Formal solutions

A formal solution is one which contains a formula weight of a solute in a litre of the solution. It is denoted by F. In most of the cases formula weights and molecular weights are identical but sometimes the true molecular weight of a compound is a multiple of the weight expressed by its formula as ordinarily written in a chemical reaction. 

Because g,(0) = 0, we find the statistical average of g, 0 is zero for all modes  

Before we look at the correlation of the end-to-end vector and the center-of-mass diffusion for the Rouse model, we derive general formulas for them by using Eqs. 3.135 and 3.138 only. These equations are also valid in the modified versions of the Rouse model. The assumptions specific to the Rouse model, such asksp = ik T/b and the neglect of the hydrodynamic interactions, show up only in the expressions for the parameters k, and Ti. 

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This permits us to denote the QCMD equations (1) in the form i = The formal solution can now be written as... 

The idea is now to replace the formal solution of the Liouville equation by the discretized version. The middle term gf the propagator in Eq. (51) can be further decomposed by an additional Trotter factorization to obtain... 

Power Series Expansions and Formal Solutions (a) Helium Atom. If the method of superposition of configurations is based on the use of expansions in orthogonal sets, the method of correlated wave functions has so far been founded on power series expansions. The classical example is, of course, Hyl-leraas expansion (Eq. III.4) for the ground state of the He atom, which is a power series in the three variables... 

Equation (10-22) will be a (formal) solution of (10-1) if the Heisenberg operator in(x) satisfies the free field equation... 

In a manner analogous to the above we can define the out Heisenberg operators rout(x) by the formal solution... 

The results of the studies.discussed in Section II,C permit calculations to be made of the time required for the flame to spread to the entire propellant surface. Once this phase of the motor-ignition process has been completed, the time required to fill the combustion chamber and establish the steady-state operating conditions must be computed. This can be done by the formal solution of Eq. (7). Because this equation is a Bernoulli type of nonlinear equation, the formal solution becomes... 

When perturbation is fast enough in comparison with the molecular response to it, the averaging procedure proposed in [91] is justified. After substitution of a formal solution of Eq. (2.20)... 

The first step in developing the numerical method is to And a formal solution to Equation (8.63). Observe that Equation (8.63) is variable-separable ... 

The application of OCT requires the repeated solution of the time-dependent Schrddinger equation [see Eqs. (4.a, b)]. Our ability to solve this equation is therefore central to the application of the theory. If the Hamiltonian is independent of time, then the formal solution to the time-dependent Schrddinger... 

To define the language and comment on some limitations usually overlooked in the theoretical chemical literature, let us outline the standard procedure leading to a formal solution for such an equation. 

At this stage, we are in a position which allows us to make very easily a detailed analysis of the exact formal solutions (41) and (42). 

We write its formal solution, take the mean over the ensemble,... 

The elements (m Mz(t) ny appearing in Q(t) are then expressed by means of the formal solution... 

Fwolution of the mass states is given by the equation idvynafiAj dx — H x)vma/lfi with the Hamiltonian given in (6). Its formal solution - the evolution matrix from the initial point Xq to the final point Xj - can be written as... 

So far we have assumed that the electronic structure of the crystal consists of one band derived, in our approximation, from a single atomic state. In general, this will not be a realistic picture. The metals, for example, have a complicated system of overlapping bands derived, in our approximation, from several atomic states. This means that more than one atomic orbital has to be associated with each crystal atom. When this is done, it turns out that even the equations for the one-dimensional crystal cannot be solved directly. However, the mathematical technique developed by Baldock (2) and Koster and Slater (S) can be applied (8) and a formal solution obtained. Even so, the question of the existence of otherwise of surface states in real crystals is diflBcult to answer from theoretical considerations. For the simplest metals, i.e., the alkali metals, for which a one-band model is a fair approximation, the problem is still difficult. The nature of the difficulty can be seen within the framework of our simple model. In the first place, the effective one-electron Hamiltonian operator is really different for each electron. If we overlook this complication and use some sort of mean value for this operator, the operator still contains terms representing the interaction of the considered electron with all other electrons in the crystal. The Coulomb part of this interaction acts in such a way as to reduce the effect of the perturbation introduced by the existence of a free surface. A self-consistent calculation is therefore essential, and the various parameters in our theory would have to be chosen in conformity with the results of such a calculation. 

After the energy is expressed as a functional of the 2-RDM, a systematic hierarchy of V-representabihty constraints, known as p-positivity conditions, is derived [17]. We develop the details of the 2-positivity, 3-positivity, and partial 3-positivity conditions [21, 27, 34, 33]. In Section II.E the formal solution of V-representability for the 2-RDM is presented through a convex set of two-particle reduced Hamiltonian matrices [7, 21]. It is shown that the positivity conditions correspond to certain classes of reduced Hamiltonian matrices, and consequently, they are exact for certain classes of Hamiltonian operators at any interaction strength. In Section II.F the size of the 2-RDM is reduced through the use of spin and spatial symmetries [32, 34], and in Section II.G the variational 2-RDM method is extended to open-shell molecules [35]. 

The formal solution of Al-representability for the 2-RDM is developed in terms of a convex set of two-particle reduced Hamiltonian matrices. To complement the derivation of the positivity conditions from the metric matrices, we derive them from classes of these two-particle reduced Hamiltonian matrices. This interpretation allows us to demonstrate that the 2-positivity conditions are exact for certain classes of Hamiltonian operators for any interaction strength. In this section all of the ROMs are normalized to unity. Much of this discussion appeared originally in Refs. [21, 29]. 

A quantum system of N particles may also be interpreted as a system of (r — N) holes, where r is the rank of the one-particle basis set. The complementary nature of these two perspectives is known as the particle-hole duality [13, 44, 45]. Even though we treated only the iV-representability for the particles in the formal solution, any p-hole RDM must also be derivable from an (r — A)-hole density matrix. While the development of the formal solution in the literature only considers the particle reduced Hamiltonian, both the particle and the hole representations for the reduced Hamiltonian are critical in the practical solution of N-representability problem for the 1-RDM [6, 7]. The hole definitions for the sets and are analogous to the definitions for particles except that the number (r — N) of holes is substituted for the number of particles. In defining the hole RDMs, we assume that the rank r of the one-particle basis set is finite, which is reasonable for practical calculations, but the case of infinite r may be considered through the limiting process as r —> oo. 

VI. An Exact Formal Solution to the Contracted Schrodinger Equation s Indeterminacy... 

VI. AN EXACT FORMAL SOLUTION TO THE CONTRACTED SCHRODINGER EQUATION S INDETERMINACY... 

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 ... 

The formal solution to the coupled set of differential equations presented in equation (14) is given by equation (17) [75]. 

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