The LCAO Expansion

The LCAO expansion coefficients are found from the matrix eigenvalue problem... 

I means the valence state ionization potential for the atomic orbital n, stands for the core charge, and Cj, and Xm are coefficients and atomic orbitals in the LCAO expansion... 

Hence, through the LCAO expansion we have translated the non-linear optimization problem, which required a set of difficult to tackle coupled integro-differential equations, into a linear one, which can be expressed in the language of standard linear algebra and can easily be coded into efficient computer programs. 

Mulliken s formula for Nk implies the half-and-half (50/50) partitioning of all overlap populations among the centers k,l,... involved. On one hand, this distribution is perhaps arbitrary, which invites alternative modes of handling overlap populations. On the other hand, Mayer s analysis [172,173] vindicates Mulliken s procedure. So we may suggest a nuance in the interpretation [44] departures from the usual halving of overlap terms could be regarded as ad hoc corrections for an imbalance of the basis sets used for different atoms. But one way or another, the outcome is the same. It is clear that the partitioning problem should not be discussed without explicit reference to the bases that are used in the LCAO expansions. 

Which properties are least well determined by the variational method The basis functions in the LCAO expansion are either Slater orbitals with an exponential factor e r or gaussians, e ar2 r appears explicitly only as a denominator in the SCF equations thus matrix elements are of the form < fc/r 0i> these have the largest values as r->0. Thus the parts of the wavefunction closest to the nuclei are the best determined, and the largest errors are in the outer regions. This corresponds to the physical observation that the inner-shell orbitals contribute most to the molecular energy. It is unfortunate in this respect that the bonding properties depend on the outer shells. 

Schmidtke et al. [30] presented a simplified model of the K integrals, which retains the Stevens delocalization coefficients featuring in the LCAO expansion of the metal d-orbitals over the ligands but does not allow for differences in the radial parts of the metal function. More-center and ligand-centered contributions are neglected. In this approximation the K integrals can be expressed as follows ... 

N+/is the number of unit cells of the polymer chain. The LCAO expansion coefficient matrices C(k) and the corresponding energy eigenvalues, e(k), are obtained by solving the eigenvalue problem for each value of k-... 

Inserting the LCAO expansions for %x and of the form of (10) in (11), the matrix element of the angular momentum operators, and hence the matrix element of hso on the left hand side of (11) is easily seen to be a linear combination of terms of the form... 

The energy expression (14) is a function of the molecular orbitals, appearing in the one- and two-electron integrals, and of the Cl expansion coefficients through the first- and second-order reduced density matrices (15). In an MCSCF optimization procedure the Cl coefficients and the parameters determining the MOs (normally the LCAO expansion coefficients) are varied until the energy reaches a stationary value. A number of procedures for performing the optimization have been described in the literature (see Ref. 26 for an extensive review). Here only the basic features of these procedures will be outlined, and the reader is referred to the literature for further details . ... 

It is obvious by symmetry that the coefficients are related ca = cb, a = /b and Ca = =teB, but what about the ratios of ca to a to epP. I ll just mention for now that there is a systematic procedure called the Hartree-Fock self consistent field method for solving this problem. In the special case of the hydrogen molecular ion, which only has a single electron, we can calculate the variational integral and find the LCAO expansion coefficients by requiring that the variational integral is a minimum. Dickinson (1933) first did this calculation using Is and 2porbital exponents to be is = 1.246 and 2pa = 2.965 (See Table 3.2.)... 

For a basis set of atomic (a) orbitals which are nodeless in the plane perpendicular to the radial vector (see Fig. 16a), the required Harmonics are simply the Scalar Spherical Harmonics (YLM(0, cj))146. The LCAO expansion, therefore, takes the following form ... 

The zero subscripts in Eqs (5.127) and (5.128) emphasize that the initial-guess c s, with no iterative refinement, were used to calculate G in the subsequent iterations of the SCF procedure H will remain constant while G will be refined as the c s, and thus the P s, change from SCF cycle to cycle. The change in the electron repulsion matrix G corresponds to that in the molecular wavefiinction as the c s change (recall the LCAO expansion) it is the wavefunction (squared) which represents the time-averaged electron distribution and thus the electron/charge cloud repulsion (sections 5.2.3.2,... 

The Cl strict analogue of the iterative refinement of the coefficients that we saw in HF calculations (section 5.2.3.6e) would refine just the weighting factors of the determinants (the c s of Eqs (5.168), but in the MCSCF version of Cl the spatial MOs within the determinants are also optimized (by optimizing the c s of the LCAO expansion, Eq. (5.164)). A widely-used version ofthe MCSCF method is the CASSCF method, in which one carefully chooses the orbitals to be used in forming the various Cl determinants. These active orbitals, which constitute the active space, are the MOs that... 

Expressions (7)-(10) are given in a molecular-orbital basis. If we adopt the LCAO expansions of and bg, then we obtain similar types of expressions in an atomic-orbital basis. In particular, (9) then gives the familiar charge-bond-order matrix of Hiickel theory. 

It is, however, natural to ask if these orbitals have any meaning do they have any physical interpretation for example, are they approximations to any orbitals associated with the system under investigation or are they just an artifact of the use of the LCAO technique In the case of the orbitals of the single determinant — the occupied orbitals — the LCAO expansion is a more or less good approximation to the solutions of the differential Hartree-Fock equation depending on the quality and length of the AO expansion. As the quality and size of the basis is improved, the occupied MOs presumably become better and better approximations to these orbitals. What about the unoccupied molecular orbitals ... 

As we noted much earlier in Chapter 3, making the LCAO expansion independently of the linear variation method is usually invalid in the sense that the operation of an arbitrary differential operator (h, say) on a linear expansion wiU... 

Here Er stands for the term which can be called the "Roothaan energy because its form in the LCAO expansion... 

Here as a reminder, / indexes tlie molecular orbital level while v, and k index the /Lith, i ih, and Kth atomic orbitals, respectively, in the LCAO expansion of equation 1.9. Equation 1.28 is satisfied for k = 1, 2, , m and the explicit form of these m equations is... 

The preceding equilibrium of alkylallenes shows a formal analogy with the initial reaction slep of an acidic hydrolysis of saturated carboxylic acid esters R (RO)C—O or protonation reactions of carbonyl compounds RiR2C=0. This similarity concerns the position of the attack of the proton relative to the positions of the substituents and the kinds of atoms involved in that reactions. In both the cases (C and O) the attacked atoms have perpendicular p AOs in the LCAO expansions of their two outermost occupied orbitals. 

If the LCAO expansion is introduced to the expression for flie total energy, Eq. (8.43) gives... 

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