Density matrix one-particle

This is the desired result which may be substituted into the scattering amplitude formula (6). The resulting scattering formula is the same as found by other authors [5], except that in this work SI units are used. The contributions to the Fourier component of magnetic field density are seen to be the physically distinct (i) linear current JL and (ii) the magnetisation density Ms associated with the spin density. A concrete picture of the physical system has been established, in contrast to other derivations which are heavily biased toward operator representations [5]. We note in passing that the treatment here could be easily extended to inelastic scattering if transition one particle density matrices (x x ) were used in Equations (12)—(14). 

Here we will present the formulae needed for calculating the reduced one-particle density matrices from the floating correlated Gaussians used in this work. The first-order density matrix for wave function T (ri,r2,..., r ) for particle 1 is defined as... 

In the normal (probability theory) use of the term, two probability distributions are not correlated if their joint (combined) probability distribution is just the simple product of the individual probability distributions. In the case of the Hartree-Fock model of electron distributions the probability distribution for pairs of electrons is a product corrected by an exchange term. The two-particle density function cannot be obtained from the one-particle density function the one-particle density matrix is needed which depends on two sets of spatial variables. In a word, the two-particle density matrix is a (2 x 2) determinant of one-particle density matrices for each electron ... 

A time reversible BO molecular dynamics scheme based on the propagation of one-particle density matrices has been proposed by Niklasson [8]. The equation of motion for the density matrix is... 

At the expense of an extra summation, which introduces a step of order M, we have reduced the two-particle density matrix to a sum involving one-particle density matrices. The Cl problem therefore reduces to the problem of evaluating the one-particle density matrices efficiently. 

Muller AMK (1984) Explicit approximate relation between reduced two- and one-particle density matrices. Phys Lett A 105 446-452... 

The equation shows that, similar to the calculation of vibrational frequencies, the response of the one-particle density matrix to a perturbation is necessary, which is in the case of NMR shieldings the magnetic field B,. Therefore, the CPSCF equations need to be solved for the perturbed one-particle density matrices (short P ). In the context of NMR shieldings, the computational effort of conventional schemes " " scales cubically with molecular size. 

Locality and Sparsity of ab initio One-Particle Density Matrices and Localized Orbitals. 

The two updates differ only by a factor of one-half before the first-order change from A and the second-order change. Unlike the wavefunction power method, the A -particle density matrices from each iteration in Eq. (Ill) are not exactly positive semidehnite until convergence. 

For the formulation of the generalized Wick theorem corresponding to the generalized normal ordering, we need the matrix element rf, Eq. (65), of the one-hole density matrix and the cumulants kjc, Eqs. (39)-(47), of the fc-particle density matrices. 

Consequently, we can carry out the BCH expansion to arbitrarily high order without any increase in the complexity of the terms in the effective Hamiltonian. In practice, the expansion is carried out until convergence in a suitable norm of the operator coefficients is achieved, as illustrated in Table I. Rapid convergence is usually observed. Note that through the decomposition (23), the effective Hamiltonian depends on the one- and two-particle density matrices and therefore becomes state specific, much like the Fock operator in Hartree-Fock theory. 

We now discuss (ii), the evaluation of operator expectation values with the reference ho- We are interested in multireference problems, where may be extremely complicated (i.e., a very long Slater determinant expansion) or a compact but complex wavefunction, such as the DMRG wavefunction. By using the cumulant decomposition, we limit the terms that appear in the effective Hamiltonian to only low-order (e.g., one- and two-particle operators), and thus we only need the one- and two-particle density matrices of the reference wavefunction to evaluate the expectation value of the energy in the energy expression (7). To solve the amplitude equations, we further require the commutator of which, for a two-particle effective Hamiltonian and two-particle operator y, again involves the expectation value of three-particle operators. We therefore invoke the cumulant decomposition once more, and solve instead the modihed amplitude equation... 

Consequently, with the simplihcations above, all the working equations of the canonical transformation theory can be evaluated entirely in terms of a limited number of reduced density matrices (e.g., one- and two-particle density matrices) and no explicit manipulation of the complicated reference function is required. 

Finally, px and rY, with X = A or B, are the conventional one- and two-particle density matrices for monomer X, normalized to Nx and NX(NX — 1), respectively. In Eqs. (1-75) and (1-76) qj = (r Sj) denotes the space and spin coordinates of the /th electron. Since theoretical methods for the evaluation of the density matrices px and Yx for many-electron molecules are well developed, Eqs. (1-75) and (1-76) enable practical calculations of the first-order exchange energy using accurate electronic wave functions of the monomers A and B116. 

The other terms of Eq. (1-225) follow by permutation of A, B and C. Similarly as in the two-body case the first-order exchange nonadditivity can be expressed through one- and two-particle density matrices of the isolated monomers314,... 

The Schrodinger equation provides a way to obtain the A-electron wave function of the system, and the approximate methods described in the previous section permit reasonable approaches to this wave function. From the approximate wave function the total energy can be obtained as an expectation value and the different density matrices, in particular the one-particle density matrix, can be obtained in a straighforward way as... 

All matrix elements in the Newton-Raphson methods may be constructed from the one- and two-particle density matrices and transition density matrices. The linear equation solutions may be found using either direct methods or iterative methods. For large CSF expansions, such micro-iterative procedures may be used to advantage. If a micro-iterative procedure is chosen that requires only matrix-vector products to be formed, expansion-vector-dependent effective Hamiltonian operators and transition density matrices may be constructed for the efficient computation of these products. Sufficient information is included in the Newton-Raphson optimization procedures, through the gradient and Hessian elements, to ensure second-order convergence in some neighborhood of the final solution. 

The integral-driven procedure indicated above is practicable only if the elements of the two-particle density matrix can be rapidly accessed. In the closed-shell Hartree-Fock case, the two-particle density matrix can be easily constructed from the one-particle density. The situation is similar for open-shell and small multiconfigurational SCF wavefunctions the two-particle density matrix can be built up from a few compact matrices. In most open-shell Hartree Fock theories (Roothaan, 1960), the energy expression (Eq. (23))... 

In the formulation we are going to present here, all the quantities of interest, viz. the hamiltonian, the cluster amplitudes, the one- and many- particle density matrices are all expressed with respect to the entire reference function j/ o- Just as in the traditional many-body formulations, it becomes convenient... 

The higher order particle density matrices are adl factorizable into antisymmetric products of one body a s. Mixed densities with particle as well as valence have a similar factorizability involving one-body cr s and R s with valence labels. 

For SCF methods, p represents elements of the one- and two-particle density matrices. In correlated methods, the one-particle part of p is the sum of the aaual reduced density and a contribution that is proportional to the derivative of the energy with respect to orbital rotations. For the MCSCF method, the orbitals are variationally optimum, and this latter term vanishes. Similarly, for FCI there is no orbital contribution. However, it is required for other Cl, CC, and MBPT correlated methods, because the energy in these approaches is not stationary with respect to first-order changes in the molecular orbitals. This... 

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