TRANSFORMATION OF ab initio FORCE FIELDS IN CALCULATIONS OF MOLECULAR VIBRATIONS USING REGUNDAND VIBRATIONAL COORDINATES G. A. Pitsevich, a* A. V. Kostopravova, a D. S. Umreiko, b and M. A. Ksenofontov b UDC 539.19 A technique was suggested to transform ab initio molecular force fields calculated using a set of independent vibrational coordinates into a form corresponding to a complete set of regundand coordinates and reflecting the molecular symmetry. Conditions necessary for the appropriate transformations to be possible are formu- lated. The possibility of transforming the force field for the simplest fragment containing regundand coordi- nates was demonstrated using ethylene as an example. Keywords: force field, vibrational spectrum, sets of regundand and independent vibrational coordinates. Introduction. The increased processing power of modern computers enables the acceleration and improvement of ab initio methods for calculating molecular force fields [1]. This includes the expansion of the basis set of func- tions [2] and consideration of increasingly finer effects from the interaction of electrons with each other and the nu- cleus [3–7]. However, vibrational spectra of only comparatively small molecules can be analyzed using the currently most accurate ab initio methods. Force constants that are calculated in several quantum-chemical programs using a set of independent vibrational coordinates cannot be translated into calculated force fields for more complicated molecules. It is well known that force fields that are constructed for a set of regundand natural coordinates do not have the aforementioned shortcomings and reflect fully the molecular symmetry. Such fields, similar to the empirical ones used earlier, can be readily translated to molecules of arbitrary complexity and can be used rather reliably to interpret vi- brational spectra by using slight variations of the off-diagonal force constants. Therefore, finding the force field for a set of regundand coordinates by using the force-constant matrix for a set of independent coordinates is a very timely problem. Calculation Methods. Designating the force-constant matrix for a set of regundand natural coordinates by K and the corresponding matrix for a set of independent ones by k, it can be confirmed that if K is known then k can always be found unambiguously if matrix A that expresses a set of regundand natural coordinates through a set of in- dependent ones is defined [8]. We will attempt to utilize the pseudo-inverse matrix A −1 to solve the inverse problem. Let [α reg ] = A[α ind ]. In the simplest instance for a planar configuration of four atoms (Fig. 1), the coordinates of the three bond angles are related by the obvious relationship: α 1 + α 2 + α 3 = 0 . Therefore, the sets of regundand and independent coordinates are related by the equation: ⎡ ⎢ ⎣ ⎢ ⎢ α 1 α 2 α 3 ⎤ ⎥ ⎦ ⎥ ⎥ = ⎡ ⎢ ⎣ ⎢ ⎢ 1 0 − 1 0 1 − 1 ⎤ ⎥ ⎦ ⎥ ⎥ ⎡ ⎢ ⎣ α 1 α 2 ⎤ ⎥ ⎦ , (2) a Belarusian State University, 4 Nezavisimosti Ave., Minsk, 220030, Belarus; e-mail: pitsevich@bsu.by; b A. N. Sevchenko Institute of Applied Physics Problems, Belarusian State University, Minsk. Translated from Zhurnal Priklad- noi Spektroskopii, Vol. 78, No. 5, pp. 661–667, September–October, 2011. Original article submitted March 12, 2011. 0021-9037/11/7805-0617 ©2011 Springer Science+Business Media, Inc. 617 Journal of Applied Spectroscopy, Vol. 78, No. 5, November, 2011 (Russian Original Vol. 78, No. 5, September–October, 2011) ∗ To whom correspondence should be addressed.