Cartesian spherical harmonics

In a different form, the traceless moment operators can be written as the Cartesian spherical harmonics c,mp multiplied by r, which defines the spherical harmonic electrostatic moments ... 

The factor r1 enters because the Cartesian spherical harmonics clmp are defined in terms of the direction cosines in a Cartesian coordinate system. The expressions for clmp are listed in appendix D. As an example, the c2mp functions have the form 3z2 — 1, xz, yz, (x2 — y2)/2 and xy, where x, y and z are the direction cosines of the radial vector from the origin to a point in space. 

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. 

It does not make a significant difference that in practice one uses cartesian Gaussians rather than Gaussians with explicit inclusion of spherical harmonics. One... 

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

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

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... 

The spherical harmonics in real form therefore exhibit a directional dependence and behave like simple functions of Cartesian coordinates. Orbitals using real spherical harmonics for their angular part are therefore particularly convenient to discuss properties such as the directed valencies of chemical bonds. The linear combinations still have the quantum numbers n and l, but they are no longer eigenfunctions for the z component of the angular momentum, so that this quantum number is lost. 

The applicability of Eq. (21) rests on the validity of the assumption that the averages over internal and external variables are uncorrelated and thus can be calculated separately. Furthermore, theexpression of Eq. (21) emphasizes the close similarity of the irreducible Cartesian representation to the expression of the problem in terms of polar angles and the normalized 2nd rank spherical harmonics Y (see Eq. (7)). The corresponding polar angles ( (1), (t)) and (C(t), (t)), shown in Fig. 2B, describe the orientation of the internuclear vector and the magnetic field relative to the arbitrary reference frame, respectively. The different representations are related according to the following relationships.37... 

The angular functions 7<(0, ), called spherical harmonics, are common to all atoms. They are listed in Table 11-2.1 (in this table the normalizing constants have been omitted) together with the well-known symbols for the orbitals to which they correspond, i.e. s, pr, pr, p0, etc. The subscripts in these symbols are directly related to the angular functions if, for example, the angular function is ain 0 sin 2, then changing to the Cartesian coordinates x, y, and z where x r sin Q cos y = r sin 6 sin and z — r cos 0 gives us ... 

The parity of atomic states is important in spectroscopy. A radial function is an even function [see (1.113)] the spherical harmonic Y(m is found to be an even or odd function of the Cartesian coordinates according to whether / is an even or odd number. For a many-electron atom, it follows that states arising from a configuration for which the sum of the / values of all the electrons is an even number are even functions when 2,/, is odd, the state has odd parity. 

The form of equation (3) is most suitable for the evaluation of the effects of the CFP using (/-orbital wave functions. However, it is possible to recast it in the form of equation (6), where functions of the Cartesian axes rather than the spherical harmonics are employed... 

The Onk and Q k are operators which are respectively linear combinations of spherical harmonics and expansions in terms of Cartesian coordinates, 1 = 2 for d-orbitals, 3 for /-orbitals. The parameters B k and A k are, of course, specialized forms of the general form given in equation (2), but including the evaluation of the relevant radial integrals. 

In practical calculations, the AOs occur in shells, where members of a shell have the same radial dependence about a centre but differ in their angular behaviour. As discussed elsewhere, efficiency of integral evaluation usually involves using complete shells of spherical harmonic or Cartesian functions p, d, etc. We shall regard as a shell those functions which transform into themselves or their images on other centres under all operations in the group. Thus... 

One vexed question concerning polarization sets is the number of functions in a given shell, as discussed elsewhere. The early quantum chemistry codes employed Cartesian Gaussian functions, so that a d set actually comprises the five spherical harmonic d functions and a 3s function generally termed a contaminant. The reason for this emotive terminology is that with multiple polarization sets the contaminants,... 

The components may be expressed in either a space-fixed axis system (p) ora molecule-fixed system (q). The early literature used cartesian coordinate systems, but for the past fifty years spherical tensors have become increasingly common. They have many advantages, chief of which is that they make maximum use of molecular symmetry. As we shall see, the rotational eigenfunctions are essentially spherical harmonics we will also find that transformations between space- and molecule-fixed axes systems, which arise when external fields are involved, are very much simpler using rotation matrices rather than direction cosines involving cartesian components. 

The components C2qi (9, spherical harmonics, with the angles 9 and < /> defined in figure 8.52, shown in appendix 8.1. Equation (8.229) is similar to (8.10), except that we have chosen to couple the vectors differently because of the basis set used in the present problem. Clearly the components of the cartesian tensor T are related to those of the spherical tensor T2(C) these relationships are derived in appendix 8.2. 

Matrix dements between atomic orbitals can be defined in terms of spherical harmonics based on an axis along the inteinuclear separation, as indicated at (a) in the upper left. There, tine lower state is an hi = 0 p orbital and the upper state is an m = 0 d orbital, which has angular dependence as 3 f(2 /r ) — l/3]/2. For other d orbitals (hi 0), it is easy to visualize other cartesian forms, such as the symmetry z.v/c shown at (e) or the xy/r shown at (f), which lead to the same values for the matrix elements as do the spherical harmonics. There is only one independent sd matrix element for it, the different cases indicated at (c) in the upper right can be related by algebraic manipulation or the transformations described in Eq. (19-21). The two independent pd matrix elements are shown at (a) and (b) and the three independent dd matrix elements arc shown at (d), (e), and (f). Signs of the wave functions arc chosen such that all but and Vjj, are expected to be negative. 

Figure 3.2 Comparison of the local Cartesian and transformed coordinate systems at a general point on the unit sphere the standard spherical harmonic functions as the angular parts of the appropriate atomic orbitals are defined with reference to the local Cartesian set ex(j), ey(j) and ez(j) for each atomic position (j) with radius vector R on the unit sphere. Then the transformation of equation 3.1 is applied to construct the new local coordinate system cr(j), and JT (j).

Figure 3.8 The cr-type group orbitals on the vertices of an O3 structure orbit exhibiting D31J point symmetry displayed on the elliptical projections of Figure 3.7. The circular icons, filled and open circles, identify cr-oriented orbital components at the vertices, sized to reflect the coefficients of the linear combinations, equation 3.20, for the spherical harmonics in Table 3.11. The icons and identify the distinct group orbitals transforming as the irreducible components of the reducible character over the decorated orbit, unnecessary repetitions of these components and central functions for which no group orbital can be constructed owing to the locations of the decorated vertices of the orbit in the Cartesian coordinate system.

The quantities relevant to the rotationally averaged situation of randomly oriented species in solution or the gas phase must necessarily be invariants of the rotational symmetry. Accordingly, they must transform under the irreducible representations of the rotation group in three dimensions (without inversion), R3, just like the angular momentum functions of an atom. The polarisability, po, is a second-rank cartesian tensor and gives rise to three irreducible tensors (5J), (o), a(i),o(2), corresponding in rotational behaviour to the spherical harmonics, with / = 0,1,2 respectively. The components W, - / < m < /, of the irreducible tensors are given below. 

Kaijser and Smith [17] have presented analytic forms for many of the Slater-type orbitals and to Gaussian-type orbitals in both spherical harmonic and Cartesian form. The total momentum-space electron density p(p) is given by... 

The work [5], to be reviewed in this section, makes use of the asymmetric distribution Hamiltonian, as well as the cartesian component, ladder and square of the angular momentum operators and their actions on the chosen spherical harmonic basis lnij, for (L/,k) = cyc x,y,z). Here, we start from its Eqs. (26-28) for the matrix elements of H in the alternative bases ... 

To relate Dff to the anisotropic motion of a molecule in a liquid crystalline solvent, we employ the function P(d, < ), defined as the probability per unit solid angle of a molecular orientation specified by the angles 6 and <3>, the polar coordinates of the applied magnetic field direction relative to a molecule-fixed Cartesian coordinate system. We expand P(0, ) in real spherical harmonics ... 

Since the transformation properties of spherical harmonics are well known, the spherical-tensor notation has some advantages, particularly in the derivation of general theorems however, the reality of cartesian tensors also has its attractions, especially for small values of /. Normally, the moments of a particular three-dimensional molecule are most conveniently given in an x,y,z frame. 

Ni is the normalization factor and Yjjm denotes the usual spherical harmonics. Note that only Is and 2p Gaussian Cartesian functions are used. 

In the Cartesian scheme (Eq. (19)), there are (/+1)(/+ 2)/2 components of a given /, whereas the number of independent spherical harmonics is only 21+ 1. Usually, therefore, the Cartesian GTOs are not used individually but instead are combined linearly to give real solid harmonics (see Ref. 1). In addition, for a more compact and accurate description of the electronic structure, the GTOs (Eq. (19)) are not used individually as primitive GTOs but mostly as contracted GTOs (i.e., as fixed, linear combinations of primitive GTOs with different exponents a). 

If the group is rotational or helical and ij> is not 5-type, then the />, on each site become linear combinations of basis functions related by the rotation matrix of the appropriate angular momentum and the appropriate rotational or helical step angle [27]. It is traditional to use Cartesian-Gaussian orbital basis sets in quantum-chemical calculations [28], but solid-spherical-harmonic Gaussians [29] are best for symmetry adaption and matrix element evaluation. Including an extra factor of (-)M in the definition of the solid spherical harmonics [30]... 

The transition to a Cartesian basis is easily performed by expanding the spherical harmonics in the ) in Cartesian components. However,... 

By definition the components of the second-rank Cartesian tensor ax transform under rotation just like the product of coordinates xy (e.q., see Jeffreys, 1961) The motivation for what ensues springs from the observation that the spherical harmonics Ym (0, ft) (where 6, ft) are the polar and azimuthal angles of the unit vector (r/1 r )) can be written in terms of the coordinates (x, y, z) of the vector r, for example,... 

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