Orthonormality radial functions

The radial solutions are here distinguished by two quantum numbers - recognizing that, for each / > 0, we may solve (6.3.11) and obtain a complete set of orthonormal radial functions rR i(r) in L2(R+)... 

Thus, the operator Hi is hermitian and the radial functions REi(r) constitute an orthonormal set with a weighting function w(r) equal to r ... 

Coefficients fk and gk are defined by (19.73) and (20.30), respectively, whereas Nxt denotes the number of electrons in subshell Quantities e, proportional to Lagrange multipliers, are in charge of orthonormality (2.17) of the radial functions. 

Inserting (11.6) into the time-independent Schrodinger equation and utilizing the orthonormality of the expansion functions leads to the following set of coupled equations for the radial functions Xjn(R JMp)J... 

Note that the radial functions Rn i also form an orthonormal set of functions ... 

It is to be expected that the orthonormal function formed by applying equation 3.10, should be a good fit, when normalized, to the numerical radial function obtained by direct solution of the Schrodinger equation for the lithium atom. This proposition is examined in fig3-3.xls, in which the chart, displayed in Figure 3.3, incorporates the Herman-Skillman... 

Figure 3.8 The two worksheets in the spreadsheet, fig3-8.xls, for the calculation of the orthonormal double-zeta Slater radial function of the lithium 2s orbital. The comparison graphs in this figure show the possible starting situation, with both C coefficients equal 1 and the C2 equal 0. In both cases, changing the value of cj in cell SCSI 1 leads to the best fit results shown in the next diagram.

Complete the construction of the orthonormal 2s radial function, by calculating the overlap integral between the two double-zeta Slater functions and apply equation 3.10. So, using the projections of the Is and 2s functions on the radial array calculate their overlap integral in G 17 of dz2s , with... 

Table 3.1 The coefficients of the Slater functions for the Is and 2s radial functions in lithium after 16 cycies of Mitroy s scf program (64) to convergence. The best fit overall coefficients based on the orthonormal functions are in the last two rows. ...

Figure 3.17 Symmetric orthonormalization of the sto-3g) basis sets for hydrogen Is and 2s radial functions and the comparisons with the exact radial functions from solution of the Schrodinger equation.

Figure 4.20b Comparisons of the exact radial projections of the hydrogen Is and 2s radial functions with those of the two linear combinations resulting from the calculations in fig4-20ab.xls. The second linear combination is a primitive approximation to the hydrogen 2s radial function to the extent that it is orthonormal to the Is function [cells canonical K 24 to SM 24].

The orthonormality restriction for the spinors, Eq. (8.108), may be split into two parts in the case of atoms. The product ansatz for the spinor automatically yields orthonormal angular parts (coupled spherical harmonics cf. chapter 9). But these do not contain information about the principal quantum numbers in the composite indices i and j. For this reason, the restriction to orthonormal spinors results in the orthonormality restriction for radial functions... 

This holds true only if Kj = Kj since the spin and angular parts guarantee orthogonality in all other cases. If one wants to calculate excited electronic states using the same set of radial functions optimized for both the ground state and the excited state within the same symmetry, the orthonormality requirement for these states leads to orthogonal vectors of Cl coefficients. 

Hol0ien, E., Phys. Rev. 104, 1301, Radial configurational interaction in He and similar atomic systems." An orthonormal set of associated Laguerre functions is used. 

This work introduced the concept of a vibronic R-matrix, defined on a hypersurface in the joint coordinate space of electrons and intemuclear coordinates. In considering the vibronic problem, it is assumed that a matrix representation of the Schrodinger equation for N+1 electrons has been partitioned to produce an equivalent set of multichannel one-electron equations coupled by a matrix array of nonlocal optical potential operators [270], In the body-fixed reference frame, partial wave functions in the separate channels have the form p(q xN)YL(0, radial channel orbital function i/(q r) and antisymmetrized in the electronic coordinates. Here 0 is a fixed-nuclei A-electron target state or pseudostate and Y] is a spherical harmonic function. Both and i r are parametric functions of the intemuclear coordinate q. It is assumed that the target states 0 for each value of q diagonalize the A-electron Hamiltonian matrix and are orthonormal. 

The spin functions vanish from the formalism due to their orthonormality. The exchange term however has a factor (v v ) restricting its effect to pairs of electrons with the same spin projection. Equn. (5.26) becomes an integrodifferential equation in coordinate space, which is reduced to an equation in the radial variable by the methods of sections 3.3 and 4.3. The coordinate-space Hartree—Fock equation for a closed-shell structure is... 

The diagonalisation of the hydrogen Hamiltonian in a Slater-function (4.38) basis has been reviewed by Callaway (1978) in the context of variational solutions of the integrodifferential equations. This basis has useful features. The inclusion of all the Slater functions necessary for the radial eigenstate u r) produces exact eigenstates for principal quantum numbers up to n in the / manifold. The remainder are pseudostates, which represent the higher discrete states and the continuum. Since Slater functions are not orthonormal there are linear-dependence difficulties that severely limit the size of the basis for which the diagonalisation is numerically feasible. 

Let us consider now, the optimization - by a combined intra- and inter-orbit variation - of the energy density functional given by Eq. (Ill). Clearly, when we perform an intra-orbit variation we modify the one-particle density p(x). But when we perform an inter-orbit variation at fixed density, we just modify the orbitals For simplicity, we take these orbitals to be locally-scaled Raffenetti-type functions. The Raffenetti Rno(r) radial orbitals [91] are expanded in terms of set of Is Slater-type orbitals, i.e., orbitals having norbital exponents are defined by aQj = a/31. In the present case we do not consider the orbital coefficients as variational parameters these are obtained by Schmidt orthonormalization of the Raffenetti initial values [91]. Hence, the energy density functional becomes explicitly ... 

In evaluating the one-electron integral the radial R(r) and the angular T( , wave function are specified. As the angular functions are orthonormal, viz. 

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]

Note, carefully, the entries for the normalization constants, the squares of which, render the integrals to be of unit value. Distinct normalization constants have been included for radial and angular parts. To get the overall constant, it is necessary to multiply the two partial constants together. The table has been constructed, in this manner, to draw attention to a possible confusion in basis set theory. Often, the normalization condition is not clear for particular basis sets. Moreover, only rarely are basis sets mutually orthogonal, one with respect to another. Thus, it will be important to check the normalization data in Table 1.1 as an exercise in using the numerical integration techniques developed in Chapter 2. Orthonormalization is the subject of Chapter 3 because, in the end, all calculations in quantum chemistry require the rendering of approximate wave functions mutually orthonormal. 

Construct the projections of the orthonormal linear combination on the radial mesh using the overlap integral, cell onorm I 8, and the normalization integral of the Is functions, onorm I 6, with, for example,... 

Figure 3.5 The forming of the orthonormal 2s radial wave function in lithium from the Is and 2s sto-3g) Gaussian basis sets of Table 1.6, using the two-variable table procedure of fig2-9.xls. Figure 3.6 reproduces the chart demonstrating the match with the Herman-Skillman data.

Use the CHART wizard to construct comparison graphs of the exact hydrogen Is and 2s functions and the canonically orthonormal set of columns H and I, as is reproduced in Figure 3.15. Use the coarser radial mesh set out in cells N 18 to N49. 

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