Density operator matrix elements

The density operator (matrix) is Hermitian and for an arbitrary countable basis may be represented by a square matrix, that may be infinite, and with elements... 

It is seen from (60)-(61) that there are two alternative ways to calculate the density variation i) through the transition density and matrix elements of Qfc-operator and ii) through the ground state density. The second way is the most simple. It becomes possible because, in atomic clusters, Vres has no T-odd Ffc-operators and thus the commutator of Qk with the full Hamiltonian is reduced to the commutator with the kinetic energy term only ... 

In this section, we will show how the thermal density matrix is used in PIMC to compute quantum viiial coefficients. Consider the Hamiltonian of a monatomic molecule like helium with mass m (Eq. 7). Using the primitive approximation (Eq. 4), Trotter formula (Eq. 5), and following the procedure outlined in Ref. [9], we can obtain the kinetic-energy operator matrix elements as ... 

Let us apply the Redfield theory to a deuteron with its quadrupole moment experiencing a fluctuating electric field gradient arising from anisotropic molecular motions in liquids. When the static average of quadrupole interaction is nonzero, i.e. 0, it can be included in the static Hamiltonian Hq. The density operator matrix for a deuteron spin is of the dimension 3x3 and the corresponding Redfield relaxation supermatrix has the dimension V- x 3. When only nuclear spin-lattice relaxation is considered, the spin precession term in Equation [22] is set to zero and the diagonal elements 2, 3)... 

When considering spin-spin relaxation, it is necessary to examine the off-diagonal elements of the density operator matrix and there are three independent spin-spin relaxation times (T2, T213, and 2d) ... 

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. 

An alternative relax-and-drive procedure can be based on a strictly unitary treatment where the advance from Iq to t is done with a norm-conserving propagation such as provided by the split-operator propagation technique.(49, 50) This however is more laborious, and although it conserves the norm of the density matrix, it is not necessarily more accurate because of possible inaccuracies in the individual (complex) density matrix elements. It can however be used to advantage when the dimension of the density matrix is small and exponentiation of matrices can be easily done.(51, 52)... 

In general, the equations for the density operator should be solved to describe the kinetics of the process. However, if the nondiagonal matrix elements of the density operator (with respect to electron states) do not play an essential role (or if they may be expressed through the diagonal matrix elements), the problem is reduced to the solution of the master equations for the diagonal matrix elements. Equations of two types may be considered. One of them is the equation for the reduced density matrix which is obtained after the calculation of the trace over the states of the nuclear subsystem. We will consider the other type of equation, which describes the change with time of the densities of the probability to find the system in a given electron state as a function of the coordinates of heavy particles Pt(R, q, Q, s,...) and Pf(R, q, ( , s,... ).74,77 80... 

Construction of the density operator can also not be achieved without assumption of an additional axiom All quantum states of a system compatible with the knowledge revealed by macroscopic measurement have equal a priori probabilities and random a priori phases. This axiom implies that for a system as defined above all diagonal elements of the density matrix q belonging to the ith cell must be equal. Hence... 

There is another widely used method of obtaining the Fock operator, namely to obtain its matrix elements F lv as the derivative of the energy functional with respect to the density. In our case that yields... 

Here, Flffl are matrix elements of a zeroth-order Hamiltonian, which is chosen as a one-electron operator in the spirit of MP2. is an overlap matrix The excited CFs are not in general orthogonal to each other. Finally, Vf)(i represents the interaction between the excited function and the CAS reference function. The difference between Eq. [2] and ordinary MP2 is the more complicated structure of the matrix elements of the zeroth-order Hamiltonian in MP2 it is a simple sum of orbital energies. Here H is a complex expression involving matrix elements of a generalized Fock operator F combined with up to fourth-order density matrices of the CAS wave function. Additional details are given in the original papers by Andersson and coworkers.17 18 We here mention only the basic principles. The zeroth-order Hamiltonian is written as a sum of projections of F onto the reference function 0)... 

Let us investigate the change of the various components of the density matrix under propagation from time t to t + 5t. We first consider the action of jSfo- Due to this operator, the density matrix element p (t) attains a phase factor... 

Here the generalized Fock operator/with matrix elements/ appears for the first time. It looks familiar and resembles the Fock operator of Hartree-Fock theory. However, now the yf are matrix elements of the exact one-particle density matrix... 

R is called the relaxation superoperator. Expanding the density operator in a suitable basis (e.g., product operators [7]), the a above acquires the meaning of a vector in a multidimensional space, and eq. (2.1) is thereby converted into a system of linear differential equations. R in this formulation is a matrix, sometimes called the relaxation supermatrix. The elements of R are given as linear combinations of the spectral density functions (a ), taken at frequencies corresponding to the energy level differences in the spin system. 

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