Molecular potential coordinates

CM Becker. Principal coordinate maps of molecular potential energy surfaces. J Comput Chem 19 1255-1267, 1998. 

Related to the coordination chemistry of P4S3 is that of a-P4S3l2, which also displays multiple potential coordination sites in a semi-open (nido) structure, but with fewer steric constraints and no P3-unit. The coordination abilities of a-P4S3l2, however, have also proven rather limited. Baudler et al reported a series of molecular low-valent metal adducts of type M(a-P4S3l2)jj (CO)n-ni(M = Ni, Fe m = 1-3) (98). It is remarkable - and frus-... 

Representing the molecular potential energy as an analytic function of the nuclear coordinates in this fashion implicitly invokes the Born-Oppenheimer approximation in separating the very fast electronic motions from the much slower ones of the nuclei. 

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. 

The structure of this chapter is as follows. In Section II, after the concept of potential-energy surface and the coordinate systems in which the potential can be represented have been introduced, we describe the most important topographical characteristics of the molecular potential function. The general aspects, which refer to the calculation of the potential energy by ab initio methods, are analyzed in Section III. The need to develop efficient methods for the calculation of the potential function and the corresponding gradient in... 

The symmetry coordinates show themselves to be particularly useful for the functional representation of the molecular potential. For example, the potential function of a X3-type molecule must be invariant with respect to the interchange of any internal coordinate ft, (/ = 1, 2, 3) hence it must be totally symmetric in relation to those coordinates. Thus, in terms of the coordinates Qi (/ =1,2, 3), such a function can only be written in terms of or totally symmetric combinations of Q2 and Q3. Such combinations may in fact be obtained by using the projection-operator technique.16"27 In fact, one can demonstrate16 27 that any totally symmetric function of three variables is representable in terms of the integrity basis,28... 

However, the rest of the program which embraces 99% of the work can be efficiently vectorized. This remainder is exclusively concerned with the evaluation of the molecular potential energy consequent upon changes in the orthogonal coordinates, and with matrix inversion and matrix by vector multiplication. 

K.C. Thompson, M.J.T. Jordan, M.A. CoUins, Polyatomic molecular potential energy surfaces by interpolation in local internal coordinates, /. Chem. Phys. 108 (20) (1998) 8302-8316. 

Computer simulation of molecular dynamics is concerned with solving numerically the simultaneous equations of motion for a few hundred atoms or molecules that interact via specified potentials. One thus obtains the coordinates and velocities of the ensemble as a function of time that describe the structure and correlations of the sample. If a model of the induced polarizabilities is adopted, the spectral lineshapes can be obtained, often with certain quantum corrections [425,426]. One primary concern is, of course, to account as accurately as possible for the pairwise interactions so that by carefully comparing the calculated with the measured band shapes, new information concerning the effects of irreducible contributions of inter-molecular potential and cluster polarizabilities can be identified eventually. Pioneering work has pointed out significant effects of irreducible long-range forces of the Axilrod-Teller triple-dipole type [10]. Very recently, on the basis of combined computer simulation and experimental CILS studies, claims have been made that irreducible three-body contributions are observable, for example, in dense krypton [221]. 

While the model employed in the present work provides a reasonable picture of a unimolecular reaction involving a large molecule in solution, other ingredients not considered here may play a role in some systems. The possible role played by intramolecular friction (nonlinear coupling between the reaction coordinate and other nonreactive modes near the barrier) has been discussed in Section IV. Also, the dependence of the molecular potential surface, in particular the activation barrier on the molecule-solvent interaction, may dominate in some cases the observed solvent effect on the rate. Such may be the case (see Section VIII) in a polar solvent when the reaction involves a change in the molecular dipole moment (such as a charge transfer reaction). 

The second symmetry requirement that the expression for the inter-molecular potential has to meet is that it must be invariant under any rotation of the global coordinate frame. The transformation properties of the symmetry-adapted functions Gj Hw) under such a rotation are easily obtained from Eqs. (10) and (5) ... 

Thus Eq. (4.7) for the optical activity tensors of the molecule are again employed, but now the summation is over all occupied LMOs and the vectors Ri define the positions of the orbital centroids. Once the wavefunctions are known, the polarizability of the ith LMO and the position of its centroid R( can be determined. The derivatives of a, and Rj with respect to the normal coordinates are calculated using the electric field perturbation approach recently shown to be very effective for the calculation of conventional infrared and Raman intensities61) the required derivatives of R= and a. are determined from the first and second derivatives, respectively, of the gradient of the molecular potential energy with respect to a small applied electric field. One important aspect of this method is that both infrared CD and ROA can be determined from the same conceptual and calculational method, which will enhance the study of the relationship between these two forms of vibrational optical activity. So far, only one ROA calculation using LMO methods has been reported59), and since that was for the model compound NHDT there has been no comparison with experimental data. 

Here t/M( ry ) is the molecular potential operator that depend on all electronic and nuclear coordinates, and py, fy, and qj are respectively the momentum and... 

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