Method Development

 1. Statically Screened Potentials

 2. Electron Density and Electron Distribution

 3. Natural Orbital Functional Theory

 4. Hookean systems and Quantum Dots


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1. Statically Screened Potentials

Finding the energies of bound states of screened Coulomb potentials has raised considerable interest for many years. There are many problems for which the reduction of the long-range Coulomb interactions due to the screening can drastically affect the results emerging from the consideration of bare Coulomb potentials. Thus, the calculation of thermodynamic properties of many-body systems in partially ionized gases, i.e., plasmas, has seen a rebirth since the inclusion of screening effects. The screened Coulomb potential can be represented by different models, the most famous of which is the analytic exponentially decaying potential of Yukawa type.

Please refer to the following publications for more information:

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2. Electron Density and Electron Distribution

The rapid progress in electronic structure theory and computer technology during the last two decades has made possible the determination of accurate wave functions for small and medium size molecules. However, this is only the first step in solving problems of chemical interest as most application of quantum mechanics in chemistry deal with the computation of expectation values or density functions in terms of which the properties of sought are rationalized. This requirement is closely related with one of the major challenges of quantum chemistry: the development of practical procedures for the extraction of chemically interesting information from N-electron wavefunctions.

We have succeeded in generating a full package of programs that allow for the analysis of Configuration Interaction and Hartree-Fock molecular and atomic wavefunctions in terms of one- and two-electron density functions in coordinate space. Drop us a note if you are interested in downloading the package.

Please refer to the following publications for more information:

  • J. M. Ugalde, R. J. Boyd, Angular aspects of exchange correlation and the fermi hole , Int. J. Quantum Chem. 27, 439 (1985).
  • J. M. Ugalde, R. J. Boyd, The radius of the Fermi hole in atoms, J. Phys. B 18, L701 (1985).
  • J.M. Ugalde, R.J. Boyd, On the relationship between the electron-pair distribution function and the electron correlation, Int. J. Quantum Chem. 29, 1 (1986).
  • J. M. Ugalde, R. J. Boyd, J. S. Perkyns, Angular aspects of electron correlation and the Coulomb hole, J. Chem. Phys. 87, 1216 (1987).
  • C. Sarasola, J.M. Ugalde , R.J. Boyd, The evaluation of extracule and intracule densities in the first row hydrides LiH, BeH, BH, CH, NH, OH and FH, from self-consistent field molecular wave functions, J. Phys. B 23, 1095 (1990).
  • J. M. Ugalde, C. Sarasola, L. Dominguez, R. J. Boyd, The evaluation of electronic extracule and intracule densities and related probability functions in terms of Gaussian basis functions, J. Math. Chem. 6, 51 (1991)
  • J. M. Ugalde, C. Sarasola, Upper bounds to the electron-electron coalescence density in terms of the one-electron density function, Phys. Rev. A 49, 3081 (1994).
  • J. M. Mercero, J. E. Fowler, C. Sarasola, J. M. Ugalde, Atomic configuration-interaction electron-electron counterbalance densities, Phys. Rev. A 59, 4255 (1999).
  • E. Valderrama, J. M. Mercero, J. M. Ugalde, The separation of the dynamical and non-dynamical electron correlation effects, J. Phys. B 34, 275 (2001).
  • E. Valderrama, X. Fradera, J. M. Ugalde, Electron-electron counterbalance density for molecules: exchange and correlation effects, J. Chem. Phys. 115, 1987 (2001).
  • E. Valderrama, J. M. Ugalde, Role of electron-electron coalescence density in density functional theory, Int. J. Quantum Chem. 86, 40 (2002).
  • E. V. Ludeña, J. M. Ugalde, X. Lopez, J. Fernández-Rico, G. Ramírez, A reinterpretation of the nature of the Fermi hole, J. Chem. Phys. 120, 540 (2004).
  • E.G. Valderrama, J.M. Ugalde, Electron correlation studies by means of local-scaling transformations and electron-pair density functions, J. Math. Chem. 37, 211 (2005).
  • M. Piris, X. Lopez, J.M. Ugalde, Electron-pair relaxation holes, J. Chem. Phys. 128, 214105 (2008).
  • M. Piris, X. Lopez, J.M. Ugalde, Correlation holes for the helium dimer, J. Chem. Phys. 128, 134102 (2008).

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3. Natural Orbital Functional Theory

The most accurate electronic structure methods based on N-particle wave functions are too expensive to be applied to large systems. It is clearer every day the need for treatments of electron correlation that scale favorably with the number of electrons. Among them, the Kohn-Sham formulation of the Density Functional Theory (DFT) has become very popular thanks to its relatively low computational cost. However, present-day functionals have several problems due mainly to the so-called “correlation kinetic energy”, but most importantly, currently available functionals are not N-representable.

A direction for improving DFT lies in the development of a functional theory based upon the one-particle reduced density matrix (1-RDM) rather than on the one-electron density. The 1-RDM is a much simpler object than the N-particle wave function, and the ensemble N-representability conditions are well-known. The existence and properties of the total energy functional of the 1-RDM are well-established. Its development may be greatly aided by imposition of multiple constraints that are more strict and abundant than their DFT counterparts. The major advantage of a 1-RDM formulation is that the kinetic energy is explicitly defined and does not require the construction of a functional. The unknown functional in a 1-RDM-based theory only needs to incorporate electron correlation. The 1-RDM functional incorporates fractional occupations in a natural way, which may provide a correct description of both dynamical and nondynamical correlation.

The 1-RDM functional is called Natural Orbital Functional (NOF) when it is based upon the spectral expansion of the 1-RDM. We have recently proposed an explicit antisymmetric form for the cumulant of the two-particle RDM in terms of two symmetric matrices, Δ and Π. The so-called D, Q and G positivity conditions have allowed us to propose the functional forms for the Δ and Π matrices. We refer to this approach as PNOF (Int. J. Quantum Chem. 106, 1093, 2006). This functional reduces to the exact expression for the total energy in two-electron systems. The PNOF depends only on the Coulomb (J), exchange (K) and exchange and time-inversion (L) integrals, thus can be referred to as JKL-only approximation. A spin-conserving NOF theory has been also formulated (J. Chem. Phys. 131, 021102, 2009). Validation tests of PNOF for predicting several atomic and molecular properties have been performed using the PNOFID code based on a novel iterative diagonalization method to obtain the natural orbitals (J. Comp. Chem. 30, 2078, 2009).

Please refer to the following publications for more information:

Review Article on NOFT:

  • M. Piris, Natural Orbital Functional Theory, in Reduced-Density-Matrix Mechanics: With Applications to Many-electron Atoms and Molecules, edited by D. Mazziotti, Adv. Chem. Phys. 134, 387 (2007).

Review Article on PNOF:

PNOF5:

  • M. Piris, X. Lopez, F. Ruipérez, J. M. Matxain, J.M. Ugalde, A natural orbital functional for multiconfigurational states, J. Chem. Phys. 134, 164102 (2011).
  • J. M. Matxain, M. Piris, F. Ruipérez, X. Lopez, J. M. Ugalde, Homolytic molecular dissociation in natural orbital functional theory, Phys. Chem. Chem. Phys. 13, 20129 (2011).
  • J. M. Matxain, M. Piris, J. M. Mercero, X. Lopez, J. M. Ugalde, “sp3 hybrid orbitals and ionization energies of methane from PNOF5″, Chem. Phys. Lett. 531, 272 (2012).
  • J. M. Matxain, M. Piris, J. Uranga, X. Lopez, G. Merino, J. M. Ugalde, “The Nature of the Chemical Bonds from PNOF5 calculations″, ChemPhysChem. (2012).
  • M. Piris, J. M. Matxain, X. Lopez, J. M. Ugalde, “The extended Koopmans’ theorem: vertical ionization potentials from Natural Orbital Functional Theory”,  J. Chem. Phys. 136, 174116 (2012).

PNOF4:

  • M. Piris, J. M. Matxain, X. Lopez and J. M. Ugalde, The role of the positivity N-representability conditions in Natural Orbital Functional Theory, J. Chem. Phys. 133, 111101 (2010).
  • X. Lopez, F. Ruipérez, M. Piris, J. M. Matxain, J. M. Ugalde, Diradicals and diradicaloids in Natural Orbital Functional Theory, ChemPhysChem 12, 1061 (2011) .
  • X. Lopez, M. Piris, J. M. Matxain, F. Ruipérez, J. M. Ugalde, Natural orbital functional theory and reactivity studies of diradical rearrangements : ethylene torsion as a case study, ChemPhysChem 12, 1673 (2011) .

PNOF3:

PNOF2:

PNOF1:

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4. Hookean systems and Quantum Dots

Hookean systems and Quantum DotsExact solutions of the Schrodinger equation for multiparticle systems with interparticle Coulombic interactions are unknown. One can try to find such exact solutions by modeling the interparticle interaction potential. Within this context, a much studied model is the Hookean two-electron atom, a system possessing a nucleus with charge +2 interacting through a harmonic potential with the electrons which, in turn, repel each other through the usual Coulomb interaction.

The replacement of the Coulombic central confining electron-nucleus potential occurring in real systems by a harmonic potential makes the problem separable in terms of center-of-mass and relative coordinates. For the relative motion, the ensuing equation has analytic solutions only for discrete values of the harmonic confinement strength parameter. We have extended these models to  solve exactly a general three-body problem depending on the relation of masses and force constant of the harmonic potential. In addition, we have also developt quasi-exact solutions for the non-Born-Oppenheimer Hookean H2 molecule, in which we model the confining electron–nucleus Coulombic potential with a harmonic potential, while keeping the remaining interparticle interactions. The corresponding electron and nuclei pair densities are calculated from these solutions, and they are used to get insoght various phenomena, such as the relation between the different correlation regimes between electrons and nuclei and the emergence of molecular structure in 3-body systems as a function of the relative masses. In addition, we have observed that the inclusion of the proper Coulomb potential for the electron-electron interaction is the main relevant aspect affecting the dynamics of electronic motion, which pinpoints to the suitability of this toy-system as fas as modelization is concerned.

A particularly interesting  extension and application of these models is the description of the electronic structure in Quantum Dots, in which  a small number of electrons confined artifically using nanodevices can be simulated with harmonic potentials. There are various areas in which Quantum Dots show potential applications, from quantum computing to nano-medicine. However, the characteristics  of the electronic structure of  Quantum Dots is yet not fully understood. In particular, the effect of external magnetic fields on the spin transitions in QD’s, and the treatment of the screening effects caused by the external medium are two aspects of these articial atoms that need a careful investigation. On the other hand, the magnetic coupling of two or more localized QD’s has not been studied yet. For the 2 electron spherical Quantum Dots, we can use our exact solutions developed for the three body problem to understand how single-tripet transition in QD’s are dependent on the magnetic fields. However, the extension of this work to systems containing more than 2 electrons, incorporation of screening effects, and the consideration of multiple QD’s, requires the use of standard approximate quantum chemical methods. For this purpose,  the corresponding one electron confinement integrals over gaussian-type functions, and the one- and two-electron Yukawa integrals have been developed and interfaced with the GAMESS-US program, which opens the possibility to study various aspects of the electronic structure of Quantum Dots.

Please refer to the following publications for more information:

  • E. V. Ludeña, X. Lopez, J. M. Ugalde, Non-Born-Oppenheimer Treatment of the H2 Hookean Molecule, J. Chem. Phys. 123, 024102 (2005).
  • X. Lopez, J. M. Ugalde, E. V. Ludeña, Extracular Densities of the Non–Born–Oppenheimer Hookean H2 Molecule, Chem. Phys. Lett. 412, 381 (2005).
  • E. V. Ludeña, X. Lopez, J. M. Ugalde, Pair Densities for the Hooke and Hooke-Calogero models of the non-Born-Oppenheimer hydrogen molecule, Lecture Series on Computer and Computational Sciences 4, 1217 (2005).
  • X. Lopez, J. M. Ugalde, E. V. Ludeña, Exact non-Born-Oppenheimer wave function for the Hooke-Calogero model of the H2 molecule. An useful exact molecular model for elecrton correlation studies. European Journal of Physics D 37, 351 (2006).
  • X. Lopez, J. M. Ugalde, L. Echevarria, E. V. Ludeña, Exact non-Born-Oppenheimer wave functions for three-particle Hookean systems with arbitrary masses, Phys. Rev. A 74, 042504 (2006).
  • V. V. Karasiev, X. Lopez, J. M. Ugalde, E. V. Ludeña, Kinetic Energy Functionals: Exact Ones From Analytic Model Wave Functions and Appoximate Ones in Orbital-Free Molecular Dynamics, Int. J. Mod. Phys. B 24, 5139 (2010).
  • E. V. Ludeña, L. Echevarria, J. M. Ugalde, X. Lopez, A. Corella-Madueño, Model for a biexciton in a lateral quantum dot based on exact solutions for the Hookean H2 molecule, Int. J. Quantum Chem.  111 1808 (2011).
  • E. V. Ludeña, L. Echevarria, X. Lopez, J. M. Ugalde, Non-Born-Oppenheimer electronic and nuclear densities for a Hooke-Calogero three-particle model: Non-uniqueness of density-derived molecular structure?, J. Chem. Phys. , (2012, in press).