Hamiltonian electrostatic interaction

EM was quite extensively and successfully applied to model optical spectra of molecular crystals and aggregates. Extensions were discussed [18] to account for disorder, whose effects are particularly important in aggregates, and to include the coupling between electronic degrees of freedom and molecular vibrations [48], needed to properly describe the absorption and emission bandshapes. However, as it was already recognized in original papers [7, 46], other terms enter the excitonic Hamiltonian. Electrostatic interactions between local excitations can in fact be introduced as ... 

Racah parameters for f configurations = electronic charge = single-electron operators = direct Slater parameters = two-electron operators = exchange Slater parameters = one of the special Cartan s groups = Planck s constant /2ir = hamiltonian = electrostatic interaction = spin-orbit interaction = spin-spin interaction = spin-other-orbit interaction... 

Considering, for simplicity, only electrostatic interactions, one may write the solute-solvent interaction temi of the Hamiltonian for a solute molecule surrounded by S solvent molecules as... 

There is a very convenient way of writing the Hamiltonian operator for atomic and molecular systems. One simply writes a kinetic energy part — for each election and a Coulombic potential Z/r for each interparticle electrostatic interaction. In the Coulombic potential Z is the charge and r is the interparticle distance. The temi Z/r is also an operator signifying multiply by Z r . The sign is - - for repulsion and — for atPaction. 

The individual terms in (5.2) and (5.3) represent the nuclear-nuclear repulsion, the electronic kinetic energy, the electron-nuclear attraction, and the electron-electron repulsion, respectively. Thus, the BO Hamiltonian is of treacherous simplicity it merely contains the pairwise electrostatic interactions between the charged particles together with the kinetic energy of the electrons. Yet, the BO Hamiltonian provides a highly accurate description of molecules. Unless very heavy elements are involved, the exact solutions of the BO Hamiltonian allows for the prediction of molecular phenomena with spectroscopic accuracy that is... 

There are two ways of handling the interaction between the QM region and MM region one way is to calculate electrostatic QM-MM interaction with the MM method (sometimes called mechanical embedding, or ME) and the other is to include the QM-MM interaction in the QM Hamiltonian (called electronic embedding or EE). The major difference is that in the ME scheme the QM wave function is the same in the gas phase and the electrostatic interaction is included classically, while in the EE scheme the QM wave function is polarized by the MM charges. The EE scheme is substantially more expensive than ME scheme, as the SCF iteration needs to be performed until self-consistency is achieved for QM electron distribution. Although the polarization effects are called important, as we will show later,... 

Molecules are described in terms of a Hamiltonian operator that accounts for the movement of the electrons and the nuclei in a molecule, and the electrostatic interactions among the electrons and the electrons and the nuclei. Unlike the theory of the nucleus, there are no unknown potentials in the Hamiltonian for molecules. Although there are some subtleties, for all practical purposes, this includes relativistic corrections, [2] although for much of light-element chemistry those effects are... 

The second area we review is that of computational efficiency of energy evaluations including special considerations for cutoffs and long-range treatment of electrostatic interactions. We provide a brief overview of the required bookkeeping given a Hamiltonian of a certain complexity. Many of these issues are unique to MC calculations due to the fundamentally different ways in which systems evolve when compared to MD calculations. 

For the electro-nuclear model, it is the charge the only homogeneous element between electron and nuclear states. The electronic part corresponds to fermion states, each one represented by a 2-spinor and a space part. Thus, it has always been natural to use the Coulomb Hamiltonian Hc(q,Q) as an entity to work with. The operator includes the electronic kinetic energy (Ke) and all electrostatic interaction operators (Vee + VeN + Vnn)- In fact this is a key operator for describing molecular physics events [1-3]. Let us consider the electronic space problem first exact solutions exist for this problem the wavefunctions are defined as /(q) do not mix up these functions with the previous electro-nuclear wavefunctions. At this level. He and S (total electronic spin operator) commute the spin operator appears in the kinematic operator V and H commute with the total angular momentum J=L+S in the I-ffame L is the total orbital angular momentum, the system is referred to a unique origin. 

The nuclear spins give rise to additional terms in the Breit-Pauli Hamiltonian due to the interaction of the electrons with the magnetic moment of the nuclei and the electrostatic interaction with the electric quadrupole interaction of the nuclei. The magnetic interaction term of the spins with the nuclei is of the same type as the spin-spin interaction and following Abragam and Pryce (61) can be written as... 

Figure 13.7 Example of potential energy curves for separate VB Hamiltonians and the curve for the lowest energy eigenvalue when the separate Hamiltonians are coupled by an off-diagonal term in a 2x2 Hamiltonian matrix. Note that the difference between the minima for and 7/22 is the term r2 in the example given in the text only if all non-bonded and electrostatic interactions are identical in the two VB representations, and thus the quotes around the label...

Figure 5. Schematic representation of a two-ion resonance-coupled system. Energy gaps Ea and Eb are equal. The coupling Hamiltonian // represents the electrostatic interaction of the electron of ion A with the electron of ion B.

For a system of n electrons and N nuclei, and considering only electrostatic interactions between the particles, we have for the total Hamiltonian... 

Let us also recall that for a relativistic Hamiltonian and wave function only jj coupling is possible due to the prevailing spin-orbit interactions over the non-spherical part of the electrostatic interaction. 

Operators corresponding to physical quantities, in second-quantization representation, are written in a very simple form. In the quantum mechanics of identical particles we normally have to deal with two types of operators symmetric in the coordinates of all particles. The first type includes N-particle operators that are the sum of one-particle operators. An example of such an operator is the Hamiltonian of a system of noninteracting electrons (e.g. the first two terms in (1.15)). The second type are iV-particle operators that are the sum of two-particle operators (e.g. the energy operator for the electrostatic interaction of electrons - the last term in (1.15)). In conventional representations these operators are... 

One of the easiest ways to improve the results is through the replacement of the matrix element of the energy operator of electrostatic interaction by some effective interaction, in which, together with the usual expression of the type (19.29), there are also terms containing odd k values. This means that we adopt some effective Hamiltonian, whose matrix elements of the... 

Let us emphasize that in single-configurational approach the terms of the Hamiltonian describing kinetic and potential energies of the electrons as well as one-electron relativistic corrections, contribute only to average energy and, therefore, are not contained in, which in the non-relativistic approximation consists only of the operators of electrostatic interaction e and the one-electron part of the spin-orbit interaction so, i.e. 

These are produced by autoionization transitions from highly excited atoms with an inner vacancy. In many cases it is the main process of spontaneous de-excitation of atoms with a vacancy. Let us recall that the wave function of the autoionizing state (33.1) is the superposition of wave functions of discrete and continuous spectra. Mixing of discrete state with continuum is conditioned by the matrix element of the Hamiltonian (actually, of electrostatic interaction between electrons) with respect to these functions. One electron fills in the vacancy, whereas the energy (in the form of a virtual photon) of its transition is transferred by the above mentioned interaction to the other electron, which leaves the atom as a free Auger electron. Its energy a equals the difference in the energies of the ion in initial and final states ... 

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