Occupied subspace

As a consequence, such examples show that the orthogonality relations (between vectors in different subspaces) alone, do not fix the S subspace. To do so, one would need some previous additional information on the basis which spans S and Sk That is to say, one would need to constrain the set of recovered O s to form a basis of the occupied subspace. This would then make additional orthogonality constraints within the subspace to take into account in the search for a K formula,... 

Our first way of answering the last question will be based on the fundamental theorems on Hilbert space [14], Indeed, the theorem on separability tells us that any subspace of h is also a separable Hilbert space. As a consequence, the inner product defined on, say, the occupied subspace is hermitian irrespectively of the choice of the basis x f (/)], as long as this latter satisfies the fundamental requirements of Quantum Mechanics. One should therefore not have to impose this property as a constraint when counting the number of conditions arising from the constraint CC+ =1 but, on the contrary, can take it for granted. 

A second way of resolving this question is provided by examining the constraint itself. Indeed, the condition CC+ = N is equivalent to requiring the orthonormalization of the basis functions 33(0 of the occupied subspace. That is to say, 33(/) s must satisfy... 

The energy corresponding to the single determinant wave function with the occupied subspace ImP is given by [36,37] ... 

Eq. (40) yields the following equation in the occupied subspace for the new unknowns U ... 

The inclusion of the electronic degrees of freedom in the molecular dynamics step is an important innovation by Car and Parrinello. Since this scheme operates only on the electronic coordinates belonging to the occupied subspace of the electronic eigenvectors, it speeds up the... 

The propagation of wavefunctions expanded in atom centered basis functions needs special care. It is best to use an extrapolated contra-covariant density matrix PS as a projector on to the occupied subspace... 

Consider a general case of the Af-electron systan and the AO basis set / = (Xi,X2>- > Xj. Examine the ground-state configuration defined by the singly occupied subspace V = of N lowest spin MO, with the spatial (MO) parts

Hartree-Fock (HF) or KS SCF calculations [63]. In these MO approximations, the nonadditive Fisher information density in the AO resolution for the ground-state electron configuration. 

This proposal is based on an analysis of E s + E x with the first order correction E in the SA perturbation theories (see Eqs. (5) and (9)). The double use of the occupied subspaces is accepted here, of course. Using the terminology employed here for this proposal, our 3 correction is split into two components which we could call 3 and 3 ... 

These considerations call our attention that individual molecular orbitals have no direct physical significance in a many-electron system. It is merely the occupied subspace which has physical meaning in the Hartree-Fock theory. Any transformation among occupied MOs is permitted without affecting the validity of the Brillouin condition, the value of the ground-state energy, or that of any physical observable obtained as an expectation value by the Hartree-Fock wave function. 

The occupied-occupied block of the fock matrix, fy, is known. The POOs are orthogonal to the occupied subspace of the original basis set, they satisfy the local Brillouin theorem, i.e., the occnpied-virtual block is zero. As a resnlt, in the local excitation approximation, we need only to deal with the fock matrix elements f p. 

In a matrix representation c = Rc should be a column of coefficients representing a function of the occupied subspace (i.e. one that can be expressed as a linear combination of the occupied MOs). If c represents an arbitrary function in the space of the m AOs, it may certainly be written as a linear combination of the columns Ca, Cb,. .., representing a full set of MOs—including the n occupied MOs and any furffier m-n independent orthonormal functions, which we may refer to as the unoccupied MOs ... 

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