Total electronic hamiltonian

In this work, relativistic effects are included in the no-pah or large component only approximation [13]. The total electronic Hamiltonian is H (r R) = H (r R) + H (r R), where H (r R) is the nom-elativistic Coulomb Hamiltonian and R) is a spin-orbit Hamiltonian. The relativistic (nomelativistic) eigenstates, are eigenfunctions of R)(H (r R)). Lower (upper)... 

With the localized basis set—that is, xe) = et) gm) where ee) ( gm)) is the electronically excited (ground) state of the Bchl Be (Bm)—the total electronic Hamiltonian for the model system is given by [56]... 

The second quantization formalism also greatly simplifies the treatment of the Hamiltonian and allows its analysis pertinent to the GF approximation for the wave function.23 Indeed, the total electron Hamiltonian H can be rewritten using the second quantization formalism according to the division of the orbital basis set into carrier subspace basis sets as introduced above ... 

We are now in a position to present the total electronic Hamiltonian by summing over all possible electrons i. We must be very careful, however, not to count the various interactions twice. Thus on summing over i, we modify all terms which are symmetric in i and j by a factor of 1 /2. These terms are the electron-electron Coulomb interaction (3.141), the orbit-orbit interaction (3.145), the spin-spin interaction (3.151) and the spin-other-orbit interaction from (3.144) and (3.153). 

The total electronic Hamiltonian (6.1) is rewritten in the second quantized form... 

To begin, let us assume that the total electronic Hamiltonian H is decomposed into two pieces... 

The Hartree-Fock energy is the expectation value of the total electronic hamiltonian evaluated over the determinantal wavefunction (42). This is just the sum of the zero order and first-order energy coefficients. Explicitly, they take the form... 

Since the wave function is a good approximation of the exact ground state wave function at high values of R, we may calculate what is called the Heitler-London interaction energy (R ) as the mean value of the total (electronic) Hamiltonian minus the energies of the isolated subsystems... 

As usual, the total electronic Hamiltonian H we consider can be written as... 

In hydrogenlike atoms, the total electronic Hamiltonian now becomes... 

We begin with the correct total electronic Hamiltonian for the system, written as the sum of two t3q)es of terms. The first type contains terms pertaining only to the individual groups the second type represents the potential energy of interaction between the groups... 

In the above discussion we have been concerned with the exact electronic Hamiltonian, energies and wave functions of a supersystem consisting of an array of well-separated subsystems. We now turn our attention to the description afforded by some independent particle model, in which the electrons move in some mean field. The most commonly used approximation of this type is the Hartree-Fock model, but the discussion presented in this section is not restricted to this particular method. In particular, we write the total electronic Hamiltonian operator in the form... 

With the current development of quantum chemistry, it is routine to evaluate Eq. [17] for the QM + MM model and the application of QM + MM direct dynamics is described in the section on trajectory initial conditions. However, in many situations the QM/MM boundary must cut through a chemical bond in a molecule. In such a case, the total electronic Hamiltonian cannot be divided as for the QM + MM model. Different approaches have been developed to treat QM/MM interactions when the boundary cuts through a chemical bond. Gao et al. identified a criterion for treating a covalent bond at the QM/MM boundary. In general, a reasonable boundary method should be able to mimic the real physical properties of the model system as closely as possible. The obtained properties such as vibrational frequencies, energies, and electronegativities, etc. should be comparable to experiment or accurate ab initio calculations. 

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