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- Molecular_Hamiltonian abstract "In atomic, molecular, and optical physics and quantum chemistry, the molecular Hamiltonian is the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule. This operator and the associated Schrödinger equation play a central role in computational chemistry and physics for computing properties of molecules and aggregates of molecules, such as thermal conductivity, specific heat, electrical conductivity, optical, and magnetic properties, and reactivity.The elementary parts of a molecule are the nuclei, characterized by their atomic numbers, Z, and the electrons, which have negative elementary charge, −e. Their interaction gives a nuclear charge of Z + q, where q = −eN, with N equal to the number of electrons. Electrons and nuclei are, to a very good approximation, point charges and point masses. The molecular Hamiltonian is a sum of several terms: its major terms are the kinetic energies of the electrons and the Coulomb (electrostatic) interactions between the two kinds of charged particles. The Hamiltonian that contains only the kinetic energies of electrons and nuclei, and the Coulomb interactions between them, is known as the Coulomb Hamiltonian. From it are missing a number of small terms, most of which are due to electronic and nuclear spin.Although it is generally assumed that the solution of the time-independent Schrödinger equation associated with the Coulomb Hamiltonian will predict most properties of the molecule, including its shape (three-dimensional structure), calculations based on the full Coulomb Hamiltonian are very rare. The main reason is that its Schrödinger equation is very difficult to solve. Applications are restricted to small systems like the hydrogen molecule.Almost all calculations of molecular wavefunctions are based on the separation of the Coulomb Hamiltonian first devised by Born and Oppenheimer. The nuclear kinetic energy terms are omitted from the Coulomb Hamiltonian and one considers the remaining Hamiltonian as a Hamiltonian of electrons only. The stationary nuclei enter the problem only as generators of an electric potential in which the electrons move in a quantum mechanical way. Within this framework the molecular Hamiltonian has been simplified to the so-called clamped nucleus Hamiltonian, also called electronic Hamiltonian, that acts only on functions of the electronic coordinates.Once the Schrödinger equation of the clamped nucleus Hamiltonian has been solved for a sufficient number of constellations of the nuclei, an appropriate eigenvalue (usually the lowest) can be seen as a function of the nuclear coordinates, which leads to a potential energy surface. In practical calculations the surface is usually fitted in terms of some analytic functions. In the second step of the Born–Oppenheimer approximation the part of the full Coulomb Hamiltonian that depends on the electrons is replaced by the potential energy surface. This converts the total molecular Hamiltonian into another Hamiltonian that acts only on the nuclear coordinates. In the case of a breakdown of the Born–Oppenheimer approximation—which occurs when energies of different electronic states are close—the neighboring potential energy surfaces are needed, see this article for more details on this.The nuclear motion Schrödinger equation can be solved in a space-fixed (laboratory) frame, but then the translational and rotational (external) energies are not accounted for. Only the (internal) atomic vibrations enter the problem. Further, for molecules larger than triatomic ones, it is quite common to introduce the harmonic approximation, which approximates the potential energy surface as a quadratic function of the atomic displacements. This gives the harmonic nuclear motion Hamiltonian. Making the harmonic approximation, we can convert the Hamiltonian into a sum of uncoupled one-dimensional harmonic oscillator Hamiltonians. The one-dimensional harmonic oscillator is one of the few systems that allows an exact solution of the Schrödinger equation.Alternatively, the nuclear motion (rovibrational) Schrödinger equation can be solved in a special frame (an Eckart frame) that rotates and translates with the molecule. Formulated with respect to this body-fixed frame the Hamiltonian accounts for rotation, translation and vibration of the nuclei. Since Watson introduced in 1968 an important simplification to this Hamiltonian, it is often referred to as Watson's nuclear motion Hamiltonian, but it is also known as the Eckart Hamiltonian.".
- Molecular_Hamiltonian wikiPageID "7148738".
- Molecular_Hamiltonian wikiPageLength "37634".
- Molecular_Hamiltonian wikiPageOutDegree "103".
- Molecular_Hamiltonian wikiPageRevisionID "665561198".
- Molecular_Hamiltonian wikiPageWikiLink Adiabatic_theorem.
- Molecular_Hamiltonian wikiPageWikiLink Atomic,_molecular,_and_optical_physics.
- Molecular_Hamiltonian wikiPageWikiLink Atomic_nucleus.
- Molecular_Hamiltonian wikiPageWikiLink Atomic_number.
- Molecular_Hamiltonian wikiPageWikiLink Atomic_orbital.
- Molecular_Hamiltonian wikiPageWikiLink Born–Oppenheimer_approximation.
- Molecular_Hamiltonian wikiPageWikiLink Canonical_quantization.
- Molecular_Hamiltonian wikiPageWikiLink Category:Molecular_physics.
- Molecular_Hamiltonian wikiPageWikiLink Category:Quantum_chemistry.
- Molecular_Hamiltonian wikiPageWikiLink Category:Spectroscopy.
- Molecular_Hamiltonian wikiPageWikiLink Center_of_mass.
- Molecular_Hamiltonian wikiPageWikiLink Centrifugal_force.
- Molecular_Hamiltonian wikiPageWikiLink Chain_rule.
- Molecular_Hamiltonian wikiPageWikiLink Chapman_&_Hall.
- Molecular_Hamiltonian wikiPageWikiLink Chirality_(chemistry).
- Molecular_Hamiltonian wikiPageWikiLink Computational_chemistry.
- Molecular_Hamiltonian wikiPageWikiLink Computational_physics.
- Molecular_Hamiltonian wikiPageWikiLink Coriolis_force.
- Molecular_Hamiltonian wikiPageWikiLink Coulombs_law.
- Molecular_Hamiltonian wikiPageWikiLink Curvilinear_coordinates.
- Molecular_Hamiltonian wikiPageWikiLink Del.
- Molecular_Hamiltonian wikiPageWikiLink Distance.
- Molecular_Hamiltonian wikiPageWikiLink Dover_Publications.
- Molecular_Hamiltonian wikiPageWikiLink Eckart_conditions.
- Molecular_Hamiltonian wikiPageWikiLink Eigenvalues_and_eigenvectors.
- Molecular_Hamiltonian wikiPageWikiLink Electric_potential.
- Molecular_Hamiltonian wikiPageWikiLink Electrical_resistivity_and_conductivity.
- Molecular_Hamiltonian wikiPageWikiLink Electron.
- Molecular_Hamiltonian wikiPageWikiLink Elementary_charge.
- Molecular_Hamiltonian wikiPageWikiLink Enantiomer.
- Molecular_Hamiltonian wikiPageWikiLink Energy.
- Molecular_Hamiltonian wikiPageWikiLink Euler_angles.
- Molecular_Hamiltonian wikiPageWikiLink Fermi-contact_term.
- Molecular_Hamiltonian wikiPageWikiLink Fictitious_force.
- Molecular_Hamiltonian wikiPageWikiLink Frame_of_reference.
- Molecular_Hamiltonian wikiPageWikiLink Franck–Condon_principle.
- Molecular_Hamiltonian wikiPageWikiLink Function_(mathematics).
- Molecular_Hamiltonian wikiPageWikiLink GF_method.
- Molecular_Hamiltonian wikiPageWikiLink Hamiltonian_(quantum_mechanics).
- Molecular_Hamiltonian wikiPageWikiLink Harmonic_approximation.
- Molecular_Hamiltonian wikiPageWikiLink Harmonic_oscillator.
- Molecular_Hamiltonian wikiPageWikiLink Heat_capacity.
- Molecular_Hamiltonian wikiPageWikiLink Hessian_matrix.
- Molecular_Hamiltonian wikiPageWikiLink Inertial_frame_of_reference.
- Molecular_Hamiltonian wikiPageWikiLink Kinetic_energy.
- Molecular_Hamiltonian wikiPageWikiLink Laplace_operator.
- Molecular_Hamiltonian wikiPageWikiLink Laplace–Beltrami_operator.
- Molecular_Hamiltonian wikiPageWikiLink Least_squares.
- Molecular_Hamiltonian wikiPageWikiLink Levi-Civita_symbol.
- Molecular_Hamiltonian wikiPageWikiLink List_of_quantum_chemistry_and_solid-state_physics_software.
- Molecular_Hamiltonian wikiPageWikiLink Lorentz_covariance.
- Molecular_Hamiltonian wikiPageWikiLink Magnetic_dipole–dipole_interaction.
- Molecular_Hamiltonian wikiPageWikiLink Magnetism.
- Molecular_Hamiltonian wikiPageWikiLink Molecule.
- Molecular_Hamiltonian wikiPageWikiLink Non-inertial_reference_frame.
- Molecular_Hamiltonian wikiPageWikiLink Nuclear_quadrupole_coupling.
- Molecular_Hamiltonian wikiPageWikiLink Optics.
- Molecular_Hamiltonian wikiPageWikiLink Paul_Dirac.
- Molecular_Hamiltonian wikiPageWikiLink Point_particle.
- Molecular_Hamiltonian wikiPageWikiLink Potential_energy_surface.
- Molecular_Hamiltonian wikiPageWikiLink Quadratic_function.
- Molecular_Hamiltonian wikiPageWikiLink Quadrupole.
- Molecular_Hamiltonian wikiPageWikiLink Quantum_chemistry.
- Molecular_Hamiltonian wikiPageWikiLink Reactivity_(chemistry).
- Molecular_Hamiltonian wikiPageWikiLink Rigid_rotor.
- Molecular_Hamiltonian wikiPageWikiLink Rotation.
- Molecular_Hamiltonian wikiPageWikiLink Schrödinger_equation.
- Molecular_Hamiltonian wikiPageWikiLink Self-adjoint.
- Molecular_Hamiltonian wikiPageWikiLink Semi-rigid_molecule.
- Molecular_Hamiltonian wikiPageWikiLink Spin-other-orbit_coupling.
- Molecular_Hamiltonian wikiPageWikiLink Spin_(physics).
- Molecular_Hamiltonian wikiPageWikiLink Spin–orbit_interaction.
- Molecular_Hamiltonian wikiPageWikiLink Standard_Model.
- Molecular_Hamiltonian wikiPageWikiLink Thermal_conductivity.
- Molecular_Hamiltonian wikiPageWikiLink Translation.
- Molecular_Hamiltonian wikiPageWikiLink Translation_(geometry).
- Molecular_Hamiltonian wikiPageWikiLink Uncertainty_principle.
- Molecular_Hamiltonian wikiPageWikiLink Vibration.
- Molecular_Hamiltonian wikiPageWikiLink Wigner_D-matrix.
- Molecular_Hamiltonian wikiPageWikiLinkText "Hamiltonian".
- Molecular_Hamiltonian wikiPageWikiLinkText "Molecular Hamiltonian".
- Molecular_Hamiltonian wikiPageWikiLinkText "Schrödinger equation".
- Molecular_Hamiltonian wikiPageWikiLinkText "clamped nucleus Hamiltonian".
- Molecular_Hamiltonian wikiPageWikiLinkText "hydrogen atom Hamiltonian".
- Molecular_Hamiltonian wikiPageWikiLinkText "molecular Hamiltonian".
- Molecular_Hamiltonian wikiPageUsesTemplate Template:Cite_book.
- Molecular_Hamiltonian wikiPageUsesTemplate Template:Cite_journal.
- Molecular_Hamiltonian wikiPageUsesTemplate Template:Reflist.
- Molecular_Hamiltonian wikiPageUsesTemplate Template:Use_dmy_dates.
- Molecular_Hamiltonian subject Category:Molecular_physics.
- Molecular_Hamiltonian subject Category:Quantum_chemistry.
- Molecular_Hamiltonian subject Category:Spectroscopy.
- Molecular_Hamiltonian hypernym Operator.
- Molecular_Hamiltonian type Company.
- Molecular_Hamiltonian type Mechanic.