Finite basis functions completeness

As an example, one of more well-known constraints on the basis functions is the so-called kinetic balance condition [60, 61]. Specifically, most of the finite basis functions do not form complete basis sets in the Hilbert space. If the large- and small-component radial wave functions are expanded in terms of one of these orthonormal basis sets ip such that P r) = and Q r) = Ylj then the operator identity (cr p) [Pg.168]

When the wave function is completely general and pennitted to vary in the entire Hilbert space the TDVP yields the time-dependent Schrodinger equation. However, when the possible wave function variations are in some way constrained, such as is the case for a wave function restricted to a particular functional form and represented in a finite basis, then the corresponding action generates a set of equations that approximate the time-dependent Schrodinger equation. 

In developing perturbation theory it was assumed that the solutions to the unpermrbed problem formed a complete set. This is general means that there must be an infinite number of functions, which is impossible in actual calculations. The lowest energy solution to the unperturbed problem is the HF wave function, additional higher energy solutions are excited Slater determinants, analogously to the Cl method. When a finite basis set is employed it is only possible to generate a finite number of excited determinants. The expansion of the many-electron wave function is therefore truncated. 

To find the true Hartree-Fock orbitals, one must use a complete set in (1.295), which means using an infinite number of gk s. As a practical matter, one must use a finite number of basis functions, so that one gets approximations to the Hartree-Fock orbitals. However, with a well-chosen basis set, one can approach the true Hartree-Fock orbitals and energy quite closely with a not unreasonably large number of basis functions. Any MOs (or AOs) found by iterative solution of the Hartree-Fock-Roothaan equations are called self-consistent-field (SCF) orbitals, whether or not the basis set is large enough to give near-Hartree-Fock accuracy. 

Use of a complete (necessarily infinite) set of basis orbitals in the expansion for the molecular orbitals would insure absolute convergence to the Hartree-Fock limit. In practice, this is both impossible and unnecessary, and only a finite number of basis functions is employed in the expansion. The selection of the finite basis set is, therefore, of crucial importance in determining how closely one approximates the true Hartree-Fock solution. 

Because computers can represent numbers, but not functions, the molecular orbitals at each stage of the SCF procedure have to be represented by an expansion in a finite set of basis functions < >j(r), i = 1,2,... N. If the set is mathematically complete, the result of the SCF procedure is termed the HF or KS limit otherwise the result is dependent on the basis set used. Many types of basis funtion have been explored, and several are currently used in routine applications. However, their interrelationships and relative strengths and weaknesses are not often clarified and it may be instructive to do so here. 

The condition 0 < w < 1 is necessary, though not sufficient, for the periodic surface to be free of self-intersections. The function w (m, u) must be found to complete the characterization of the surface. Traditionally one solves the partial differential equation Vj-n = — 2H, which is a second-order p.d.e. for w (u,v). A finite-difference or finite-element solution for w u,v) is sought using basis functions (j)j[u, v) and finding coefficients such that... 

Three known facts are essentially important in the development of a divide-and-conquer strategy. First, the KS Hamiltonian is a single particle operator that depends only on the total density, not on individual orbitals. This enables one to project the energy density in real space in the same manner in which one projects the density (see below). Second, any complete basis set can solve the KS equation exactly no matter where the centers of the basis functions are. Thus, one has the freedom to select the centers. It is well known that for a finite basis set the basis functions can be tailored to better represent wavefunctions, and thus the density, of a particular region. The inclusion of basis functions at the midpoint of a chemical bond is the best known example. Finally, the atomic centered basis functions used in almost all quantum chemistry computations decay exponentially. Hence both the density and the energy density contributed by atomic centered basis functions also decrease rapidly. All these... 

We assume for the moment that we know the true wave function (if the spin-orbital basis is complete) or the best variational function in the given finite basis in the form of Eq. (48). Then it holds that... 

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