Vector representation of large-amplitude vibrations for the determination of kinetic energy functions

Vibrational coordinates and their gradients: A geometric algebra approach

“…On the other hand, the covariant approach of the Ref. 25 is restricted to numerical work. In that approach, one forms the covariant metric tensor g q i q j ϭ ͚ ␣ m ␣ (dx ␣ /dq i )•(dx ␣ /dq j ), which must be inverted to obtain the reciprocal metric tensor g (q i q j ) that appears in the kinetic energy operator.…”

“…25 In the latter case, the ring coordinates are typically symmetrized linear combinations of some already familiar internal coordinates ͑for example, the breathing coordinate is a sum of ring bond lengths͒. An important property for any ring structure is the ring closure; the sum of all the ring bond vectors is zero, ͚ r ␣␤ ϭ0.…”

“…The absolute value of the ring puckering coordinate Z ␣␤␥ is defined 25 as the half of the distance between the two ring diagonals r ␣␤ ϭx ␤ Ϫx ␣ and r ␥ ϭx Ϫx ␥ , i.e.,…”

Solving the vibrational Schrödinger equation on an arbitrary multidimensional potential energy surface by the finite element method

“…Chiang and Laane [88] derived the G-matrix element functions for the kinetic energy operator based on the vector method [37,89]: (2) cos(2φ 1 ) + cos(2φ 2 ) + G (4) cos(4φ 1 ) + cos(4φ 2 ) + G (c) cos(2φ 1 ) cos(2φ 2 ) + G (s) sin(2φ 1 ) sin(2φ 2 ). (40) Since φ 1 and φ 2 are equivalent, G 11 and G 22 are identical functions.…”

Perspective: Computing (ro-)vibrational spectra of molecules with more than four atoms

“…9,17,18 Often it is advantageous not to derive a KEO but to write a computer program that numerically evaluates coefficients of derivatives in the KEO; the coefficient values depend on the shape of the molecule. [19][20][21][22] Although new tricks that facilitate deriving KEOs are welcome, it is safe to say that when doing a variational calculation, the derivation of an appropriate KEO is not a problem. Although the ideas are largely unexplored, it is even possible to compute a spectrum using a space-fixed KEO and thereby obviate the need to derive an internal-coordinate KEO or even to numerically obtain values of coefficients of derivatives at points.…”