Real-Valued Harmonics

The scaled solid harmonics are decomposed into their real (c) and imaginary (s) components = + iRf r) from which 

One can construct a real-valued STGO by replacing the Cartesian coordinate arguments of C (r) with their Cartesian gradients. 

By using the complex-valued STGO differentiation rules and the relationship between the complex- and real-valued harmonics, one obtains the gradients [33] 

The gradients of W(r) can be expressed in terms of the matrix elements themselves in a manner analogous to Eq. (1.47)-(1.49), e.g. d/dzWf y/ r) = The translation matrix is efficiently 

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The assumption of harmonic vibrations and a Gaussian distribution of neighbors is not always valid. Anharmonic vibrations can lead to an incorrect determination of distance, with an apparent mean distance that is shorter than the real value. Measurements should preferably be carried out at low temperatures, and ideally at a range of temperatures, to check for anharmonicity. Model compounds should be measured at the same temperature as the unknown system. It is possible to obtain the real, non-Gaussian, distribution of neighbors from EXAFS, but a model for the distribution is needed and inevitably more parameters are introduced. 

A more detailed discussion of the complex and real spherical harmonic functions, with explicit expressions and numerical values for the normalization factors, can be found in appendix D. 

Fig. A.I. Real spherical harmonics. The first one, Y , is a constant. The coordinate system attached to the unit sphere is shown. The two zonal harmonics, IT and II, section the unit sphere into vertical zones. The unshaded area indicates a positive value for the harmonics, and the shaded area indicates a negative value. The four sectoral harmonics are sectioned horizontally. The two tesserai harmonics have both vertical and horizontal nodal lines on the unit sphere. The corresponding "chemists notations," such as (3z — r ), are also marked.

The special case of equation (7.45) with P x) = Q x) — 0 and 5(x) equal to a positive constant, n2 (the choice of n2 as the constant ensures that it is positive quantity for any real value for n), gives rise to equation (7.46) for simple harmonic motion, the solution of which can be used to model nuclear motion in molecules ... 

The fast Fade transform successfully overcomes the problems of the FFT. As a nonlinear transform, the FFT suppresses noise from the analyzed time signals. Most importantly, the FFT completely obviates postprocessing via fitting or any other subjective adjustments. This is accomplished by a direct quantification of the time signal under study through exact spectral analysis, which provides the unique solutions for the inverse harmonic (HI) problem [1, 2]. The solution of the HI problem contains four real-valued spectral parameters (one complex frequency and one complex amplitude) for each resonance or peak in the associated frequency spectrum. From these spectral parameters, the metabolite concentrations are unequivocally extracted. This bypasses the ubiquitous uncertainties inherent to fitting, such... 

We can now summarize our discussion on nuclear structure as follows A stationary state of the atomic nucleus can be represented, in general, by a real-valued non-negative charge density distribution p r) (a scalar function of coordinates), and by a real-valued current density distribution j r) (a vector function of coordinates). The former can be expanded into a series with standard spherical harmonics i i(r) [ the unit vector r = r/r is equivalent to the angles Q = 9,(f)) ],... 

The description of the mDC method in the present work is supplemented with mathematical details that we Have used to introduce multipolar densities efficiently into the model. In particular, we describe the mathematics needed to construct atomic multipole expansions from atomic orbitals (AOs) and interact the expansions with point-multipole and Gaussian-multipole functions. With that goal, we present the key elements required to use the spherical tensor gradient operator (STGO) and the real-valued solid harmonics perform multipole translations for use in the Fast Multipole Method (FMM) electrostatically interact point-multipole expansions interact Gaussian-multipoles in a manner suitable for real-space Particle Mesh Ewald (PME) corrections and we list the relevant real-valued spherical harmonic Gaunt coefficients for the expansion of AO product densities into atom-centered multipoles. 

Table 1.5 The unique nonzero real-valued spherical harmonic Gaunt coefficients for expanding atomic orbital products to quadrupole. where...

Real-Valued Spherical Harmonic Gaunt Coefficients... 

A real-valued spherical harmonic Gaunt coefficient corresponds to the integral... 

Equation 10.15 is the general form of Fourier s mthesis which sums harmonics from —to It can be shown that for all real valued signals (that is ones like speech with no imaginary part), the complex amplitudes are conjugate symmetric such that a t = Ok. In this case, the negative... 

In the Rouse model N monomers (beads) are coupled to each other via harmonic springs [16, 17, 20], As is well-known, the forces are of entropic origin. It is customary to revert to a continuous picture in which n, the bead s running number, takes real values. For a detailed discussion see Doi and Edwards [17]. The Langevin equations of motion for such a polymer in the MdM flow field are... 

There is a lot compressed in this expression so it is weU that we spend some time unravelling it First of aU, the subscripts t and u label the angular momenta of the real spherical harmonics and take the Im values 00,10,11c,11s,20,21c,21s,22c,22s,--- (the labels c and s stand for cosine and sine respectively). Qf is the real form of the multipole moment operator of rank t centered on A and expressed in the local-axis system of A, while T/J is the so-called T-tensor that carries the distance and angular dependence. The T-function of ranks h and I2 has a distance dependence where R is the separation between the centers of A and B. 

This distribution can be used to approximately represent a homogeneous Gaussian random field by a superposition of harmonics. For a real-valued random field, one obtains the representation... 

The two-electron integrals are for our purposes real entities, so it is clear that using complex terms (soHd harmonics) in the multipole expansion is unnecessary and only makes the computer implementation slower and more difficult. A reformulation in terms of real multipole integrals and interaction matrix elements is possible by splitting each term into a real and a complex part and dealing with them separately. After some algebra, one can see that the imaginary part drops out and one obtains the multipole expansion in terms of real-valued multipole moments and interaction matrix elements. The realvalued multipole expansion may be cast into exactly the same form as that of the complex series. As both formulations are formally very similar, we do not introduce the real formulation here but instead refer the interested reader to the literature, e.g.. Ref. 13. 

Complex algebra is avoided by rewriting the spherical harmonics in terms of the real solid harmonics of Section 6.4.2 (which then absorb the monomials in r), leaving us with the following simple set of real-valued spherical-harmonic GTOs... 

As discussed in Chapter 8, the primitive Cartesian GTOs (9.1.3) are mostly used in fixed linear combinations X/i(r). A typical AO thus consists of a linear combination of primitive Cartesian GTOs of the same angular-momentum quantum number / but of different Cartesian quantum numbers i, j and k and of different exponents a. In Section 9.1.2, we shall discuss how Cartesian GTOs of the same / but different i, j and k are combined to yield the real-valued herical-harmonic GTOs next, in Section 9.1.3, we shall see how the GTOs of different exponents are combined to yield the final AOs as contracted spherical-harmonic GTOs. 

A real-valued spherical-harmonic GTO of quantum numbers / and m. with exponent a and centred on A is given by the expression... 

Taking into account that Bq parameters represent the coefficient of an operator related to the spherical harmonic ykq then the ranges of k and q are limited to a maximum of 27 parameters (26 independent) Bq with k = 2,4,6 and q = 0,1,. .., k. The B°k values are real and the rest are complex. Due to the invariance of the CF Hamiltonian under the operations of the symmetry groups, the number of parameters is also limited by the point symmetry of the lanthanide site. Notice that for some groups, the number of parameters will depend on the choice of axes. In Table 2.1, the effect of site symmetry is illustrated for some common ion site symmetries. 

The MP2/TZDP optimized structures were then used to calculate the stationary state geometry force constants and harmonic vibrational frequencies, also at the MP2 level. These results serve several purposes. Firstly, they test that the calculated geometry is really an energy minimum by showing all real frequencies in the normal coordinate analysis. Secondly, they provide values of the zero-point energy (ZPE) that can be used... 

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