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Spherical Harmonics

Questaal objects with basis sets defined by spherical harmonics, such as LMTOs or smooth Hankel functions, order the harmonics as m=−l, −l+1, … 0, … l.

The Questaal codes use real harmonics , real versions of the usual (complex) spherical harmonics . The are functions of solid angle on a unit sphere, while the “solid real harmonics” are polynomials in x, y, and z. (1)  There is a corresponding set of “solid spherical harmonics” or spherical harmonic polynomials .

Caution: Questaal source codes typically use the symbol Ylm to refer to either or to the , depending on the context. In places they can be rotated to the true spherical harmonics .

The , and are shown for l=0…3 in the order used by the Questaal code:

indexlmpolynomial spherical harmonics spherical harmonic polynomials                       
100                            
21-1 
310 
411 
52-2 
62-1 
720  
821 
922 
103-3 
113-2 
123-1  
1330  
1431  
1532 
1633 

The and are related as follows, using standard conventions(2), as in e.g. Jackson

where . The inverse operation is

Standard definition of spherical harmonics in terms of Legendre polynomials

The [standard definition](2) of is

The and functions are related by [see Jackson (3.51) and (3.53)]

  and  

Matrix relations between linear combinations of real and spherical harmonics

Real and spherical harmonics combine states of and differently. For a particular pair, relations (1-6) for can be expressed as a 2×2 matrix as

Note that is unitary.

Rotations between linear combinations of spherical harmonics and real harmonics

Rotations mix only and , so it is sufficient to consider only a pair of functions at a time, though the rotation used below does not make use of that fact.

Express a function in both real spherical harmonics representations and compare:

Then

Eigenfunctions have the form Eq. (10a). Thus Eq. (10) gives the transformation of an eigenvector between real and spherical harmonics.

Structure Matrices

For definiteness, consider a matrix connecting functions at two points, e.g. a structure matrix expanding a function around another site,

(,) are in real harmonics; (,) are in spherical harmonics.

Consider the segment of a vector of coefficients relating linear combinations of and in real harmonics:

For any point r, and correspond to a particular term in the first expression, Eq. (10a). Each term can be re-expressed in spherical harmonics through the rotation in the first expression, Eq. (10). The same argument applies to and . Thus the preceding relation expressed as coefficients to functions in spherical harmonics is:

This establishes that

Gradients of Spherical harmonics

This section addresses the evaluation of the gradient operator acting on a function .

Spherical components of a vector

It is convenient to define the “spherical components” of a vector as (See Edmonds Sec 5.1)

The two components and () are related by , i.e. for odd :

The spherical components are convenient because the “solid spherical harmonics” are polynomials in the spherical components of .
This is shown explicitly in the table above. In particular, .

Products of the can be expanded in linear combinations of them, using Clebsch Gordan or Gaunt coefficients or Wigner 3-j symbols, e.g.

Spherical components of Gradients of Spherical Harmonics

Since , the gradient can be written as and its action on some can be evaluated as an “operator product” in terms of these expansion coefficients.

The gradient operator expressed in spherical harmonic components is then

can be written in spherical coordinates as

Expressing in spherical coordinates, and making use of properties of the Legendre polynomials, it is straightforward to show that

which establishes that

so that there are two nonzero matrix elements, of .

The general matrix elements are (Edmonds, Section 5.7)

Looking up Wigner 3-j symbols, explicit forms of the matrix elements can be obtained:

and

with

Gradients in terms of Vector Spherical Harmonics

Gradients are sometimes expressed as in terms of “vector spherical harmonics” , a special case of “tensorial spherical harmonics” which refer to products of two spherical harmonics.

As two special cases, and , note that the first or second term vanishes and the gradient becomes

Note that the gradient can be written

Rotations of functions of the gradient operator

We consider a pair linear combinations of spherical harmonics derived from the operator acting on a single function. For clarity we switch to Greek indices when referring to Cartesian coordinates and Roman indices when referring to spherical harmonics. Using Eqs. (13) and (10)

References

See
(1) Jackson, Electrodynamics.
(2) A.R.Edmonds, Angular Momentum in quantum Mechanics, Princeton University Press, 1960.
(3) M.E.Rose, Elementary Theory of angular Momentum, John Wiley & Sons, 1957.

Footnotes

(1) For example,

(2) Definitions (7) and (8) of spherical harmonics are the same as in Jackson. Jackson’s definition differs from that of Edmonds and Rose, by a phase factor . This phase is sometimes referred to as the “Condon–Shortley phase.” Wikipedia follows Jackson’s convention for . Some authors, e.g. Abramowitz and Stegun omit the Condon–Shortley phase.

Wikipedia also refers to Jackson’s definition as the “standard definition,” and refers to the definition which omits the Condon–Shortley phase in both the and the as a “quantum mechanics” defnition of spherical harmonics, since some texts in quantum mechanics (e.g. Messiah) use that convention. Since the phase is omitted in both places, “standard” and “quantum mechanics” definitions are identical for the , but differ by the Condon–Shortley phase for the .

This is apparently also the definition K. Haule uses in his CTQMC code. However, his code may scale some by .

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