Valence effective

Fig. 2-18. Electron energy and state density in p-type semiconductors ca = acceptor level Nf, = acceptor concentrati

The ACSE has important connections to other approaches to electronic structure including (i) variational methods that calculate the 2-RDM directly [36-39] and (ii) wavefunction methods that employ a two-body unitary transformation including canonical diagonalization [22, 29, 30], the effective valence Hamiltonian method [31, 32], and unitary coupled cluster [33-35]. A 2-RDM that is representable by an ensemble of V-particle states is said to be ensemble V-representable, while a 2-RDM that is representable by a single V-particle state is said to be pure V-representable. The variational method, within the accuracy of the V-representabihty conditions, constrains the 2-RDM to be ensemble N-representable while the ACSE, within the accuracy of 3-RDM reconstruction, constrains the 2-RDM to be pure V-representable. The ACSE and variational methods, therefore, may be viewed as complementary methods that provide approximate solutions to, respectively, the pure and ensemble V-representabihty problems. 

Both the effective valence Hamiltonian method [31, 32] and unitary coupled cluster [33-35] employ a single two-body unitary transformation. In the effective valence Hamiltonian method [31, 32], the unitary transformation, selected by perturbation theory, is applied to the Hamiltonian to produce an effective... 

M. G. Sheppard and K. F. Freed, Effective valence shell Hamiltonian calculations using third-order quasi-degenerate many-body perturbation theory. J. Chem. Phys. 75, 4507 (1981). 

First, we note that the determination of the exact many-particle operator U is equivalent to solving for the full interacting wavefunction ik. Consequently, some approximation must be made. The ansatz of Eq. (2) recalls perturbation theory, since (as contrasted with the most general variational approach) the target state is parameterized in terms of a reference iko- A perturbative construction of U is used in the effective valence shell Hamiltonian theory of Freed and the generalized Van Vleck theory of Kirtman. However, a more general way forward, which is not restricted to low order, is to determine U (and the associated amplitudes in A) directly. In our CT theory, we adopt the projection technique as used in coupled-cluster theory [17]. By projecting onto excited determinants, we obtain a set of nonlinear amplitude equations, namely,... 

Effective valence-shell Hamiltonian [19, 25], Assumes degenerate valence space. 

Figure 6.4 shows the effective valence as a function of 0-0 distance for a variety of known and unknown cation environments, the unknown environments being shown in italics. The observed environments, shown in bold type, mostly lie to the right of the solid line which is given by eqn (6.2). This line can therefore be taken as the closest distance that two oxygen atoms can approach... 

In eqn (6.2) i o = 220 pm, Z) = 34pm, and i min is the smallest 0-0 distance that can be achieved with an effective valence of i. The maximum coordination number that can be observed must therefore satisfy the inequality (6.3) obtained from eqns (6.1) and (6.2) ... 

There are some anomalies in Fig. 6.4, particularly for cations with s < 0.2 vu, and it is instructive to ask why these occur. There are several possible reasons. If a given pair of oxygen atoms is bonded to more than one cation, the effective valence is the sum of the effective valences of each of the cation O bonds, reducing the minimum length of the 0-0 contact, an elfect that has not been taken into account in preparing Fig. 6.4. For example, a weak cation may be chelated by N03 ions so that the shared 0-0 edges are only 215 pm long rather... 

Fig. 6.4. Effective valence, s versus 0-0 distance for various regular MO coordination environments. Coordination environments shown in bold are known, those in italics are not known. The solid line represents the observed itmin given by eqn (6.3). The broken line represents Ztunstrained> the distance at which the anion-anion repulsion becomes negligible.

Anion-anion repulsion places an upper limit on the coordination number that a cation can adopt but, since the 0 ions do not behave like hard spheres, the size of the limiting 0-0 distance depends on the effective valence of the bonds. Either Fig. 6.4 or eqn (6.3) can be used to decide whether or not a particular coordination number is physically possible. 

Fig. 7.2. The effective valence, s, as a function of 0-0 distance (7 oo)- AB is Rmir. and A B is iJunstrained from Fig. 6.4 ED shows the distance expected for a linear hydrogen bond uncorrected for 0-0 repulsion (based on the thin line of Fig. 7.1) CD shows the distance expected for a linear hydrogen bond corrected for 0-0 repulsion (based on the heavy line of Fig. 7.1) CF is the minimum value of iJoo observed for maximally bent hydrogen bonds (based on the broken line in Fig. 7.4).

P. Strodel and P. Tavan. A revised MRCI-algorithm coupled to an effective valence-shell Hamiltonian. 11. Application to the valence excitations of butadiene, J. Chem. Phys., 117 4677 683 (2002). 

Compound Metal Electronegativity102 Metal Effective Valency Epc redi Epc red2 Em oxi... 

For a closed-shell system the effective valence hamiltonian (we follow the work of Weeks, Hazi, and Rice13) can be written in the general form... 

A 100 a realistic [FED77] proton shell is Z=38-50 for N<60 and Z=28-50 for N>60. Np Nn plots, subject to these definitions, are given in Figs. 1-2, They reveal that, again, a remarkable simplification results. Only the N 90 points in Fig. 1 deviate from a smooth curve this simply reflects the fact that the Z 64 gap is still partly intact for N 90. Indeed, one can exploit the otherwise smooth systematics by shifting these N 90 points to this smooth curve and extracting effective valence proton numbers that reflect the evolving proton subshell structure. 

However, in this form the equation is too limited and does not take into account that anion-anion bonds as well as cation clusters may occur. Such bonds reduce the effective valency of the anions with respect to the cations and vice-versa, so that we may write for the heteropolar part... 

---