Constrained internal coordinates

The two structures with D3/e/3 and C3/e/3 conformations have different orientations of the caps. The remaining conformer (Z)3/e/3) is only a stable energy minimum for relatively large metal ions (metal-amine distances larger than ca. 2 A, see below). Thus, this structure can be obtained from the initial conformer (Z)3/e/3) by constraining the six C03-NT bonds to >2.2 A (see Section 17.15 for the technique used to constrain internal coordinates). 

Equation [73] has the same form as the equations of motion for molecules with constrained internal coordinates, and we already know that such equations can be solved effectively using the SHAKE algorithm4 ° Equations [72] and [73] play a key role in the Car-Parrinello method and enable one to run the dynamics for both ionic and electronic degrees of freedom in parallel. With carefully chosen effective mass p and a small time step, the electronic state adjusts itself instanteously to the nuclear configuration (Born-Oppenheimer principle), and, therefore, the atomic dynamics is computed along the system s Born-Oppenheimer surface. Note that there is no need to carry out the costly matrix-diagonalization procedure for performing electronic structure calculations. 

The choice of the constrained internal coordinate relies heavily on chemical intuition and experience. Consequently, the method has not yet been incorporated in most available quantum chemical programs and perhaps never will be. Despite this, studies have used this procedure with much success. One can construct a software interface to currently existing programs that would effect the constrained optimization algorithm in a semi-automated manner. 

By combining the Lagrange multiplier method with the highly efficient delocalized internal coordinates, a very powerfiil algoritlun for constrained optimization has been developed [ ]. Given that delocalized internal coordinates are potentially linear combinations of all possible primitive stretches, bends and torsions in the system, cf Z-matrix coordinates which are individual primitives, it would seem very difficult to impose any constraints at all however, as... 

Baker J 1997 Constrained optimization in delocalized internal coordinates J. Comput. Chem. 18 1079... 

Equations (56) and (57) give six constrains and define the BF-system uniquely. The internal coordinates qk(k = 1,2, , 21) are introduced so that the functions satisfy these equations at any qk- In the present calculations, 6 Cartesian coordinates (xi9,X29,xi8,Xn,X2i,X3i) from the triangle Og — H9 — Oi and 15 Cartesian coordinates of 5 atoms C2,C4,Ce,H3,Hy are taken. These 21 coordinates are denoted as qk- Their explicit numeration is immaterial. Equations (56) and (57) enable us to express the rest of the Cartesian coordinates (x39,X28,X38,r5) in terms of qk. With this definition, x, ( i, ,..., 21) are just linear functions of qk, which is convenient for constructing the metric tensor. Note also that the symmetry of the potential is easily established in terms of these internal coordinates. This naturally reduces the numerical effort to one-half. Constmction of the Hamiltonian for zero total angular momentum J = 0) is now straightforward. First, let us consider the metric. 

The constrained mobility defined in this chapter is given by the inverse of Cab within the entire /-dimensional soft subspace. In any translationally invariant problem, in which there is no mechanical force or overall concentration gradient present to induced a generalized elastic force conjugate to the central position, one is typically interested only in the projection of onto the f — 3)-dimensional subspace of internal coordinates, which we will... 

The expression given by BCAH for elements of the constrained mobility within the internal subspace is based on inversion of the projection of the modified mobility within the internal subspace, rather than inversion of the projection (at of the mobility within the entire soft subspace. BCAH first define a tensor given by the projection of the modified friction tensor onto the internal subspace, which they denote by the symbol gat and refer to as a modified covariant metric tensor, which is equivalent to our CaT - They then define an inverse of this quantity within the subspace of internal coordinates, which they denote by g and refer to as a modified contravariant metric tensor, which is equivalent to our for afi = 1,..., / — 3. It is this last quantity that appears in their diffusion equation, given in Eq. (16.2-6) of Ref. 4, in place of our constrained mobility Within the space of internal coordinates, the two quantities are completely equivalent. 

The simple CSL model is directly applicable to the cubic crystal class. The lower symmetry of the other crystal classes necessitates the more sophisticated formalism known as the constrained coincidence site lattice, or CCSL (Chen and King, 1988). In this book we treat only cubic systems. Interestingly, whenever an even value is obtained for E in a cubic system, it will always be found that an additional lattice point lies in the center of the CSL unit cell. The true area ratio is then half the apparent value. This operation can always be applied in succession until an odd value is obtained thus, E is always odd in the cubic system. A rigorous mathematical proof of this would require that we invoke what is known as O-lattice theory (Bollman, 1967). The O-lattice takes into account all equivalence points between two neighboring crystal lattices. It includes as a subset not only coinciding lattice points (the CSL) but also all nonlattice sites of identical internal coordinates. However, expanding on that topic would take us well beyond the scope of this book. The interested reader is referred to Bhadeshia (1987) or Bollman (1970). 

Below is another important theorem that ensures that once the integration interval has been specified the zeros of the orthogonal polynomials must be constrained inside this interval. This is an important feature of orthogonal polynomials because, as will become clearer below, it guarantees that the internal coordinates are always sampled within the support of the NDF. 

Since this transformation to normal coordinates is invertible, one can readily determine the functional dependencies of the terms in Eq. (1) using either the normal or internal coordinates. Interestingly, in our study of vibrational states of the well-known local mode molecule H20 and its deuterated analogs we found only minor differences between the results of CVPT in the internal and normal mode representations (46). The normal mode calculations, however, required significantly less computer time to run, since many terms in the Hamiltonian are constrained to zero by symmetry. For this reason we chose to use the normal mode coordinates for all subsequent studies. 

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