Electronic Hamilton operator

Here Hg is the electronic Hamilton operator and H p is called the mass-polarization (Mtot is the total mass of all the nuclei and the sum is over all electrons). We note that He depends only on the nuclear positions (via Vne and Vnn, see eq. (3.23)) and not on their momenta. 

Ho is the normal electronic Hamilton operator, and the perturbations are described by the operators Pi and P2, with A determining the strength. Based on an expansion in exact wave functions, Rayleigh-Schrddinger perturbation theory (section 4.8) gives the first- and second-order energy collections. 

In wave mechanics the electron density is given by the square of the wave function integrated over — 1 electron coordinates, and the wave function is determined by solving the Schrddinger equation. For a system of M nuclei and N electrons, the electronic Hamilton operator contains the following tenns. 

The notation used in this book is in terms of first quantization. The electronic Hamilton operator, for example, is written as (eq. (3.23))... 

The second quantization representation of the electronic hamilton operator is thus... 

The inclusion of point charges is achieved by adding the contributions of all MM point charges as a perturbational potential term Vpc to the electronic Hamilton operator Ha ... 

A wave function of an electron configuration is generally approximated as a product of molecular orbital functions, which are eigenfunctions of a one-electron Hamilton operator. When 2n electrons occupy n molecular orbitals, the wave function of electron configuration of the lowest energy is written as... 

In the Bom-Oppenheimer picture the nuclei move on a potential energy surface (PES) The electronic Hamilton operator is normally written as... 

Electronic chemical potential, 353 Electronic Hamilton operator, 54 Electrostatic energy, 23 Electrostatic Potential (ESP) 220, Elimination reactions, 370 Embedding, 344... 

The question now is what this functiony(/ yv) looks like. The answer leads to the most fundamental, general QM model, in which the energy is basically the expectation value of the electronic Hamilton operator H that parametrically depends on the nuclear positions (eq. (2)), where W is the multi electron wave function. 

There still is a point to be discussed the calculation of energy expectation values, within the EH space framework. This can be done, in practice, using Bom-Oppenheimer approximation, defining in this context an electronic Hamilton operator, adopting some diagonal matrix structure and, in addition, supposing the original scalar wavefunction j normalised ... 

A serious deficiency is that neither a Hartree product nor a Slater determinant can be an eigenfunction of the fV-electron Hamilton operator. Therefore cannot be a solution of the time-independent electronic Schrodinger equation. The reason is that the A-electron Hamiltonian cannot be written as a sum of N one-electron Hamiltonians, due to the repulsive Coulomb interactions between the electrons. Nevertheless, in practice it turns out that we can work rather well with an approximate wave function consisting of only one Slater determinant if we choose that particular Slater determinant that yields the lowest energy expectation value 7 ). In other words, we must vary the spin orbitals in P until we have reached the lowest value... 

This can be compared with the true A -electron Hamilton operator,... 

The Hohenberg-Kohn theorem, however, leaves two crucial questions unresolved (1) what is the form of Fhk. and (2) how can the exact charge density p(r) be determined To make applications of DFT possible, one certainly needs a more explicit form. These requirements were met by the formulation of a set of one-particle eigenvalue equations by Kohn and Sham (KS), employing an effective one-electron Hamilton operator, H s, which incorporates both Fhk[p1 and K(r) ... 

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