Basis functions, construction

The basis functions constructed in this manner automatically satisfy the necessary boundary conditions for a magnetic cell. They are orthonormal in virtue of being eigenfunctions of the Hermitian operator Ho, therefore the overlapping integrals(6) take on the form... 

Using these tr-orbitals as basis functions, construct a seven-dimensional representation of the point group. 

This result has the following consequence for the completeness of basis function constructed from the eigenvectors obtained by stability analysis of external flows. It has been clearly shown by Mack (1976) that internal flows, like the channel flow, has denumerable infinite number of eigen modes and any arbitrary applied disturbance can be expressed in terms of this complete basis set. However, for external flows, as we have seen for the Blasius flow in Table 2.1 that there are only a few discrete eigenmodes and it is not possible to express any arbitrary functions in terms of these only, in the absence of any other singularities for this flow. 

Figure 2.5 (a) Adaptive FDDI reconstruction of phantom square jet data (6) adaptive FDDI reconstruction utilizing the new global basis function in (c), notice that while the structure on top of the peak has become more complex, the sides more closely approximate that of a square top-hat and (c) normalized glob[Pg.16]

The linear methods devised by Andersen [1.19] are characterised by using fixed basis functions constructed from partial waves and their first energy derivatives obtained within the muffin-tin approximation to the potential. 

In SCF problems, there are some cases where the wave function must have a lower symmetry than the molecule. This is due to the way that the wave function is constructed from orbitals and basis functions. For example, the carbon monoxide molecule might be computed with a wave function of 41 symmetry even though the molecule has a C-xt symmetry. This is because the orbitals obey C41 constraints. 

It is a well-known fact that the Hartree-Fock model does not describe bond dissociation correctly. For example, the H2 molecule will dissociate to an H+ and an atom rather than two H atoms as the bond length is increased. Other methods will dissociate to the correct products however, the difference in energy between the molecule and its dissociated parts will not be correct. There are several different reasons for these problems size-consistency, size-extensivity, wave function construction, and basis set superposition error. 

LCAO (linear combination of atomic orbitals) refers to construction of a wave function from atomic basis functions LDA (local density approximation) approximation used in some of the more approximate DFT methods... 

The basis of constructing the MOs is the linear combination of atomic orbitals (LCAO) method. This takes account of the fact that, in the region close to a nucleus, the MO wave function resembles an AO wave function for the atom of which the nucleus is a part. It is reasonable, then, to express an MO wave function 1/ as a linear combination of AO wave functions Xi on both nuclei ... 

The columns to the right of the first vertical line of asterisks hold the exponents (a above) and the coefficients (the d p s) for each primitive gaussian. For example, basis function 1, an s function, is a linear combination of six primitives, constructed with the exponents and coefficients (the latter are in the column labeled S-COEF ) listed in the table. Basis function 2 is another s function, comprised of three primitives using the exponents and S-COEF coefficients from the section of the table corresponding to functions 2-5. Basis function 3 is a p function also made up or three primitives constructed from the exponents and P-COEF coefficients in the same section of the table ... 

Basis functions 4 and 5 are Py and p functions constructed in the same manner using the same exponents and coefficients. 

The coefficients specified for the component primitive gaussians are chosen so that the resulting constructed basis functions are normalized. This means that one coefficient in each set is effectively constrained so that this condition is fulfilled. 

The phrase symmetry adapted basis functions refers to those linear combinations of basis functions (on several atoms) that transform like the particular irreducible representation of the appropriate point group. Molecular symmetry is used at various points in these calculations twenty years ago I would have had to write several chapters on molecular symmetry, point groups, constructing symmetry-adapted combinations of basis functions, factoring a Hamiltonian matrix using symmetry and related topics. The point is that twenty... 

To construct the Fock matrix, eq. (3.51), integrals over all pairs of basis functions and the one-electron operator h are needed. For M basis functions there are of the order of of such one-electron integrals. These one-integrals are also known as core integrals, they describe the interaction of an electron with the whole frame of bare nuclei. The second part of the Fock matrix involves integrals over four basis functions and the g two-electron operator. There are of the order of of these two-electron integrals. In conventional HF methods the two-electron integrals are calculated and saved before the... 

How are the additional determinants beyond the HF constructed With N electrons and M basis functions, solution of the Roothaan-Hall equations for the RHF case will yield N/2 occupied MOs and M — N/2 unoccupied (virtual) MOs. Except for a minimum basis, there will always be more virtual than occupied MOs. A Slater detemfinant is determined by N/2 spatial MOs multiplied by two spin functions to yield N spinorbitals. By replacing MOs which are occupied in the HF determinant by MOs which are unoccupied, a whole series of determinants may be generated. These can be denoted according to how many occupied HF MOs have been replaced by unoccupied MOs, i.e. Slater determinants which are singly, doubly, triply, quadruply etc. excited relative to the HF determinant, up to a maximum of N excited electrons. These... 

In addition, it is possible that differences in the BSR may arise due to choice of the s-basis from which the higher angular momentum basis functions are constructed. However, on comparison of the Bethe sum rule for basis C, with the sum rule generated by the same method described above, from another energy optimized, frequently used basis (that of van Duijneveldt [16]), we find only small differences in the BSR, and only at large q values. 

Molecules and Clusters. The local nature of the effective Hamiltonian in the LDF equations makes it possible to solve the LDF equations for molecular systems by a numerical LCAO approach (16,17). In this approach (17), the atomic basis functions are constructed numerically for free atoms and ions and tabulated on a numerical grid. By construction, the molecular basis becomes exact as the system dissociates into its atoms. The effective potential is given on the same numerical grid as the basis functions. The matrix elements of the effective LDF Hamiltonian in the atomic basis are given by... 

The conclusion is that for every particular set of basis functions and given data, there exists an appropriate size of G that can approximate both accurately and smoothly this data set. A decisive advantage would be if there existed a set of basis functions, which could probably represent any data set or function with minimal complexity (as measured by the number of basis functions for given accuracy). It is, however, straightforward to construct different examples that acquire minimal representations with respect to different types of basis functions. Each basis function for itself is the most obvious positive example. A Gaussian (or discrete points... 

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