Basis sets symmetry-adapted

In the C3V group symmetry adapted basis set (see Ref. [4] for details), the matrix of the effective Hamiltonian defined by Eqs. (3)-(5) is block-diagonal. The blocks and the diagonalized ones for two nearest vibrational levels are ... 

The UHF formalism becomes inconvenient for open-shell configurations of atoms or molecules with point-group symmetry. Unless specific restrictions are imposed, the self-consistent occupied orbitals fall into sets that are nearly but not quite transformable into each other by operations of the symmetry group. By imposing equivalence and symmetry restrictions, these sets become symmetry-adapted basis states for irreducible representations of the symmetry group. This makes it possible to construct symmetry-adapted /V-clcctron functions, as described in Section 4.4. The constraints in general invalidate the theorems of Brillouin and Koopmans. This restricted theory (RHF) is described in detail for atoms by Hartree [163] and by Froese Fischer [130],... 

Symmetry considerations play a role on several levels in the analysis of Hartmann-Hahn experiments. In the presence of rotational symmetry and permutation symmetry, the effective Hamiltonian often can be simplified by using symmetry-adapted basis functions (Banwell and Primas, 1963 Corio, 1966). For example, any zero-quantum mixing Hamiltonian can be block-diagonalized in a set of basis functions that have well defined magnetic quantum numbers. Block-diagonalization of the effective Hamiltonian simplifies the analysis of Hartmann-Hahn experiments (Muller and... 

If these are symmetry-adapted basis functions, then the local potential must transform according to the invariant representation. Thus for use with localized basis sets it must be expressible as a superposition of identical single-center terms,... 

Band Structures. - Conceptually the simplest approximation amounts to assuming that the crystalline solid is infinite and periodic in all three dimensions. Then, when expanding the solutions to the Kohn-Sham (or Hartree-Fock) single-particle equations in a set of atom-centered basis functions [cf. Eq. (13)], the translational symmetry can be utilized in constructing symmetry-adapted basis functions from the identical ones of different unit cells, i.e.,... 

Most texts on the use of group representation theory in physical science list the character tables for the commonly occurring point groups euid the better ones will list the full standard irreducible representation matrices. If one has the full representation matrices it is possible to use these to effect a complete solution of the problem of the formation of symmetry-adapted basis from a given set of basis functions. 

The minimal-basis-set AOs for ethane are the hydrogen I5 orbitals and the carbon Ij, 2s, and 2p orbitals, a total of 6(1) + 2(5) = 16 basis AOs. To calculate the barrier in ethane, we must calculate the energy of the staggered and the eclipsed conformations, which requires two separate SCF calculations. One first forms appropriate linear combinations of the hydrogen AOs and of the carbon AOs to get symmetry-adapted basis functions. The Roothaan equations are then solved iteratively to give the basis-function coefficients and orbital energies, and the total molecular energy is found. 

As a second step, symmetry type 2 can be applied to the set of the s k points, which allows one to further reduce the Fock matrix into a block-diagonal form. By transforming the basis set into an equivalent set of symmetry adapted basis functions, every block of the transformed matrix in Figure 4, which corresponds to one particular point ky, reveals, in turn, a block-diagonal structure, for example, of the kind depicted in Figure 20. 

We call these new basis functions symmetry orbitals or symmetry-adapted basis functions. The labels on these linear combinations are explained in Appendix 1. The new basis set consists of the oxygen orbitals r/ris, fis, ipy, fip, and the symmetry orbitals and. The canonical orbitals will be simpler linear combinations of these basis orbitals than if we used the original basis set. 

Construct symmetry-adapted basis orbitals for the BeH2 molecule from the minimal basis set used in Section 21.1. How do the LCAOMOs in Table 21.1 relate to these basis orbitals ... 

List the symmetry operations that belong to rrans-1,3-butadiene in its equilibrium nuclear conformation. Construct a set of symmetry-adapted basis... 

These linear combinations are called symmetry-adapted basis functions. Pitzer and Merrifield carried out a Hartree-Fock-Roothaan calculation on H2O using a minimal basis set of Slater-type orbitals and obtained the orbitals displayed in Table 21.2. The a orbitals are numbered from lower to higher energy, as are the b and b2 orbitals. 

The importance of the characters of the symmetry operations lies in the fact that they do not depend on the specific basis used to form them. That is, they are invariant to a unitary or orthorgonal transformation of the objects used to define the matrices. As a result, they contain information about the symmetry operation itself and about the space spanned by the set of objects. The significance of this observation for our symmetry adaptation process will become clear later. 

ADF uses a STO basis set along with STO fit functions to improve the efficiency of calculating multicenter integrals. It uses a fragment orbital approach. This is, in essence, a set of localized orbitals that have been symmetry-adapted. This approach is designed to make it possible to analyze molecular properties in terms of functional groups. Frozen core calculations can also be performed. 

This paper considers the hyperspherical harmonics of the four dimensional rotation group 0(4) in the same spirit ofprevious investigations [2,11]), where the possibility is considered of exploiting different parametrizations of the 5" hypersphere to build up alternative Sturmian [12] basis sets. Their symmetry and completeness properties make them in fact adapt to solve quantum mechanical problems where the hyperspherical symmetry of the kinetic energy operator is broken by the interaction potential, but the corresponding perturbation matrix elements can be worked out explicitly, as in the case of Coulomb interactions (see Section 3). A final discussion is given in Section 4. 

To illustrate the advantage of spin- and spatial-symmetry adaptation, consider the BH molecule in a minimal basis set. If only is considered, the largest block of the two-electron RDM (i.e., is of dimension 36. Spin adaptation divides into two blocks, with sizes 15 and 21,... 

The factor lj/h, where h is the order of the group and b 18 the dimension of the y th irreducible representation, has been included in (9.67) for convenience. Application of this procedure to the functions / gives us (unnormalized) symmetry-adapted functions g,. This procedure is applicable to generating sets of functions that form bases for irreducible representations from any set of functions that form a basis for a reducible representation. The proof of the procedure (9.67) for one-dimensional representations is outlined in Problem 9.22 we omit its general proof.5 Symmetry-adapted functions produced by (9.67) that belong to the same irreducible representation are not, in general, orthogonal. 

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