Internal coordinate derivatives

If we calculate, ab initio, the Cartesian gradients and second derivatives of the PES, this data can be transformed into internal coordinate derivatives by solving the following linear equations which follow from a change of... 

Z-matriccs arc commonly used as input to quantum mechanical ab initio and serai-empirical) calculations as they properly describe the spatial arrangement of the atoms of a molecule. Note that there is no explicit information on the connectivity present in the Z-matrix, as there is, c.g., in a connection table, but quantum mechanics derives the bonding and non-bonding intramolecular interactions from the molecular electronic wavefunction, starting from atomic wavefiinctions and a crude 3D structure. In contrast to that, most of the molecular mechanics packages require the initial molecular geometry as 3D Cartesian coordinates plus the connection table, as they have to assign appropriate force constants and potentials to each atom and each bond in order to relax and optimi-/e the molecular structure. Furthermore, Cartesian coordinates are preferable to internal coordinates if the spatial situations of ensembles of different molecules have to be compared. Of course, both representations are interconvertible. 

The treatment described above (which was introduced in Ref. 1) is much simpler than the standard treatment (which uses internal coordinates b, 0, large molecules or small proteins, evaluating the second derivative matrix F numerically, using analytical first derivatives. 

The vibrational kinetic energy can also be expressed in terms of the velocities in internal coordinates by taking the partial derivatives of Eq. (49). Thus, S = GP and, as G is square and nonsingular, P G lS and its transpose... 

The final step in the molecular-mechanics calculation of molecular conformation involves the minimization of the energy Approximations are involved whose importance is not always clear. Usually, all first derivatives with respect to the various internal coordinates are set equal to zero - although these coordinates are often not independent (see Section 10.6). Furthermore, the final conformation obtained depends on the assumed initial structure. Therefore, (he method must be applied with care and a certain amount of chemical intuition. In spite of these uncertainties the molecular mechanics method has been employed with considerable success, particularly in the conformational analysis of branched alkanes. For molecules containing hetero-atoms, it can be applied, but with somewhat less confidence. 

The subscript e means that the second derivatives are taken for internal parameters of the known or assumed molecular geometry.) It then follows that, in principle, to every molecule a F-matrix is assigned with molecule-specific force constants Fd.Transferability can only be expected for molecules which contain qualitatively and quantitatively very similar internal coordinates For instance, a force field which reproduces very well frequencies of n-al-kanes (from which it was derived) gives significantly worse results for branched alkanes... 

We shall not discuss all the numerous energy minimisation procedures which have been worked out and described in the literature but choose only the two most important techniques for detailed discussion the steepest descent process and the Newton-Raphson procedure. A combination of these two techniques gives satisfactory results in almost all cases of practical interest. Other procedures are described elsewhere (1, 2). For energy minimisation the use of Cartesian atomic coordinates is more favourable than that of internal coordinates, since for an arbitrary molecule it is much more convenient to derive all independent and dependent internal coordinates (on which the potential energy depends) from an easily obtainable set of independent Cartesian coordinates, than to evaluate the dependent internal coordinates from a set of independent ones. Furthermore for our purposes the use of Cartesian coordinates is also advantageous for the calculation of vibrational frequencies (Section 3.3.). The disadvantage, that the potential energy is related to Cartesian coordinates in a more complex fashion than to internals, is less serious. 

The constraint of F = Ft s = O in these derivatives implies that the remaining internal coordinates of the nuclear frame are free to relax the atomic positions until the forces associated with these geometrical degrees-of-freedom vanish, thus marking the minimum of the system energy with respect to Qtys -... 

As is well known, the vibrational Hamiltonian defined in internal coordinates may be written as the sum of three different terms the kinetic energy operator, the Potential Energy Surface and the V pseudopotential [1-3]. V is a kinetic energy term that arises when the classic vibrational Hamiltonian in non-Cartesian coordinates is transformed into the quantum-mechanical operator using the Podolsky trick [4]. The determination of V is a long process which requires the calculation of the molecular geometry and the derivatives of various structural parameters. 

These equations differ from the previous definitions of the X and Y matrix elements since the derivatives of the internal coordinates with respect to vibrational coordinate are considered. 

The derivatives of these equations require one to obtain the first and second derivatives of the G matrix elements. This, in turn, requires to obtain the first, second and third derivatives of the d, R, a and P (equations 5) internal coordinates with respect to Y, and the first, second and third derivatives of the Cartesian coordinates with respect to the internal coordinates. 

As an initial molecular system of reference a system centered on the 01 atom, has been selected. The x axis coincides with the 01-02 bond and the three atoms 01, 02 and HI lie in the xy plane. Appendix 1 shows the equations that connect Cartesian and internal coordinates and their derivatives. From the initial Cartesian coordinates, the X, Y and Z center-of-mass coordinates and its X, Y yZ. . . derivatives are calculated. The positions of the atoms have to be referred to the center of mass ... 

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