Basis spherical harmonic functions

For sites of cubic symmetry the point-group symmetry elements mix the spherical harmonic basis functions. As a result, linear combinations of spherical harmonic functions, referred to as Kubic harmonics (Von der Lage and Bethe 1947), must be used. 

Each component of generally contracted with Gaussian type spherical harmonics functions. Contraction coefficients of the basis sets are determined by four-component atomic calculations [5],... 

Orbitals (GTO). Slater type orbitals have the functional form e, if) = NYi, d, e- -- (5.1) is a normalization constant and T are the usual spherical harmonic functions. The exponential dependence on the distance between the nucleus and the electron mirrors the exact orbitals for the hydrogen atom. However, STOs do not have any radial nodes. centre of a bond. 5.2 Classification of Basis Sets Having decided on the type of function (STO/GTO) and the location (nuclei), the most important factor is the number of functions to be used. The smallest number of functions... 

A free atom belongs to the continuous rotation group R3. The irreducible representations of group R3 are labelled by the quantum number /. The spherical harmonic functions Yl m form the basis of the irreducible representation of R3 with the dimension 2/ + 1. 

The results of the action of the operations R(ct/ y) on the set of spherical harmonic functions (representing basis elements of the reducible representation within the subgroup G) are obtained as follows... 

For the simplest case of a one-electron configuration dl the term functions are identical with the d-orbitals, and thus the formulae for the pertinent matrix elements listed in Table 8.10 are directly applicable. Then the 5 x 5 secular determinant is solved. For the case of an octahedral complex the matrix elements of the crystal field potential in the basis set of spherical harmonic functions Yi m form the matrix... 

The spherical harmonics functions form an orthonormal basis set for Lq, ... 

The most important feature of the matrices and vectors in Eq. 5 is that they are constructed in a vector space, or basis set, that consist of the direct product of generalized spherical harmonic functions representing the rotational degrees of freedom of the probe, and spin functions that represent its spin degrees of freedom. Proper specification of the basis set is therefore an important requirement for accurate calculation of the slow-motional spectrum. 

Each basis function in the EPRLL program is specified by five quantum numbers L, K, M, pi, Qi, with L, K, and M specifying the generalized spherical harmonic function, and the transition indices pi and ft specifying the spin functions. Note here that pi and ft refer only to nuclear spin states. In the general formulation of the SLE, it would also be necessary to include the indices ps and for the electronic spin states however, the EPRLL programs make use of the high-field approximation, which implicitly restricts the calculation to the ps = 1. s = 0 subspace. The basis set indices and the physical quantities they represent are summarized in Table 5. 

The basis sets that we have considered thus far are sufficient for most calculations. However, for some high-level calculations a basis set that effectively enables the basis set limit to be achieved is required. The even-tempered basis set is designed to achieve this each function m this basis set is the product of a spherical harmonic and a Gaussian function multiplied... 

The presence of a single polarization function (either a full set of the six Cartesian Gaussians dxx, d z, dyy, dyz and dzz, or five spherical harmonic ones) on each first row atom in a molecule is denoted by the addition of a. Thus, STO/3G means the STO/3G basis set with a set of six Cartesian Gaussians per heavy atom. A second star as in STO/3G implies the presence of 2p polarization functions on each hydrogen atom. Details of these polarization functions are usually stored internally within the software package. 

Here, n corresponds to the principal quantum number, the orbital exponent is termed and Ylm are the usual spherical harmonics that describe the angular part of the function. In fact as a rule of thumb one usually needs about three times as many GTO than STO functions to achieve a certain accuracy. Unfortunately, many-center integrals such as described in equations (7-16) and (7-18) are notoriously difficult to compute with STO basis sets since no analytical techniques are available and one has to resort to numerical methods. This explains why these functions, which were used in the early days of computational quantum chemistry, do not play any role in modem wave function based quantum chemical programs. Rather, in an attempt to have the cake and eat it too, one usually employs the so-called contracted GTO basis sets, in which several primitive Gaussian functions (typically between three and six and only seldom more than ten) as in equation (7-19) are combined in a fixed linear combination to give one contracted Gaussian function (CGF),... 

Appendix B Expansion of Cartesian Gaussian Basis Functions Using Spherical Harmonics... 

APPENDIX B EXPANSION OF CARTESIAN GAUSSIAN BASIS FUNCTIONS USING SPHERICAL HARMONICS... 

Spherical harmonics, Cartesion Gaussian basis functions, 261... 

To incorporate the angular dependence of a basis function into Gaussian orbitals, either spherical harmonics or integer powers of the Cartesian coordinates have to be included. We shall discuss the latter case, in which a primitive basis function takes the form... 

To find the irreducible representations of 0(3) it is necessary to find a set of basis functions which transform into their linear combinations on operating with the elements of 0(3). The set of 21 + 1 spherical harmonics Y[m(d, ), where l = 0,1, 2... and —l[Pg.91]

Then the moment induced by the electric vector of the incident light is parallel to that vector resulting in complete polarization of the scattered radiation. The A lg i>(CO) mode of the hexacarbonyls provides a pertinent example08. Suppose we have a set of coupled vibrators, equidistant from some origin. Then it must be possible to express the basis functions for the vibrations in terms of spherical harmonics, for the former are orthogonal and the latter comprise a complete set. The polarization of a totally symmetric vibration will be determined by its overlap with the spherically symmetrical term which may be taken as r2 = x2 + y1 + z2. Because of the orthogo-... 

In Appendix A2, we have formally applied the perturbation method to find the energy levels of a d ion in an octahedral environment, considering the ligand ions as point charges. However, in order to understand the effect of the crystalline field over d ions, it is very illustrative to consider another set of basis functions, the d orbitals displayed in Figure 5.2. These orbitals are real functions that are derived from the following linear combinations of the spherical harmonics ... 

Equations (4.30) and (4.31) have been developed and dehned within a time-dependent framework. These equations are identical to Eqs. (35) and (32), respectively, of Ref. 80. They differ only in that a different, more appropriate, normalization has been used here for the continuum wavefunction and that the transition dipole moment function has not been expanded in terms of a spherical harmonic basis of angular functions. All the analysis given in Ref. 80 continues to be valid. In particular, the details of the angular distributions of the various differential cross sections and the relationships between the various possible integral and differential cross sections have been described in that paper. 

Variational one-center restoration. In the variational technique of one-center restoration (VOCR) [79, 80], the proper behavior of the four-component molecular spinors in the core regions of heavy atoms can be restored as an expansion in spherical harmonics inside the sphere with a restoration radius, Rvoa, that should not be smaller than the matching radius, Rc, used at the RECP generation. The outer parts of spinors are treated as frozen after the RECP calculation of a considered molecule. This method enables one to combine the advantages of two well-developed approaches, molecular RECP calculation in a gaussian basis set and atomic-type one-center calculation in numerical basis functions, in the most optimal way. This technique is considered theoretically in [80] and some results concerning the efficiency of the one-center reexpansion of orbitals on another atom can be found in [75]. 

The major Re and P spherical harmonic basis functions. Highest occupied orbital. 

While we have chosen to proceed here by reducing representations for the full group D3h, it would have been simpler to take advantage of the fact that D3h is the direct product of C3u and C where the plane in the latter is perpendicular to the principal axis of the former. The behaviour of any atomic basis functions with respect to the C3 subgroup is trivial to determine, and there are only two classes of non-trivial operations in C3v. In more general cases, it is often worthwhile to look for such simplifications. It is seldom useful, for instance, to employ the full character table for a group that contains the inversion, or a unique horizontal plane, since the symmetry with respect to these operations can be determined by inspection. With these observations and the transformation properties of spherical harmonics given in the Supplementary Notes, it should be possible to determine the symmetries spanned by sets of atomic basis functions for any molecular system. Finally, with access to the appropriate literature the labour can be eliminated entirely for some cases, since... 

In either case it is helpful to have a table of transformation properties of spherical harmonics like that given in the Supplementary Notes. We shall illustrate the procedure by finding symmetry-adapted basis functions arising from an s function on each F atom in BF3, using full matrix projection operators. We already know that the three atomic basis functions transform as a 0 s, and since the behaviour with respect to the horizontal plane is already known, we can, without loss of generality, work with the subgroup C3u only. We denote the s functions on Fi, F2, and F3 as si, S2, and s3, respectively, and apply our C3u projection operators... 

We shall examine the behaviour of these functions under various symmetry operations. We need only consider the behaviour of functions centred at the origin, as functions centred elsewhere display the same behaviour together with a possible translation of the centre that is easily determined. Cubic groups are excluded as the spherical harmonics are less well suited as basis functions for these cases. 

A nondegenerate irrep that is symmetric with respect to the principl axis is denoted A, while B indicates antisymmetry with respect to this axis. In point groups with a horizontal plane of reflection, primes and " respectively indicate symmetry and antisymmetry with respect to the plane, while g and u indicate symmetry and antisymmetry with respect to inversion. For doubly degenerate irreps a subscript m indicates which spherical harmonics VJ, m form basis functions for that irrep. Numerical subscripts are used on nondegenerate irreps to distinguish them where necessary the numbers indicate the first of the vertical planes or perpendicular twofold axes (in the order specified in the character table) with respect to which the irrep is antisymmetric. 

We list here full matrix representations for several groups. Abelian groups are omitted, as their irreps are one-dimensional and hence all the necessary information is contained in the character table. We give C3v (isomorphic with D3) and C4u (isomorphic with D4 and D2d). By employing higher 1 value spherical harmonics as basis functions it is straightforward to extend these to Cnv for any n, even or odd. We note that the even n Cnv case has four nondegenerate irreps while the odd n Cnv case has only two. 

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