Corrections to order

First-order corrected energy, correct to order a2, is... 

Then W = — U 2 is negative semidefinite and correct to order n. It also contains higher order terms, which are not the correct ones, but serve to make W negative semidefinite. 

In addition, V 2 should include corrections due to the finite speed of the electromagnetic interaction, as well as magnetic contributions present due to the electron spin. An approximate way to account for these effects, correct to order a2 a.u., is provided by the Breit operator [115]... 

We have therefore achieved our objective in that equation (3.83), which is correct to order 1 /c2, contains even operators only. It would, of course, be possible to proceed further with the Foldy Wouthuysen transformation but there is little point in doing so, since the theory is inaccurate in other respects. For example, we have treated the electromagnetic field classically, instead of using quantum field theory. Furthermore, we shall ultimately be interested in many-electron diatomic molecules, for which it will be necessary to make a number of assumptions and approximations. 

In section 3.4 we carried out a Foldy-Wouthuysen transformation on the Dirac Hamiltonian and obtained the result (3.84), correct to order c 2,... 

These expressions now correspond to a Coulomb gauge (3.205) to order 1/c2 note that the very simple result for 0 is correct to order 1 /c2. 

After carrying out the same procedure as employed for the case of the prolate spheroid, we find that i/ (r, ff) and the ff-ij/o relationship, both correct to order e, are given by... 

These agree very satisfactorily with the theoretical value 2 0337 X 105 Mc/s (equation 12.6) which includes the quantum electrodynamic corrections to order J a3. The uncorrected value, 2 044 x 106 Mc/s, is clearly inadequate. The importance of the annihilation term is demonstrated beyond doubt. 

With these definitions of the perturbation and the model space, we can now solve successively the Bloch equation (16) for the energy corrections (e ) and wave operator (w ). As discussed above, we shall need the wave operator up to order (n — 1) if we wish to determine the energy correction to order n ... 

III. Mobility Expression Correct to Order (Henry s Formula). 28... 

III. MOBILITY EXPRESSION CORRECT TO ORDER f (HENRY S FORMULA)... 

Henry s equation (2.6) assumes that is low, in which case the double layer remains spherically symmetrical during electrophoresis. For high zeta potentials, the double layer is no longer spherically symmetrical. This effect is called the relaxation elfect. Henry s equation (2.6) does not take into account the relaxation effect, and thus this equation is correct to the first order of Ohshima et al. [19] derived an accurate analytic mobility expression correct to order 1/ka in a symmetrical electrolyte of valence z and bulk concentration (number density) n with the relative error less than 1% for 10 < Ka < 00, which is... 

An approximate analytic mobility equation apphcable for arbitrary values of was daived by Levich [28], and Ohshima et al. [30] derived a more accurate mobility expression correct to order 1 /ka for the case of symmetrical electrolytes of valence z. The leading term of their expression is given by [30]... 

As in the case of rigid particles, the general solution of the problem must be obtained numerically, but in the same paper [96] the authors obtained approximate analytical solutions correct to order or or valid for large Ka. For instance, their expression valid for low zeta potential and arbitrary Ka reads... 

The resulting Dirac-Coulomb-Breit many-electron Hamiltonian is now correct to order a. The Breit interaction can be rewritten as a sum of two terms the magnetic interaction... 

The discretization scheme, which leads to an error 0 h ) for second-order differential equations (without first derivative) with the lowest number of points in the difference equation, is the method frequently attributed to Nu-merov [494,499]. It can be efficiently employed for the transformed Poisson Eq. (9.232). In this approach, the second derivative at grid point Sjt is approximated by the second central finite difference at this point, corrected to order h, and requires values at three contiguous points (see appendix G for details). Finally, we obtain tri-diagonal band matrix representations for both the second derivative and the coefficient function of the differential equation. The resulting matrix A and the inhomogeneity vector g are then... 

In lowest-order perturbation theory, the QED potential arises from one-photon exchange. This potential contains a static part (i.e. a velocity independent part), which is the Coulomb potential, and non-static corrections. These non-static terms are most commonly treated in the Fermi-Breit approximation, which gives the corrections to order w /c. The Fermi-Breit terms include a spin-spin interaction, a spin-orbit interaction, and a tensor interaction. They also include a spin-independent part which depends on the particle momenta. 

The procedure for obtaining the potential in QCD is as follows One divides the potential into two intervals of r. In the short-distance interval one uses QCD perturbation theory to calculate the potential. This may be done either analogously to the case of QED or by other suitable procedures. The resulting potential, which in lowest-order perturbation theory arises from one-gluon exchange, is a Coulomb potential plus Fermi-Breit corrections to order... 

These are exactly the relations that van der Waals assumed in the generalization of his equation of state to mixtures. Within the mean-density approximation, we see that these mixing rules are correct to order T in the compressibility factor terms that have higher-order temperature dependence in the reference fluid are not correctly mapped by the van der Waals mixing rules. 

Equation (IIIB-18) is again correct to order h. It shows that each transition in a polymer can be primarily assigned to one group or set of identical groups, but it also contains contributions from all other permanent and induced moments. The magnitude of these contributions depends on the distance between the contributor and the primary group and the energy difference between the transitions. 

In this section we will derive an expression for the configurational distribution function, correct to order A , for a system of interacting linear molecules. 

In Section III we have found a formal expression for the distribution function with accuracy up to terms in P. By an analogous procedure we will presently determine the expression of the distribution function correct to order P, for a system of interacting rigid symmetric top molecules. 

From (V-20) we obtain the reduced equation of state correct to order... 

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