Fermi-sea wavefunctions

Another Example of the Possible Utility of the Small Fermi-Sea Wavefunctions in Atoms The Case of the Low-Lying States of Be... 

The Fermi-sea wavefunctions (Section 8) were restricted to the largest components of >1 ° of Eq. (8). They were computed self-consistently. They are as follows ... 

As in the case of the D-ND above, for the sake of completeness I close the present subsection by quantifying the H-ND in the Be isoelectronic sequence. The energy for Be will be used in Section 9 on the Fermi-sea wavefunctions and energies of the low-lying states of Be. 

ANOTHER EXAMPLE OF THE POSSIBLE UTILITY OF THE SMALL FERMI-SEA WAVEFUNCTIONS IN ATOMS THE CASE OF THE LOW-LYING STATES OF Be... 

Table 2.7 contains five columns of numbers for excitation energies in Be. The first two are from Ref. [106]. The acronyms SC-SF-CIS and FCI stand for "spin-complete, spin-flip, configuration-interaction singles" and "full configuration interaction," respectively. In this case, the application of SPSA employs the differences of only the Fermi-sea energies given in the previous subsection (15a, 16c, and 19). Also, the Fermi-sea wavefunction for the... 

As explained in the previous sections, in the framework of the SPSA, the response of atomic or molecular states to external perturbations is best described, heuristically and quantitatively, by employing as reference the Fermi-sea wavefunction, followed by the necessary additional corrections. For example, this was outlined in Section 4.3 with the FOTOS. 

In conclusion, the results presented in the previous paragraphs and sections as to the choice and significance of Fermi-sea wavefunctions of states of Be and Be2 symmetry) support the argument that the unusual Be2 bond is produced predominantly by ND-type correlations, whose characteristics originate from the participation in the bonding of a few low-lying... 

In a mixed system like (Sm/Gd) S we have seen that, in phase region B, the Sm d electrons are excluded from the Fermi sea. Metallic conduction persists essentially using only the electrons of the Gd subarray with the corresponding wavefunctions tending to avoid the Sm sites. The resulting metal though not a good one does show a positive temperature coefficient. This same feature is found also for the mixed systems in the M phase at all temperatures, and in this there arises a marked contrast to the behaviour of a pure material in its metallic IGF condition. 

Excitation energies of the iow-iying vaience states of Be. Comparison of resuits from conventionai quantum chemistry caicuiations and from the use of the Fermi-sea energies obtained from numericai MCHF wavefunctions... 

It is such types of heavy mixing that, together with additional evidence (see following sections), led to the proposal of opfimizing fhe quantitative description of electronic structures by computing appropriately chosen multiconfigurational Fermi-sea zero-order wavefunctions (Sections 3 and 8). 

In general, the steps of the SPSA toward the computation of the correlated wavefunction for each state and the property under consideration are as follows (adjustments in special cases are inevitable) Once is established with self-consistent orbitals that are numerically accurate, one should seek the form of the part of the remaining wavefunction that results from the action on the Fermi-sea of two operators The Hamiltonian and the operator of the property that is being studied. This provides the information to first order beyond the MCHF (or nearly so) on the symmetry and the spatial characteristics of the function space that is created by the action of the two operators. The final result for the total wavefunction is obtained to all orders via diagonalization of the total matrix after judicious choices and... 

For molecules, especially in excited states, the choice of the proper, for each problem, extended set of zero-order orbitals and corresponding configurations that would allow, to a good approximation, the recognition, in quantitative terms, of the main features of the wavefunction and the bonds constitutes a challenging problem. For example, such a problem is discussed in Sections 9 and 10, where the exceptional bond of Be2 X E+ is examined in the framework of ND versus D correlation using as reference points the Fermi-seas of the low-lying states of Be. 

Photoabsorption transition probabilities and cross-sections for the two categories contain different satellite peaks due to the presence (or absence) of different zero-order Fermi-sea SACs in initial and final sfafes, for example, Ref. [101]. This is in accordance wifh FOTOS, where, as explained in Section 4, the essential features of the transition probability and the related phenomena are explained by using in the zero-order transition matrix element the Fermi-sea multiconfigurational wavefunctions of the initial and the final states of the transition [26b, 45]. 

The case of fhe ground state, S, was already discussed in Section 5, with numerical results for different values of Z. Bofh fhe H-F sea and the Fermi-sea zero-order wavefunctions are described by a superposition of fhe ls 2s IS and W2p SACs. 

Furthermore, and this is important for fhe inferprefafion of the bonding in Be2 to be discussed below, the results (17-20) show that primarily the state D and to a lesser degree the state carry with them a significant d — wave component in their Fermi-sea zero-order wavefunctions. This means that such orbitals should also be considered in the self-consistent, zero-order description of molecules containing Be. 

Thus, according to Eq. (8), I focused on the calculation of ao h -p where two choices of were made. The first is a CASSCF wavefunction obtained in terms of the 2s,2pj set of active orbitals, which is the choice of the earlier publications by other researchers. This calculation was carried out only for reasons of comparison. The second choice for is the CASSCF solution that is obtained when the Fermi-sea set (2s, 2p, 3s, 3p, 3d is adopted, according to the analysis of the Fermi-seas of the low-lying states of Be presented in the earlier sections. For this case, both the and the = aoT ° -F 0 " wavefunctions of Eq. (8) were obtained, the latter at the level of MRCISD. On the other hand, I deemed it unnecessary for the economy and the purposes of the problem to account for the (small) changes of electron correlation along the PEC, which are due to the inner electrons, Is. ... 

Figure 2.1 A hierarchy of CASSCF calculations that reveal the critical significance of the presence of d orbitals in the zero order, Fermi-sea set of orbitals. The curve for CASSCF (4, 26)/+3s, +3d, +3p is in essential agreement with that of the MRCISD, whose reference wavefunction is the standard CASSCF (4, 8)/2s -1- 2p. (The aug-cc-pVDZ basis set was used).

Finally, the last PEC represents = ao T + and is obtained from a MRCISD calculation based on the CASSCF (4, 8)/2s + 2p zero-order wave-function. In such a calculation, the corrections that are taken into account by are considered part of the D correlation. The difference befween the last two curves is very small. Obviously, when additional correlations of the D type (in terms of configurations and basis sets) are included in a full Cl compulation, the inner minimum goes down close to its true value while the outer one disappears [95,103,115]. The same occurs in the present work using the larger CASSCF (4, 26) reference wavefunction with the Be Fermi-sea of (2s, 2p, 3s, 3p, 3d] orbitals. 

Table 2.8 Satisfaction of criterion for Eq. (8), oq 1, in the case of the lowest four Be2 states along the whole PEC, when 1 ° is the Fermi-sea (CASSCF) wavefunction with active set the Be 2s, 2p, 3s, Ip, Id orbitals ...

In symbols, the general compass is the form + 0 " of the wave-function of Eq. (8). In principle, the two parts are represented by different function spaces, whose elements and size depend on the problem. The zero-order wavefunction, is normally obtained self-consistently. Its orbitals belong to the state-specific Fermi-sea, see below. 

The introduction in the early 1970s of the concept and the methodology of the Fermi-sea as the zero-order orbital set for the construction of the state-specific multiconfigurational wavefunction played on the themes... 

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