1,721,157 research outputs found
Many-Body Interatomic Potentials for the Study of RadiationEffects in Nuclear Structural Materials.
No full textFaddeev random phase approximation applied to molecules
Full text linkThe Faddeev Random Phase Approximation (FRPA) is a Green’s function method which couples collective degrees of freedom to the single particle motion by resumming an infinite number of Feynman diagrams. The Faddeev technique is applied to describe the two-particle-one-hole (2p1h) and two-hole-one-particle (2h1p) Green’s function in terms of non-interacting propagators and kernels for the particle-particle (pp) and particle-hole (ph) interactions. This results in an equal treatment of the intermediary pp and ph channels. In FRPA both the pp and ph phonons are calculated on the random phase approximation (RPA) level. In this work the equations that lead to the FRPA eigenvalue problem are derived. The method is then applied to atoms, small molecules and the Hubbard model, for which the ground state energy and the ionization energies are calculated. Special attention is directed to the RPA instability in the dissociation limit of diatomic molecules and in the Hubbard model. Several solutions are proposed to overcome this problem
Variational determination of the two-particle density matrix : the case of doubly-occupied space
Full text linkThe 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
The Atom-In-Molecule concept from a density-matrix perspective
No full textDe elektronenstructuur bepaalt het chemisch gedrag van moleculen. Deze structuur kan worden gesimuleerd door toepassing van de wetten van de kwantummechanica op een systeem van “elektronen en kernen”. Het is echter niet eenvoudig om deze fysische informatie te vertalen naar de klassieke chemische concepten, die uitgaan van een model gebaseerd op “atomen en bindingen”, zoals partiële atoomladingen, het karakteristieke gedrag van functionele groepen (nucleofiliciteit, basiciteit, elektronegativiteit), substituenteffecten, regio-selectiviteit, sterische effecten, ... Dergelijke vertaling vereist dat de elektronische structuur wordt verdeeld in bijdragen van atomen in moleculen. Dit is het onderwerp van het “quantum Atoms in Molecules” (AIM) project.
De eenvoudigste weergave van de elektronen-structuur is de elektronendichtheid. De courante AIM procedures zijn gebaseerd op deze grootheid. Volgens de Hohenberg-Kohn theorema's bepaalt de elektronendichtheid de elektronische golffunctie en aldus alle elektronische eigenschappen van de molecule. Moleculaire eigenschappen zijn universele functionalen van de elektronendichtheid. In de praktijk zijn er echter geen computationeel haalbare (exacte) expliciete uitdrukkingen voor deze functionalen bekend. Dit levert problemen op voor het bepalen van AIM eigenschappen.
De elektronische eigenschappen van een molecule kunnen echter wel expliciet worden geschreven als een functionaal van zogenaamde “dichtheids-matrices”. Voor de chemisch meest relevante eigenschappen (bv. energie, elektronegativiteit,...) wordt deze dichtheidsmatrix bepaald door de coördinaten van hooguit twee elektronen of – wanneer een gemiddeld-veld benadering wordt toegepast - zelfs door de coördinaten van een enkel elektron. Dit proefschrift behandelt AIM technieken die werden ontwikkeld op basis van een verdeling van de volledige ééndeeltjes dichtheidsmatrix (1DM) over de atomen in een molecule.
AIM technieken die gebaseerd zijn op de volledige ééndeeltjesdichtheidsmatrix zijn nuttig om de beschrijving van atomen en bindingen in de molecule te verbeteren zodat bekomen AIM waarden meer accuraat zijn. Ze vermijden ook de omslachtige numerieke integraties die nodig zijn in de conventionele AIM methoden. Daarnaast laten ze toe om waarden te berekenen voor AIM grootheden die verband houden met belangrijke concepten in de chemie (bv. substituenteffecten), maar die niet kunnen worden bekomen met de conventionele methoden. Dit doctoraatswerk illustreert het potentieel van dergelijke AIM technieken en laat daardoor toe om op een meer fundamentele wijze fysische informatie te vertalen en chemische concepten te kwantificeren
Accurate variational electronic structure calculations with the density matrix renormalization group
Full text linkDuring 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
Variational optimization of second order density matrices for electronic structure calculation
Full text linkThe exponential growth of the dimension of the exact wavefunction with the size of a chemical system makes it impossible to compute chemical properties of large chemical systems exactly. A myriad of ab initio methods that use simpler mathematical objects to describe the system has thrived on this realization, among which the variational second order density matrix method. The aim of my thesis has been to evaluate the use of this method for chemistry and to identify the major theoretical and computational challenges that need to be overcome to make it successful for chemical applications.
The major theoretical challenges originate from the need for the second order density matrix to be N-representable: it must be derivable from an ensemble of N-electron states. Our calculations have pointed out major drawbacks of commonly used necessary N-representability conditions, such as incorrect dissociation into fractionally charged products and size-inconsistency. We have derived subspace energy constraints that fix these problems, albeit in an ad-hoc manner. Additionally, we have found that standard constraints on spin properties cause serious problems, such as false multiplet splitting and size-inconsistency. The subspace constraints relieve these problems as well, though only in the dissociation limit.
The major computational challenges originate from the method’s formulation as a vast semidefinite optimization problem. We have implemented and compared several algorithms that exploit the specific structure of the problem. Even so, their slow speed remains prohibitive. Both the second order methods and the zeroth order boundary point method that we tried performed quite similar, which suggests that the underlying problem responsible for their slow convergence, ill-conditioning due to the singularity of the optimal matrix, manifests itself in all these algorithms even though it is most explicit in the barrier method.
Significant progress in these theoretical and computational aspects is needed to make the variational second order density matrix method competitive to comparable wavefunction based methods
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