HLSP functions

Of these two schemes, it appears that the standard tableaux functions have properties that allow more efficient evaluation. This is directly related to the occurrence of the J f on the outside of OAfVAf. Tableau functions have the most antisymmetry possible remaining after the spin eigenfunction is formed. The HLSP functions have the least. Thus the standard tableaux functions are closer to single determinants, with their many properties that provide for efficient manipulation. Our discussion of evaluation methods will therefore be focused on them. Since the two sets are equivalent, methods for writing the HLSP functions in terms of the others allow us to compare results for weights (see Section 1.1) of bonding patterns where this... 

We note finally that if P = for a particular product function the standard tableaux function and HLSP function are the same. 

Transformations between standard tableaux and HLSP functions... 

Since the standard tableaux functions and the HLSP functions span the same vector space, a linear transformation between them is possible. Specifically, it would appear that the task is to determine the a,y s in... 

After this digression we now return to the problem of determining the HLSP functions in terms of the standard tableaux functions. We solve Eq. (5.118) by... 

To obtain the symmetry functions in terms of HLSP functions we can transform the standard tableaux functions using the methods of Chapter 5. The transformation matrix is given in Eq. (5.128) ... 

The transformation from standard tableaux functions to HLSP functions is independent of the spatial symmetry and so we need the 4-matrix in Eq. (6.24) again. This time the results are... 

On the other hand, no such invariance of G1 or HLSP functions occurs, so the orthogonality constraint has a real impact on the calculated energy. 

This happens when we consider the most important configuration using HLSP functions. The two Rumer diagrams are shown with dots to indicate the extra electron. 

Transforming our wave function to the HLSP function basis, we obtain... 

The HLSP function form of this wave function is easily obtained with the method of Section 5.5.5,... 

The reader will recall that a given configuration has different standard tableaux functions and HLSP functions if and only if it supports more than one standard tableaux function (or HLSP function). 

Example of transformation to HLSP functions The permutations we use are based upon the particle label tableau... 

The corresponding normalization and overlap integrals for the HLSP functions are then obtained with the transformation of Eq. (10.22),... 

Because of the spatial symmetry there is only one configuration (as with allyl), and in this case the HLSP function function is the simpler of the two. We have for... 

The principal configurations in the wave function are shown as HLSP functions in Table 10.8 and as standard tableaux functions in Table 10.9. Considering the HLSP functions, the first is the ground state configuration of the separated atoms, the next two are bonding functions with the s-p hybrid of Be and the fourth contributes polarization to the Be2/ z component. The corresponding entries in the third and fourth columns of Table 10.9 do not include the tableau function with the orbital. 

Table 10.11. Coefficients and tableaux for standard tableaux functions and HLSP functions for SCVB treatment of BeH.

We have collected some results for standard tableaux functions and HLSP functions in Tables 11.5 and 11.6. The structure ofthese tables will be repeated in several later sections, and we describe it here. 

When the two atoms are at the geometry of the energy minimum the results are as shown in Tables 11.7 and 11.8, where, as before, we give both the standard tableaux function and HLSP function results. It is clear that the wave function is now a more complicated mixture of many structures. In addition, the apparent importance of the structures based upon the values of the coefficients is somewhat different for the... 

Table 11.8. Principal HLSP function structures for B2 at the energy minimum bond distance.

First, however, we examine the asymptotic geometry. The principal structures are shown in Tables 11.9 and 11.10. In the standard tableaux function case structure 1 is one of the possible couplings of two P atoms, and stmctures 2 and 3 produce electron correlation in the closed 2s shell. The results with HLSP functions are essentially the same with some differences in the coefficients. The apparently smaller coefficients in the latter case result mainly from the larger number of terms in the S3munetry functions. 

The results for HLSP functions in Table 11.12 show a somewhat different picture. In this case the dominant (but not by much) structure is the one with two n bonds and structures 3 and 4 provide a o bond. Structure 2 is the double structure, but, since HLSP functions do not have a close relationship to the actual state as above, there is less importance to just one Rumer coupling scheme. 

The situation is not so simple with HLSP functions. They do not have the antisymmetry characteristic mentioned above, and the asymptotic state requires a sum of three of them as shown in Table 11.14. 

When two N atoms form a molecule we have the possibility that there could be three bonds, one from the two Pa orbitals, and two from the four pj orbitals. Some mixing of the 2s with the p orbitals might lead to hybridization. No other possibilities seem likely. We show the principal configurations in the HLSP function and standard tableaux function cases in Tables 11.15 and 11.16, respectively. We see that the same orbitals are present in both main structures. The situation with... 

The results for the standard tableaux functions at the energy minimum are shown in Table 11.16. Structures 1,2, 4, and 5 are different standard tableaux corresponding to two ground state atoms and represent mixing in different states from the ground configurations. The standard tableaux functions are not so simple here since they do not represent three electron pair bonds as a single tableau. Structure 3 represents one of the atoms in the first excited valence state and contributes to s-p hybridization in the cr bond as in the HLSP function case. 

As we pass to F2, with a minimal basis the amount of flexibility remaining is small. The only unpaired orbital in the atom is a 2/ one, and these are expeeted to form a o electron pair bond and a S+ molecular state. In fact, with 14 electrons and 8 orbitals (outside the core) there can be, at most, one unpaired orbital set in any structure. Therefore, in this case there is no distinction between the standard tableaux and HLSP function representations of the wave functions, and we give only one set of tables. As is seen from Table 11.21, there is oifly one configuration present at asymptotic distances. That shown is one of the combinations of two P atoms. 

We give tables of the important structures in the full wave function using spherical AOs and using the s-p hybrids, 2s 2pz. The energies are, of course, the same for these alternatives, but the apparent importance of the standard tableaux functions or HLSP functions differs. We also discuss EGSO results for the series. 

Table 12.2. N2 The most important terms in the wave function when spherical AOs are used as determined by the magnitudes of the coefficients. Results for standard tableaux and HLSP functions are given. See text.

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