Basis states representations

Using GTO bases, it cannot be expected that the variational representations of the electron waves are snfficiently accnrate far ontside the so-called molecular region , i.e. the rather limited region of space where the potential clearly deviates from the asymptotic Conlomb form. Therefore the phaseshifts of the pwc basis states cannot be obtained from the analysis of their long-range behaviour, as was done in previous works with the STOCOS bases. In the present approach, this analysis may be avoided since the K-matrix techniqne allows to determine, by equation [3] below, the phase-shift difference between the eigenfunctions of Hp and the auxiliary basis functions... 

The next important problem in algebraic theory is the construction of the basis states (the representations) on which the operators X act. A particular role is played by the irreducible representations (Appendix A), which can be labeled by a set of quantum numbers. For each algebra one knows precisely how many quantum numbers there are, and a list is given in Appendix A. The quantum numbers are conveniently arranged in patterns (or tableaux), called Young tableaux. Tensor representations of Lie algebras are characterized by a set of integers... 

In what follows we introduce the model Hamiltonian using both diabatic and adiabatic representations. Adopting diabatic electronic basis states /j ), the molecular model Hamiltonian can be written as [162, 163]... 

In chemical reactions there is an electronic reordering in which some bonds are broken to form new ones. A full description of a chemical process thus requires the understanding of the electronic change involved since it will determine the main forces appearing along the process. Using the electronic states of reactants and products as a diabatic basis set representation, the reactions take place when... 

The essential concept in the definition of the CDF is the use of time-dependent basis states in place of stationary basis states in the representation of the time evolution of a system, with the constraint that both sets of states are orthonormal. Consider a complete set of orthonormal stationary states S and a complete set of orthonormal time-dependent basis states D t) related by the unitary transformation U t) ... 

The success of this extended STIRAP scheme can be traced to the fact that the basis of the subset of dressed eigenstates of the coupled matter-radiation field is a stationary state representation. In this representation, all couplings are already taken into account via the identity of and the locations of the energy levels. The contribution of the background states to the population transfer process is then limited to effects associated with nonresonant coupling to the field, and if these background states are far off resonance such effects are small. 

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). 

The results of this transformation are the p vectors (C). These vectors represent the basis coherences therefore they remain untouched during the exchange. The reaction itself is interpreted in basis coherence representation at this state. The simulation of the next (r + 1th) time slice is started from this point. At the beginning of time slice r + 1, the p vector is converted to the eigenfunction representation again (D) with the only difference that the parameters of the conformer in time slice r + 1 should be used in Equation (66) ... 

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],... 

We now apply these general results to the specific problem of HF. Apart from the Stark effect, we shall otherwise ignore the very small matrix elements which are off-diagonal in J, and evaluate the terms for J = 1, Mj = 0, 1, /H = h = 1 /2. As Weiss pointed out, the total magnetic component Mz = Mj + MF + Mh is a good quantum number and may be used to set up a decoupled representation in which states of different Mz value are diagonalised separately. For J = 1 there are twelve primitive basis states, which we write below in the form MF, Mh, Mj). 

Examples of representations in common use in atomic reaction theory are the coordinate and momentum representations where, if the system under study is a single electron, the basis states are the eigenstates (r and (pi of the position and momentum of the electron respectively. Examples of discrete representations are also important. They will be left until later. 

The use of the function theorem can be seen in conjunction with the representation theorem. We choose the spectral representation of the observable a, that is the representation in which the basis states are the eigenstates (corresponding to the eigenvalue spectrum) of a. 

The ith electron is characterised by its position—spin coordinates. The corresponding basis state in the coordinate—spin representation is abbreviated thus. 

We use the representation theorem for the basis states j) and form the matrix element with the bra vector (i. ... 

Note that the spectral representation of the Green s function has the basis states for two reasons. First, it is necessary to have the small... 

In addition, the number of basis functions required to obtain a satisfactory representation depends on the choice of the harmonic frequency for the basis. Stated in an-... 

Hence, given potentials of all interactions and all basis states ipi, we can, in principle, obtain its representation in terms of creations and annihilations, e.g., in the language of occupation numbers. 

It was shown in [30] that the so,(3) subalgebra of u,(3) can be defined only in the space of completely symmetric irreducible representations (SIRs) of u,(3), the basis states of which have the following form ... 

Construction of a matrix representation of the microscopic Hamiltonian appropriate to the specific structural problem under consideration. This matrix, constructed from a limited number of basis states, is of finite dimension. This is equivalent to treating the effect of all other energetically remote basis states by second- (or higher) order perturbation theory. The effect of these remote levels is introduced, either implicitly or explicitly, upon construction of an effective Hamiltonian He by partitioning the true Hamiltonian H into two parts, and (Section 4.2). 

To illustrate the significance of measurements of internal state branching ratios, we will turn once again to the example of the photodissociation of the hydrogen halides, HX. The fine structure ratio is the branching ratio of X /X populations 2Pi/2/2P3/2- In the non-relativistic adiabatic representation, this branching ratio would be predicted to be zero because the only case (a) basis state which has a non-zero transition moment from the X1E+ state is the 1IIi state which correlates adiabatically with the X(2P3/2) +H(2S) separated atom limit. However, in the more realistic relativistic adiabatic representation, Afl = 0 3E/, 1n3 3ni, and 3ni 3Ei" spin-orbit matrix elements... 

---