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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Molecular Hamiltonian ： ウィキペディア英語版 Molecular Hamiltonian 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. == Coulomb Hamiltonian == The algebraic form of many observables—i.e., Hermitian operators representing observable quantities—is obtained by the following quantization rules: * Write the classical form of the observable in Hamilton form (as a function of momenta p and positions q). Both vectors are expressed with respect to an arbitrary inertial frame, usually referred to as ''laboratory-frame'' or ''space-fixed frame''. * Replace p by $-i\backslash hbar\backslash boldsymbol$ and interpret q as a multiplicative operator. Here $\backslash boldsymbol$ is the nabla operator, a vector operator consisting of first derivatives. The well-known commutation relations for the p and q operators follow directly from the differentiation rules. Classically the electrons and nuclei in a molecule have kinetic energy of the form ''p''2/''(2 m)'' and interact via Coulomb interactions, which are inversely proportional to the distance ''r''ij between particle ''i'' and ''j''. :$r\_\; \backslash equiv\; |\backslash mathbf\_i\; -\backslash mathbf\_j|$ = \sqrt_j)\cdot(\mathbf_i -\mathbf_j)} = \sqrt . In this expression r''i'' stands for the coordinate vector of any particle (electron or nucleus), but from here on we will reserve capital R to represent the nuclear coordinate, and lower case r for the electrons of the system. The coordinates can be taken to be expressed with respect to any Cartesian frame centered anywhere in space, because distance, being an inner product, is invariant under rotation of the frame and, being the norm of a difference vector, distance is invariant under translation of the frame as well. By quantizing the classical energy in Hamilton form one obtains the a molecular Hamilton operator that is often referred to as the Coulomb Hamiltonian. This Hamiltonian is a sum of five terms. They are # The kinetic energy operators for each nucleus in the system; # The kinetic energy operators for each electron in the system; # The potential energy between the electrons and nuclei – the total electron-nucleus Coulombic attraction in the system; # The potential energy arising from Coulombic electron-electron repulsions # The potential energy arising from Coulombic nuclei-nuclei repulsions – also known as the nuclear repulsion energy. See electric potential for more details. # $\backslash hat\_n\; =\; -\; \backslash sum\_i\; \backslash frac\; \backslash nabla^2\_$ # $\backslash hat\_e\; =\; -\; \backslash sum\_i\; \backslash frac\; \backslash nabla^2\_$ # $\backslash hat\_\; =\; -\; \backslash sum\_i\; \backslash sum\_j\; \backslash frac\_j\; \backslash right\; |\; \}$ # $\backslash hat\_\; =\; \backslash sum\_i\; \backslash sum\_\; \backslash frac\_j\; \backslash right\; |\; \}\; =$ \sum_i \sum_ \frac_j \right | } # $\backslash hat\_\; =\; \backslash sum\_i\; \backslash sum\_\; \backslash frac\_j\; \backslash right\; |\; \}\; =$ \sum_i \sum_ \frac_j \right | }. Here ''M''i is the mass of nucleus ''i'', ''Z''''i'' is the atomic number of nucleus ''i'', and ''m''e is the mass of the electron. The Laplace operator of particle ''i'' is :$\backslash nabla^2\_\; \backslash equiv\; \backslash boldsymbol\_\backslash cdot\; \backslash boldsymbol\_$ = \frac + \frac + \frac . Since the kinetic energy operator is an inner product, it is invariant under rotation of the Cartesian frame with respect to which ''x''i, ''y''i, and ''z''i are expressed. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Molecular Hamiltonian」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.