Density matrix terms

Evaluate the remaining Fock and density matrix terms needed to build the perturbed polarizability. 

The dipole moment operator is a sum of one-electron operators r , and as such the electronic conlribution to the dipole moment can be written as a sum of one-electron contributions. The eleclronic contribution can also be written in terms of the density matrix, P, as follows ... 

More accurately, one rewrites the problem in terms of the coordinates Q+ and Q. The probability to be at a certain point is given by the diagonal element of the density matrix... 

Note that, since the von Neumann equation for the evolution of the density matrix, 8 j8t = — ih H, / ], differs from the equation for a only by a sign, similar equations can be written out for p in the basis of the Pauli matrices, p = a Px + (tyPy -t- a p -t- il- In the incoherent regime this leads to the master equation [Zwanzig 1964 Blum 1981]. For this reason the following analysis can be easily reformulated in terms of the density matrix. 

When working with atomic orbitals, it is usual to write the electron density in terms of a certain matrix called (not surprisingly) the electron density matrix. For the simple dihydrogen VB wavefunction, we have... 

Since the exact density matrix is not known, the (approximate) density is written in terms of a set of auxiliary one-electron functions, orbitals, as... 

The original definition of natural orbitals was in terms of the density matrix from a full Cl wave function, i.e. the best possible for a given basis set. In that case the natural orbitals have the significance that they provide the fastest convergence. In order to obtain the lowest energy for a Cl expansion using only a limited set of orbitals, the natural orbitals with the largest occupation numbers should be used. 

The density matrix can be written in terms of blocks of basis functions belonging to a specific centre as... 

The first two terms involve products of the density matrix with derivatives of the atomic integrals, while the two last terms can be recognized as derivatives of the density matrix times the Fock matrix (eq. (3.51)). 

The first four terms only involve derivatives of operators and AO integrals however, for the last three terms we need the derivative of the density matrix and MO energies. These can be obtained by solving the first-order CPHF equations (Section 10.5). 

The expectation value of the density operator, and, indeed, all the components of the density matrix, are stationary in time for an ensemble set up in terms of energy eigenstates. IT we use occupation number representation to set up the density matrix, it is at once seen from Eq. (8-187) that it also is independent of time ... 

The treatment developed here is based on the density matrix of quantum mechanics and extends previous work using wavefunctions.(42 5) The density matrix approach treats all energetically accessible electronic states in the same fashion, and naturally leads to average effective potentials which have been shown to give accurate results for electronically diabatic collisions. 19) The approach is taken here for systems where the dynamics can be described by a Hamiltonian operator, as it is possible for isolated molecules or in models where environmental effects can be represented by terms in an effective Hamiltonian. 

Here the matrix V contains the effect of the nuclear displacements therefore the inhomogeneous first term to the right is a driving term the second term to the right is of second order in the driving effect, and could be dropped in calculations. Formally, the solution for the configuration density matrix correction is... 

Approximations have been reviewed in the case of short deBroglie wavelengths for the nuclei to derive coupled quantal-semiclassical computational procedures, by choosing different types of many-electron wavefunctions. Time-dependent Hartree-Fock and time-dependent multiconfiguration Hartree-Fock formulations are possible, and lead to the Eik/TDHF and Eik/TDMCHF approximations, respectively. More generally, these can be considered special cases of an Eik/TDDM approach, in terms of a general density matrix for many-electron systems. 

Dead time A very short delay introduced before the start of acquisition that allows the transmitter gate to close and the receiver gate to open. Density matrix A description of the state of nuclei in quantum mechanical terms. 

By identifying the first term in round brackets with the charge of atom B -which is plausible given the assumption that the sum over density matrix... 

This step is similar to what we have done in equation (7-7) where we obtained the matrix representation of the Kohn-Sham operator. If we insert expression (7-14) for the charge density in terms of the LCAO functions and make use of the density matrix P defined in equation (7-15), we arrive at... 

While the computational work for setting up the matrix representation R of p(r) scales formally as N4, this can be cut down to N3 using again the trick introduced in section 7-3 by expanding the density in terms of an atom centered, orthonormalized auxiliary basis set cok (recall equation (7-25)). Let us review this simplification under a slightly different perspective. The starting point is again... 

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