Quantum mechanics force field

Quantum mechanics is essential for studying enzymatic processes [1-3]. Depending on the specific problem of interest, there are different requirements on the level of theory used and the scale of treatment involved. This ranges from the simplest cluster representation of the active site, modeled by the most accurate quantum chemical methods, to a hybrid description of the biomacromolecular catalyst by quantum mechanics and molecular mechanics (QM/MM) [1], to the full treatment of the entire enzyme-solvent system by a fully quantum-mechanical force field [4-8], In addition, the time-evolution of the macromolecular system can be modeled purely by classical mechanics in molecular dynamicssimulations, whereas the explicit incorporation... 

Williams, R. W., V. F. Kalasinsky, and A. H. Lowrey. 1993. Scaled Quantum Mechanical Force Field for Cis- and Trans-glycine in Acidic Solution. J. Mol. Struct. (Theochem) 281, 157-171. 

Over the last decade, ab initio quantum-mechanical force fields have begun to be applied in theoretical stable isotope studies of molecules and dissolved species (Bochkarev et al. 2003 Driesner et al. 2000 Oi 2000 Oi and Yanase 2001). This method shows great promise for future studies, because ab initio calculations accurately describe chemical properties such as force constants without the necessity of assuming allowed force-constant types (which may not be universally applicable). Ab initio calculations are also ideally suited to molecules with... 

S, Cl and Si-isotope fractionations for gas-phase molecules and aqueous moleculelike complexes (using the gas-phase approximation) are calculated using semi-empirical quantum-mechanical force-field vibrational modeling. Model vibrational frequencies are not normalized to measured frequencies, so calculated fractionation factors are somewhat different from fractionations calculated using normalized or spectroscopically determined frequencies. There is no table of results in the original pubhcation. 

Characterization of amide vibrational modes as seen in IR and Raman spectra has developed from a series of theoretical analyses of empirical data. The designation of amide A, B, I, II, etc., modes stem from several early studies of the (V-methyl acetamide (NMA) molecule vibrational spectra which continues to be a target of theoretical analysis. 15 27,34 162 166,2391 Experimental frequencies were originally fitted to a valence force field using standard vibrational analysis techniques and subsequently were compared to ab initio quantum mechanical force field results. 

Another hmitation is inherent to the harmonic approximation on which standard quantum mechanical force-field calculations are invariably based. Due to a fortui-tious (but surpisingly systematic) cancellation of errors, the harmonic frequencies calculated by modem density functional methods often match very well with the experimental ones, in spite of the fact that the latter involve necessarily more or less anharmonic potentials. Thus one is tempted to forget that the harmonic approx-imaton can become perilous when strong anharmonicity prevails along one or another molecular deformation coordinate. 

J. Wang, R. J. Boyd, and A. Laaksonen, J. Cbem. Pbys., 104, 7261. A Hybrid Quantum Mechanical Force Field Molecular Dynamics Simulation of Liquid Methanol Vibrational Frequency Shifts as a Probe of the Quantum Mechanical/Molecular Mechanical Coupling. 

Nowadays a wide variety of quantum-chemical programs are disposable, which permit to calculate with high accuracy the equilibrium geometry of the molecules and their energy of formation. Theoretical methods have been developed for analytical calculation of the first and second derivatives of energy [8,9], so that the force-constant matrix FHT and the harmonic frequencies can be extracted from the quantum-mechanical calculations. Since as a rule the molecular orbitals (MO) obtained by the quantum-mechanical methods are spread around the entire molecule, the corresponding quantum-mechanical force fields incorporate the important effects of the off-diagonal interactions. 

One modern theoretical approach consists in using the quantum-mechanical force-constants as atrial force field in the regularization functional (5), instead of empirical force-constants [2,10]. It is obvious that the reliability of this method depends crucially on the choice of a realistic quantum-mechanical force field. Unfortunately even the advanced quantum-chemical methods often fail to reproduce the molecular frequencies with sufficient accuracy to assign the experimental spectroscopic features to the corresponding normal vibrations. 

The reasons for this discrepancies will be discussed shortly in the following sections. As a rule frequencies calculated on the Hartree-Fock (HF) or density functional theory (DFT) levels are upshifted with respect to the experimental values, and differences sometimes overwhelm 100 cm-1 [11-13], This makes the pristine quantum-mechanical force fields inadequate as a trial approximations in the regularization technique. Consequently some empirical corrections to the quantum-mechanical force-constants are unavoidable when analyzing the normal vibrations of the molecules. 

The principal idea, which underlies different schemes of empirical corrections to Fq"m is that even if the quantum mechanical force fields fail to reproduce the experimental vibrational frequencies in absolute scale, they still keep the right information about the relative magnitude and sign of the different elements in the force-constant matrix. 

However, the most reliable and widely used is the method developed by Puley [16], which is now referred to as Scaled Quantum Mechanical Force Field (SQMF). In SQMF a scale constant Xs is ascribed to each internal coordinate qH such that the corrected (scaled) force-constants are calculated according to the equation ... 

A theoretical justification of the scaling procedure was given by Pupyshev et al [14]. They compared the force field Fhf obtained in the Hartree-Fock (HF) limit with the force-field Fa obtained in the configuration interaction (Cl) technique, where the electron correlation effects are taken into account by mixing the HF ground state function with electronic excitations from the occupied one-electron HF states to the virtual (unoccupied) HF states. It was assumed that the force constants F01 obtained in the Cl approximation simulate the exact harmonic force field while those, extracted from the HF approximation FHF cast the quantum-mechanical force field F1-"1. It was demonstrated that under conditions listed below ... 

Therefore the scaling transformation of the quantum-mechanical force field is an empirical way to account for the electronic correlation effects. As far as the conditions listed above are not always satisfied (e.g. in the presence of delocalized 7r-electron wavefunctions) the real transformation is not exactly homogeneous but rather of Puley s type, involving n different scale constants. The need of inhomogeneous Puley s scaling also arises due to the fact that the quantum-mechanical calculations are never performed in the perfect Hartree-Fock level. The realistic calculations employ incomplete basis sets and often are based on different calculation schemes, e.g. semiempirical hamiltonians or methods which account for the electronic correlations like Cl and density-functional techniques. In this context we want to stress that the set of scale factors for the molecule under consideration is specific for a given set of internal coordinates and a given quantum-mechanical method. 

As a rule the quantum-mechanical force-fields and the corresponding normal frequencies are calculated in a harmonic approximation, while the experimentally accessible frequencies are influenced by anharmonic contributions. The Puley s scaling factors are also found to incorporate the relevant empirical corrections for the vibrational anharmonicity. 

Normally the scaling factors are extracted by minimizing the squared deviation (4) considered as a functional R A) of the variable set A, - The frequency parameters z alc now correspond to the harmonic normal frequencies calculated with the scaled quantum-mechanical force-field (6). The first and second derivatives of R( A) with respect to the scaling factors can be calculated analytically [17,18], which permits to implement rapidly converging minimization procedures of the Newton-Gauss type. Alternative iterative minimization methods were also proposed [19]. 

The quantum-mechanical force fields of the two different molecules are obtained by means of the same method (the same level of theoretical approximation and the same basis set of one-electron wavefunctions). 

Obviously, these structural changes make the transfer of force-constants from the neutral B3 molecule to the B3+ radical inadequate. Instead, we tentatively transferred the scale factors optimized for the neutral molecule to the quantum-mechanical force-field of B3+ and calculated the corresponding scaled normal frequencies. We obtained a clear correspondence between many of the frequencies experimentally observed in Cl doped B3 crystals (Fig. 3(a)) and the calculated scaled frequencies (Fig. 3(b)). We also observed that some of the calculated scaled frequencies in the neutral B3 molecule are present in the spectra of the Cl doped crystals (Fig. 3(c)). This fact tells us that there is some portion of unoxidized B3 molecules in the sample and gives additional proof for the validity of the SQMF calculations performed on the neutral B3 molecule. 

A. Aamouche et al., Neutron inelastic scattering, optical spectroscopies and scaled quantum mechanical force fields for analyzing the vibrational dynamics of pyrimidine nucleic acid bases. 1. Uracil. J. Phys. Chem. 100, 5224-5234 (1996)... 

McNamara JP, AM Muslim, H Abdel-Aal, H Wang, M Mohr, IH Hillier, RA Bryce (2004) Towards a quantum mechanical force field for carbohydrates a reparametrized semi-empirical MO approach. Chem. Phys. Lett. 394 (4-6) 429-436... 

V. Scaled Quantum Mechanical Force-Field Method.240... 

Scaled quantum-mechanical force fields for furan (and thiophene) and its isotopomers have been calculated with the B3LYP/6-31G method. Corresponding MP2 and FIE calculations gave less satisfactory results. Excellent agreement... 

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