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Related Concept Videos

The Quantum-Mechanical Model of an Atom02:45

The Quantum-Mechanical Model of an Atom

Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, Erwin Schrödinger extended de Broglie’s work by deriving what is now known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra. Schrödinger...
Electronic Structure of Atoms02:28

Electronic Structure of Atoms


An atom comprises protons and neutrons, which are contained inside the dense, central core called the nucleus, with electrons present around the nucleus. Taking into account the wave–particle duality of electrons and the uncertainty in position around the nucleus, quantum mechanics provides a more accurate model for the atomic structure. It describes atomic orbitals as the regions around the nucleus where electrons of discrete energy exist, characterized by four quantum numbers:  n, l, ml, and...
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Atomic Orbitals

An atomic orbital represents the three-dimensional regions in an atom where an electron has the highest probability to reside. The radial distribution function indicates the total probability of finding an electron within the thin shell at a distance r from the nucleus. The atomic orbitals have distinct shapes which are determined by l, the angular momentum quantum number. The orbitals are often drawn with a boundary surface, enclosing densest regions of the cloud.
The Uncertainty Principle04:08

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Werner Heisenberg considered the limits of how accurately one can measure properties of an electron or other microscopic particles. He determined that there is a fundamental limit to how accurately one can measure both a particle’s position and its momentum simultaneously. The more accurate the measurement of the momentum of a particle is known, the less accurate the position at that time is known and vice versa. This is what is now called the Heisenberg uncertainty principle. He mathematically...
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Molecular Orbital Theory I

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Path integrals for electronic densities, reactivity indices, and localization functions in quantum systems.

Mihai V Putz1

  • 1Laboratory of Computational and Structural Physical Chemistry, Chemistry Department, West University of Timişoara, Pestalozzi Street No.16, Timişoara, RO-300115, Romania.

International Journal of Molecular Sciences
|January 21, 2010
PubMed
Summary

Path Integral (PI) formalism offers a quantum mechanical framework for many-electronic systems, extending density matrix theory. This versatile approach models quantum fluctuations and chemical reactivity for accurate computational chemistry.

Keywords:
Feynman integralFokker-Planck equationchemical action and hardnessdensity matrix and functionalselectronegativityelectronic localizationpartition function

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Area of Science:

  • Quantum Mechanics and Computational Chemistry
  • Theoretical Physics and Physical Chemistry

Background:

  • Density matrix theory, a precursor to density functional theory, provides a foundation for Path Integral (PI) development.
  • Extending canonical density to many-electronic systems requires a density functional closure relationship.

Purpose of the Study:

  • To explore the advantages of using path integral formalism for electronic density prescription in quantum mechanics.
  • To present four distinct levels of path integral formalism for theoretical modeling.
  • To demonstrate the reliability of PI formalism for modeling fundamental physical and chemical concepts.

Main Methods:

  • Development and application of four levels of path integral formalism: Feynman quantum mechanical, semiclassical, Feynman-Kleinert effective classical, and Fokker-Planck non-equilibrium.
  • Rigorous definition and presentation of density matrix and canonical density within each formalism.
  • Practical specializations including quantum free/harmonic motions, high/low temperature limits, and quantum chemical calculations.

Main Results:

  • Path integral formalism provides an inner quantum mechanical description, averages quantum fluctuations, and acts as a propagator for quantum information.
  • Demonstrated applications include justifying Bohr's quantum stability postulate, calculating semiclassical electronegativity and hardness, and generalizing electronic focalization functions.
  • The formalism resembles the Schrödinger equation and allows for quantum statistical descriptions via partition function computation.

Conclusions:

  • Path Integral (PI) formalism is a versatile quantum mechanical framework suitable for analytical and computational modeling.
  • It effectively models diverse physical and chemical reactivity concepts in many-electronic systems.
  • The presented levels and applications underscore the reliability and broad applicability of PI methods in theoretical chemistry and physics.