Convergence of the perturbational series

The perturbational approach is applicable when the perturbation only slightly changes the energy levels, therefore not changing their order. This means that the unperturbed energy level separations have to be much larger than a measure of perturbation such as However, even in this case we may 

The subsequent perturbational corrections need not be monotonically decreasing. However, if the perturbational series eq. (5.19) converges, for any e 0 we may choose such Nq that for N Nov/e have i-C- the vectors 

Unfortunately, perturbational series are often divergent in a sense known as asymptotic convergence. A divergent series called an asymptotic series of a function/(z), if the function/ (z) = z [/(z) -S (z)], where S (z) = Y k=o satisfies the following condition limj oo Rn(z) = Q for any fixed n. In other words, the error of the summation, i.e. f(z) — S (z)] tends to 0 as or faster. 

Despite the fact that the series used in physics and chemistry are often asymptotic, i.e. divergent, we are abfe to obtain results of high accuracy with them provided we limit ourselves to appropriate number of terms. The asymptotic character 

In perturbation theory we assume that t( ) and if/kW are analytical functions of A (p. 205). In this mathematical aspect of the physical problem we may treat A as a complex number. Then the radius of convergence p of the perturbational series on the complex plane is equal to the smallest A, /or which one has a pole of E IX) or ipkW- The convergence radius pk for the energy perturbational series may be computed as (if the limit exists ) 

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As we have already pointed out, the theoretical basis of free energy calculations were laid a long time ago [1,4,5], but, quite understandably, had to wait for sufficient computational capabilities to be applied to molecular systems of interest to the chemist, the physicist, and the biologist. In the meantime, these calculations were the domain of analytical theories. The most useful in practice were perturbation theories of dense liquids. In the Barker-Henderson theory [13], the reference state was chosen to be a hard-sphere fluid. The subsequent Weeks-Chandler-Andersen theory [14] differed from the Barker-Henderson approach by dividing the intermolecular potential such that its unperturbed and perturbed parts were associated with repulsive and attractive forces, respectively. This division yields slower variation of the perturbation term with intermolecular separation and, consequently, faster convergence of the perturbation series than the division employed by Barker and Henderson. 

Indeed for the Neel-based theory to work best it is better to have a bipartite system (i.e., a system with two sets of sites all of either set having solely only members from the other set as neighbors). Of course, when there is a question about the adequacy of the zero-order description questions about the (practical) convergence of the perturbation series arises. But for favorable systems these [38] or closely related [41] expansions can now be made through high orders to obtain very accurate results. 

If any of the functions of A above is not analytic for all arguments A 1, the perturbation series of that function does not converge. We will be concerned only with the conventional situation where the wavefunc-tion is contained in a finite (but potentially huge) space spanned by Slater determinant functions constructed by a finite set of orbitals, or equivalently, such determinants precombined into configuration functions (CFs). From the point of view of this article, the Hamiltonian H is thus not the exact electronic Hamiltonian but is its projection in the CF space. Our aim is thus to construct a perturbation theory that approximates the full Cl result in this space. The FCI equations are algebraic equations, and lack of convergence of the perturbation series is then caused by near coincidence of energies of model functions with functions outside the model space, which are in this context called intruders. 

Since beryllium only has four electrons, CISDTQ is a full Cl treatment and completely equivalent to a CCSDTQ calculation. The multi-reference character displays itself as a relatively slow convergence of the perturbation series, with millihartree accuracy being attained at the MP6 level and inclusion of terms up to MP20 is required in order to converge the energy to within 10 au of the exact answer. Note also that the correlation energy is overestimated at order seven, i.e. the perturbation series oscillates at higher orders. The contribution from triply excited states is minute, as expected... 

Tosio Kato (1917-1999) was an outstanding Japanese physicist and mathematician. His studies at the University of Tokyo were interrupted by Wortd War it. After the war, he got his Ph.D. at this university (his thesis was about convergence of the perturbational series), and obtained the title of professor in 1958. 

We now have an apparently well-behaved operator, in which no expansion has been made that is invalid at small r. That is not to say that the magnitude of the perturbation is always small, because we still have V jc as the perturbation, which becomes infinite at r = 0 for a point nucleus. The rate of convergence of the perturbation series will still be related to this feature. However, the fact that the perturbation operator is large in some region of space does not mean that the integrals over the operator are also large in value, as discussed above in relation to the Pauli Hamiltonian. 

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