Symmetrized density matrix

Ramasesha, S., Pati, S.K., Krishnamurthy, H.R., Shuai, Z., Bredas, J.L. Low-lying electronic excitations and nonlinear optic properties of polymers via symmetrized density matrix renormalization group method. Synth. Met. 1997, 85(1-3), 1019. 

In two recent papers144,145 a symmetrized Density Matrix Renormalization145... 

The symmetrized Density Matrix Renormalisation Group Procedure145 using a Hubbard-Peierls model Hamiltonian has given for a linear polyene chain quite satisfactory results for aL(static) and for yL(coa co, co2, ) (coa = co + a>2 + ) and for (yL/jV)(static). 

As in the case of the propagator, we shall be applying a symmetrical version of the Trotter fomuila [48] to the high-temperature density matrix... 

Let us consider the simple case of the H atom and its variational approximation at the standard HF/3-21G level, for which we can follow a few of the steps in terms of corresponding density-matrix manipulations. After symmetrically orthogonalizing the two basis orbitals of the 3-21G set to obtain orthonormal basis functions A s and dA, we obtain the corresponding AO form of the density operator (i.e., the 2 x 2 matrix representation of y in the... 

Since the Hartree-Fock wavefunction 0 belongs to the totally symmetric representation of the symmetry group of the molecule, it is readily seen that the density matrix of Eq. (10) is invariant under all symmetry operations of that group, and the same holds, therefore, for the Hartree-Fock operator 7. 

Examples for non-totally-symmetric components in the decomposition of density matrix into irreducible tensor components are the one-particle spin density matrices ... 

Here, Y(j is the 15-element column vector of the angular part of the < (df)0(dj) orbital products, Py is the row vector of the 15 unique elements of the symmetric 5x5 matrix of the coefficients in Eq. (10.3), and Pimp is the row vector containing the coefficients of the 15 spherical harmonic density functions d,mp with / = 0, 2, or 4. Density functions with other / values do not contribute to the d-orbital density. 

Poliak and Eckhardt have shown that the QTST expression for the rate (Eq. 52) may be analyzed within a semiclassical context. The result is though not very good at very low temperatures, it does not reduce to the low temperature ImF result. The most recent and best resultthus far is the recent theory of Ankerhold and Grabert," who study in detail the semiclassical limit of the time evolution of the density matrix and extract from it the semiclassical rate. Application to the symmetric one dimensional Eckart barrier gives very good results. It remains to be seen how their theory works for asymmetric and dissipative systems. 

Using the fact that the symmetrized flux operator commutes with the density matrix, and representing the latter as exp(-/3//) = exp[-(/3 -... 

With symmetric boundary conditions at the chosen time t = 0, the microscopic formulation conforms to time reversible laws as expected. The same conclusion follows from an analogous examination of the Liouville equation. In this setting, the initial data at time, t = 0, is a statistical density distribution or density matrix. Although there are celebrated discussions on the problem of the approach to equilibrium, we nevertheless observe that without course graining or any other simplifying approximations the exact subdynamics would submit to the same physical laws as above, i.e., time reversibility and therefore constant entropy. 

This is nothing but the electron-vibration interaction in the chosen notation. The quantity h is the three index supervector acting on the vector of nuclear shifts they form the scalar product (.... ..) giving a 10 x 10 matrix, next forming a Liouville scalar product with matrix V. On the other hand, acting on the variations V of the density matrix by forming the Liouville scalar product h produces a vector to be convoluted with that of nuclear shifts 5q. With use of this set of variables the energy in the vicinity of the symmetric equilibrium point becomes ... 

It should be noted that as we progress down the sequence, each density matrix is formed from fewer and fewer electrons. Thus Z>(1> consists of N— 1 electrons [and hence (N— 1) terms], Z><2> of N— 2 electrons,. . . , and Z)( v) of no electrons. The AT-electron density matrix depends only upon N, S, and consists of just the representation matrices US(P) defined in equation (16) in symmetrized form ... 

A linear transformation of a configuration vector cb thus requires the construction of a configuration gradient with B> as the reference state [Eq. (94)], and the construction of an orbital gradient with a symmetric transition density matrix [Eq. (95)]. A linear transformation on an orbital vector °b requires the construction of a configuration gradient [Eq. (96)] and an orbital gradient [Eq. (97)] from the one-index transformed Hamiltonian K. 

From (12) it can be seen that either the symmetric (X + Y) or the antisymmetric (Y X) part of the perturbed density matrix is needed. An expression for either one is readily obtained by adding and subtracting the two equations obtained from (13) to yield... 

For a perturbing electric field in the v-direction we have V = W = Dv and W — Y = 0, while for a magnetic field in the v-direction we have for the imaginary magnetic moment operator W = —V = +MV and V + W = 0. A nonzero frequency couples the symmetric and the antisymmetric part of the perturbed density matrix, whereas in the static case the two equations in (16) are not coupled. For comments on the apparent lack of symmetry for the perturbation equations for static electric and magnetic fields see [46]. 

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