Matrices density matrix eigenvalues

Truncation scheme baised on density matrix eigenvalues... 

The density matrix eigenvalues sum to unity and the truncation error, which is defined as the sum of the density matrix eigenvalues corresponding to the discarded DMEV, gives a qualitative estimate as to the accuracy of the calculation as well as providing a framework... 

The orbital occupation numbers n, (eigenvalues of the density matrix) will be between 0 and 1, corresponding to the number of electrons in the orbital. Note that the representation of the exact density normally will require an infinite number of natural orbitals. The first N occupation numbers N being the total number of electrons in the system) will noraially be close to 1, and tire remaining close to 0. 

Each pair of atoms (AB, AC, BC,...) is now considered, and the two-by-two subblocks of the density matrix (with the core and lone pair contributions removed) are diagonalized. Natural bond orbitals are identified as eigenvectors which have large eigenvalues (occupation numbers larger than say 1.90). 

Density matrix element in AO basis Matrix eigenvalue van der Waals parameter Dielectric constant... 

Suppose that the density matrix has been brought to diagonal form, whereby it is immediately apparent that all its eigenvalues should be either zero or one. More precisely, since trg=l, only one of the eigenvalues can be equal to one and all the others are zero. 

The eigenvalue equation corresponding to the Hamiltonian of Eq. [57] can be solved self-consistently by an iterative procedure for each orientation of the spin magnetization (identified as the z direction). The self-consistent density matrix is then employed to calculate the local spin and orbital magnetic moments. For instance, the local orbital moments at different atoms i are determined from... 

M. Rosina, Large eigenvalues of the G-matrix and collective states, in Report of the Density Matrix Seminar (A. J. Coleman and R. M. Erdahl, eds.), Queen s Press, Kingston, Ontario, 1968, p. 51. 

Let the eigenvalue w be fixed and assume that fit is nondegenerate and unit-normalized. The restriction to nondegenerate eigenstates will be relaxed in Section V, but for now we consider only pure-state density matrices. The A -electron density matrix for the pure state fit is... 

There is an interesting pairing relation between the eigenvalues of the p-particle density matrix (with eigenfunctions, i.e., namral p-states, and those of y p ... 

The above mentioned positivity conditions state that the 2-RDM D, the electron-hole density matrix G, and the two-hole density matrix Q must be positive semidefinite. A matrix is positive semidefinite if and only if all of its eigenvalues are nonnegative. The solution of the corresponding eigenproblems is readily carried out [73]. For D, it yields the following set of eigenvalues ... 

Because the Slater hull constraints are insufficient to ensure A -representability, it is important to find additional methods for constraining the off-diagonal elements of the density matrix. Obtaining constraints that supersede the Slater hull requires considering Hamiltonians with a more general form than polynomials of number operators. As discussed in Section UFA, matters are especially simple if the Hamiltonian has nonnegative eigenvalues, because then the necessary conditions for A-representability take the form... 

The correlation entropy is a good measure of electron correlation in molecular systems [5, 7]. It is defined using the eigenvalues of the one-particle density matrix IPDM,... 

Equation (40) is a generalized eigenvalue equation. The eigenvalue j is interpreted as the excitation energy from the ground state to the Jth excited state. The vectors Xj and Yj are the first-order correction to the density matrix at an excitation and describe the transition density between the ground state and the excited state J. 

In the absence of a radio-frequency magnetic field Bx, the density matrix components concerned with an eigenvalue Ff (=k) change with time according to u... 

This can be seen to have a macroscopically large eigenvalue signifying a condensate of electron pairs. All yj (Mt) low-lying states have the same density matrix eigenvector and eigenvalue. 

To advance further, the resonance property of the exact density response at the true excitation energies (0k [9, 20, 32] is used, namely, that a finite external perturbation 8vext(ri,OJk) leads to an infinite change in the density. Then Eq. 29 can be satisfied only if the matrix in the square brackets possesses a zero eigenvalue at cok. This leads, after a unitary transformation, to the master matrix eigenvalue equations for the excitation energies cok [9]... 

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