Janne Pesonen

New inversion coordinate for ammonia: Application to a CCSD(T) bidimensional potential energy surface

A new inversion coordinate is defined for ammonia as a function of the valence angles. Its square is similar to the often used totally symmetric bending displacement coordinate for the pyramidal XY3–type molecules. We have used this in a two-dimensional calculation including the totally symmetric stretching and the inversion mode. A conventional symmetrized internal coordinate is employed for the symmetric stretch. A two-dimensional potential energy surface is calculated using the ab initio CCSD(T) method together with the aug-cc-pVTZ, cc-pVQZ, and aug-cc-pVQZ basis sets. The corresponding eigenvalues are calculated variationally using a Morse oscillator basis set for the stretch and a harmonic oscillator basis set for the inversion. A good agreement is obtained between the calculated and 22 experimental inversion levels, 9 of 14NH3 and the others involving 4 other isotopomers (14ND3, 15NH3, 15ND3, and 14NT3). With the aug-cc-pVTZ basis, a mean absolute error of 5.0 cm−1 is obtained whereas with the aug-c...

Vibrational coordinates and their gradients: A geometric algebra approach

The gradients of vibrational coordinates are needed in order to form the exact vibrational kinetic energy operator of a polyatomic molecule. The conventional methods used to obtain these gradients are often quite laborious. However, by the methods of geometric algebra, the gradients for any vibrational coordinate can be easily calculated. Examples are given, and special attention is directed to ring coordinates.

Vibration–rotation kinetic energy operators: A geometric algebra approach

The elements of the reciprocal metric tensor g(qiqj), which appear in the exact internal kinetic energy operators of polyatomic molecules can, in principle, be written as the mass-weighted sum of the inner products of measuring vectors associated to the nuclei of the molecule. In the case of vibrational degrees of freedom, the measuring vectors are simply the gradients of the vibrational coordinates. It is more difficult to find these vectors for the rotational degrees of freedom, because the components of the total angular momentum operator are not conjugated to any rotational coordinates. However, by the methods of geometric algebra, the rotational measuring vectors are easily calculated for any geometrically defined body-frame, without any restrictions to the number of particles in the system. In order to show that the rotational measuring vectors produced by the present method agree with the known results, the general formulas are applied to the triatomic bond-z, and to the triatomic angle bisector fr...

Six-dimensional variational calculations for vibrational energy levels of ammonia and its isotopomers

Results of six-dimensional variational calculations of vibrational energy levels are presented for ammonia using a Hamiltonian expressed in curvilinear internal bond coordinates. A two-dimensional potential energy surface, which was introduced in a preliminary study on the inversion motion, is combined with a surface by Martin et al. Both surfaces are calculated at the aug-cc-pVTZ/CCSD(T) ab initio level. The exact kinetic energy operator is an enlargement of the one used in the previous two-dimensional calculations. Eigenvalues are computed variationally using successive basis set contractions for some symmetric and asymmetric isotopomers of ammonia.

Volume-elements of integration: A geometric algebra approach

In this work, geometric algebra is applied to obtain the volume-element of integration for the 3 Cartesian coordinates of the center-of-mass, 3 Euler angles, and 3N−6 shape coordinates needed to describe the position, orientation, and shape of an N-atomic molecule. The volume-element is obtained as a product of N volume-elements, each associated with a set of three coordinates. The method presented has several advantages. For example, one does not need to expand any determinants, and all calculations are performed in the three-dimensional physical space (not in some 3N-dimensional abstract configuration space). Several examples and applications are given.

Eckart frame vibration-rotation Hamiltonians: Contravariant metric tensor

Eckart frame is a unique embedding in the theory of molecular vibrations and rotations. It is defined by the condition that the Coriolis coupling of the reference structure of the molecule is zero for every choice of the shape coordinates. It is far from trivial to set up Eckart kinetic energy operators (KEOs), when the shape of the molecule is described by curvilinear coordinates. In order to obtain the KEO, one needs to set up the corresponding contravariant metric tensor. Here, I derive explicitly the Eckart frame rotational measuring vectors. Their inner products with themselves give the rotational elements, and their inner products with the vibrational measuring vectors (which, in the absence of constraints, are the mass-weighted gradients of the shape coordinates) give the Coriolis elements of the contravariant metric tensor. The vibrational elements are given as the inner products of the vibrational measuring vectors with themselves, and these elements do not depend on the choice of the body-frame. The present approach has the advantage that it does not depend on any particular choice of the shape coordinates, but it can be used in conjunction with all shape coordinates. Furthermore, it does not involve evaluation of covariant metric tensors, chain rules of derivation, or numerical differentiation, and it can be easily modified if there are constraints on the shape of the molecule. Both the planar and non-planar reference structures are accounted for. The present method is particular suitable for numerical work. Its computational implementation is outlined in an example, where I discuss how to evaluate vibration-rotation energies and eigenfunctions of a general N-atomic molecule, the shape of which is described by a set of local polyspherical coordinates.

A rotamer energy level study of sulfuric acid

It is a common approach in quantum chemical calculations for polyatomic molecules to rigidly constrain some of the degrees of freedom in order to make the calculations computationally feasible. However, the presence of the rigid constraints also affects the kinetic energy operator resulting in the frozen mode correction, originally derived by Pesonen [J. Chem. Phys. 139, 144310 (2013)]. In this study, we compare the effects of this correction to several different approximations to the kinetic energy operator used in the literature, in the specific case of the rotamer energy levels of sulfuric acid. The two stable conformers of sulfuric acid are connected by the rotations of the O-S-O-H dihedral angles and possess C2 and Cs symmetry in the order of increasing energy. Our results show that of the models tested, the largest differences with the frozen mode corrected values were obtained by simply omitting the passive degrees of freedom. For the lowest 17 excited states, this inappropriate treatment introduces an increase of 9.6 cm(-1) on average, with an increase of 8.7 cm(-1) in the zero-point energies. With our two-dimensional potential energy surface calculated at the CCSD(T)-F12a/VDZ-F12 level, we observe a radical shift in the density of states compared to the harmonic picture, combined with an increase in zero point energy. Thus, we conclude that the quantum mechanical inclusion of the different conformers of sulfuric acid have a significant effect on its vibrational partition function, suggesting that it will also have an impact on the computational values of the thermodynamic properties of any reactions where sulfuric acid plays a role. Finally, we also considered the effect of the anharmonicities for the other vibrational degrees of freedom with a VSCF-calculation at the DF-MP2-F12/VTZ-F12 level of theory but found that the inclusion of the other conformer had the more important effect on the vibrational partition function.

Polymer conformations in internal (polyspherical) coordinates

The small‐amplitude conformational changes in macromolecules can be described by the changes in bond lengths and bond angles. The descriptors of large scale changes are torsions. We present a recursive algorithm, in which a bond vector is explicitly written in terms of these internal, or polyspherical coordinates, in a local frame defined by two other bond vectors and their cross product. Conformations of linear and branched molecules, as well as molecules containing rings can be described in this way. The orientation of the molecule is described by the orientation of a body frame. It is parametrized by the instantaneous rotation angle, and the two angles that parametrize the orientation of the instantaneous rotation axis. The reason not to use more conventional Euler angles is due to the fact that Euler angles are not well‐defined in gimbal lock (i.e., when a body axis becomes aligned with its space fixed counter part). The position of the molecule is parametrized by its center of mass. Original and calculated positions are compared for several proteins, containing up to about 100,000 atoms.

Gradients of vibrational coordinates from the variation of coordinates along the path of a particle

The gradients of vibrational coordinates are needed in order to form the exact vibrational kinetic energy operator of a polyatomic molecule. In my previous work [J. Chem. Phys. 112, 3121 (2000)], it was demonstrated that they can be easily obtained for any geometrically defined shape coordinates by the direct vectorial differentiation. However, there is in some cases a more practical way to obtain these gradients from the variation of coordinates along the path of a particle. This approach can be used effectively to find the gradients of the shape coordinates, which are given as implicit functions of the nuclear positions. As a new application, I use this method to obtain the gradients of the eigenvalues of the moment tensor.

Normal mode analysis of molecular motions in curvilinear coordinates on a non-Eckart body-frame: an application to protein torsion dynamics

Normal mode analysis (NMA) was introduced in 1930s as a framework to understand the structure of the observed vibration-rotation spectrum of several small molecules. During the past three decades NMA has also become a popular alternative to figuring out the large-scale motion of proteins and other macromolecules. However, the “standard” NMA is based on approximations, which sometimes are unphysical. Especially problematic is the assumption that atoms move only “infinitesimally”, which, of course, is an oxymoron when large amplitude motions are concerned. The “infinitesimal” approximation has the further unfortunate side effect of masking the physical importance of the coupling between vibrational and rotational degrees of freedom. Here, we present a novel formulation of the NMA, which is applied for finite motions in non-Eckart body-frame. Contrary to standard normal mode theory, our approach starts by assuming a harmonic potential in generalized coordinates, and tries to avoid the linearization of the coordinates. It also takes explicitly into account the Coriolis terms, which couple vibrations and rotations, and the terms involving Christoffel symbols, which are ignored by default in the standard NMA. We also computationally explore the effect of various terms to the solutions of the NMA equation of motions.