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En optique et en chimie quantique, « l'ham … En optique et en chimie quantique, « l'hamiltonien moléculaire » est l'opérateur hamiltonien de l'énergie des électrons et des noyaux d'une molécule. Cet opérateur hermitien et l'équation de Schrödinger associée sont à la base du calcul des propriétés des molécules et des agrégats de molécules, comme la conductivité, les propriétés optiques et magnétique, ou encore la réactivité.es et magnétique, ou encore la réactivité.
, Na física atômica, molecular e óptica e na … Na física atômica, molecular e óptica e na química quântica, o hamiltoniano molecular é o operador hamiltoniano que representa a energia dos elétrons e núcleos de uma molécula. Esse operador e a equação de Schrödinger associada desempenham um papel central na química e na física computacional para calcular propriedades de moléculas e agregados de moléculas, como condutividade térmica, calor específico, condutividade elétrica, propriedades ópticas e magnéticas e reatividade. O Hamiltoniano molecular é uma soma de vários termos: seus principais termos são as energias cinéticas dos elétrons e as interações Coulomb (eletrostática) entre os dois tipos de partículas carregadas. O Hamiltoniano que contém apenas as energias cinéticas dos elétrons e núcleos, e as interações de Coulomb entre eles, é conhecido como Hamiltoniano de Coulomb. Nele faltam alguns termos pequenos, a maioria dos quais são devidos a spin eletrônico e nuclear.s são devidos a spin eletrônico e nuclear.
, En física atómica, molecular y óptica, así … En física atómica, molecular y óptica, así como en química cuántica, hamiltoniano molecular es el nombre dado al operador hamiltoniano que representa la energía del sistema constituido por los electrones y el conjunto de núcleos de una molécula. Este es una operador autoadjunto, es decir hermítico, cuya ecuación de Schrödinger asociada juega un papel central en la química computacional y en la física computacional para calcular propiedades de moléculas y agregados de las mismas, tales como conductividad térmica, calor específico, conductividad eléctrica, propiedades ópticas y magnéticas, y reactividad.dades ópticas y magnéticas, y reactividad.
, In atomic, molecular, and optical physics … 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 , 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.t is also known as the Eckart Hamiltonian.
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En optique et en chimie quantique, « l'ham … En optique et en chimie quantique, « l'hamiltonien moléculaire » est l'opérateur hamiltonien de l'énergie des électrons et des noyaux d'une molécule. Cet opérateur hermitien et l'équation de Schrödinger associée sont à la base du calcul des propriétés des molécules et des agrégats de molécules, comme la conductivité, les propriétés optiques et magnétique, ou encore la réactivité.es et magnétique, ou encore la réactivité.
, Na física atômica, molecular e óptica e na … Na física atômica, molecular e óptica e na química quântica, o hamiltoniano molecular é o operador hamiltoniano que representa a energia dos elétrons e núcleos de uma molécula. Esse operador e a equação de Schrödinger associada desempenham um papel central na química e na física computacional para calcular propriedades de moléculas e agregados de moléculas, como condutividade térmica, calor específico, condutividade elétrica, propriedades ópticas e magnéticas e reatividade.edades ópticas e magnéticas e reatividade.
, In atomic, molecular, and optical physics … 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., and magnetic properties, and reactivity.
, En física atómica, molecular y óptica, así … En física atómica, molecular y óptica, así como en química cuántica, hamiltoniano molecular es el nombre dado al operador hamiltoniano que representa la energía del sistema constituido por los electrones y el conjunto de núcleos de una molécula. Este es una operador autoadjunto, es decir hermítico, cuya ecuación de Schrödinger asociada juega un papel central en la química computacional y en la física computacional para calcular propiedades de moléculas y agregados de las mismas, tales como conductividad térmica, calor específico, conductividad eléctrica, propiedades ópticas y magnéticas, y reactividad.dades ópticas y magnéticas, y reactividad.
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Hamiltoniano molecular
, Molecular Hamiltonian
, Hamiltonien moléculaire
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