Independent Force Constants Research Articles

A minimal coarse-grained model for the low-frequency normal mode analysis of icosahedral viral capsids.

The main goal of this work is the design of a coarse-grained theoretical model of minimal resolution for the study of the physical properties of icosahedral virus capsids within the linear-response regime. In this model the capsid is represented as an interacting many-body system whose composing elements are capsid subunits (capsomers), which are treated as three-dimensional rigid bodies. The total interaction potential energy is written as a sum of pairwise capsomer-capsomer interactions. Based on previous work [Gomez Llorente et al., Soft Matter, 2014, 10, 3560], a minimal and complete anisotropic binary interaction that includes a full Hessian matrix of independent force constants is proposed. In this interaction model, capsomers have rotational symmetry around an axis of order n > 2. The full coarse-grained model is applied to analyse the low-frequency normal-mode spectrum of icosahedral T = 1 capsids. The model performance is evaluated by fitting its predicted spectrum to the full-atom results for the Satellite Tobacco Necrosis Virus (STNV) capsid [Dykeman and Sankey, Phys. Rev. Lett., 2008, 100, 028101]. Two capsomer choices that are compatible with the capsid icosahedral symmetry are checked, namely pentamers (n = 5) and trimers (n = 3). Both subunit types provide fair fits, from which the magnitude of the coarse-grained force constants for a real virus is obtained. The model is able to uncover latent instabilities whose analysis is fully consistent with the current knowledge about the STNV capsid, which does not self-assemble in the absence of RNA and is thermally unstable. The straightforward generalisability of the model beyond the linear regime and its completeness make it a promising tool to theoretically interpret many experimental data such as those provided by the atomic force microscopy or even to better understand processes far from equilibrium such as the capsid self-assembly.

The SPASIBA force field of aldehydes. Part II: structure and vibrational wavenumbers of ethandial, propenal and 2-methylpropenal

The SPASIBA potential energy function has been extended to conjugated aldehydes. Molecular structures, conformational energies, moments of inertia, dipole moments and vibrational wavenumbers have all been examined. The tested molecules are ethandial (glyoxal), propenal (acrolein), 2-methylpropenal (methacrolein) and some of their deuterated analogs. The parameters of the force field were developed in order to reproduce experimental values: structures, conformational energies and vibrational wavenumbers (minimization of the standard deviation between observed and calculated vibrational wavenumbers). A set of 30 independent force constants was found to be sufficient to describe correctly the structure and vibrational wavenumbers. The average r.m.s errors is 15.25 cm −1.

Calculation of internal valence force constants for XY 3Z ( C3v) molecules

In this paper, a pure vibrational force field [K. Kuczera and R. Czerminski, J. Mol. Struct., 105 (1983) 269; K. Kuczera, J. Mol. Struct., 160 (1987) 159] is calculated for XY 3Z ( C 3v) molecules from the most reliable empirical and ab initio force constants in symmetry coordinates available in the literature. The molecules studied include the monohalomethanes H 3CZ and trihalomethanes HCY 3 (Z and Y = F, Cl, Br and I). This method allows us to reduce the 16 F R dependent constants to 12 F∗ R independent constants, the same as the number of F S ° independent symmetry force constants. This can be explained by the fulfillment of four relationships between the F∗ R values obtained from the sum rule in Kuczera's treatment. It is thus possible to assign numerical values for bending, bending-bending and bending-stretching force constants. The above-mentioned relationships can also be obtained from the interactions among the displacement coordinates associated with each force constant. In addition, we study the behaviour of the force constants as a function of the bond lengths, obtaining some empirical relations. We also analyze the effect of replacing the central atom by Si, Ge and Sn in the case of H 3CZ molecules.

The SPASIBA force field of esters

The SPASIBA potential energy function has been extended to include esters in order to investigate biomolecules related to lipids. The parameters of the potential energy surface were derived by minimizing the average error between the observed and calculated structures, conformational energy differences, vibrational frequencies and predicted quantities for a series of esters, including some deuterated derivatives when available. For several molecules, the dipole moment was also calculated and compared with the experimental data or ab initio values. A set of 25 independent force constants was found to be sufficient to describe correctly the structures, potential energy surfaces and vibrational frequencies. The average error between the predicted and observed vibrational frequencies was about 13.5 cm −1.

The structures and vibrational frequencies of a series of aliphatic alcohols as obtained using the spectroscopic potential SPASIBA

The SPASIBA potential energy function has been extended to include alcohols, in order to investigate biomolecules related to saccharides. The parameters of the potential energy surface were derived by minimizing the average error between observed and calculated structures, conformational energy differences, vibrational frequencies and the predicted quantities for a series of alcohols including some deuterated derivatives when available. For several molecules, the dipole moment was also calculated and compared with the experimental data or ab initio values. A set of 28 independent force constants was found to be sufficient to describe correctly the structures, potential energy surfaces and vibrational frequencies. The average error between the predicted and the observed frequencies was about 15 cm −1.

The structures and vibrational frequencies of a series of aliphatic alcohols as obtained using the spectroscopic potential SPASIBA

The SPASIBA potential energy function has been extended to include alcohols, in order to investigate biomolecules related to saccharides. The parameters of the potential energy surface were derived by minimizing the average error between observed and calculated structures, conformational energy differences, vibrational frequencies and the predicted quantities for a series of alcohols including some deuterated derivatives when available. For several molecules, the dipole moment was also calculated and compared with the experimental data or ab initio values. A set of 28 independent force constants was found to be sufficient to describe correctly the structures, potential energy surfaces and vibrational frequencies. The average error between the predicted and the observed frequencies was about 15 cm −1.

The structures and vibrational frequencies of a series of aliphatic ethers obtained using the spectroscopic potential SPASIBA

The SPASIBA potential energy function has been extended to include linear ethers, in order to investigate biomolecules related to saccharides. The parameters of the potential energy function were derived from minimization of the average error between the observed structures, conformational energy differences, vibrational frequencies and the predicted quantities for a series of aliphatic ethers, including some deuterated analogues. For several molecules the dipole moment was also calculated and compared with experimental or ab initio values. The molecules under investigation are dimethyl ether, trans and gauche forms of methyl ethyl ether, trans-trans and trans-gauche forms of diethyl ether, methyl isopropyl ether and ethyl isopropyl ether. A set of 32 independent force constants was found to be sufficient to correctly describe their structures, potential energy surfaces and vibrational frequencies. The average deviation between the predicted and observed frequencies is about 11 cm −1.

The vibrational spectra of natural and perdeuterated monochlorogallane, H2Ga(μ-Cl)2GaH2, and an empirical harmonic potential function

The infrared and Raman spectra of monochlorogallane and its fully deuterated isotopomer are recorded and assigned on the basis of the dimeric structures. H2Ga(μ-Cl)2GaH2 and D2Ga(μ-Cl)2GaD2, conforming to D2 symmetry. The observed frequencies are corrected for anharmonicity and fitted to a potential function in which 19 of the 33 independent force constants are refined.

Local geometry and resonant vibrations of Cu+:NaF. Results of ab initio perturbed ion, cluster-in-the-lattice calculations involving clusters of 179 ions

The ground state electronic structure and energy of a CuF92Na5−86 cluster (Cu+ plus 12 shells of neighbors) embedded into a quantum lattice representing the NaF crystal are determined by using the ab initio perturbed ion (aiPI) method, with unrelaxed Coulomb–Hartree–Fock (uCHF) correlation energy corrections. Parallel calculations are performed on the NaF92Na5−86 cluster of the pure crystal in order to identify the changes induced by the impurity and to estimate the systematic errors in our calculations. The geometry of the first four shells (32 ions) is allowed to relax by following symmetric breathing modes. An inwards relaxation of −0.12 Å is predicted for the nearest neighbors (nn) shell, but negligible relaxations are found for the outer shells. The substitution of the Na+ ion by the Cu+ impurity is favored by −1.03 eV. The Cu+ ion is found to occupy an on-center octahedral position. The 138 independent Oh force constants corresponding to the vibration of the Cu+ and its first four shells of neighbors are then numerically computed from the aiPI energy using a Richardson iterated, finite-difference limit formula. These force constants give the vibrational modes of the impurity center. Strong couplings are found among vibrational modes of adjacent shells. Vibration frequencies characteristic of the doped system are obtained at 206 cm−1 (1a1g), 108 cm−1 (1t1u), and 173 cm−1 (1eg) determined mainly by the motions of the CuF6 octahedron.

The structures and vibrational frequencies of a series of alkanes using the SPASIBA force field

The spectroscopic potential energy function SPASIBA is applied to a series of alkanes. SPASIBA is actually a hybrid of the AMBER force field and the Urey-Bradley-Shimanouchi force field. The parameters of the energy function are optimized to give best fit to the observed vibrational spectra and structures of the series of alkanes, including a few deuterated analogs. The test molecules are methane, ethane and ethane- d 6 , propane, trans- and gauche-n-butane, cyclohexane and its fully deuterated analog, isobutane, 2-methylbutane, neopentane, 2,2,3,3-tetramethylpentane, n-octane and n-decane. A set of 18 independent force constants is found to be sufficient to represent the vibrational frequencies accurately, and the standard deviation between the observed and calculated frequencies is 13 cm −1. Although such an accuracy was obtained using a consistent valence force field with a comparable number of adjustable parameters, 21, this potential has the merit to be expressed in a more physical way and, consequently, can be applied very easily and successfully to all the molecules without delocalized electrons and highly strained energies.

Calculation of internal valence force constants for XY 4 ( Td) tetrahedral molecules

Valence force constants have been calculated in terms of simple dependent rectilinear internal coordinates for a series of XY 4 tetrahedral molecules that have been studied from a vibrational point of view. These molecules include the tetrahydrides of carbon, silicon, germanium and tin, and the tetrafluorides of carbon, silicon and germanium. Using the most recent and reliable force constants in symmetry coordinates, the pure vibrational force field has been arrived at by applying Kuczera's treatment. The results show that the seven F R dependent constants are reduced to five F * R independent constants; the same occurs for the F 0 S independent force constants. The relationships f * rα = − f *′ rα and f *′ αα = − f * α − 4 f * αα are obtained from the sum rule in Kuczera's treatment. On the basis of Wilson's S vectors and from the interactions between the displacement coordinates, a method is proposed here to obtain the above relationships. This allows individual values to be assigned to the bending, bending—bending and bending—stretching force constants for the above-mentioned molecules, without the need for recourse to any model whatsoever. Likewise, in each case we have attempted to obtain an empirical function to fit the force constant values dependent on the bond lengths. This method allows us to some extent to predict the values of the force constants for some tetrahydride molecules of this group, which has not yet been studied experimentally.

Cartesian Dynamics of Simple Molecules VI Centrosymmetric Linear Pentatomics

Abstract A simple spring model for molecular vibrations, which uses Cartesian co-ordinates for both longitudinal and transverse displacements, is applied to centresymmetric linear pentatomic molecules such as carbon suboxide and carbon subsulphide. Analytical expressions for the four stretching and three bending mode frequencies are derived in terms of seven independent force constants. By substitution of Raman and infrared frequencies, values of these force constants are obtained and briefly discussed. Eigenvectors of the normal modes and eigenfrequencies of various isotopic species are also calculated.

Cartesian Dynamics of Simple Molecules IV Centrosymmetric Linear Quadratomics

Abstract A simple spring model for molecular vibrations, which uses Cartesian co-ordinates for both longitudinal and transverse displacements, is applied to centrosymmetric linear quadratomic molecules, such as cyanogen and acetylene and its halogenated derivatives. Analytical expressions for the three stretching and two bending mode frequencies are obtained in terms of five independent force constants. By substitution of Raman and infrared frequencies, values of these force constants are obtained and briefly discussed. The validity of the model is confirmed by the satisfactory agreement between calculated and observed frequencies of isotopic species.

Calculation of internal valence force constants for XY 6( Oh) octahedral molecules

The valence force constants have been calculated in terms of simple dependent rectilinear internal co-ordinates for a series of XY 6 octahedral molecules that have been amply studied from a vibrational point of view, i.e. the hexafluorides of sulphur, selenium, molybdenum, tungsten and uranium. The calculations have been carried out using the most recent force constants in symmetry co-ordinates to appear in the literature and following Kuczera's treatment, according to which the indetermination of the internal valence force constants involved in redundancies can be solved by using the transformation F* R= W T WF R W T W, which he calls the pure vibrational force field. The results show that the eleven F R dependent constants are reduced to seven F* R independent constants, the same number as the F S independent force constants. This is because of the zero value of the ƒ′* dα constants and the relationships ƒ* dα= −ƒ″* dα ƒ* αα = −ƒ″* αα ƒ* α=−2ƒ′* αα−ƒ′″* αα that are obtained from the combination of the above mentioned F* R transformation and the sum rule in Kuczera's treatment. These latter relationships can be obtained from the interactions between the displacement coordinates with which they are associated. This has allowed individual values to be assigned to the bending, bending—bending and stretching—bending constants for the above mentioned molecules without the need for recourse to any model whatsoever.

Inertial defects of planar symmetric top molecules

A general expression for the inertial defects of planar symmetric top molecules has been derived and applied to H 3 +, CH 3 +, C 3H 3 +, and C 6H 6-type molecules. It has been found that the inertial derivatives a n αα for these molecules can be expressed in terms of Coriolis constants. This allowed us to find a sufficient number of relations among Coriolis constants that the number of independent Coriolis constants is equal to the number of independent force constants. A numerical calculation of the inertial defect has been conducted for benzene, using Wilson's GF matrix method, and compared with experimental values.

A revised application of the effective nuclear charge model to the prediction of force constants in trigonal-bipyramid XY5 molecules

A revision of the application of the effective nuclear charge model used in the prediction of valence force constants in trigonal-bipyramid XY 5 ( D 3 h ) molecules has been carried out. In this revision the contribution to the force constants, and thus to the potential function, of the tree-redundancy relationships existing between the displacement co-ordinates has been taken into account. In this way the potential function is expressed in terms of independent force constants and can be said to be almost a general valence potential function.

The prediction of valence force constants and vibration frequencies for XY6 (Oh) molecules, using the effective nuclear charge model

Ohwada's effective nuclear charge model has been extended to the calculation of valence force constants for XY6 (Oh) molecules. To achieve this the potential energy function has had to be expressed in terms of independent force constants for which it has been necessary to eliminate the gem- and tree-redundancies that exist between the displacement coordinates. The vibration frequencies predicted by the use of these constants agree well in general with experimental observations.

An out-of-plane force field for some benzene derivatives having unsaturated substituents

A simplified out-of-plane valence force field has been developed for some benzene derivatives that have unsaturated substituents. The molecules treated in this study are benzene, styrene, benzaldehyde, ethynylbenzene and m- and p-diethynylbenzenes. In the force field we have used two different approximations. The first one includes 27 independent force constants, 12 of which are directly transferred from analogous molecules and 15 optimized on 242 experimental frequencies, giving an rms frequency deviation of 6.69 cm−1. The second and somewhat simpler one deals with 12 transferred and 12 optimized force constants giving an rms deviation of 6.95 cm−1. For comparison a force field for some methyl substituted benzenes has also been derived using the same approximations. The transferability of force constants is discussed in the light of the results achieved. On the basis of the calculations certain revisions in previous assignments have been made.

ChemInform Abstract: CALCULATION OF THE CUBIC SYMMETRY FORCE CONSTANTS OF METHYL BROMIDE AND METHYL CHLORIDE

The cubic symmetry force constants have been calculated for methyl bromide and methyl chloride through the method developed by Hoy et al. (Mol. Phys. 24, 1265–1290 (1972)). The spectroscopic constants recently reported in the literature have been utilized for the present analysis. Out of a total of 38 independent cubic force constants, 6 and 10 constants were constrained to zero for a methyl bromide and methyl chloride, respectively, in which a diagonal force constant of F666 was included for both molecules. However, it is noted that those force constants which are only concerned with the A1 symmetry coordinates have been determined independent of the constraint for both of these molecules.

High‐temperature infrared and Raman spectra of aluminium chloride dimer and monomer in the vapour phase

AbstractThe infrared spectra of aluminium chloride vapour (Al2Cl6 and AlCl3) including AlCl6 and AlCl6 at 473–843 K were measured in the region 700–50 cm−1 with an evacuable Fourier transform spectrometer by transmission and emission techniques. Evacuable cells of nickel were employed having windows of type IIa diamond and sealed with gold O‐rings. Raman spectra of Al2Cl6 vapour at ca 500 K and of AlCl3 at 1075 K with pressures from 0.3 to 10 atm were recorded, polarization measurements were carried out and additional spectra of the 35Cl and 37Cl compounds were obtained. The dimer spectra were interpreted in terms of D2h symmetry. All the nine Raman‐active and seven of the infrared‐active dimer fundamentals were assigned. The IR active mode ν10(B1u), expected below 40 cm−1, and the inactive mode ν5(Au) were left unassigned. The monomer spectra were interpreted in terms of D3h symmetry and all four fundamentals were assigned. A dimer force field, involving 12 independent force constants, was derived. The calculated frequencies were fitted with a least‐squares programme to 30 observed fundamentals of AlCl6 and AlCl6 and 14 additional isotopic shifts transferred from previous matrix isolation spectra. A complete force field for the monomer was derived, using the four observed fundamentals and seven isotopic shifts transferred from previously reported matrix isolation spectra.