One-electron density operator

The calculation of the density operators over time requires integration of the sets of coupled differential equations for the nuclear trajectories and for the density matrix in a chosen expansion basis set. The density matrix could arise from an expansion in many-electron states, or from the one-electron density operator in a basis set of orbitals for a given initial many-electron state a general case is considered here. The coupled equations are... 

Thus if u, is the displacement of an ion at site j arising from phonons, then the one-electron density operator is approximated by... 

For this limited Cl problem, it is convenient to diagonalize the one-electron density matrix, defining a set of natural orbitals that vary at each step of an iterative procedure. The one-electron density operator is defined by... 

Beyond symmetry, the eigenstate vectors also contain information about orbital occupations with respect to the one-electron basis functions (f>k, which can be very useful for chemists. AOMX computes orbital occupation numbers ( m) in a state function P, from the diagonal elements of the one-electron density operator p,- ... 

Electron density from 2c-spinors. Most of the time, the V-electron ground state wave functions are approximated by an antisymmetrized product of N orthonormal single-electron functions (spin-orbitals) and are expressed in terms of a Slater determinant y/>. The electron density is then the expectation value of the one-electron density operator ... 

To evaluate the energy change associated with interatomic orthogonalization, we note that the eigenorbitals of the one-electron density operator allow exact evaluation of the formal one-electron kinetic energy operator (Sidebar 5.4). For HF or DFT wavefunc-tions, where a one-electron effective Hamiltonian operator (Fock or Kohn-Sham operator Fop) is available, these eigenorbitals of the density operator are the NLMOs... 

As a formal one-electron property, x can be evaluated exactly from the one-electron density operator (see V B, p. 2Iff). For single-configuration SCF-MO or DFT description, this implies in turn that p, can be simply evaluated (and visualized) as a sum of localized NEMO bond dipoles, namely, for NEMO (cf. Section 5.4),... 

For every electronic wavefunction that is an eigenfunction of the electron spin operator S, the one-electron density function always comprises an spin part... 

Because of the dimerization, the one electron density is more pronounced on the double bonds than on the single bonds this is more and more apparent when the dimerization increases. More generally, this is the case for every conjugated polymers where the one-electron density is peaked on the monomer region. In this sense, conjugated polymers are not strictly one dimensional systems but rather intermediate between quasi-zero and quasi-onedimensional systems [36, 37, 38, 39]. It seems then convenient to perform a unitary transformation which favours the orbitals localized on the monomers, in our case, on the double bonds. The new operators are then given by... 

With the superscript R we indicate that the corresponding operator is related to the solvent reaction potential, and with the subscripts r and rr the one- or two-electron nature of the operator. The convention of summation over repeated indices followed by integration has been adopted, p is the electron density operator and is the operator which describes the two components of the interaction energy we have previously called t/en and f/ne. In more advanced formulations of continuum models going beyond the electrostatic description, other components are collected in this term. yR is sometimes called the solvent permanent potential, to emphasize the fact that in performing an iterative calculation of P > in the BO approximation this potential remains unchanged. 

It leaves intact the fermion operators related to the /1-th group itself. By virtue of this the two-electron operators WBA result in a renormalization of one-electron terms in the Hamiltonians for each group. <4 = 1,..., M. The expectation values ((b+b ))B are the one-electron densities. The Schrodinger equation eq. (1.193) can be driven close to the standard HFR form. This can be done if one defines generalized Coulomb and exchange operators for group A by their matrix elements in the carrier space of group A ... 

The one-electron density matrix corresponding to the solution of the Hartree-Fock problem in the CLS is, like any Hartree-Fock density matrix, an operator (matrix) P... 

Here K(r) is the one-particle potential in which the electron moves and VA — r ) is the bare Coulomb potential between the electrons. Fe(rt) is the external space-and time-dependent potential which acts as a source coupled to the electron density operator, a is the spin index, and is such that... 

Finite-field methods were first used to calculate dipole polarizabilities by Cohen and Roothaan [66]. For a fixed field strength V, the Hamiltonian potential energy term for the interaction between the electric field and ith electron is just The induced dipole moment with the applied field can be calculated from the Hartree-Fock wavefunction by integrating the dipole moment operator with the one-electron density since this satisfies the Hellmann-Feyman theorem. With the usual dipole moment expansion. 

This equation represents the general solution of the non-orthogonality problem , since the construction of the matrix G " (A) does not involve N operations. All the other density matrices needed in the evaluation of the matrix of the Hamiltonian (36) may be derived similarly. Thus, for example, the required one-electron density matrix elements are given by... 

The FCI state (Eq. (43)) has been generated from the Hartree-Fock state (Eq. (42)) by application of the double excitation operator as in Eq. (41), followed by a variational optimization. Whereas the one-electron density p( 7) represents the overall probability of finding an electron at a given point 7 in space, the two-electron density p(7, 77) represents the probability of finding one electron at position 7 when the other electron is known to be at 7z-... 

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