Orthonormalization space model

The eigenvalue problem of the Hamiltonian operator (1) is defined in an infinite-dimensional Hilbert space Q and may be solved directly only for very few simple models. In order to find its bound-state solutions with energies not too distant from the ground-state it is reduced to the corresponding eigenvalue problem of a matrix representing H in a properly constructed finite-dimensional model space, a subspace of Q. Usually the model space is chosen to be spanned by TV-electron antisymmetrized and spin-adapted products of orthonormal spinorbitals. In such a case it is known as the full configuration interaction (FCI) space [8, 15]. The model space Hk N, K, S, M) may be defined as the antisymmetric part of the TV-fold tensorial product of a one-electron space... 

The matrix defined in Eq.(47) represents a general, non-relativistic spin-independent TV-electron Hamiltonian given by Eq.(2) in a specifically defined model space. The one-electron orbital space is spanned by N orthonormal localized orbitals 4>j, j = 1,2,..., TV and the TV-electron orbital space is one-dimensional with the basis function... 

There is a convenient interpretation of the Tucker model in terms of a new basis to express a three-way array. Consider the SVD of X X = USV. The matrix V is an orthonormal basis for the row-space of X. X can be expressed on this new basis by orthogonally projecting X on this basis the new coordinates are found by regressing X on V ... 

Scatter plots in PCA have special properties because the scores are plotted on the base P, and the columns of P are orthonormal vectors. Hence, the scores in PCA are plotted on an orthonormal base. This means that Euclidean distances in the space of the original variables, apart from the projection step, are kept intact going to the scores in PCA. Stated otherwise, distances between two points in a score plot can be understood in terms of Euclidian distances in the space of the original variables. This is not the case for score plots in PARAFAC and Tucker models, because they are usually not expressed on an orthonormal base. This issue was studied by Kiers [2000], together with problems of differences in horizontal and vertical scales. The basic conclusion is that careful consideration should be given to the interpretation of scatter plots. This is illustrated in Example 8.3. 

It is useful to go back to two-way visualization in principal component analysis to find what really is seen in a plot. A score plot for a two-way PCA model has an orthonormal basis, because the loadings are orthonormal. This can be compared to projecting all points in multidimensional space on a movie screen using a strong light source at a large distance. What is seen in this projection is true Euclidean distance in the reduced space, if both... 

We will now define 5° as the model space generated by the zero-order description of the states 1 ). These states are denoted n)° and are eigenvectors of the H° operator. The basic functions n)°, n = 1, N form an orthonormal basis set. We will now define... 

The abstract formalism introduced in this chapter builds the fundament of the theory of extended two-particle Green s functions. Our approach is very general in order to allow for a unified treatment of the different species of extended Green s functions discussed in the main part of this paper. Since the discussed propagators can be applied to a wide variety of physical situations, the emphasis of this chapter lies on the unifying mathematical structure. The formalism is developed simultaneously for (projectile) particles of fermionic and bosonic character. We will define the general extended states which serve to define the primary or model space of the extended Green s functions. We also define the /.j-product under which the previously defined extended states fulfil peculiar orthonormality conditions. Finally we introduce a canonical extension of common Fock-space operators and super-operators to the space of the extended states. 

The set of states Vr,), where the indices r and s are both taken from the full set of single-particle indices, will be called primary set of states which spans the so-called model space. For each of the three choices this set of states is p-orthonormal (in the sense of Sec. IIB) since the following orthonormality relations hold ... 

We will now derive a Dyson equation by expressing the inverse matrix of the extended two-particle Green s function Qr,y, u ) by a matrix representation of the extended operator H. We already mentioned that the primary set of states l rs) spans a subspace (the model spaice) of the Hilbert space Y. Since the states IVrs) are /r-orthonormal they are also linearly independent and thus form a basis of this subspace. Here and in the following the set of pairs of singleparticle indices (r, s) has to be restricted to r > s for the pp and hh cases (b) and (c) where the states are antisymmetric under permutation of r and s. No restriction applies in the ph case (a). The primary set of states Yr ) can now be extended to a complete basis Qj D Yr ) of the Hilbert space Y. We may further demand that the states Qj) are /r-orthonormal ... 

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... 

The first application of the effective Liouvillians [9,65] models the regression of a fluctuation. We consider one observable A and its flux A. It is convenient to introduce an orthonormal base in the Liouville space of the operators A and A ... 

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... 

Orthonormalized projection of on a model space Biorthogonal projection of on a model space Normalized biorthogonal projection of on a model space... 

Select the N eigenfunctions of the full Hilbert space (e.g. obtained in an ab initio calculation) with the largest projection onto the model space. (Bi-)orthonormalize the projections of these vectors and take the total energy of one of the roots as zero of energy. 

Bloch-equation-based multistate PT formulations, termed quasidegenerate PT (QDPT) [22], largely assume an orthonormal set of vectors in the configuration interaction (Cl) space that is partitioned for a model space and its complement. This restricts applicability to model spaces easily separable from the rest, e.g., formed by simple determinants. While determinants facilitate a transparent derivation of many-body QDPT formulae [23, 24], identifying the determinants that need to be included in the model space is not always trivial. Though complete active space (CAS) appears a simple way out, CAS-based QDPT is unfortunately prone to the so-called intruder problem, especially for large active spaces. 

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