Symmetry-adapted linear combinations calculations

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-... 

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. 

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. 

These same programs can also be used to transform any localized atomic basis into a symmetry-adapted linear-combination-of-atomic-orbitals (LCAO) basis. For methods that begin with an atomic basis set and include multi-center overlap, however, symmetry adaption can actually increase the complexity of the calculation as will be shown. Nevertheless, even with localized basis functions, symmetry can be used to significantly reduce the cost of the calculation if it is carefully applied. How to do so is the subject of the next section. On the other hand, sometimes one chooses not to symmetrize 14c to approximate real physical effects that cannot yet be treated exactly within density functional theory. 

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-... 

Now to the problem in hand the use of symmetry-adapted linear combinations of the basis functions in the SCF procedure. The method is now obvious, it is simply one more additional step in the chain of transformations from the raw basis to the actual working basis. Deferring the actual technique of calculation of the elements of the transformation between the raw basis and the symmetry-adapted basis until later in this chapter, we assume that the matrix U effects this transformation. The matrix is actually unitary or, more commonly in the real case, orthogonal so that, unlike the monomial/harmonic transformation above, it does not disturb the orthogonality properties of the basis and it is, of course, square. 

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. 

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 electronic charge density in an MO extends over the whole molecule, or at least over a volume containing two or more atoms, and therefore the MOs must form bases for the symmetry point group of the molecule. Useful deductions about bonding can often be made without doing any quantum chemical calculations at all by finding these symmetry-adapted MOs expressed as linear combinations of AOs (the LCAO approximation). So we seek the LCAO MOs... 

From these frequencies and with the help of the corresponding G-Matrix elements (Wilson et al., 1955), the symmetry-adapted force constants (F) can be calculated directly. In the vibrations discussed here, F is a linear combination of stretching and interaction force constants / and f,r . 

Force constant calculations are facilitated by applying symmetry concepts. Group theory is used to find the appropriate linear combination of internal coordinates to symmetry-adapted coordinates (symmetry coordinates). Based on these coordinates, the G matrix and the F matrix are factorized, which makes it possible to carry out separate calculations for each irreducible representation (c.f. Secs. 2.133 and 5.2). The main problem in calculating force constants is the choice of the potential function. Up until now, it has not been possible to apply a potential function in which the number of force constants corresponds to the number of frequencies. The number of remaining constants is only identical with the number of internal coordinates (simple valence force field SVFF) if the interaction force constants are neglected. If this force field is applied to symmetric molecules, there are often more frequencies than force constants. However, the values are not the same in different irreducible representations, a fact which demonstrates the deficiencies of this force field (Becher, 1968). 

In calculations based on the MO-LCAO technique [32-34], the one-electron Kohn-Sham equations Eq. (11) are solved by expanding the molecular orbital wavefunctions V i(r) in a set of symmetry adapted functions Xj(r), which are expanded as a linear combination of atomic orbitals i.e. 

By forming suitable linear combinations ot basts tunctions ( symmetry adapted Tunctions), many one- and iwo-electron integrals need not be calculated as tliey ai e... 

At R=5.14 A also in RHF a symmetry breaking occurs where on one site a charge of e and one of -e on the other occurs. Linear combination of the two possible solutions of this kind leads to the symmetry adapted HF wave function. As calculations up to R= 100 A show this solution converges also to -23 eV. The... 

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. 

If the group is rotational or helical and ij> is not 5-type, then the />, on each site become linear combinations of basis functions related by the rotation matrix of the appropriate angular momentum and the appropriate rotational or helical step angle [27]. It is traditional to use Cartesian-Gaussian orbital basis sets in quantum-chemical calculations [28], but solid-spherical-harmonic Gaussians [29] are best for symmetry adaption and matrix element evaluation. Including an extra factor of (-)M in the definition of the solid spherical harmonics [30]... 

Studies of this type are frequently performed to investigate configurational variations, e.g. at metal centres. The obvious initial parameters are the relevant valence angles L-M-L, where L represents a ligating atom. Such studies may then be broken down chemically according to the nature of L. However, it is often more informative to study the deviations of observed coordination geometries from some idealized symmetric form [7, 8, 9], as described in Chapter 2. This requires use of symmetry-adapted deformation coordinates which can readily be calculated, using the CSD System, for instance, as simple linear combinations of standard internal coordinates. 

As a preliminary step in finding the MOs of a molecule, it is helpful (but not essential) to construct linear combinations of the original basis AOs such that each linear combination does transform according to one of the molecular symmetry species. Such linear combinations are called symmetry oihitals or symmetry-adapted basis functions. The symmetry orbitals are used as the basis functions Xs n the expansions i = c iXs [Eq. (13.156)] of the MOs The use of basis functions that transform according to the molecular symmetry species simplifies the calculation by putting the secular determiruint in block-diagonal form this will be illustrated below. 

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. 

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. 

---