The Sum Rule

In addition, it is possible that differences in the BSR may arise due to choice of the s-basis from which the higher angular momentum basis functions are constructed. However, on comparison of the Bethe sum rule for basis C, with the sum rule generated by the same method described above, from another energy optimized, frequently used basis (that of van Duijneveldt [16]), we find only small differences in the BSR, and only at large q values. 

P is an empirical parameter that was determined to 0.0042 by a least-squares fit to the exactly known exchange energies of the rare gas atoms He through Rn. In addition to the sum rules, this functional was designed to recover the exchange energy density asymptotically far from a finite system. 

Equation (2.2) is just the Rutherford cross section for scattering of, strictly speaking, free charges. To apply this to atomic electrons that are not free but can be excited4 with energy En, Bohr surmised the sum rule... 

Even though the number ( nmb ) of NMB functions is much smaller than that ( nrb) of NRB functions, the former are seen to make overwhelmingly greater percentage contributions (>99%) to the total electron density, and thus, to the occupied bonding h, s. This conclusion is also confirmed by the accuracy of the sum rules to be described below, or by examination of individual hybrids. 

How accurate are the sum rules (3.37a) and (3.37b) for the molecules considered in Example 3.1 The table below summarizes the hybridization parameters A.,- and sum-rule values for C in each species ... 

To estimate the hybridization of hydride bonds and lone pairs, we first rewrite the sum rule (3.37a) for bond hydride bonds of hybridization AH and one lone pair of hybridization Anb,... 

We now assume that hydride bond p character /p(An can be adequately approximated with Eq. (3.7lb) (i.e., that L substitution primarily affects /P(LA ) rather than /P(AH). The A—L hybridization /P,AI) of the mono-substituted hydride can then be determined from the sum rule (3.37b) as... 

We can also generalize the sum rules (3.37) for a general set of inequivalent sp d hybrids. Each of the nine possible hybrids can be expressed as... 

For completeness, let us also briefly summarize the hybrid-conservation sum rules and bond angles for inequivalent sdM hybrids. For pure sd/2 hybridization, the sum rules (4.32) reduce to the simplified form... 

Equations (3.38) and (3.48) are the analytical expressions for the band and adstate contributions to the total reduced-energy spectrum of the adatom DOS, pa(x), respectively. Their graphs are displayed in Fig. 3.4 for 2/3 = 1 and the parameter values indicated. As can be seen, the presence of the large adstate spike at X = Xa markedly reduces the area under the in-band portion of the DOS, in accordance with the sum rule (cf. (3.34))... 

Instead of one resonance frequency per individual electron, Bethe recovered the spectrum of resonance frequencies for the atom, weighted by dipole oscillator strengths satisfying the sum rule... 

Finally - and equally important - Jens contribution to the formal treatment of GOS based on the polarization propagator method and Bethe sum rules has been shown to provide a correct quantum description of the excitation spectra and momentum transfer in the study of the stopping cross section within the Bethe-Bloch theory. Of particular interest is the correct description of the mean excitation energy within the polarization propagator for atomic and molecular compounds. This motivated the study of the GOS in the RPA approximation and in the presence of a static electromagnetic field to ensure the validity of the sum rules. 

As discussed in the previous sections, LSD works well, and GGA does better, by modelling the system-averaged hole in a way which respects the sum rules, amongst other things. A much better way to construct a hybrid functional [54,68] would be to construct a hybrid hole, in which the small-separation contributions would be modelled in LSD (or GGA), which works well here, and the large-separation contributions would be modelled by some approximation designed to work at large distances, e.g., the random phase approximation [35], or the correct asymptotic limit [19],... 

It is readily demonstrated that the sum rules (24) and (25) are compatible with the Eq. (14). The traces of these two-matrix components read... 

Taking into account the sum rule, Eq. (70), the expectation value of is likewise obtained from Eq. (29),... 

A very different functional form is expected for A in molecules with more than two electrons. Without any calculations it is clear that H2 is atypical, since vanishes for this system. = nqUp (q 7 p), taken from the A = 2 case, violates the sum rule, Eq. (89), in the general case of A > 2. This means that the functional form of nondiagonal elements is unknown for A-electron systems, as yet. Nevertheless, some constraints can be achieved for these quantities using known necessary conditions of 2-RDM A-representability. 

We assume further that the ON of the open shell p is always one (tip = 1). This assumption is trivial for a doublet, but it is more restrictive for higher multi-plet with a corresponding underestimation of the energy. Remember that matrix elements of are nonvanishing only if aU its labels refer to partially occupied NOs therefore A = 0 and II = 0 if we consider a cumulant made up of at least one open-shell level. Since A and II refer only to closed shells, we consider them spin-independent. The sum rule, Eq. (89), becomes... 

Electronic structure computations would be greatly simplified by the finding of practical NOFs. One may attempt to approximate the unknown off-diagonal elements of A considering the sum rule (89) and analytic constraints (101) imposed by the D-, G-, and Q-conditions. However, it is not evident how to approach A, for p q, in terms of the ONs. Due to this fact, let s rewrite the energy term, which involves A, as... 

Other moments of momentum can also be obtained directly from the Compton profile without first going through the numerical differentiation of Eq. (5.64), which is prone to roundoff and truncation errors. In particular, several groups [9,188-191] independently reported one or more of the sum rules... 

The factor in the Womskian cancels the factor of the surface integral. As expected, the value of the surface integral is independent of the actual location of the surface. Therefore, we obtain a general expression for the tunneling matrix elements, the sum rule,... 

Chen, C. J. (1990a). Tunneling matrix elements in three-dimensional space The derivative rule and the sum rule. Phys. Rev. B 42, 8841-8857. 

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