Subsidiary conditions

Bohm, D., Huang, K., and Pines, D., Phys. Rev. 107, 71, Role of subsidiary conditions in the collective description of electron interactions. ... 

However, an acceptable wave function must satisfy the subsidiary condition (9-475), i.e., p- = 0. 

However, clearly the solution for IF = 0 does not satisfy the subsidiary condition k-g0>k = 0. There exists, therefore, only one acceptable solution of Eq. (9-482) and it corresponds to a one-photon state of momentum k and negative helicity (s-k = —k). The g+k for that situation satisfies... 

We shall adopt Eqs. (9-510) and (9-511) as the covariant wave equation for the covariant four-vector amplitude 9ttf(a ) describing a photon. The physically realizable amplitudes correspond to positive frequency solutions of Eq. (9-510), which in addition satisfy the subsidiary condition (9-511). In other words the admissible wave functions satisfy... 

It is possible to formulate the Coulomb gauge theory in terms of radiation operators which satisfy the subsidiary condition... 

Now suppose that we have solved the one-electron equations (26) and suppose that the subsidiary conditions ... 

Making use of the potential-weighted orthonormality relation (58) and the subsidiary conditions (62) and (63), we obtain ... 

The matrix Wf,x will then be given in terms of the parameter s = kgR by equation (45), and the one-electron problem closely resembles the example discussed in connection with this equation, the only difference being that we now relax the condition k bfi = 1, and we instead impose the subsidiary conditions (62) and (63). These respectively require that... 

Using the subsidiary condition (75) and the fact that= p + p, we can rewrite this requirement in the form ... 

We have used for the row vectors of the respective entities, while we denote by ( ) and O the orbitals and many-electron functions, and by O and T(0) the two corresponding linear transformations, respectively. Various types of many-electron space for which such transformations may be carried out have been described by Malmqvist [34], In general, O may be non-unitary, possibly with subsidiary conditions imposed for ensuring that the corresponding transformation of the V-electron space exists e.g. a block-diagonal form according to orbital subsets or irreducible representations). 

Thus the singlet spatial function is symmetric and the triplet one antisymmetric. If we use the variation theorem to obtain an approximate solution to the ESE requiring symmetry as a subsidiary condition, we are dealing with the singlet state for two electrons. Alternatively, antisymmetry, as a subsidiary condition, yields the triplet state. 

With e as a given elementary electric charge, there is also a condition on the quantization of magnetic flux. This could be reinterpreted as a subsidiary condition in an effort to quantize the electron charge and deduce its absolute value by means of the present theory [ 13,18,20], but the details of such an analysis are not yet available. Magnetic flux quantization is discussed in further detail in Appendix B. 

Distance least squares (DLS), a method developed by Meier and Vill-iger (1) for generating model structures (DLS models) of prescribed symmetry and optimum interatomic distances, can supply atomic coordinates which closely approach the values obtained by extensive structure refinement. DLS makes use of the available information on interatomic distances, bond angles, and other geometric features. It is primarily based on the fact that the number of crystallographically non-equivalent interatomic distances exceeds the number of coordinates in framework-type structures. A general DLS program is available (8) which allows any combination of prescribed parameters (interatomic distances, ratios of distances, unit cell constants etc). In addition, subsidiary conditions (as discussed in Refs. 1 and 8) can also be prescribed. 

We can remove the problem with this subsidiary condition by instead using as the variational space the orthogonal complement to the MCSCF state I0>. This variational space is defined as a set of states IK> expanded in the same set of basis states m> as I0> ... 

NextT wc must impose boundary conditions on our chainftf atoms as wc did with thp eontinnons solid. We are assuming N atoms, with undisplaced positions at x = d, 2d,. . . Nd. We shall assume that the chain is held at the ends, and to be precise we assume hypothetical atoms at x = 0, x = (N + l)d, which are held fast. As with the continuous solid, the precise nature of the boundary conditions is without effect on the higher harmonics. In Eq. (1.3), then, we assume that the equations can be extended to include terms f0 and fcv+i, but with the subsidiary conditions... 

For a two-parameter treatment of solvent effects (with two independent solvent vectors), only two critical subsidiary conditions must be defined in order to force the two solvent parameters to represent physically significant solvent properties. Four other trivial arbitrary conditions have to be defined in order to fix zero reference points and scale-unit sizes. However, for a three-parameter treatment (with three independent solvent vectors), already six critical subsidiary conditions must be defined, in addition to the six trivial reference or scale-factor conditions. On the contrary, singleparameter treatments require no definition of critical subsidiary conditions, but only one reference (zero) condition and one standard (unit) condition, whose arbitrary assignment changes only the reference solvent and the scale-unit size (265, 276). 

The subsidiary condition implies charge conservation and all quantities involved are supposed to be fully renormalised. 

The framework within which condensed phases are studied is provided by the phase rule, which gives the number of independent thermodynamic variables required to determine completely the state of the system. This number, called the number of degrees of freedom or the variance of the system, will be denoted by F. The number of phases present (that is, the number of homogeneous, physically distinct parts) will be denoted by P, and the number of independently variable chemical constituents will be called C. By independently variable constituents, we mean those whose concentrations are not determined by the concentrations of other constituents through chemical-equilibrium equations or other subsidiary conditions. The phase rule states that... 

In order to prove equation (51), it is convenient to consider first the case in which N chemical components are present in each of the P phases and there are no chemical reactions or subsidiary conditions. In writing the thermodynamic properties, the phase will be identified by superscripts and the species will be identified by subscripts. If p, 7, and are known for i 1,..., and for y = 1,..., P, then the state of the system is completely determined (except for the total mass of material in each phase, knowledge of which is seldom wanted). Since... 

Waser, J. Least-squares refinement with subsidiary conditions. Acta Cryst. 16, 1091-1094 (1963). 

These two equations together with Equation (25) constitute the necessary subsidiary conditions to properly define the characteristics of the interior and exterior densities. Furthermore, from Equations (28) and (29) and Eqs. (23), (24) and (26), (27), the following coupling relations among the variational parameters (cn, fa, y) are found to be satisfied ... 

Hence, considering these subsidiary conditions, the energy functional ( ,. fa, y R, Vo) given by Equation (17) is variationally optimized for a given value of R and Vo relative to only two sets of variational parameters, i.e. ... 

While the result just deduced for the distribution of the density of the molecules was to be expected from the start, the same method, applied to the distribution of the velocity of the molecules, leads to a new result. The calculations in this case are exactly analogous to those above. We construct a velocity space by drawing lines from a fixed point as origin, representing as vectors the velocities of the individual molecules in magnitude and direction. We then investigate the distribution of the ends of these vectors in the velocity space. In this case as before we can make a partition into cells, and consider the question of the number of vectors whose ends fall in a definite cell. There is, however, one essential dilierence as compared with the former case, in that there are now two subsidiary conditions, viz. besides the condition... 

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