Matrix charge density

It is helpful to simplify Equation (2.157) by introducing the charge density matrix, F, whose elements are defined as ... 

The calculation proceeds as illustrated in Table 2.2, which shows the variation in the coefficients of the atomic orbitals in the lowest-energy wavefunction and the energy for the first four SCF iterations. The energy is converged to six decimal places after six iterations and the charge density matrix after nine iterations. 

In the electronic ground state, the six rr-electrons occupy the three lowest-energy orbitals (the first three from Table 7.2). The Huckel rr-electron charge density matrix is... 

Some molecular descriptors related to the charge density matrix have been proposed [Salem, 1966]. Bond index B,y was proposed to measure the multiplicity of bonds between two atoms [Wiberg, 1968]. It is defined between two atoms i and j as the square of the off-diagonal elements of the charge density matrix between atomic orbitals p on the fth atom and v on the j th atom, summed over all such distinct orbitals ... 

The bond index is a function of the square of the charge density matrix elements, whereas the bond order is defined in terms of the charge density matrix elements themselves. It is a measure of the extent of electron sharing between two atoms, but it has the disadvantage that it is always positive and hence cannot describe antibonding situations. 

Other molecular descriptors derived from the charge density matrix are the average atom charge density Pi and the average bond charge density P2, defined as ... 

For closed shell systems, it is useful to define a charge density matrix P as... 

Some other molecular descriptors related to the charge density matrix have been proposed [Salem, 1966 Mayer, 2007]. Some of them are listed below. 

The charge density matrix is obtained by integrating over the spin-variable... 

These equations should be solved by an iterative procedure since the Fock matrix depends upon the LCAO coefficients through the charge-density matrix. 

In an alternative formulation, the first-order charge-density matrix is the subject of determination through the CHF equations [18]... 

The meaning of the matrices reported in the eqs.(12) is the following h° and G°(P) are the usual one- and two-electron matrices used in the HF problem for M in vacuo here we have indicated that the two-electron matrix comes from a contraction of the supermatrix I of the two-electron integrals involving the one-electron charge density matrix P. 

The term deriving from this elaboration of the theory is divided into two sets, one depending on the solute charge density matrix and the other not depending on it. The formal expressions of the matrix elements are ... 

First, we provide the main EUE definitions using the conventional reduced density matrix (RDM) methods. In singlet states, the first-order RDM (1-RDM) can be defined as a spin-free matrix which is also termed the charge density matrix. Throughout the paper, the capital letter D will be denoted the charge density matrix. In Dirac s bra-ket notation, D conveniently takes a compact form of a spectral resolution, that is the following diagonal form ... 

Now turn to computational aspects. For matrices / and pp, a block representation is easy to find by simple algebra (see Appendix B). As a result, we get charge density matrix of the QCTB model, Eq. (B4), and the respective NOON spectrum, Eq. (B5). It comes to a suitable working formula for the main EUE index ... 

We now have to sum over spin indices, making spin trace in Eq. (A8). As a result, the dual charge density matrix is yielded, viz. 

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