Approximations hermitian

The possibility to write down an (even if approximate) Hermitian Hamiltonian representing the sum of monoatomic and diatomic terms has a significant conceptual importance, in particular because the expectation values of these terms of the Hamiltonian reproduce the one- and two-center energy components in the CECA analysis [9], We hope that this way of writing the Hamiltonian will permit to accomphsh some a priori approaches to molecular structure problems, and not only a posteriori ones like the energy decomposition. In the next section, we shall consider the application of our approach at the SCF level of theory it is not utilizing explicitly the detailed form (32)-(34) of the Hamiltonian. 

The fact that the projective integral approximations discussed in Sect. 2 lead to the approximate Hermitian Hamiltonian (30), opens a quite straightforward way to introduce the respective approximate SCF equations. As the Hamiltonian in the second quantized framework is defined by the integrals over the basis orbitals, one should simply introduce the same integral approximations in the SCF equations as were used for the Hamiltonian. 

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. 

It contains the entire description of the bath behavior in our approximation. Finally, we define a Hermitian matrix F with elements... 

Since He is Hermitian the matrix is a real symmetric matrix. This N x N determinant gives an ATh-order polynomial in E. The lowest root is the best approximation to the ground-state energy within the framework of the trial function in Eq. (3.7). 

Applying the approximate expansion for U from the right and the Hermitian conjugate from the left to the perturbed Hamiltonian matrix yields ... 

High-Order Compact (Hermitian) Current Approximation... 

Lastly, in Chap. 9, Sect. 9.2.7, an improvement in the above is described, based on Hermitian schemes for a better gradient approximation... 

As already described in some detail in Chap. 3, a one-sided first derivative such as the current approximation G can be raised to higher-order by a Hermitian scheme, as introduced by Bieniasz [108], This can then be used both to obtain better current approximations, and also in those cases where G enters a boundary condition. For the simpler case of the current approximation on a concentration grid already calculated, see the relevant Sect. 3.6 in Chap. 3. Here we need to go into some detail on the boundary conditions application. 

There are simulation cases (for example using unequal intervals) where it is desirable to use a two-point approximation for G, both for the evaluation of a current, and as part of the boundary conditions. In that case, an improvement over the normally first-order two-point approximation is welcomed, and Hermitian formulae can achieve this. Two cases of such schemes are now described that of controlled current and that of an irreversible reaction, as described in Chap. 6, Sect. 6.2.2, using the single-species case treated in that section, for simplicity. The reader will be able to extend the treatment to more species and other cases, perhaps with the help of Bieniasz seminal work on this subject [108]. Both the 2(2) and 2(3) forms are given. It is assumed that we have arrived at the reduced didiagonal system (6.3) and have done the u-v calculation (here, only v and iq are needed). 

We must also specify the time integration method used, because the Hermitian scheme makes use of terms in dC/dT, which must be consistent with the time integration. We assume the three-point BDF method, second-order in time, so that an improvement in the usual two-point G-approximation to second or perhaps third-order (in space) will be appropriate. 

Within the quadrupole approximation, the magnetic field is irrotational and the Hermitian operator representing the Lorentz magnetic force on the electrons is... 

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