206 research outputs found
Disentangling the marginal problem in quantum chemistry
It is well known that determining the energy of molecules and other quantum many-body systems reduces in the standard approximation to optimizing a simple linear functional of a 12-variable object, the two-electron reduced density matrix (2-RDM). The difficulty is that the variation ensemble for that functional has never been satisfactorily determined. This is known as the N-representability problem of quantum chemistry (which to a large extent is a problem of quantum information theory). The situation has given rise to competing research programs, typically trading more complicated functionals for simpler representability conditions. Chief among them is density functional theory, based on a three-variable object for which the N-representability is trivial, whereas the exact functional is very strange indeed, and probably forever unknowable. An intermediate position is occupied by 1-RDM functional theory. Postulated by Pauli to explain the electronic structure of atoms and molecules, the exclusion principle establishes an upper bound of 1 for the fermionic natural occupation numbers, accordingly allowing no more than one electron in each quantum state. This is a necessary and sufficient condition for a 1-RDM to be the contraction of an ensemble N-body density matrix. The fermionic one-body quantum marginal problem asks whether given natural occupation numbers can arise from an antisymmetric (ensemble or pure) N-particle state. The configuration interaction method affords optimal descriptions of quantum states of atoms and molecules by expanding the wave function in terms of orbital-based configurations of Slater determinants. For these systems, the dimension of the Hilbert space grows binomially with m, the number of spin-orbitals in the basis of the one-particle Hilbert space, and N, the number of electrons of the system. It has been observed that for the rank-six approximation of a pure-state N=3 system, the occupation numbers satisfy some additional constraints, stronger than the Pauli exclusion principle. The recent analysis by Alexander Klyachko and coworkers of the one-body marginal problem of the pure N-fermion state establishes a systematic approach to this type of constraints. In fact, for a pure quantum system of N electrons arranged in m spin-orbitals the occupation numbers satisfy a set of linear inequalities, known as generalized Pauli constraints (GPC). These inequalities define a convex polytope of allowed states in R^m. They are conditions for a 1-RDM to be the contraction of a N-body state. When one of the GPC is completely saturated, the system is said to be pinned, and it lies on one of the facets of the polytope. The nature of those conditions has been explored till now only in a few systems: a model of three spinless fermions confined to a one-dimensional harmonic potential, the lithium isoelectronic series and ground and excited states of some three- and four-electron molecules for the rank being at least twice the number of electrons. For all these systems the inequalities are (quite often) nearly saturated. This is the so-called quasipinning phenomenon. In this PhD thesis we have taken up the challenge of using numerical and analytical methods to examine pinned and quasipinned states, for atoms and molecules, starting from scratch with configuration-interaction and multiconfiguration self-consistent methods. This procedure serves to study the occurrence of quasipinning in realistic systems. A second goal is to show how the subsets of pinned states defined by GPC give rise to the most efficient approach from a computational viewpoint, yielding the leading order of the electron-electron correlations. As a consequence, we underline in this thesis the theoretical and practical importance of Klyachko's approach to the quantum marginal problem and its impact on the competing research programs to determine feasible electronic densities, 1-RDM, 2-RDM, intracular distributions or Wigner density quasiprobabilities. In relation with the above, our research provides a new variational optimization method for few-fermion ground states. We quantitatively confirm its high accuracy for quasipinned systems and derive an upper bound on the error of the correlation energy given by the ratio of the numerical value of the Klyachko inequality and the distance to the Hartree-Fock point. Depending on the details of the algorithm, we are able to reach 98%-99% of the correlation energy for such systems
Variational determination of the two-particle density matrix : the case of doubly-occupied space
The world at the level of the atom is described by the branch of science called quantum mechanics. The crown jewel of quantum mechanics is given by the Schrödinger equation which describes a system of indistinguishable particles, that interact with each other. However, an equation alone is not enough: the solution is what interests us. Unfortunately, the exponential scaling of the Hilbert space makes it unfeasible to calculate the exact wave function.
This dissertation concerns itself with one of the many ab initio methods that were developed to solve this problem: the variational determination of the second-order density matrix. This method already has a long history.
It is not considered to be on par with best ab initio methods.
This work tries an alternative approach. We assume that the wave function has a Slater determinant expansion where all orbitals are doubly occupied or empty. This assumption drastically reduces the scaling of the N-representability conditions. The downside is that the energy explicitly depends on the used orbitals and thus an orbital optimizer is needed. The hope is that by using this approximation, we can capture the lion's share of the static correlation and that any missing dynamic correlation can be added through perturbation theory.
We developed an algorithm based on Jacobi rotations. The scaling is much more favorable compared to the general case. The method is then tested on a array of benchmark systems
Ab Initio Calculations of Hydrocarbon Thermochemistry and Reaction Kinetics
In the framework of the SFB 551 "Carbon from the Gas Phase: Elementary Reactions, Structures, Materials" several areas of carbon related chemistry have been studied with help of computational tools. They include the exploration
of different ways of building PAHs, the attempt to check the limits of quantum chemistry methods in hydrocarbon chemistry using explicitly-correlated methods and the calculation of accurate reaction rates
Advances in the theoretical determination of molecular structure with applications to anion photoelectron spectroscopy
This Dissertation is focussed primarily on development of methods aiming at the determination
of molecular structures with application to systems with intra and intermolecular
hydrogen bonds. I have developed and demonstrated usefulness of Potential Energy
Surface Scanning Tool (PESST) by performing a systematic search for the most stable
structures of neutral and anionic phenylalanine and tyrosine molecules using electronic
structure methods. I have found out that tautomers resulting from the proton transfer
from the carboxylic OH to phenyl ring determine the structure of the most stable anions
of phenylalanine, but double proton transfer from the carboxylic and hydroxyl groups
determine structures of the most stable anions of tyrosine. The most stable conformer
of these valence anions remained adiabatically unbound with respect to the canonical
neutral in case of phenylalanine but bound in case of tyrosine. Valence anions identified
in this report have recently been observed experimentally.
Acetoacetic acid (AA), equipped with neighbouring carboxylic and keto groups, is a
promising system for studies of intramolecular proton transfer. The results of my computational
search for the most stable tautomers and conformers of the neutral and
anionic AA were used to interpret anion photoelectron and electron energy-loss spectroscopy
measurements. The valence anion was identi ed in photoelectron spectroscopy
experiments and the measured electron vertical detachment energy is in good agreement
with my computational predictions. My computational results allow rationalizing these
experimental findings in terms of the co-existence of various conformers of AA.
I considered stability of dimers formed by molecules that can exist in different conformational
states. I have developed a protocol that allows the dissection of the total
stabilisation energy into one-body conformational and deformational components and
the two-body interaction energy term. Interplay between these components determines
the overall stability of the dimer. The protocol has been tested on the dimers of oxalic
acid. The global minimum stability results from a balancing act between a moderately
attractive two-body interaction energy and small repulsive one-body terms. I have analysed
zero-point vibrational corrections to the stability of various conformers of oxalic
acid and their dimers. I have found that minimum energy structures with the most stabilising
sets of hydrogen bonds have the largest zero-point vibrational energy, contrary
to a naive anticipation based on red shifts of OH stretching modes involved in hydrogen
bonds.
My computational results demonstrated an unusual electrophilicity of oxalic acid (OA),
the simplest dicarboxylic acid. The electrophilicity results primarily from the bonding
carbon-carbon interaction in the SOMO orbital of the anion, but it is further enhanced
by intramolecular hydrogen bonds. The well-resolved structure in the photoelectron
spectrum has been reproduced theoretically, based on Franck-Condon factors for the
vibronic anion!neutral transitions. The excess electron binding energies in the dimer
and trimer of OA become very signi cant due to intermolecular proton transfer, with
the corresponding vertical detachment energy (VDE) values of approximately 3.3 and
4.6 eV. I have postulated a mechanism of excess electron mobility along molecular linear
chains supported by cyclic hydrogen bonds.
Searches for the most stable molecular conformer are frustrated by energy barriers separating
minima on the potential energy surface (PES). I have suggested that the barriers
might be suppressed by subtracting selected force field terms from the original PES.
The resulting deformed PES can be used in standard molecular dynamics (MD) or
Monte Carlo simulations. The MD trajectories on the original and deformed PESs of
ethanolamine differ markedly. The former gets stuck in a local minimum basin while
the latter moves quickly to the global minimum basin.(US) National Science Foundation grant CHE-111169
Accurate variational electronic structure calculations with the density matrix renormalization group
During the past 15 years, the density matrix renormalization group (DMRG) has
become increasingly important for ab initio quantum chemistry. The underlying
matrix product state (MPS) ansatz is a low-rank decomposition of the full
configuration interaction tensor. The virtual dimension of the MPS controls the
size of the corner of the many-body Hilbert space that can be reached.
Whereas the MPS ansatz will only yield an efficient description for
noncritical one-dimensional systems, it can still be used as a variational
ansatz for other finite-size systems. Rather large virtual dimensions are then
required. The two most important aspects to reduce the corresponding
computational cost are a proper choice and ordering of the active space
orbitals, and the exploitation of the symmetry group of the Hamiltonian. By
taking care of both aspects, DMRG becomes an efficient replacement for exact
diagonalization in quantum chemistry.
DMRG and Hartree-Fock theory have an analogous structure. The former can be
interpreted as a self-consistent mean-field theory in the DMRG lattice sites,
and the latter in the particles. It is possible to build upon this analogy to
introduce post-DMRG methods. Based on an approximate MPS, these methods provide
improved ans\"atze for the ground state, as well as for excitations.
Exponentiation of the single-particle (single-site) excitations for a Slater
determinant (an MPS with open boundary conditions) leads to the Thouless
theorem for Hartree-Fock theory (DMRG), an explicit nonredundant
parameterization of the entire manifold of Slater determinants (MPS
wavefunctions). This gives rise to the configuration interaction expansion for
DMRG. The Hubbard-Stratonovich transformation lies at the basis of auxiliary
field quantum Monte Carlo for Slater determinants. An analogous transformation
for spin-lattice Hamiltonians allows to formulate a promising variant for MPSs.Comment: PhD thesis (225 pages). PhD thesis, Ghent University (2014), ISBN
978946197194
Accurate variational electronic structure calculations with the density matrix renormalization group
During the past fifteen years, the density matrix renormalization group (DMRG) has become increasingly important for ab initio quantum chemistry. Its underlying wavefunction ansatz, the matrix product state (MPS), is a low-rank decomposition of the full configuration interaction tensor. The virtual dimension of the MPS, the rank of the decomposition, controls the size of the corner of the many-body Hilbert space that can be reached with the ansatz. This parameter can be systematically increased until numerical convergence is reached.
Whereas the MPS ansatz can only capture exponentially decaying correlation functions in the thermodynamic limit, and will therefore only yield an efficient description for noncritical one-dimensional systems, it can still be used as a variational ansatz for finite-size systems. Rather large virtual dimensions are then required. The two most important aspects to reduce the corresponding computational cost are a proper choice and ordering of the active space orbitals, and the exploitation of the symmetry group of the Hamiltonian. By taking care of both aspects, DMRG becomes an efficient replacement for exact diagonalization in quantum chemistry. For hydrogen chains, accurate longitudinal static hyperpolarizabilities were obtained in the thermodynamic limit. In addition, the low-lying states of the carbon dimer were accurately resolved.
DMRG and Hartree-Fock theory have an analogous structure. The former can be interpreted as a self-consistent mean-field theory in the DMRG lattice sites, and the latter in the particles. It is possible to build upon this analogy to introduce post-DMRG methods. Based on an approximate MPS, these methods provide improved ansätze for the ground state, as well as for excitations. Exponentiation of the single-particle excitations for a Slater determinant leads to the Thouless theorem for Hartree-Fock theory, an explicit nonredundant parameterization of the entire manifold of Slater determinants. For an MPS with open boundary conditions, exponentiation of the single-site excitations leads to the Thouless theorem for DMRG, an explicit nonredundant parameterization of the entire manifold of MPS wavefunctions. This gives rise to the configuration interaction expansion for DMRG. The Hubbard-Stratonovich transformation lies at the basis of auxiliary field quantum Monte Carlo for Slater determinants. An analogous transformation for spin-lattice Hamiltonians allows to formulate a promising variant for matrix product states
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A Variational Principle for Modeling Electronic Excitations in Gas and Condensed Phase
Accurate modeling of electronic excited states is one of the most important and challenging problems in electronic structure theory. This thesis focuses on a recently developed excited state variational principle and its applications in gas and condensed phase. In contrast to the widely used excited states method such as linear response (LR) and many-body perturbation theory (MBPT), which find excited states by perturbing around the ground state wave function or a zeroth order particle-hole excitation picture, the new excited state variational principle directly targets excited states with the full flexibility of an approximate wave function ansatz. Due to its non-perturbative nature, this method offers balanced and systematically improvable descriptions to excited states. We will also discuss the efficient implementation of the new excited state variational principle through variational Monte Carlo and the Linear Method optimization algorithm.The new excited state variational principle is applied to predict both the excitation energies of low lying excited states in small molecules and optical gaps in solids. In molecules, the new method yields order-of-magnitude of improvements over the state-of-art excited state methods based on LR theory in double excitations. In solids, not only is the new method demonstrated to be more accurate than the commonly used MBPT method, but it could also be used to analyze and provide insights into MBPT. In order to further extend the method’s applicability, we introduce a modified optimization method that addresses a fatal memory bottleneck in the original algorithm. With only minor lose in accuracy, the modified algorithm reduces the required memory per parallel process from tens of gigabytes to hundreds of megabytes. With the aid of the new optimization method, we show that the new excited state variational principle could systematically converge the excitation energy in a strongly correlated, Mott-insulating hydrogen ring with respect to increasing flexibility in the wave function ansatzes
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 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
sufficient.
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
and NWHFI.
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
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