ZCAM Course

The last school of 2017 took place at Zaragoza during the first week of October (2nd-6th), the city was preparing for its annual city Fest called “la Fiesta del Pilar” where the inhabitants celebrate the day of their Patron Saint, “La virgen del Pilar”, and that ambiance could be felt on their streets. An excellent atmosphere to organize the ZCAM school at the Campus Rio Ebro, a brand new campus of the Zaragoza University dedicated to scientific fields. The main focus of the ZCAM School was, as suggested by PhD Candidates during the development of their thesis, to have a School more dedicated to the theoretical basis of main electronic structure computational methods from its formalism to the application of the knowledge. This way the school was divided in the Following topics.

– Relativistic Quantum Chemistry by Prof. Dr. Timo Fleig

– Correlation methods by Prof. Dr. Stefano Evangelisti

– Configuration Interaction by Prof. Dr. Nadia ben Amor

– Density Functional Theory by Prof. Dr. Arjan Berger

– Time Dependent Density Functional Theory by Prof. Dr. Alberto Castro

The school was also one of the 10 mandatory activities for ITN researchers.

Day 1. Special relativity in atoms and molecules

The first day of the school started in the morning with an introduction to the foundations and Lorentz covariant formalism of Einsteinian special relativity. An analysis of the different frames, from classical to relativistic was done and the formalism of the transformations among them were also explained.

The second part concerned the deduction of the free-fermion Dirac equation and — including external scalar potentials — its approximate rendering in the framework of Pauli theory for applications in atomic physics.

The lab course treated scalar and non-scalar effects of special relativity in the series of 5 neutral pnictogen atoms by applying the Kramers-Restricted Configuration Interaction (KRCI) module of the DIRAC program package, the currently most widely used code for relativistic electronic-structure calculations.

Day 2. Second Quantization and configuration interaction

The second day started with the introduction to second quantization; After a presentation of the second quantization operators (creation, annihilation, and excitation) the Hamiltonian and density matrices have been expressed using these operators.

The Hückel Hamiltonian was introduced using a discretization scheme for position and momenta operators. The Hubbard was also introduced, which makes use of creation and annihilation operators.

Once the theoretical basis was introduced in its mathematical formalism, the next step was presenting the configuration interaction (CI) theory and its different methods. The first part of the presentation concerned the essential ideas of configuration interaction theory in a mathematical framework. The Full CI method and the specific cases of the complete, restricted and generalized active space self consistent field methods (CASSCF, RASSCF, GASSCF) have been presented, as well as the truncated CI ones.

During the afternoon the activity continued by doing a Hands-on session on truncated CI, using molcas and casdi codes. In a second part, the size-extensivity error of truncated CI as well as different corrections (Davidson, CEPA, ACPF,AQCC, (SC) 2 , Class Dressing) have also been treated. The last part concerned the localization of the orbitals and the linear scaling of the CI code.

Day 3. Fermi and Coulomb Holes

The morning of the third day was centered on the correlation due to Pauli exclusion principle and Coulombic interaction. These were formulated as the Antisymmetry (Fermi) Hole, which describes a decrease of the probability of presence of an electron (fermion) in the proximity of another equivalent electron with same spin.

The Correlation Hole (Coulomb, Hubbard, etc.) describes a decrease of the probability of presence of an electron in the proximity of another electron (of any spin type) because of their mutual interactions. The theory was exemplified with a study o the H2 system.

The following lesson consisted on Size-extensivity error and Corrections and Localization of the orbitals and linear scaling CI.

During the afternoon an Applied session had taken place, on which wich the knowledge was applied by implementing it onto a FORTRAN code.

Day 4. Density Functional Theory (DFT)

The fourth day was centered on Density Functional Theory (DFT). During the morning the theoretical basis was introduced, the main advantage of the theory, substituting the 3N degrees of freedom wavefunction by the 3 degrees of freedom of its density was the starting point to introduce the DFT theorems. The theory develops all magnitudes as functionals of the density of the system, the method relies greatly on the choice for the exchange-correlation functional which is nowadays the main field of both critics and investigation within the DFT theory itself.

The different exchange-correlation functional theories were introduced from the Local Density Approximation based functional, the Geneneralized Gradient approximation to the most lately and system ad hoc developed.

The Afternoon was devoted to apply the theory by running calculations.

Day 5. Time Dependent Density Functional Theory (TDDFT)

The last day also was devoted to the introduction to Density Functional Theory (DFT) again and the and the Time Dependent Density Functional Theory (TDDFT). The last one tries to deal with how a quantum mechanical system responds to external perturbations this is, response properties using a density functional theory scheme. And thus calculating the excited states of the systems of interest that can explain processes like polarizabilities or photo-absorption. The TDDFT is based on the assumption that there exists a unique relationship between time-dependent densities and external potentials, and therefore, any property of the system can be written, in principle, as a functional of the time-dependent density. There exists a potential for this non-interacting system such that it reproduces the density of the interacting system. This is the so-called time-dependent Kohn-Sham potential. The evolution of the non-interacting system may be easily obtained by propagating single-particle equations. TDDF Response Theory à la Casida was also explained.

Finally some examples in bio physics were also explained and the Hands-on session with the octopus code was carried out to calculate some basic optical absorption calculations.

Carles Martí

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