Hamiltonian operator average

You should remember the basic physical idea behind the HF model each electron experiences an average potential due to the other electrons (and of course the nuclei), so that the HF Hamiltonian operator contains within itself the averaged electron density due to the other electrons. In the LCAO version, we seek to expand the HF orbitals i/ in terms of a set of fixed basis functions X X2 > and write... 

The treatment developed here is based on the density matrix of quantum mechanics and extends previous work using wavefunctions.(42 5) The density matrix approach treats all energetically accessible electronic states in the same fashion, and naturally leads to average effective potentials which have been shown to give accurate results for electronically diabatic collisions. 19) The approach is taken here for systems where the dynamics can be described by a Hamiltonian operator, as it is possible for isolated molecules or in models where environmental effects can be represented by terms in an effective Hamiltonian. 

We have bracketed the potential energy U with vib because both quantities depend on the coordinates of the nuclei relative to one another, whereas Trot averaged over the vibrations does not. The quantum-mechanical Hamiltonian operator is obtained by replacing the classical quantities with operators ... 

The energy of such a determinantal wavefunction, ( R), is obtained by the quantum mechanical averaging of the electronic Hamiltonian operator given in equation (6)... 

The form of this correction can be appreciated by comparison with the expression for the /t-doubling terms themselves, equation (8.400). There is, however, a problem with this form for the Hamiltonian operator because the two operator factors, such as (J2 + J2) and (J — S )2 do not commute with each other. The Hamiltonian (8.421) is therefore not Hermitian and so has complex eigenvalues. The operator can be made to have Hermitian form by taking the so-called Hermitian average,... 

The unperturbed Hamiltonian operator is based on an independent particle model, that is, a model in which each particle, nucleus or electron, experiences an averaged interaction with the other particles in the system. The unperturbed Hamiltonian operator is a sum of a kinetic energy term and an effective potential energy term... 

Finally, hc i) is the one-electron Hamiltonian operator for electron i in the average field produced by the Nc core electrons,... 

All NMR interactions are represented by tensors of rank 2, and enter the static NMR Hamiltonian after averaging over the fast degree of freedoms. The lineposition and lineshape are determined by this mean tensor, which should reflect the local symmetry. How the NMR interaction tensors are transformed by the symmetry operations is consequently highly relevant to structural phase transition that are characterized by a loss of some symmetry operations. 

To motivate the Kohn-Sham method, we return to molecular Hamiltonian [Eq. (2)] and note that, were it not for the electron-electron repulsion terms coupling the electrons, we could write the Hamiltonian operator as a sum of one-electron operators and solve Schrodinger equation by separation of variables. This motivates the idea of replacing the electron-electron repulsion operator by an average local representation thereof, w(r), which we may term the internal potential. The Hamiltonian operator becomes... 

The MO method used with the hydrogen molecule can be applied to other diatomic molecules after some modifications have been made. Each electron is considered to move in the potential field produced by the nuclei, plus some additional electrical field which represents the average effect of all the other electrons. This gives rise to the following one-electron Hamiltonian operator ... 

Here, it is usual to make the Bom-Oppenheimer approximation that allows a classical treatment of the nuclei to be separated from a quantum mechanical description of the electrons. In this case, the wave function becomes just that of the electrons, and the nuclear-nuclear interaction is added to the energy as a sum over point particles. Consequently, the Hamiltonian operator H includes the kinetic energy of the electrons, the electron-electron interactions, and the electron-nuclei interactions. The wave function determined by solving this eigenproblem consists of a Slater determinant of the molecular orbitals for a molecule or, alternatively, the band structure of a solid. Unfortunately, direct solution of this equation is complicated by the electron-electron interactions. Often, it is necessary to introduce a mean-field approximation that neglects the individual dynamical electron-electron correlations but instead treats the electrons as moving in the average field created by the other electrons. Various corrections have been developed to improve upon this approximation [160, 167, 168]. 

These are important points for any quantitative work, electron-electron interactions must be taken into account, and the theories underpinning computation of MOs do this at various levels of accuracy. Approaches such as Hartree-Fock or density functional theory adapt the Hamiltonian operator to include electron-electron terms in an averaged way so electrons see the Coulomb field of each other averaged over the calculated density associated with each MO (see the Further Reading section in this chapter). 

In a dilute gas the molecules are on the average far enough from each other that the intermolecular forces are negligible, as can be seen in Problem 25.9. In this case the system Hamiltonian operator can be written as a sum of Hamiltonian operators for independent molecules ... 

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