Uncoupled basis set

The picture here is of uncoupled Gaussian functions roaming over the PES, driven by classical mechanics. The coefficients then add the quantum mechanics, building up the nuclear wavepacket from the Gaussian basis set. This makes the treatment of non-adiabatic effects simple, as the coefficients are driven by the Hamiltonian matrices, and these elements couple basis functions on different surfaces, allowing hansfer of population between the states. As a variational principle was used to derive these equations, the coefficients describe the time dependence of the wavepacket as accurately as possible using the given... 

As is apparent from the above definitions, each of these effective matrices depend on basis sets and molecular orbitals of both fragments. It is also important to observe that these matrices possess a correct asymptotic behavior as at large interfragment distances they become the usual overlap and Fock matrices of the separate fragments, while the paired secular systems uncouple and converge to the separate Roothaan equations for the single monomers. Finally, as it is usual in a supermolecular approach, the interaction energy is expressed as... 

The main conclusion of this section is that the matrix elements of all terms in the collision Hamiltonian in the fully uncoupled space-fixed representation can be reduced to simple products of integrals of the type (8.46). Such matrix elements are very easy to evaluate numerically. The fiilly uncoupled representation is therefore very convenient for the development of the coupled channel codes for collision problems involving open-shell molecules with many angular momenta that need to be accounted for. The price for simplicity is a very large number of basis states that need to be included in the expansion of the eigenstates of the full Hamiltonian to achieve full basis set convergence (see Section 8.3.4). 

In order to appreciate the size of the basis sets required for fully converged calculations, consider the interaction of the simplest radical, a molecule in a electronic state, with He. The helium atom, being structureless, does not contribute any angular momentum states to the coupled channel basis. If the molecule is treated as a rigid rotor and the hyperfine structure of the molecule is ignored, the uncoupled basis for the collision problem is comprised of the direct products NMf ) SMg) lnii), where N = is the quantum number... 

Several papers have dealt with the evaluation of wave functions including correlation in various ways. Bimstock34 has calculated the 13C shielding constants in CH4 and several other small molecules using an approximate form of uncoupled Hartree-Fock theory and the minimal basis set wave functions of Palke and Lipscomb.35 The results were similar to those obtained earlier by Ditchfield et al.33... 

Calculations of Higher Polarizabilities and Semi-empirical Calculations.—O Hare and Hurst77 have calculated the first hyperpolarizabilities of some first-row diatomics by the uncoupled method. They find that a, and to a greater extent 0, are very sensitive to the basis set. Table 8 shows the effect of basis set on a for LiCl. Gupta et al.7 have... 

Note that we are using here a fully uncoupled treatment of the eleetronie angular momentum of the X atom [44] (case 2A in the notation of Dubemet and Hutson [45]). By contrast, in the work of Schatz and eo-workers [36, 37] on the Cl+HCl exchange reaction, a partially-coupled basis set is used in which the total angular momentum... 

A simple choice for a complete basis set of functions for obtaining a matrix representation F is the uncoupled set... 

The initial vector v must also be evaluated. Instead of computing directly the vector representation of the given rotational function (i.e., the spherical harmonic in ftj, which is an element of the uncoupled basis set), one can evaluate the matrix representation of the function, which we call M, and then multiply it by the vector representation of which we call Vq, whose calculation is relatively easy utilizing the coupled basis set (see Appendix C.2). That is, let... 

In contrast, 3-j coeficients (e.g. Section 3.2.1.4) are useful in transforming between coupled abccz) and uncoupled aazbbz) basis sets. 

Alternatively, the matrix elements of the Zeeman operator are included in the spin Hamiltonian on the original basis set (of uncoupled angular momenta kets)... 

We have assumed that no angular momentum contribution assists. Then the basis set of spin functions consists of the uncoupled set Sa,Msa) Sb, Msb), or the coupled set SA,SB,S,Ms), its size is N = (2Sa + l)(2Ss + 1). Additionally, the orbital angular momentum can be added and then the basis set becomes a direct product of all orbital and spin functions. In a special case, spin delocalisation (double exchange) operates. 

The basis set consists of the kets assigned to the uncoupled microstates it transforms into the set of uncoupled atomic terms first, and then into the set of molecular eigenfunctions... 

As a first strategy, all matrix elements are evaluated in the basis set of uncoupled spin functions, i.e. 

The expressions for the matrix elements of the exchange interaction operators in the uncoupled basis set are collected in Table 11.1. 

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