Slater sum

Up to this point, the theory has been presented in a form suitable only for adsorption systems which can be treated by classical statistical mechanics. When the quantization of the motion of the atoms is important, one must replace the Boltzmann factors in the integrals for ZN by the appropriate Slater sums. After integrating over the coordinates, one obtains... 

While the result in Eq. (8) depends on the gauge chosen for the vector potential in Eq. (9), its diagonal element, C(r, r, P), the so-called Slater sum, denoted by 5oB(r> P), does not. Indeed, for free electrons, this becomes independent of position r and is given, in suitable units, by... 

Pfalzner and March [14] have performed numerically the Laplace transform inversion referred to above to obtain the density p( ) from the Slater sum in Eq. (10). Below, we shall rather restrict ourselves to the extreme high field limit of Eq. (10), where analytical progress is again possible. Using units in which the Bohr magneton is put equal to unity, the extreme high field limit amounts to the replacement of the sinh function in Eq. (10) by a single exponential term, to yield... 

Using again the Laplace transform relation between density and Slater sum, one readily obtains... 

However, for many purposes, the essential physics required is contained in the Slater sum S(r, P), and therefore Lehmann and March [19] and subsequently Amovilli and March [20] have attempted to study this quantity directly. In particular, Lehmann and March obtain an equation for a potential energy K(r) corresponding to a uniform electric field, in D dimensions. This reads... 

For D = 3, and putting zq = z in Eq. (24) to obtain the Slater sum S, use of the explicit form of V in Eq. (22) readily allows one to verify that the diagonal form of Eq. (24) is indeed an exact solution of Eq. (25). Later, Amovilli and March [20] made similar progress on central field problems. It remains of interest to treat atoms in intense electric fields by direct use of the Slater sum rather than by use of the off-diagonal canonical density matrix. 

To gain orientation, let us first neglect the self-consistent field (see below for its inclusion). The TF solution for the diagonal element of the canonical density matrix or the Slater sum S(r, fi) can be written (compare Eq. (20)) as SoB(P)exp(— PF(r)) with F(r) = — Ze /r. This is readily shown to yield, via the ground-state electron density p(r, B)=L (5(r, p, B)/P where L denotes the inverse Laplace transform discussed earlier ... 

As to future directions, the problem of the canonical density matrix, or equivalently the Feynman propagator, for hydrogen-like atoms in intense external fields remain an unsolved problem of major interest. Not unrelated, differential equations for the diagonal element of the canonical density matrix, the important Slater sum, are going to be worthy of further research, some progress having already been made in (a) intense electric fields and (b) in central field problems. Finally, further analytical work on semiclassical time-dependent theory seems of considerable interest for the future. 

Two points associated with Eq. (B4) are noteworthy. First, the function cj)(p), which appears in the phase of the density matrix in Eq. (B3) does not enter the Slater sum at Eq. (B4). Secondly, 5(r, P) falls off in Gaussian fashion with the distance from the origin of the confining potential. However, since h(P) has the form... 

Let us turn then to the calculation of the current density J(r, P), following the work of Amovilli and March [51] see also below. One can write J(r, P) in terms of the canonical density and its diagonal-the Slater sum, as (in suitable units)... 

As in the spin-polarized NR case, the convenience of having only two potentials to represent magnetic interactions is obtained at a price. This price includes some contamination of the SCF solutions with a mixture of multiplets, which can sometimes be resolved by projection techniques, including for example, the Slater Sum Rule of atomic theory. The ease of calculation of an R potential which treats exchange in open-shell heavy atom systems reasonably well, without introducing artificial (and incorrect) spin-polarization is a considerable advantage. 

When quantum effects must be taken into account, the Boltzmann factor must be replaced by its quantum generalization, the Slater sum, which will be denoted by S(r1 ..., rN). Thus... 

Both sides of Eq. 11.27 are quantum mechanical generalizations Sfo,..., r,) of the Boltzmann factor, since they indicate the probability of position in configuration space normalized to unity when 0 vanishes. They are called Slater sums. 

Dawson, K. A., March, N. H. (1984). Slater sum in one dimension explicit kinetic energy functional. Phy. Lett. 106A, 158-160. 

The BS plus spin projection method discussed here is closely connected to the simple open-shell singlet method for optical excitations based on the Slater sum rule and ASCF (self-consistent-field total energy difference method). The mixed spin excited state is like the BS state, also of mixed spin. The Slater sum rule method" " is also quite effective for multiplet problems for excited states of transition metal complexes as shown in the work of Dahl and Baerends. ... 

Equations (4.4) and (4.5) present an interesting physical picture of nonrelativistic quantum mechanics. A similar approach can be employed in the treatment of thermal properties in quantum statistical mechanics. - In quantum statistical mechanics, the Slater sum, or thermal Green s function, is written in Dirac notation as... 

Another semiclassical approach to deal with the QHS system is due to Yoon and Scheraga [145], It is based on a superposition of Slater sums > a) corresponding to every pair of hard spheres in the systan (overlaps imply S" (r jtHigher-order terms may also be incorporated in S ... 

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