Symmetry-adapted linear combinations determination

The determination of molecular orbitals in terms of symmetry-adapted linear combinations of atomic orbitals is analogous to the determination of normal vibrational modes by forming symmetry-adapted linear combinations of displacements. Both calculations are in reality the reduction of a representa-... 

Symmetry adapted linear combinations (SALCs) of the determinants... 

This expression is an approximation since the numerical factor of 1/6 was omitted. The coefficient (the normalization factor) in the symmetry-adapted linear combinations can be determined at a later stage by normalization. In an actual calculation this is necessary, whereas here we are interested only in the symmetry aspects, which are well represented by the relative values. In fact, the normalization factors will be ignored throughout our discussions. 

Various methods (described in Chapter 4) can be used to determine the symmetry of atomic orbitals in the point group of a molecule, i. e., to determine the irreducible representation of the molecular point group to which the atomic orbitals belong. There are two possibilities depending on the position of the atoms in the molecule. For a central atom (like O in H20 or N in NH3), the coordinate system can always be chosen in such a way that the central atom lies at the intersection of all symmetry elements of the group. Consequently, each atomic orbital of this central atom will transform as one or another irreducible representation of the symmetry group. These atomic orbitals will have the same symmetry properties as those basis functions in the third and fourth areas of the character table which are indicated in their subscripts. For all other atoms, so-called group orbitals or symmetry-adapted linear combinations (SALCs) must be formed from like orbitals. Several examples below will illustrate how this is done. 

The MODPOT/VRDDO LCA0-M0-SCF programs have been meshed in with the configuration interaction programs we use. In this Cl (71) program each configuration is a spin- and symmetry-adapted linear combination of Slater determinants in the terms of the spin-bonded functions of Boys and Reeves ( 2, 3> 7 0 as formu-... 

When the EOM excitation basis operators (al,a, al,ala a,...] or EOM ionization basis operators [a, a al,a,...] act on the ground-state wave function (either approximate or exact), the many-electron basis wave functions (9/10> that are formed are linear combinations of many determinants. (Since 0> is taken to include correlation effects, it is a linear combination of many determinants.) This is in contrast to the basis of configurations employed in Cl calculations, which are generally symmetry-adapted linear combinations of only a few determinants. Since in large-scale EOM or Cl calculations, it is necessary to truncate the basis set of operators or configurations employed, respectively, the EOM method often includes the effects of many more excited-state or ionized-state configura-... 

In order to determine which harmonics are emitted by nanotubes during the interaction with circularly polarized radiation, it is sufficient to check the invariance of the symmetry-adapted linear combination operators... 

If the CMOs of cyclopropane, stemming from an ab initio SCF calculation, are subjected to a localization routine, one obtains six LMOs A h three banana LMOs Acc- From these one can form the symmetry-adapted linear combinations shown in Figure 18, which belong, under symmetry, to the irreducible representations A, E, A and E". Of these the orbitals 3a and the pair 3e are the FCM orbitals depicted in diagram 75. They can interact with the Ach linear combinations 2a and the pair 2e, respectively, to form the valence-shell CMOs of corresponding symmetry. In contrast the linear combinations la" and le" are uniquely determined by symmetry and identical to the CMOs. The linear combinations la and le of the carbon Is AOs are very much lower in energy than the other ones, and practically identical to the CMOs. 

Determine the symmetry-adapted linear combinations for the molecular orbitals of H2O using the atomic orbitals of HI, H2, and O. (The numbers on the H s are labels for identification purposes only.)... 

Despite the huge increase in computational effort, this direct symmetry-adapted LCAO method was used to study ozone [22], tetrahedral Ni4 [23], and D5h-symmetric ferrocene (Fe(C5H5)2) [24] using molecular orbital (MO) contraction coefficients in the linear-combination-of-Gaussian-type orbital (LCGTO) computer code of [25]. Obviously, symmetry-adapted calculations are important enough to pay an order-TV computational price. The reasons are first, and foremost, that the calculations converge, and second that the wavefunction and one-electron orbitals can be used to address experiment, which typically must first determine the symmetry of the molecule. 

This wave function thus uses different orbitals for different spin (DODS). The total wave function is a determinant with one electron in each of the two SOs. We can express the SOs in terms of the symmetry adapted orbitals CTg and cr . The corresponding wave function will be a linear combination of four determinants. After some algebra we obtain ... 

As we are discussing here, it may be useful to work in terms of symmetry-adapted basis functions which are linear combinations of the raw basis functions in fixed amounts determined by the representations of the molecular point group these combinations are discussed in the next section. 

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