucose or D-galactose and are fitted so as to obtain a good agreement between the calculated and the observed frequencies. The obtained results reproduce the experimental frequencies and in order to test the validity of the obtained force field, it has been applied to beta-D-allose O-D(5).     

A tensor representation of the real spherical functions for the continuous group O in cartesian coordinates: Example of eulerian rotations

The aim of this article is to give a practical way for the use of real spherical functions in another frame than the frame in which they have been defined. For instance, we calculate physical properties from a local frame and use them in the general frame, deduced one from the other by Eulerian rotations on the coordinates. The power of the method is in the use of cartesian coordinates and in the definition of a scalar product between these functions to set up the complete vectorial space generated by these representations of the group O.