Accurate Prediction of Core Properties for Chiral Molecules using Pseudo Potentials
Pseudo potentials (PPs) constitute perhaps the most common way to treat relativity, often in a formally non-relativistic framework, and reduce the electronic structure to the chemically relevant part. The drawback is that orbitals obtained in this picture (called pseudo orbitals (POs)) show a redu...
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|Pseudo potentials (PPs) constitute perhaps the most common way to treat relativity, often in a formally non-relativistic framework, and reduce the electronic
structure to the chemically relevant part. The drawback is that orbitals obtained
in this picture (called pseudo orbitals (POs)) show a reduced nodal structure
and altered amplitude in the vicinity of the nucleus, when compared to the
corresponding molecular orbitals (MOs). Thus expectation values of operators
localized in the spatial core region that are calculated with POs, deviate significantly from the same expectation values calculated with all-electron (AE)
MOs. This study describes the reconstruction of AE MOs from POs, with a
focus on POs generated by energy consistent pseudo Hamiltonians. The method
reintroduces the nodal structure into the POs, thus providing an inexpensive
and easily implementable method that allows to use nonrelativistic, efficiently
calculated POs for good estimates of expectation values of core-like properties.
The discussion of the method proceeds in two parts: Firstly, the reconstruction scheme is developed for atomic cases. Secondly, the scheme is discussed in
the context of MO reconstruction and successfully applied to numerous numerical examples.
Starting from the equations of the state-averaged multi-configuration self-
consistent field method, used for the generation of energy consistent pseudo
potentials, the electronic spectrum of the many-electron Hamiltonian is linked
to the spectrum of the effective one-electron Fock operator by means of various
models systems. This relation and the Topp–Hopfield–Kramers theorem, are
used to show the shape-consistency of energy-consistent POs for atomic systems.
Shape-consistency describes POs that follow distinct AOs exactly outside a core-radius r_core . In the cases presented here, shape-consistency holds to a high degree
and it follows that in atomic systems every PO has one distinct partner in the
set of AOs. The overlap integral between these two orbitals is close to one, as it
is determined mainly by the spatial orbital parts outside r_core . Expanding, e.g.,
a 5s PO in occupied AOs, the 5s AOs will have the highest contribution. The
POs itself contains contributions from high-energy unoccupied AOs as well (e.g.
15s), which damp the nodal structure of the POs near the nucleus. Consequently,
neglecting contributions from unoccupied orbitals in a projection of the POs
reintroduces the nodal structure.
This approach is not directly suitable for the reconstruction of MOs, as they
often need to be expanded in a full set of AOs at each atomic center, including all
unoccupied orbitals, to properly account for the electron density distribution in
the molecule. However, it is shown that the occupied MOs are well described by
occupied and low-energy unoccupied AOs only and a mapping of the POs onto
a basis containing only these orbitals reconstructs the nodal structure of the MO.
The approach uses only standard integrals available in most quantum chemistry
programs. The computational cost of these integrals scales with N^2 , where N is
the number of basis functions. The most time consuming step is a Gram-Schmidt
orthogonalization, which scales in this implementation with MN^2 , M being the
number of reconstructed orbitals.
The reconstruction method is subsequently tested: Valence orbitals of atomic,
closed-shell systems were reconstructed numerically exactly. The influence of
numerical parameters is investigated using the molecule BaF . It is shown that
the method is basis set dependent: One has to ensure that the PO basis can be
expanded exactly in the basis of AOs. Violating this rule of thumb may degrade
the quality of reconstructed orbitals. Additionally, the representation of MOs by
a linear combination of occupied and unoccupied AOs is investigated. For the
exemplary systems, the shells included in the fitting procedure of the PP were
Reconstruction of the alkaline earth monofluorides showed that periodic
trends can be reconstructed as well. Scaling of hyperfine structure parameters
with increasing atomic number is discussed. For hydrogenic atoms, the scaling should be linear, whereas small deviations from the linear behavior were
observed for molecules. The scaling laws computed from reconstructed and
reference orbitals were almost identical. In this context, the failure of commonly
used relativistic enhancement factors beyond atomic number 100 is discussed.
Applicability of the method is also tested on parity violating properties for which
the main contribution is generated by the valence orbitals near the nucleus.
Symmetry-independence of the method is shown by successful reconstruction of
orbitals of the tetrahedral PbCl_4 and chiral NWHClF. The reliable reconstruction
of chemical trends is shown with the help of the NWHClF derivatives NWHBrF
The study of chiral compounds as, e.g., NWHClF and its group 17 derivatives, which have been proposed as paradigm for the detection of parity-violation
in chiral molecules, remains of great importance. Especially the direct determination of absolute configuration of chiral centers is still non-trivial. The author
contributed to this field with a self-written molecular dynamics (MD) program
to simulate Coulomb explosions and thus to provide an insight especially into
the early explosion stages directly after an instantaneous multi-ionization of
the molecule CHBrClF, comparable to experiments using the Cold Target Recoil Ion Momentum Spectroscopy (COLTRIMS) technique. An algorithm for
the determination of the investigated molecule’s absolute configuration from
time-of-flight data and detection locations of molecular fragments is included
in the program. The program was used to generate experiment-equivalent data
which allowed for the first time the investigation of non-racemic mixtures by
the analysis routines of the experiment. The MD program includes harmonic
and anharmonic bond potentials. A charge-exchange model can model partial
charges in early phases of the Coulomb explosion.
Furthermore, Born–Oppenheimer MD simulations and statistical models
are used to explain the relative abundance of products belonging to competing
reaction channels, as obtained by photoion coincidence measurements. Additionally, qualitative statements about reaction branching ratios are made by
comparing the partition functions of involved degrees of freedom. Analytic
equations for partition functions of simple models are used to provide a simple
formula allowing fast estimates of reaction branching ratios.