This paper presents the derivation of a kinetic-balance condition for explicitly correlated basis functions employed in semi-classical relativistic calculations. Such a condition is important to ensure variational stability in algorithms based on the first-quantized Dirac theory of 1/2-fermions. We demonstrate that the kinetic-balance condition can be obtained from the row reduction process commonly applied to solve systems of linear equations. The resulting form of kinetic balance establishes a relation for the 4 N components of the spinor of an N -fermion system to the non-relativistic limit, which is in accordance with recent developments in the field of exact decoupling in relativistic orbital-based many-electron theory.

In this paper we present a simple strategy for the elimination of the translational kinetic energy contamination of the total energy in pre-Born-Oppenheimer calculations carried out in laboratory-fixed Cartesian coordinates (LFCCs). The simple expressions for the coordinates and the operators are thus preserved throughout the calculations, while the mathematical form and the parametrisation of the basis functions are chosen so that the translational and rotational invariances are respected. The basis functions are constructed using explicitly correlated Gaussian functions (ECGs) and the global vector representation.First, we observe that it is not possible to parametrise the ECGs so that the system is at rest in LFCCs and at the same time the basis functions are square-integrable with a non-vanishing norm. Then, we work out a practical strategy to circumvent this problem by making use of the properties of the linear transformation between the LFCCs and translationally invariant and center-of-mass Cartesian coordinates as well as the transformation properties of the corresponding basis function parameter matrices. By exploiting these formal mathematical relationships we can identify and separate the translational contamination terms in the matrix representation of the kinetic energy operator in the LFCC formalism.We present numerical examples for the translational contamination and its elimination for the two lowest rotational energy levels of the singlet hydrogen molecule, corresponding to para-and ortho-H 2 , respectively, treated as four-particle quantum systems.

To study the capabilities and limitations of Raman optical activity, (-)-(M)σ-[10]helicene and (-)-(M)σ-[4]helicene serve as scaffold molecules on which new chiral centers are introduced by substitution of hydrogen atoms with other functional groups. These functional groups are deuterium atoms, fluorine atoms, and methyl groups. Multiply deuterated species are compared. Then, results of singly deuterated derivatives are compared against results obtained from singly fluorinated and methylated derivatives. The analysis required the calculation of a total of 2433 Raman optical activity spectra. The method we propose for the comparison of the various Raman optical activity spectra is based on the total intensity of squared difference spectra. This allows a qualitative comparison of pairs of Raman optical activity spectra and the extraction of the pair of most similar Raman optical activity spectra for each group of stereoisomers. Different factors were accounted for, such as the spectral resolution (modeled by line broadening) and the range of vibrational frequencies considered. In the case of σ-[4]helicene all generated stereoisomers in each group can be distinguished from one another by Raman optical activity spectroscopy. For σ-[10]helicene this holds except for the lower one of the two resolutions considered. Here, the group consisting of stereoisomers with five chiral centers contains at least one pair of derivatives whose Raman optical activity spectra cannot be distinguished from one another. This indicates that an increased molecular size has a negative effect on the number of chiral centers which can be distinguished by Raman optical activity spectroscopy. Regarding the different substituents, stereoisomers are the better distinguishable in Raman optical activity spectroscopy, the more distinct the signals of the substituent are from the rest of the spectrum.

This paper presents the calculation of the electric transition dipole moment in a pre-Born-Oppenheimer framework. Electrons and nuclei are treated equally in terms of the parametrization of the non-relativistic total wave function, which is written as a linear combination of basis functions constructed from explicitly correlated Gaussian functions and the global vector representation. The integrals of the electric transition dipole moment are derived corresponding to these basis functions in both the length and the velocity representation. The calculations are performed in laboratory-fixed Cartesian coordinates without relying on coordinates which separate the center of mass from the translationally invariant degrees of freedom. The effect of the overall motion is eliminated through translationally invariant integral expressions. The electric transition dipole moment is calculated between two rovibronic levels of the H2 molecule assignable to the lowest rovibrational states of the X (1)Σ(g)(+) and B (1)Σ(u)(+) electronic states in the clamped-nuclei framework. This is the first evaluation of this quantity in a full quantum mechanical treatment without relying on the Born-Oppenheimer approximation.

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