Density matrix construction

The macromolecular density matrix constructed from the fragment density matrices within the ADMA framework represents the same level of accuracy as the electron densities obtained with the MEDLA and ALDA methods. The effects of interactions between local fragment representations are determined to the same level of accuracy within the ADMA, the MEDLA, and the ALDA approaches. The ADMA direct density matrix technique allows small readjustments of nuclear geometries, in a manner similar to the ALDA technique however, within the ADMA framework, the geometry readjustment can be carried out directly on the macromolecule. 

The parameters NLW and NUW are the number of lower walks and upper walks for a particular Shavitt loop. (The mapping vector R( ) is loop-independent and gives the correspondence between the full set of upper and lower walks.) The walk offsets YB and YK are determined by the loop shape, and the coefficient T is a linear combination of the loop values and integrals associated with the Shavitt loop. Since only unique Shavitt loops are constructed, the innermost DO loop must be repeated with the bra and ket values interchanged, resulting in even more arithmetic operations for each Shavitt loop. The transition density matrix construction involves an analogous DO loop structure with the last statement replaced by... 

The generalized fractional occupation Up is related to diagonal matrix elements of the first-order reduced density matrix constructed in natural... 

In the basis of the Hamiltonian eigenstates, the thermal equilibrium density matrix constructed from Equation (2.5.7) is diagonal ... 

Currently, work is underway to split up the subsystem level computations across a large number nodes on multiprocessor machines. Each node is assigned the tasks of matrix diag-onalization and density matrix construction for one or more subsystems. Sample timings on the protein crambin for a number of parallel platforms are given in Table 6. From these pilot studies it is clear that some parallelization can be achieved, but that load balancing and parallelization of other parts of the code is required prior to developing an efficient parallel implementation of DivCon. 

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

Form an initial guess for the molecular orbital coefficients, and construct the density matrix. 

McWeeny, R., Proc. Roy. Soc. [London) A235, 496, (i) The density matrix in self-consistent field theory. I. Iterative construction of the density matrix." Beryllium atom is studied. Steepest descent method is described. 

We may now construct the density matrix for the polarization of a one-photon state. If we choose for our basic states the states of right and left circular polarization then for an arbitrary pure state... 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

Density functions can be obtained up to any order from the manipulation of the Slater determinant functions alone as defined in section 5.1 or from any of the linear combinations defined in section 5.2. Density functions of any order can be constructed by means of Lowdin or McWeeny descriptions [17], being the diagonal elements of the so called m-th order density matrix, as was named by Lowdin the whole set of possible density functions. For a system of n electrons the n-th order density function is constructed from the square modulus of any n-electron wavefunction attached to the n-electron system somehow. 

The implimentation of quantum statistical ensemble theory applied to physically real systems presents the same problems as in the classical case. The fundamental questions of how to define macroscopic equilibrium and how to construct the density matrix remain. The ergodic theory and the hypothesis of equal a priori probabilities again serve to forge some link between the theory and working models. 

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... 

By construction Eq. (38) associated with observables, describes the mass-shell condition, while Eq. (43) is a density matrix equation giving the time-evolution of the state l/. That this is so can be seen in the following way. Considering Eqs. (41), let us multiply Eq. (43) by ), that is... 

In order to construct a spinorial density-matrix equation, we can introduce an equation like... 

Now, using Equation 20.4, one can construct the reduced second-order density matrix. For / cr2, P2>det is quite boring ... 

Projection of the density matrix onto the internal basis, which is, due to the way the internal basis is constructed, no extra approximation, and build the full Coulomb matrix. A further approximation can be made by calculating only the intra-atomic components of this matrix, which is not expected to produce a serious loss in accuracy, since V l Ic is only large near the nuclei. We will refer to these alternatives as the full and atomic Coulomb ZORA option. Of course these two options are equivalent in the atomic calculations presented in the next section. 

Clearly then, the closed expression for the density-transformed 1-matrix in Eq. (52) (i.e., the 1-matrix constructed from transformed orbitals) is a function of the single-electron density p(r). Hence, when we express the total energy of an N-particle system (in the single-determinantal approximation) in terms of the transformed one-matrix described by Eq. (52), one can readily obtain an energy functional which depends on p. This fact, which has been exploited by several authors [59-62,85], is considered below with particular reference to the work of Ludeha [60]. 

The A-representability constraints presented in this chapter can also be applied to computational methods based on the variational optimization of the reduced density matrix subject to necessary conditions for A-representability. Because of their hierarchical structure, the (g, R) conditions are also directly applicable to computational approaches based on the contracted Schrodinger equation. For example, consider the (2, 4) contracted Schrodinger equation. Requiring that the reconstmcted 4-matrix in the (2, 4) contracted Schrodinger equation satisfies the (4, 4) conditions is sufficient to ensure that the 2-matrix satisfies the rather stringent (2, 4) conditions. Conversely, if the 2-matrix does not satisfy the (2, 4) conditions, then it is impossible to construct a 4-matrix that is consistent with this 2-matrix and also satisfies the (4, 4) conditions. It seems that the (g, R) conditions provide important constraints for maintaining consistency at different levels of the contracted Schrodinger equation hierarchy. 

To complete the definition of the renormalization step for the left block, we also need to construct the new matrix representations of the second-quantized operators. In the product basis Z <8> p, matrix representations can be formed by the product of operator matrices associated with left, p j and the partition orbital p separately. Then, given such a product representation of O say, the renormalized representation O in the reduced M-dimensional basis / of LEFIi. p is obtained by projecting with the density matrix eigenvectors L defined above,... 

Moritz, G., Reiher, M. Construction of environment states in quantum-chemical density-matrix renormalization group calculations. J. Chem. Phys. 2006, 124(3), 034103. 

S v are elements of the overlap matrix. Similar types of expressions may be constructed for density functional and correlated models, as well as for semi-empirical models. The important point is that it is possible to equate the total number of electrons in a molecule to a sum of products of density matrix and overlap matrix elements. ... 

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