Orthonormal states

Following the well-established coupled-channel theory [17], we represent the eigenstates of the full Hamiltonian by an expansion in terms of some orthonormal states. 

Fig. 1. Double interacting chains of orthonormal states, with definite values ofy, representing the Renner-Teller system for orbital p-like functions. N represents the total number of phonons.

Once we have obtained y/0 and y/t, we can construct a set of orthonormal states y/2,y/3.in succession defined by the following relation... 

The projection operator in the manifold of the orthonormal states is given by... 

Consider an operator ff (we suppose here it is Hermitian) and a normalized seed state /o). We construct a hierarchy of orthonormal states according to the recursion scheme ... 

The problem is to find I codewords yi) which meet the conditions (15a) and (15b). To this end, we employ an iterative method. First, we randomly pick a set of I orthonormal state vectors which we take as the starting point. Then we repeatedly optimize a conveniently chosen functional which shows us the direction to follow at each step in the (I x IV) -dimensional space of parameters (coordinates of the I orthonormal vectors), in order to get arbitrarily close to the desired solution. 

Let us, for the sake of argument, assume now that only two surfaces 1 and 2 approach each other and that all other surfaces are well separated the argument is easily generalized to more surfaces. We then have to solve a set of two coupled nuclear Schrodinger equations with non-negligible coupling element i(r R)l7 nl 2(r R))(r)-Define two new orthonormal states by a rotation of xi and xi (for clarity reasons we suppress the coordinates)... 

III Effective two-particle description with orthonormal states... 

III. EFFECTIVE TWO-PARTICLE DESCRIPTION WITH ORTHONORMAL STATES... 

This equation expresses the fact that the extended Green s function Q u>) is the projection of an operator resolvent onto a set of )U-orthonormal states [10]. Note that the matrix is hermitian if the Hamiltonian H of the many-body system is hermitian (which is assumed throughout this paper). By matrix partitioning we can write for the inverse of the Green s function... 

The dynamic part DJ E), which can be found in the literature [8], possesses poles related to resonances of two-particle-hole and higher excitations and a branch cut above the ionisation threshold. It has an analytic structure similar to the dynamic self energy (42) from the formal point of view. This is not surprising since the projection method for deriving the Feshbach Hamiltonian is formally similar to the derivation of Dyson s equation of Sec. Ill or in Refs. [26,9]. There, a partitioning was performed yielding a projection to the primary space spanned by the orthonormal T-states while Feshbach, on the other hand, projected onto the space spanned by the non-orthonormal states aj 5 ). 

The model describes n quasi-bound states interacting with m quasi-continua. The quasi-bound states i) i = 1,1,. .., n) span the inner space whose projector is P. The orthonormalized states ka) (a = 1, 2,. .., m) span the complementary outer space whose projector is... 

In the following it is assumed that H and H are Hermitian. Let us suppose that H is acting in a basis of orthonormalized states denoted />. The simplest definition of the distance between and H is... 

The most straightforward numerical technique for the solution of Eq. (1) is based on the expansion of the state vector (t)) in a complete set of time-independent basis functions. Such a complete basis can be constructed as the direct product of diabatic electronic basis states l n) and suitable orthonormal states xiyj) for each nuclear degree of freedom (see Chapter 7)... 

The first three states of the method of moments [8] characterize the space of the three relevant quasi-bound states (the model space). Since these states are non-orthogonal, the Gram-Schmidt procedure applied to 1), H l) and provide the orthonormalized states i), i — 1,2,3). In this basis the matrix representation of the exact energy-dependent Hamiltonian (13) may be written as... 

We assume that we have a set of states P) that constitutes an orthonormal basis in the configuration space of dimension m, spanned, for example, by a set of m Slater determinants. From the states P), we may generate a new set of orthonormal states P) by a unitary transformation... 

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