Second-order matrices

Owing to permutational symmetry, at most six second-order matrices are independent. To accoimt for point molecnlar symmetry let ns introdnce the symmetrized Kronecker square of T, with matrix elements [4]... 

The 2-RDM, the 2-HRDM, and the G-matrix are the only three second-order matrices which (by themselves) are Hermitian and positive semidefinite thus they are at the center of the research in this field. Recently, a formally exact solution of the A -representability problem was published [12] but this solution is unfeasable in practice [40]. 

Unitary Decomposition of Antisymmetric Second-Order Matrices... 

Although a formal solution of the A-representability problem for the 2-RDM and 2-HRDM (and higher-order matrices) was reported [1], this solution is not feasible, at least in a practical sense [90], Hence, in the case of the 2-RDM and 2-HRDM, only a set of necessary A-representability conditions is known. Thus these latter matrices must be Hermitian, Positive semidefinite (D- and Q-conditions [16, 17, 91]), and antisymmetric under permutation of indices within a given row/column. These second-order matrices must contract into the first-order ones according to the following relations ... 

Recently, a unitarily invariant decomposition of Hermitian second-order matrices of arbitrary symmetry under permutation of the indices within the row or column subsets of indices has been reported by Alcoba [77]. This decomposition, which generalizes that of Coleman, also presents three components that are mutually orthogonal with respect to the trace scalar product [77] ... 

In order to get significant results, the initial data must be formed by a set of clearly non-A -representable second-order matrices, which would generate upon contraction a closely ensemble A -representable 1-RDM. It therefore seemed reasonable to choose as initial data the approximate 2-RDMs built by application of the independent pair model within the framework of the spin-adapted reduced Hamiltonian (SRH) theory [37 5]. This choice is adequate because these matrices, which are positive semidefinite, Hermitian, and antisymmetric with respect to the permutation of two row/column indices, are not A -representable, since the 2-HRDMs derived from them are not positive semidefinite. Moreover, the 1-RDMs derived from these 2-RDMs, although positive semidefinite, are neither ensemble A -representable nor 5-representable. That is, the correction of the N- and 5-representability defects of these sets of matrices (approximated 2-RDM, 2-HRDM, and 1-RDM) is a suitable test for the two purification procedures. Attention has been focused only on correcting the N- and 5-representability of the a S-block of these matrices, since the I-MZ purification procedure deals with a different decomposition of this block. 

In 1974 Coleman [73] proposed to decompose any Hermitian antisymmetric second-order matrix as... 

The three parts of this decomposition reveal the structure of the matrix with respect to the contraction operations. These parts have been called the 0-, 1- and 2-body part of the second-order matrix A, respectively. Following the notation introduced in Ref. [73], each of these parts have been identified by a left-lower index. 

As has been mentioned, the MZ purification procedure is based on Coleman s unitary decomposition of an antisymmetric Hermitian second-order matrix described earlier. When applied to singlet states of atoms and molecules, the computational cost of this purification procedure is reduced, since the 2-RDM (and thus the 1-RDM obtained by contraction) presents only two different spin-blocks, the aa- and a/i-blocks (and only one spin-block for the 1-RDM). For the remaining part of this section only this type of state will be treated. 

It must be noted that, due to the arbitrary symmetry under permutation of indices of this second-order matrix, a larger set of contractions into the 0- and 1 -body space must be taken into account. 

Morphine sulphate- sodium alginate PDMS Second order matrix swelling may be the reason [12]... 

The polarizability in (1) is conveniently expressed in terms of the second-order matrix element of the electric dipole moment component Dz... 

If the Hartree-Fock determinant dominates the wavefunction, some of the occupation numbers will be close to 2. The corresponding MOs are closely related to the canonical Hartree-Fock orbitals. The remaining natural orbitals have small occupation numbers. They can be analysed in terms of different types of correlation effects in the molecule . A relation between the first-order density matrix and correlation effects is not immediately justified, however. Correlation effects are determined from the properties of the second-order reduced density matrix. The most important terms in the second-order matrix can, however, be approximately defined from the occupation numbers of the natural orbitals. Electron correlation can be qualitatively understood using an independent electron-pair model . In such a model the correlation effects are treated for one pair of electrons at a time, and the problem is reduced to a set of two-electron systems. As has been shown by Lowdin and Shull the two-electron wavefunction is determined from the occupation numbers of the natural orbitals. Also the second-order density matrix can then be specified by means of the natural orbitals and their occupation numbers. Consider as an example the following simple two-configurational wavefunction for a two-electron system ... 

The expansion of the second-order matrix R2 in terms of products of the zeroth-order eigenvectors. 

Use the formulas for the second-order matrix P2( ) appearing in Eq. (6.79) to express the 2x2 matrix relevant to evaluating the ionization potential and electron affinities of the minimal-basis HeH problem. 

Modes that obey this selection rule, are said to be JT active. The evaluation of the second-order matrix elements requires two steps. One first couples the two distortion modes to a composite tensor operator 2co. ... 

The second-order matrix element then becomes ... 

Thus, the operator ad carries the two-dimensional real plane spanned by the vectors Ea + E aji(Ea — E a) into itself and is given in this plane by the following skew-symmetric second-order matrix... 

Two successive contact transformations remove from the expression of the Hamiltonian the first and second degree terms. Thus, the wave functions of die zero order term which, in our case, are the standard linear harmonic oscillator wave functions, are also eigenfunctions of the Hamiltonian H up to the second order. If a standard perturbation dieory is applied, there will be an extensive number of off-diagonal matrix elements of the first-order perturbation Hamiltonian appearing in the expressions for any molecular quantity estimated from second order matrix elements. By the contact transformations the matrix elements will be diagonal through second order which greatly simplifies the calculations. If the linear harmonic oscillator wave function is denoted by ( n i, die matrix element < n I H" I m) may be expressed as [Eq. (6.10)]... 

Associated with every square matrix is a real number called the determinant of the matrix. The determinant of a second-order matrix is defined as... 

Lengsfield B H III 1980 General second-order MC-SCF theory a density matrix directed algorithm J. Chem. Phys. 73 382... 

To calculate the matrix elements of second-order derivatives, we have... 

Once there is an estimate for the error in calculating the adiabatic-to-diabatic tiansfomiation matrix it is possible to estimate the error in calculating the diabatic potentials. For this purpose, we apply Eq. (22). It is seen that the error is of the second order in , namely, of 0( ), just like for the adiabatic-to-diabatic transformation matrix. 

Obviously, the fact that the solution of the adiabatic-to-diabatic transformation matrix is only perturbed to second order makes the present approach rather attractive. It not only results in a very efficient approximation but also yields an estimate for the error made in applying the approximation. 

Th c Newton-Raph son block dingotial method is a second order optim izer. It calculates both the first and second derivatives of potential energy with respect to Cartesian coordinates. I hese derivatives provide information ahont both the slope and curvature of lh e poten tial en ergy surface, Un like a full Newton -Raph son method, the block diagonal algorilh m calculates the second derivative matrix for one atom at a lime, avoiding the second derivatives with respect to two atoms. 

To fin d a first order saddle poiri t (i.e., a trail sition structure), a m ax-imiim must be found in on e (and on/y on e) direction and minima in all other directions, with the Hessian (the matrix of second energy derivatives with respect to the geometrical parameters) bein g varied. So, a tran sition structu re is ch aracterized by th e poin t wh ere all th e first derivatives of en ergy with respect to variation of geometrical parameters are zero (as for geometry optimization) and the second derivative matrix, the Hessian, has one and only one negative eigenvalue. 

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

Components of a second-order tensor T in a three-dimensional frame of reference are written as the following 3x3 matrix... 

It follows that by using component fomis second-order tensors can also be manipulated by rules of matrix analysis. 

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