Expectation value density matrix form

This procedure would generate the density amplitudes for each n, and the density operator would follow as a sum over all the states initially populated. This does not however assure that the terms in the density operator will be orthonormal, which can complicate the calculation of expectation values. Orthonormality can be imposed during calculations by working with a basis set of N states collected in the Nxl row matrix (f) which includes states evolved from the initially populated states and other states chosen to describe the amplitudes over time, all forming an orthonormal set. Then in a matrix notation, (f) = (f)T (t), where the coefficients T form IxN column matrices, with ones or zeros as their elements at the initial time. They are chosen so that the square NxN matrix T(f) = [T (f)] is unitary, to satisfy orthonormality over time. Replacing the trial functions in the TDVP one obtains coupled differential equations in time for the coefficient matrices. 

The preceding is a rather comprehensive—but not exhaustive— review of N-representability constraints for diagonal elements of reduced density matrices. The most general and most powerful V-representability conditions seem to take the form of linear inequalities, wherein one states that the expectation value of some positive semidefinite linear Hermitian operator is greater than or equal to zero, Tr [PnTn] > 0. If Pn depends only on 2-body operators, then it can be reduced into a g-electron reduced operator, Pq, and Tr[Pg vrg] > 0 provides a constraint for the V-representability of the g-electron reduced density matrix, or 2-matrix. Requiring that Tr[Pg Arrg] > 0 for every 2-body positive semidefinite linear operator is necessary and sufficient for the V-representability of the 2-matrix [22]. 

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 ... 

Perturbative estimate of ESVs with respect to noncorrelated bare Hamiltonian. The specificity of each bond and molecule in the approach based on the SLG expressions for the wave function is taken into account perturbatively by using the linear response approximation [25]. We need perturbative estimates of the expectation values of the pseudospin operators which, in their turn, give values of the density matrix elements according to eq. (3.5). According to the general theory (Section 1.3.3.2) the linear response 5(A) of an expectation value of the operator A to the time independent perturbation AB of the Hamiltonian (A is the parameter characterizing the intensity of the perturbation) has the form ... 

The energy expectation value for the crystal field model in the ensemble form is supplemented by the subsidiary conditions that diagonal elements of the density matrix are between 0 and 1, that their sum equals q, and that the matrix is nonnegative. A variational form is then... 

Introductions to the density matrix formalism in magnetic resonance can be found in [28a] and [28b]. An expectation value <0> of any operator O is obtained by multiplying the density matrix with the matrix of the operator and forming the trace,... 

It was noted in Section 5.3 that when the 1-electron density matrix is written in the form (5.3.12) the difference between the a and /3 components allows us to define a resultant spin density, essentially as the excess density of up-spin electrons compared with down-spin. In recent years, with the development of magnetic resonance techniques, this quantity has acquired great importance. To indicate its origin we note tiiat fhe expectation value of the z component of spin angular momentum may be written, using for clarity the explicit form (5.2.12),... 

The density matrix in the spin-orbital representation was introduced in second quantization for the evaluation of one-electron expectation values in the following form... 

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