Constructing a symmetry adapted function

In the Bom-Oppenheimer approximation the electronic ground-state wave function of H2 has to be the eigenfunction of the nuclear inversion symmetry operator i interchanging nuclei a and h (cf. Appendix C). Since P = 1, the eigenvalues can be either —1 (called u symmetry) or -1-1 g symmetry). The ground-state is of g symmetry, therefore the projection operator will take care of that (it 

We obtain as a zero-order approximation to the wave function N ensures normalization) 

This is precisely the Heitler-London wave function from p. 521, where its important role in chemistiy has been highlighted  

The function is of the same qrmmetry as the exact solution to the Schrodinger equation (antisymmetric with respect to the exchange of elections and symmetric with respect to the exchange of protons). It is easy to calculate, that normaliza- 

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Polarization Approximation Is Illegal Constructing a Symmetry Adapted Function The Perturbation is Always Large in Polarization Approximation Iterative Scheme of SAPT Symmetry Forcing... 

To identify the states which arise from a given atomic configuration and to construct properly symmetry-adapted determinental wave functions corresponding to these symmetries, one must employ L and Ml and S and Ms angular momentum tools. One first identifies those determinants with maximum Ms (this then defines the maximum S value that occurs) within that set of determinants, one then identifies the determinant(s) with maximum Ml (this identifies the highest L value). This determinant has S and L equal to its Ms and Ml values (this can be verified, for example for L, by acting on this determinant with f2 in the form... 

Note, however, that since we now work with only the trace of the matrix, we have no information about off-diagonal elements of the irrep matrices and hence no way to construct shift operators. The business of establishing symmetry-adapted functions therefore involves somewhat more triad and error than the approach detailed above. Character projection necessarily yields a function that transforms according to the desired irrep (or zero, of course), but application of character projection to different functions will be required to obtain a set of basis functions for a degenerate irrep, and the resulting basis functions need not be symmetry adapted for the full symmetry species (irrep and row) obtained above. 

Whatever method is used in practice to generate spin eigenfunctions, the construction of symmetry-adapted linear combinations, configuration state functions, or CSFs, is relatively straightforward. First, we note that all the methods we have considered involve 7V-particle functions that are products of one-particle functions, or, more strictly, linear combinations of such products. The application of a point-group operator G to such a product is... 

We may disregard the closed-shell cores of the atoms since these play no role in the construction of symmetry-adapted wavefunctions, and concentrate attention upon the valence electrons. In the simplest case, with one valence electron per atom, we have a configuration 0102 n of N singly-occupied, non-degenerate valence orbitals which is then said to form a covalent structure for the molecule. Then under any spatial symmetry operation (%, a VB function Vsu-.k transforms as... 

The symmetry-adapted functions themselves may then be constructed directly from equations (59), (60), and (61). The same procedure can be applied without any difficulty to the other types of ionic configuration. The results show that there are altogether a total of 268 multiplets of which 22 correspond to xAig states. A similar result, of course, would be obtained from MO theory with full configuration interaction.33... 

The usually well-localised nature of the orbitals appearing in VB wavefunction makes spatial symmetry more difficult to use than in the MO case. In MO theory, symmetry can be introduced and utilised at the orbital level Each delocalised MO can be constructed as a symmetry-adapted linear combination (SALQ of basis functions, which is straightforward to implement in program code and can be exploited to achieve substantial computational savings. As a rule, the individual localised orbitals from VB wavefunctions are not S5mimetry-adapted, but transform into one another under the symmetry operations of the molecular point group. The use of symmetry of this type normally requires prior knowledge of the orbital shapes and positions and is very difficult to handle without human intervention. 

A hypothetical molecule A 3 with the geometry of an equilateral triangle. The basis set consisted of one function from each atom. Symmetry adapted functions (SAF), constructed ... 

At least two semiempirical methods, MINDO/3 and MNDO, have been applied successfully to the study of linear polymers using conventional solid-state theoretical techniques. The MINDO/3 calculation, showed how the band-structure of polyethylene could be calculated, while in the MNDO calculation, the optimized geometry, electronic band structure, and vibrational frequencies for polyethylene were calculated. These calculation used conventional methods, which rely on the factorization of the infinite Hamiltonian into complex symmetry adapted functions, followed by the use of those functions in the construction of a real density matrix. A more general solid-state method has been developed, but like the other conventional methods, it is very slow, and these methods have not been used to any great extent. 

To summarize it is often convenient to assume that the basis functions in an expansion such as (3.1.4) are all of the same pure symmetry species with respect to both space and spin symmetry operations. The symmetry-adapted functions are no longer necessarily single Slater determinants, and may often be linear combinations of a considerable number. Such functions, constructed within the individual configurations, are often called configurational functions (CFs), a terminology that we frequently adopt.t... 

The phrase symmetry adapted basis functions refers to those linear combinations of basis functions (on several atoms) that transform like the particular irreducible representation of the appropriate point group. Molecular symmetry is used at various points in these calculations twenty years ago I would have had to write several chapters on molecular symmetry, point groups, constructing symmetry-adapted combinations of basis functions, factoring a Hamiltonian matrix using symmetry and related topics. The point is that twenty... 

The kind of functions we need may be called symmetry-adapted linear combinations (SALCs). It is the purpose of this chapter to explain and illustrate the methods for constructing them in a general way. The details of adaptation to particular classes of problems will then be easy to explain as the needs arise. 

If the TV-particle basis were a complete set of JV-electron functions, the use of the variational approach would introduce no error, because the true wave function could be expanded exactly in such a basis. However, such a basis would be of infinite dimension, creating practical difficulties. In practice, therefore, we must work with incomplete IV-particle basis sets. This is one of our major practical approximations. In addition, we have not addressed the question of how to construct the W-particle basis. There are no doubt many physically motivated possibilities, including functions that explicitly involve the interelectronic coordinates. However, any useful choice of function must allow for practical evaluation of the JV-electron integrals of Eq. 1.7 (and Eq. 1.8 if the functions are nonorthogonal). This rules out many of the physically motivated choices that are known, as well as many other possibilities. Almost universally, the iV-particle basis functions are taken as linear combinations of products of one-electron functions — orbitals. Such linear combinations are usually antisymmetrized to account for the permutational symmetry of the wave function, and may be spin- and symmetry-adapted, as discussed elsewhere ... 

The Hamiltonian (58) commutes with the operator that displaces all the spins by one unit cell cyclically. Therefore, the eigenfunctions of (58) must be characterized by hole quasi-impulse k = 2nmlN (m=l,2...N). The symmetry adapted basis functions corresponding to a fixed value of k can be constructed by the usual group theory technique. 

The equations (59)—(63) solve the formal problem of which and how many multiplets 25+1/I arise from a given set of covalent and ionic configurations, and of how to construct the corresponding symmetry-adapted VB functions. However, one seeks to express at least some part of an expansion in many VB functions in a more compact and, hopefully, physically suggestive form. This is essentially the motivation behind the introduction of hybrid orbitals. 

To test your understanding of the MO model for a typical octahedral coordination complex, construct an appropriate, qualitative MO diagram for Oh SHg (a model for known SF6). Hint first calculate the total number of MOs you should end up with from the number of available basis functions (AOs). Second, compare the valence AO functions of S with those of a transition metal (refer to Figure 1.9 and realize that, for a coordinate system with the H atoms on the x, y and z axes, the AO functions of the central atom and the symmetry-adapted linear combinations of ligand functions transform as s, aig p, tiu djey d dy, t2g dx2-y2 dz2, eg in the Oh point group). Now count the number of filled MOs and the number of S-H bonding interactions. 

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