A quartic force field coordinate substitution scheme using hyperbolic sine coordinates

Abstract

Quartic force fields (QFF) are currently the most cost-effective method for the approximation of potential energy surfaces for the calculation of anharmonic vibrational energies. It is known, although, that its performance can be less than satisfactory due to limitations related to slow convergence of the series. In this article, we present a coordinate substitution scheme using a combination of Morse and sinh coordinates, well adapted for its use with cartesian normal coordinates. We derive expressions for analytical integrals for use in VSCF and VCI calculations and show that the simultaneous substitution of symmetric and antisymmetric normal coordinates by Morse and sinh coordinates, respectively, significantly improves the vibrational transition frequencies for these modes in a well-balanced fashion. The accuracy of this substitution scheme is demonstrated by comparing one and two-dimensional sections of substituted and unsubstituted QFF with ab initio potential energy grids, as well as with vibrational energy calculations using as test cases two well-studied benchmark molecules: water and formaldehyde. We conclude that the coordinate substitution scheme presented constitutes a very attractive alternative to simple QFFs in the context of cartesian normal coordinates.quantum chemistry, quartic force fields, vibrational structure | I N T R O D U C T I O NThe calculation of accurate potential energy surfaces (PES) is certainly a crucial part of the computation of anharmonic vibrational energies and wavefunctions of molecular systems. Grid representations [1] of the PES are usually the most accurate option, although its high computational cost limits its application to very small systems, as the number of energy evaluations necessary for a complete grid rises exponentially with the number of normal modes. This limitation can be reduced somewhat by the use of N-mode coupling representations, [2][3][4] or more recently, adaptive sparse grid expansions. [5] Grids may be used directly, or converted into an analytical expression either by fitting [6] or interpolation [7] procedures. For example, very high accuracy have been obtained using reproducing kernel Hilbert space interpolation coupled to a grid representation for triatomic reactive systems. [8] Grid methods, even the most efficient examples of them, are still limited to small systems, so alternatives are needed for bigger molecules.Quartic force fields (QFF), this is, fourth order Taylor expansions of the potential with respect to the normal coordinates of the system, currently represent the best trade-off of accuracy and computational cost for the representation of molecular potential energy surfaces (PES) for anharmonic vibrational structure applications. The efficiency of the QFF scheme comes from the possibility of calculating the expansion coefficients by numerical differentiation of the potential, analytical gradients, or Hessians, if available. Moreover, recently Ramakrishnan and Rauhut presented an efficient scheme for computing QFFs from multimode exp...

Spectroscopy in Complex Environments from QM–MM Simulations

“…This can always be realized numerically in the equilibrium structure, although analytical expressions have been devised for the most common electronic structure methods. Additionally, anharmonicity can be introduced into the potential representation using a fourth-order Taylor series (a quartic force field) , or resorting to grids, in which case more advanced methods are necessary to compute the vibrational wave functions and energies. Such methods can be based on variational (VSCF, VCI), , perturbative (VMP2), or coupled pair (VCC) theories.…”

Picking up Good Vibrations through Quartic Force Fields and Vibrational Perturbation Theory

“…The use of Morsecosine coordinate transformations have been shown to provide such limiting behavior, 37,107 brought to bear for the angles and torsions may be better suited for certain molecular systems. 108 The use of continuous functions can be favorable for molecular coordinate definitions in certain QFF applications, as well. However, their use is much less standardized with failures in their implementations littleunderstood as they are little-used.…”

“…Vibrational configuration interaction (VCI) computations can employ QFFs if the QFFs can be extrapolated to the natural limits of their functions, i.e., they must have proper limiting behavior. The use of Morse-cosine coordinate transformations have been shown to provide such limiting behavior, , and similar trigonometric functions brought to bear for the angles and torsions may be better suited for certain molecular systems . The use of continuous functions can be favorable for molecular coordinate definitions in certain QFF applications, as well.…”